Quantum Spin Chains and von Neumann Algebra Lieb-Schultz-Mattis - - PowerPoint PPT Presentation

quantum spin chains and von neumann algebra
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Quantum Spin Chains and von Neumann Algebra Lieb-Schultz-Mattis - - PowerPoint PPT Presentation

Quantum Spin Chains and von Neumann Algebra Lieb-Schultz-Mattis type theorem without continuous symmetry Hal Tasaki Quantum Information and String Theory 2019 (YITP , June 10) Yoshiko Ogata and Hal Tasaki, LiebSchultzMattis Type


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SLIDE 1

Yoshiko Ogata and Hal Tasaki, “Lieb–Schultz–Mattis Type Theorems for Quantum Spin Chains Without Continuous Symmetry” 
 arXiv:1808.08740, Commnun. Math. Phys.

Lieb-Schultz-Mattis type theorem without continuous symmetry

Quantum Information and String Theory 2019 (YITP , June 10)

Hal Tasaki

Quantum Spin Chains and 
 von Neumann Algebra

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SLIDE 2

Lieb-Schultz-Mattis Theorem and its Generalizations

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SLIDE 3

Lieb-Schultz-Mattis (LSM) type theorem

∆E = O(1)

NO HALDANE GAP!!

No-go theorem which states that certain quantum many- body systems CANNNOT have a gapped unique ground state the original theorem ˆ H = PL

j=1 ˆ

Sj · ˆ Sj+1 antiferromagnetic Heisenberg chain with

Lieb, Schultz, Mattis 1961, Affmeck, Lieb 1986

L ↑ ∞

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S = 1 2, 3 2, 5 2, . . .

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for any , there exists an energy eigenvalue

` < L

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such that EGS < E ≤ EGS + const.

`

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E

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SLIDE 4

gradual non-uniform rotation to g.s. g.s. is rotation invariant exp h i

L

X

j=1

θ Sz

j

i |GSi = |GSi ˆ H = PL

j=1 ˆ

Sj · ˆ Sj+1 |GSi unique ground state

Proof of the original theorem

2π j `

ˆ V` = exp ⇥ i P`

j=0 2⇡ j

` ˆ Sz

j

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|Ψ`i = ˆ V`|GSi

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Lieb, Schultz, Mattis 1961, Affmeck, Lieb 1986

hΨ`| ˆ H|Ψ`i EGS  const. `

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uniform rotation about z ( 1 ) variational estimate from an elementary estimate

slide-5
SLIDE 5

ˆ H = PL

j=1 ˆ

Sj · ˆ Sj+1 |GSi unique ground state

Proof of the original theorem

Lieb, Shultz, Mattis 1961, Affmeck, Lieb 1986

hΨ`| ˆ H|Ψ`i EGS  const. `

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hΨ`|GSi = 0

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it can be shown (by symmetry) that ( 1 ) variational estimate (2) orthogonality for any , there exists an energy eigenvalue

` < L

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such that EGS < E ≤ EGS + const.

`

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E

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for S = 1

2, 3 2, 5 2, . . .

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there cannot be a unique gapped ground state! |Ψ`i = ˆ V`|GSi

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slide-6
SLIDE 6

Lieb-Schultz-Mattis (LSM) type theorem

No-go theorem which states that certain quantum many- body systems CANNNOT have a gapped unique ground state the original theorem and its extensions U(1) invariance is essential

Lieb, Shultz, Mattis 1961, Affmeck, Lieb 1986 Oshikawa, Yamanaka, Affmeck 1997 Oshikawa 2000, Hastings 2004, Nachtergaele, Sims 2007 Chen, Gu, Wen 2011 Watanabe, Po, Vishwanath, Zaletel 2013

recent “extensions” projective representation of the symmetry is inconsistent with the existence of unique disordered state

Matsui 2001

similar no-go statements for models without continuous symmetry, but with some discrete symmetry

the argument appears already in

slide-7
SLIDE 7

Rx

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ˆ Sx

j → ˆ

Sx

j

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ˆ Sy

j → − ˆ

Sy

j

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ˆ Sz

j → − ˆ

Sz

j

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ˆ Sx

j → − ˆ

Sx

j

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ˆ Sy

j → ˆ

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Ry

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ˆ Sz

j → − ˆ

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j

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Rz

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ˆ Sz

j → ˆ

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j

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ˆ Sy

j → − ˆ

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j

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ˆ Sx

j → − ˆ

Sx

j

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ˆ H = P

j

  • Jx ˆ

Sx

j ˆ

Sx

j+1 + Jy ˆ

Sy

j ˆ

Sy

j+1 + Jz ˆ

Sz

j ˆ

Sz

j+1 + K ˆ

Sx

j ˆ

Sy

j ˆ

Sz

j

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Z2 × Z2

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transformation invariant Hamiltonian (an example) THEOREM 1: Consider a quantum spin chain with
 and a short-ranged Hamiltonian that is invariant under translation and

  • transformation. Then it can never be the case that

the corresponding ground state is unique and accompanied by a nonzero gap.

Z2 × Z2

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S = 1

2, 3 2, 5 2, . . .

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π-rotations about the three axes

A Typical Theorem

slide-8
SLIDE 8

transformation of a single spin

Z2 × Z2

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ˆ S = ( ˆ Sx, ˆ Sy, ˆ Sz)

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ˆ S

2 = S(S + 1)

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spin operator

ˆ uα = exp[−iπ ˆ Sα]

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α = x, y, z

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ˆ uxˆ uy = ˆ uz

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ˆ uyˆ uz = ˆ ux

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ˆ uzˆ ux = ˆ uy

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π-rotation about the -axis

α

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ˆ uzˆ uyˆ ux = ˆ 1

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ˆ Sx, ˆ Sy, ˆ Sz

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(2S + 1) × (2S + 1)

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su(2)

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S = 1/2

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S = 1/2

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ˆ ux = −iX

<latexit sha1_base64="WmqQrb1phEPOr3jwFYodn8ai w4=">A CD3icbVC7SgNBFL3rM4mv1ZQ2g0GwMezGQhshaGMZwTw WcPsZDYZMvtgZlYSlvyBjaUfkFqwsVDE1tbOr9HJo9DEA8OcOede7tzjRpxJZVlfxsLi0vLKaiqdWVvf2Nwyt3cqMowFoWUS8lDUXCwpZwEtK6Y4rUWCYt/ltOp2z0d+9ZYKycLgSvUj6vi4HTCPEay0FJpZGEIHMChI YB3Oh7CAJ8QNDT71M4BAa1p mz8tY aJ7YU5IrpqOH68fed6lpfjZaIYl9GijCsZR124qUk2ChGOF0kGnEk aYdHGb1jUNsE+lk4z3GaB9rbSQFwp9AoXG6u+OBPtS9n1XV/pYdeSsNxL/8+qx8k6chAVRrGhAJoO8mCMVolE4qMUEJYr3NcFEMP1XRDpY KJ0hBkdgj278jypFPL2Ub5wqdM4gwlSsAt7cA 2HEMRLqAEZSBwB0/wAq/GvfFsvBnvk9IFY9qThT8wPn4AvpaZ4g= </latexit>

ˆ uy = −iY

<latexit sha1_base64="OLzZcgAY+HVTgtR+Ttob1d3vHkU=">A CD3icbVC5TgMxFHzLmYRrISWNRYREQ7QbCmiQImgog0QOSJbI6ziJFe8h24tYrfIHNJR8QGokGgoQoqWl42vAOQpIGMnyeOY9Pb9xQ86ksqwvY25+YXFpOZXOrKyurW+Ym1sVGUSC0DIJeCBqLpaUM5+WFVOc1kJBsedyWnV7p0O/ekOFZIF/oeKQOh7u+KzNCFZaCswsDKALGBQkE frvU9A EeI j1+xj2gcFl08xZeWsENEvsCckV0+HD1ePtd6lpfjZaAYk86ivCsZR12wqVk2ChGOG0n2lEkoaY9HCH1jX1sUelk4z26aNdrbRQOxD6+AqN1N8dCfakjD1XV3pYdeW0NxT/8+qRah85CfPDSFGfjAe1I45UgIbhoBYTlCgea4KJYPqviHSxwETpCDM6BHt65VlSKeTtg3zhXKdxAmOkYBt2YA9sOIQinE JykDgDp7gBV6Ne+PZeDPex6VzxqQnC39gfPwAwaiZ5A= </latexit>

ˆ uz = −iZ

<latexit sha1_base64="Cu24OFSclFKswUTv6+sTkwAJPZQ=">A CD3icbVC7TgJBFL2L 8DXKqXNRGJiI9nFQhsTo 0lJvI sJLZY AJs4/MzBpxwx/YWPoB1CY2Fhpja2vn1+jwKBQ8yWTOnHNv7tzjhpxJZVlfRmJhcWl5JZlKr6 tb2yaW9tlGUSC0BIJeC qLpaUM5+WF OcVkNBsedyWnF7ZyO/ck2FZIF/qfohdTzc8VmbEay0FJgZGEIXMCiI YIBXOl7CAI8QHCr3ydwA xqT Nr5awx0DyxpyRbSIUPtceb72LT/Gy0AhJ51FeEYynrthUqJ8ZCMcLpIN2IJA0x6eEOrWvqY49KJx7vM0B7Wm hdiD08RUaq787YuxJ2fdcXelh1ZWz3kj8z6tHqn3sxMwPI0V9MhnUj hSARqFg1pMUKJ4XxNMBN /RaSLBSZKR5jWIdizK8+Tcj5nH+byFzqNU5g CTuwC/tgwxEU4ByKUAICd/AEL/Bq3BvPxpvxPilNGNOeDPyB8fEDxLqZ5g= </latexit>

(representation of the ) generators of matrices

ˆ Sx = X 2 = ✓0

1 2 1 2

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ˆ Sy = Y 2 = ✓ − i

2 i 2

<latexit sha1_base64="XBgFcpfQoLRKYKoQe1fkbp3CtsQ=">A CmXicdVFPb9MwFH8JjI0MWAGJCe1iMVHtQpWUAz2ANECbJk6bRrdObTc57ktr1XEi20FU T9Ov8+0G7vsuo/B6x8htsKTrPz+ PfsPMe5ktaF4S/Pf/Bw5dHq2uNg/cnTZxuV5y9ObFY gU2Rqcy0Ym5RSY1NJ53CVm6Qp7HC03j4deqf/kBjZa /u1GO3ZT3tUyk4I6krNKC QyAg4MSjmEM5/SdgIEUGIyIfyKWEOcgyDkjpYT6Qo8BoQ8SNGk5JaZdDPGf5AcQ hXe3UnLP+kJcfMfr0rJgDykvr3lzheV7bAWzo tg2gBtncbN5eb17evDy8qV51eJo UtROKW9uOwtx1S26cFArHQaewmHMx5H1sE9Q8RdstZ5Mds7ek9FiSGVrasZn6d6LkqbWjNKadKXcDe9+biv/y2oVLGt1S6rxwqMX8oKRQzGVs+kysJw0Kp0YEuDCS7srEgBsuHD1mQEOI7v/yMjip16L3tfoRTeMLzGsNtuAN7EAEH2AXDuAQmiC8V95Hb8/b97f8z/6B/2 +1fcWmZdwp/zj36WDsro=</latexit>

ˆ Sz = Z 2 = ✓ 1

2

1 − 1

2

<latexit sha1_base64="lKQFjidQm/Ri8KZvayqLXSz Lpo=">A CmXicdVHBbhMxEJ3dAi0LpaFIXHqxGrXqpdFuemgPrSg UMUpK SNmobK68wmVrzele1FhFW+BvVbuHNB/A2zSYTaBkay/Oa9eWN7HOdKWheGvz1/5cHDR6trj4MnT9efbdSeb57brDACOyJTmenG3K SGjtO oXd3CBPY4UX8fhtpV98QWNlpj+5SY79lA+1TKTgjqis1oUbGAEHByW0YQqfab8BAykw+Eb5CWUJ5RwEKZfElNBc8DEgDEGCJi4nR9XFUP6V9OCOL/r 24WQFEFKRHj/P1WVG6nvYLnzda0eNsJZsGUQLUD91Y/gJP/+K2hd135eDTJRpKidUNzaXhTmrl9y46RQOA2uCos5F2M+xB5BzVO0/XI2 SnbIWbAkszQ0o7N2NuOkqfWTtKYKlPuRva+VpH/0nqFS476pdR54VCL+UFJoZjLWPVNbCANCqcmBLgwku7KxIgbLhx9ZkBDiO4/eRmcNxvRQaP5MayfvoF5rMEWbM ejfoQTuEMWtAB4b30jr13 nt/y3/tn/kf5qW+t/C8gDvht/8AzrexdQ= </latexit>

the simplest (but an important) case with for we have

S = 1

2, 1, 3 2, . . .

<latexit sha1_base64="s/wXgFkLHpD4vZWnei2wuc6Iqo4=">A CLXicbVC7SgNBFL0bXzG+opY2S4KgEMJuUmgjRAWxjGgekIQwO5lNhszuLDOzQljyKZ pbPwVESwiYqu1X+DkUeThgbkczrmXO/c4AaNSWdbQiK2srq1vxDcTW9s7u3vJ/YOy5KHApIQ546LqIEkY9UlJUcVINRAEeQ4jFad7PfIrj0RIyv0H1QtIw0Ntn7oUI6UlnkzBPVzA FwQgABDBDb0dc3pmtE8M+flZ7wBMGgB wWymUxbW sMc5nYU5IunFo3P79Pl8Vm8q3e4j 0iK8wQ1LWbCtQjQgJRTEj/UQ9lCRAuIvapKapjzwiG9H42r5 rJW 6XKhn6/MsTo7ESFPyp7n6E4PqY5c9Ebif14tVO5 I6J+ECri48kiN2Sm4uYoOrNFBcGK9TRBWFD9VxN3kEBY6YATOgR78eRlUs5l7Xw2d2enC1cwQRyOIAUnOuwzKMAtFKGkw36GVxjCh/FivBufxtekNWZMZw5hDsb3H9PJoBM=</latexit>
slide-9
SLIDE 9

transformation of a single spin

Z2 × Z2

<latexit sha1_base64="JHQfJqko41NQTU+UlgobuCZc8Gk=">A CJXicdVC7SgNBFL0bXzG+VgUbQRaDYBV2Y6FliI1lAuaByRJmJ5NkyOyDmbtCWPIzglj5FxY2FgYRbPRXnDwKTfQMA4dz mXmHi8SXKFtfxip eWV1bX0emZjc2t7x9zdq6owlpRVaChCWfeIYoIHrI cBatHkhHfE6zm9S/Hfu2WScXD4BoHEXN90g14h1OCWgrNI7gH wg 9MDTJ4EbGEIL8lpH4NpjoP7LtMysnbMnsBaJMyPZwkH5kz8Wn0otc9RshzT2WYBUEKUajh2hmxCJnAo2zDRjxSJC+6TLGpoGxGfKTSZbDq0TrbStTij1DdCaqD8nEuIrNfA9nfQJ9tS8Nxb/8hoxdi7chAdRjCyg04c6sbAwtMaVW 0uGU x0IRQyfVfLdojklDUxWZ0Cc78youkms85Zzm7rNsowhRpOIRjOAUHzqEAV1C lC4g2d4hZHxYLwYb8b7NJoyZjP78AvG1ze+rp9H</latexit>

(ˆ uα)2 = ˆ 1

<latexit sha1_base64="Gl05AWeBRKZtahOQKoVBjzq3dTY=">A CHXicbVDLSsNAFL2pr1pfUXe6CS2CIpSkLnQjFN24rGAf0MYymU7aoZMHMxMh PyFKzeu/A83LhRx4Ub6N07TLmr gYEz5 zLzD1OyKiQpjnSckvLK6tr+fXCxubW9o6+u9cQ cQxqeOABbzlIE Y9UldUslIK+QEeQ4jTWd4PfabD4QLGvh3Mg6J7aG+T12KkVRSoB/AMbzA B ISC FLrqjoB mKkncA8VuJzJWJB29ZJZNjMYi8SaklK12Dl9HFXjWlf/7vQCH nEl5ghIdqWGUo7QVxSzEha6ESChAgPUZ+0FfWR 4SdZNulxpFSeoYbcHV8aWTq7ESCPCFiz1FJD8mBmPfG4n9eO5LuhZ1QP4wk8fHkITdihgyMcV Gj3KCJYsVQZhT9VcD xBHWKpC 6oEa37lRdKolK2zsnmr2riC fJwCEV uwXnUIUbqE dMDzBK7zDh/asvWmf2tckmtOmM/vwB9rPLxX9nDg=</latexit>

ˆ uαˆ uβ = ˆ uβ ˆ uα

<latexit sha1_base64="szUobgIVkMtXQrPqxsEDwHBUauc=">A CV3icfVG7TgJBFL27ICL4AC1tNqKJFdnVAhsTo 0lJvJIgJC7w ATZh+ZmSUhGz7HX/AH/ANjw4doYaPDQqFgOMk 5 z7z OuCFnUtn23DBT6Z3MbnYvl98/ODwqFI8bMogEoXUS8EC0XJSUM5/WFVOctkJB0XM5b rj+4XfnFAhWeA/qWlIux4OfTZgBJW gkIFnmE CApi GAGPV0jcAgTd Nzgeoa4XaLt2XHXqFkl+0E1iZxVqRUPf94eZ3kP2u9wlunH5DIo74iHKVsO3aoujEKxQins1wnkjREMsYhbWvqo0dlN05ymVkXWulbg0Do5SsrUX9PxOhJOfVc3emhGsl1byH+57UjNbjpxswPI0V9sjxoEHFLBdYiZKvPBCWKTzVBIpi+q0VGKJAo/RU5HYKz/uRN0rgqO9dl+1GncQdLZOEUzuASHKhAFR6gBnUg8A5fRspIG3Pj28yY2W ra xmTuAPzOIPVcuo6Q= </latexit>

ˆ uαˆ uβ = −ˆ uβ ˆ uα

<latexit sha1_base64="n6dG rB47i eX qoIrDcYcQ+qX0=">A CWXicfVG7SgNBFL27Go0bH9GUNotRsDHsamEaIWhjGcE8IFnC3ck GTL7YGY2EJZ8jr/gD/gHFoL4HVpZOHkUmkgODJx7zr3zO PHnEnlO +GubGZ2drO7li53b39g/zhUV1GiSC0RiIeia PknIW0p i tNmLCgGPqcNf3g39RsjKiSLwkc1jqkXYD9kPUZQaSnKl+EJBoCgI UEJtDRNQKHeKauej5QXSPcwMUad82enXzRKTkz2KvEXZBi5fTz+W U+6p28q/tbkS gIaKcJSy5Tqx8lIUihFOJ1Y7kTRGMsQ+bWkaYkCl 86SmdhnWunavUjoFSp7pv6eSDGQchz4ujNANZDL3lT8z2slqlf2UhbGiaIhmR/US7itInsas91lghLFx5ogEUzf1SYDFEiU/gxLh+AuP3mV1C9L7lXJedBp3MIcWTiGEzgHF6 hAvdQhRoQeINvI2NsGR+mYWZNa95qGouZAvyBWfgB4SeoKg= </latexit>

(ˆ uα)2 = −ˆ 1

<latexit sha1_base64="+XglZKJz0tjtTB2pX4Z3x6kJwBA=">A CH3icbVDLSsNAFL2pr1pfUZfdhBZBEUtSF7oRim5cVrAPaGOZTCft0MmDmYkQ v7CpRsX/ogbF4qIu/6N07SL2npg4Mw5 zJzjxMyKqRpjrXcyura+kZ+s7C1vbO7p+8fNEUQcUwaOGABbztIE Z90pBUMtIO UGew0jLGd1M/NYj4YIG/r2MQ2J7aOBTl2IklRToRTiGVxgCAgkJRJBCT90RMAgz9Q eoApXcDaXsiDt6W zYmYwlok1I+VaqXv6NK7F9Z7+0+0HOPKILzFDQnQsM5R2grikmJG0 I0ECREeoQHpKOoj wg7yfZLjSOl9A034Or40sjU+YkEeULEnqOSHpJDsehNxP+8TiTdSzuhfh J4uPpQ27EDBkYk7KMPuUESxYrgjCn6q8GHiKOsFSVFlQJ1uLKy6RZrVjnFfNOtXENU+ShC V vAUXUINbqEMDMDzDG3zAp/aivWtf2vc0mtNmM4fwB9r4F7SbnHg=</latexit>

α 6= β

<latexit sha1_base64="1JLukjNCl/tjL3Kj65ks6MbCQeA=">A CA3icbVC7SgNBFL0bXzG+opaKLAbBKuxqoWXQxjIB84BkCbOTSTJkdnaZuSuEJWCT 4mNhSIWafwJO7/Bn3DyKDTxwNw5nHMvM/f4keAaHefLSq2srq1vpDczW9s7u3vZ/YOKDmNFWZmGIlQ1n2gmuGRl5ChYLVKMBL5gVb93O/GrD0xpHsp7 EfMC0hH8janBI0kYQ EBETQNfcIJDBTfVMRSDObc/LOFPYyceckVzgel76HJ+NiM/vZaIU0DphEKojWd eJ0EuIQk4FG2QasWYRoT3SYXVDJQmY9pLpDgP7zCgtux0qcyTaU/X3REICrfuBbzoDgl296E3E/7x6jO1rL+EyipFJOnuoHQsbQ3sSiN3i lEUfUMIVdz81aZdoghFE1vGhOAur xMKhd59zLvlEwaNzBDGo7gFM7BhSsowB0UoQwUHuEJXuDVGlrP1pv1PmtNWfOZQ/gD6+MHDcCXoQ= </latexit>

integer S

(S = 1, 2, . . .)

<latexit sha1_base64="w cMxUs8o6WVw1eDKxsQAHBz2Dg=">A CA3icbVDLSgNBEOyNrxhfUY9elgRBUcJuPOhFCHrxGNE8IFnC7OwkGTI7s8zMCs S8OIf+At68aCIV3/CW/7GyeOg0YKGoq b7i4/YlRpx lZmYXFpeWV7GpubX1jcyu/vVNXIpaY1LBgQjZ9pAijnNQ01Yw0I0lQ6DPS8AeXY79xR6Sigt/qJCJeiHqcdilG2kgcDuAGzsGFYyibegIGAQjQoOCwky86JWcC+y9xZ6RYKbSPHkeVpNrJf7UDgeOQcI0ZUqrlOpH2UiQ1xYwMc+1YkQjhAeqRlqEchUR56eSHob1vlMDuCm Ka3ui/pxIUahUEvqmM0S6r+a9sfif14p198xLKY9iT ieLurGzNbCHgdiB1QSrFliCMKSmlt 3EcSYW1iy5kQ3PmX/5J6ueSelJxrk8YFTJGFPSiYeF04hQpcQRVqgOEenuEV3qwH68V6tz6mrRlrNrMLv2B9fgMUZ UI</latexit>

half-odd-integer S (S = 1

2, 3 2, . . .)

<latexit sha1_base64="XtCA3v5JBbX4kmZICxKBD7PFxfY=">A CLXicbVHLSgMxFL3js9ZX1aWbwSJYkDLTLnQjVAVxWdE+oB1KJpNpQzOTIckIZeinuHTjxl8RwUVF3OraLzB9LGzrgVwO59xLck/ciFGpLGtgLCwuLa+sptbS6xubW9uZnd2q5LHApI 546LuIk YDUlFUcVIPRIEBS4jNbd7OfRr90RIysM71YuIE6B2SH2KkdISzxzAEdzCGTyCDwIQYEjAhr6uBV2Pp/TilM7A w4KJORamayVt0Yw54k9IdlSzr 6/nk4L7cyr02P4zg ocIMSdmwrUg5CRK Ykb6 WYsSYRwF7VJQ9MQBUQ6yWjbvnmoFc/0udAnVOZI/TuRoEDKXuDqzgCpjpz1huJ/XiNW/qmT0DCKFQnx+CI/Zqbi5jA606OCYMV6miAsqH6riTtI Kx0wGkdgj278jypFvJ2MW/d2NnSBYyRgn0YfoMNJ1C ayhDRcf9BC8wgHfj2XgzPozPceuCMZnZgykYX7/CSaAF</latexit>

give a genuine representation of Z2 × Z2

<latexit sha1_base64="JHQfJqko41NQTU+UlgobuCZc8Gk=">A CJXicdVC7SgNBFL0bXzG+VgUbQRaDYBV2Y6FliI1lAuaByRJmJ5NkyOyDmbtCWPIzglj5FxY2FgYRbPRXnDwKTfQMA4dz mXmHi8SXKFtfxip eWV1bX0emZjc2t7x9zdq6owlpRVaChCWfeIYoIHrI cBatHkhHfE6zm9S/Hfu2WScXD4BoHEXN90g14h1OCWgrNI7gH wg 9MDTJ4EbGEIL8lpH4NpjoP7LtMysnbMnsBaJMyPZwkH5kz8Wn0otc9RshzT2WYBUEKUajh2hmxCJnAo2zDRjxSJC+6TLGpoGxGfKTSZbDq0TrbStTij1DdCaqD8nEuIrNfA9nfQJ9tS8Nxb/8hoxdi7chAdRjCyg04c6sbAwtMaVW 0uGU x0IRQyfVfLdojklDUxWZ0Cc78youkms85Zzm7rNsowhRpOIRjOAUHzqEAV1C lC4g2d4hZHxYLwYb8b7NJoyZjP78AvG1ze+rp9H</latexit>

give a projective representation of Z2 × Z2

<latexit sha1_base64="JHQfJqko41NQTU+UlgobuCZc8Gk=">A CJXicdVC7SgNBFL0bXzG+VgUbQRaDYBV2Y6FliI1lAuaByRJmJ5NkyOyDmbtCWPIzglj5FxY2FgYRbPRXnDwKTfQMA4dz mXmHi8SXKFtfxip eWV1bX0emZjc2t7x9zdq6owlpRVaChCWfeIYoIHrI cBatHkhHfE6zm9S/Hfu2WScXD4BoHEXN90g14h1OCWgrNI7gH wg 9MDTJ4EbGEIL8lpH4NpjoP7LtMysnbMnsBaJMyPZwkH5kz8Wn0otc9RshzT2WYBUEKUajh2hmxCJnAo2zDRjxSJC+6TLGpoGxGfKTSZbDq0TrbStTij1DdCaqD8nEuIrNfA9nfQJ9tS8Nxb/8hoxdi7chAdRjCyg04c6sbAwtMaVW 0uGU x0IRQyfVfLdojklDUxWZ0Cc78youkms85Zzm7rNsowhRpOIRjOAUHzqEAV1C lC4g2d4hZHxYLwYb8b7NJoyZjP78AvG1ze+rp9H</latexit>

ˆ uα = exp[−iπ ˆ Sα]

<latexit sha1_base64="+B71zaX5YxsShUfgVUMy9FzIosw=">A COXicbVDLSgMxFL1TX7W+Rl26GVqEglhmdKEboejGZUX7gOlYMmnahmYeJBlxGPoX/oK/4Ma/cFdw40IRt/6A6bSCb 2QcM6595Dc4 aMCm aQy2zsLi0vJ dza2tb2xu6ds7NRFEHJMqDljAGy4ShFGfVCWVjDRCTpDnMlJ3+xejfv2OcE D/0bGIXE81PVph2IklRToRXiEHiCQkEAEA2gpjoB mKpnihG4V8yGQ6CKhen967hWjtsph9PSC2bJTMuYB9YEFMr5 sHDsBxXWvpLsx3gyCO+xAwJYVtmKJ0EcUkxI4NcMxIkRLiPusRW0EceEU6Sbj4w9pXSNjoBV8eXRqr+dSTIEyL2XDXpIdkTs72R+F/PjmTn1EmoH0aS+Hj8UCdihgyMUYxGm3KCJYsVQJhT9VcD9xBHWKqwcyoEa3bleVA7KlnHJfNKpXEO48rCHuShCBacQBkuoQJVwPAEr/AOH9qz9qZ9al/j0Yw28ezCVGnfP2knpEo=</latexit>

α = x, y, z

<latexit sha1_base64="8oRqSte2pP1ZctXcxFnfR s/HyQ=">A C XicbVC7SgNBFL0bXzG+Vi0VGQyChYRdLbQRgjaWCZgHJEuYncwmQ2YfzMyK65JSG7/DzsZC b +gZ3f4E84eRSaeOAyZ865l5l73IgzqSzry8jMzS8sLmWXcyura+sb5uZWVYaxILRCQh6Kuosl5SygFcU p/VIUOy7nNbc3uXQr91QIVkYXKsko 6POwHzGMFKS6FpwhNg4B V5/n+ibABwS3cASJr uWmbcK1gholtgTki/uDsrf93uDUsv8bLZDEvs0UIRjKRu2FSknxUIxwmk/14wljTDp4Q5taBpgn0onHW3SRwda SMvFLoChUbq74kU+1Imvqs7fay6ctobiv95jVh5Z07KgihWNCDjh7yYIxWiYSyozQ li eaYCKY/isiXSw UTq8nA7Bnl5 l SPC/ZJwSr NC5gjCzswD4cg 2nUIQrKE FCDzAM7zCm/FovBgD43 cmjEmM9vwB8bHDwnl +M=</latexit>

ˆ uxˆ uy = ˆ uz

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ˆ uyˆ uz = ˆ ux

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ˆ uzˆ ux = ˆ uy

<latexit sha1_base64="6HprY41W0FXiLe0+fkLaAcW6F04=">A CRXicfVC5TgMxFHwbrpBwBChpLAISVbQLBTRIETSUQSKHF aR1/EmVryHbG/Es rn5Bf4ACjo6fgDGgoQpAXnKCABRrI8b+Y9PXuckDOpTP JSM3NLywupZcz2ZXVtfXcxmZFBpEgtEwCHoiagyXlzKdlxRSntVBQ7DmcVp3O2dCvdqmQLPAvVRxS28Mtn7mMYKWlIFeAPrQBg4IEIuhBQ9 EOABghtd/+1e6/rkHz+GXiOXNwvmCGiW BOSL+5+3N53s4NSI/d41QxI5F fEY6lrFtmqOwEC8UIp73MVSRpiEkHt2hdUx97VNrJKIUe2tNKE7mB0MdXaKR+n0iwJ2XsObrTw6otp72h+JtXj5R7bCfMDyNFfTJe5EYcqQANI0VNJihRPNYE 8H0WxFpY4GJ0sFndAjW9JdnSeWgYB0WzAudximMkYZt2IF9sOAIinAOJSgDgTt4hld4Mx6MF+PdGIxbU8ZkZgt+wPj8AkBpqPI=</latexit>

π-rotation about the -axis

α

<latexit sha1_base64="7JPOQpg8DxhCw5kPw6YTxPsSrDc=">A B83icbVDJSgNBEK2JWxy3qEcvg0HwFGb0oBcx6MVjBLNAMoSaTk/SpKen6e4RQshvePGguBz9Du9exL+xsxw0+qDg8V4V fUiyZk2v /l5BYWl5ZX8qvu2vrG5lZhe6em0 wRWiUpT1UjQk05E7RqmOG0IRXFJOK0HvUvx379lirNUnFjBpKGCXYFixlBY6UYXgGBg4QeYLtQ9Ev+BN5fEsxI8fzdPZP n26lXfhodVKSJVQYwlHrZuBLEw5RGUY4HbmtTFOJpI9d2rRUYEJ1OJzcP IOrNLx4lTZEsabqD8nhphoPUgi25mg6el5byz+5zUzE5+GQyZkZqg 0 Vx j2TeuMAvA5TlBg+sASJYvZWj/RQITE2JteGEMy/ JfUjkrBc m/9ovlC5giD3uwD4cQwAmU4QoqUAVig72DB3h0Mufe XJepq05ZzazC7/gvH0DUHyS5g= </latexit>

ˆ uzˆ uyˆ ux = ˆ 1

<latexit sha1_base64="C0Fcwc R3bGJnipAq+wCna7SWqw=">A CU3icfVFNTwIxFHy7oiK oh69NBIT 2QXNXoxIXrxiIl8JEBItxRo6G43bdeIm/05/B9j4sE/4sWDFtgDgjpJ03kz76Xt1As5U9px3i17LbO+sZndym3v5Hf3CvsHdSUiSWiNC 5k08OKchbQm a 02YoKfY9Thve6HbqNx6pVEwED3oc0o6PBwHrM4K1kUThAiYwBAwaYog a7ZJyDB wTPpv7bHf/rPpn6esF3IekWik7JmQGtEjclRUhR7RZe2z1BIp8GmnCsVMt1Qt2JsdSMcJrk2pGiISYjPKAtQwPsU9WJZ5k 6MQoPdQX0qxAo5m6OBFjX6mx75lOH+uhWvam4m9eK9L9q07MgjDSNCDzg/oR 1qgacCoxyQlmo8NwUQyc1dEhlhios035EwI7vKTV0m9XHLPSuX782LlJo0jC0dwDKcmyEuowB1UoQYEXuADviyw3qxP27Yz81b SmcO4Qfs/DfqHqRD</latexit>

ˆ 1, ˆ ux, ˆ uy, ˆ uz

<latexit sha1_base64="peiWcSohTPnQRabWUnbExer65zM=">A CV3ichVFNTwIxEJ1dEBG/QI9eGomJB0N20QSPRC8eMZGPBAjpli40tLubtmvEzf4c/pDxwl/xouXjgGLCJE3fvDeTad94EWdKO87csjPZvdx+/qBweHR8closnbVUGEtCmyTkoex4WFHOAtrUTHPaiSTFwuO07U0eF3r7lUrFwuBFTyPaF3gUMJ8RrA0VFmswgzFg0JCACyncbOSxyQfmnoE AQjedujTHfo7pINi2ak4y0DbwF2DMqyjMSh+9IYhiQUN OFYqa7rRLqfYKkZ4TQt9GJFI0wmeES7BgZYUNVPlr6k6MowQ+SH0pxAoyW72ZFgodRUeKZSYD1Wf7UF+Z/WjbV/309YEMWaBmQ1yI850iFamIyGTFKi+dQATCQzb0VkjCUm2qyiYExw/35 G7SqFfe2Un2+K9cf1nbk4QIu4dosqgZ1eI GNIHAJ3xZGStrza1vO2fnV6W2te45h19hl34A4iWjsQ= </latexit>

ˆ 1, ˆ ux, ˆ uy, ˆ uz

<latexit sha1_base64="peiWcSohTPnQRabWUnbExer65zM=">A CV3ichVFNTwIxEJ1dEBG/QI9eGomJB0N20QSPRC8eMZGPBAjpli40tLubtmvEzf4c/pDxwl/xouXjgGLCJE3fvDeTad94EWdKO87csjPZvdx+/qBweHR8closnbVUGEtCmyTkoex4WFHOAtrUTHPaiSTFwuO07U0eF3r7lUrFwuBFTyPaF3gUMJ8RrA0VFmswgzFg0JCACyncbOSxyQfmnoE AQjedujTHfo7pINi2ak4y0DbwF2DMqyjMSh+9IYhiQUN OFYqa7rRLqfYKkZ4TQt9GJFI0wmeES7BgZYUNVPlr6k6MowQ+SH0pxAoyW72ZFgodRUeKZSYD1Wf7UF+Z/WjbV/309YEMWaBmQ1yI850iFamIyGTFKi+dQATCQzb0VkjCUm2qyiYExw/35 G7SqFfe2Un2+K9cf1nbk4QIu4dosqgZ1eI GNIHAJ3xZGStrza1vO2fnV6W2te45h19hl34A4iWjsQ= </latexit>

ˆ ux = −iX

<latexit sha1_base64="WmqQrb1phEPOr3jwFYodn8ai w4=">A CD3icbVC7SgNBFL3rM4mv1ZQ2g0GwMezGQhshaGMZwTw WcPsZDYZMvtgZlYSlvyBjaUfkFqwsVDE1tbOr9HJo9DEA8OcOede7tzjRpxJZVlfxsLi0vLKaiqdWVvf2Nwyt3cqMowFoWUS8lDUXCwpZwEtK6Y4rUWCYt/ltOp2z0d+9ZYKycLgSvUj6vi4HTCPEay0FJpZGEIHMChI YB3Oh7CAJ8QNDT71M4BAa1p mz8tY aJ7YU5IrpqOH68fed6lpfjZaIYl9GijCsZR124qUk2ChGOF0kGnEk aYdHGb1jUNsE+lk4z3GaB9rbSQFwp9AoXG6u+OBPtS9n1XV/pYdeSsNxL/8+qx8k6chAVRrGhAJoO8mCMVolE4qMUEJYr3NcFEMP1XRDpY KJ0hBkdgj278jypFPL2Ub5wqdM4gwlSsAt7cA 2HEMRLqAEZSBwB0/wAq/GvfFsvBnvk9IFY9qThT8wPn4AvpaZ4g= </latexit>

ˆ uy = −iY

<latexit sha1_base64="OLzZcgAY+HVTgtR+Ttob1d3vHkU=">A CD3icbVC5TgMxFHzLmYRrISWNRYREQ7QbCmiQImgog0QOSJbI6ziJFe8h24tYrfIHNJR8QGokGgoQoqWl42vAOQpIGMnyeOY9Pb9xQ86ksqwvY25+YXFpOZXOrKyurW+Ym1sVGUSC0DIJeCBqLpaUM5+WFVOc1kJBsedyWnV7p0O/ekOFZIF/oeKQOh7u+KzNCFZaCswsDKALGBQkE frvU9A EeI j1+xj2gcFl08xZeWsENEvsCckV0+HD1ePtd6lpfjZaAYk86ivCsZR12wqVk2ChGOG0n2lEkoaY9HCH1jX1sUelk4z26aNdrbRQOxD6+AqN1N8dCfakjD1XV3pYdeW0NxT/8+qRah85CfPDSFGfjAe1I45UgIbhoBYTlCgea4KJYPqviHSxwETpCDM6BHt65VlSKeTtg3zhXKdxAmOkYBt2YA9sOIQinE JykDgDp7gBV6Ne+PZeDPex6VzxqQnC39gfPwAwaiZ5A= </latexit>

ˆ uz = −iZ

<latexit sha1_base64="Cu24OFSclFKswUTv6+sTkwAJPZQ=">A CD3icbVC7TgJBFL2L 8DXKqXNRGJiI9nFQhsTo 0lJvI sJLZY AJs4/MzBpxwx/YWPoB1CY2Fhpja2vn1+jwKBQ8yWTOnHNv7tzjhpxJZVlfRmJhcWl5JZlKr6 tb2yaW9tlGUSC0BIJeC qLpaUM5+WF OcVkNBsedyWnF7ZyO/ck2FZIF/qfohdTzc8VmbEay0FJgZGEIXMCiI YIBXOl7CAI8QHCr3ydwA xqT Nr5awx0DyxpyRbSIUPtceb72LT/Gy0AhJ51FeEYynrthUqJ8ZCMcLpIN2IJA0x6eEOrWvqY49KJx7vM0B7Wm hdiD08RUaq787YuxJ2fdcXelh1ZWz3kj8z6tHqn3sxMwPI0V9MhnUj hSARqFg1pMUKJ4XxNMBN /RaSLBSZKR5jWIdizK8+Tcj5nH+byFzqNU5g CTuwC/tgwxEU4ByKUAICd/AEL/Bq3BvPxpvxPilNGNOeDPyB8fEDxLqZ5g= </latexit>

S = 1/2

<latexit sha1_base64="x42E5TdlUbe+ULU0M0GIF3VqUX4=">A B7HicbVBNSwMxEJ2tX7V+VT16CRbBU92tgl6EohePFd2 0C4lm2b 0Gy JFmhLP0NXjwo4tUf5M1/Y9ruQVsfD zem2FmXphwpo3rfjuFldW19Y3iZmlre2d3r7x/0NQyVYT6RHKp2iHWlDNBfcM p+1EURyHnLbC0e3Ubz1RpZkUj2ac0CDGA8EiRrCxkv9w7Z3VeuWKW3VnQMvEy0kFcjR65a9uX5I0psIQjrXueG5ig wrw ink1I31T BZIQHtGOpwDHVQTY7doJOrNJHkVS2hE z9fdEhmOtx3FoO2NshnrRm4r/eZ3URFdBxkS GirIfFGUcmQkmn6O+kxRYvjYEkwUs7ciMsQKE2PzKdkQvMWXl0mzVvXOq7X7i0r9Jo+jCEdwDKfgwSXU4Q4a4AMB s/wCm+OcF6cd+dj3lpw8plD+APn8wd/fY3S</latexit>

for we have

ˆ S = ( ˆ Sx, ˆ Sy, ˆ Sz)

<latexit sha1_base64="Y 8xo9VuWoD+Q1Br5mVkZwpwemI=">A CdXichVHLSgMxFL0zvmp9Vd0I sT6QFHqjC50IxQFcVnRqtCOk nTNjQzGZKMOA79C3/BH3LX3 Dj1nTahS/0hsC5 zLTe71I86UdpyeZY+Mjo1P5CbzU9Mzs3OF+YVrJWJ aJUILuStjxXlLKRVzTSnt5GkOPA5vfE7p3 95oFKxUR4pZOIegFuhazJCNaGEoUzeIE2YNCQGuSDA 4NUJBAM xSuISuOcew9cnb5+6yGm cCB5Nv uHnvyjP5l8+76w5pScLNBP4A7BWrlY3 nulZPKfeG13hAkDmioCcdK1Vwn0l6KpWaE026+HisaYdLBLVozM QBV 6aTa2LNgzTQE0hzQ01ytjPFSkOlEoC3zgDrNvqu9Ynf9NqsW4e SkLo1jTkAwaNWO tED9FaAGk5RonhiAiWTmrYi0scREm0XlzRDc71/+Ca73S+5Bybkw0ziBQeRgGYpmS 4cQhnOoQJVIPBmLVmrVtF6t1fsdXtzYLWtYc0ifAl7 wMflavI</latexit>

ˆ S

2 = S(S + 1)

<latexit sha1_base64="vJ/TPrASQdg0bOYCo+lmiMpFRX4=">A CI3icbZDLSsNAFIZPvNZ6i7oUIbQISqEkdaEboejGZUV7gTaWyXTSDp1kwsxECKFv4QO46au4caEUN134Lk4vC209w4Fv/nMOM+f3Ikalsu2xsbK6tr6xmdnKbu/s7u2bB4c1yWOBSRVzxkXDQ5IwGpKqo qR iQICjxG6l7/dlKvPxMhKQ8fVRIRN0DdkPoUI6Ulbp7AEHqAQEGqyQMOD ogIYFgfkvhAQb6PE JrjWf6SyA +dtM28X7WlYy+DMIV/OtQov43JSaZujVofjOC hwgxJ2XTsSLkpEopiRgbZVixJhHAfdUlTY4gCIt10u PAOtVKx/K50Bkqa6r+nkhRIGUSeLozQKonF2sT8b9aM1b+lZvSMIoVCfHsIT9mluLWxDCrQwXBi UaEBZU/9XCPSQ VtrWrDbBWVx5GWqlonNRtO+1Gzcwiw cQ06b6cAl OEOKlAFDK/wBh/waQyNd2NkfM1aV4z5zBH8CeP7BzlpnRA=</latexit>

spin operator

S = 1

2, 1, 3 2, . . .

<latexit sha1_base64="s/wXgFkLHpD4vZWnei2wuc6Iqo4=">A CLXicbVC7SgNBFL0bXzG+opY2S4KgEMJuUmgjRAWxjGgekIQwO5lNhszuLDOzQljyKZ pbPwVESwiYqu1X+DkUeThgbkczrmXO/c4AaNSWdbQiK2srq1vxDcTW9s7u3vJ/YOy5KHApIQ546LqIEkY9UlJUcVINRAEeQ4jFad7PfIrj0RIyv0H1QtIw0Ntn7oUI6UlnkzBPVzA FwQgABDBDb0dc3pmtE8M+flZ7wBMGgB wWymUxbW sMc5nYU5IunFo3P79Pl8Vm8q3e4j 0iK8wQ1LWbCtQjQgJRTEj/UQ9lCRAuIvapKapjzwiG9H42r5 rJW 6XKhn6/MsTo7ESFPyp7n6E4PqY5c9Ebif14tVO5 I6J+ECri48kiN2Sm4uYoOrNFBcGK9TRBWFD9VxN3kEBY6YATOgR78eRlUs5l7Xw2d2enC1cwQRyOIAUnOuwzKMAtFKGkw36GVxjCh/FivBufxtekNWZMZw5hDsb3H9PJoBM=</latexit>
slide-10
SLIDE 10

Theorem for 
 Matrix Product States (MPS)

Watanabe, Po, Vishwanath, Zaletel 2013

slide-11
SLIDE 11

Matrix Product States (MPS)

Fannes, Nachtergaele, Werner 1991, 1992

quantum spin system with spin on {1, 2, . . . , L}

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S

<latexit sha1_base64="GaAj9Cg siMSYT4U81avqCQzmek=">A B6HicbZC7SgNBFIbPxltcb1FLm8EgWIXdWGgjBm0sEzQXSJYwOzmbjJm9MDMrhJAnsLFQxFYfxt5GfBsnl0ITfxj4+P9zmHO nwiutON8W5ml5ZXVtey6vbG5tb2T292rqTiVDKs FrFs+FSh4BFWNdcCG4lEGvoC637/apzX71EqHke3epCgF9JuxAPOqDZW5a dyzsFZyKyCO4M8hcf9n y/mWX27nPVidmaYiRZoIq1XSdRHtDKjVnAkd2K1WYUNanXWwajGiIyhtOBh2RI+N0SB L8yJNJu7vjiENlRqEvqkMqe6p+Wxs/pc1Ux2ceUMeJanGiE0/ClJBdEzGW5MOl8i0GBigTHIzK2E9KinT5ja2OYI7v/Ii1IoF96RQrDj50iVMlYUDOIRjcOEUSnANZagCA4QHeIJn6856tF6s12lpxpr17M fW 8/DwmQGA= </latexit>

ˆ Sz|σi = σ|σi

<latexit sha1_base64="uWkFyX1pW3/bilrdAPMCuDR5gSQ=">A CTXicfVFNSyMxGH6m6 pbV7fuHr0Ei7CwUGbqYb0IRS8elbUqtFXeSdM2mMkMSUaoY/+Ff8E/s5cFb/6LvexBWcS09eAXPhDyfLwvSd7EmZLWheFNUPow83F2bv5Te Hz4tKXyvLXA5vmhosmT1VqjmKyQk tmk46JY4yIyiJlTiMT7fH+eGZMFamet8NM9FJqK9lT3Jy3kordVxhAIJDgV8Y4djvVzBIwHDu9YVXFhJ979AkIWivFAQ2n2XvVJ5UqmEtnIC9JtEjqTZW2z8ubxrD3ZPKdbub8jwR2nF 1raiMHOdgoyTXIlRuZ1bkRE/pb5oeaopEbZT KYxYmve6bJeavzSjk3cpx0FJdYOk9hXJuQG9mU2Nt/KWrnrbXQKqbPcCc2nB/VyxVzKxqNlXWkEd2roCXEj/V0ZH5Ah7vwHlP0QopdPfk0O6rVovVbfi6qNLUwxjxWs4jsi/EQDO9hFExy/8Re3uAv+BP+C/8H9tLQUP Z8wzOU5h4A6cqotg= </latexit>

standard basis states

D × D

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matrices with

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σ = −S, . . . , S

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|σ1, . . . , σLi = NL

j=1 |σjij

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|Φi = PS

σ1,...,σL=−S Tr[Mσ1 . . . MσL] |σ1, . . . , σLi

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translation invariant state (MPS) it is known that disordered states (area-law states) 
 can be approximated by MPS

|Φi

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PS

σ=−S Mσ(Mσ)† = I

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|Φi

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is said to be injective if , and span the whole space of matrices

D × D

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is injective if it is disordered, and not a “ cat” heuristic there is such that with all possible

Mσ1Mσ2 · · · Mσ`

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σ1, . . . , σ`

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`

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slide-12
SLIDE 12

Theorem for MPS

THEOREM 1’: There cannot be a translation invariant and invariant injective MPS for

Z2 × Z2

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S = 1

2, 3 2, 5 2, . . .

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|Φi = PS

σ1,...,σL=−S Tr[Mσ1 . . . MσL] |σ1, . . . , σLi

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|Φi

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PS

σ=−S Mσ(Mσ)† = I

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|Φi

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is said to be injective if , and span the whole space of matrices

D × D

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is injective if it is disordered, and not a “ cat” heuristic there is such that with all possible

Mσ1Mσ2 · · · Mσ`

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σ1, . . . , σ`

<latexit sha1_base64="Aed nLe9rlhZaJpa GCD/ 0vBZI=">A CH3icbZC7SgNBFIbPxluMt1XLiCwGwSKE3VhoGbSxTMBcIFnC7OwkGTK7s8zMCmFJ6VvYWOQdUt YKCJ2eQZfwsml0MQDAz/ dw5nzu9FjEpl2xMjtba+sbmV3s7s7O7tH5iHRzXJY4FJFXPGRcNDkjAakq i pFGJAgKPEbqXv92yusPREjKw3s1iIgboG5IOxQjpS1uZmE ih0IQAEbXAgrx0GPnBQmuSX+AiIpqxt5uyCPStrVTgLkSudjCvfj6fjctv8avkcxwEJFWZIyqZjR8pNkFAUMzLMtGJ IoT7qEuaWoYoINJNZvcNrXPt+FaHC/1CZc3c3xMJCqQcBJ7uDJDqyWU2Nf9jzVh1rt2EhlGsSIjnizoxsxS3pmFZPhUEKzbQAmFB9V8t3EMCYaUjzegQnOWTV0WtWHAuC8WKkyvdwLzSkIUzuNBhX0EJ7qAMVcDwBC/wBu/Gs/FqfBif89aUsZg5hj9lTH4Am5adyA= </latexit>

`

<latexit sha1_base64="RPyH4CmM5HvzBnDXIxj1ByH+cS8=">A B93icbVDLSgNBEOz1GeMr6tHLYBA8SNiNgh6DXjxGMA9IljA725sMmX0wMys S37Cq4Inxaufo1/jJNmDJhY0FNVd Hd5ieBK2/aXtbK6tr6xWdoqb+/s7u1XDg7bKk4lwxaLRSy7HlUoeIQtzbXAbiKRhp7Aj e+nfY7jygVj6MHnSXohnQY8YAzqo3kwxsgCBCDStWu2TOQZeIUpAoFmoPKd9+PWRpipJmgSvUcO9FuTqXmTOCk3E8VJpSN6RB7hkY0ROXms3sn5NQoPgliaSrSZKb+duQ0VCoLvfM0GUrEsXGEVI/U4sxU/K/XS3Vw7eY8SlKNEZsvDFJBdEymIRCfS2RaZIZQJrm5mbARlZRpE1XZhOEsvr5M2vWac1Gr319WGzdFLCU4h M4AweuoAF30IQWMBPvEz Di5VZr9a79TEfXbEKzxH8gfX5A8Pzkgs=</latexit>

translation invariant state (MPS)

slide-13
SLIDE 13

Proof of Theorem 1’

THEOREM 1’: There cannot be a translation invariant and invariant injective MPS for

Z2 × Z2

<latexit sha1_base64="JHQfJqko41NQTU+UlgobuCZc8Gk=">A CJXicdVC7SgNBFL0bXzG+VgUbQRaDYBV2Y6FliI1lAuaByRJmJ5NkyOyDmbtCWPIzglj5FxY2FgYRbPRXnDwKTfQMA4dz mXmHi8SXKFtfxip eWV1bX0emZjc2t7x9zdq6owlpRVaChCWfeIYoIHrI cBatHkhHfE6zm9S/Hfu2WScXD4BoHEXN90g14h1OCWgrNI7gH wg 9MDTJ4EbGEIL8lpH4NpjoP7LtMysnbMnsBaJMyPZwkH5kz8Wn0otc9RshzT2WYBUEKUajh2hmxCJnAo2zDRjxSJC+6TLGpoGxGfKTSZbDq0TrbStTij1DdCaqD8nEuIrNfA9nfQJ9tS8Nxb/8hoxdi7chAdRjCyg04c6sbAwtMaVW 0uGU x0IRQyfVfLdojklDUxWZ0Cc78youkms85Zzm7rNsowhRpOIRjOAUHzqEAV1C lC4g2d4hZHxYLwYb8b7NJoyZjP78AvG1ze+rp9H</latexit>

S = 1

2, 3 2, 5 2, . . .

<latexit sha1_base64="ZhW2WS+CyP2QBxz u6GcZX+r6lc=">A CQXicdZFLSwMxEMdnfdb6WvXoZbEICqXstohehKogHivaB7SlZN sG5rdLElWKEs/ike/h e/gTfvHvSgSK9eTB8H2+pAhj+/mWGSf9yQUals+8WYm19YXFpOrCRX19Y3Ns2t7ZLk cCkiDnjouIiSRgNSF RxUglFAT5LiNlt3MxqJfviJCUB7eqG5K6j1oB9ShGSiNupuEGTuEBPBCA EM DvR0zuqcnuC5f/jRBGfQBA4KZMNM2Rl7GNascMYilT+0L9/692eFhvlca3Ic+SRQmCEpq4 dqnqMhK YkV6yFk SItxBLVLVMkA+kfV46EDP2tekaXlc6BMoa0h/T8TIl7Lru7rTR6otp2sD+FetGinvpB7TI wUCfBokRcxS3FrYKfVpIJgxbpaICyovquF20g rLTpSW2CM/3kWVHKZpxcJnvtpPLnMIoE7MIeHOjPOIY8XE BitrsR3iFD/g0nox348voj1rnjPHMDkyE8f0DnuqlOA= </latexit>

α = x, y, z

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|Φi = PS

σ1,...,σL=−S Tr[Mσ1 . . . MσL] |σ1, . . . , σLi

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assume that is injective, and invariant, i.e., for

Z2 × Z2

<latexit sha1_base64="JHQfJqko41NQTU+UlgobuCZc8Gk=">A CJXicdVC7SgNBFL0bXzG+VgUbQRaDYBV2Y6FliI1lAuaByRJmJ5NkyOyDmbtCWPIzglj5FxY2FgYRbPRXnDwKTfQMA4dz mXmHi8SXKFtfxip eWV1bX0emZjc2t7x9zdq6owlpRVaChCWfeIYoIHrI cBatHkhHfE6zm9S/Hfu2WScXD4BoHEXN90g14h1OCWgrNI7gH wg 9MDTJ4EbGEIL8lpH4NpjoP7LtMysnbMnsBaJMyPZwkH5kz8Wn0otc9RshzT2WYBUEKUajh2hmxCJnAo2zDRjxSJC+6TLGpoGxGfKTSZbDq0TrbStTij1DdCaqD8nEuIrNfA9nfQJ9tS8Nxb/8hoxdi7chAdRjCyg04c6sbAwtMaVW 0uGU x0IRQyfVfLdojklDUxWZ0Cc78youkms85Zzm7rNsowhRpOIRjOAUHzqEAV1C lC4g2d4hZHxYLwYb8b7NJoyZjP78AvG1ze+rp9H</latexit>

Watanabe, Po, Vishwanath, Zaletel 2013 (arranged by H.T.)

exp[iπ P

j ˆ

j ]|Φi = const |Φi

<latexit sha1_base64="3OKB1vldN IrIdkfb1 4svkdZns=">A CeXicdZE9TxtBEIbnDpKA82VCmRSbWEQoCs4dFKFBspKGEgQGJHOx9tZje8N+nHb3ENbhmr+RJn+EMmW6/JE0aTJnUySQjHSnd5 3 vuYzQslfUiSH1G8sHjv/oOl5cbDR4+fPG2uPDvytnQCu8Iq605y7lFJg90g 8KTwiHXucLj/Oxj7R+fo/PSmsMwKTDTfGTkUAoeCNnmLnwFhAso AcbIKkrZncPJWjow2fSY+AQoI DmMIn6jkomqp 7WdwSWyP+jrniBoY0QRCA3YoVTMNDARYcjw9aUrs7f9S/WYraSezYndFeiNanXfX 6 +DXb2+s3vpwMrSo0mCMW976VJEbK uyCFwmnjtPRYcH GR9gjabhGn1Wz U3ZGpEBG1pHlwlsRv9MVFx7P9E5TWoexv62V8N/eb0yDLezSpqiDGjE/EXDUrFgWX0MbCAdiqAmJLhwkr6ViTF3XAQ6rAYtIb39y3fF0WY73Wpv7qetzgeY1xI8h1ewDim8hw7s0oK7tPif0YtoLXod/Ypfxuvxm/loHN1kVuGvird+Azj2rfk=</latexit>

P Tr[ ˜ Mσ1 . . . ˜ MσL] |σ1, . . . , σLi = const P Tr[Mσ1 . . . MσL] |σ1, . . . , σLi

<latexit sha1_base64="QHRSUkKJFQ/2PtqQ1cjFbsBgRco=">A Dq3icrVJNb9NAEJ3afBTzFeDYy6oRUg8hs BLkgRPbQHQEU0TURiovVmnay6H9bu lJkcutfyR/ixpF/wsTpoa0RqhAjWX7 5r2dmdVkhRTOx/HPrSC8c/fe/e0H0cNHj58 bT17fupMaRnvMyONHWbUcSk073vhJR8Wl OVST7Izg7W+cE5t04YfeIXBU8VnWmRC0Y9Uqb1C1bgoAQFEVSILSICJ/hfwgjPHgRImAKvswo MnN05Hj+iJolfKszDnWzOj+B NlV7TKodv94ywdkU+xqBR343qjQuVah0/CuJ6GgkZFYNYJ3V6Zj6NKo93Wf 5u+2eltpr2N6z9ON2m1425cB2mC5BK0e7vjw53Di/bxpPVjPDWsVFx7JqlzoyQufFpR6wWTfBmNS8cLys7ojI8Qaq 4S6t615bkJTJTkhuLn/akZq86Kq cW6gMlYr6ubuZW5N/yo1Kn79NK6GL0nPN oXyUhJvyHpxyVRYzrxcIKDMCuyVsDm1lHlc7wgfIbk5chOc7neT1939z0m79x42sQ07sAt7+OhvoAdHcAx9YMFe8CkYBMPwVfgl/BqON9Jg69LzAq5FyH8DSubsoQ= </latexit>

˜ Mσ = P

σ0hσ|ˆ

uα|σ0iMσ0

<latexit sha1_base64="glwj2S8H/Oet ugYrdEqw9rJu1A=">A Cu3icbVG7TsMwFL0J7/AqMLJEVAimKoGBLkgVLCxI F AKm3luE5icJzIdpCq0E9h7A+xAB/Czm2KRGm5VqTzuOfasYNMcG0879Oy5+YXFpeWV5zVtfWNzcrW9q1Oc0VZk6YiVfcB0UxwyZqG 8HuM8VIEgh2Fzydj/y7Z6Y0T+WN6WesnZBI8pBTYlBK xyGYICDgB4wKJAlQFCJQUOI/BIGuDqoa+yKSvcUnJLnyLpl5tc7wO4hTiMgURE4c9J9QRaX8wtMDzA9RCYgK9WXqUlDUBNznH/P1pndv1upejWvLHcW+D+g2tj/+nh/daKrbuXtoZfSPGHSUEG0bvleZtoFUYZTwQbOQ65ZRugTiVgLoSQJ0+2ivPuBu49Kzw1ThZ80bqlOJgqSaN1PAuxMiIn1tDcS/ NauQnr7YL DdM0vFGYS5ck7qjh3R7XDFqRB8BoYrjWV0aE0Wowed28BL86V+eBbdHNf+4dnTtVxtnMK5l2IU9OAQfTqABF3AFTaBW3epYkRXbpza1H20xbrWtn8wO/Ck7/wZ+6r0Q</latexit>

with

ˆ uα = exp[−iπ ˆ Sα]

<latexit sha1_base64="+B71zaX5YxsShUfgVUMy9FzIosw=">A COXicbVDLSgMxFL1TX7W+Rl26GVqEglhmdKEboejGZUX7gOlYMmnahmYeJBlxGPoX/oK/4Ma/cFdw40IRt/6A6bSCb 2QcM6595Dc4 aMCm aQy2zsLi0vJ dza2tb2xu6ds7NRFEHJMqDljAGy4ShFGfVCWVjDRCTpDnMlJ3+xejfv2OcE D/0bGIXE81PVph2IklRToRXiEHiCQkEAEA2gpjoB mKpnihG4V8yGQ6CKhen967hWjtsph9PSC2bJTMuYB9YEFMr5 sHDsBxXWvpLsx3gyCO+xAwJYVtmKJ0EcUkxI4NcMxIkRLiPusRW0EceEU6Sbj4w9pXSNjoBV8eXRqr+dSTIEyL2XDXpIdkTs72R+F/PjmTn1EmoH0aS+Hj8UCdihgyMUYxGm3KCJYsVQJhT9VcD9xBHWKqwcyoEa3bleVA7KlnHJfNKpXEO48rCHuShCBacQBkuoQJVwPAEr/AOH9qz9qZ9al/j0Yw28ezCVGnfP2knpEo=</latexit>
slide-14
SLIDE 14

there are unitary matrices which form a projective representation of , and constants
 with for , such that

Proof of Theorem 1’

Fannes, Nachtergaele, Werner 1992
 Perez-Garcia, Wolf, Sanz, Verstraete, and Cirac 2008
 Pollmann, Turner, Berg, Oshikawa 2010

α = x, y, z

<latexit sha1_base64="NvRZe4um2dVyv DpvaoGHSWPvAw=">A C XicbVC7SgNBFL0bXzG+Vi0VGQyChYTdWGgjBG0sEzAPSJYwO5lNhsw+mJkV1yWlNn6HnY2FEmz9Azu/wZ9w8ig0euAyZ865l5l73IgzqSzr08jMzS8sLmWXcyura+sb5uZWTYaxILRKQh6Khosl5SygVcU p41IUOy7nNbd/sXIr19TIVkYXKk o 6PuwHzGMFKS6FpwiNg4B T59n+ibABwQ3cASJrtu2mbcK1hjoL7GnJF/aHVa+7vaG5b 50eqEJPZpoAjHUjZtK1JOioVihN BrhVLGmHSx13a1DTAPpVO t5kgA60 kFeKHQFCo3VnxMp9qVMfFd3+lj15Kw3Ev/zmrHyTp2UBVGsaEAmD3kxRypEo1hQhwlKFE80wUQw/VdEelhgonR4OR2CPbvyX1IrFuzjQrFi50vnME WdmAfDsG EyjBJZShCgTu4Qle4NV4MJ6NofE2ac0Y05lt+AXj/RsK2Zfm</latexit>

P Tr[ ˜ Mσ1 . . . ˜ MσL] |σ1, . . . , σLi = const P Tr[Mσ1 . . . MσL] |σ1, . . . , σLi

<latexit sha1_base64="QHRSUkKJFQ/2PtqQ1cjFbsBgRco=">A Dq3icrVJNb9NAEJ3afBTzFeDYy6oRUg8hs BLkgRPbQHQEU0TURiovVmnay6H9bu lJkcutfyR/ixpF/wsTpoa0RqhAjWX7 5r2dmdVkhRTOx/HPrSC8c/fe/e0H0cNHj58 bT17fupMaRnvMyONHWbUcSk073vhJR8Wl OVST7Izg7W+cE5t04YfeIXBU8VnWmRC0Y9Uqb1C1bgoAQFEVSILSICJ/hfwgjPHgRImAKvswo MnN05Hj+iJolfKszDnWzOj+B NlV7TKodv94ywdkU+xqBR343qjQuVah0/CuJ6GgkZFYNYJ3V6Zj6NKo93Wf 5u+2eltpr2N6z9ON2m1425cB2mC5BK0e7vjw53Di/bxpPVjPDWsVFx7JqlzoyQufFpR6wWTfBmNS8cLys7ojI8Qaq 4S6t615bkJTJTkhuLn/akZq86Kq cW6gMlYr6ubuZW5N/yo1Kn79NK6GL0nPN oXyUhJvyHpxyVRYzrxcIKDMCuyVsDm1lHlc7wgfIbk5chOc7neT1939z0m79x42sQ07sAt7+OhvoAdHcAx9YMFe8CkYBMPwVfgl/BqON9Jg69LzAq5FyH8DSubsoQ= </latexit>

˜ Mσ = P

σ0hσ|ˆ

uα|σ0iMσ0

<latexit sha1_base64="glwj2S8H/Oet ugYrdEqw9rJu1A=">A Cu3icbVG7TsMwFL0J7/AqMLJEVAimKoGBLkgVLCxI F AKm3luE5icJzIdpCq0E9h7A+xAB/Czm2KRGm5VqTzuOfasYNMcG0879Oy5+YXFpeWV5zVtfWNzcrW9q1Oc0VZk6YiVfcB0UxwyZqG 8HuM8VIEgh2Fzydj/y7Z6Y0T+WN6WesnZBI8pBTYlBK xyGYICDgB4wKJAlQFCJQUOI/BIGuDqoa+yKSvcUnJLnyLpl5tc7wO4hTiMgURE4c9J9QRaX8wtMDzA9RCYgK9WXqUlDUBNznH/P1pndv1upejWvLHcW+D+g2tj/+nh/daKrbuXtoZfSPGHSUEG0bvleZtoFUYZTwQbOQ65ZRugTiVgLoSQJ0+2ivPuBu49Kzw1ThZ80bqlOJgqSaN1PAuxMiIn1tDcS/ NauQnr7YL DdM0vFGYS5ck7qjh3R7XDFqRB8BoYrjWV0aE0Wowed28BL86V+eBbdHNf+4dnTtVxtnMK5l2IU9OAQfTqABF3AFTaBW3epYkRXbpza1H20xbrWtn8wO/Ck7/wZ+6r0Q</latexit>

with uniqueness of
 injective MPS

D × D

<latexit sha1_base64="Z1WpvYhVUntjG8KQYecvH2JEXNI=">A B+XicbVC7SgNBFL3rM4mvqKXNYBCswm4stAyawjKCeWCyhNnJbDJkZnaZmQ2GJR8i2FgoYmPhn9j5NTp5FJp4 MLhnHu5954g5kwb1/1yVlbX1jc2M9nc1vbO7l5+/6Cuo0QRWiMRj1QzwJpyJmnNM NpM1YUi4DTRjC4mviNIVWaRfLWjGLqC9yTLGQEGysxqMAzG Ag I GBJVOvuAW3SnQMvHmpFDOxg937/f 1U7+s92NSCKoNIRjrVueGxs/xcow uk41040jTEZ4B5tWSqxoNpPp5eP0YlVui MlC1p0FT9PZFiofVIBLZTYNPXi95E/M9rJSa8 FMm48RQSWaLwoQjE6FJDKjLFCWGjyzBRDF7KyJ9rDAxNqycDcFbfHmZ1EtF76xYurFpXMIMGTiCYzgFD86hDNdQhRoQGMKj fnFSZ0n59V5m7WuOPOZQ/gD5+MHI36Udg= </latexit>

Ux, Uy, Uz

<latexit sha1_base64="m2fmrZuDBEHSopNJd3eTRxS/5YI=">A CX icjVHLTgIxFL0zivJSR124cNOIJi4MmcGFxhXRjUtM5JEAIZ3SgYbOI2 HiBM+h1/wB/wDd7rxK3RteSwUTPQmzT0959y0PXUjzqSy7VfDXFtPbWymM9lcfmt7x9rdq8kwFoRWSchD0XCxpJwFtKqY4rQRCYp9l9O6O7iZ6vUhFZKFwb0aRbTt417APEaw0lRoXcE fMCgoA8SPEigCmPo6D4BoRUED3p/pvtfvtE/fY8w7lgFu2jPCq0CZwEK5eOPp+dh7rPSsV5a3ZDEPg0U4VjKpmNHqp1goRjhdJxtxZJGmAxwjzY1DLBPZTuZpTNGJ5rpIi8UegUKzdjvEwn2pRz5rnb6WPXlsjYlf9OasfIu2wkLoljRgMwP8mKOVIimUaMuE5QoPtIAE8H0XRHpY4GJ0h+S1SE4y09eBbVS0Tkvlu6cQvka5pWGQziCU3DgAspwCxUdMYE3A4yMkTXezZSZN7fnVtNYzOzDjzIPvgBW+6at</latexit>

Z2 × Z2

<latexit sha1_base64="JHQfJqko41NQTU+UlgobuCZc8Gk=">A CJXicdVC7SgNBFL0bXzG+VgUbQRaDYBV2Y6FliI1lAuaByRJmJ5NkyOyDmbtCWPIzglj5FxY2FgYRbPRXnDwKTfQMA4dz mXmHi8SXKFtfxip eWV1bX0emZjc2t7x9zdq6owlpRVaChCWfeIYoIHrI cBatHkhHfE6zm9S/Hfu2WScXD4BoHEXN90g14h1OCWgrNI7gH wg 9MDTJ4EbGEIL8lpH4NpjoP7LtMysnbMnsBaJMyPZwkH5kz8Wn0otc9RshzT2WYBUEKUajh2hmxCJnAo2zDRjxSJC+6TLGpoGxGfKTSZbDq0TrbStTij1DdCaqD8nEuIrNfA9nfQJ9tS8Nxb/8hoxdi7chAdRjCyg04c6sbAwtMaVW 0uGU x0IRQyfVfLdojklDUxWZ0Cc78youkms85Zzm7rNsowhRpOIRjOAUHzqEAV1C lC4g2d4hZHxYLwYb8b7NJoyZjP78AvG1ze+rp9H</latexit>

ζα ∈ C

<latexit sha1_base64="gEe6VA1wYVwFEuvjshGp8eosypA=">A CG3icbZC7SgNBFIbPeo3xFrUSR aDYBV2tdAymMYyAXOBJITZySQZMju7zJwV4pLSd7CxzDuksrFQxEqw8Bl8CSeXQhP MDMf/38OM+d4oeAaHefLWlhcWl5ZTawl1zc2t7ZTO7slHUSKsiINRKAqHtFMcMmKyFGwSqgY8T3Byl43N/L t0xpHsgb7IWs7pO25C1OCRopSO3DAO6A QKBhmECAkLomHsAHKQ5fcNoFM+sGHLQb6TSTsYZhz0P7hTS2cNh4fv+aJhvpD5qzYBGPpNIBdG6 joh1mOikFPB+slapFlIaJe0WdWgJD7T9XjcW98+MUrTbgXKbIn2WP1dERNf657vmUyfYEfPeiPxP68aYeuyHnMZRsgknTzUioSNgT0alN3kilEUPQOEKm7+atMOUYSiGWfSDMGdbXkeSmcZ9z jFNx09gomkYADOIZTcOECsnANeSgChQd4ghd4tR6tZ+vNep+kLljTmj34E9bnD6oLnW0=</latexit>

|ζα| = 1

<latexit sha1_base64="3Mr8AevR6Vzsdhje4S5HlgnfbFw=">A C XicbZC7TsMwFIZPyq2UW4CRJWqFhIRUJTDAghTBwlgkepHaqHJcp7XqOJHtI W0Kwsr 8DGwgBCrLwBW98G9zJAy5Esf/r/c2Sf348Zlcq2R0ZuaXl dS2/XtjY3NreMXf3ajJKBCZVHLFINHwkCaOcVBV jDRiQVDoM1L3+1djv35HhKQRv1VpTLwQdTkNKEZKS5FpwgCe4R4IKEDQ1oyAQ w9fQ/gApy2WbL 9qSsRXBmUHKLreOnkZtW2uZ3qxPhJCRcY akbDp2rLwMCU xI8NCK5EkRriPuqSpkaOQSC+b DK0DrXSsYJI6MOVNVF/T2QolDINfd0ZItWT895Y/M9rJio49zLK40QRjqcPBQmzVGSNY7E6VBCsWKoBYUH1Xy3cQwJhpcMr6BCc+ZUXoXZSdk7L9o1Tci9hWnk4gCIcgQNn4MI1VKAKGB7gBd7g3Xg0Xo0P43PamjNmM/vwp4yvH/pLl8I=</latexit>

˜ Mσ = ζα U†

αMσUα

<latexit sha1_base64="RrQBP8zG5kKVB1+Tfgn1Q+ySUtc=">A CtXichVHBat AEB0pTZu4Sesmx1xETCGHYCTn4F4Kpr3kUkihTgyOYkarkb1kpRW7q4Aj/CU9 2N6za20H5OxHGgTBzrLsm/evJnZnU1KJa0Lw1+ev/Fi8+Wr e3W653dN2/b7/bOra6MoKHQSptRgpaULGjopFM0Kg1hni 6SK4/L+MXN2Ss1MU3Ny8pznFayEwKdEzpdgY/wIE BSkQ1OzlgMzMwELG/hdY8Lpi3rJq2kQ/snfLasd4whg5u+QMZHwMrbUaQ67wWLesl/I5 UVgnu161dzmb9fF/+tO2p2wGzYWrIPoAXQGh0d/fvd/fj+btO8uUy2qnAonF o7jsLSxTUaJ4WiReuyslSiuMYpjRkWmJON62bqi+A9M2mQacO7cEHD/ptRY27tPE9YmaOb2aexJflcbFy57ENcy6KsHBVi1SirVOB0sPzCIJWGhFNzBi M5LsGYoYGheOPbvEQoqdPXgfnvW50 u19jTqDT7CyLTiAQziC PowgFM4 wELr+eNP QSv+/HfupnK6nvPeTswyPz9T34WbvC</latexit>

thus the matrices satisfy nontrivial constraints

Mσ = ζα P

σ0hσ|ˆ

u†

α|σ0iU† αMσ0Uα

<latexit sha1_base64="62muYBchXOx1lbIfVE6jPV7r/UE=">A DJXichVJNa9tAEB0p/UjVj7jpsRcRE9qTkdJDcymY5tJLIYE6CTiuGa1X8pLVSuyuCq7in5IeCqZ/pZceEkrSnvIrcs9YDtSRDd1F8GbmvTc7q41yKYwNgr+Ou3Lv/oOHq4+8x0+ePltrPF/fN1mhGe+wTGb6MELDpVC8Y4WV/D XHN I8oPoeGdaP/jCtRGZ+mRHOe+lmCgRC4aWUlnjG0wgBQ LQzAQ wkfYQyfKWtAQFLV3lH0FThxEPqE STkxMeKVRCnT7p5xSvymBALQVFGkna+ekLRsOpZknrWbUBxQpuDrvU4qTlPiPHP1 s4f4c 7zrU/ZdPvGyC/zj3G82gFVTLXwThLWi2N6/ XJ56yW6/cX40yFiRcmWZRGO6YZDbXonaCib52DsqDM+RHWPCuwQVptz0yuovj/1Nygz8ONP0KetX2XlFiakxozQiZop2aOq1aXJZrVvYeLtXCpUXlis2axQX0reZP30y/kBozqwcEUCmBZ3VZ0PUyCw9LI8uIayPvAj2t1rhm9bWXthsv4fZWoWXsAGvIYS30IYPsEsXzJzvzk/nzDl3f7i/3N/uxYzqOreaF3BnuVc3f6nYgQ= </latexit>

α = x, y, z

<latexit sha1_base64="NvRZe4um2dVyv DpvaoGHSWPvAw=">A C XicbVC7SgNBFL0bXzG+Vi0VGQyChYTdWGgjBG0sEzAPSJYwO5lNhsw+mJkV1yWlNn6HnY2FEmz9Azu/wZ9w8ig0euAyZ865l5l73IgzqSzr08jMzS8sLmWXcyura+sb5uZWTYaxILRKQh6Khosl5SygVcU p41IUOy7nNbd/sXIr19TIVkYXKk o 6PuwHzGMFKS6FpwiNg4B T59n+ibABwQ3cASJrtu2mbcK1hjoL7GnJF/aHVa+7vaG5b 50eqEJPZpoAjHUjZtK1JOioVihN BrhVLGmHSx13a1DTAPpVO t5kgA60 kFeKHQFCo3VnxMp9qVMfFd3+lj15Kw3Ev/zmrHyTp2UBVGsaEAmD3kxRypEo1hQhwlKFE80wUQw/VdEelhgonR4OR2CPbvyX1IrFuzjQrFi50vnME WdmAfDsG EyjBJZShCgTu4Qle4NV4MJ6NofE2ac0Y05lt+AXj/RsK2Zfm</latexit>

for

ˆ uα = exp[−iπ ˆ Sα]

<latexit sha1_base64="+B71zaX5YxsShUfgVUMy9FzIosw=">A COXicbVDLSgMxFL1TX7W+Rl26GVqEglhmdKEboejGZUX7gOlYMmnahmYeJBlxGPoX/oK/4Ma/cFdw40IRt/6A6bSCb 2QcM6595Dc4 aMCm aQy2zsLi0vJ dza2tb2xu6ds7NRFEHJMqDljAGy4ShFGfVCWVjDRCTpDnMlJ3+xejfv2OcE D/0bGIXE81PVph2IklRToRXiEHiCQkEAEA2gpjoB mKpnihG4V8yGQ6CKhen967hWjtsph9PSC2bJTMuYB9YEFMr5 sHDsBxXWvpLsx3gyCO+xAwJYVtmKJ0EcUkxI4NcMxIkRLiPusRW0EceEU6Sbj4w9pXSNjoBV8eXRqr+dSTIEyL2XDXpIdkTs72R+F/PjmTn1EmoH0aS+Hj8UCdihgyMUYxGm3KCJYsVQJhT9VcD9xBHWKqwcyoEa3bleVA7KlnHJfNKpXEO48rCHuShCBacQBkuoQJVwPAEr/AOH9qz9qZ9al/j0Yw28ezCVGnfP2knpEo=</latexit>
slide-15
SLIDE 15

S is a half-odd integer

−ˆ 1

<latexit sha1_base64="6jVdSep1RFZxU9YoFaM/eXM9gTQ=">A B93icbZDLSsNAFIZPvNZ6q7p0Ey CG0tSBV0W3bisYC/QhjKZTpqhk0mYORFC6Iu4ceEFt76KO9/GaZuFtv4w8POfczhnPj8RXKPjfFsrq2vrG5ulrfL2zu7efuXgsK3jVFHWorGIVdcnmgkuWQs5CtZNFCORL1jH 9 O651HpjSP5QNmCfMiMpI84JSgiUI4hzcIgQBCDi5MBpWqU3NmspeNW5gqFGoOKl/9YUzTiEmkgmjdc50EvZwo5FSwSbmfapYQOiYj1jNWkohpL5/dPbFPT K0g1iZJ9Gepb8nchJpnUW+6YwIhnqxNg3/q/VSDK69nMskRSbpfFGQChtjewrBHnLFKIrMGEIVN7faNCSKUDSoygaCu/jlZdOu19yLWv3+stq4KXCU4BhO4MyAvI G3E TWkAN2id4gVcrs56td+tj3rpiFTNH8EfW5w9gDpBe</latexit>

=

<latexit sha1_base64="MWbL6R2hZsQThSAdMNvF0orSXwY=">A B6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQS9C0IvHBMwDkiXMTnqTMbOzy8ysE K+wIsHRbz6Sd78GyfJHjSxoKGo6qa7K0gE18Z1v53c2vrG5lZ+u7Czu7d/UDw8auo4VQwbLBaxagdUo+ASG4Ybge1EIY0Cga1gdDfzW0+oNI/lgxkn6Ed0IHnIGTVWqt/0i W37M5BVomXkRJkqPWKX91+zNI pWGCat3x3MT4E6oMZwKnhW6qMaFsRAfYsVTSCLU/mR86JWdW6ZMwVrakIXP198SERlqPo8B2RtQM9bI3E/ zOqkJr/0Jl0lqUL FojAVxMRk9jXpc4XMiLEl ClubyVsSBVlxmZTsCF4y +vkmal7F2UK/XLUvU2iyMPJ3AK5+DBFVThHmrQA YIz/AKb86j8+K8Ox+L1pyTzRzDHzifP4 bjMU=</latexit>

Proof of Theorem 1’

= ζxζyζz Mσ

<latexit sha1_base64="zJrc3usi3Xe+rxZgck7fbrTDt3M=">A CY3icdZHLSgMxFIbPjPd6Gy87EYJFcCFlRgXdCEU3bgQFa4VaSybNtMFkZkjOiO3Qx/GF3Llz43uYTrtQaw8E/vzfOST5E6ZSGPT9D8edmZ2bX1hcKi2vrK6texub9ybJNOM1lshEP4TUcCliXkOBkj+kmlMVSl4Pny+HvP7CtRFJfIe9lDcV7cQiEoyitRLvHM7hDfrA YFC 3K706CAwCsMp DeVNIvyKFdyhKELhiIL +2/lPRZ0BAp6CDl f2K35RZFIEY1G cd20vPfHdsIyxWNk hrTCPwUmznVKJjkg9JjZnhK2TPt8IaVMVXcNPMiowHZt06bRIm2K0ZSuD8ncq M6anQdiqKXfOXDc3/WCPD6KyZizjNkMdsdFCUSYIJGQZO2kJzhrJnBWVa2LsS1qWaMrTfUrIhBH+fPCnujyrBceXo9qRcvRjHsQg7sAcHEMApVOEKbqAGD 6deWfd8Zwvd9nd LdHra4zntmCX+XufgNBOaU7</latexit>

(ζα)2 = −1

<latexit sha1_base64="R9RCx/0S DoE9MTp3GkZQjfAG5M=">A CD3icbZDLSgMxFIbP1Fut 9Eu3YQWQRHLTF3oRi 6cVnBXqAdSybNtKGZC0lG Ie+gRvBV/AF3LhQxK1bd30b08tCqwdCPv7/HJLzuxFnUlnWyMgsLC4tr2RXc2vrG5tb5vZOXYaxILRGQh6Kposl5SygNcU p81IUOy7nDbcwcXYb9xSIVkYXKsko 6PewHzGMFKS6GZh314gjugoABDRzMGDhH09X0AN1CGMzgCu2MWrZI1KfQX7BkUK4X24eOoklQ75le7G5LYp4EiHEvZsq1IOSkWihFOh7l2LGmEyQD3aEtjgH0qnXSyzxDta WLvFDoEyg0UX9OpNiXMvFd3elj1Zfz3lj8z2vFyjt1UhZEsaIBmT7kxRypEI3DQV0mKFE80YCJYPqviPSxwETpCHM6BHt+5b9QL5fs45J1ZRcr5zCtLOxCQ dtw lU4BKqUAMC9/AMr/BmPBgvxrvxMW3NGLOZP wq4/MbEmuYEQ= </latexit>

ζxζyζz = 1

<latexit sha1_base64="QMP6W75ng/LlKSQzTJQ/f5LDa8E=">A CO3icdZC7TgJBFIbP4g3BC2p MxFNtCG7WmhjQrSx EQuCWzI7DALE2YvmZklwobH8RWMjZVvYGdjY6Extlg7XAoF+ZNJ/vm/czJzjhNyJpVpvhiJhcWl5ZXkaiq9tr6xmdnaLskgEoQWScADUXGwpJz5tKiY4rQSCo 9h9Oy074c8nKHCskC/0Z1Q2p7uOkzlxGsdBRkjuAOekB AY 6xPomwAMEt9CfQ7pzSU+Tc7DqmayZM0dCs8a mGx+f3D/1El/F+qZ51ojIJFHfU 4lrJqmaGyYywUI5z2U7VI0hCTNm7SqrY+9qi049HsfXSgkwZyA6GPr9Ao/d0RY0/KrufoSg+rlpxmw/A/Vo2Ue2bHzA8jRX0yfsiNOFIBGi4SNZigRPGuNpgIpv+KSAsLTJRed0ovwZoe daUjnPWSc68trL5CxgrCbuwB4dgwSnk4QoKUAQCD/AK7/BhPBpvxqfxNS5NGJOeHfgjY/AD+U6l4w= </latexit>

contradiction!

Mσ0

<latexit sha1_base64="UMLFfmKyz Y1GTHJljHxgCNMJKw=">A CEXicbVC7TsMwFL3hWcorPDaWiArBVCUFCcYKFhakItGH1IbKcZ3Wqp1EtoNURfkF r6hf8DCAEKsbGz8DW6aAVqOZPn4nHt1fY8XMSqVbX8bC4tLy urhbXi+sbm1ra5s9uQYSw qeOQhaLlIUkYDUhdUcVIKxIEcY+Rpje8mvjNByIkDYM7NYqIy1E/oD7FSGkpNPdhDBwQKBiAB 8SuIEU7vU91m8K/cw9hrRrluy ncGaJ05OSpCj1jW/Or0Qx5wECjMkZduxI+UmSCiKGUmLnViSCOEh6pO2pgHiRLpJtlFqHWmlZ/mh0CdQVqb+7kgQl3LEPV3JkRrIW 8i/ue1Y+VfuAkNoliRAE8H+TGzVGhN4rF6VBCs2EgThAXVf7XwA mElQ6xqENwZle J41K2TktV27PStXLPI4CHMAhnIAD51CFa6hBHTA8wjO8wpvxZLwY78bHtHTByHv24A+Mzx8Iq5b </latexit>

=

<latexit sha1_base64="MWbL6R2hZsQThSAdMNvF0orSXwY=">A B6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQS9C0IvHBMwDkiXMTnqTMbOzy8ysE K+wIsHRbz6Sd78GyfJHjSxoKGo6qa7K0gE18Z1v53c2vrG5lZ+u7Czu7d/UDw8auo4VQwbLBaxagdUo+ASG4Ybge1EIY0Cga1gdDfzW0+oNI/lgxkn6Ed0IHnIGTVWqt/0i W37M5BVomXkRJkqPWKX91+zNI pWGCat3x3MT4E6oMZwKnhW6qMaFsRAfYsVTSCLU/mR86JWdW6ZMwVrakIXP198SERlqPo8B2RtQM9bI3E/ zOqkJr/0Jl0lqUL FojAVxMRk9jXpc4XMiLEl ClubyVsSBVlxmZTsCF4y +vkmal7F2UK/XLUvU2iyMPJ3AK5+DBFVThHmrQA YIz/AKb86j8+K8Ox+L1pyTzRzDHzifP4 bjMU=</latexit>

=

<latexit sha1_base64="MWbL6R2hZsQThSAdMNvF0orSXwY=">A B6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQS9C0IvHBMwDkiXMTnqTMbOzy8ysE K+wIsHRbz6Sd78GyfJHjSxoKGo6qa7K0gE18Z1v53c2vrG5lZ+u7Czu7d/UDw8auo4VQwbLBaxagdUo+ASG4Ybge1EIY0Cga1gdDfzW0+oNI/lgxkn6Ed0IHnIGTVWqt/0i W37M5BVomXkRJkqPWKX91+zNI pWGCat3x3MT4E6oMZwKnhW6qMaFsRAfYsVTSCLU/mR86JWdW6ZMwVrakIXP198SERlqPo8B2RtQM9bI3E/ zOqkJr/0Jl0lqUL FojAVxMRk9jXpc4XMiLEl ClubyVsSBVlxmZTsCF4y +vkmal7F2UK/XLUvU2iyMPJ3AK5+DBFVThHmrQA YIz/AKb86j8+K8Ox+L1pyTzRzDHzifP4 bjMU=</latexit>

ˆ 1

<latexit sha1_base64="y1iKw8v6jIU3nQUiB41IuGtK0j4=">A B9XicbVBNS8NAEJ34WetX1aOXxSJ4Kk V9Fj04rGC/YA2lM120y7dbMLuRCmh/8OLB0W9+l+8+W/ctjlo64OBx3sz MwLEikMu 63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LRzdRvPXBtRKzucZxwP6IDJULBKFp AO8wBAoIGXgw6ZXKbsWdgSwTLydlyFHvlb6 /ZilEVfIJDWm47kJ+hnVKJjk 2I3NTyhbEQHvGOpohE3fja7ekJOrdInYaxtKSQz9fdERiNjxlFgOyOKQ7PoTcX/vE6K4ZWfCZWkyBWbLwpTSTAm0whIX2jOUI4toUwLeythQ6opQxtU0YbgLb68TJrVindeqd5dlGvXeRwFOIYTOLNBXkINbqEODWCg4Qle4NV5dJ6dN+dj3r i5DNH8AfO5w/Q2ZAe</latexit>

Mσ0

<latexit sha1_base64="UMLFfmKyz Y1GTHJljHxgCNMJKw=">A CEXicbVC7TsMwFL3hWcorPDaWiArBVCUFCcYKFhakItGH1IbKcZ3Wqp1EtoNURfkF r6hf8DCAEKsbGz8DW6aAVqOZPn4nHt1fY8XMSqVbX8bC4tLy urhbXi+sbm1ra5s9uQYSw qeOQhaLlIUkYDUhdUcVIKxIEcY+Rpje8mvjNByIkDYM7NYqIy1E/oD7FSGkpNPdhDBwQKBiAB 8SuIEU7vU91m8K/cw9hrRrluy ncGaJ05OSpCj1jW/Or0Qx5wECjMkZduxI+UmSCiKGUmLnViSCOEh6pO2pgHiRLpJtlFqHWmlZ/mh0CdQVqb+7kgQl3LEPV3JkRrIW 8i/ue1Y+VfuAkNoliRAE8H+TGzVGhN4rF6VBCs2EgThAXVf7XwA mElQ6xqENwZle J41K2TktV27PStXLPI4CHMAhnIAD51CFa6hBHTA8wjO8wpvxZLwY78bHtHTByHv24A+Mzx8Iq5b </latexit>

=

<latexit sha1_base64="MWbL6R2hZsQThSAdMNvF0orSXwY=">A B6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQS9C0IvHBMwDkiXMTnqTMbOzy8ysE K+wIsHRbz6Sd78GyfJHjSxoKGo6qa7K0gE18Z1v53c2vrG5lZ+u7Czu7d/UDw8auo4VQwbLBaxagdUo+ASG4Ybge1EIY0Cga1gdDfzW0+oNI/lgxkn6Ed0IHnIGTVWqt/0i W37M5BVomXkRJkqPWKX91+zNI pWGCat3x3MT4E6oMZwKnhW6qMaFsRAfYsVTSCLU/mR86JWdW6ZMwVrakIXP198SERlqPo8B2RtQM9bI3E/ zOqkJr/0Jl0lqUL FojAVxMRk9jXpc4XMiLEl ClubyVsSBVlxmZTsCF4y +vkmal7F2UK/XLUvU2iyMPJ3AK5+DBFVThHmrQA YIz/AKb86j8+K8Ox+L1pyTzRzDHzifP4 bjMU=</latexit>

Mσ = (ζα)2 P

σ0hσ|(ˆ

u†

α)2|σ0i(U† α)2Mσ0(Uα)2 = (ζα)2 Mσ

<latexit sha1_base64="VunvQhqfSakFD9X1df6NzAvBAnE=">A DkXicjVJda9RAFL1N/KhR2619 CW4FBV0SVbB9mHLoi+C BXctrBdl5vZSXboZBJmJoU13Z/Sx/4hEdQf4rt3syvuF9gJgZNz 0nd2aiXApjg+DXhuPeun3n7uY97/6Dh1vbtZ1HxyYrNOMdlslMn0ZouBSKd6ywkp/m MaSX4Snb+b1E8u DYiU5/tKOe9FBMlYsHQEpXVvsM1pIBgYQgGYijhI4zhC7EGBCRVrQXP6PsrcFIh9AkjSMipA+E5aZuVuiBtn/rnO5+Sl0eMJKyIk+QxX7+snIdVfk O0+QBfSf0cNBr0y6XMq5J98/fqzwXZ+qQ86LTcs7fOdbtxepMN0mYOrbg5Q1278X/T6FfqweNoFr+KghnoN7e+/3zx5WXHPVr384G StSriyTaEw3DHLbK1FbwSQfe2eF4Tmyc0x4l6DClJteWd2osb9HzMCPM02vsn7FzneUmBozSiNSpmiHZrk2IdfVuoWN93ulUHlhuWLToLiQvs38yfX0B0JzZuWIADIt6F9 NkSNzNIl9mgTwuWRV8FxsxG+ajQ/hfX2W5iuTXgMT+g QngDbXgPR3RgzNlyXjst59Dd Q/ctjvTOhuznl1YWO6HP4L45ho=</latexit>

Mσ = ζx P

σ0hσ|ˆ

u†

x|σ0iU† xMσ0Ux

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= ζxζy P

σ0hσ|(ˆ

uyˆ ux)†|σ0i(UyUx)†Mσ0UyUx

<latexit sha1_base64="+suSdkPo01qHFndNL dUvukA2p8=">A DlXicrVJNT9tAEB1s2lL3g5QeOHBZNUJtL5FND+WCFChCvSCo1ABSCNF6s3ZW7K6t3X VYPJTOPYP9dCPH9J7xw5SaRI4MStLb957s7MeTZxLYV0Y/l7w/MUHDx8tPQ6ePH32fLnxYuXIZoVhvM ymZmTmFouheYdJ5zkJ7nhVMWSH8fnHyr9+As3VmT6sxvlvKdoqkUiGHVIZY0fsAXf4AI4OKDQhxIzAwoIfIXxLcqoViwUmE14CwJSzCi8Ri1ARiLWyEmsv6lfwhvMh4gcVhbonr35NrV60Vs4w3yAjhQPR+Vyqn/l/tc7qPup+sYh+hK8rzO3692e+b2nq/bRdTZnIvf0gn6jGb COsgsiK5Bs73+59fPqyA97De+nw4yVi uHZPU2m4U5q5XUuMEk3wcnBaW5 Sd05R3EWq uO2V9VaNyToyA5JkBj/tSM3erCipsnakYnQq6oZ2WqvIeVq3cMlmrxQ6LxzXbNIoKSRxGalWlAyE4czJEQLKjMC3EjakhjKHixzgEKLpX54FRxut6F1r41PUbO/AJ ZgDV7hOkTwHtrwEQ5x Mxb8Ta9bW/HX/W3/F1/b2L1Fq5rXsJ/4R/8BS/06uM=</latexit>

= ζxζyζz P

σ0hσ|(ˆ

uzˆ uyˆ ux)†|σ0i(UzUyUx)†Mσ0UzUyUx

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Mσ = ζα P

σ0hσ|ˆ

u†

α|σ0iU† αMσ0Uα

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α = x, y, z

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for thus the matrices satisfy nontrivial constraints we then find and

ˆ uα = exp[−iπ ˆ Sα]

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slide-16
SLIDE 16

Theorem for MPS

THEOREM 1’: There cannot be a translation invariant and invariant injective MPS for

Z2 × Z2

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S = 1

2, 3 2, 5 2, . . .

<latexit sha1_base64="ZhW2WS+CyP2QBxz u6GcZX+r6lc=">A CQXicdZFLSwMxEMdnfdb6WvXoZbEICqXstohehKogHivaB7SlZN sG5rdLElWKEs/ike/h e/gTfvHvSgSK9eTB8H2+pAhj+/mWGSf9yQUals+8WYm19YXFpOrCRX19Y3Ns2t7ZLk cCkiDnjouIiSRgNSF RxUglFAT5LiNlt3MxqJfviJCUB7eqG5K6j1oB9ShGSiNupuEGTuEBPBCA EM DvR0zuqcnuC5f/jRBGfQBA4KZMNM2Rl7GNascMYilT+0L9/692eFhvlca3Ic+SRQmCEpq4 dqnqMhK YkV6yFk SItxBLVLVMkA+kfV46EDP2tekaXlc6BMoa0h/T8TIl7Lru7rTR6otp2sD+FetGinvpB7TI wUCfBokRcxS3FrYKfVpIJgxbpaICyovquF20g rLTpSW2CM/3kWVHKZpxcJnvtpPLnMIoE7MIeHOjPOIY8XE BitrsR3iFD/g0nox348voj1rnjPHMDkyE8f0DnuqlOA= </latexit>

Chen, Gu, Wen 2011 Watanabe, Po, Vishwanath, Zaletel 2013

nontrivial projective representation of the on-site
 symmetry is inconsistent with the existence of an injective MPS

Matsui 2001

Z2 × Z2

<latexit sha1_base64="JHQfJqko41NQTU+UlgobuCZc8Gk=">A CJXicdVC7SgNBFL0bXzG+VgUbQRaDYBV2Y6FliI1lAuaByRJmJ5NkyOyDmbtCWPIzglj5FxY2FgYRbPRXnDwKTfQMA4dz mXmHi8SXKFtfxip eWV1bX0emZjc2t7x9zdq6owlpRVaChCWfeIYoIHrI cBatHkhHfE6zm9S/Hfu2WScXD4BoHEXN90g14h1OCWgrNI7gH wg 9MDTJ4EbGEIL8lpH4NpjoP7LtMysnbMnsBaJMyPZwkH5kz8Wn0otc9RshzT2WYBUEKUajh2hmxCJnAo2zDRjxSJC+6TLGpoGxGfKTSZbDq0TrbStTij1DdCaqD8nEuIrNfA9nfQJ9tS8Nxb/8hoxdi7chAdRjCyg04c6sbAwtMaVW 0uGU x0IRQyfVfLdojklDUxWZ0Cc78youkms85Zzm7rNsowhRpOIRjOAUHzqEAV1C lC4g2d4hZHxYLwYb8b7NJoyZjP78AvG1ze+rp9H</latexit>

Mσ = ζα P

σ0hσ|ˆ

u†

α|σ0iU† αMσ0Uα

<latexit sha1_base64="62muYBchXOx1lbIfVE6jPV7r/UE=">A DJXichVJNa9tAEB0p/UjVj7jpsRcRE9qTkdJDcymY5tJLIYE6CTiuGa1X8pLVSuyuCq7in5IeCqZ/pZceEkrSnvIrcs9YDtSRDd1F8GbmvTc7q41yKYwNgr+Ou3Lv/oOHq4+8x0+ePltrPF/fN1mhGe+wTGb6MELDpVC8Y4WV/D XHN I8oPoeGdaP/jCtRGZ+mRHOe+lmCgRC4aWUlnjG0wgBQ LQzAQ wkfYQyfKWtAQFLV3lH0FThxEPqE STkxMeKVRCnT7p5xSvymBALQVFGkna+ekLRsOpZknrWbUBxQpuDrvU4qTlPiPHP1 s4f4c 7zrU/ZdPvGyC/zj3G82gFVTLXwThLWi2N6/ XJ56yW6/cX40yFiRcmWZRGO6YZDbXonaCib52DsqDM+RHWPCuwQVptz0yuovj/1Nygz8ONP0KetX2XlFiakxozQiZop2aOq1aXJZrVvYeLtXCpUXlis2axQX0reZP30y/kBozqwcEUCmBZ3VZ0PUyCw9LI8uIayPvAj2t1rhm9bWXthsv4fZWoWXsAGvIYS30IYPsEsXzJzvzk/nzDl3f7i/3N/uxYzqOreaF3BnuVc3f6nYgQ= </latexit>

projective 
 representation genuine 
 representation contradiction!

Z2 × Z2

<latexit sha1_base64="JHQfJqko41NQTU+UlgobuCZc8Gk=">A CJXicdVC7SgNBFL0bXzG+VgUbQRaDYBV2Y6FliI1lAuaByRJmJ5NkyOyDmbtCWPIzglj5FxY2FgYRbPRXnDwKTfQMA4dz mXmHi8SXKFtfxip eWV1bX0emZjc2t7x9zdq6owlpRVaChCWfeIYoIHrI cBatHkhHfE6zm9S/Hfu2WScXD4BoHEXN90g14h1OCWgrNI7gH wg 9MDTJ4EbGEIL8lpH4NpjoP7LtMysnbMnsBaJMyPZwkH5kz8Wn0otc9RshzT2WYBUEKUajh2hmxCJnAo2zDRjxSJC+6TLGpoGxGfKTSZbDq0TrbStTij1DdCaqD8nEuIrNfA9nfQJ9tS8Nxb/8hoxdi7chAdRjCyg04c6sbAwtMaVW 0uGU x0IRQyfVfLdojklDUxWZ0Cc78youkms85Zzm7rNsowhRpOIRjOAUHzqEAV1C lC4g2d4hZHxYLwYb8b7NJoyZjP78AvG1ze+rp9H</latexit>

if S is an integer, there are translation and 
 invariant injective MPS, such as the AKLT state

slide-17
SLIDE 17

Toward the Full Theorem

slide-18
SLIDE 18

From Theorem 1’ to Theorem 1

THEOREM 1’: There cannot be a translation invariant and invariant injective MPS for

Z2 × Z2

<latexit sha1_base64="JHQfJqko41NQTU+UlgobuCZc8Gk=">A CJXicdVC7SgNBFL0bXzG+VgUbQRaDYBV2Y6FliI1lAuaByRJmJ5NkyOyDmbtCWPIzglj5FxY2FgYRbPRXnDwKTfQMA4dz mXmHi8SXKFtfxip eWV1bX0emZjc2t7x9zdq6owlpRVaChCWfeIYoIHrI cBatHkhHfE6zm9S/Hfu2WScXD4BoHEXN90g14h1OCWgrNI7gH wg 9MDTJ4EbGEIL8lpH4NpjoP7LtMysnbMnsBaJMyPZwkH5kz8Wn0otc9RshzT2WYBUEKUajh2hmxCJnAo2zDRjxSJC+6TLGpoGxGfKTSZbDq0TrbStTij1DdCaqD8nEuIrNfA9nfQJ9tS8Nxb/8hoxdi7chAdRjCyg04c6sbAwtMaVW 0uGU x0IRQyfVfLdojklDUxWZ0Cc78youkms85Zzm7rNsowhRpOIRjOAUHzqEAV1C lC4g2d4hZHxYLwYb8b7NJoyZjP78AvG1ze+rp9H</latexit>

S = 1

2, 3 2, 5 2, . . .

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THEOREM 1: Consider a quantum spin chain with
 and a short-ranged Hamiltonian that is invariant under translation and

  • transformation. Then it can never be the case that

the corresponding ground state is unique and accompanied by a nonzero gap.

Z2 × Z2

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S = 1

2, 3 2, 5 2, . . .

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we have proved this seems to imply the desired

slide-19
SLIDE 19

From Theorem 1’ to Theorem 1

assume that the GS is unique and gapped this contradicts Theorem 1’ this “proof” looks plausible, but does not work!!! the approximation by MPS is not that precise Hamiltonian has translation and symmetry

Z2 × Z2

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the GS is disordered, and translationally and 
 invariant

Z2 × Z2

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there exists an injective MPS that is translationally and invariant

Z2 × Z2

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disordered states can be approximated by MPS

the proof of Theorem 1 makes an essential use of

  • perator algebraic formulation
slide-20
SLIDE 20

Opinions of a mathematical physicist on

  • perator algebraic approaches to spin systems

mid 20’s (posdoc) early 20’s (student)

In most cases physically interesting results are proved in finite systems without operator algebra… It’s useful for formulating various concepts of infinite systems, but not for proving concrete

  • results. We can work within finite systems to prove

important and interesting results! Hey! Here’s a formulation that allows us to treat infinite systems as they are! Probably we can solve phase transitions, renormalization, and everything!

slide-21
SLIDE 21

Opinions of a mathematical physicist on

  • perator algebraic approaches to spin systems

late 50’s (old guy)

WOW! 
 IT’S USEFUL!!!!!

Ogata 2018, 2019 Ogata, Tasaki 2018

index theorems for SPT phases

slide-22
SLIDE 22

projective 
 representation genuine 
 representation

the core of the proof of Theorem 1

σ = −S, . . . , S

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ζα ∈ C

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|ζα| = 1

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α = x, y, z

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transformation of

Z2 × Z2

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exactly the same transformation rule as in MPS! the same argument leads to contradiction Matsui 2001 representation of the Cuntz algebra if the g.s. is unique and accompanied by a gap, there is a

cσ ∈ B( ˜ HR)

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Mσ = ζα P

σ0hσ|ˆ

u†

α|σ0iU† αMσ0Uα

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cσ = ζα P

σ0hσ|ˆ

u†

α|σ0i ˜

Rα(cσ0)

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˜ Rx, ˜ Ry, ˜ Rz

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B( ˜ HR)

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*-automorphisms on give a genuine representation of Z2 × Z2

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cσ = ζα P

σ0hσ|ˆ

u†

α|σ0i ˜

Rα(cσ0)

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ˆ uα = exp[−iπ ˆ Sα]

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slide-23
SLIDE 23

= |σ, σ1, σ2, σ3, σ4, . . .i

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cσ |σ1, σ2, σ3, σ4, . . .i

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“infinite dimension version” of matrices for MPS

(cσ)σ=−S,...,S

related to the shift in 
 a half-finite chain

(cσ)∗cσ0 = δσ,σ0ˆ 1

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πR(|σihσ0| ⌦ ˆ 1[1,1)) = cσ(cσ0)⇤

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P

σ cσπR( ˆ

A)(cσ)∗ = πR(τ( ˆ A))

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the core of the proof of Theorem 1

σ = −S, . . . , S

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representation of the Cuntz algebra cσ ∈ B( ˜

HR)

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if the g.s. is unique and accompanied by a gap, there is a non-rigorous picture! r e m i n d s u s

  • f

H i l b e r t ’ s h

  • t

e l 
 w i t h i n fi n i t e l y m a n y r

  • m

s !

slide-24
SLIDE 24

Outline of the Formulation and the Proof

slide-25
SLIDE 25

Operator algebraic formulation of 
 an infinite quantum spin chain 1

/ 4

a state is a linear function the set of all polynomials of states on

C∗

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  • algebra

Aloc

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ˆ S(α)

j

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j ∈ Z

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α = x, y, z

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A := Aloc

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A

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(completion w.r.t. the operator norm)

ρ : A → C

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A

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ρ(ˆ 1) = 1

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ρ( ˆ A∗ ˆ A) ≥ 0

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ˆ A ∈ A

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such that and for any the set of all local operators + a little bit more

ρ( ˆ A)

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ˆ A

<latexit sha1_base64="4ujTzdT7UyWrB0dge315ATWE6o0=">A B9XicbVDLTgJBEOzF +IL9ehlIjHxRHbR I+oF4+YyCOBDZkdBpgwO7uZ6dWQDf/hxYNGvfov3vwbB9iDgpV0UqnqTndXE th0HW/ndzK6tr6Rn6zsLW9s7tX3D9omCjRjNdZJCPdCqjhUiheR4GSt2LNaRhI3gxGN1O/+cC1EZG6x3HM/ZAOlOgLRtFKA3iHIVBASOEKJt1iyS27M5Bl4mWkB lq3eJXpxexJOQKmaTGtD03Rj+lGgWTfFLoJIbHlI3ogLctVT kxk9nV0/IiV 6pB9pWwrJTP09kdLQmHEY2M6Q4tAselPxP6+dYP/ST4WKE+SKzRf1E0kwItMISE9ozlCOLaFMC3srYUOqKUMbVMG 4C2+vEwalbJ3Vq7cnZeq1 kceTiCYzgFDy6gCrdQgzow0PAEL/DqPDrPzpvzMW/NOdnMIfyB8/kD6TmQLg= </latexit>

the expectation value of in the state (Rem: the set of all states is weak-* compact)

slide-26
SLIDE 26

Operator algebraic formulation of 
 an infinite quantum spin chain 2/

4

formal Hamiltonian Hamiltonian and commutator is well-defined for in a finite system

ˆ H = P

j∈Z ˆ

hj

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ˆ A ∈ Aloc

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[ ˆ H, ˆ A] = [P`

j=−` ˆ

hj, ˆ A]

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`

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ground states

ω( ˆ A∗[ ˆ H, ˆ A]) ≥ 0

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ˆ A ∈ Aloc

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hGS| ˆ A∗[ ˆ H, ˆ A]|GSi = hGS| ˆ A∗ ˆ H ˆ A|GSi EGShGS| ˆ A∗ ˆ A|GSi 0

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unique gapped ground states

ω( ˆ A∗[ ˆ H, ˆ A]) ≥ γ ω( ˆ A∗ ˆ A)

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ω( ˆ A) = 0

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with for sufficiently large commutator

ω

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a state is a g.s. iff for any a unique g.s. is accompanied by a nonzero gap iff there exists such that

γ > 0

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ω

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ˆ A ∈ Aloc

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for any with

ˆ hj ∈ Aloc

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slide-27
SLIDE 27

Operator algebraic formulation of 
 an infinite quantum spin chain 3/

4

GNS (Gelfand-Naimark-Segal) construction given a state on , one can define

A

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ρ

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H

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a separable Hilbert space a representation of on , i.e., s.t.

π

<latexit sha1_base64="wF8isoF ZDRzkbcx6qDWr JEFCQ=">A B+XicbVBNSwMxEJ31s9avqkcvwSJ4kLJbBT0WvXisYD+gXUo2nW1Ds9klyQpl6b/wquB EK/+Gv01pu0etPUNA483b8jkBYng2rjul7Oyura+sVnYKm7v7O7tlw4OmzpOFcMGi0Ws2gHVKLjEhuFGYDtRSKNAYCsY3U7nrUdUmsfywYwT9CM6kDzkjBorIbxBCokt3iuV3Yo7A1kmXk7KkKPeK313+zFLI5SGCap1x3MT42dUGc4ETordVGNC2YgOsGOp BFqP5tdPCGnVumTMFa2pSEz9fdGRiOtx1FgnRE1Q704m4rnaTJQiKP/PJ3UhNd+xmWSGpRs/mCYCmJiMo2B9LlCZsTYEsoUtzcTNqSKMmPDKtowvMWvL5NmteJdVKr3l+XaTR5LAY7hBM7AgyuowR3UoQEMJDzBM7w4mfPqvDsfc+uKk+8cwR84nz/W/pKf</latexit>

π : A → B(H)

<latexit sha1_base64="r9EA0fbJi2S2x2J5irhGHtYf79Y=">A COXicbZB axNBFMf VmtjtDa1Ry+DUbBQwm4UKj3V9uIxBdMWkqXMTt4mQ2Z3lpm3Qljycbz6BfwaXiwo9CZe/QJ9SfYQW98wzJ/ B8z75cURnsKwx/BxoOHm4+2Go+bT5 uP9tp7T4/97Z0Cv KGusuE+nR6Bz7pMngZeFQZonBi2R6uri/+IzOa5t/olmBcSbHuU61ksSWb 2CL1BCwUvDEesMJB MIAXHagoVfIA5+wQWBJzAm7WM4tNw4iMn9q9a7bATLkvcF1Et2lBX76p1PRxZVWaYkzLS+0EUFhRX0pFWBufNYemxkGoqxzhgmcsMfVwtJ56L1+yMRGod75zE0l3vqGTm/SxLOJlJmvi7dwvzoCzGDnH6v8ygpPR9XOm8KAlztXowLY0gKxY xUg7VGRmLKRymv8s1EQ6qYh NxlGdHf0+ K824nedrpn79rHJzW BryAlw 5gkM4ZrA96DPmr/AdfsKv4FtwE/wO/qyiG0Hdswf/VPD3Fl1ToX0=</latexit>

H

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A

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π(α ˆ A + β ˆ B) = α π( ˆ A) + β π( ˆ B)

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π( ˆ A ˆ B) = π( ˆ A) π( ˆ B)

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π( ˆ A∗) = π( ˆ A)∗

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π(ˆ 1) = ˆ 1

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the set of all bounded operators

a vector s.t. for any

Ω ∈ H

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ρ( ˆ A) = hΩ, π( ˆ A)Ωi

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ˆ A ∈ A

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(Hρ, πρ, Ωρ)

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(H, π, Ω)

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  • r, more precisely , is the GNS triple

is a physical Hilbert space that consists of the state (which is now ) and other states “ close” to it

ρ

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H

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what is the Hilbert space of the model?

H∞ := N

j∈Z C2S+1

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is too large (physically and mathematically)

slide-28
SLIDE 28

for we define the representation by

Operator algebraic formulation of 
 an infinite quantum spin chain 4/

4

GNS (Gelfand-Naimark-Segal) construction the idea of the construction given a state on , one can define

A

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ρ

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define an inner product in by h ˆ

A, ˆ Bi := ρ( ˆ A∗ ˆ B)

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H

<latexit sha1_base64="FsN4O3CoBnpvgEKcEvKL3L8ej3k=">A B/XicbZDLSgMxFIbP1Fut 6pLN8EiuCozVdBl0Y3LCvYC7VAyadqGZjJDckaoQ/FV3HShiFvfw51vY9rOQlsPhPz8/znk5AtiKQy67reTW1vf2NzKbxd2dvf2D4qHRw0TJZrxOotkpFsBNVwKxesoUPJWrDkNA8mbweh2ljcfuTYiUg84jrkf0oESfcEoWmsEUwiBAsIQmL0lpHAHk26x5JbdeZFV4W iBFnVusWvTi9iScgVMkmNaXtujH5KNQom+aTQSQyPKRvRAW9bqWjIjZ/Ot5+QM+v0SD/S9igkc/f3REpDY8ZhYDtDikOznM3M/7J2gv1rPxUqTpArtnion0iCEZmhID2hOUM5toIyLeyuhA2p gwtsIKF4C1/eVU0KmXvoly5vyxVbzIceTiBUzgHD6 ganHWoG7hPsELvMKb8+xMnXfnY9Gac7KZY/hTzucPpdOSHg= </latexit>

a separable Hilbert space a representation of on , i.e., s.t.

π

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π : A → B(H)

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H

<latexit sha1_base64="FsN4O3CoBnpvgEKcEvKL3L8ej3k=">A B/XicbZDLSgMxFIbP1Fut 6pLN8EiuCozVdBl0Y3LCvYC7VAyadqGZjJDckaoQ/FV3HShiFvfw51vY9rOQlsPhPz8/znk5AtiKQy67reTW1vf2NzKbxd2dvf2D4qHRw0TJZrxOotkpFsBNVwKxesoUPJWrDkNA8mbweh2ljcfuTYiUg84jrkf0oESfcEoWmsEUwiBAsIQmL0lpHAHk26x5JbdeZFV4W iBFnVusWvTi9iScgVMkmNaXtujH5KNQom+aTQSQyPKRvRAW9bqWjIjZ/Ot5+QM+v0SD/S9igkc/f3REpDY8ZhYDtDikOznM3M/7J2gv1rPxUqTpArtnion0iCEZmhID2hOUM5toIyLeyuhA2p gwtsIKF4C1/eVU0KmXvoly5vyxVbzIceTiBUzgHD6 ganHWoG7hPsELvMKb8+xMnXfnY9Gac7KZY/hTzucPpdOSHg= </latexit>

A

<latexit sha1_base64="0s3NyOFBu nr3d39nAXIGRPs1+8=">A B/3icbZBNSwMxEIZn/az1q+rRS7AInspuFfRY9eKxgv2AtpRsm 1Dk90lmRXK2oN/R APinj1b3jz35i2e9DWFwIP78w k9ePpTDout/O0vLK6tp6biO/ubW9s1vY26+bKNGM1 gkI930qeFShLyGAiVvxp T5Uve8IfXk3rjnmsjovAORzHvKNoPRSAYRWtJeAYF BAGEIC2NIQULmHcLRTdkjsVWQ vgyJkqnYLX+1exBLFQ2S GtPy3Bg7KdUomOTjfDsxPKZsSPu8ZTGkiptO r1/TI6t0yNBpO0LkUzd3xMpVcaMlG87FcWBma9NzP9qrQSDi04qwjhBHrLZoiCRBCMyCYP0hOYM5cgCZVrYWwkbUE0Z2sjyNgRv/suLUC+XvN S+fasWLnK4sjBIRzBCXhwDhW4gSrUgMEDPMErvDmPzovz7nzMWpecbOYA/sj5/AGdipKe</latexit>

the set of all bounded operators

a vector s.t. for any

Ω ∈ H

<latexit sha1_base64="5YJDeRs8XtXMyaCdovl1eDzoKFA=">A CD3icbZDLSgMxFIbP1Futl46 dBMsg spM1XQZdFNd1awF2hLyaSnbWgmMyQZoZTufAEfwo1bBXfi1kfQpzG9L T1h4SP/5xDcv4gFlwbz/tyUiura+sb6c3M1vbObtbd26/qKFEMKywSkaoHVKPgEiuG 4H1WCENA4G1YHA9qdfuUWkeyTszjLEV0p7kXc6osVbkZuEJbiAEhB5QyxykvZl AQRKbTfn5b2pyDL4c8jBXOW2+93sRCwJURomqNYN34tNa0SV4UzgON MNMaUDWgPGxYlDVG3RtM9xuTYOh3SjZQ90pCp+3tiREOth2FgO0Nq+nqxNjFPk7inEAf/9TQS071sjbiME4OSzR7sJoKYiEzCIR2ukBkxtECZ4vbPhPWposzYCDM2DH9x9W oFvL+Wb5we54rXs1jScMhHMEJ+HABRShBGSo24gd4h d4dR6dN+fd+Zi1p z5zAH8kfP5A4C0l3I=</latexit>

ρ( ˆ A) = hΩ, π( ˆ A)Ωi

<latexit sha1_base64="w9by/RISQeI+a+OWIALaW1Clhrk=">A CX3icbVHNThsxEJ5dwk9TSBd6qnpZESGBiKJdeqAXpNBe muQGoiURJHXmWyseO2V7a0UrfIWvAJvxAVufY2eOklQBYSxrPnm x/bn5NcCu i6MHzNyqbW9s7 6rvd/dqH4L9g2urC8Oxw7XUp swi1Io7DjhJHZzgyxLJN4k0+ L/M1vNFZo9cvNchxkLFViLDhzROngHO7AwAQ0HBOaA MHJVzCHE7g h JjIKUPFL0EzLyKXENigrIaYk3Op9XLub/nzEM6lEzWlq4DuInUG8d9k9vH1qz9jB47I80LzJUjktmbS+OcjcomXGCS5xX+4XFnPEpS7FHULEM7aBc6jIPj4gZhWNtaCsXLtn HSXLrJ1lCV mzE3s69yCbBR5ahCnb9X0Cjf+Oi FyguHiq8OHBcydDpciB2OhEHu5IwA40bQnUM+Y ZxR19SJTHi109fB9dnzfhL8+yKVPkGK9uBz3BIs dwDi34AW3oAId7+Ot eBXvj7/t1/xgVep7Tz0f4YX5n/4BkyqoGA= </latexit>

ˆ A ∈ A

<latexit sha1_base64="7j/Stbn7DQSPajRVBOe6inswZnQ=">A CHXicbVC7SgNBFL0bXzG+o KNzWIQLCTsxkI7ozaWEcwDkhBmJ3eTIbOzy8ysEJb8hD9g5T/YKtiJreif2DnZpNDEAzOcOfdc7tzjRZwp7TifVmZhcWl5JbuaW1vf2NzKb+/UVBhLilUa8lA2PK QM4FVzT HRiSRB 7Huje4GtfrdygVC8WtHkbYDkhPMJ9Ro 0U5vfgEfpAQEMCFzAyLwbC3EGq9cEHadhgUu3kC07RSWHPE3dKCuf
  • pKJ/ V6oY0DlBoyolSTdeJdDshUjPKcZRrxQojQgekh01DBQlQtZN0q5F9aJSu7YfSHKHtVP3dkZBAqWHgGWdAdF/N1sbicRz1JOLgP08z1v5ZO2Ei jUKOhnox9zWoT2Oyu4yiVTzoSGESmb+bNM+kYRqE2jOhOHOrj5PaqWie1Is3TiF8iVMkIV9OIAjcOEUynANFagChXt4gmd4sR6sV+vNep9YM9a0Zxf+wPr4AVDEnws=</latexit>

A

<latexit sha1_base64="0s3NyOFBu nr3d39nAXIGRPs1+8=">A B/3icbZBNSwMxEIZn/az1q+rRS7AInspuFfRY9eKxgv2AtpRsm 1Dk90lmRXK2oN/R APinj1b3jz35i2e9DWFwIP78w k9ePpTDout/O0vLK6tp6biO/ubW9s1vY26+bKNGM1 gkI930qeFShLyGAiVvxp T5Uve8IfXk3rjnmsjovAORzHvKNoPRSAYRWtJeAYF BAGEIC2NIQULmHcLRTdkjsVWQ vgyJkqnYLX+1exBLFQ2S GtPy3Bg7KdUomOTjfDsxPKZsSPu8ZTGkiptO r1/TI6t0yNBpO0LkUzd3xMpVcaMlG87FcWBma9NzP9qrQSDi04qwjhBHrLZoiCRBCMyCYP0hOYM5cgCZVrYWwkbUE0Z2sjyNgRv/suLUC+XvN S+fasWLnK4sjBIRzBCXhwDhW4gSrUgMEDPMErvDmPzovz7nzMWpecbOYA/sj5/AGdipKe</latexit>

ˆ A ⇠ ˆ B , h ˆ A ˆ B, ˆ A ˆ Bi = 0

<latexit sha1_base64="fi6KOv0c7gLTkFVfikJ aP9lbP0=">A Ck3ichVHbahsxEJ3d9OK6NzelT30RCYVCU7ObPiQ Co7bhz60kECdBGwTtPKsLayVFmk2xSz+i/xCPir9mo7tQBOn0BGCM+ecGUmjrDQ6UJ cR/HGg4ePHje NJ8+e/7iZevV5klwlVfYU84 f5bJgEZb7JEmg2elR1lkBk+z6ZeFfnqBPmhnf9KsxGEhx1bnWkliyrWO4QomI GghkOYcxZAQ3GL7S7Z74CQc+5ZHbNGrHpeDn6BYN1wblkx7Lvb8eNar53/6Ffc9W+vz5Cct7aTdrIMcR+kN2C7szX4cHndmR2dt34PRk5VBVpSRobQT5OShrX0pJXBeXNQBSylmsox9hlaW AY1stZzsU7ZkYid563JbFkb1fUsgh VmTsLCRNwrq2IHeqcuwRp/ y9CvK94e1tmVFaNXqwLwygpxYfJAYaY+KzIyBVF7znYWaSC8V8Tc2eRjp+tPvg5Pd vqpvXvMU+nCKhrwFrbgPaSwBx34BkfQAxU1ona0F+3Hb+KDuBt/XVnj6KbmNdyJ+Mcfbt+yeg= </latexit>

H := A/ ∼

<latexit sha1_base64="XF2aFEdtTbXlfhxtLgJKwunQ/3U=">A CPXicbZHLSiNBFIZPO86omYvRWboplBFh ky3LpQBIY4blwpGhSRIdeV0UqQuTVW1EJq8xbzCvIAv4Gvo7EZwJ27depK48Hag4Oc7f1F1/pPmSvoQx1fR1Lvp9x9mZucqHz9 /jJfXVg8 rZwAhvCKutOUu5RSYONI PCk9wh16nC47S/O+ofn6Hz0prDM ixrXnXyEwKHgjZ6iqU8BcEcFDAYA+G8Au2iVg4AwRHVI hNXJpcgXoQUacQ5/YDvl/UseTS8PwtLoS1+JxsdcieRQr9eXW9z9X9cH+afVfq2NFodE obj3zSTOQ7vkLkihcFhpFR5zLvq8i02Shmv07XI89ZB9I9JhmXV0TGBj+vRGybX3A52SU/PQ8y97I/ijyLsOsf+Wp1mEbKtdSpMXAY2YPJgVigXLRlGyjnQoghqQ4MJ +jMTPe64CBR4hcJIXo7+Whyt15KN2voBpfIbJjULS7AMa5DAJtRpJfvQoAWdwyX8h+voIrqJbqO7iXUqerz FZ5VdP8AKD+l4Q= </latexit>

ψ ˆ

A ∈ H

<latexit sha1_base64="u1NBwRBcy5YblaVm8EzfVnZF5Jo=">A CH3icbZA7SwNBEMfnfMb4ilpY2CyKaCHhTgstfTQWFgpGhRjC3maSLNnbO3b3hHDkW1gLFvkStgoWglrY6Kdxklj4m WH /+ZYXf+YaKkdb7/7g0Nj4yOjecm8pNT0zOzhbn5MxunRmBJxCo2FyG3qKTGkpNO4UVikEehwvOwd Crn1+hsTLWp6 dYCXiDS3rUnBHUlxYhC4kYEFCFTLiJnBwRHvQodMlXVMWpCpgcFgtrPhFvx/sLwRfsLK7dvT2dPNSO64WPi5rsUgj1E4obm058BNXybhxUijs5C9TiwkXLd7AMqHmEdpK1t+rw1ZJqbF6bOhqx/rq94mMR9a2o5A6I+6a9netJ26kScMgtv7rKaeuvlPJpE5Sh1oMHqynirmY9cxiNWlQONUm4MJI+jMT W64cGRpnswIfq/+F842i8FWcfOEXNmHQeRgCZ hHQLYhl04hGMokcX cAf38ODdeo/es/c6aB3yvmYW4Ed475/x4Z/+</latexit>

π( ˆ B)ψ ˆ

A = ψ ˆ B ˆ A

<latexit sha1_base64="404MmDzH+5ZOtGzogmygzCeAdKc=">A CVXicdVFNSwMxEJ2uX7V+VT16WR QcpuPehF0Hrx4EHBqlBLyabTNjS7G5KsUJb+FW/+GT14EUR/hVcFwWkrqFUnBF7e 0OSN4GSwljPe8o4I6Nj4xPZydzU9MzsXH5+4czEieZY5rGM9UXADEoRYdkK /FCaWRhIPE8aB/09PMr1EbE0antK yGrBmJhuDMEhXni3ADCShaAtYJt4CBhR K0IUNOiswpNSI+dL2SevC7j9qr3PIW8uveAWvX+5v4H+Clb21o4e76/v6cS3/fFmPeRJiZLlkxlR8T9lqyrQVXGI3d5kYVIy3WRMrBCMWoqm /TS67ioxdbcRa9qRdfvs946UhcZ0woCcIbMtM6z1yM1ENTVi+y9PJbGNnWoqIpVYjPjgwkYiXRu7vYjdutDIrewQYFwLerPLW0wzbmkQOQrDH/76b3BWLPhbheIJpVKCQWVhCZ pSD5swx4cwjGUgcMtvMArvGUeM+/OqDM+sDqZz5 F+FHO3Af2eKrV</latexit>

we already have , which is a vector space

A

<latexit sha1_base64="0s3NyOFBu nr3d39nAXIGRPs1+8=">A B/3icbZBNSwMxEIZn/az1q+rRS7AInspuFfRY9eKxgv2AtpRsm 1Dk90lmRXK2oN/R APinj1b3jz35i2e9DWFwIP78w k9ePpTDout/O0vLK6tp6biO/ubW9s1vY26+bKNGM1 gkI930qeFShLyGAiVvxp T5Uve8IfXk3rjnmsjovAORzHvKNoPRSAYRWtJeAYF BAGEIC2NIQULmHcLRTdkjsVWQ vgyJkqnYLX+1exBLFQ2S GtPy3Bg7KdUomOTjfDsxPKZsSPu8ZTGkiptO r1/TI6t0yNBpO0LkUzd3xMpVcaMlG87FcWBma9NzP9qrQSDi04qwjhBHrLZoiCRBCMyCYP0hOYM5cgCZVrYWwkbUE0Z2sjyNgRv/suLUC+XvN S+fasWLnK4sjBIRzBCXhwDhW4gSrUgMEDPMErvDmPzovz7nzMWpecbOYA/sj5/AGdipKe</latexit>

make into a Hilbert space by

A

<latexit sha1_base64="0s3NyOFBu nr3d39nAXIGRPs1+8=">A B/3icbZBNSwMxEIZn/az1q+rRS7AInspuFfRY9eKxgv2AtpRsm 1Dk90lmRXK2oN/R APinj1b3jz35i2e9DWFwIP78w k9ePpTDout/O0vLK6tp6biO/ubW9s1vY26+bKNGM1 gkI930qeFShLyGAiVvxp T5Uve8IfXk3rjnmsjovAORzHvKNoPRSAYRWtJeAYF BAGEIC2NIQULmHcLRTdkjsVWQ vgyJkqnYLX+1exBLFQ2S GtPy3Bg7KdUomOTjfDsxPKZsSPu8ZTGkiptO r1/TI6t0yNBpO0LkUzd3xMpVcaMlG87FcWBma9NzP9qrQSDi04qwjhBHrLZoiCRBCMyCYP0hOYM5cgCZVrYWwkbUE0Z2sjyNgRv/suLUC+XvN S+fasWLnK4sjBIRzBCXhwDhW4gSrUgMEDPMErvDmPzovz7nzMWpecbOYA/sj5/AGdipKe</latexit>

we set Ω := ψˆ

1

<latexit sha1_base64="qt1Wda1bQRikzxaZEeY4Apw6AQ =">A CFXicbZDLSgMxFIbP1Futl4 KbtwMloKrMlMXi AU3bizgr1AO5RMmrahmWRIMkIZ5iXcuHPnO7hxoYhbwZ1vY3pZaOsPgS/ OYfk/EHEqNKu+21l pZXVtey67mNza3tvL2zW1cilpjUsGBCNgOkCKOc1DTVjDQjSVAYMNI hpfjeuO SEUFv9WjiPgh6nPaoxhpYwl7H57gGkIg0AcEZ3Bu7hEo NCBxPDAuNqQBymkHbvgltyJnEXwZlCoFBv5 H Vrnbsr3ZX4DgkXGOGlGp5bqT9BElNMSNprh0rEiE8RH3SMshRSJSfTLZKnaJxuk5PSHO4dibu74kEhUqNwsB0hkgP1HxtbP5Xa8W6d+onlEexJhxPH+rFzNHCGUfkdKk WLORAYQlNX918ABJhLUJMmdC8OZX oR6ueQdl8o3XqFyAVNl4QAO4chEeQIVuI q1AD PTzDK7xZD9aL9W59TFsz1mxmD/7I+vwBcguZsA= </latexit>
slide-29
SLIDE 29

depends only on with

Setup for Theorem 1

spin operators ˆ

S(α)

j

<latexit sha1_base64="GP/ks9hXutX5JE VoBh726kzQ7M=">A CEXicbZC7TsMwFIZPuJZyC5eNJaJCgqVKChKMFSyMRdCL1IbKcd3W1LEj20Gqor4C 8/AG7AwgBArGxtvg5tmgJYjWf71/efIPn8QMaq0635bc/MLi0vLuZX86tr6xqa9tV1TIpaYVLFgQjYCpAijnFQ1 Yw0IklQGDBSDwYXY79+T6Sigt/oYUT8EPU47VKMtEHC3oUn6AMCDQlcw huzX1oGAIGUeocGdqGu7ZdcItuWs6s8DJRgKwqbfur1RE4DgnXmCGlmp4baT9BUlPMyCjfihWJEB6gHmkayVFIlJ+kG42cA0M6TldIc7h2Uvp7IkGhUsMwMJ0h0n017Y3hf14z1t0zP6E8ijXhePJQN2aOFs4 HqdDJcGaDY1AWFLzVwf3kURYmxDzJgRveuVZUSsVveNi6eqkUD7P4sjBHuybYD04hTJcQgWqgOEBnuEV3qxH68V6tz4mrXNWNrMDf8r6/AF9WpaD</latexit>

j ∈ Z

<latexit sha1_base64="xLy8n5ifslXIcS1Ypnes7UMC DQ=">A CA3icbVDLSgMxFL1TX7W+qi7dBIvgqsxUQZdFNy4r2Ae2Q8mkmTY2yQxJRihDwY2fohsXirj1J9z5N6btL T1hFwO59xLck8Qc6aN6347uaXl dW1/HphY3Nre6e4u9fQUaI rZOIR6oVYE05k7RumOG0FSuKRcBpMxheTvzmPVWaRfLGjGLqC9yXLGQEGytJuIMnYCBtFYDBwA Ce1K4hXG3WHL 7hRokXgZKUG Wrf41elFJBFUGsKx1m3PjY2fYmUY4XRc6CSaxpgMcZ+2LZVYUO2n0x3G6MgqPR Gyl5p0FT9PZFiofVIBLZTYDPQ895E/M9rJyY891Mm48RQSWYPhQlHJkKTQFCPKUoMH1mCiWL2r4gMsMLE2NgKNgRvfuVF0qiUvZNy5fq0VL3I4sjDARzCMXhwBlW4ghrUgcADPM rvDmPzovz7nzMWnNONrMPf+B8/gD7LpPj</latexit>

α = x, y, z

<latexit sha1_base64="NvRZe4um2dVyv DpvaoGHSWPvAw=">A C XicbVC7SgNBFL0bXzG+Vi0VGQyChYTdWGgjBG0sEzAPSJYwO5lNhsw+mJkV1yWlNn6HnY2FEmz9Azu/wZ9w8ig0euAyZ865l5l73IgzqSzr08jMzS8sLmWXcyura+sb5uZWTYaxILRKQh6Khosl5SygVcU p41IUOy7nNbd/sXIr19TIVkYXKk o 6PuwHzGMFKS6FpwiNg4B T59n+ibABwQ3cASJrtu2mbcK1hjoL7GnJF/aHVa+7vaG5b 50eqEJPZpoAjHUjZtK1JOioVihN BrhVLGmHSx13a1DTAPpVO t5kgA60 kFeKHQFCo3VnxMp9qVMfFd3+lj15Kw3Ev/zmrHyTp2UBVGsaEAmD3kxRypEo1hQhwlKFE80wUQw/VdEelhgonR4OR2CPbvyX1IrFuzjQrFi50vnME WdmAfDsG EyjBJZShCgTu4Qle4NV4MJ6NofE2ac0Y05lt+AXj/RsK2Zfm</latexit>

τ( ˆ S(α)

j

) = ˆ S(α)

j+1

<latexit sha1_base64="oWD7JWtTO7YzPI7zTFK8Gtu4x c=">A CT3icjZHLSgMxFIbP1Ftb 1WXbgaLoAhlphV0IxTduKxoL9DWk nTNjZzITkjlKGP49u40Z2v4caFImbGW jrwgPhfPz/CUn+OIHgCi3rxcgsLC4tr2Rz+dW19Y3NwtZ2Q/mhpKxOfeHLlkMUE9xjdeQoWCuQjLiOYE1nfBH7zXsmFfe9G5wErOuSoc HnBLUkl+owAMgEAjhQN IE0IE1zCFW91j YCAIHEOtdqDO93P/jkb6ekjsGHaKxStkpWUOQ92CkVIq9YrPHf6Pg1d5iEVRKm2bQXYjYhETgWb5juhYgGhYzJkbY0ecZnqRk eU3NfK31z4Eu9PDQT9e OiLhKTVxHT7oER2rWi8W/vHaIg9NuxL0gRObR74MGoTDRN+NwzT6XjK YaCBUcn1Xk46IJBT1F+R1CPbsk+ehUS7ZlVL56rhYPU/jyMIu7OmAbTiBKlxCDepA4RFe4R0+jCfjzfjMpKMZI4Ud+FWZ3BcHA6Lr</latexit>

translation automorphism Hamiltonian ˆ

H = P

j∈Z ˆ

hj

<latexit sha1_base64="QXOotaNvW+RyBWQ2GHNLOv2i4oU=">A CNXicbVG5TgMxEJ3lDOEKUNKsiEBUYTcU0ESKoKEMEjlEiCKv47AG27uyvUjRKjW/QcMP0NBTUlJBQ FCtPwCk4SCayxb 957I3vGQSy4sZ735IyNT0xOTWdmsrNz8wuLuaXlmokSTVmVRiLSjYAYJrhiVcutYI1YMyIDwerB+f5Ar18wbXikjmwvZi1JThXvckosUlFuA64hBAIWUjiAPpQwN5CAhDYyZ5hxUHjKoSeEAFcKx+jsf6sM WvDWTuX9wreMNy/wP8C+fLW3dXlfadUaeceTjoRTSRTlgpiTNP3Yt KibacCtbPniSGxYSek1PWRKiIZKaVDrvu +vIdNxupHEr6w7Z7xUpkcb0ZIBOSWxofmsD8j+tmdjubivlKk4sU3R0UTcRro3cwQjdDteMWtFDQKjm+FaXhkQTanHQWRyC/7vlv6BWLPjbheKhny/vwSgysAprsAk+7EAZP6MCVaBwA4/wAq/OrfPsvDnvI+uY81WzAj/C+fgExPGkVw= </latexit>

etc. translation invariant: short ranged: ˆ

hj

<latexit sha1_base64="DAcp9MAuQS1XjlmuZce pVltYz0=">A B+XicbVBNTwIxEJ3FL8Qv1KOXRmLi eyi R6JXjxiIkgCG9ItXah0u5t2loRs+CdePGiMV/+JN/+NBfag4Esm8/LeTDp9QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZeJUM95ksYx1O6CGS6F4EwVK3k40p1Eg+WMwup35j2OujYjVA04S7kd0oEQoGEUrCXiFIVBAyGyfQg+e uWKW3XnIKvEy0kFcjR65a9uP2ZpxBUySY3peG6CfkY1Cib5tNRNDU8oG9EB71iqaMSNn80vn5Izq/RJG tbCslc/b2R0ciYSRTYyYji0Cx7M/E/r5NieO1nQiUpcsUWD4WpJBiTWQykLzRnKCeWUKaFvZWwIdWUoQ2rZEPwlr+8Slq1qndRrd1fVuo3eRxFOIFTOAcPrqAOd9CAJjAYw7MN+s3JnBfn3flYjBacfOcY/sD5/AH0J5FE</latexit>

ˆ S(α)

i

<latexit sha1_base64="pJa3Z4oKoetUd89ZFTtQwkL3YrA=">A CEXicbZC7TsMwFIZPyq2UW7hsLBYVEixVUpBgrGBhLIJepDZUju 2Vp04sh2kKuorsPAMvAELAwixsrHxNrhpBmg5kuVf3 +O7P 7EWdKO863lVtYXFpeya8W1tY3Nrfs7Z26ErEktEYEF7LpY0U5C2lNM81pM5IUBz6nDX94OfEb91QqJsJbPYqoF+B+yHqMYG2QsPfgCQaAQUMCNzCGO3MfGYaBQ5Q6x4Z2gHXsolNy0kLzws1E bKqduyvdleQOKChJhwr1XKdSHsJlpoRTseFdqxohMkQ92nLyBAHVHlJutEYHRrSRT0hzQk1SunviQ HSo0C3 QGWA/UrDeB/3mtWPfOvYSFUaxpSKYP9WKOtECTeFCXSUo0HxmBiWTmr4gMsMREmxALJgR3duV5US+X3JNS+fq0WLnI4sjDPhyY F04gwpcQRVqQOABnuEV3qxH68V6tz6mrTkrm9mFP2V9/gB71paC</latexit>

|i − j| ≤ r

<latexit sha1_base64="WXYu6iXt857ytUaVoQ5VFQRrOv0=">A B+3icbVDLSgNBEOyNrxhfUY9eBoPgxbAbBT0GvXiMYB6QhDA76U3GzD6YmRXDml/xIqKIV3/Em3/jbLIHTSxouqjqZnrKjQRX2ra/rdzS8srqWn69sLG5tb1T3N1rqDCWDOs FKFsuVSh4AHWNdcCW5FE6rsCm+7oKvWb9ygVD4NbPY6w69NBwD3OqDbSHTwChxNI+wsIQCAge8WSXbanI vEyUgJMtR6xa9OP2Sxj4FmgirVduxIdxMqNWcCJ4VOrDCibEQH2DY0oD6qbjK9fUKOjNInXihNBZpM1d8bCfWVGvu mfSpHqp5LxX/89qx9i6 CQ+iWGPAZg95sSA6JGkQpM8lMi3GhlAmubmVsCGVlGkTV8GE4Mx/eZE0KmXntFy5OStVL7M48nA h3AMDpxDFa6hBnVg8ABP8Apv1sR6t 6tj9lozsp29uEPrM8fLxKRTA= </latexit>

τ(ˆ hj) = ˆ hj+1

<latexit sha1_base64="NLBj9OpO8t/czPJPR473OJA6nqs=">A CI3icbZDLSsNAFIZPvNZ6i7oUIVgERShJFXQjFN24rGAv0IYymU7bsZMLMydC XkXN30VNy6U4saF7+K0zUJbDwz 8f/nMHN+LxJcoW1/GUvLK6tr67mN/ObW9s6u bdfU2EsKavSUISy4RHFBA9YFTkK1ogkI74nWN0b3E38+jOTiofBIw4j5vqkF/AupwS1FJpHMAIEAjGcauprQkj0nUIbnuAMbhbUROvn4EDaNgt20Z6WtQhOBgXIqtI2x61OSGOfBUgFUarp2BG6CZHIqWBpvhUrFhE6ID3W1BgQnyk3me6YWida6VjdUOoToDV f08kxFdq6Hu60yfYV/PeRPzPa8bYvXYTHkQxsoDOHurGwsLQmgRmdbhkFMVQA6GS679atE8koahjzesQnPmVF6FWKjoXxdLDZaF8m8WRg0M41qE7cAVluIcKVIHC 7zCO3wYI+PNGBufs9YlI5s5gD9lfP8AQiOazQ= </latexit>

j ∈ Z

<latexit sha1_base64="xLy8n5ifslXIcS1Ypnes7UMC DQ=">A CA3icbVDLSgMxFL1TX7W+qi7dBIvgqsxUQZdFNy4r2Ae2Q8mkmTY2yQxJRihDwY2fohsXirj1J9z5N6btL T1hFwO59xLck8Qc6aN6347uaXl dW1/HphY3Nre6e4u9fQUaI rZOIR6oVYE05k7RumOG0FSuKRcBpMxheTvzmPVWaRfLGjGLqC9yXLGQEGytJuIMnYCBtFYDBwA Ce1K4hXG3WHL 7hRokXgZKUG Wrf41elFJBFUGsKx1m3PjY2fYmUY4XRc6CSaxpgMcZ+2LZVYUO2n0x3G6MgqPR Gyl5p0FT9PZFiofVIBLZTYDPQ895E/M9rJyY891Mm48RQSWYPhQlHJkKTQFCPKUoMH1mCiWL2r4gMsMLE2NgKNgRvfuVF0qiUvZNy5fq0VL3I4sjDARzCMXhwBlW4ghrUgcADPM rvDmPzovz7nzMWnNONrMPf+B8/gD7LpPj</latexit>

for any

Z2 × Z2

<latexit sha1_base64="QGsH6AVHbdcoeQlURKgozyT3FBQ=">A CJXicdVDLSgMxFL3js9bXqEtB ovgqsxUQZdFNy4r2Ae2Q8mkaRuazAzJHaEM/RlB/BU3LiwiuPJXTNtZaKsnBA7n EtyTxALrtF1P62l5ZXVtfXcRn5za3tn197br+koUZRVaSQi1QiIZoKHrIocBWvEihEZCFYPBtcTv/7AlOZReIfDmPmS9ELe5ZSgkSL7CJ5A gGEPgTmpHAPI2hDyegI3HgM9H+Ztl1wi+4UziLxMlKADJW2PW51Ip IFiIVROum58bop0Qhp4KN8q1Es5jQAemxpqEhkUz76XTLkXNilI7TjZS5ITpT9edESqTWQxmYpCTY1/PeRPzLaybYvfRTHsYJspDOHuomwsHImVTmdLhiFMXQE IVN391aJ8oQtEUmzclePMrL5JaqeidFUu354XyV ZHDg7hGE7Bgwsow 1UoAoUHuEF3mBsPVuv1rv1MYsuWdnMAfyC9fUN4KqbrA= </latexit>

Rα(ˆ hj) = ˆ hj

<latexit sha1_base64="Xd3HvP19z8w GvW+cxd1h3IKmPU=">A CNXicbVG7TsMwFL3hWcorwMhiUYFgqZKCBAtSBQtjQfQhtVXluG5r6jiR7SBVUT6H 2DhP5hgYA hVn4BJ+0ALVey7rn 3Cv7HnshZ0o7zqs1N7+wuLScW8mvrq1vbNpb2zUVRJLQKgl4IBseVpQzQaua U4boaTY9zite8PLVK/fU6lYIG71K RtH/cF6zGCtaEC+wBieA CGDg uIE OqZOqxAGJh+aKs3a9A0y9Q6O4HyGjQ2fdOyCU3SyQLPAnYACTKLSsZ9b3YBEPhWacKxU03VC3Y6x1IxwmuRbkaIhJkPcp0 DBfapasfZ1gnaN0wX9QJpjtAoY39PxNhXauR7ptPHeqCmtZT8T2tGunfWjpkI 0 FGV/UizjSAUotRF0mKdF8ZA mkpm3IjLAEhNtjM4bE9zplWdBrVR0j4ul65NC+WJiRw52Yc8Y7sIplOEK lA1H/MIL/AOH9aT9WZ9Wl/j1jlrMrMDf8L6/gHPyp92</latexit>

invariant:

j ∈ Z

<latexit sha1_base64="xLy8n5ifslXIcS1Ypnes7UMC DQ=">A CA3icbVDLSgMxFL1TX7W+qi7dBIvgqsxUQZdFNy4r2Ae2Q8mkmTY2yQxJRihDwY2fohsXirj1J9z5N6btL T1hFwO59xLck8Qc6aN6347uaXl dW1/HphY3Nre6e4u9fQUaI rZOIR6oVYE05k7RumOG0FSuKRcBpMxheTvzmPVWaRfLGjGLqC9yXLGQEGytJuIMnYCBtFYDBwA Ce1K4hXG3WHL 7hRokXgZKUG Wrf41elFJBFUGsKx1m3PjY2fYmUY4XRc6CSaxpgMcZ+2LZVYUO2n0x3G6MgqPR Gyl5p0FT9PZFiofVIBLZTYDPQ895E/M9rJyY891Mm48RQSWYPhQlHJkKTQFCPKUoMH1mCiWL2r4gMsMLE2NgKNgRvfuVF0qiUvZNy5fq0VL3I4sjDARzCMXhwBlW4ghrUgcADPM rvDmPzovz7nzMWnNONrMPf+B8/gD7LpPj</latexit>

for any

α = x, y, z

<latexit sha1_base64="NvRZe4um2dVyv DpvaoGHSWPvAw=">A C XicbVC7SgNBFL0bXzG+Vi0VGQyChYTdWGgjBG0sEzAPSJYwO5lNhsw+mJkV1yWlNn6HnY2FEmz9Azu/wZ9w8ig0euAyZ865l5l73IgzqSzr08jMzS8sLmWXcyura+sb5uZWTYaxILRKQh6Khosl5SygVcU p41IUOy7nNbd/sXIr19TIVkYXKk o 6PuwHzGMFKS6FpwiNg4B T59n+ibABwQ3cASJrtu2mbcK1hjoL7GnJF/aHVa+7vaG5b 50eqEJPZpoAjHUjZtK1JOioVihN BrhVLGmHSx13a1DTAPpVO t5kgA60 kFeKHQFCo3VnxMp9qVMfFd3+lj15Kw3Ev/zmrHyTp2UBVGsaEAmD3kxRypEo1hQhwlKFE80wUQw/VdEelhgonR4OR2CPbvyX1IrFuzjQrFi50vnME WdmAfDsG EyjBJZShCgTu4Qle4NV4MJ6NofE2ac0Y05lt+AXj/RsK2Zfm</latexit>

Z2 × Z2

<latexit sha1_base64="QGsH6AVHbdcoeQlURKgozyT3FBQ=">A CJXicdVDLSgMxFL3js9bXqEtB ovgqsxUQZdFNy4r2Ae2Q8mkaRuazAzJHaEM/RlB/BU3LiwiuPJXTNtZaKsnBA7n EtyTxALrtF1P62l5ZXVtfXcRn5za3tn197br+koUZRVaSQi1QiIZoKHrIocBWvEihEZCFYPBtcTv/7AlOZReIfDmPmS9ELe5ZSgkSL7CJ5A gGEPgTmpHAPI2hDyegI3HgM9H+Ztl1wi+4UziLxMlKADJW2PW51Ip IFiIVROum58bop0Qhp4KN8q1Es5jQAemxpqEhkUz76XTLkXNilI7TjZS5ITpT9edESqTWQxmYpCTY1/PeRPzLaybYvfRTHsYJspDOHuomwsHImVTmdLhiFMXQE IVN391aJ8oQtEUmzclePMrL5JaqeidFUu354XyV ZHDg7hGE7Bgwsow 1UoAoUHuEF3mBsPVuv1rv1MYsuWdnMAfyC9fUN4KqbrA= </latexit>

automorphism Rα( ˆ

S(β)

j

) = ( ˆ S(β)

j

α = β ˆ S(β)

j

α 6= β

<latexit sha1_base64="oWzJTo7UuhRFU6xFNzk6KX03U0A=">A C93icnVJLixNBEK4ZX7vjK+rRS+OiKGiYWQ+KsLDoxeOumt2FbAw9nUrSbk/P0F0jhCFn/4J48aC7eBUE/4c3/43VSQ7JrgexoKmv q uvqRV0Z7StPfUXzu/IWLl9bWk8tXrl673rpxc8+XtVPYUaUp3UEuPRptsUOaDB5UDmWRG9zPj16E/P57dF6X9g1NKuwVcmT1UCtJTJWtD9DAMSiQYEDAK5hCn+MQVTBmf5+j4Il1rzn7ln3gckDmJDyYVbxjvwXJgh+B su60NVz7FmT/GOfeyu7by0pwpSO+z 6r07HPBEuqcI8yNxgdc5+ayNtpzMTZ0G2ABvb7Wc/fj7c/bjTb/06HJSqLtCSMtL7bpZW1GukI60MTpPD2mMl1ZEcYZehlQX6XjN7t6m4y8xADEvHy5KYscsVjSy8nxQ5KwtJY386F8i/5bo1DZ/2Gm2rmtCq+UbD2g qRfgEYqAdKjITBlI5zbMKNZ OKuKvkvAlZKePfBbsb azx+3NXb6N5zC3NbgNd/gxMngC2/ASdqADKqLoU/Ql+hpP4s/xSfxtLo2jRc0tWLH4+x+97se </latexit>

ˆ Sj = ( ˆ S(x)

j

, ˆ S(y)

j

, ˆ S(z)

j )

<latexit sha1_base64="5a1EuFxDSxtYZknf0R1DTIxj+nY=">A CkXicjZHdSsMwFMdP6/f8mnrpTVAEBRmtCn7AZOiN4M1ENwfbHGmWubi0KUkq1rL38AV8A8H38M63Md2KqBPxhJB/ ud3SHLihZwp7Tjvlj02PjE5NT2Tm52bX1jMLy1XlYgkoRUiuJA1DyvKWUArm lOa6Gk2Pc4vfZ6p2n+ p5KxURwpeOQNn18G7AOI1gbS+Sv4Bm6gEFDYpQHAji0QUEMfrZL4BL6ZrTgDoqw+YVP/Ruzp 40PI H2MrI7T+5+J/c4ye31cqvOwVnEGhUuJlYLxWOXl63L57Krfxboy1I5N AE46VqrtOqJsJlpoRTvu5RqRoiEkP39K6kQH2qWomg4720YZx2qgjpJmBRgP3a0WCfaVi3zOkj3VX/cyl5m+5eqQ7B82EBWGkaUCGB3UijrRA6fegNpOUaB4bgYlk5q6IdLHERJtPzJkmuD+fPCq OwV3t7BzYbpxAsOYhlVYM41 YR9KcAZlqACxFqw9q2gd2yv2oV2yM9a2spoV+Bb2+QdZS68G</latexit>

(ˆ Sj)2 = S(S + 1) ˆ 1

<latexit sha1_base64="iWk4eXkar0OFuZOeBzd5BYRLSDY=">A CPXicbVC5TgMxFJzlJlwBSpoVh8SlaDcU0CAhaChBE AKIfI6DjHxrle2Fyla5XP4BQoaCv6Ajo6GAoTogBbnKLjGetJ43mG/CWLBtfG8B6ent69/YHBoODMyOjY+kZ2cOtIyUZQVqBRSnQREM8EjVjDcCHYSK0bCQLDjoL7Tyh9fMqW5jA5NI2alkJxHvMopMVaS2WUs4go1EBiklgWQEKhAo4Gwe0txgKY9ZVxgCWfIY9MqizZW4FvlCqvfZvholrNzXs5rw/1L/C6Z25p/v767HPnYK2fvTyuSJiGLDBVE6 Lvxa UEmU4FayZOU0 iwmtk3NWtDQiIdOltL19012wSsWtSmUjMm5b/d6RklDrRhjYypCYmv6da4n/5YqJqW6U h7FiWER7TxUTYRrpNuy0q1wxagRDUsIVdz+1aU1og 1 vCMNcH/vfJfcpTP+Wu5/L51YxsdDGEGs9ZeH+vYwi72UADFDR7xjBfn1nlyXp23Tm P0+2Zxg84n19t9KQ5</latexit>

S = 1

2, 3 2, 5 2, . . .

<latexit sha1_base64="ZhW2WS+CyP2QBxz u6GcZX+r6lc=">A CQXicdZFLSwMxEMdnfdb6WvXoZbEICqXstohehKogHivaB7SlZN sG5rdLElWKEs/ike/h e/gTfvHvSgSK9eTB8H2+pAhj+/mWGSf9yQUals+8WYm19YXFpOrCRX19Y3Ns2t7ZLk cCkiDnjouIiSRgNSF RxUglFAT5LiNlt3MxqJfviJCUB7eqG5K6j1oB9ShGSiNupuEGTuEBPBCA EM DvR0zuqcnuC5f/jRBGfQBA4KZMNM2Rl7GNascMYilT+0L9/692eFhvlca3Ic+SRQmCEpq4 dqnqMhK YkV6yFk SItxBLVLVMkA+kfV46EDP2tekaXlc6BMoa0h/T8TIl7Lru7rTR6otp2sD+FetGinvpB7TI wUCfBokRcxS3FrYKfVpIJgxbpaICyovquF20g rLTpSW2CM/3kWVHKZpxcJnvtpPLnMIoE7MIeHOjPOIY8XE BitrsR3iFD/g0nox348voj1rnjPHMDkyE8f0DnuqlOA= </latexit>

we only consider models with we assume that the g.s. is unique and accompanied by a nonzero energy gap

ω

<latexit sha1_base64="ikgU8pvgNP3u74Edg2wXP SVsVw=">A B83icbVC7SgNBFL0bX3F9RS1tBoNgFXZjoY0YtLGMYB6QLGF2MpsMmZ0dZmaFsOQ3bCwUH6XfYW8j/o2TR6GJBy4czrmXe+8J WfaeN63k1taXl dy6+7G5tb2zuF3b26TlJFaI0kPFHNEGvKmaA1w ynTakojkNOG+Hgauw37qjSLBG3ZihpEO eYBEj2FgpgjdI AYKPcCdQtEreROgReLPSPHiwz2XL19utVP4bHcTksZUGMKx1i3fkybIsDKMcDpy26m EpMB7tGWpQLHVAfZ5OYROrJKF0WJsiUMmqi/JzIcaz2MQ9sZY9PX895Y/M9rpSY6CzImZGqoIN FUcqRSdA4ANRlihLDh5Zgopi9FZE+VpgYG5NrQ/DnX14k9XLJPymVb7xi5RKmyM BHMIx+HAKFbiGKtSAgIR7eIQnJ3UenGfnd qac2Yz+/AHzvsPVfS 6w= </latexit>

(linear *-automorphism) (linear *-automorphism)

slide-30
SLIDE 30

we assume that the g.s. is unique and accompanied by a nonzero energy gap

ω

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α = x, y, z

<latexit sha1_base64="mu4/Z2oS2Z+pwvasCR2KIQo jXw=">A C XicbVC7SgNBFL0bXzG+Vi0VGQyChYRdLbQRgjaWCZgHJEuYncwmQ2YfzMyK65JSG7/DzsZC b +gZ3f4E84eRSaeOAyZ865l5l73IgzqSzry8jMzS8sLmWXcyura+sb5uZWVYaxILRCQh6Kuosl5SygFcU p/VIUOy7nNbc3uXQr91QIVkYXKsko 6POwHzGMFKS6FpwhNg4B V5/n+ibABwS3cASJr uWmbcK1gholtgTki/uDsrf93uDUsv8bLZDEvs0UIRjKRu2FSknxUIxwmk/14wljTDp4Q5taBpgn0onHW3SRwda SMvFLoChUbq74kU+1Imvqs7fay6ctobiv95jVh5Z07KgihWNCDjh7yYIxWiYSyozQ li eaYCKY/isiXSw UTq8nA7Bnl5 l SPC/ZJwS7b+eIFjJGFHdiHQ7DhFIpwBSWoAIEHeIZXeDMejRdjYLyPWzPGZGYb/sD4+AEKh5fl</latexit>

projective 
 representation genuine 
 representation

cσ = ζα P

σ0hσ|ˆ

u†

α|σ0i ˜

Rα(cσ0)

<latexit sha1_base64="h58D5o07zRHq43QxyVX L7Do/Ls=">A C2XicbVI7j9NAEB6b1xFeAUqaFdEJaCL7roAGKYKG8kAkd1IuROvNxFndem3tjhHBSUEBQrTUlPeHaIAfQs/YuSIPZrXyNzPfNzO76 Qw2lMU/QnCS5evXL2 d71 4+at23fad+8NfF46hX2Vm9ydJNKj0Rb7pMngSeFQZonB4+TsZ 0/fo/O69y+pXmBo0ymVk+1ksShvP0BFLyDc/CgIYUMJDxn7yMgEOMxYwkGCpjxt2aVzBlDtaF4BEv2DSPLEcPa9eyCvVpNrCqZWXebsJ/yQnBbPRZblc+ZsV6XOGdYj80MqlEKeMN1N+s8bs61O+eTcbsTdaPGxC6IL0Cnt/ 3968frfRo3P5 OslVmaElZaT3wzgqaFRJR1oZXLZOS4+FVGcyxSFDKzP0o6p5maXY58hETHPH25Jo u KSmbez7OEmZmkmd/O1cH/5Y lTZ+NKm2LktCqVaNpaQTlon5mMdEOFZk5A6mc5lmFmk nFfHP0OJLiLePvAsGB934sHvwOu70XsDK9uABPOTLjeEp9OAVHE fVDAIFsHn4Es4D +FX8NvK2oYXGjuw4aF3/8BQW7Edw= </latexit>

contradiction!

THEOREM 1: Consider a quantum spin chain with
 and a short-ranged Hamiltonian that is invariant under translation and

  • transformation. Then it can never be the case that

the corresponding ground state is unique and accompanied by a nonzero gap.

Z2 × Z2

<latexit sha1_base64="JHQfJqko41NQTU+UlgobuCZc8Gk=">A CJXicdVC7SgNBFL0bXzG+VgUbQRaDYBV2Y6FliI1lAuaByRJmJ5NkyOyDmbtCWPIzglj5FxY2FgYRbPRXnDwKTfQMA4dz mXmHi8SXKFtfxip eWV1bX0emZjc2t7x9zdq6owlpRVaChCWfeIYoIHrI cBatHkhHfE6zm9S/Hfu2WScXD4BoHEXN90g14h1OCWgrNI7gH wg 9MDTJ4EbGEIL8lpH4NpjoP7LtMysnbMnsBaJMyPZwkH5kz8Wn0otc9RshzT2WYBUEKUajh2hmxCJnAo2zDRjxSJC+6TLGpoGxGfKTSZbDq0TrbStTij1DdCaqD8nEuIrNfA9nfQJ9tS8Nxb/8hoxdi7chAdRjCyg04c6sbAwtMaVW 0uGU x0IRQyfVfLdojklDUxWZ0Cc78youkms85Zzm7rNsowhRpOIRjOAUHzqEAV1C lC4g2d4hZHxYLwYb8b7NJoyZjP78AvG1ze+rp9H</latexit>

S = 1

2, 3 2, 5 2, . . .

<latexit sha1_base64="ZhW2WS+CyP2QBxz u6GcZX+r6lc=">A CQXicdZFLSwMxEMdnfdb6WvXoZbEICqXstohehKogHivaB7SlZN sG5rdLElWKEs/ike/h e/gTfvHvSgSK9eTB8H2+pAhj+/mWGSf9yQUals+8WYm19YXFpOrCRX19Y3Ns2t7ZLk cCkiDnjouIiSRgNSF RxUglFAT5LiNlt3MxqJfviJCUB7eqG5K6j1oB9ShGSiNupuEGTuEBPBCA EM DvR0zuqcnuC5f/jRBGfQBA4KZMNM2Rl7GNascMYilT+0L9/692eFhvlca3Ic+SRQmCEpq4 dqnqMhK YkV6yFk SItxBLVLVMkA+kfV46EDP2tekaXlc6BMoa0h/T8TIl7Lru7rTR6otp2sD+FetGinvpB7TI wUCfBokRcxS3FrYKfVpIJgxbpaICyovquF20g rLTpSW2CM/3kWVHKZpxcJnvtpPLnMIoE7MIeHOjPOIY8XE BitrsR3iFD/g0nox348voj1rnjPHMDkyE8f0DnuqlOA= </latexit>

Setup for Theorem 1

we shall derive the transformation rule

slide-31
SLIDE 31

algebras for the half-infinite chain

AR

<latexit sha1_base64="9xUemq7z3u5plpxIYvh0leT3b8o=">A CEXicbZC7TsMwFIZPuJZyCxexsFhUSExVUgY S1kYW0QvUltVju 0Vu0ksh2kKuorsPAMHdlYGEAIsbGxsfAsuJcBWo5k+dP/nyP7/F7EmdKO82UtLC4tr6ym1tLrG5tb2/bObkWFsS 0TEIeypqHFeUsoGXN Ke1SFIsPE6rXu9y5FdvqVQsDG50P6JNgTsB8xnB2kihvQ9DEIB Qxd8kIZ6kMAFDKBl7qFRBC 4hkHLzjhZ 1xoHtwpZPIHpW/2UHgvtuzPRjsksaCBJhwrVXedSDcTLDUjnA7SjVjRCJMe7tC6wQALqprJeKMBOjZKG/mhNCfQaKz+nkiwUKovPNMpsO6qW 8k/ufVY+2fNxMWRLGmAZk85Mc 6RCN4kFtJinRvG8AE8nMXxHpYomJNiGmTQju7MrzUMl 3dNsrmTSKMCkUnAIR3ACLpxBHq6gCGUgcAeP8Awv1r31ZL1ab5PWBWs6swd/yvr4AaiUmlE=</latexit>

ωR

<latexit sha1_base64="X74VmaTzZzjVUu+igX+zl7ao08I=">A CA3icbVC7SgNBFL0bXzG+o KNzWAQrMJuL QMsbFMxDwgCWF2MpsMmZldZmaFsARs/BRtRBSxtfML7Gz8FiePQhMPDJx7zr3cucePONPGdb+c1NLy upaej2zsbm1vZPd3avpMFaEVknIQ9Xwsa cSVo1zHDaiBTFwue07g8uxn79hirNQnlth FtC9yTLGAEGytJuIcQBFDoAY OJLZWtkZwBaNONufm3QnQIvFmJFc8qHyzp9JHuZP9bHVDEgsqDeFY6 bnRqadYGUY4XSUacWaRpgMcI82LZVYUN1OJjeM0LFVuigIlX3SoIn6eyLBQu h8G2nwKav572x+J/XjE1w3k6YjGJDJZkuCmKOTIjGgaAuU5QYPrQE 8XsXxHpY4WJsbFlbAje/MmLpFbIe6f5QsWmUYIp0nAIR3ACHpxBES6hDFUgcAsP8Awvzp3z6Lw6b9PWlDOb2Yc/cN5/AF3slzA=</latexit>

ω

<latexit sha1_base64="ikgU8pvgNP3u74Edg2wXP SVsVw=">A B83icbVC7SgNBFL0bX3F9RS1tBoNgFXZjoY0YtLGMYB6QLGF2MpsMmZ0dZmaFsOQ3bCwUH6XfYW8j/o2TR6GJBy4czrmXe+8J WfaeN63k1taXl dy6+7G5tb2zuF3b26TlJFaI0kPFHNEGvKmaA1w ynTakojkNOG+Hgauw37qjSLBG3ZihpEO eYBEj2FgpgjdI AYKPcCdQtEreROgReLPSPHiwz2XL19utVP4bHcTksZUGMKx1i3fkybIsDKMcDpy26m EpMB7tGWpQLHVAfZ5OYROrJKF0WJsiUMmqi/JzIcaz2MQ9sZY9PX895Y/M9rpSY6CzImZGqoIN FUcqRSdA4ANRlihLDh5Zgopi9FZE+VpgYG5NrQ/DnX14k9XLJPymVb7xi5RKmyM BHMIx+HAKFbiGKtSAgIR7eIQnJ3UenGfnd qac2Yz+/AHzvsPVfS 6w= </latexit>

(HR, πR, ΩR)

<latexit sha1_base64="4mydYOopb9hHVByeUnzy9uq FmA=">A CT3icbVHJSgNBEK2Je9yiHr0 iqAY4owe9Bj04s0oxgS EHo6laRJz0J3jxCG/IW/4J/4Ax71E7wqeBMry0Gj1TS8eq+ql3p+rKSxrv iZGZm5+YXFpey ura+u5jc07EyVaYFlEKtJVnxtUMsSylVZhNdbIA19hxe9dDPXKPWojo/DW9mNsBLwTyrYU3BIV5Y5gH1J4BAEcFDC4hAE0R4yGgPIbyvOUJRDTkv9qV5QhdOiEafWgmdt1C+4o2F/gTcBucad+ PBS7Jeaud 6KxJ gKEVihtT89zYNlKurRQKB9l6YjDmosc7WCMY8gBNIx3NYcD2iGmxdqRph5aN2J8dKQ+M6Qd+Pok7GrFH QG3XTNdMyT/02qJbZ81UhnGicVQjC9sJ4rZiA2Hy1pSo7CqT4AL enNTHS5 sKSBVkahjf9 b/g7rjgnRSOr2kq5zCORdiGHTLKg1Mok UlKJNhT/AG7/DhPDufzldmUp xJmALfkVm6Rv5VqdO</latexit>

M ⊂ B(H)

<latexit sha1_base64="52J7ozPh4EbXhl WdZ2TRvUb8mg=">A CL3icbVDLSgNBEOz1bXxFPXpwiAiKEnbjQY8hXrwICiYKSZDZSW8yZPbBzKwQlvyFn2A+wN/wqiB6EK/+hZ3Eg6+G Wq unp8hMljX dF2dicmp6ZnZuPrewuLS8kl9dq5k41QKrIlaxv K5QSUjrFp FV4lGn oK7z0u8dD/fIGtZFxdGF7CTZD3o5kIAW3RMX5TRhACBwsdCA TagLGZxCn3gDKfh0I6kMKrBDygAEeRS9T8ize53fcovuqNhf4H2BrXKhsXf7Uu6dXedfG61YpCFGVihuTN1zE9vMuLZSKOznGqnBhIsub2OdYMRDNM1stGefbRPTYkGs6USWjdjvHRkPjemF/n6atDVilzpCbjvmt2dI/qfVUxscNTMZJanFSIwHBqliNmbD8FhLahRW9QhwoSX9mYkO1 xYijhHYXi/V/8LaqWid1AsnVMqFRjXHGxAgeL14BDKFOwZVCnmO3iAR3hy7p1n5815H1sn K+edfhRzscn3JChwQ= </latexit>

M0 := { ˆ A ∈ B(H) | [ ˆ A, ˆ B] = 0 for any ˆ B ∈ M}

<latexit sha1_base64="yJOyLpfmAe24EdcCwiZh67w/T8U=">A Cz3icdVJNbxMxEJ3d8lHCV4AjF6sVAkQU7ZYDCAkpDRy4ILUSaSslq8r eBMTr72yZ6GrbRBX7vyF/Kjya5hsKhFSG msN/Pe2GOP0 Irj1F0EYRb167fuLl9q3X7zt179 sPHh5 WzohB8Jq605S7qVWRg5QoZYnhZM8T7U8TmfvlvzxF+m8suYTVoVMcj4xKlOCI6Vs28ICcuCAMIUMHKEZ1PAR5vAU3sBbYmvya OoYZ/yC1Bg E fnjWcIE5T/IG45xR34LxZhxt1nbW4T3FCu0eNcumMVgQJZw2fgaVeGKkNVM2Zf1Tre6x6+d8NFjA/be9G3agxdhXEl2C3tzN68fOiVx2ctn+NxlaUuTQoNPd+GEcFJjV3qISW89ao9L gYsYnckjQ8Fz6pG7mMGdPKDNm X kBlmTXa+oe 59laedspg4KWdUkXOc+k3NMvkvblhi9jqplSlKlEasDsxKzdCy5XDZWDkpUFcEuHCKemZiyh0XSF+gRY8Rb179Kj a68Yvu3uH9Cp9WNk2PIYdGncMr6BHgz6A Yjgf A58AG h+HX8Fv4fSUNg8uaR/CXhT9+A+rJwAE=</latexit>

πR(AR)

<latexit sha1_base64="4GSbIV8ZXbT7zfhW9ALyT3SAz/o=">A CN3icbVDLSitBEK3xbfRq1KWbxnB 8RJmdKHLqBuXKkaFJISeTk3SpHtm6O4RwpC/8Bfc+QP+hjsf4E7c+gdWEhcabzUNp06dQ3edMFXSOt9/9CYmp6ZnZufmCwuLf5aWiyurFzbJjMCqSFRirkJuUckYq046hVepQa5DhZdh92gwv7xGY2USn7teig3N27GMpOCOqKRYglvI KUjoQk5dQY0MDiDPmxSp4GDgw5ExHPokuKAJuPKrWax5Jf9YbHfIPgCpcpGf vmsdI7aRaf6q1EZBpjJxS3thb4qWvk3DgpFPYL9cxiykWXt7FGMOYabSMf7t nf4lpsSgxdGPHhux3R861tT0d/svStkHsk Nz17HjmgH5v1ktc9F+I5dxmjmMxejBKFPMJWwQImtJg8KpHgEujKQ/M9Hh gtHURcojGB89d/gYqc 7JZ3TimVQxjVHKzDBsUewB5U4BhOoAoC7uABnuHFu/devTfvfS d8L48a/CjvI9PvTGkLA= </latexit>

πR(AR)00 =: MR

<latexit sha1_base64="mQk/ceb2vceLaxL4csGrqbBTdbU=">A CZ3icbVHRShtBFL271jZGramC H2ZKqLSEnb1waI sX3xRdDSqJCEMDu5mwyZ2V1mZgthyV/4C35UxB/wL7xJfKiJ9zJw7rn MDP3RpmS1gXByPMXPix+/FRaKi+vrH5eq3xZv7FpbgTWRapScxdxi0omWHfSKbzLDHIdKbyN+r/H/dt/aKxMk79ukGFL824iYym4IyqtnMID5JBRSmhDQZUBDQz+wBD2qdLAwUEPYuI59ElxTp1Z5QHsUZ7BybuOy3lHu7ITVINJsHkQvoKd2nbz+/2oNrhqVx6bnVTkGhMnFLe2EQaZaxXcOCkUDsvN3GLGRZ93sUEw4Rptq5jMZ8h2iemwODV0Escm7P+OgmtrBzr6kWd g9gnh+auZ2c1Y/K9XiN38c9WIZMsd5iI6YVxrphL2XjorCMNCqcGBLgwkt7MRI8bLhytpkzDCGe/Pg9uDqvhUfXwmqbyC6ZRgq+wTWsK4RhqcAFXUAcBT57vLXsr3rO/5m/6W1Op7 16NuBN+N9eAJOVqEU=</latexit>

bicommutant

ˆ S(α)

j

<latexit sha1_base64="GP/ks9hXutX5JE VoBh726kzQ7M=">A CEXicbZC7TsMwFIZPuJZyC5eNJaJCgqVKChKMFSyMRdCL1IbKcd3W1LEj20Gqor4C 8/AG7AwgBArGxtvg5tmgJYjWf71/efIPn8QMaq0635bc/MLi0vLuZX86tr6xqa9tV1TIpaYVLFgQjYCpAijnFQ1 Yw0IklQGDBSDwYXY79+T6Sigt/oYUT8EPU47VKMtEHC3oUn6AMCDQlcw huzX1oGAIGUeocGdqGu7ZdcItuWs6s8DJRgKwqbfur1RE4DgnXmCGlmp4baT9BUlPMyCjfihWJEB6gHmkayVFIlJ+kG42cA0M6TldIc7h2Uvp7IkGhUsMwMJ0h0n017Y3hf14z1t0zP6E8ijXhePJQN2aOFs4 HqdDJcGaDY1AWFLzVwf3kURYmxDzJgRveuVZUSsVveNi6eqkUD7P4sjBHuybYD04hTJcQgWqgOEBnuEV3qxH68V6tz4mrXNWNrMDf8r6/AF9WpaD</latexit>

j = 0, 1, 2, . . .

<latexit sha1_base64="Qclo sRm+HyZF+83B3vBoKcUMmM=">A CA3icbVDLSgMxFL3js9ZX1aUig0VwIW mLnQjFN24bME+oB1KJpO2sZlkSDJCGQpu+im6caGIi278CXd+gz9h+lho64EDh3PuJbnHjxhV2nG+rIXFpeWV1dRaen1jc2s7s7NbUSKWmJSxYELWfKQIo5yUNdWM1CJ UOgzUvW716O8ek+ko Lf6l5EvBC1OW1RjLSxONzBJThwCq5h3vARGAQgQINqZrJOzhnDnhfuVGQLB8PS9+BwWGxmPhuBwHFIuMYMKV 3nUh7CZKaYkb6 UasSIRwF7VJ3UiOQqK8ZHxD3z42TmC3hDTk2h67vzcSFCrVC30zGSLdUbPZyPwvq8e6deEl EexJhxPHmrFzNbCHhViB1QSrFnPCIQlNX+1cQdJhLWpLW1KcGdPnheVfM49y+VLpo0rmCAF+3AEJ6bYcyjADRShDBge4Ale4NUaWM/Wm/U+GV2wpjt78AfWxw8a0ZXI</latexit>

C∗

<latexit sha1_base64="VIz7p7H7QpaUyLaiZfe/OoKk+Y0=">A B6nicbZC7SgNBFIbPxltcb6uWNoNBEIuwq4U2YjCNZURzgWQNs5PZ Mjs7DIzK4Qlj2BjoYilvou9jfg2Ti6FJv4w8PH/5zDn CDhTGnX/bZyC4tLy v5VXt fWNzy9neqak4lYRWScxj2QiwopwJWtVMc9pIJMVRwGk96JdHef2eSsVicasHCfUj3BUsZARrY92U747aTsEtumOhefCmULj4sM+Tty+70nY+W52YpBEVmnCsVN zE+1nWGpGOB3arVTRBJM+7tKmQYEjqvxsPOoQHRing8JYmic0Gru/OzIcKTWIAlMZYd1Ts9nI/C9rpjo8 zMmklRTQSYfhSlHOkajvVGHSUo0HxjARDIzKyI9LDHR5jq2OYI3u/I81I6L3knRvXYLpUuYKA97sA+H4MEplOAK lAFAl14gCd4trj1aL1Yr5PSnDXt2YU/st5/A 8CkKI=</latexit>
  • algebra generated from with

restriction of the unique g.s. onto AR

<latexit sha1_base64="9xUemq7z3u5plpxIYvh0leT3b8o=">A CEXicbZC7TsMwFIZPuJZyCxexsFhUSExVUgY S1kYW0QvUltVju 0Vu0ksh2kKuorsPAMHdlYGEAIsbGxsfAsuJcBWo5k+dP/nyP7/F7EmdKO82UtLC4tr6ym1tLrG5tb2/bObkWFsS 0TEIeypqHFeUsoGXN Ke1SFIsPE6rXu9y5FdvqVQsDG50P6JNgTsB8xnB2kihvQ9DEIB Qxd8kIZ6kMAFDKBl7qFRBC 4hkHLzjhZ 1xoHtwpZPIHpW/2UHgvtuzPRjsksaCBJhwrVXedSDcTLDUjnA7SjVjRCJMe7tC6wQALqprJeKMBOjZKG/mhNCfQaKz+nkiwUKovPNMpsO6qW 8k/ufVY+2fNxMWRLGmAZk85Mc 6RCN4kFtJinRvG8AE8nMXxHpYomJNiGmTQju7MrzUMl 3dNsrmTSKMCkUnAIR3ACLpxBHq6gCGUgcAeP8Awv1r31ZL1ab5PWBWs6swd/yvr4AaiUmlE=</latexit>

the corresponding GNS triple representation of the C* algebra von Neumann algebra

πR(AR) ⊂ πR(AR)00 = MR ⊂ B(HR)

<latexit sha1_base64="7M4eInBKn6OTY1KGYWIZTurSq9U=">A C73icrVLBbhMxEJ1dCi0p0FCOXKxGqC haLcgtZdKpVx6QSqoS OlUeT1ziZW7N2t7UWKVhF3riBxQ730 E+iX9PZdA9tmiNj2Xrz5o3H jvKlbQuCP5 /qOVx09W15421p89f7HRfLnZtVlhBHZEpjLTi7hFJVPsO kU9nKDXEcKT6PJ5yp+ h2NlVl64qY5DjQfpTKRgjuisuYPuIQCchoShlCSZ0ADg28wg7fkaeDgYAwJ8RwmpPhEkUXlO/Is7RPRiqT/P3tu09hfqv+yRH/ BAwOqValEJSjyD9aVmPYbAXtYG7sIQhr0ILajofN67M4E4XG1AnFre2HQe4GJTdOCoWzxl hMediwkfYJ5hyjXZQzt9pxt4QE7MkMzRTx+bs3YySa2unOnpf5CODOKEMzd3YLmoqclmsX7hkb1DKNC8cpuK2YFIo5jJWPT6LpUHh1JQAF0bSmZkYc8OFoy/SoGaEi1d/CLo7 fBDe+frx9bBYd2WNXgNW9TuEHbhgBp9DB0QXuz9 H5 v/1z/4/ 17+4lfpenfMK7pl/dQN8q8Qz</latexit>

πR(AR)

<latexit sha1_base64="7F1pU2ne5HCmkGkFPiBVyFqRGCY=">A CN3icbVDLSgMxFL3j2/q unQTLIKClJkq6NLHxmUVq4W2lEx6pw2TzAxJRihDP8edP+BvuPMB7sStf2DazkKrNwTOPfc knv8RHBtXPfZmZqemZ2bX1gsLC2vrK4V1zdudJwqhjUWi1jVfapR8Ahrh uB9UQhlb7AWz8 H85v71BpHkfXp 9gS9JuxAPOqLFUXCzBPaSQ2MOhDZntFEg cAUD2LWdBAoGehBYnkJoFad2MqncaxdLbtkdFfkLvByUIK9qu/jS7MQslRgZJqjWDc9NTCujynAmcFBophoTykLaxYaFEZWoW9lo3wHZsUyHBLGyNzJkxP50ZFRq3Zf+fp 0FWJoHZKanp7UDMn/Zo3UBMetjEdJajBi4weDVBATk2GIpM VMiP6FlCmuP0zYT2qKDM26oINw5tc/S+4qZS9g3Ll8rB0cpbHsgBbsG1j9+AITuACqlADBg/wBK/w5jw6786H8zmWTjm5ZxN+lfP1DYoQoSE=</latexit>

closure of w.r.t. the weak topology

  • Def. of commutant
slide-32
SLIDE 32

a unique gapped g.s. satisfies the split property

shift on the von Neumann algebra

ω

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“Wigner’s theorem” guarantees that there are operators
 such that for any

Θσ X = cσX(cσ)∗

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X ∈ MR

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cσ ∈ B( ˜ HR)

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Hastings 2007, Matsui 2013

MR

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this means the von Neumann algebra is a type-I factor

MR ∼ = B( ˜ HR)

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˜ HR

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then there is a separable Hilbert space and

Θσ(πR( ˆ A)) = πR(|σihσ| ⌦ τ( ˆ A))

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we define

ˆ A ∈ AR

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for σ = −S, . . . , S

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, and

ˆ A

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τ( ˆ A)

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|σihσ|

<latexit sha1_base64="Cuk1n QRyCPhB4BC2+VJx6J5iYA=">A CKXicbVDLTgIxFL2DL8TXqEtdTCAmJhoygwtdEt24xEQeCRDSKRdo6HQmbceEAH/h2p0rf8OtJu6ErT9ieSQKepM25 7Ttp7/IgzpV13ZCVWVtfWN5Kbqa3tnd09e/+gpMJYUizSkIey4hOFnAksaqY5ViKJ PA5lv3uzWRefkCpWCjudS/CekDagrUYJdpQoX0MA3gGBQzaEA xWJpbmI4Dmo4vdD+6QcPOuFl3Ws5f4M1BJp+unT2O8r1Cwx7XmiGNAxSacqJU1XMjXe8TqRnlOEzVYoURoV3Sxq Bg So6v3phkPnxDBNpxVKc4R2puxvR58ESvUC/zyO2hKxaxwB0R21rJmQ/82qsW5d1ftMRLFGQWcPtmLu6NCZxOY0mUSqec8AQiUzf3Zoh0hCtQk3ZcLwl f/C0q5rHeRzd2ZVK5hVk 4gjScg eXkIdbKEARKDzBK7zBu/VifVif1ngmTVhz yEslPX1DS58oc =</latexit>

πR( ˆ A) ∈ πR(AR) ⊂ MR

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since can be extended to a unital endomorphism on

Θσ

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MR

<latexit sha1_base64="VbrHN7nAWp3cfLdFlOKG1VPL1eM=">A CGXicbZDLSgMxFIbPeLfexsvOTbAILqTMVEGXRTduB WrQjuUTHqmDU1mhiQjlKGv4Au49g3cKrgTt670aUzrLNR6IOTjP+cnOX+YCq6N5304E5NT0zOzc/OlhcWl5RV3de1KJ5liWGeJSNRNSDUKHmPdcCPwJlVIZSjwOuwdD/vXt6g0T+JL0 8xkLQT84gzaqyUuBvwABIoGOhCBMpSD3I4hQG07P1gFQkELmDQcstexRsVGQe/gDIUd ZyP5vthGUSY8ME1brhe6kJcqoMZwIHpWamMaWsRzvYsBhTiTrIRxsNyLZV2iRKlD2xISP1pyOnUu +DHeztKMQe9YhqenqvzND8b9eIzPRYZDzOM0Mxuz7wSgTxCRkGBNpc4XMiL4FyhS3fyasSxVlxoZ smH4f1cfh6tqxd+rVM/3y7WjIpY52IQt2AEfDqAGJ3AGdWBwB4/wBM/OvfPivDpv36MT uFZh1/lvH8Br9uaDw= </latexit>

translation invariance is essential!

slide-33
SLIDE 33

representation of the Cuntz algebra

= |σ, σ1, σ2, σ3, σ4, . . .i

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cσ |σ1, σ2, σ3, σ4, . . .i

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gives a representation of the Cuntz algebra they roughly correspond to “Wigner’s theorem” guarantees that there are operators
 such that for any

Θσ X = cσX(cσ)∗

<latexit sha1_base64="Y980Akw8mBPYK5ecjPmYasAdjZ4=">A CP3icbZDJTgJBEIZrcEPcRj16 UCMGyEzetCLCdGLR0wASQBNT1NAh54l3T0mhPgWvgJv4Av4GHDVxJvx6s1mOShYS d/fVWV6vq9SHClHWdgJRYWl5ZXkquptfWNzS17e6eswlgyL FQhL iUYWCB1jSXAusRBKp7wm8 zrXo/rdI0rFw6CouxHWfdoKeJMzqg0K7QPoQxHagKCBwr3JFHBogW+yPmShApfAZjgx9HCOHpn8+MHO DlnHGReuFORyadrJ8+DfLfwYA9rjZDFPga CapU1XUiXe9RqTkT+JSqxQojyjq0hVUjA+qjqvfGdz+RfUMapBlK8wJNxvT3RI/6SnV9LxtHLYnYMRM+1W012zOC/9WqsW5e1Hs8iGKNAZs bMaC6JCMzCQNLpFp0TWCMsnNnwlrU0mZNpanjBnu7Onzonyac89yp7fGlSuYRBL2IG0sduEc8nADBSgZu19gCG/wbr1aH9an9TVpTVjTmV34E9b3D5kYpVI=</latexit>

X ∈ MR

<latexit sha1_base64="S/RQt 8/oeATn+KGRgwZPMy73 Q=">A CIXicbVDLSgMxFL3js9ZXVRDBTVAEF1Jm6kKXohs3gorVQi0lk95pQ5PMkGSEMvQ3/AHp0j9wqYI7caeC32LauvB1ILmHc+8huSdMBDfW91+9kdGx8YnJ3FR+emZ2br6wsHhu4lQzL NYxLoSUoOCKyxb gVWEo1UhgIvwvZBv39xhdrwWJ3ZToI1SZuKR5xR6 S4sAIV6AEH5W4JFCy0IALtWBsyOI u1F3tOU CgVPo1gvrftEfgPwlwRdZ31s+ eC3+/fH9cLbZSNmqURlmaDGVAM/sbWMasuZwG7+MjWYUNamTaw6q hEU8sGm3XJhlMaJIq1O8qSgfrdkVFpTEeGW2nS1Iht5 DUtszvmb74X6+a2mi3lnGVpBYVGz4YpYLYmPTjIg2ukVnRcYQyzd2fCWtRTZl1oeZdGMHv1f+S81Ix2C6WTlwq+zBEDlZhDTYhgB3Yg0M4hjIwuIY7eIBH78Z78p69l+HoiPflWYIf8N4/AfI9n8w=</latexit>

cσ ∈ B( ˜ HR)

<latexit sha1_base64="WgVuc3r5Wx+JfaC2HXBjI3b2di0=">A CPXicbVA9SwNBEJ3z2/gVtbRZDIJahLsoaBliY6lijB j2NtsksW9vWN3TghHfo5/wsb/YGdnY6GIra2TmEKNsy 8eW8eu/PCRCuHv /kTUxOTc/Mzs3nFhaXl fyq2sXLk6tkFUR69hehtxJrYysokItLxMreR qWQtvjgZ67VZap2Jzjr1ENiLeMaqtBEei4vwuCLiGO3CgoAMRcMIKD CowDZhpE5DCyRk1A10hC5 OLEZHEOfTnOoWVIZnFG/08wX/KI/LDYOghEowKhOmvnHq1Ys0kgaFJo7Vw/8B sZt6iElv3cVepkwsUN78g6QcMj6RrZcPs+2yKmxdqxpWuQDdmfjoxHzvWikCYj l3 VxuQ/2n1FNuHjUyZJEVpxPdD7VQzjNkgStZSVgrUPQJcWEV/ZaL RdIgecohODvyuPgolQM9oql0/1CuTK Yw42YJPCD+A yhTxCVQp8Ht4hld48x68F+/d+/genfBGn X4Vd7nF8I2oT4=</latexit>

ˆ A

<latexit sha1_base64="ozjQdsINdq+GDF/dlUkm0jNQlLs=">A B/XicbVDJSgNBEK1xjeMW9ehlMAgeJMzEg17EqBePEcwCyRB6OjWTJj0L3T1CGIK/4VUlN/HqJ/gJol9jZzlo4oOCx6t6VNXzEs6ksu0vY2FxaXl Nbdmrm9sbm3nd3ZrMk4FxSqNeSwaHpHIWYRVxRTHRiKQhB7Hute7HvXr9ygki6M71U/QDUkQMZ9RorQUwBC6QEB pcwaOcLdtEew5onzpQULj7M8+Tl06y089+tTkzTECNFOZGy6diJcjMiFKMcB2YrlZgQ2iMBNjWNSIjSzcZXD6xDrXQsPxa6ImWN1d+OjIRS9kPvOE0CgdjTjpCorpydGYn/9Zqp8s/cjEVJqjCik4V+yi0VW6MorA4TSBXva0KoYPpmi3aJIFTpwEwdhjP7+jyplYrOSbF0axfKVzB DvbhAI7AgVMow 1UoAoUBDzCEzwbD8bQeDXeJqMLxtSzB39gvP8ACQ6Wug= </latexit>

τ( ˆ A)

<latexit sha1_base64="6LVDCk5j/ALZ/SpRSfPwbWXuXB0=">A C XicbZDJSgNBEIZr4h63qEcvgyIoSpjRgx6jXjwqmAWSIfR0apImPQvdNUIY4gt49RH0qOBNvPoU+jR2loMm/tDw8VcV1fX7iRSaHOfLys3Mzs0vLC7l 1dW19YLG5sVHaeKY5nHMlY1n2mUIsIyCZJYSxSy0JdY9buXg3r1DpUWcXRLvQS9kLUjEQjOyFghPAMBgxT2DXUMEWRwDn04aBZ2naIzlD0N7h 2SzuNw8evUu+6WfhutGKeh gRl0zru sk5GVMkeAS+/lGqjFhvMvaWDcYsRC1lw0v6Nt7xmnZQazMi8geur8nMhZq3Qv9ozRpK8SumQgZdfRkz8D8r1ZPKTjzMhElKWHERwuDVNoU24NY7JZQyEn2D CuhPmz TtM U4mvLwJw508fRoqx0X3pHh8Y1K5gJEWYRt2TLgunEIJruAaysDhHp7gBV6tB+vNerc+Rq05azyzBX9kf 4AIzOZOg= </latexit>

|σihσ|

<latexit sha1_base64="Cuk1n QRyCPhB4BC2+VJx6J5iYA=">A CKXicbVDLTgIxFL2DL8TXqEtdTCAmJhoygwtdEt24xEQeCRDSKRdo6HQmbceEAH/h2p0rf8OtJu6ErT9ieSQKepM25 7Ttp7/IgzpV13ZCVWVtfWN5Kbqa3tnd09e/+gpMJYUizSkIey4hOFnAksaqY5ViKJ PA5lv3uzWRefkCpWCjudS/CekDagrUYJdpQoX0MA3gGBQzaEA xWJpbmI4Dmo4vdD+6QcPOuFl3Ws5f4M1BJp+unT2O8r1Cwx7XmiGNAxSacqJU1XMjXe8TqRnlOEzVYoURoV3Sxq Bg So6v3phkPnxDBNpxVKc4R2puxvR58ESvUC/zyO2hKxaxwB0R21rJmQ/82qsW5d1ftMRLFGQWcPtmLu6NCZxOY0mUSqec8AQiUzf3Zoh0hCtQk3ZcLwl f/C0q5rHeRzd2ZVK5hVk 4gjScg eXkIdbKEARKDzBK7zBu/VifVif1ngmTVhz yEslPX1DS58oc =</latexit>

(cσ)σ=−S,...,S (cσ)∗cσ0 = δσ,σ0ˆ 1

<latexit sha1_base64="gyrXhLY5rNb3U4AY3Xm1VsZ eMY=">A CVXicbZHLSgMxFIbPjLfaehl16WbwghekzCioG6HoxqWCrUKtJZOmbTBzITkjlKGP01fwBXyDbsQHUdwInrYutPVA4M93/pDkP0GipEHPe7PsqemZ2bncfL6wsLi07KysVkycai7KPFaxvguYEUpGo wSlbhLtGBhoMRt8Hgx6N8+CW1kHN1gJxG1kLUi2ZScIaHYOYZd4PA PTAgoQUhMNij/f6QZn/4DnThjEgDBChAIvUx 8GEvwdtUkg+H7p1Z9MresNyJ4X/IzZLW+/PL0+Fj6u6079vxDwNRYRcMWOqvpdgLWMaJVeim79PjUgYf2QtUSUZsVCYWjZMpetuE2m4zVjTitAd0t8nMhYa0wkDcoYM2 a8N4D/9aopNk9rmYySFEXERxc1U+Vi7A4idhtSC46qQ4JxLemtLm8z TjSIPIUgj/+5UlROSz6R8XDa0rjHEaVg3XYoH 5cAIluIQrKNOQ+vBpWZ tvVpf9rQ9O7La1s+ZNfhT9vI36DOl/g= </latexit>

πR(|σihσ0| ⌦ ˆ 1[1,1)) = cσ(cσ0)⇤

<latexit sha1_base64="MGb0J7reNKMobAlGfM+nU0U0kRk=">A Cr3icbVFLb9NAEB4bKCVACXDksiIgWlRFdloJLkhVuSBOBZqmUpqE9WacrLIPa3eNFJn8H 5U+TWM3UiQlLFW+h7z8M5mhZI+JMl1FN+5e2/n/u6D1sNHj/e tJ8+u/C2dAL7wir LjPuU mD/SCDwsvCIdeZwkG2+Fj7gx/ovLTmPCwLHGk+MzKXg eSbHsCv6CEgj4JE6iIOdDA4CusYB9+EvfkzEj cfBEFOAxNQG+5v3pqmzE jR5Hlic9ID9U+pbz1nSOiQdEkdcnKWcEDOAXwA eONbvuNUm1NqHPH8HbS7iTdpAl2G6Rr0IF1nE3av6+mVpQaTRCKez9MkyKMKu6CFApXravSY8HFgs9wSNBwjX5UNXtesdekTFluHR0TWKP+W1Fx7f1SZ4dlMXOIC6rQPMz9dk4t/s8bliF/P6qkKcqARtwMzEvFgmX147GpdCiCWhLgwkn6Zybm3HER6IlbtIx0+ q3wUWvmx51e1+O yen67Xswgt4SYtO4R2cwCc4gz6I6FX0OfoWncdpPIjH8feb1Dha1zyHjYjlH7j7tVE=</latexit>
  • ne can show that

P

σ cσπR( ˆ

A)(cσ)∗ = πR(τ( ˆ A))

<latexit sha1_base64="X5zemFfZLr o0r2Vqfrb4fNTSBw=">A Cl3ichVFNbxMxEJ3dUihpgbScEBeLqFKDULQbDlSCigAS6rGFpq2UhsjrTDZWvB+yx5WiVX4Kx/4mPv4HdyZJDzStxFiW3jy/N7ZnktJoR1H0KwjX7q3f 7DxsLa59ejxk/r2zqkrvFXYVYUp7HkiHRqdY5c0GTwvLcosMXiWTD7Nz8 u0Tpd5Cc0LbGfyT XI60kMVXUv8IVOPCQwWCBNKSMJQhQ8O0Gc8WqkpdmZcWZ VbAF5jBHmdjVhDzHzhvMrPqbnL+Eg7+U4VY6e+o1xzUG1ErWoS4DeJr0Ojs/vn543stPRrUf18MC+Uz EkZ6VwvjkrqV9KSVgZntQv sJRqIlPsMcxlhq5fLfo5E7vMDMWosLxzEgv2X0clM+emWfLKl6lFnLAjkzR2q5o5ed Zz9Nov1/pvPSEuVpeOPJGUCHmQxJDbVGRmTKQymp+s1Bja UiHmWNmxGvfv02OG234tet9nHc6HyEZWzAc3jB7Y3hDXTgEI6gCyrYCtrB2+Bd+Cx8H34OD5fSMLj2PIUbER7/BY6+sv8=</latexit>
slide-34
SLIDE 34

transformation on

Z2 × Z2

<latexit sha1_base64="QGsH6AVHbdcoeQlURKgozyT3FBQ=">A CJXicdVDLSgMxFL3js9bXqEtB ovgqsxUQZdFNy4r2Ae2Q8mkaRuazAzJHaEM/RlB/BU3LiwiuPJXTNtZaKsnBA7n EtyTxALrtF1P62l5ZXVtfXcRn5za3tn197br+koUZRVaSQi1QiIZoKHrIocBWvEihEZCFYPBtcTv/7AlOZReIfDmPmS9ELe5ZSgkSL7CJ5A gGEPgTmpHAPI2hDyegI3HgM9H+Ztl1wi+4UziLxMlKADJW2PW51Ip IFiIVROum58bop0Qhp4KN8q1Es5jQAemxpqEhkUz76XTLkXNilI7TjZS5ITpT9edESqTWQxmYpCTY1/PeRPzLaybYvfRTHsYJspDOHuomwsHImVTmdLhiFMXQE IVN391aJ8oQtEUmzclePMrL5JaqeidFUu354XyV ZHDg7hGE7Bgwsow 1UoAoUHuEF3mBsPVuv1rv1MYsuWdnMAfyC9fUN4KqbrA= </latexit>

AR

<latexit sha1_base64="9xUemq7z3u5plpxIYvh0leT3b8o=">A CEXicbZC7TsMwFIZPuJZyCxexsFhUSExVUgY S1kYW0QvUltVju 0Vu0ksh2kKuorsPAMHdlYGEAIsbGxsfAsuJcBWo5k+dP/nyP7/F7EmdKO82UtLC4tr6ym1tLrG5tb2/bObkWFsS 0TEIeypqHFeUsoGXN Ke1SFIsPE6rXu9y5FdvqVQsDG50P6JNgTsB8xnB2kihvQ9DEIB Qxd8kIZ6kMAFDKBl7qFRBC 4hkHLzjhZ 1xoHtwpZPIHpW/2UHgvtuzPRjsksaCBJhwrVXedSDcTLDUjnA7SjVjRCJMe7tC6wQALqprJeKMBOjZKG/mhNCfQaKz+nkiwUKovPNMpsO6qW 8k/ufVY+2fNxMWRLGmAZk85Mc 6RCN4kFtJinRvG8AE8nMXxHpYomJNiGmTQju7MrzUMl 3dNsrmTSKMCkUnAIR3ACLpxBHq6gCGUgcAeP8Awv1r31ZL1ab5PWBWs6swd/yvr4AaiUmlE=</latexit>

ωR

<latexit sha1_base64="X74VmaTzZzjVUu+igX+zl7ao08I=">A CA3icbVC7SgNBFL0bXzG+o KNzWAQrMJuL QMsbFMxDwgCWF2MpsMmZldZmaFsARs/BRtRBSxtfML7Gz8FiePQhMPDJx7zr3cucePONPGdb+c1NLy upaej2zsbm1vZPd3avpMFaEVknIQ9Xwsa cSVo1zHDaiBTFwue07g8uxn79hirNQnlth FtC9yTLGAEGytJuIcQBFDoAY OJLZWtkZwBaNONufm3QnQIvFmJFc8qHyzp9JHuZP9bHVDEgsqDeFY6 bnRqadYGUY4XSUacWaRpgMcI82LZVYUN1OJjeM0LFVuigIlX3SoIn6eyLBQu h8G2nwKav572x+J/XjE1w3k6YjGJDJZkuCmKOTIjGgaAuU5QYPrQE 8XsXxHpY4WJsbFlbAje/MmLpFbIe6f5QsWmUYIp0nAIR3ACHpxBES6hDFUgcAsP8Awvzp3z6Lw6b9PWlDOb2Yc/cN5/AF3slzA=</latexit>

ω

<latexit sha1_base64="ikgU8pvgNP3u74Edg2wXP SVsVw=">A B83icbVC7SgNBFL0bX3F9RS1tBoNgFXZjoY0YtLGMYB6QLGF2MpsMmZ0dZmaFsOQ3bCwUH6XfYW8j/o2TR6GJBy4czrmXe+8J WfaeN63k1taXl dy6+7G5tb2zuF3b26TlJFaI0kPFHNEGvKmaA1w ynTakojkNOG+Hgauw37qjSLBG3ZihpEO eYBEj2FgpgjdI AYKPcCdQtEreROgReLPSPHiwz2XL19utVP4bHcTksZUGMKx1i3fkybIsDKMcDpy26m EpMB7tGWpQLHVAfZ5OYROrJKF0WJsiUMmqi/JzIcaz2MQ9sZY9PX895Y/M9rpSY6CzImZGqoIN FUcqRSdA4ANRlihLDh5Zgopi9FZE+VpgYG5NrQ/DnX14k9XLJPymVb7xi5RKmyM BHMIx+HAKFbiGKtSAgIR7eIQnJ3UenGfnd qac2Yz+/AHzvsPVfS 6w= </latexit>

(HR, πR, ΩR)

<latexit sha1_base64="4mydYOopb9hHVByeUnzy9uq FmA=">A CT3icbVHJSgNBEK2Je9yiHr0 iqAY4owe9Bj04s0oxgS EHo6laRJz0J3jxCG/IW/4J/4Ax71E7wqeBMry0Gj1TS8eq+ql3p+rKSxrv iZGZm5+YXFpey ura+u5jc07EyVaYFlEKtJVnxtUMsSylVZhNdbIA19hxe9dDPXKPWojo/DW9mNsBLwTyrYU3BIV5Y5gH1J4BAEcFDC4hAE0R4yGgPIbyvOUJRDTkv9qV5QhdOiEafWgmdt1C+4o2F/gTcBucad+ PBS7Jeaud 6KxJ gKEVihtT89zYNlKurRQKB9l6YjDmosc7WCMY8gBNIx3NYcD2iGmxdqRph5aN2J8dKQ+M6Qd+Pok7GrFH QG3XTNdMyT/02qJbZ81UhnGicVQjC9sJ4rZiA2Hy1pSo7CqT4AL enNTHS5 sKSBVkahjf9 b/g7rjgnRSOr2kq5zCORdiGHTLKg1Mok UlKJNhT/AG7/DhPDufzldmUp xJmALfkVm6Rv5VqdO</latexit>

ˆ S(α)

j

<latexit sha1_base64="GP/ks9hXutX5JE VoBh726kzQ7M=">A CEXicbZC7TsMwFIZPuJZyC5eNJaJCgqVKChKMFSyMRdCL1IbKcd3W1LEj20Gqor4C 8/AG7AwgBArGxtvg5tmgJYjWf71/efIPn8QMaq0635bc/MLi0vLuZX86tr6xqa9tV1TIpaYVLFgQjYCpAijnFQ1 Yw0IklQGDBSDwYXY79+T6Sigt/oYUT8EPU47VKMtEHC3oUn6AMCDQlcw huzX1oGAIGUeocGdqGu7ZdcItuWs6s8DJRgKwqbfur1RE4DgnXmCGlmp4baT9BUlPMyCjfihWJEB6gHmkayVFIlJ+kG42cA0M6TldIc7h2Uvp7IkGhUsMwMJ0h0n017Y3hf14z1t0zP6E8ijXhePJQN2aOFs4 HqdDJcGaDY1AWFLzVwf3kURYmxDzJgRveuVZUSsVveNi6eqkUD7P4sjBHuybYD04hTJcQgWqgOEBnuEV3qxH68V6tz4mrXNWNrMDf8r6/AF9WpaD</latexit>

j = 0, 1, 2, . . .

<latexit sha1_base64="Qclo sRm+HyZF+83B3vBoKcUMmM=">A CA3icbVDLSgMxFL3js9ZX1aUig0VwIW mLnQjFN24bME+oB1KJpO2sZlkSDJCGQpu+im6caGIi278CXd+gz9h+lho64EDh3PuJbnHjxhV2nG+rIXFpeWV1dRaen1jc2s7s7NbUSKWmJSxYELWfKQIo5yUNdWM1CJ UOgzUvW716O8ek+ko Lf6l5EvBC1OW1RjLSxONzBJThwCq5h3vARGAQgQINqZrJOzhnDnhfuVGQLB8PS9+BwWGxmPhuBwHFIuMYMKV 3nUh7CZKaYkb6 UasSIRwF7VJ3UiOQqK8ZHxD3z42TmC3hDTk2h67vzcSFCrVC30zGSLdUbPZyPwvq8e6deEl EexJhxPHmrFzNbCHhViB1QSrFnPCIQlNX+1cQdJhLWpLW1KcGdPnheVfM49y+VLpo0rmCAF+3AEJ6bYcyjADRShDBge4Ale4NUaWM/Wm/U+GV2wpjt78AfWxw8a0ZXI</latexit>

C∗

<latexit sha1_base64="VIz7p7H7QpaUyLaiZfe/OoKk+Y0=">A B6nicbZC7SgNBFIbPxltcb6uWNoNBEIuwq4U2YjCNZURzgWQNs5PZ Mjs7DIzK4Qlj2BjoYilvou9jfg2Ti6FJv4w8PH/5zDn CDhTGnX/bZyC4tLy v5VXt fWNzy9neqak4lYRWScxj2QiwopwJWtVMc9pIJMVRwGk96JdHef2eSsVicasHCfUj3BUsZARrY92U747aTsEtumOhefCmULj4sM+Tty+70nY+W52YpBEVmnCsVN zE+1nWGpGOB3arVTRBJM+7tKmQYEjqvxsPOoQHRing8JYmic0Gru/OzIcKTWIAlMZYd1Ts9nI/C9rpjo8 zMmklRTQSYfhSlHOkajvVGHSUo0HxjARDIzKyI9LDHR5jq2OYI3u/I81I6L3knRvXYLpUuYKA97sA+H4MEplOAK lAFAl14gCd4trj1aL1Yr5PSnDXt2YU/st5/A 8CkKI=</latexit>
  • algebra generated from with

restriction of the unique g.s. onto AR

<latexit sha1_base64="9xUemq7z3u5plpxIYvh0leT3b8o=">A CEXicbZC7TsMwFIZPuJZyCxexsFhUSExVUgY S1kYW0QvUltVju 0Vu0ksh2kKuorsPAMHdlYGEAIsbGxsfAsuJcBWo5k+dP/nyP7/F7EmdKO82UtLC4tr6ym1tLrG5tb2/bObkWFsS 0TEIeypqHFeUsoGXN Ke1SFIsPE6rXu9y5FdvqVQsDG50P6JNgTsB8xnB2kihvQ9DEIB Qxd8kIZ6kMAFDKBl7qFRBC 4hkHLzjhZ 1xoHtwpZPIHpW/2UHgvtuzPRjsksaCBJhwrVXedSDcTLDUjnA7SjVjRCJMe7tC6wQALqprJeKMBOjZKG/mhNCfQaKz+nkiwUKovPNMpsO6qW 8k/ufVY+2fNxMWRLGmAZk85Mc 6RCN4kFtJinRvG8AE8nMXxHpYomJNiGmTQju7MrzUMl 3dNsrmTSKMCkUnAIR3ACLpxBHq6gCGUgcAeP8Awv1r31ZL1ab5PWBWs6swd/yvr4AaiUmlE=</latexit>

the corresponding GNS triple

Z2 × Z2

<latexit sha1_base64="45Gg98zcqmVv4QA875dMxIV80R8=">A CLXicdVDJSgNBEK2JW4xb1KMg 0HwIGEmCnoMevEYwSyYhNDTqSRNeha6a4Qw5GcEz/6GVwUPinj1N+wsB020mobHW+iu50VSaHKcNyu1sLi0vJ ezaytb2xuZbd3KjqMFc yD2Woah7TKEWAZRIksRYpZL4nser1L0d69Q6VFmFwQ4MImz7rBqIjOCNDhdl9eA fGBD0wDMngVsYQgsKhicQRkPQ/3la2ZyTd8ZjzwN3CnIwnVIr+95ohz 2MSAumdZ1 4momTBFgkscZhqx ojxPuti3cCA+aibyXjLoX1omLbdCZW5Adlj9mciYb7WA987jqOuQuybhM+op2c9I/IvrR5T57yZiC KCQM+ebATS5tCe1Sd3RYKOcmBAYwrYf5s8x5TjJMpOGPKcGdXnweVQt49yReuT3PFi2ktadiDAzgCF86gCFdQgjJwuIcneIYX69F6tT6sz4k1ZU0zu/Br K9v1VGe+w= </latexit>

invariance of the g.s.

ˆ A ∈ AR

<latexit sha1_base64="+DB/5Pr+ubu2SylTHTubHv1W5E8=">A CLXicbVDLSgMxFL3j2/q uhQkWAQXUmZU0KWPjUsV+4BaSia904Ym SHJCGXozwiu/Q23Ci4UcetvmNZ aPVAkpNz7iW5J0wEN9b3X72Jyanpmdm5+cLC4tLySnF1rWriVDOs FjEuh5Sg4Ir FhuBdYTjVSGAmth72zo125RGx6ra9tPsClpR/GIM2qdFBc34R6 QMFCBicwcDcOyu1ypHUhAu1YL3db7rx3igQCVzBoFUt+2R+B/CVBTkqQ46JVfLtpxy VqCwT1JhG4Ce2mVFtORM4KNykBhPKerSD UcVlWia2WjKAdl2SptEsXZLWTJSf3ZkVBrTl+FumnQ0Ys91SGq7ZrxmKP7nNVIbHTUzrpLUomLfD0apIDYmw+hIm2tkVvQdoUxz92fCulRTZl3ABRdGMD76X1LdKwf75b3Lg9LxaR7LHGzAFuxA IdwDOdwARVgcAeP8ATP3oP34r17H9+lE17esw6/4H1+AXKWn1g=</latexit>

α = x, y, z

<latexit sha1_base64="Q/vRWLX7Kqr6qisApC8syQM1IxI=">A CEXicbVDLSgMxFL3js9bXqEs3wSK4KGWmCroRim5cVrAPaIeS W/b0MyDJCPWoVt/wG9w51bBnbj1C/RrTB8LbT1wycm595Dc48eCK+04X9bC4tLy mpmLbu+sbm1be/sVlWUSIYVFolI1n2qUPAQK5prgfVYIg18gTW/fznq125RKh6FN3oQoxfQbsg7nF tpMi24QkoCIihZ85zc5MQAIE7yMPA1H3LzjkFZw yT9wpycEU5Zb93WxHLAkw1ExQpRquE2svpVJzJnCYbSYKY8r6tIsNQ0MaoPLS8SZDcmiUNulE0lSoyVj97UhpoNQg8PNJ3JWIfeMIqO6p2ZmR+F+vkejOmZfyME40hmzyYCcR EdkFA9pc4lMi4EhlElu/kxYj0rKtAkxa8JwZ1efJ9ViwT0uFK9PcqWLaSwZ2IcDOAIXTqE V1CGCjB4gGd4gVfr0Xqz3q2PyeiCNfXswR9Ynz/ekpeN</latexit>

for any and the invariance of the GNS inner product

MR

<latexit sha1_base64="VbrHN7nAWp3cfLdFlOKG1VPL1eM=">A CGXicbZDLSgMxFIbPeLfexsvOTbAILqTMVEGXRTduB WrQjuUTHqmDU1mhiQjlKGv4Au49g3cKrgTt670aUzrLNR6IOTjP+cnOX+YCq6N5304E5NT0zOzc/OlhcWl5RV3de1KJ5liWGeJSNRNSDUKHmPdcCPwJlVIZSjwOuwdD/vXt6g0T+JL0 8xkLQT84gzaqyUuBvwABIoGOhCBMpSD3I4hQG07P1gFQkELmDQcstexRsVGQe/gDIUd ZyP5vthGUSY8ME1brhe6kJcqoMZwIHpWamMaWsRzvYsBhTiTrIRxsNyLZV2iRKlD2xISP1pyOnUu +DHeztKMQe9YhqenqvzND8b9eIzPRYZDzOM0Mxuz7wSgTxCRkGBNpc4XMiL4FyhS3fyasSxVlxoZ smH4f1cfh6tqxd+rVM/3y7WjIpY52IQt2AEfDqAGJ3AGdWBwB4/wBM/OvfPivDpv36MT uFZh1/lvH8Br9uaDw= </latexit>

ωR(Rα( ˆ A)) = ωR( ˆ A)

<latexit sha1_base64="KxNXsf+I0X2tinUb3TsgcbJS8JQ=">A CcXichVHdShtBFD67rX8x1rSiULwZDEJKJexqQW+EVG96aYtRIQlhdnKSDJndW ZmhbDkcXwofQBfwBfo2bgXNil4hmG+7zvnzM83UaqkdUHw6PkfPq6srq1vVDarW5+2a5+/3FidGYFtoZU2dxG3qGSCbSedwrvUI 8jhbfR5L I396jsVIn126aYi/mo0QOpeCOJF1rwQNoiAFhB z6kBM3xBn8gRk05lxQRpVKn3jBUhjT2iBWrI7qflL2G43zd3Zc6OjX6kEzmAdbBmEJ6lDGVb/21B1okcWYOKG4tZ0wSF0v58ZJoXBW6WYWUy4mfIQdg mP0fbyuVMzdkjKgA21oZk4Nlf duQ8tnYaR0dZOjKIE+qIuRvbxZpC/F+uk7nhWS+XSZo5TMTrgcNM adZYT8bSIPCqSkBLoykOzMx5oYLR59UITPCxacvg5vjZnjSP 79o96 KG1Zh304IHNDOIUW/I raNO3PXtVb9fb8178rz7zD15Lfa/s2YF/wv/+F6jTp3o=</latexit>

hψ ˆ

A, ψ ˆ Bi = ωR( ˆ

A∗ ˆ B) = ωR(Rα( ˆ A∗)Rα( ˆ B)) = hψRα( ˆ

A), ψRα( ˆ B)i

<latexit sha1_base64="ylZG+vaZVQ76S/981sQT38371uo=">A Dr3iclVJNb9NAEJ3afJTw0RSOXFYEJIKqyC5I9IJUwgVxKtA0ldIkrDcbZ5X17mp3jR Z+Tn9UXDhrzB2jNQmjYC1L +ZN292/DSJkcL5KPqxE4S3bt+5u3uvcf/Bw0d7zf3HZ07nlvEe01Lb84Q6LoXiPS+85OfGcpolkveT+YeS73/n1gmtTv3C8GFGUyWmglGPKd38BZcg YKCFL8cIwMOBIyhQDxDxiN6D0t8Draw3Yq9BHulz uMNWSIUsyu6i3GBL5g7cu13iN4tdavDY2/9ihjhoysM2OMy8hUnW6 o72hGW3VrKb4M8c2j/5vhvaGi/+u79b6az6Pm62oE1WHbIK4Bi2oz8m4+fNiolmeceWZpM4N4sj4YUGtF0zyZeMid9xQNqcpHyBUNONuWFR7tiQvMDMhU23xVZ5U2auKgmbOLbLkIDep5XyOioz6mVuvKZM3cYPcT4+GhVAm91yx1YXTXBKvSbm8ZCIsZ14uEFBmBc5M2IxayjyueAPNiNd/fROcHXbi153Dz29ax93al 14Cs/Q7BjewjF8hBPoAQueB5+Cr8FpGIf9cBR+W5UGO7XmCVw7ofgNjpbqRw= </latexit>

unitary on can be defined by

HR

<latexit sha1_base64="QdgXfyBRrMSgXvpmn7bdqvLZ7Ro=">A C 3icbVDLSgNBEOyNrxhfUY9eBoPgQcJuFPQY9OIxinlAEsLspJM mZ1dZmaFsAT8AT9DrwrexKsfoV/jJNmDJhYMVFd30dPlR4Jr47pfTmZpeWV1Lbue29jc2t7J7+7VdBgrhlUWilA1fKpRcIlVw43ARqSQBr7Auj+8mvTr96g0D+WdGUXYDmhf8h5n1FhJwhMwoC AwDV0ILG1gsBWtzDu5Atu0Z2CLBIvJQVIUenkv1vdkMUBSsME1brpuZFpJ1QZzgSOc61Y 0TZkPaxa mkAep2Mr1hTI6s0iW9UNknDZmqvx0JDbQeBf5JHPUV4tA6AmoGen5mIv7Xa8amd9FOuIxig5LNFvZiQUxIJsGQLlfIjBhZQpni9s+EDai zNj4cjYMb/70RVIrFb3TYunmrFC+TGPJwgEcwjF4cA5lG3IFqjbyB3iGF3h1Hp035935mI1mnNSzD3/gfP4Aha6WcQ= </latexit>

invariance is essential!

Z2 × Z2

<latexit sha1_base64="45Gg98zcqmVv4QA875dMxIV80R8=">A CLXicdVDJSgNBEK2JW4xb1KMg 0HwIGEmCnoMevEYwSyYhNDTqSRNeha6a4Qw5GcEz/6GVwUPinj1N+wsB020mobHW+iu50VSaHKcNyu1sLi0vJ ezaytb2xuZbd3KjqMFc yD2Woah7TKEWAZRIksRYpZL4nser1L0d69Q6VFmFwQ4MImz7rBqIjOCNDhdl9eA fGBD0wDMngVsYQgsKhicQRkPQ/3la2ZyTd8ZjzwN3CnIwnVIr+95ohz 2MSAumdZ1 4momTBFgkscZhqx ojxPuti3cCA+aibyXjLoX1omLbdCZW5Adlj9mciYb7WA987jqOuQuybhM+op2c9I/IvrR5T57yZiC KCQM+ebATS5tCe1Sd3RYKOcmBAYwrYf5s8x5TjJMpOGPKcGdXnweVQt49yReuT3PFi2ktadiDAzgCF86gCFdQgjJwuIcneIYX69F6tT6sz4k1ZU0zu/Br K9v1VGe+w= </latexit>

ˆ Uα

<latexit sha1_base64="c34xQTIHcxapPAmPym896MSu8Qk=">A C 3icbVDLSsNAFL3xWeur6tJNsAgupCRV0GXRjcsK9gFtKJPpT t0MgkzE6GEgj/gZ+hWwZ249SP0a5y0W jrgQtnzr2Huf 4MWdKO86XtbS8srq2Xtgobm5t7+yW9vabKkokxQaNeCTbPlHImcCGZp jO5ZIQp9jyx9dZ/3WPUrFInGnxzF6IRkIFjBKtJE PMEQCGhIoQET6Jk3AQ5xpvZKZafiTGEvEjcnZchR75W+u/2IJiEKT lRquM6sfZSIjWjHCfFbqIwJnREBtgxVJAQlZdOb5jYx0bp20EkTQltT9XfjpSESo1D/zSJBxJxZBwh0UM1P5OJ/ U6iQ4uvZSJONEo6OzDIOG2juwsGLvPJFLNx4YQKpnZ2aZDIgnVJr6iCcOdP32RNKsV96xSvT0v167yWApwCEdwAi5cQA1uoG6CpvA z/ACr9aj9Wa9Wx+z0SUr9xzAH1ifP53xlyA=</latexit>

ˆ Uαψ ˆ

A = ψRα( ˆ A)

<latexit sha1_base64="7 dmhu/l2Ri+inViZOGM9WqSp9o=">A CaXicbVHLSgMxFL0zvqutU92IboJFUJAyUwVdKPjYuFSxt CWk lv29DMgyQjlGE+x4/ShT/gT5iOXdTWCyEn5 6Tx4kfC6 0635Y9tLy ura+kZhc6tY2nbKO68qSiTDOotEJ s+VSh4iHXNtcBmLJEGvsCGP7qf9BtvKBWPwhc9jrET0EHI+5xRbajIuYJ3GAIFDSnUIYOuWVMQEOfsu5kVcMOmM7pbo8vg+k930me5k8Dzwj7Hc+4TyLpOxa26eZF 4E1B ab12HU+272IJQG mgmqVMtzY91JqdScCcwK7URhTNmIDrBlYEgDVJ0 TygjR4bpkX4kzQg1ydlZR0oDpcaBf5rEA4k4Mo6A6qGa10zI/3qtRPcvOykP40RjyH4P7CeC6IhMYic9LpFpMTaAMsnNnQkbUkmZNp9TMGF4809fBK+1qndWrT2dV27uprGsw Ecmng9uIAbeIBH84kMvqxlq2iVrG+7bO/Z+79S25p6duFP2ZUf 16nXw= </latexit>

ˆ Ux, ˆ Uy, ˆ Uz

<latexit sha1_base64="+8YsnMhLf7bg+xKx32OzBklZfgI=">A CS3ichVDLSgMxFL1Trdb6GnXpJlgEF1Jmq DLohuXFewD2lIya YNzTxIMuI4zOf0X9y4cedPuHGhiAvTdhbaKh4I9+Tc 7nJcULOpLKsZyO3tJxfWS2sFdc3Nre2zZ3dhgwiQWidBDwQLQdLyplP64opTluhoNhzOG06o8tJv3lLhWSBf6PikHY9P CZywhW gpMG8YwBAwKEqhDCj1dxyDA wR3+n6s69+O+F/HPaQ9s2SVrSnQIrEzUoIMtZ751OkHJPKorwjHUrZtK1TdBAvFCKdpsRNJGmIywgPa1tTH pXdZJpFig610kduIPTxFZq 3ycS7EkZe452elgN5XxvIv7Wa0fKPe8mzA8jRX0yW+RGHKkATYJFfSYoUTzWB PB9FsRGWKBidLxF3UI9vyXF0mjUrZPypXr01L1IoujAPtwAEdgwxlU4QpqOmICD/ACb/BuPBqvxofxObPmjGxmD34gl/8CinCj5A= </latexit>

Z2 × Z2

<latexit sha1_base64="45Gg98zcqmVv4QA875dMxIV80R8=">A CLXicdVDJSgNBEK2JW4xb1KMg 0HwIGEmCnoMevEYwSyYhNDTqSRNeha6a4Qw5GcEz/6GVwUPinj1N+wsB020mobHW+iu50VSaHKcNyu1sLi0vJ ezaytb2xuZbd3KjqMFc yD2Woah7TKEWAZRIksRYpZL4nser1L0d69Q6VFmFwQ4MImz7rBqIjOCNDhdl9eA fGBD0wDMngVsYQgsKhicQRkPQ/3la2ZyTd8ZjzwN3CnIwnVIr+95ohz 2MSAumdZ1 4momTBFgkscZhqx ojxPuti3cCA+aibyXjLoX1omLbdCZW5Adlj9mciYb7WA987jqOuQuybhM+op2c9I/IvrR5T57yZiC KCQM+ebATS5tCe1Sd3RYKOcmBAYwrYf5s8x5TjJMpOGPKcGdXnweVQt49yReuT3PFi2ktadiDAzgCF86gCFdQgjJwuIcneIYX69F6tT6sz4k1ZU0zu/Br K9v1VGe+w= </latexit>

form a genuine representation of

slide-35
SLIDE 35

transformation on

Z2 × Z2

<latexit sha1_base64="QGsH6AVHbdcoeQlURKgozyT3FBQ=">A CJXicdVDLSgMxFL3js9bXqEtB ovgqsxUQZdFNy4r2Ae2Q8mkaRuazAzJHaEM/RlB/BU3LiwiuPJXTNtZaKsnBA7n EtyTxALrtF1P62l5ZXVtfXcRn5za3tn197br+koUZRVaSQi1QiIZoKHrIocBWvEihEZCFYPBtcTv/7AlOZReIfDmPmS9ELe5ZSgkSL7CJ5A gGEPgTmpHAPI2hDyegI3HgM9H+Ztl1wi+4UziLxMlKADJW2PW51Ip IFiIVROum58bop0Qhp4KN8q1Es5jQAemxpqEhkUz76XTLkXNilI7TjZS5ITpT9edESqTWQxmYpCTY1/PeRPzLaybYvfRTHsYJspDOHuomwsHImVTmdLhiFMXQE IVN391aJ8oQtEUmzclePMrL5JaqeidFUu354XyV ZHDg7hGE7Bgwsow 1UoAoUHuEF3mBsPVuv1rv1MYsuWdnMAfyC9fUN4KqbrA= </latexit>

X ∈ πR(AR)

<latexit sha1_base64="7zXl790iIY lN4woqtzi+D+fZUY=">A CP3icbVDNThsxEJ6F8hf+UjhysUAIECjahQMc0/bCEaoGIiVR5HVmEyu2d2V7kaJV3qKv0DfoC/QxkitIvVW9 sYk4VACY3n0zTf yJ4vzpR0PgxHwcLih6Xl dW10vrG5tZ2+ePOnUtzK7AmUpXaeswdKm w5qVXWM8sch0rvI/7Xyb9+we0Tqbm x9k2NK8a2QiBfdEpeUjqM PkGAo5 DRkdCGgioLGh 8hSEcU6WBg4ceJMRz6JPiE3XmlSft8kFYCafB3oLoBRxU95un30fVwU27PG52UpFrNF4o7lwjCjPfKrj1Uigclpq5w4yLPu9ig6DhGl2rmO49ZIfEdFiSWr Gsyn7/0TBtXMDHZ/lWdci9mlCc9 z85oJ+V6vkfvkqlVIk+UejZg9mOSK+ZRNzGQdaVF4NSDAhZX0ZyZ63HLhyfISmRHNr/4W3J1Xo vK+S258hlmsQp7sE+2R3AJVbiG 6iBgJ8whkd4Cn4Fv4M/wd+ZdCF4mdmFVxH8ewY7uqZM</latexit>

for , we define is then extended to which satisfies

πR(AR)00 = MR

<latexit sha1_base64="8sPr9NxqiV8TVuDvT/6eLc6TNxk=">A CZXicbVHbihNBEK0ZdU2iWc Lv hgY1hWYQkzcWEXYSHqiy9CFHOBJISeTk3SpHtm6O5ZCEM+x48yP7C/YeXyoMlW0XDq1Cm6+1ScK2ldGP7x/AcPH508rlRrT57WT58Fz1/0bFY gV2RqcwMYm5RyRS7TjqFg9wg17HCfrz4u n3b9FYmaW/3DLHseazVCZScEdUFnyC31BATilhAiV BjQw+AkreE+VBg4O5pAQz2FBis/UOVR+gHPKm3v134/1k6ARNsNtsGMQ7UED9tGZBOvRNBOFxtQJxa0dRmHuxiU3TgqFq9qosJhzseAzHBJMuUY7LrfurNgZMVOWZIZO6tiW/Xei5NrapY4vinxmEBc0obmb20PNhryvNyxc j0uZ oXDlOxuzApFHMZ21jOptKgcGpJgAsj6c1MzLnhwtFiamRGdPj1Y9BrNaOPzdaPy0b7y96WCryBd7SkCK6gDd+gA10QsPbAq3o1786v+6/81zup7+1nXsJ/4b/9C5YFpO0=</latexit>

MR

<latexit sha1_base64="VbrHN7nAWp3cfLdFlOKG1VPL1eM=">A CGXicbZDLSgMxFIbPeLfexsvOTbAILqTMVEGXRTduB WrQjuUTHqmDU1mhiQjlKGv4Au49g3cKrgTt670aUzrLNR6IOTjP+cnOX+YCq6N5304E5NT0zOzc/OlhcWl5RV3de1KJ5liWGeJSNRNSDUKHmPdcCPwJlVIZSjwOuwdD/vXt6g0T+JL0 8xkLQT84gzaqyUuBvwABIoGOhCBMpSD3I4hQG07P1gFQkELmDQcstexRsVGQe/gDIUd ZyP5vthGUSY8ME1brhe6kJcqoMZwIHpWamMaWsRzvYsBhTiTrIRxsNyLZV2iRKlD2xISP1pyOnUu +DHeztKMQe9YhqenqvzND8b9eIzPRYZDzOM0Mxuz7wSgTxCRkGBNpc4XMiL4FyhS3fyasSxVlxoZ smH4f1cfh6tqxd+rVM/3y7WjIpY52IQt2AEfDqAGJ3AGdWBwB4/wBM/OvfPivDpv36MT uFZh1/lvH8Br9uaDw= </latexit>

˜ Rα(X) := ˆ UαX ˆ U ∗

α

<latexit sha1_base64="yrVIONHLQp9fauqAdGL5fgpPesU=">A CZXicfVFdSxtBFL27Vqvxo2ktv TBwSCoSNi1QiVQkPrioxWjgRjD7OQmGTL7wcxdISz5Of6o5g/0b/RuzIM14oVhzjn3XGbmTJQZ7SgI/nj+0ofl Y+ra5X1jc2tT9XPX25dmluFTZWa1LYi6dDoBJukyWArsyj yOBdNLo +3ePaJ1OkxsaZ9iJ5SDRfa0ksZRWG/AEB oM9AChYKZAMhNwDRPoMi9ZBkPeD6AFh9CAn6yWnNjfXHAJdr3Xf4CjbrUW1INZiU QzkEN5nXVrU7ve6nKY0xIGelcOw y6hTSklYGJ5X73GEm1UgOsM0wkTG6TjFLZyL2WemJfmp5JSRm6suJQsbOjePoOM8GFnHE 7GkoXvtKcW3eu2c+medQidZTpio5wP7uRGUijJy0dMWFZkxA6ms5jsLNZRWKuKPqXAY4eunL4Lbk3r4vX7y+7R2/msey p8gz3+lB +wDlcwhWHrWDqgbfmVby/ qb/1d95tvrefGYb/it/9x+Ya6T1</latexit>

˜ Rα(πR( ˆ A)) = πR(Rα( ˆ A))

<latexit sha1_base64="q b4/6Jb/iSIdF/qLQJ3fjEf4ac=">A Cm3ichVFdaxNBFL27rVrj17Y+ijAYBAUJu1XavpRG+yKlD1XctpCGMDu5SYbM7g4zdwthyc/pj9Jf4910HzSVeodhzjn3 Pm4k1mjPcXxzyDc2Hzw8NHW486Tp8+ev4i2d859WTmFqSpN6S4z6dHoAlPSZPDSOpR5ZvAimx83+YtrdF6XxQ9aWBzmclroiVaSWCqjFG6AQIOBMSDUzBRIZgK+wxJGzBtmYcbrO2YVY8v+0cr IG+dTa7xEOufmb/ncXiv/ 9nre03irpxL16FuAuSFnShjbNR9OtqXKoqx4KUkd4PktjSsJaOtDK47FxVHq1UcznFAcNC5uiH9aqnS/GWlbGYlI5nQWKl/l Ry9z7RZ59qOzUIc65Ipc08+ueRvxXblDR5GBY68JWhIW6PXBSGUGlaD5KjLVDRWbBQCqn+c5CzaSTivg7O9yMZP3pd8H5bi/52Nv9 qnb/9K2ZQtewRtubwL70IevcAYpqCAK9oKjoB+ Do/Dk/D01hoGbc1L+CvC9DcKHK/G</latexit>

˜ Rα

<latexit sha1_base64="rD0fn+KAYm/GcChD 1 M05FXGeU=">A CG3icbVDLSgNBEOz1GeMrKp68DAbBg4TdKOgx6MVjFPOAZAmzk95kyOyDmVkhLPkHf8Cjv+BVwZt49aBf4yTZgyY2DFRVd9HT5cWCK23bX9bC4tLy mpuLb+ sbm1XdjZrasokQxrLBKRbHpUoeAh1jTXApuxRBp4Ahve4Grcb9yjVDwK7/QwRjegvZD7nF tpKiwD0+g YOALiCkhjGgh G4hRF0DB+zGPpAO4WiXbInReaBk4EiZFXtFL7b3YglAYa CapUy7Fj7aZUas4EjvLtRGFM2YD2sGVgSANUbjq5aUSOjNIlfiTNCzWZqL8dKQ2UGgbeSRL3JOLAOAKq+2p2Ziz+12sl2r9wUx7GicaQTRf6iSA6IuOgSJdLZFoMDaBMcvNnwvpU qZNnHkThjN7+jyol0vOa l8c1asXGax5OA DuEYHDiHClxDFWom8gd4h d4tR6tN+vd+piOLliZ w/+lPX5A8ksmp0=</latexit>

unitary on can be defined by

HR

<latexit sha1_base64="QdgXfyBRrMSgXvpmn7bdqvLZ7Ro=">A C 3icbVDLSgNBEOyNrxhfUY9eBoPgQcJuFPQY9OIxinlAEsLspJM mZ1dZmaFsAT8AT9DrwrexKsfoV/jJNmDJhYMVFd30dPlR4Jr47pfTmZpeWV1Lbue29jc2t7J7+7VdBgrhlUWilA1fKpRcIlVw43ARqSQBr7Auj+8mvTr96g0D+WdGUXYDmhf8h5n1FhJwhMwoC AwDV0ILG1gsBWtzDu5Atu0Z2CLBIvJQVIUenkv1vdkMUBSsME1brpuZFpJ1QZzgSOc61Y 0TZkPaxa mkAep2Mr1hTI6s0iW9UNknDZmqvx0JDbQeBf5JHPUV4tA6AmoGen5mIv7Xa8amd9FOuIxig5LNFvZiQUxIJsGQLlfIjBhZQpni9s+EDai zNj4cjYMb/70RVIrFb3TYunmrFC+TGPJwgEcwjF4cA5lG3IFqjbyB3iGF3h1Hp035935mI1mnNSzD3/gfP4Aha6WcQ= </latexit>

invariance is essential!

Z2 × Z2

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ˆ Uα

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ˆ Uαψ ˆ

A = ψRα( ˆ A)

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ˆ Ux, ˆ Uy, ˆ Uz

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Z2 × Z2

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form a genuine representation of

˜ Rx, ˜ Ry, ˜ Rz

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form a genuine representation of Z2 × Z2

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˜ Rα ˜ Rβ = ˜ Rβ ˜ Rα

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˜ Rα ˜ Rα = id

<latexit sha1_base64="yCjSxeXd5Go38MheOinL6j0flGY=">A CX3icjVFNSwMxEJ1dv6vWVU/iJVgED1J2VdCLIHrxqGKrUEvJZqdtaPaDJCuUpT/Hf+RFb/4NT87WHrT14ECS92bmkeRNmClprO+/Oe7c/MLi0vJKZXVtvbrhbW41TZprgQ2RqlQ/htygkgk2rLQKHzONPA4VPoSDq7L+8IzayDS5t8M 2zHvJbIrBbeUSr1TeAELEhREgFAQE8CJMbiDEXSIly DPp1lTYKm/f+aCpwT1xBTVULU8Wp+3R8HmwXB NRgEjcd7/0pSkUeY2KF4sa0Aj+z7YJrK4XCUeUpN5hxMeA9bBFMeIymXYx9GbF9ykSsm2paiWXj7E9FwWNjhnF4mGc9jTg Rcxt30z3lMm/aq3cds/ahUy 3GIivi/s5orZlJVms0hqF YNCXChJb2ZiT7X FgaSYXMCKa/PguaR/XguH50e1K7uJzYsgy7sAcHEMApXMA13ECDRvAKn86cM+98uEtu1fW+W1 notmGX+HufAG596Un</latexit>

˜ Rx ˜ Ry = ˜ Rz

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˜ Ry ˜ Rz = ˜ Rx

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˜ Rz ˜ Rx = ˜ Ry

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slide-36
SLIDE 36

transformation of the Cuntz algebra

= |σ, σ1, σ2, σ3, σ4, . . .i

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cσ |σ1, σ2, σ3, σ4, . . .i

<latexit sha1_base64="FH+/C/cHPrn9lYIb1/IdnVqcPW8=">A CYXicbZHNTgIxFIXvjKiAgiMu2UwgJiYaMgMmunB dOMSE/lJAEmnFGjotJO2Y0KQt/AVeCF3bNz4IpafhYA3aXLynXvT9p4gYlRpz1tY9kHi8Og4mUqfnGayZ85 rqFELDGpY8GEbAVIEUY5qWuqGWlFkqAwYKQZjJ+WfvOdSEUFf9WTiHRDNOR0QDHSBgn ATC8wRwU BhC MjoG/jYIj3wDdsm5T1S2SO3K8KgDwK0ceYgDefGZ0B6TtEreaty94W/EcVqoXP9uahOaj3nq9MXOA4J15ghpdq+F+nuFElNMSOzdCdWJEJ4jIakbSRHIVHd6WpDM/fSkL47ENIcrt0V/TsxRaFSkzAwnSHSI7XrLeF/XjvWg/vulPIo1oTj9UWDmLlauMt1u30qCdZsYgTCkpq3uniEJMLahJI2S/B3v7wvGuWSXyn5L36x+gjrSkIeCnBlgrmDKjxD eomyG8rYW srPVjp2zHzq1b WszcwFbZed/AbeZpcA=</latexit>

fix , and let

cσ ∈ B( ˜ HR) ∼ = MR = πR(AR)00

<latexit sha1_base64="ayZFxb37a38q/F+HfjhiyXDjOgo=">A Cw3icbVHbat AEB2pt9S9Oe1jX5aEkpQWI6UPzUvATSn0pZCWOgnYjlmtR/Ki1a7YHRWM8F/0F/pR6d 0bOchjT3Lwjlz5uxlJquNDpQk1 F87/6Dh492Hne PH32/EV39+V5cI1XOFDO H+ZyYBGWxyQJoOXtUdZ QYvsvLzUr/4hT5oZ3/SvMZxJQurc60kc p1NSi4gj8Q EMBFUjG iwIOIVDxsTMwBQ WmZLnWDGHsnZFr7CgtdkpXlWBfxg/paZAsenFLc8OVdIKLn2 xbPCbMGal56QzvcesqnrTcfwMGku5/0klWITZDegP3+3ujd7+v+/GzS/TuaOtVUaEkZGcIwTWoat9KTVgYXnVETsJaqlAUOGVpZYRi3q94vxBvOTEXuPG9LYpW97WhlFcK8yt43deERS3ZUkmbhbs0yuU0bNpQfj1t 64bQqvWFeWMEObEcqJhqj4rMnIFUXvObhZpJLxXx2DvcjPTu1zfB+VEv/dA7+s5dOYV17MBr2OPWp/AR+jzoMxiAio6jq6iIZvGXuIx9TOvSOLrxvIL/Il78AxP vSQ=</latexit>

(cσ)∗cσ0 = δσ,σ0ˆ 1

<latexit sha1_base64="gyrXhLY5rNb3U4AY3Xm1VsZ eMY=">A CVXicbZHLSgMxFIbPjLfaehl16WbwghekzCioG6HoxqWCrUKtJZOmbTBzITkjlKGP01fwBXyDbsQHUdwInrYutPVA4M93/pDkP0GipEHPe7PsqemZ2bncfL6wsLi07KysVkycai7KPFaxvguYEUpGo wSlbhLtGBhoMRt8Hgx6N8+CW1kHN1gJxG1kLUi2ZScIaHYOYZd4PA PTAgoQUhMNij/f6QZn/4DnThjEgDBChAIvUx 8GEvwdtUkg+H7p1Z9MresNyJ4X/IzZLW+/PL0+Fj6u6079vxDwNRYRcMWOqvpdgLWMaJVeim79PjUgYf2QtUSUZsVCYWjZMpetuE2m4zVjTitAd0t8nMhYa0wkDcoYM2 a8N4D/9aopNk9rmYySFEXERxc1U+Vi7A4idhtSC46qQ4JxLemtLm8z TjSIPIUgj/+5UlROSz6R8XDa0rjHEaVg3XYoH 5cAIluIQrKNOQ+vBpWZ tvVpf9rQ9O7La1s+ZNfhT9vI36DOl/g= </latexit>

πR(|σihσ0| ⌦ ˆ 1[1,1)) = cσ(cσ0)⇤

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P

σ cσπR( ˆ

A)(cσ)∗ = πR(τ( ˆ A))

<latexit sha1_base64="X5zemFfZLr o0r2Vqfrb4fNTSBw=">A Cl3ichVFNbxMxEJ3dUihpgbScEBeLqFKDULQbDlSCigAS6rGFpq2UhsjrTDZWvB+yx5WiVX4Kx/4mPv4HdyZJDzStxFiW3jy/N7ZnktJoR1H0KwjX7q3f 7DxsLa59ejxk/r2zqkrvFXYVYUp7HkiHRqdY5c0GTwvLcosMXiWTD7Nz8 u0Tpd5Cc0LbGfyT XI60kMVXUv8IVOPCQwWCBNKSMJQhQ8O0Gc8WqkpdmZcWZ VbAF5jBHmdjVhDzHzhvMrPqbnL+Eg7+U4VY6e+o1xzUG1ErWoS4DeJr0Ojs/vn543stPRrUf18MC+Uz EkZ6VwvjkrqV9KSVgZntQv sJRqIlPsMcxlhq5fLfo5E7vMDMWosLxzEgv2X0clM+emWfLKl6lFnLAjkzR2q5o5ed Zz9Nov1/pvPSEuVpeOPJGUCHmQxJDbVGRmTKQymp+s1Bja UiHmWNmxGvfv02OG234tet9nHc6HyEZWzAc3jB7Y3hDXTgEI6gCyrYCtrB2+Bd+Cx8H34OD5fSMLj2PIUbER7/BY6+sv8=</latexit>

α = x, y, z

<latexit sha1_base64="NvRZe4um2dVyv DpvaoGHSWPvAw=">A C XicbVC7SgNBFL0bXzG+Vi0VGQyChYTdWGgjBG0sEzAPSJYwO5lNhsw+mJkV1yWlNn6HnY2FEmz9Azu/wZ9w8ig0euAyZ865l5l73IgzqSzr08jMzS8sLmWXcyura+sb5uZWTYaxILRKQh6Khosl5SygVcU p41IUOy7nNbd/sXIr19TIVkYXKk o 6PuwHzGMFKS6FpwiNg4B T59n+ibABwQ3cASJrtu2mbcK1hjoL7GnJF/aHVa+7vaG5b 50eqEJPZpoAjHUjZtK1JOioVihN BrhVLGmHSx13a1DTAPpVO t5kgA60 kFeKHQFCo3VnxMp9qVMfFd3+lj15Kw3Ev/zmrHyTp2UBVGsaEAmD3kxRypEo1hQhwlKFE80wUQw/VdEelhgonR4OR2CPbvyX1IrFuzjQrFi50vnME WdmAfDsG EyjBJZShCgTu4Qle4NV4MJ6NofE2ac0Y05lt+AXj/RsK2Zfm</latexit>
  • π rotation in

the spin space π rotation in the

  • perator space

tσ = P

σ0hσ|ˆ

u†

α|σ0i ˜

Rα(cσ0)

<latexit sha1_base64="GORLM3zNW7woiforXlI5dXrsqks=">A CwXicbVFNb9NAEB2bj5bwFeDIZUVUAZfIbg/0AorohWNBpK2UptF6PXFW3V1bu2OkyM1P4Zg/xAX4IdyZOD0 acey9s28N8/jnawyOlCS/I3ie/cfPNzZfdR5/OTps+fdFy9PQl 7hUNVmtKfZTKg0Q6HpMngWeVR2szgaXZ5tOJPf6APunTfaV7h2MrC6alWkrhUdg0QXMASAmgowIKEj21WM5 As8G8hQXnhpHjigHcYK84m/FJ3FWzcuWac17wg+DZbcmZgapVXW05L1lx05eYM9yP7Qyq7RTwjX03fd4xd3H nO8n3V7ST9oQt0F6DXqDvX9/fv/sFMeT7q/zvFS1RUfKyB GaVLRuJGetDK46JzXASupLmWBI4ZOWgzjpt3AQux JRfT0vPrSLTVmx2NtCHMbcZK 2kWtrlV8S5uVNP0cNxoV9WETq0/NK2NoFKs1ily7VGRmTOQymueVaiZ9FIRL73Dl5Bu/ JtcL fTw/6+1/T3uAzrGMX sMbvtwUPsA vsAxDEF n6I8spGLj2IdV7FfS+Pou cVbETc/Af/175B</latexit>

(tσ)∗tσ0 = δσ,σ0ˆ 1

<latexit sha1_base64="0DbeKiPT2dBh6fkxuh6aqzcg52M=">A CVXicbVFdSwJBFL27WZl9aPXYy5JEH4TsWlQvgdRLjwb5AbrJ7Djq4OwHM3cDWfw5/iFfon/S 9C4+pDahYEz5 zLnTnXiwRXaNtfhrmR2dzazu7kdvf2D/KFw6O6CmNJWY2GIpRNjygmeMBqyFGwZiQZ8T3BGt7weaY3PphUPAzecBQx1yf9gPc4JaipsHAHF4DwDhNQwKEP hC41PerlE2W+HMYw6NmusBAaJ1AZ8VxveafwEAj1D4Hxp1C0S7ZaVnrwFmAIiyq2ilM292Qxj4LkAqiVMuxI3QTIpFTwca5dqxYROiQ9FlLw4D4TLlJmsrYOtNM1+qFUp8ArZT925EQX6mR72mnT3CgVrUZ+Z/WirH34CY8iGJkAZ0P6sXCwtCaRWx1uWQUxUgDQiX b7XogEhCUS8ip0NwVr+8DurlknNTKr/eFitPiziycAKnel0O3EMFXqAKNaAwhW/DMEzj0/gxM+bW3Goai5 jWCoz/wtmVKHb</latexit>

πR(|σihσ0| ⌦ ˆ 1[1,1)) = tσ(tσ0)⇤

<latexit sha1_base64="2o+Xg1sm6EXsMsVaUJEnlZjk0oM=">A Cr3icbVFLb9NAEB4bKCVACXDksiIgWlRFdloJLkhVuSBOBZqmUpqE9WacrLIPa3eNFJn8H 5U+TWM3UiQlLFW+h7z8M5mhZI+JMl1FN+5e2/n/u6D1sNHj/e tJ8+u/C2dAL7wir LjPuU mD/SCDwsvCIdeZwkG2+Fj7gx/ovLTmPCwLHGk+MzKXg eSbHsCv6CEgj4JE6iIOdDA4CusYB9+EvfkzEj cfBEFOAxNQG+5v3pqmzE jR5Hlic9ID9U+pbz1nSOiQdEkdcnKWcEDOAXwgPN7ot 8o1daEOncMbyftTtJNm C3QboGHVjH2aT9+2pqRanRBKG498M0KcKo4i5IoXDVuio9Flws+AyHBA3X6EdVs+cVe03KlOXW0TGBNeq/FRX 3i91dlgWM4e4oArNw9xv59Ti/7xhGfL3o0qaogxoxM3AvFQsWFY/HptKhyKoJQEunKR/ZmLOHReBnrhFy0i3r34bXPS6 VG39+W4c3K6Xs uvICXtOgU3sEJfI z6IOIXkWfo2/ReZzGg3gcf79JjaN1zXPYiFj+AfI5tXM=</latexit>

P

σ tσπR( ˆ

A)(tσ)∗ = πR(τ( ˆ A))

<latexit sha1_base64="eMAcYNrPr6rsxDsAZlylkv8xmWM=">A Cl3ichVFNbxMxEJ3dUihpgbScEBeLqFKDULQbDlSCigAS6rGFpq2UhsjrTDZWvB+yx5WiVX4Kx/4mPv4HdyZJDzStxFiW3jy/N7ZnktJoR1H0KwjX7q3f 7DxsLa59ejxk/r2zqkrvFXYVYUp7HkiHRqdY5c0GTwvLcosMXiWTD7Nz8 u0Tpd5Cc0LbGfyT XI60kMVXUv8IVOPCQwWCBNKSMJQg +HaDuWJVyUuzsuLM ivgC8xgj7MxK4j5D5w3mVl1Nzl/CQf/qUKs9HfUaw7qjagVLULcBvE1aHR2/ z8 b2WHg3qvy+GhfIZ5qSMdK4XRyX1K2lJK4Oz2oV3WEo1kSn2GOYyQ9evFv2ciV1mhmJUWN45iQX7r6OSmXPTLHnly9QiTtiRSRq7Vc2cvOus52m03690XnrCXC0vH kjqBDzIYmhtqjITBlIZTW/WaixtFIRj7LGzYhXv34bnLZb8etW+zhudD7CMjbgObzg9sbwBjpwCEfQBRVsBe3gbfAufBa+Dz+Hh0tpGFx7nsKNCI/ AtHQsyE=</latexit>

the we can show also gives a representation of the Cuntz algebra

(tσ)σ=−S,...,S

<latexit sha1_base64="6NX8mT9SZj3icV R+Aa6UlRYqpc=">A CJ3icbVHLSgMxFL1TX7X1MerShYNVUNAyUxe6EYpuXFZqH9DWk nTNjQzGZJMoQz9G134A/6AOzeCiujSj9C16WNhrRcCJ+fcw809cQNGpbLtDyM2Mzs3vxBfTCSXl dWzbX1ouShwKSAOeOi7CJ GPVJQVHFSDkQBHkuIyW3cz7QS10iJOX+leoFpOahlk+bFCOlKW5uwR4ouIZbkEChBR4g2Ic6RBPMKRxCHg40x6ABXDukvuWhXzdTdtoeljUNnDFIZXe+7h+6ye9c3XyuNjgOPeIrzJCUFc OVC1CQlHMSD9RDSUJEO6gFqlo6COPyFo03LNv7WqmYTW50MdX1pD97YiQJ2XPc3Wnh1Rb/tUG5H9aJVTNk1pE/SBUxMejQc2QWYpbg9CsBhUEK9bTAGFB9Vst3EYCYaWjTegQnL8rT4NiJu0cpTOXTip7BqOKwyZs6w9w4BiycAE5KACG 3iEF3g17own4814H7XGjLFnAybK+PwB9uOfaw= </latexit>

ˆ uα = exp[−iπ ˆ Sα]

<latexit sha1_base64="+B71zaX5YxsShUfgVUMy9FzIosw=">A COXicbVDLSgMxFL1TX7W+Rl26GVqEglhmdKEboejGZUX7gOlYMmnahmYeJBlxGPoX/oK/4Ma/cFdw40IRt/6A6bSCb 2QcM6595Dc4 aMCm aQy2zsLi0vJ dza2tb2xu6ds7NRFEHJMqDljAGy4ShFGfVCWVjDRCTpDnMlJ3+xejfv2OcE D/0bGIXE81PVph2IklRToRXiEHiCQkEAEA2gpjoB mKpnihG4V8yGQ6CKhen967hWjtsph9PSC2bJTMuYB9YEFMr5 sHDsBxXWvpLsx3gyCO+xAwJYVtmKJ0EcUkxI4NcMxIkRLiPusRW0EceEU6Sbj4w9pXSNjoBV8eXRqr+dSTIEyL2XDXpIdkTs72R+F/PjmTn1EmoH0aS+Hj8UCdihgyMUYxGm3KCJYsVQJhT9VcD9xBHWKqwcyoEa3bleVA7KlnHJfNKpXEO48rCHuShCBacQBkuoQJVwPAEr/AOH9qz9qZ9al/j0Yw28ezCVGnfP2knpEo=</latexit>
slide-37
SLIDE 37

transformation of the Cuntz algebra

fix , and let

cσ ∈ B( ˜ HR) ∼ = MR = πR(AR)00

<latexit sha1_base64="ayZFxb37a38q/F+HfjhiyXDjOgo=">A Cw3icbVHbat AEB2pt9S9Oe1jX5aEkpQWI6UPzUvATSn0pZCWOgnYjlmtR/Ki1a7YHRWM8F/0F/pR6d 0bOchjT3Lwjlz5uxlJquNDpQk1 F87/6Dh492Hne PH32/EV39+V5cI1XOFDO H+ZyYBGWxyQJoOXtUdZ QYvsvLzUr/4hT5oZ3/SvMZxJQurc60kc p1NSi4gj8Q EMBFUjG iwIOIVDxsTMwBQ WmZLnWDGHsnZFr7CgtdkpXlWBfxg/paZAsenFLc8OVdIKLn2 xbPCbMGal56QzvcesqnrTcfwMGku5/0klWITZDegP3+3ujd7+v+/GzS/TuaOtVUaEkZGcIwTWoat9KTVgYXnVETsJaqlAUOGVpZYRi3q94vxBvOTEXuPG9LYpW97WhlFcK8yt43deERS3ZUkmbhbs0yuU0bNpQfj1t 64bQqvWFeWMEObEcqJhqj4rMnIFUXvObhZpJLxXx2DvcjPTu1zfB+VEv/dA7+s5dOYV17MBr2OPWp/AR+jzoMxiAio6jq6iIZvGXuIx9TOvSOLrxvIL/Il78AxP vSQ=</latexit>

α = x, y, z

<latexit sha1_base64="NvRZe4um2dVyv DpvaoGHSWPvAw=">A C XicbVC7SgNBFL0bXzG+Vi0VGQyChYTdWGgjBG0sEzAPSJYwO5lNhsw+mJkV1yWlNn6HnY2FEmz9Azu/wZ9w8ig0euAyZ865l5l73IgzqSzr08jMzS8sLmWXcyura+sb5uZWTYaxILRKQh6Khosl5SygVcU p41IUOy7nNbd/sXIr19TIVkYXKk o 6PuwHzGMFKS6FpwiNg4B T59n+ibABwQ3cASJrtu2mbcK1hjoL7GnJF/aHVa+7vaG5b 50eqEJPZpoAjHUjZtK1JOioVihN BrhVLGmHSx13a1DTAPpVO t5kgA60 kFeKHQFCo3VnxMp9qVMfFd3+lj15Kw3Ev/zmrHyTp2UBVGsaEAmD3kxRypEo1hQhwlKFE80wUQw/VdEelhgonR4OR2CPbvyX1IrFuzjQrFi50vnME WdmAfDsG EyjBJZShCgTu4Qle4NV4MJ6NofE2ac0Y05lt+AXj/RsK2Zfm</latexit>

tσ = P

σ0hσ|ˆ

u†

α|σ0i ˜

Rα(cσ0)

<latexit sha1_base64="GORLM3zNW7woiforXlI5dXrsqks=">A CwXicbVFNb9NAEB2bj5bwFeDIZUVUAZfIbg/0AorohWNBpK2UptF6PXFW3V1bu2OkyM1P4Zg/xAX4IdyZOD0 acey9s28N8/jnawyOlCS/I3ie/cfPNzZfdR5/OTps+fdFy9PQl 7hUNVmtKfZTKg0Q6HpMngWeVR2szgaXZ5tOJPf6APunTfaV7h2MrC6alWkrhUdg0QXMASAmgowIKEj21WM5 As8G8hQXnhpHjigHcYK84m/FJ3FWzcuWac17wg+DZbcmZgapVXW05L1lx05eYM9yP7Qyq7RTwjX03fd4xd3H nO8n3V7ST9oQt0F6DXqDvX9/fv/sFMeT7q/zvFS1RUfKyB GaVLRuJGetDK46JzXASupLmWBI4ZOWgzjpt3AQux JRfT0vPrSLTVmx2NtCHMbcZK 2kWtrlV8S5uVNP0cNxoV9WETq0/NK2NoFKs1ily7VGRmTOQymueVaiZ9FIRL73Dl5Bu/ JtcL fTw/6+1/T3uAzrGMX sMbvtwUPsA vsAxDEF n6I8spGLj2IdV7FfS+Pou cVbETc/Af/175B</latexit>

also gives a representation of the Cuntz algebra

(tσ)σ=−S,...,S

<latexit sha1_base64="6NX8mT9SZj3icV R+Aa6UlRYqpc=">A CJ3icbVHLSgMxFL1TX7X1MerShYNVUNAyUxe6EYpuXFZqH9DWk nTNjQzGZJMoQz9G134A/6AOzeCiujSj9C16WNhrRcCJ+fcw809cQNGpbLtDyM2Mzs3vxBfTCSXl dWzbX1ouShwKSAOeOi7CJ GPVJQVHFSDkQBHkuIyW3cz7QS10iJOX+leoFpOahlk+bFCOlKW5uwR4ouIZbkEChBR4g2Ic6RBPMKRxCHg40x6ABXDukvuWhXzdTdtoeljUNnDFIZXe+7h+6ye9c3XyuNjgOPeIrzJCUFc OVC1CQlHMSD9RDSUJEO6gFqlo6COPyFo03LNv7WqmYTW50MdX1pD97YiQJ2XPc3Wnh1Rb/tUG5H9aJVTNk1pE/SBUxMejQc2QWYpbg9CsBhUEK9bTAGFB9Vst3EYCYaWjTegQnL8rT4NiJu0cpTOXTip7BqOKwyZs6w9w4BiycAE5KACG 3iEF3g17own4814H7XGjLFnAybK+PwB9uOfaw= </latexit>

from the uniqueness of representation of the Cuntz algebra

ζα ∈ C

<latexit sha1_base64="gEe6VA1wYVwFEuvjshGp8eosypA=">A CG3icbZC7SgNBFIbPeo3xFrUSR aDYBV2tdAymMYyAXOBJITZySQZMju7zJwV4pLSd7CxzDuksrFQxEqw8Bl8CSeXQhP MDMf/38OM+d4oeAaHefLWlhcWl5ZTawl1zc2t7ZTO7slHUSKsiINRKAqHtFMcMmKyFGwSqgY8T3Byl43N/L t0xpHsgb7IWs7pO25C1OCRopSO3DAO6A QKBhmECAkLomHsAHKQ5fcNoFM+sGHLQb6TSTsYZhz0P7hTS2cNh4fv+aJhvpD5qzYBGPpNIBdG6 joh1mOikFPB+slapFlIaJe0WdWgJD7T9XjcW98+MUrTbgXKbIn2WP1dERNf657vmUyfYEfPeiPxP68aYeuyHnMZRsgknTzUioSNgT0alN3kilEUPQOEKm7+atMOUYSiGWfSDMGdbXkeSmcZ9z jFNx09gomkYADOIZTcOECsnANeSgChQd4ghd4tR6tZ+vNep+kLljTmj34E9bnD6oLnW0=</latexit>

|ζα| = 1

<latexit sha1_base64="3Mr8AevR6Vzsdhje4S5HlgnfbFw=">A C XicbZC7TsMwFIZPyq2UW4CRJWqFhIRUJTDAghTBwlgkepHaqHJcp7XqOJHtI W0Kwsr 8DGwgBCrLwBW98G9zJAy5Esf/r/c2Sf348Zlcq2R0ZuaXl dS2/XtjY3NreMXf3ajJKBCZVHLFINHwkCaOcVBV jDRiQVDoM1L3+1djv35HhKQRv1VpTLwQdTkNKEZKS5FpwgCe4R4IKEDQ1oyAQ w9fQ/gApy2WbL 9qSsRXBmUHKLreOnkZtW2uZ3qxPhJCRcY akbDp2rLwMCU xI8NCK5EkRriPuqSpkaOQSC+b DK0DrXSsYJI6MOVNVF/T2QolDINfd0ZItWT895Y/M9rJio49zLK40QRjqcPBQmzVGSNY7E6VBCsWKoBYUH1Xy3cQwJhpcMr6BCc+ZUXoXZSdk7L9o1Tci9hWnk4gCIcgQNn4MI1VKAKGB7gBd7g3Xg0Xo0P43PamjNmM/vwp4yvH/pLl8I=</latexit>

tσ = ζα cσ

<latexit sha1_base64="xSKjVJkzNZ5rt7KyFX621zfaChg=">A CJ3icbVC7SgNBFL3rM8bXq Wi 0GwkLAbC2 EoI1lAuYBSQx3J5NkyOyDmVkh pT+iSn8g3yDjaAiWlr7E04ehUk8M yZc8/lzj1uyJlUtv1lzM0vLC4tx1biq2vrG5vm1nZeBpEgNEcCHoi 5Jy5tOcYorTYigoei6nBbd1NagX7qiQLPBvVDukFQ8bPqszgkpLgXkACm6hBxIYNMADhAv9ugeqdYSq5g cQmjquwcnQCbdVTNhJ+0hrFnijEkivdfP/jzs9zNV87VcC0jkUV8RjlKWHDtUlQ4KxQin3Xg5kjRE0sIGLWnqo0dlpTPcs2sda Vm1QOhj6+sofq3o4OelG3P1U4PV NO1wbif7VSpOrnlQ7zw0hRn4wG1SNuqcAahGbVmKBE8bYmSAT f7VIEwUSpaON6xCc6ZVnST6VdE6TqayTSF/C DHYhUM4BgfOIA3XkIGcDvgRnuEN3o0n48X4MD5H1jlj3LMDEzC+fwFSM6Ak</latexit>

with projective 
 representation genuine 
 representation S is a half-odd integer we finally get the desired transformation rule contradiction!

α = x, y, z

<latexit sha1_base64="NvRZe4um2dVyv DpvaoGHSWPvAw=">A C XicbVC7SgNBFL0bXzG+Vi0VGQyChYTdWGgjBG0sEzAPSJYwO5lNhsw+mJkV1yWlNn6HnY2FEmz9Azu/wZ9w8ig0euAyZ865l5l73IgzqSzr08jMzS8sLmWXcyura+sb5uZWTYaxILRKQh6Khosl5SygVcU p41IUOy7nNbd/sXIr19TIVkYXKk o 6PuwHzGMFKS6FpwiNg4B T59n+ibABwQ3cASJrtu2mbcK1hjoL7GnJF/aHVa+7vaG5b 50eqEJPZpoAjHUjZtK1JOioVihN BrhVLGmHSx13a1DTAPpVO t5kgA60 kFeKHQFCo3VnxMp9qVMfFd3+lj15Kw3Ev/zmrHyTp2UBVGsaEAmD3kxRypEo1hQhwlKFE80wUQw/VdEelhgonR4OR2CPbvyX1IrFuzjQrFi50vnME WdmAfDsG EyjBJZShCgTu4Qle4NV4MJ6NofE2ac0Y05lt+AXj/RsK2Zfm</latexit>

for

cσ = ζα P

σ0hσ|ˆ

u†

α|σ0i ˜

Rα(cσ0)

<latexit sha1_base64="h58D5o07zRHq43QxyVX L7Do/Ls=">A C2XicbVI7j9NAEB6b1xFeAUqaFdEJaCL7roAGKYKG8kAkd1IuROvNxFndem3tjhHBSUEBQrTUlPeHaIAfQs/YuSIPZrXyNzPfNzO76 Qw2lMU/QnCS5evXL2 d71 4+at23fad+8NfF46hX2Vm9ydJNKj0Rb7pMngSeFQZonB4+TsZ 0/fo/O69y+pXmBo0ymVk+1ksShvP0BFLyDc/CgIYUMJDxn7yMgEOMxYwkGCpjxt2aVzBlDtaF4BEv2DSPLEcPa9eyCvVpNrCqZWXebsJ/yQnBbPRZblc+ZsV6XOGdYj80MqlEKeMN1N+s8bs61O+eTcbsTdaPGxC6IL0Cnt/ 3968frfRo3P5 OslVmaElZaT3wzgqaFRJR1oZXLZOS4+FVGcyxSFDKzP0o6p5maXY58hETHPH25Jo u KSmbez7OEmZmkmd/O1cH/5Y lTZ+NKm2LktCqVaNpaQTlon5mMdEOFZk5A6mc5lmFmk nFfHP0OJLiLePvAsGB934sHvwOu70XsDK9uABPOTLjeEp9OAVHE fVDAIFsHn4Es4D +FX8NvK2oYXGjuw4aF3/8BQW7Edw= </latexit>
slide-38
SLIDE 38

S is a half-odd integer

−ˆ 1

<latexit sha1_base64="6jVdSep1RFZxU9YoFaM/eXM9gTQ=">A B93icbZDLSsNAFIZPvNZ6q7p0Ey CG0tSBV0W3bisYC/QhjKZTpqhk0mYORFC6Iu4ceEFt76KO9/GaZuFtv4w8POfczhnPj8RXKPjfFsrq2vrG5ulrfL2zu7efuXgsK3jVFHWorGIVdcnmgkuWQs5CtZNFCORL1jH 9 O651HpjSP5QNmCfMiMpI84JSgiUI4hzcIgQBCDi5MBpWqU3NmspeNW5gqFGoOKl/9YUzTiEmkgmjdc50EvZwo5FSwSbmfapYQOiYj1jNWkohpL5/dPbFPT K0g1iZJ9Gepb8nchJpnUW+6YwIhnqxNg3/q/VSDK69nMskRSbpfFGQChtjewrBHnLFKIrMGEIVN7faNCSKUDSoygaCu/jlZdOu19yLWv3+stq4KXCU4BhO4MyAvI G3E TWkAN2id4gVcrs56td+tj3rpiFTNH8EfW5w9gDpBe</latexit>

=

<latexit sha1_base64="MWbL6R2hZsQThSAdMNvF0orSXwY=">A B6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQS9C0IvHBMwDkiXMTnqTMbOzy8ysE K+wIsHRbz6Sd78GyfJHjSxoKGo6qa7K0gE18Z1v53c2vrG5lZ+u7Czu7d/UDw8auo4VQwbLBaxagdUo+ASG4Ybge1EIY0Cga1gdDfzW0+oNI/lgxkn6Ed0IHnIGTVWqt/0i W37M5BVomXkRJkqPWKX91+zNI pWGCat3x3MT4E6oMZwKnhW6qMaFsRAfYsVTSCLU/mR86JWdW6ZMwVrakIXP198SERlqPo8B2RtQM9bI3E/ zOqkJr/0Jl0lqUL FojAVxMRk9jXpc4XMiLEl ClubyVsSBVlxmZTsCF4y +vkmal7F2UK/XLUvU2iyMPJ3AK5+DBFVThHmrQA YIz/AKb86j8+K8Ox+L1pyTzRzDHzifP4 bjMU=</latexit>

Proof of Theorem 1

(ζα)2 = −1

<latexit sha1_base64="R9RCx/0S DoE9MTp3GkZQjfAG5M=">A CD3icbZDLSgMxFIbP1Fut 9Eu3YQWQRHLTF3oRi 6cVnBXqAdSybNtKGZC0lG Ie+gRvBV/AF3LhQxK1bd30b08tCqwdCPv7/HJLzuxFnUlnWyMgsLC4tr2RXc2vrG5tb5vZOXYaxILRGQh6Kposl5SygNcU p81IUOy7nDbcwcXYb9xSIVkYXKsko 6PewHzGMFKS6GZh314gjugoABDRzMGDhH09X0AN1CGMzgCu2MWrZI1KfQX7BkUK4X24eOoklQ75le7G5LYp4EiHEvZsq1IOSkWihFOh7l2LGmEyQD3aEtjgH0qnXSyzxDta WLvFDoEyg0UX9OpNiXMvFd3elj1Zfz3lj8z2vFyjt1UhZEsaIBmT7kxRypEI3DQV0mKFE80YCJYPqviPSxwETpCHM6BHt+5b9QL5fs45J1ZRcr5zCtLOxCQ dtw lU4BKqUAMC9/AMr/BmPBgvxrvxMW3NGLOZP wq4/MbEmuYEQ= </latexit>

ζxζyζz = 1

<latexit sha1_base64="QMP6W75ng/LlKSQzTJQ/f5LDa8E=">A CO3icdZC7TgJBFIbP4g3BC2p MxFNtCG7WmhjQrSx EQuCWzI7DALE2YvmZklwobH8RWMjZVvYGdjY6Extlg7XAoF+ZNJ/vm/czJzjhNyJpVpvhiJhcWl5ZXkaiq9tr6xmdnaLskgEoQWScADUXGwpJz5tKiY4rQSCo 9h9Oy074c8nKHCskC/0Z1Q2p7uOkzlxGsdBRkjuAOekB AY 6xPomwAMEt9CfQ7pzSU+Tc7DqmayZM0dCs8a mGx+f3D/1El/F+qZ51ojIJFHfU 4lrJqmaGyYywUI5z2U7VI0hCTNm7SqrY+9qi049HsfXSgkwZyA6GPr9Ao/d0RY0/KrufoSg+rlpxmw/A/Vo2Ue2bHzA8jRX0yfsiNOFIBGi4SNZigRPGuNpgIpv+KSAsLTJRed0ovwZoe daUjnPWSc68trL5CxgrCbuwB4dgwSnk4QoKUAQCD/AK7/BhPBpvxqfxNS5NGJOeHfgjY/AD+U6l4w= </latexit>

contradiction!

=

<latexit sha1_base64="MWbL6R2hZsQThSAdMNvF0orSXwY=">A B6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQS9C0IvHBMwDkiXMTnqTMbOzy8ysE K+wIsHRbz6Sd78GyfJHjSxoKGo6qa7K0gE18Z1v53c2vrG5lZ+u7Czu7d/UDw8auo4VQwbLBaxagdUo+ASG4Ybge1EIY0Cga1gdDfzW0+oNI/lgxkn6Ed0IHnIGTVWqt/0i W37M5BVomXkRJkqPWKX91+zNI pWGCat3x3MT4E6oMZwKnhW6qMaFsRAfYsVTSCLU/mR86JWdW6ZMwVrakIXP198SERlqPo8B2RtQM9bI3E/ zOqkJr/0Jl0lqUL FojAVxMRk9jXpc4XMiLEl ClubyVsSBVlxmZTsCF4y +vkmal7F2UK/XLUvU2iyMPJ3AK5+DBFVThHmrQA YIz/AKb86j8+K8Ox+L1pyTzRzDHzifP4 bjMU=</latexit>

=

<latexit sha1_base64="MWbL6R2hZsQThSAdMNvF0orSXwY=">A B6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQS9C0IvHBMwDkiXMTnqTMbOzy8ysE K+wIsHRbz6Sd78GyfJHjSxoKGo6qa7K0gE18Z1v53c2vrG5lZ+u7Czu7d/UDw8auo4VQwbLBaxagdUo+ASG4Ybge1EIY0Cga1gdDfzW0+oNI/lgxkn6Ed0IHnIGTVWqt/0i W37M5BVomXkRJkqPWKX91+zNI pWGCat3x3MT4E6oMZwKnhW6qMaFsRAfYsVTSCLU/mR86JWdW6ZMwVrakIXP198SERlqPo8B2RtQM9bI3E/ zOqkJr/0Jl0lqUL FojAVxMRk9jXpc4XMiLEl ClubyVsSBVlxmZTsCF4y +vkmal7F2UK/XLUvU2iyMPJ3AK5+DBFVThHmrQA YIz/AKb86j8+K8Ox+L1pyTzRzDHzifP4 bjMU=</latexit>

ˆ 1

<latexit sha1_base64="y1iKw8v6jIU3nQUiB41IuGtK0j4=">A B9XicbVBNS8NAEJ34WetX1aOXxSJ4Kk V9Fj04rGC/YA2lM120y7dbMLuRCmh/8OLB0W9+l+8+W/ctjlo64OBx3sz MwLEikMu 63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LRzdRvPXBtRKzucZxwP6IDJULBKFp AO8wBAoIGXgw6ZXKbsWdgSwTLydlyFHvlb6 /ZilEVfIJDWm47kJ+hnVKJjk 2I3NTyhbEQHvGOpohE3fja7ekJOrdInYaxtKSQz9fdERiNjxlFgOyOKQ7PoTcX/vE6K4ZWfCZWkyBWbLwpTSTAm0whIX2jOUI4toUwLeythQ6opQxtU0YbgLb68TJrVindeqd5dlGvXeRwFOIYTOLNBXkINbqEODWCg4Qle4NV5dJ6dN+dj3r i5DNH8AfO5w/Q2ZAe</latexit>

=

<latexit sha1_base64="MWbL6R2hZsQThSAdMNvF0orSXwY=">A B6HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQS9C0IvHBMwDkiXMTnqTMbOzy8ysE K+wIsHRbz6Sd78GyfJHjSxoKGo6qa7K0gE18Z1v53c2vrG5lZ+u7Czu7d/UDw8auo4VQwbLBaxagdUo+ASG4Ybge1EIY0Cga1gdDfzW0+oNI/lgxkn6Ed0IHnIGTVWqt/0i W37M5BVomXkRJkqPWKX91+zNI pWGCat3x3MT4E6oMZwKnhW6qMaFsRAfYsVTSCLU/mR86JWdW6ZMwVrakIXP198SERlqPo8B2RtQM9bI3E/ zOqkJr/0Jl0lqUL FojAVxMRk9jXpc4XMiLEl ClubyVsSBVlxmZTsCF4y +vkmal7F2UK/XLUvU2iyMPJ3AK5+DBFVThHmrQA YIz/AKb86j8+K8Ox+L1pyTzRzDHzifP4 bjMU=</latexit>

α = x, y, z

<latexit sha1_base64="NvRZe4um2dVyv DpvaoGHSWPvAw=">A C XicbVC7SgNBFL0bXzG+Vi0VGQyChYTdWGgjBG0sEzAPSJYwO5lNhsw+mJkV1yWlNn6HnY2FEmz9Azu/wZ9w8ig0euAyZ865l5l73IgzqSzr08jMzS8sLmWXcyura+sb5uZWTYaxILRKQh6Khosl5SygVcU p41IUOy7nNbd/sXIr19TIVkYXKk o 6PuwHzGMFKS6FpwiNg4B T59n+ibABwQ3cASJrtu2mbcK1hjoL7GnJF/aHVa+7vaG5b 50eqEJPZpoAjHUjZtK1JOioVihN BrhVLGmHSx13a1DTAPpVO t5kgA60 kFeKHQFCo3VnxMp9qVMfFd3+lj15Kw3Ev/zmrHyTp2UBVGsaEAmD3kxRypEo1hQhwlKFE80wUQw/VdEelhgonR4OR2CPbvyX1IrFuzjQrFi50vnME WdmAfDsG EyjBJZShCgTu4Qle4NV4MJ6NofE2ac0Y05lt+AXj/RsK2Zfm</latexit>

for

cσ = ζα P

σ0hσ|ˆ

u†

α|σ0i ˜

Rα(cσ0)

<latexit sha1_base64="h58D5o07zRHq43QxyVX L7Do/Ls=">A C2XicbVI7j9NAEB6b1xFeAUqaFdEJaCL7roAGKYKG8kAkd1IuROvNxFndem3tjhHBSUEBQrTUlPeHaIAfQs/YuSIPZrXyNzPfNzO76 Qw2lMU/QnCS5evXL2 d71 4+at23fad+8NfF46hX2Vm9ydJNKj0Rb7pMngSeFQZonB4+TsZ 0/fo/O69y+pXmBo0ymVk+1ksShvP0BFLyDc/CgIYUMJDxn7yMgEOMxYwkGCpjxt2aVzBlDtaF4BEv2DSPLEcPa9eyCvVpNrCqZWXebsJ/yQnBbPRZblc+ZsV6XOGdYj80MqlEKeMN1N+s8bs61O+eTcbsTdaPGxC6IL0Cnt/ 3968frfRo3P5 OslVmaElZaT3wzgqaFRJR1oZXLZOS4+FVGcyxSFDKzP0o6p5maXY58hETHPH25Jo u KSmbez7OEmZmkmd/O1cH/5Y lTZ+NKm2LktCqVaNpaQTlon5mMdEOFZk5A6mc5lmFmk nFfHP0OJLiLePvAsGB934sHvwOu70XsDK9uABPOTLjeEp9OAVHE fVDAIFsHn4Es4D +FX8NvK2oYXGjuw4aF3/8BQW7Edw= </latexit>

cσ0

<latexit sha1_base64="etPYbrXsOC +nLY1Hc6kRkZl+FM=">A B/XicbVDLSgNBEOyNr7i+oh69LAbRU9iNB72IQS8eI5gHJGuYncwmQ2Zml5lZIS7BXxEkB0W8+gnevYh/4+Rx0MSChqKqm+6uIGZUadf9tjILi0vLK9lVe219Y3Mrt71TV EiMangiEWyHiBFGBWkoqlmpB5LgnjASC3oXY782h2RikbiRvdj4nPUETSkG kj9QD LaQwBAU OsABwSEMWrm8W3DHcOaJNyX58w/7LH76s ut3GezHeGE 6ExQ0o1PDfWfoqkp iRgd1MFIkR7qEOaRgqECfKT8fXD5wDo7SdMJKmhHbG6u+JFHGl+jw nRzprpr1RuJ/XiPR4amfUhEnmg 8WRQmzNGRM4rCaVNJsGZ9QxCW1Nzq4C6SCGsTmG1C8GZfnifVYsE7LhSv XzpAibIwh7swxF4cAIluI yVEzM9/AIz/BiPVhD69V6m7RmrOnMLvyB9f4DxeqVMw= </latexit>

cσ = (ζα)2 P

σ0hσ|(ˆ

u†

α)2|σ0i ˜

Rα ˜ Rα(cσ0) = (ζα)2 cσ

<latexit sha1_base64="o+WLIbqDNS+iaQmvBYEL4a0BvIE=">A DWXiclVLNbtNAEB7btATzF+iRy4qo pUgs MBLkgRXDgWRNpKaYjWm42z6npt7Y6RgptH4dh34Dk48COegxsHxk4lmqQHGMvab2a+ WZ3NEmhlcMo+uH5wbWt7eutG+HNW7fv3G3fu3/o8tIKORC5zu1xwp3UysgBKtTyuLCSZ4mWR8npqzp/9EFap3LzDueFHGU8NWq BEcK5e3PIOA9nIMDBSlkwOEF7JH/ESQgeWPCHDQUMKNzn7i9hl0SdwzVSuUjWEBIEU3YUEyTxuX8WaNc6yBVlsSuO0/IT+mTYK/sdrbW45x4f/XrfkhZT qyuY9oFBi8Jf1VvTqnqFr8R81eM5/Nd+7TnJ78w6Qer893 O5E3agxtgniC9Dp7/76/u1TmB6M219OJrkoM2lQaO7cMI4KHFXcohJaLsKT0smCi1OeyiFBwzPpRlWzGQu2S5EJm+aWfoOsiV6uqHjm3DxLiJlxnLn1XB28Kjcscfp8VClTlCiNWDa lp hzuo1YxNlpUA9J8CFVXRXJmbc oG0jCENIV5/8iY47HXjp93em7jTfwlLa8EDeEhj +EZ9OE1HMA hPfV+ 1v+dv+z8ALWkG4pPreRc0OrFiw8wfx8t P</latexit>

cσ = ζx P

σ0hσ|ˆ

u†

x|σ0i ˜

Rx(cσ0)

<latexit sha1_base64="YIeGWziG3oLvqhTMXce JNzW9Ok=">A C43icbVJNb9NAEB27QIv5CnDksiKqgEtktwd6QYrg0mNBpK2UhrDeTJxV1x/aHSOCm2svPYAQ1x75Q1yAH8KdsVOpxWEsy2/emzezH4 Lox2F4W/PX7t2/cb6xs3g1u07d+917j/Yd3lpFQ5UbnJ7GEuHRmc4IE0GDwuLMo0NHsTHr2r94ANap/PsLc0LHKUy fRUK0lM5Z0FKHgH38GBhgRSkPC s0+AQIzHUHFm RfwERZNXcnZkr/0PGEtYMYwzpgz7L+qn3A24y+xr+TaeuKE84Qf5P7tOSet7rV2 bmeRKwa7oGNU7Fq2PuGve1eT5sdrq732bjTDXthE2IVRBeg29/8 +vneZDsjTs/jia5KlPMSBnp3DAKCxpV0pJWBhfBUemwkOpYJjhkmMkU3ahq7mghNpmZiGlu+c1INOxVRyVT5+ZpzJWp JlrazX5P21Y0nRnVOmsKAkztRw0LY2gXNQXLibaoiIzZyCV1bxWoWbS kX8WwR8CF 7y6tgf6sXbfe2Xkfd/ktYxgY8gsd8uBE8hz7swh4MQHnv VPvs/fFR/ M/+p/W5b63oXnIfwT/vlfXNPGJQ= </latexit>

= ζxζy P

σ0hσ|(ˆ

uyˆ ux)†|σ0i ˜ Rx ˜ Ry(cσ0)

<latexit sha1_base64="6zn49Qeg7xBD/7VJ1eU4fTEVDBw=">A DSXicjVL btNAFL12eBTzaChLNiOi lZCkV0WsEGK6IZlQaStlKbReHLj DoeWzNjRHDzKV12z1ewYc G+Ac2LIoQK6 dLkKS tyRpXP Of h0cS5ktaF4XfPb1y7fuPm2q3g9p279 ab9zf2bVY gV2Rqcwcxtyikhq7TjqFh7lBnsYKD+KT3Uo/eIfGyky/dZMc+ylPtBxJwR1RWfMjvIBz+A IDjgMoKTMQAoM3sP0CmVSKxYKyma8BQkJZRwekxYQowhr4hTVz+unsEX5mJCjyoLcy52vUquNtuGY8iE5EjpIyunC/Mo9P/tJvY8jh6I6rPsJcijq+GbFhEqVlIv/rq 23iL1eMVdbA+arbAd1sGWQXQJWp3Ni29fz4Jkb9D8cjTMRJGidkJxa3tRmLt+yY2TQuE0OCos5lyc8AR7BDVP0fbL+iVM2SYxQzbKDH3asZqdryh5au0kjcmZcje2i1pFrtJ6hRs975dS54VDLWaDRoViLmPVs2JDaVA4NSHAhZG0KxNjbrhw9PgCuoRo8ZeXwf5O 3ra3nkdtTovYRZr8BAe0eVG8Aw68Ar2oAvC+ T98C68X/5n/6f/2/8zs/reZc0D+Ccajb9aEd7a</latexit>

= ζxζyζz P

σ0hσ|(ˆ

uzˆ uyˆ ux)†|σ0i ˜ Rx ˜ Ry ˜ Rz(cσ0)

<latexit sha1_base64="ew9grCy1yitpWug6yR2AtMWibJw=">A Dt3icnVLNbtQwEJ4mQEv46QJHLhGrilZCq6QgAQekFVw4FsS2FdvtynEmWauOk9oOIk3 UTjyQlyAJ+AJuDPJVqLsD0I4ivTN92OPR4 K YwNgu9rjnvl6rX1jevejZu3bm927tzdN3mpOQ54LnN9GDGDUigcWGElHhYaWRZJPIhOXjX6wQfURuTqna0KHGUsVSIRnFmi8s4PeAGf4QwQLDAYQ02Vhgx8+AjTFUq1UjlrFQMlVTPegICUKgYPSfOIkYQVcZLyl/Vz2KZ6QshSsiT34s6r1OqvanOTHTimOiZHSh+Scj7X eP+3VnT6SP6LTk 5bDdj5ND0o5vl5zQqIJq/s+p6r9SzS 2ST1eMt+dcacb9IJ2+YsgvADd/tbPb18/e neuP lKM5 maGyXDJjhmFQ2FHNtBVc4tQ7Kg0WjJ+wFIcEFcvQjOr23U39LWJiP8k1/cr6LXs5UbPMmCqLyJkxOzHzWkMu04alTZ6NaqGK0qLis4OSUvo295tH7MdCI7eyIsC4FtSrzydM 27pqXs0hHD+yotgf7cXPu7tvgm7/ZcwWxtwHx7QcEN4Cn14DXswAO48cd473Ind5+7YTdzJzOqsXWTuwR/LPf0FezfzOg= </latexit>

= ζxζyζz cσ

<latexit sha1_base64="8b19uprAwhbRIdlUMBETR/vhgNM=">A CUXicdZHNTgIxFIUv4x+CP6Mu3TQSExeGzC JbkyIblxiImACSDqlQEOnM2k7RpjM4/g6LnTle7hxobEMLBTkJk1Oz3dv2p56IWdKO857xlpZXVvfyG7m8lvbO7v23n5dBZEktEYCHsh7DyvKmaA1zTSn96Gk2Pc4bXjD6wlvPFKpWCDu9CikbR/3BesxgrWxArsMl/AMY6CgAUMHYrOT4AOCJ0iWkNFSMk7JKRB4SF0FDPqGYEg6dsEpOm hReHORAFmVe3Yr61uQCKfCk04VqrpOqFux1hqRjhNcq1I0RCTIe7Tp EC+1S14zSRB 0bp4t6gTRLaJS6vydi7Cs18j3T6WM9UPNsYv7HmpHuXbRjJsJIU0GmB/UijnSAJvGiLpOUaD4yAhPJzF0RGWCJiTafkDMhuPNPXhT1UtE9K5Zuy4XK1SyOLBzCEZyAC+dQgRuoQs2E/AIf8AXfmbfMpwW NW21MrOZA/hTVv4Hcrukg = </latexit>

we finally get the desired transformation rule we then find and

cσ0

<latexit sha1_base64="etPYbrXsOC +nLY1Hc6kRkZl+FM=">A B/XicbVDLSgNBEOyNr7i+oh69LAbRU9iNB72IQS8eI5gHJGuYncwmQ2Zml5lZIS7BXxEkB0W8+gnevYh/4+Rx0MSChqKqm+6uIGZUadf9tjILi0vLK9lVe219Y3Mrt71TV EiMangiEWyHiBFGBWkoqlmpB5LgnjASC3oXY782h2RikbiRvdj4nPUETSkG kj9QD LaQwBAU OsABwSEMWrm8W3DHcOaJNyX58w/7LH76s ut3GezHeGE 6ExQ0o1PDfWfoqkp iRgd1MFIkR7qEOaRgqECfKT8fXD5wDo7SdMJKmhHbG6u+JFHGl+jw nRzprpr1RuJ/XiPR4amfUhEnmg 8WRQmzNGRM4rCaVNJsGZ9QxCW1Nzq4C6SCGsTmG1C8GZfnifVYsE7LhSv XzpAibIwh7swxF4cAIluI yVEzM9/AIz/BiPVhD69V6m7RmrOnMLvyB9f4DxeqVMw= </latexit>

ˆ uα = exp[−iπ ˆ Sα]

<latexit sha1_base64="+B71zaX5YxsShUfgVUMy9FzIosw=">A COXicbVDLSgMxFL1TX7W+Rl26GVqEglhmdKEboejGZUX7gOlYMmnahmYeJBlxGPoX/oK/4Ma/cFdw40IRt/6A6bSCb 2QcM6595Dc4 aMCm aQy2zsLi0vJ dza2tb2xu6ds7NRFEHJMqDljAGy4ShFGfVCWVjDRCTpDnMlJ3+xejfv2OcE D/0bGIXE81PVph2IklRToRXiEHiCQkEAEA2gpjoB mKpnihG4V8yGQ6CKhen967hWjtsph9PSC2bJTMuYB9YEFMr5 sHDsBxXWvpLsx3gyCO+xAwJYVtmKJ0EcUkxI4NcMxIkRLiPusRW0EceEU6Sbj4w9pXSNjoBV8eXRqr+dSTIEyL2XDXpIdkTs72R+F/PjmTn1EmoH0aS+Hj8UCdihgyMUYxGm3KCJYsVQJhT9VcD9xBHWKqwcyoEa3bleVA7KlnHJfNKpXEO48rCHuShCBacQBkuoQJVwPAEr/AOH9qz9qZ9al/j0Yw28ezCVGnfP2knpEo=</latexit>
slide-39
SLIDE 39

Extensions

slide-40
SLIDE 40

symmetry

THEOREM 1: Consider a quantum spin chain with
 and a short-ranged Hamiltonian that is invariant under translation and

  • transformation. Then it can never be the case that

the corresponding ground state is unique and accompanied by a nonzero gap.

Z2 × Z2

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S = 1

2, 3 2, 5 2, . . .

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Z2 × Z2

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symmetry any on-site symmetry whose representation on a single spin is projective example: time-reversal symmetry for S = 1

2, 3 2, 5 2, . . .

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ˆ Sα

j → − ˆ

j

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slide-41
SLIDE 41

state

THEOREM 1: Consider a quantum spin chain with
 and a short-ranged Hamiltonian that is invariant under translation and

  • transformation. Then it can never be the case that

the corresponding ground state is unique and accompanied by a nonzero gap.

Z2 × Z2

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S = 1

2, 3 2, 5 2, . . .

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it is only essential that the state is pure, translation invariant and satisfies the split property any translation invariant pure state with area law entanglement is excluded

Matsui 2013

slide-42
SLIDE 42

general theorem

THEOREM 2: In quantum spin chains, there can be no translation invariant pure states with area law entanglement and on-site symmetry whose representation on a single spin is projective

COROLLARY: In a translation invariant spin chain with 
 and time-reversal or symmetry, any “ scar” state must be degenerate and break symmetry

Yuji Tachikawa, private communication

Z2 × Z2

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S = 1

2, 3 2, 5 2, . . .

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slide-43
SLIDE 43

illustration by Chisato Naruse

Summary

LSM-type no-go theorem is proved for quantum spin chains with translation and on-site symmetry whose representation is projective it is surprising (at least, to me) that such a mathematically abstract object as the von Neumann algebra is useful in proving physically natural theorems the proof is based on the inconsistency between the projective symmetry and the transformation property of the Cuntz algebra

(cf. Ogata’s fully rigorous index theorem for SPT) background and related topics can be found in my book in preparation (see the workshop Slack or ask me)