Disordered systems t=10000 spin glass Tensor network approaches to - - PowerPoint PPT Presentation

disordered systems
SMART_READER_LITE
LIVE PREVIEW

Disordered systems t=10000 spin glass Tensor network approaches to - - PowerPoint PPT Presentation

Disordered systems t=10000 spin glass Tensor network approaches to high-T superconductor Disordered quantum systems Anderson localization t=10000 many-body localization Yu-Cheng Lin / National Chengchi University


slide-1
SLIDE 1

December, 2018 @ RIKEN-CCS, Kobe

Tensor network approaches to 
 Disordered quantum systems

Yu-Cheng Lin / National Chengchi University

Disordered systems

✤ spin glass ✤ high-T superconductor ✤ Anderson localization ✤ many-body localization ✤ disorder-induced spin liquid

t=10000 t=10000

slide-2
SLIDE 2

Numerical challenges

✤ long relaxation time ✤ tendency to get stuck in local minima ✤ disorder average ✤ broken translational invariance (iDMRG) ✤ difficult even in 1D

Heisenberg chain

spin-spin correlations at T=0 
 entanglement entropy

H = J X

i

~ Si · ~ Si+1

(clean chain)

C(r) = (1)rh~ Si · ~ Si+ri ⇠ ln1/2(r) r

S∞(`) = 1 3 log2 ` + c1

`

slide-3
SLIDE 3

Disordered Heisenberg chain

✤ mean spin-spin correlation at T=0
 ✤ mean entanglement entropy


H = X

i

Ji ~ Si · ~ Si+1 C(r) ∼ 1 r2

`

S∞(`) = ln 2 3 log2 ` + c1

Refael & Moore (2004)

(random chain)

  • D. Fisher (1995)

✤ Project the strongest coupled pair of spins to singlet

Strong-disorder RG
 (SDRG)

strongest coupling effective coupling Random-singlet state: a non-crossing valence-bond state

Ω ˜ J = JJ0 2Ω < Ω

Ma, Dasgupta, Hu(1979)
 Dasgupta, Ma (1980)


  • D. Fisher, (1994)

1 p 2 (|"#i |#"i)

slide-4
SLIDE 4

Random-singlet state (1D)

✤ fixed-point distribution of ✤ bond-length distribution

˜ J

Γ = − ln(Ω), Ω = max{ ˜ J} Γ → ∞ ˜ J P( ˜ J) infinite-randomness

P( ˜ J) = 1 Γ ˜ J−1+1/Γ

<latexit sha1_base64="lZCAMBFJ7r2aP2qYFi7gqCSd/R4=">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</latexit><latexit sha1_base64="lZCAMBFJ7r2aP2qYFi7gqCSd/R4=">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</latexit><latexit sha1_base64="lZCAMBFJ7r2aP2qYFi7gqCSd/R4=">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</latexit><latexit sha1_base64="lZCAMBFJ7r2aP2qYFi7gqCSd/R4=">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</latexit>

P(`) ∼ `−2

<latexit sha1_base64="AzjbeYzInXC8c/t6VevLzJeUPSE=">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</latexit><latexit sha1_base64="AzjbeYzInXC8c/t6VevLzJeUPSE=">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</latexit><latexit sha1_base64="AzjbeYzInXC8c/t6VevLzJeUPSE=">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</latexit><latexit sha1_base64="AzjbeYzInXC8c/t6VevLzJeUPSE=">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</latexit>

S ∼ log `

<latexit sha1_base64="OeGpyovap4oidef0XvUsp0CkAcI=">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</latexit><latexit sha1_base64="OeGpyovap4oidef0XvUsp0CkAcI=">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</latexit><latexit sha1_base64="OeGpyovap4oidef0XvUsp0CkAcI=">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</latexit><latexit sha1_base64="OeGpyovap4oidef0XvUsp0CkAcI=">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</latexit>
  • D. Fisher (1994)

Random-singlet state (1D)

✤ Length-energy relation 


Unconventional dynamical scaling

✤ Mean spin-spin correlation function: ✤ Magnetic susceptibility:

τ ∼ exp( p ξ) z = ∞

dynamical exponent

C(r) ∼ 1 r2

  • D. Fisher (1994)

conventional:

τ ∼ ξz

χ ∼ T −1|ln T |−2 L ∼ ln2 E

slide-5
SLIDE 5

More examples for 
 1D random singlet phase

✤ bond-disordered J1-J2 chain at the Majumdar-Ghosh

point

✤ Heisenberg chain with additional multi-spin

couplings

YR Shu, DX Yao, CW Ke, YCL, A Sandvik, PRB 94, 174442 (2016)

spinon

  • A. Lavarelo & G. Roux, PRL 110, 087204 (2016)

amorphous valence-bond solid

SDRG for quantum Ising chain

Jmax

<latexit sha1_base64="qVSmz1Xn+GXmWzV5LNQqNzVCsF4=">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</latexit><latexit sha1_base64="qVSmz1Xn+GXmWzV5LNQqNzVCsF4=">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</latexit><latexit sha1_base64="qVSmz1Xn+GXmWzV5LNQqNzVCsF4=">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</latexit><latexit sha1_base64="qVSmz1Xn+GXmWzV5LNQqNzVCsF4=">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</latexit>

hx

max

<latexit sha1_base64="P25JRS0u7NyF9xm4wUJKl2dFHlc=">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</latexit><latexit sha1_base64="P25JRS0u7NyF9xm4wUJKl2dFHlc=">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</latexit><latexit sha1_base64="P25JRS0u7NyF9xm4wUJKl2dFHlc=">AFK3icfZTPb9MwFMe9EdgovzY4comIkAaHKUmrbdwGCInjQOyH1HST476mVmMnsx2kvn/4Ap3/hpOIK78H9hp1q1diqWmr+93ve9Os+O85RK5fu/lpZvObfvrKzebd27/+Dho7X1xwcyKwSBfZKlmTiKsYSUcthXVKVwlAvALE7hMB69tfHDzyAkzfgnNc6hx3DC6YASrIzreHh8fhIpOFclw+f6ZM3zN1/tbIWdLdf9P3tIAysEW532h03MB67PFSvZN1ZyXqZ6RgwBVJsZTdwM9Vr8RCUZKCbkWFhByTEU6ga0yOGcheWbWt3efG03cHmTAfrtzKez2jxEzKMYsNybAayvmYdTbFuoUa7PRKyvNCASeTQoMidVXm2j1w+1QAUenYGJgIanp1yRALTJTZqZkqMdOtVsThjGSMYd4vo5ho84CE8pKYfw1C61kALADGag7HdJpPFTD6BW4I0FpgQTyGqQKcFtV7fHlDA2qNhUQMp/yaDsdC4HGDTkVNlBYwI1C6G/TKIWBunBL9CRoMlQRQLzpJqCmcICX+FRWiFVkjvJupj5SjX3dDy1BbJtTNBd41yEdDrGzifzuDOvGqTjOnBDUtWfRC9uQebQXJOFY1vLmOCoviIT9noNeX0IxTdIpaX6IRvwjDGreHuHq/JQKJ+Yl6XLD7JyZ7VxJM0KuafDFjQlgdLKxkxlglNPczLzuekbRCyvNM9o3Z2x+n9jlhE5z7NVxeT+4i42DcDMw9oeOt/umvkRW0VP0DG2gAG2jXfQe7aF9RJBAX9E39N354fx0fjt/JujyUp3zBM0s5+8/Sn/XFw=</latexit><latexit sha1_base64="P25JRS0u7NyF9xm4wUJKl2dFHlc=">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</latexit>

H = − X

i

Jiσz

i σz i+1 −

X

i

hx

i σx i

<latexit sha1_base64="ODeXKM2JHtEGhipC/JfXjNW8ro=">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</latexit><latexit sha1_base64="ODeXKM2JHtEGhipC/JfXjNW8ro=">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</latexit><latexit sha1_base64="ODeXKM2JHtEGhipC/JfXjNW8ro=">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</latexit><latexit sha1_base64="ODeXKM2JHtEGhipC/JfXjNW8ro=">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</latexit>

GS

slide-6
SLIDE 6

Fixed-point of random Ising chain

✤ fixed-point distributions of J and hx (infinite randomness)
 ✤ length-energy scaling
 ✤ entanglement entropy

P( ˜ J) = 1 Γ ˜ J−1+1/Γ

<latexit sha1_base64="lZCAMBFJ7r2aP2qYFi7gqCSd/R4=">AFU3icfZRb9MwEMe90bJRGzwyEtEhLSBGElWbeMBaYAQiKeB2JjUlMlxL6nV2Mlsh1E8fxE+Da/wzgOfhRecNOvWrsVS2+vd7/7nXM6O8pRK5Xl/FhavNZrXl5ZvtG7eWrl9Z3Xt7qHMCkHgGRpJo4iLCGlHA4UVSkc5QIwi1L4FA1elfFPX0BImvGPaphDl+GE05gSrKzreLW9vx4qmvZAvzMbz8NYKJ9o8M3mDFsxqHP+on/2H9au49XW/z2e520N52vE3P2/EDvzSCnfZW2/Gtp1wuqtf+8VpjKexlpGDAFUmxlB3fy1VXY6EoScG0wkJCjskAJ9CxJscMZFdXj2ech9bTc+JM2A9XTuW9nKExk3LIksyrPpyOlY6Z8U6hYp3u5ryvFDAyahQXKSOypyV06PCiAqHVoDE0HtXh3Sx7ZFynZ0okrETKsVcjglmW0Q7+kwIraLESUa2KfGoQxkwCUAFhrdji43yqgNFvcEWA1gJz4hGMFeCkqN73oysaUGvMJSI4Zd0OBYCD2foVNRIaQ4zAGU6fleHKcTqzNGub0JBk74KBeZJNQUThQW+wMO0QqokZ5R1NsXLQW46QclfQsygZld4PUM+bCPVZn4351BnXhRZzanBLVb2pEz8oN2a+tOUk4krW8PbK9UNR/k5BL86hiCbpmLR/xEz8A8Q1r+Crqs6PVjixL8nods5O9u5knaEHLvBjSsTwOiosaMZYJT3M686bhW0Q0qzVPas2dsuk/sfELHOeXVcX4/OPONw2DTt/b7trv3sr5EltF9ACtIx/toD30Fu2jA0TQd/QD/US/Gr8bf5uLzcYIXVyoc+6hidVc+QePLOLf</latexit><latexit sha1_base64="lZCAMBFJ7r2aP2qYFi7gqCSd/R4=">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</latexit><latexit sha1_base64="lZCAMBFJ7r2aP2qYFi7gqCSd/R4=">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</latexit><latexit sha1_base64="lZCAMBFJ7r2aP2qYFi7gqCSd/R4=">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</latexit>

P(˜ h) = 1 Γ ˜ h−1+1/Γ

<latexit sha1_base64="uVcJlhEKx/+NBIPrGxTYq6zwMU=">AFU3icfZTfb9MwEMe90bJRGzwyEtEhLSBGElWbeMBaYAQPA7EfkhNmRz3klqNncx2GMXzP8Jfwyu8DfwgtOmnVr12Kp7fXuc9zLmdHeUql8rw/C4s3Gs2bS8u3WrfvrNy9t7p2/1BmhSBwQLI0E8cRlpBSDgeKqhSOcwGYRSkcRYM3ZfzoCwhJM/5JDXPoMpxwGlOClXWdrLb310NF0x7ovtl4GcYCE+0bHb7DjGEzDn3Wz/yn/vPafbLqepsvdreD9rbjbXrejh/4pRHstLfajm895XJRvfZP1hpLYS8jBQOuSIql7PheroaC0VJCqYVFhJyTAY4gY41OWYgu7p6POM8tp6eE2fCfrhyKu/VDI2ZlEMWZJh1ZfTsdI5K9YpVLzb1ZTnhQJORoXiInVU5pS9cnpUAFHp0BqYCGr36pA+ti1StqMTVSJmWq2QwxnJbIN4T4cRsV2MIKFcE/vUIyZBKAEwFqzwxEd51MFjH6DawK0FpgTj2CsAKdF9b6fXNOAWmMuEcEpv6LDsRB4OEOnokZKc5gBKNPxuzpMIVbnjnZ9Ewqa9FUoME+qKZgoLPAlHqYVUiU5o6zKV4OctMJSv4KWpYJzOwCb2fIh32sysT/7gzqxMs6szklqN3S1iR6Xm7Ifm3NScKRrOXtsVWuH4rydwp6dQFNEnHpP0jZuIfIa5BV9VdX60wol9SUav287Z2c6VtCPk2A1uXJsARkeNHc0Ao5zmduZNx7WKblBpntGePWPTfWIXEzrOKa+Oi/vBmW8cBpu+tT+03b3X9SWyjB6iR2gd+WgH7aH3aB8dIK+ox/oJ/rV+N3421xsNkbo4kKd8wBNrObKP7GB4xs=</latexit><latexit sha1_base64="uVcJlhEKx/+NBIPrGxTYq6zwMU=">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</latexit><latexit sha1_base64="uVcJlhEKx/+NBIPrGxTYq6zwMU=">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</latexit><latexit sha1_base64="uVcJlhEKx/+NBIPrGxTYq6zwMU=">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</latexit>

Γ = − ln(Ω)

<latexit sha1_base64="VxFjIxs2b+rlCFVzJw2DvKv2e/0=">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</latexit><latexit sha1_base64="VxFjIxs2b+rlCFVzJw2DvKv2e/0=">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</latexit><latexit sha1_base64="VxFjIxs2b+rlCFVzJw2DvKv2e/0=">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</latexit><latexit sha1_base64="VxFjIxs2b+rlCFVzJw2DvKv2e/0=">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</latexit>

Ω = ˜ Jmax or ˜ hmax

<latexit sha1_base64="r/QKFovwjlS03aTJdCfwXdkHhIg=">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</latexit><latexit sha1_base64="r/QKFovwjlS03aTJdCfwXdkHhIg=">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</latexit><latexit sha1_base64="r/QKFovwjlS03aTJdCfwXdkHhIg=">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</latexit><latexit sha1_base64="r/QKFovwjlS03aTJdCfwXdkHhIg=">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</latexit>

` ∼ Γ2 ⇒ z = ∞

<latexit sha1_base64="/mFl7xSLE4bDxTs6/yJbF0ewXA=">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</latexit><latexit sha1_base64="/mFl7xSLE4bDxTs6/yJbF0ewXA=">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</latexit><latexit sha1_base64="/mFl7xSLE4bDxTs6/yJbF0ewXA=">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</latexit><latexit sha1_base64="/mFl7xSLE4bDxTs6/yJbF0ewXA=">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</latexit>

S∞(`) = ln 2 6 log2 ` + c1

<latexit sha1_base64="ZTRqQjsRSemY0Zd7HCax1g8Uwg=">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</latexit><latexit sha1_base64="ZTRqQjsRSemY0Zd7HCax1g8Uwg=">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</latexit><latexit sha1_base64="ZTRqQjsRSemY0Zd7HCax1g8Uwg=">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</latexit><latexit sha1_base64="1FSrqb1sOzTbDZiyKhBytr1ODJA=">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</latexit><latexit sha1_base64="rhIqnH7EGBmlxpMrjwDBTr2aItY=">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</latexit><latexit sha1_base64="rhIqnH7EGBmlxpMrjwDBTr2aItY=">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</latexit><latexit sha1_base64="iA6RwdBuG37826UfzF4zyafQ+7M=">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</latexit><latexit sha1_base64="ZTRqQjsRSemY0Zd7HCax1g8Uwg=">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</latexit><latexit sha1_base64="ZTRqQjsRSemY0Zd7HCax1g8Uwg=">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</latexit><latexit sha1_base64="ZTRqQjsRSemY0Zd7HCax1g8Uwg=">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</latexit><latexit sha1_base64="ZTRqQjsRSemY0Zd7HCax1g8Uwg=">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</latexit><latexit sha1_base64="ZTRqQjsRSemY0Zd7HCax1g8Uwg=">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</latexit><latexit sha1_base64="ZTRqQjsRSemY0Zd7HCax1g8Uwg=">AFXHicfZTfb9MwEMe90bLRMdhA4oWXiAhpA2lqOgS8TBogJB7Hj/2Q6qpy3Etq1XYy2GUzH8Nfw2v8MQLfwtOmnVrm2KpzdX3ue9dL2eHKWfatNt/VlZvNZq319bvtDbubt67v7X94EQnmaJwTBOeqLOQaOBMwrFhsNZqoCIkMNpOHpX+E+/gtIskV/MOIWeILFkEaPEuK3+1sHnPmYyMuMdDJzvHuBIEZpjA9MKZ4rGNgc+l1rM1fWsyTuN8p2Oe0H/S3/PZeu1zeohFUho+qdTfbqzhQUIzAdJQTrTuBu3U9HKiDKMcbAtnGlJCRySGrjMlEaB7eVmK9Z6nYEXJcp9pPHK3ZsRORFaj0XoSEHMUM/7is06Xzcz0etezmSaGZB0kijKuGcSr2iaN2AKqOFjZxCqmKvVo0PiOmVca2eyhMK2WljCBU2EIHKQ45C6/oUQM5lT969BWTsLQAGAs+rdIZvGMwOCfYcFAVYJLPGHMFWA86x8c8WNKDSWEqEcC5v6EiFBnX6JTURGkJMwJju0HPTRVE5tL/cBixeKhwYrIuJyCmcSKXOYl0gZ5E2iLud4PUpt1PwN9AiTcfWJ3hfI4+HxBSB/60MqsDrPWcUcyVtD+LXhYFua/9JUEk1JW8O7/GD7AqnPQmysoZDGfku6HqsU/QVTx1wfckNi9JvuM652U6NdiPkuQJ3FyZAsEljJzMgmGSpm3nb9Z2i3yk1L9jAnbH5PomrCZ3GWHd1BPMXxaJx0tkLnP3xhX/4trpE1tFj9ATtoAC9QofoAzpCx4iH+gn+oV+N/42G82N5uYEXV2pYh6imdV89A+f4uY6</latexit>
  • D. Fisher (1994)

Refael & Moore (2004)

Entanglement entropy

10 10

1

10

2

10

3

0.6 0.8 1 1.2 1.4 1.6 L = 32 L = 64 L = 128 L = 256 L = 512 L = 1024

0.2 0.4 0.6 0.8 1 0.6 0.8 1 1.2 1.4 1.6 1.8 2

ℓ S(ℓ)

ℓ/L

S + ln 2

6 log2(512/ℓ)

verified by numerics via free fermion representation

S` ' ln 2 6 ln L ⇡ sin ✓`⇡ L ◆ + const.

<latexit sha1_base64="ZyNerco4MJgdLX8+LVb1h85zYk4=">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</latexit><latexit sha1_base64="ZyNerco4MJgdLX8+LVb1h85zYk4=">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</latexit><latexit sha1_base64="ZyNerco4MJgdLX8+LVb1h85zYk4=">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</latexit><latexit sha1_base64="ZyNerco4MJgdLX8+LVb1h85zYk4=">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</latexit>

Igloi & YCL, JSTAT (2008)

slide-7
SLIDE 7

Griffiths singularities

✤ in the paramagnetic phase



 
 χloc ∼ T −1+1/z

<latexit sha1_base64="AiX9+A8F1irK2ZQt97OGEiQTG04=">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</latexit><latexit sha1_base64="AiX9+A8F1irK2ZQt97OGEiQTG04=">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</latexit><latexit sha1_base64="AiX9+A8F1irK2ZQt97OGEiQTG04=">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</latexit><latexit sha1_base64="AiX9+A8F1irK2ZQt97OGEiQTG04=">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</latexit>

✓ J hx ◆1/z = 1

<latexit sha1_base64="Rf5bUPVw6wHNxM1kpnQuvc9RpoE=">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</latexit><latexit sha1_base64="Rf5bUPVw6wHNxM1kpnQuvc9RpoE=">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</latexit><latexit sha1_base64="Rf5bUPVw6wHNxM1kpnQuvc9RpoE=">AFSHicfVRLb9QwEHbLZTl0RaOXCIipBaJstkiwQWpgJAQp4LoQ9psK8c7yVobO6nt0G5T/wp+DVe48Q/4F9wQNyaPbrsvLMWZeL7vm8l47CNuTat1q+FxaXG8o2bK7eat+/cvbe6tn5/XyeZYrDHkjhRhwHVEHMJe4abGA5TBVQEMRwEg7eF/+ALKM0T+dkMU+gKGkeckYNLh2vPfUTdBfs3I8hNBt+qCjLP9i8f3RmfcWjvtk8yr1n59a+8o7X3NZWqxzOtOHVhkvqsXu83ljyewnLBEjDYqp1x2ulptTZTiLwTb9TENK2YBG0EFTUgG6m5f/Z3HuNJzwkThI41Trl5n5FRoPRQBIgU1fT3pKxZn+TqZCV92cy7TzIBkVaAwix2TOEWRnB5XwEw8RIMyxTFXh/UpFsZgKceiBMI2m76EU5YIQWUv9wNmcYKIy5zhX4OydhwABQDQmu0O+IjPDQh+DlMCvBaY4w9gpAnWbnRT6Y0oNaYiwjgRF7TkVQpOpyhU6IqpTmYARjb8bpVg104uevVjeUrKqOyC8YCK3oF9+MSUpKcinUxgdeD1HbaBf4atAjTtrMDvJshX5D+mxXUpKsYs3FGcUxnex6USD0/YcEg10LY+H0bier4r3BOj1JSjgUTxC4oeaCf8EY03cGbKs5MbGuEG2XwDq4Z9nRqN7eNgptTuy94VdRq/wWXPMV+tx0XFd12qXnKe3i+JuskLrtzxLF4bXiTl8S0sd/e8tD+NzdeVNfICvkIXlENohHXpAd8p7skj3CyFfyjXwnPxo/G78bfxp/K+jiQs15QMbG8uI/+obhKA=</latexit><latexit sha1_base64="Rf5bUPVw6wHNxM1kpnQuvc9RpoE=">AFSHicfVRLb9QwEHbLZTl0RaOXCIipBaJstkiwQWpgJAQp4LoQ9psK8c7yVobO6nt0G5T/wp+DVe48Q/4F9wQNyaPbrsvLMWZeL7vm8l47CNuTat1q+FxaXG8o2bK7eat+/cvbe6tn5/XyeZYrDHkjhRhwHVEHMJe4abGA5TBVQEMRwEg7eF/+ALKM0T+dkMU+gKGkeckYNLh2vPfUTdBfs3I8hNBt+qCjLP9i8f3RmfcWjvtk8yr1n59a+8o7X3NZWqxzOtOHVhkvqsXu83ljyewnLBEjDYqp1x2ulptTZTiLwTb9TENK2YBG0EFTUgG6m5f/Z3HuNJzwkThI41Trl5n5FRoPRQBIgU1fT3pKxZn+TqZCV92cy7TzIBkVaAwix2TOEWRnB5XwEw8RIMyxTFXh/UpFsZgKceiBMI2m76EU5YIQWUv9wNmcYKIy5zhX4OydhwABQDQmu0O+IjPDQh+DlMCvBaY4w9gpAnWbnRT6Y0oNaYiwjgRF7TkVQpOpyhU6IqpTmYARjb8bpVg104uevVjeUrKqOyC8YCK3oF9+MSUpKcinUxgdeD1HbaBf4atAjTtrMDvJshX5D+mxXUpKsYs3FGcUxnex6USD0/YcEg10LY+H0bier4r3BOj1JSjgUTxC4oeaCf8EY03cGbKs5MbGuEG2XwDq4Z9nRqN7eNgptTuy94VdRq/wWXPMV+tx0XFd12qXnKe3i+JuskLrtzxLF4bXiTl8S0sd/e8tD+NzdeVNfICvkIXlENohHXpAd8p7skj3CyFfyjXwnPxo/G78bfxp/K+jiQs15QMbG8uI/+obhKA=</latexit>
  • F. Igloi, Phys. Rev. B 65, 064416 (2002)

hx > J

Extension of SDRG

slide-8
SLIDE 8

Extension of SDRG

✤ Basic idea: refine the perturbative approximation in

SDRG by including higher energy states

HBB0

Hikihara, Furusaki, & Sigrist, PRB 60 (1999)

HBB0 = HB + HB0 + HC

BB0

HC

BB0 = JC ~

SR

B · ~

SL

B0

Extension of SDRG

✤ Compute the energy spectrum of each pair of nearest-

neighbor blocks

∆χ ∆χ ∆χ HBB0

Hikihara, Furusaki, & Sigrist, PRB 60 (1999)

∆χ ∆χ

slide-9
SLIDE 9

Extension of SDRG

✤ Merge the pair of blocks with the largest energy gap


into a block using the lowest eigenstates

fusion

∆max χ

Hikihara, Furusaki, & Sigrist, PRB 60 (1999)

∆max

χ ✤ Formulate blocks in terms of Matrix Product Operators

Implementation using MPO

Goldsborough & Römer, PRB 89 (2014)

W W W W W W work with Yu-Ping Lin, Zheng-Lin Tsai, Ying-Jer Kao, Pochung Chen

slide-10
SLIDE 10

T ree tensor network SDRG 
 (tSDRG)

Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96 (2017) Goldsborough & Römer, PRB 89 (2014) ∆max

χ

isometry V †V = I RG Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96 (2017)

T ree tensor network SDRG 
 (tSDRG)

Goldsborough & Römer, PRB 89 (2014) disordered tree tensor network

slide-11
SLIDE 11

Energy gaps (at the IRFP)

  • 25
  • 20
  • 15
  • 10
  • 5

x = ln(ε1)

10

  • 5

10

  • 4

10

  • 3

10

  • 2

10

  • 1

10

P(x)

L = 16 L = 32 L = 64 L = 128

  • 5
  • 4
  • 3
  • 2
  • 1

x = ln(ε1)L-1/2

10

  • 3

10

  • 2

10

  • 1

10

(a) (b)

quantum Ising chain at criticality

z = ∞

Ji ∈ (0, 1], hx

i ∈ (0, h0]

<latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">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</latexit><latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">AFP3icfVRLb9QwEHZLoGV5tXDkYhEhFVRVyRYJC4FhIQ4FUQf0iasHO9s1trYSW2Hdkl95dwhTs/g1/ADXHlhpNt/vIYinJZOb7vplMxo6yhCnteT9XVq84V6+trV9v3bh56/adjc27hyrNJYUDmiapPI6IgoQJONBMJ3CcSA8SuAoGr4q40efQCqWig96lEHISxYn1Girau7gd92WcAE3vK2/XA7eI4HXfbxrPYMul7Y3XC9Ha9aeNHwa8NF9drvbjprQS+lOQehaUKU6vhepsOCSM1oAqYV5AoyQockho41BeGgwqL6FIMfWk8P91NpL6Fx5Z1mFIQrNeKRXKiB2o+VjqbYp1c95+FBRNZrkHQcaJ+nmCd4rIvuMckUJ2MrEGoZLZWTAdEqpt92ayRNy0WoGAU5pyTkSvCJq7A1iJgpqvxqkMbMAKAFgreZwxCZ8poGz7AgwGqBJfEIJgpwklf/9vGCBtQaSxERnIgpHUGkJKMGnQo1VlqCGYI2HT8sgT6+hwXrm8CyeKBDiQRcTUFM4kluYQHSQWpSHjMOp/Dq2FmOu0SPwUt07RNc4LXDfLBgOiS+N/KoCZe5mnGaclsSbuz0POyIHvbXUIikarl7RbVrh/I8jkHenEBilicTJD2RTbC30O/xms409X+KTSJ7U8yxZbtnJ3tTCs7QtgW+GhAjgbN3Y8A5wJltmZNx3XKrtSvOU9ewem+8Tv5jQCcfYo8OfPygWjcP2jm/td0/cvZf1IbKO7qMHaAv56CnaQ2/QPjpAFH1BX9E39N354fxyfjt/xtDVlZpzD80s5+8/CsLbSQ=</latexit><latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">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</latexit><latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">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</latexit>

h0 = 1

<latexit sha1_base64="P0ItRpG0snFPtFEhsEzqfN5HIU=">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</latexit><latexit sha1_base64="P0ItRpG0snFPtFEhsEzqfN5HIU=">AFInicfZTNbtNAEMe3xdASvlo4crGwkAqHKk6R4IJUQEgcCyJtJTuq1puJs4p37e6OKcHZ+AKd56G+KExMOwdtw0Hw4r2R7v/OY/4/XsRlnCNbfzY2rznXb2xt32zdun3n7r2d3fvHOs0Vgy5Lk1SdRlRDwiV0kWMCp5kCKqIETqLRm9J/8gmU5qn8iOMeoLGkg84o2inusOz9kv/bMdr7er4a4afm14pB5HZ7vOVthPWS5AIkuo1oHfzrBXUIWcJWBaYa4ho2xEYwisKakA3Suqao372M703UGq7CXRrWbnIwoqtB6LyJKC4lAv+8rJl+Q4+BFr+AyxEkmyYa5ImLqVt+utvnChgmY2tQprit1WVDqihDu0ALWSJhWq1QwgVLhaCyX4QRM/YGMZcFs18NyphFAEoArNXsjvgsniMI/gVWBHgtsMYfwUwBzvPq9z1d0YBaYy0Rwbmc05FUKTpu0KmoqdIaZgRoAr9XhAkMcOIWnm9CxeMhorKuOqChcSKXuFhUiFVkDuNmizxepSZoFPyc2iZpmOaE7xtkA+HFMvA/1YGdeBVnmYOFbclHSyik7IgeztYE0QjXcvbXYieH6ryuQS9uoQiHicz0r6oRvwDGoe4TNW+6dAGtufZIo9u3K2tzPUtoVcW+CTlQ4QfLqw0x4QXPLM9rwJPKvodSrNC963e2x5ncRlh85ijD06/OWDYtU47uz71n7/zDt8XR8i2+QheUT2iE+ek0PyjhyRLmGEk6/kG/nu/HB+Or+c31N0c6OeUAWhvP3H9eJ0eQ=</latexit><latexit sha1_base64="P0ItRpG0snFPtFEhsEzqfN5HIU=">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</latexit><latexit sha1_base64="P0ItRpG0snFPtFEhsEzqfN5HIU=">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</latexit>

Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96 (2017)

Energy gaps (Griffiths phase)

  • 5

5

x = ln(ε1Lz)

10

  • 3

10

  • 2

10

  • 1

P(x) L = 16 L = 32 L = 64 L = 128

  • 2

2 4 6

x = ln(ε2Lz)

10

  • 3

10

  • 2

10

  • 1
  • 10
  • 5

x = ln(ε1)

10

  • 3

10

  • 2

10

  • 1

P(x) (a) (b)

Frechet distribution

z > 1

in the PM phase

Ji ∈ (0, 1], hx

i ∈ (0, h0]

<latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">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</latexit><latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">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</latexit><latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">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</latexit><latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">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</latexit>

h0 = 3

<latexit sha1_base64="/Z6Sfo59A4lFQ6Sj6+SuweHDNog=">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</latexit><latexit sha1_base64="/Z6Sfo59A4lFQ6Sj6+SuweHDNog=">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</latexit><latexit sha1_base64="/Z6Sfo59A4lFQ6Sj6+SuweHDNog=">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</latexit><latexit sha1_base64="/Z6Sfo59A4lFQ6Sj6+SuweHDNog=">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</latexit>

Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96 (2017)

slide-12
SLIDE 12

2-point correlations

(a) (b)

Bulk spin correlations

10

1

10

2

L

10

  • 4

10

  • 3

10

  • 2

10

  • 1

C(L/2)

δ = 1 (tSDRG) δ = 2 (tSDRG)

~ 1/L2

Heisenberg AF S=1/2 chain

π(J) = 1 δ J1/δ−1

<latexit sha1_base64="qd2tIy927v3/s9OdeP9XyUuZs=">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</latexit><latexit sha1_base64="qd2tIy927v3/s9OdeP9XyUuZs=">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</latexit><latexit sha1_base64="qd2tIy927v3/s9OdeP9XyUuZs=">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</latexit><latexit sha1_base64="qd2tIy927v3/s9OdeP9XyUuZs=">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</latexit>

0 < J ≤ 1

<latexit sha1_base64="mr3PLjcudTXVvcoVMoQhdUgnpw=">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</latexit><latexit sha1_base64="mr3PLjcudTXVvcoVMoQhdUgnpw=">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</latexit><latexit sha1_base64="mr3PLjcudTXVvcoVMoQhdUgnpw=">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</latexit><latexit sha1_base64="mr3PLjcudTXVvcoVMoQhdUgnpw=">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</latexit>
slide-13
SLIDE 13

End-end spin correlations

10

1

10

2

L

10

  • 2

10

  • 1

C1,L

Heisenberg (binary J=2, J=1/2) Ising (critical point)

~ 1/L

End-end spin correlations

  • 30 -25 -20 -15 -10
  • 5

x = ln(C1,L)

10

  • 5

10

  • 4

10

  • 3

10

  • 2

10

  • 1

10

P(x)

L = 16 L = 32 L = 64 L = 128

  • 5
  • 4
  • 3
  • 2
  • 1

x = ln(C1,L)L-1/2

10

  • 3

10

  • 2

10

  • 1

10

(a) (b)

quantum Ising chain at criticality

Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96 (2017)

slide-14
SLIDE 14

High-order moments

✤ n-th moment of an observable O of the form



 
 
 which requires a multinomial expansion

(

L

X

i=1

Oi)n = X

k1+k2+···+kL=n

n! k1! k2! · · · kL!

L

Y

j=1

Okj

j

On = L X

i=1

Oi !n

High-order moments

✤ MPO for the n-th moment 



 
 


On = L X

i=1

Oi !n = W [1]

OnW [2] On · · · W [L] On

W [i]

On =

B B B B B B B @ 1 Oi 1 O2

i

2

1

  • Oi

1 . . . ... On

i

n

1

  • On−1

i

n

2

  • On−2

i

· · · 1 1 C C C C C C C A Yu-Ping Lin, Y.-J. Kao, P. Chen & YCL, PRB 96, 064427 (2017)

slide-15
SLIDE 15

Binder cumulant

0.8 0.85 0.9 0.95 1 1.05

Γ0

0.4 0.5 0.6 0.7 0.8 0.9

U

L = 8 L = 16 L = 32 L = 64 L = 128

0.05 0.1

1/L

0.85 0.9 0.95 1

Γ0

c

h0

random quantum Ising chain

Ji ∈ (0, 1], hx

i ∈ (0, h0]

<latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">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</latexit><latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">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</latexit><latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">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</latexit><latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">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</latexit>

Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96 (2017)

Binder cumulant

quantum Ising AF chain

add a weak longitudinal field h

0.2 0.4 0.6 0.8 1

Γ0

0.2 0.4 0.6 0.8 1

U

L = 16 L = 32 L = 64 L = 128

0.5 1

Γ0

0.5 1

<ms

2>

h0

hx

h=0.02

Ji ∈ (0, 1], hx

i ∈ (0, h0]

<latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">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</latexit><latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">AFP3icfVRLb9QwEHZLoGV5tXDkYhEhFVRVyRYJC4FhIQ4FUQf0iasHO9s1trYSW2Hdkl95dwhTs/g1/ADXHlhpNt/vIYinJZOb7vplMxo6yhCnteT9XVq84V6+trV9v3bh56/adjc27hyrNJYUDmiapPI6IgoQJONBMJ3CcSA8SuAoGr4q40efQCqWig96lEHISxYn1Girau7gd92WcAE3vK2/XA7eI4HXfbxrPYMul7Y3XC9Ha9aeNHwa8NF9drvbjprQS+lOQehaUKU6vhepsOCSM1oAqYV5AoyQockho41BeGgwqL6FIMfWk8P91NpL6Fx5Z1mFIQrNeKRXKiB2o+VjqbYp1c95+FBRNZrkHQcaJ+nmCd4rIvuMckUJ2MrEGoZLZWTAdEqpt92ayRNy0WoGAU5pyTkSvCJq7A1iJgpqvxqkMbMAKAFgreZwxCZ8poGz7AgwGqBJfEIJgpwklf/9vGCBtQaSxERnIgpHUGkJKMGnQo1VlqCGYI2HT8sgT6+hwXrm8CyeKBDiQRcTUFM4kluYQHSQWpSHjMOp/Dq2FmOu0SPwUt07RNc4LXDfLBgOiS+N/KoCZe5mnGaclsSbuz0POyIHvbXUIikarl7RbVrh/I8jkHenEBilicTJD2RTbC30O/xms409X+KTSJ7U8yxZbtnJ3tTCs7QtgW+GhAjgbN3Y8A5wJltmZNx3XKrtSvOU9ewem+8Tv5jQCcfYo8OfPygWjcP2jm/td0/cvZf1IbKO7qMHaAv56CnaQ2/QPjpAFH1BX9E39N354fxyfjt/xtDVlZpzD80s5+8/CsLbSQ=</latexit><latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">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</latexit><latexit sha1_base64="H7PwMwMVOu4rP1EKNR9b3pfhTs=">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</latexit>

Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96 (2017)

slide-16
SLIDE 16

Phase diagrams

0.5 1 1.5 2

h

0.2 0.4 0.6 0.8 1

Γ

DMRG QMC 1-0.19h2 1.4(2-h) CMP

AFM PM FM

hx clean case

2 1 IRFP AFM PM FM

PM

h Γ0

hx random case Y.-P. Lin, Y.-J. Kao, P. Chen & YCL, PRB 96, 064427 (2017)

H = X

i

Jiσz

i σz i+1 −

X

i

hx

i σx i − h

X

i

σz

i

<latexit sha1_base64="aY6mWTOk/IiM4QyBi0B8fRU7Ij8=">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</latexit><latexit sha1_base64="aY6mWTOk/IiM4QyBi0B8fRU7Ij8=">AFc3icfZRLbxMxEMfd0tASHm3h2MuqC1IBtcqmSHBKiCkilNBfUnZNPI6k80oa+/W9tKmW38xvgk3rnDnjvfRtHlhaZ1Zz2/+M+uMHSQRKt1o/FxYvLdUu7+8qD+8NHjJ6tr60+PVZxKBkcsjmJ5GlAFEQo40qgjOE0kUB5EcBIMPuX+k+8gFcbiUA8TaHMaCuwho9oudYO/YBn+a9r1LeQedLB32FIacdPLtyKjPD1545u9qumP7ZpZ1H2OV2v3TcBnbW3MZOoxjOtOFVhkuqcdBZX1r2uzFLOQjNIqpUy2skup1RqZFYOp+qiChbEBDaFlTUA6qnRWfb5wXdqXr9GJpH6GdYvVuREa5UkMeWJT3VeTvnxlq+V6t67doYiSTUIVibqpZGjYyfS6eLEpiOhtagTKt1WF9KinTdsfHsgTc1Ou+gAsWc05FN/MDZuwEIYqM2a8Gacw4ADkA1prtDnAUjxo4XsGUAFYCc/wBjBTgPC364dWUBlQac4kAzsUdHUGlpMZOgVKs1hBqBNy2tnfgQ9fe1krmd8iWFf+5KsOiCscS3uJ+VCBFkFNGXU/wapCYVjPn76B5mqaZneDzDHm/T3Ue+N/KoAq8zTOb0xJtSbvj6HVekJ125wTRQFXy9lhr1/Nl/jsBfbiBAgyjEWlf5Ez8G/QqXsOlLs5Ppmlo/ySTbdmds72daGVbyLEFvpzqAI7lxpY9wFgYnvetFyr6DYLzQvs2jM2uU/8pkNHMcZeHd7kRTFtHDd3PGt/fePufawukRWyQTbJFvHIW7JH9skBOSKM/C/yG/yZ+lvbaO2WXteosLVcwzMjZq2/8AXpnvQ=</latexit><latexit sha1_base64="aY6mWTOk/IiM4QyBi0B8fRU7Ij8=">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</latexit><latexit sha1_base64="aY6mWTOk/IiM4QyBi0B8fRU7Ij8=">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</latexit>

Heisenberg S=1 chain

gapless Haldane phase random singlet phase

δ

<latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">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</latexit><latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">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</latexit><latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">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</latexit><latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">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</latexit>

strength of bond randomness

Refael & Moore J. Phys. A (2009)

  • P. Lajko, E. Carlon. H. Rieger, F. Igloi, PRB (2005)

Hyman & Yang, PRL (1997) Months, Golinelli & Jolicoeur, PRL (1997)

slide-17
SLIDE 17

Heisenberg S=1 chain

gapless Haldane phase random singlet phase

δ

<latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">AFI3icfZTPb9MwFMe9EdgovzY4comIkAaHqemQ4DhASBwHYj+kpoc9zU1tZ3MfmGU1P8DV7jz13BDXDjwv+CkWbe2KZaSvPh93ve9OM+OM8ENt/1tavedvbGzebN26fefuva3t+0cmzTWDQ5aKVJ/E1IDgCg6Ro4CTAOVsYDjePS69B9/Am14qj7gOIOepIniA84ouqmjqA8C6elW0N5tV8NfNsLaCEg9Dk63vY2on7JcgkImqDHdsJ1hr6AaORNgW1FuIKNsRBPoOlNRCaZXVOVa/7Gb6fuDVLtLoV/NXo0oqDRmLGNHSopDs+grJ5t83RwHL3oFV1mOoNg0SAXPqZ+e1+n2tgKMbOoExzV6vPhlRThm6F5rLE0rZakYJzlkpJVb+IYmbdDRKuCua+GrS18wCUADir2R3zWTxHkPwLAnwWmCFP4aZApzl1f97uqQBtcZKIoYzdUVHUa3puEGnoqZK5gRoO2GvSISMCJXwShjTRPhpqpKqC+YSa3qJR6JCqiB/GjVZ4M0os91OyV9ByzQd25zgTYN8NKRYBv63MqgDL/M0c6i5K2lvHp2UBbnb3ogGpta3m1DMJIl8F6OUFPNEzEj3ohvx9zCoeYTPWO2fAmnifpItdtzKud7O0LgW8l2BT5Y6QPLpwk57QHLFM9fzths4xaBTaZ7zvtji+skLzp0FmPd0REuHhTLxlFnN3T2u2fB/qv6ENkD8kjskNC8pzsk7fkgBwSRj6Sr+Qb+e798H56v7zfU3R9rY5QOaG9/cfwS3S7w=</latexit><latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">AFI3icfZTPb9MwFMe9EdgovzY4comIkAaHqemQ4DhASBwHYj+kpoc9zU1tZ3MfmGU1P8DV7jz13BDXDjwv+CkWbe2KZaSvPh93ve9OM+OM8ENt/1tavedvbGzebN26fefuva3t+0cmzTWDQ5aKVJ/E1IDgCg6Ro4CTAOVsYDjePS69B9/Am14qj7gOIOepIniA84ouqmjqA8C6elW0N5tV8NfNsLaCEg9Dk63vY2on7JcgkImqDHdsJ1hr6AaORNgW1FuIKNsRBPoOlNRCaZXVOVa/7Gb6fuDVLtLoV/NXo0oqDRmLGNHSopDs+grJ5t83RwHL3oFV1mOoNg0SAXPqZ+e1+n2tgKMbOoExzV6vPhlRThm6F5rLE0rZakYJzlkpJVb+IYmbdDRKuCua+GrS18wCUADir2R3zWTxHkPwLAnwWmCFP4aZApzl1f97uqQBtcZKIoYzdUVHUa3puEGnoqZK5gRoO2GvSISMCJXwShjTRPhpqpKqC+YSa3qJR6JCqiB/GjVZ4M0os91OyV9ByzQd25zgTYN8NKRYBv63MqgDL/M0c6i5K2lvHp2UBbnb3ogGpta3m1DMJIl8F6OUFPNEzEj3ohvx9zCoeYTPWO2fAmnifpItdtzKud7O0LgW8l2BT5Y6QPLpwk57QHLFM9fzths4xaBTaZ7zvtji+skLzp0FmPd0REuHhTLxlFnN3T2u2fB/qv6ENkD8kjskNC8pzsk7fkgBwSRj6Sr+Qb+e798H56v7zfU3R9rY5QOaG9/cfwS3S7w=</latexit><latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">AFI3icfZTPb9MwFMe9EdgovzY4comIkAaHqemQ4DhASBwHYj+kpoc9zU1tZ3MfmGU1P8DV7jz13BDXDjwv+CkWbe2KZaSvPh93ve9OM+OM8ENt/1tavedvbGzebN26fefuva3t+0cmzTWDQ5aKVJ/E1IDgCg6Ro4CTAOVsYDjePS69B9/Am14qj7gOIOepIniA84ouqmjqA8C6elW0N5tV8NfNsLaCEg9Dk63vY2on7JcgkImqDHdsJ1hr6AaORNgW1FuIKNsRBPoOlNRCaZXVOVa/7Gb6fuDVLtLoV/NXo0oqDRmLGNHSopDs+grJ5t83RwHL3oFV1mOoNg0SAXPqZ+e1+n2tgKMbOoExzV6vPhlRThm6F5rLE0rZakYJzlkpJVb+IYmbdDRKuCua+GrS18wCUADir2R3zWTxHkPwLAnwWmCFP4aZApzl1f97uqQBtcZKIoYzdUVHUa3puEGnoqZK5gRoO2GvSISMCJXwShjTRPhpqpKqC+YSa3qJR6JCqiB/GjVZ4M0os91OyV9ByzQd25zgTYN8NKRYBv63MqgDL/M0c6i5K2lvHp2UBbnb3ogGpta3m1DMJIl8F6OUFPNEzEj3ohvx9zCoeYTPWO2fAmnifpItdtzKud7O0LgW8l2BT5Y6QPLpwk57QHLFM9fzths4xaBTaZ7zvtji+skLzp0FmPd0REuHhTLxlFnN3T2u2fB/qv6ENkD8kjskNC8pzsk7fkgBwSRj6Sr+Qb+e798H56v7zfU3R9rY5QOaG9/cfwS3S7w=</latexit><latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">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</latexit>

10

1

10

2

L

10

  • 2

10

  • 1

10

C1,L

binary J=1.5, J=1/1.5 binary J=2.2, J=1/2.2

~ 1/L

Preliminary

end-end spin correlation

r=L/2

Heisenberg S=1 chain

gapless Haldane phase random singlet phase

δ

<latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">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</latexit><latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">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</latexit><latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">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</latexit><latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">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</latexit>

string order parameter

Preliminary

Oz(r) = − * Sz

j exp

2 4iπ

j+r−1

X

k=j+1

3 5 Sz

j+r

+

<latexit sha1_base64="HUbCo3iyNP3ji5ibCr7CJ26vkc=">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</latexit><latexit sha1_base64="HUbCo3iyNP3ji5ibCr7CJ26vkc=">AFX3icfVRdb9MwFPU+ykaB0cET4iUiQtqYNjUdEnuZNASTeGMw9iElbeW4t6nX2Mlsh63N/If4NbyON/4JTp1a5tiqcntvece3N9bT8OqVT1+u3C4tJy5dHK6uPqk6fP1p7X1l+cyigRBE5IFEbi3McSQsrhRFEVwnksADM/hDO/ymLn/0EIWnEf6hBDE2GA067lGBlXO3a56+t4YbY3N/2DtPj9kVr6MF17IXQVS71Ymp5MmHtL9/seXoVnqxJbYd7Qka9FTzuDVsZx6t2zW7vlPlzVrOIVho2IdtdeXl7xORBIGXJEQS+k69Vg1UywUJSHoqpdIiDHp4wBcY3LMQDbT/HO19dZ4OlY3EubHlZV7HzJSzKQcMN8gGVY9OR3LnGUxN1HdvWZKeZwo4GSUqJuEloqsrHdWhwogKhwYAxNBTa0W6WGBiTIdnsjiM12tehyuSMQY5p3U84k2DwgoT4n5ajBtmwRABgBjlYd9OuZTBYwOYUaAFgJz4j6MFeAyf/3YwGFBpzET5c8gc6HAuByU6OWqkNAfTB6Vdp5nm03ZjpfbdZHkC8yCfgonEAt/DvTCH5CRrxLqZwst+rN1Ghn8AzdI0dHmCwxL5jPTfqAg3ecoxylBTm7k9CbrBjz2J1Dwr4s5M0RVrbjiew9Bfp4B/JpEI6R5o8ohX+HboFXcK3ys5MqHJgN0umG6ZqZ61hJMz6WKXBzZvcZHTV1tP+MchqbedeubRTtRq5RTvmfE3id1N5iTXRvO9CUxa5w2dhxjf3tvH+wVF8gqeo3eoA3koA/oAH1BR+gEfQL/Ua36M/y38pKZa1SG0EXFwrOSzSxKq/+AdmB5qo=</latexit><latexit sha1_base64="HUbCo3iyNP3ji5ibCr7CJ26vkc=">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</latexit><latexit sha1_base64="HUbCo3iyNP3ji5ibCr7CJ26vkc=">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</latexit>
slide-18
SLIDE 18

r=L/2

Heisenberg S=1 chain

gapless Haldane phase random singlet phase

δ

<latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">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</latexit><latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">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</latexit><latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">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</latexit><latexit sha1_base64="TeI369vwoVLhr0Ju5yo5KNlqf5w=">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</latexit>

string order parameter

Preliminary

Oz(r) = − * Sz

j exp

2 4iπ

j+r−1

X

k=j+1

3 5 Sz

j+r

+

<latexit sha1_base64="HUbCo3iyNP3ji5ibCr7CJ26vkc=">AFX3icfVRdb9MwFPU+ykaB0cET4iUiQtqYNjUdEnuZNASTeGMw9iElbeW4t6nX2Mlsh63N/If4NbyON/4JTp1a5tiqcntvece3N9bT8OqVT1+u3C4tJy5dHK6uPqk6fP1p7X1l+cyigRBE5IFEbi3McSQsrhRFEVwnksADM/hDO/ymLn/0EIWnEf6hBDE2GA067lGBlXO3a56+t4YbY3N/2DtPj9kVr6MF17IXQVS71Ymp5MmHtL9/seXoVnqxJbYd7Qka9FTzuDVsZx6t2zW7vlPlzVrOIVho2IdtdeXl7xORBIGXJEQS+k69Vg1UywUJSHoqpdIiDHp4wBcY3LMQDbT/HO19dZ4OlY3EubHlZV7HzJSzKQcMN8gGVY9OR3LnGUxN1HdvWZKeZwo4GSUqJuEloqsrHdWhwogKhwYAxNBTa0W6WGBiTIdnsjiM12tehyuSMQY5p3U84k2DwgoT4n5ajBtmwRABgBjlYd9OuZTBYwOYUaAFgJz4j6MFeAyf/3YwGFBpzET5c8gc6HAuByU6OWqkNAfTB6Vdp5nm03ZjpfbdZHkC8yCfgonEAt/DvTCH5CRrxLqZwst+rN1Ghn8AzdI0dHmCwxL5jPTfqAg3ecoxylBTm7k9CbrBjz2J1Dwr4s5M0RVrbjiew9Bfp4B/JpEI6R5o8ohX+HboFXcK3ys5MqHJgN0umG6ZqZ61hJMz6WKXBzZvcZHTV1tP+MchqbedeubRTtRq5RTvmfE3id1N5iTXRvO9CUxa5w2dhxjf3tvH+wVF8gqeo3eoA3koA/oAH1BR+gEfQL/Ua36M/y38pKZa1SG0EXFwrOSzSxKq/+AdmB5qo=</latexit><latexit sha1_base64="HUbCo3iyNP3ji5ibCr7CJ26vkc=">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</latexit><latexit sha1_base64="HUbCo3iyNP3ji5ibCr7CJ26vkc=">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</latexit><latexit sha1_base64="HUbCo3iyNP3ji5ibCr7CJ26vkc=">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</latexit>

Higher dimensions

SDRG in high dimensions

Bhatt & Lee, PRL (1982) Westernerg et al, PRB (1997) Melin , EPJB (2000) YCL, et al., PRB (2003)

taken from Kimchi, Nahum & Senthil, PRX 8, 031028 (2018)

slide-19
SLIDE 19

SDRG in higher dimensions

RG

square lattice (no frustration)

F F AF no frustration

SDRG in higher dimensions with geometrical frustration

RG formation of large spins

slide-20
SLIDE 20

2D: AF order

0.01 0.02 0.03 0.04 0.05 0.06 0.07

1/L

0.002 0.004 0.006 0.008

m

2

square lattice triangular lattice

2D SDRG

2D: AF order

1 2 3 4 5 6

δ

0.1 0.2 0.3 0.4

mAF

QMC ~ exp (-0.3 δ) spin-waves ~ exp (-0.26 δ)

1 2 3 4 5 6

δ

0.1

mAF

strength of disorder

  • N. Laflorencie, et al, PRB 73, 060403(2006)

square lattice

finite AF order !

AF order is robust against
 strong bond randomness.

π(J) = 1 δ J1/δ−1, for 0 < J ≤ 1

slide-21
SLIDE 21

2D random-singlet phase

JQ model with random J or Q

Lu Liu, Hui Shao, YCL, Wenan Guo, Anders Sandvik, PRX 8, 041040 (2018)

i j

Jx

i j

Jy

i j k l m n

Qx

i j k l m n

Qy

2D random-singlet phase

8 12 16 24 32 48 10

  • 4

10

  • 3

10

  • 2

Cs(L/2,L/2) J/Q=0 Q/J=4 Q/J=2 12 8 16 24 32 48 L 10

  • 4

10

  • 3

10

  • 2

10

  • 1

Cd(L/2,L/2) J/Q=0 (a) (b)

C(L) ∼ 1/L2

<latexit sha1_base64="RE3agLjNkhIN9M48fWdg/HPze4=">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</latexit><latexit sha1_base64="RE3agLjNkhIN9M48fWdg/HPze4=">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</latexit><latexit sha1_base64="RE3agLjNkhIN9M48fWdg/HPze4=">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</latexit><latexit sha1_base64="RE3agLjNkhIN9M48fWdg/HPze4=">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</latexit>

D(L) ∼ 1/L4

<latexit sha1_base64="klCNRJ/aOFmGeGyKxdS0xgfC8=">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</latexit><latexit sha1_base64="klCNRJ/aOFmGeGyKxdS0xgfC8=">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</latexit><latexit sha1_base64="klCNRJ/aOFmGeGyKxdS0xgfC8=">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</latexit><latexit sha1_base64="klCNRJ/aOFmGeGyKxdS0xgfC8=">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</latexit>

Lu Liu, Hui Shao, YCL, Wenan Guo, Anders Sandvik, PRX 8, 041040 (2018)

slide-22
SLIDE 22

2D random-singlet phase

10

  • 2

10 10

  • 2

10

  • 1

χu , χloc L=48, χloc L=48, χu 10

  • 2

10

  • 1

10 T/J 10

  • 2

10

  • 1

χu , χloc L=64, χloc L=64, χu Q/J=4 J/Q=0

χ ∼ T −1+2/z

<latexit sha1_base64="Hz+fdOIZHxhathvrVEmr9pVA8B0=">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</latexit><latexit sha1_base64="Hz+fdOIZHxhathvrVEmr9pVA8B0=">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</latexit><latexit sha1_base64="Hz+fdOIZHxhathvrVEmr9pVA8B0=">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</latexit><latexit sha1_base64="Hz+fdOIZHxhathvrVEmr9pVA8B0=">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</latexit>

z>2

Lu Liu, Hui Shao, YCL, Wenan Guo, Anders Sandvik, PRX 8, 041040 (2018)

Experiment

  • cf. M. Watanabe, N. Kurita, et cl. PRB 98, 054422 (2018)

1 10

T [K] 2.5 3.0 3.5 χu [10

  • 3 emu/mol]

1 10

T [K] 2.5 3.0 3.5 4.0 2.0 2.2 2.4 2.6 2.5 3.0 3.5 x = 0.2 x = 0.3 x = 0.4 x = 0.5 a1 = 0.82 a2 = 0.70 a1 = 0.81 a2 = 0.73 a1 = 0.78 a2 = 0.73 a1 = 0.73 a2 = 0.62

Sr2CuTe1-xWxO6 J1-J2 on the square lattice

χ ∼ T −a1

<latexit sha1_base64="mXVGYdwRO8/IyRkN5qjENo45qIc=">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</latexit><latexit sha1_base64="mXVGYdwRO8/IyRkN5qjENo45qIc=">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</latexit><latexit sha1_base64="mXVGYdwRO8/IyRkN5qjENo45qIc=">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</latexit><latexit sha1_base64="mXVGYdwRO8/IyRkN5qjENo45qIc=">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</latexit>
slide-23
SLIDE 23

Future work

✤ dMERA (Goldsborough & Evenbly PRB (2017)) ✤ tSDRG in 2D ✤ tSDRG for dynamics

taken from PRB 95, 024205 (2017)

✤ Vosk & Altman, PRL (2013) ✤ Pekker et. al. PRX (2014)