Critical spin chains from modular invariance Ville Lahtinen - - PowerPoint PPT Presentation
Critical spin chains from modular invariance Ville Lahtinen Teresia Mnsson Juha Suorsa Eddy Ardonne PRB 89, 014409 (2014) TPQM-ESI & 2014-09-12 in preparation Mathematics meet Physics Complete reducibility of finite dimensional
Eddy Ardonne TPQM-ESI 2014-09-12
PRB 89, 014409 (2014) & in preparation
Complete reducibility of finite dimensional representations of semi-simple Lie groups. Hendrik Casimir updates his former advisor Paul Ehrenfest in a letter:
Complete reducibility of finite dimensional representations of semi-simple Lie groups. Hendrik Casimir updates his former advisor Paul Ehrenfest in a letter:
. Ehrenfest (image: ESI - 2013)
Complete reducibility of finite dimensional representations of semi-simple Lie groups. Hendrik Casimir updates his former advisor Paul Ehrenfest in a letter:
. Ehrenfest (image: ESI - 2013)
Complete reducibility of finite dimensional representations of semi-simple Lie groups. Weyl had a proof, but it used analysis, and Casimir quotes Pauli in his letter:
Complete reducibility of finite dimensional representations of semi-simple Lie groups. Weyl had a proof, but it used analysis, and Casimir quotes Pauli in his letter: Casimir: ‘But it’s unsatisfactory that one proves a purely algebraic theorem using a transcendental detour’
Complete reducibility of finite dimensional representations of semi-simple Lie groups. Weyl had a proof, but it used analysis, and Casimir quotes Pauli in his letter: Casimir: ‘But it’s unsatisfactory that one proves a purely algebraic theorem using a transcendental detour’ Quoting Pauli: ‘da sind the Mathematiker weinend umhergegangen’
Complete reducibility of finite dimensional representations of semi-simple Lie groups. Weyl had a proof, but it used analysis, and Casimir quotes Pauli in his letter: Casimir: ‘But it’s unsatisfactory that one proves a purely algebraic theorem using a transcendental detour’ Quoting Pauli: ‘da sind the Mathematiker weinend umhergegangen’
Casimir & B.L. v.d. Waerden give an algebraic proof, using a Casimir operator
★ Low-energy description of 2-d topological phases: anyon models ★ Topological phase transitions in 2-d:
★ Analogue on the level of spin chains: Ising examples ★ Beyond condensation: parafermions
An anyon model consist of a set of particles C = {1, a, b, c, . . . , n} ‘vacuum’
Moore, Seiberg,....
An anyon model consist of a set of particles C = {1, a, b, c, . . . , n} ‘vacuum’ These particles can ‘fuse’ (like taking tensor products of spins) a × b = X
c∈C
Nabcc b × a = a × b (a × b) × c = a × (b × c) a × 1 = a fusion coefficients
Moore, Seiberg,....
An anyon model consist of a set of particles C = {1, a, b, c, . . . , n} ‘vacuum’ These particles can ‘fuse’ (like taking tensor products of spins) a × b = X
c∈C
Nabcc b × a = a × b (a × b) × c = a × (b × c) a × 1 = a fusion coefficients Anyons are represented by their ‘worldlines’ a
Moore, Seiberg,....
An anyon model consist of a set of particles C = {1, a, b, c, . . . , n} ‘vacuum’ These particles can ‘fuse’ (like taking tensor products of spins) a × b = X
c∈C
Nabcc b × a = a × b (a × b) × c = a × (b × c) a × 1 = a fusion coefficients Anyons are represented by their ‘worldlines’ a Fusion is represented as a b c
Moore, Seiberg,....
An anyon model consist of a set of particles C = {1, a, b, c, . . . , n} ‘vacuum’ These particles can ‘fuse’ (like taking tensor products of spins) a × b = X
c∈C
Nabcc b × a = a × b (a × b) × c = a × (b × c) a × 1 = a fusion coefficients Anyons are represented by their ‘worldlines’ a Fusion is represented as a b c Twisting of a particle θa = e2πiha a a = θa
Moore, Seiberg,....
An anyon model consist of a set of particles C = {1, a, b, c, . . . , n} ‘vacuum’ These particles can ‘fuse’ (like taking tensor products of spins) a × b = X
c∈C
Nabcc b × a = a × b (a × b) × c = a × (b × c) a × 1 = a fusion coefficients Anyons are represented by their ‘worldlines’ a Fusion is represented as a b c Twisting of a particle θa = e2πiha a a = θa A boson has θb = 1 (hb ∈ Z)
Moore, Seiberg,....
An anyon model consist of a set of particles C = {1, a, b, c, . . . , n} ‘vacuum’ These particles can ‘fuse’ (like taking tensor products of spins) a × b = X
c∈C
Nabcc b × a = a × b (a × b) × c = a × (b × c) a × 1 = a fusion coefficients Anyons are represented by their ‘worldlines’ a Fusion is represented as a b c Twisting of a particle θa = e2πiha a a = θa A boson has θb = 1 (hb ∈ Z)
Moore, Seiberg,....
Braiding of particles a b c a b c = Ra,b
c
Ra,b
c
= ±eπi(hc−ha−hb)
Bais, Slingerland, 2009
Condensation amounts to identifying a boson with the vacuum boson b ∼ 1
Bais, Slingerland, 2009
Condensation amounts to identifying a boson with the vacuum This has several consequences: boson b ∼ 1
Bais, Slingerland, 2009
Condensation amounts to identifying a boson with the vacuum This has several consequences: boson b ∼ 1 Anyons which ‘differ by a boson’ are identified a × b = c = ⇒ a ∼ c
Bais, Slingerland, 2009
Condensation amounts to identifying a boson with the vacuum This has several consequences: boson b ∼ 1 Anyons which ‘differ by a boson’ are identified a × b = c = ⇒ a ∼ c Anyons with non-trivial monodromy with the boson ‘draw strings in the condensate’ and are therefore ‘confined’
Bais, Slingerland, 2009
Condensation amounts to identifying a boson with the vacuum This has several consequences: boson b ∼ 1 Anyons which ‘differ by a boson’ are identified a × b = c = ⇒ a ∼ c Anyons with non-trivial monodromy with the boson ‘draw strings in the condensate’ and are therefore ‘confined’ Some of the remaning particles might ‘split’: a ∼ a1 + a2
Bais, Slingerland, 2009
Condensation amounts to identifying a boson with the vacuum This has several consequences: boson b ∼ 1 Anyons which ‘differ by a boson’ are identified a × b = c = ⇒ a ∼ c Anyons with non-trivial monodromy with the boson ‘draw strings in the condensate’ and are therefore ‘confined’ a × a = 1 + b + · · · a × a = 1 + 1 + · · · vacuum twice Some of the remaning particles might ‘split’: a ∼ a1 + a2
Bais, Slingerland, 2009
Condensation amounts to identifying a boson with the vacuum This has several consequences: boson b ∼ 1 Anyons which ‘differ by a boson’ are identified a × b = c = ⇒ a ∼ c Anyons with non-trivial monodromy with the boson ‘draw strings in the condensate’ and are therefore ‘confined’ Some of the remaning particles might ‘split’: a ∼ a1 + a2 In CFT language, one condenses a boson by adding it to the chiral algebra, and in the end, one has constructed a new modular invariant partition function
A conformal field theory splits in two pieces, a chiral and anti-chiral part. To each chiral sector (primary field), one associates a ‘character’, describing the number of states in this sector The constants aj are non-negative integers, and τ is the modular parameter, describing the shape of the torus (next slide). χφ(q) = qhφ−c/24 a0q0 + a1q1 + a2q2 + · · ·
A conformal field theory splits in two pieces, a chiral and anti-chiral part. To each chiral sector (primary field), one associates a ‘character’, describing the number of states in this sector The constants aj are non-negative integers, and τ is the modular parameter, describing the shape of the torus (next slide). The full partition function is obtained by combining the chiral halves, and summing over the primary fields: Zcft = X
j
|χφj|2 χφ(q) = qhφ−c/24 a0q0 + a1q1 + a2q2 + · · ·
One should be able to put the cft on the torus: partition function should be invariant under re-parametrization of the torus!
One should be able to put the cft on the torus: partition function should be invariant under re-parametrization of the torus!
One should be able to put the cft on the torus: partition function should be invariant under re-parametrization of the torus! Shape of the torus is encoded by the modular parameter τ. The transformations S and T do not change the shape of the torus. T : τ → τ + 1 U : τ → τ/(τ + 1) S : τ → −1/τ
image: BYB
S = T −1UT −1
The partition function should be invariant under S and T!
One should be able to put the cft on the torus: partition function should be invariant under re-parametrization of the torus! Shape of the torus is encoded by the modular parameter τ. The transformations S and T do not change the shape of the torus. T : τ → τ + 1 U : τ → τ/(τ + 1) S : τ → −1/τ
image: BYB
S = T −1UT −1
The most general way to combine the chiral halves: Zcft = X
i,j
ni,jχφiχ∗
φj
ni,j ∈ Z≥0
The most general way to combine the chiral halves: Zcft = X
i,j
ni,jχφiχ∗
φj
ni,j ∈ Z≥0 Presence of the vacuum: Invariance under T: Invariance under S: the matrix (ni,j) commutes with the modular S-matrix, that diagonalizes the fusion rules. ni,j 6= 0 ) hφi hφj = 0 mod 1 n1,1 = 1
The most general way to combine the chiral halves: Zcft = X
i,j
ni,jχφiχ∗
φj
ni,j ∈ Z≥0 Presence of the vacuum: Invariance under T: Invariance under S: the matrix (ni,j) commutes with the modular S-matrix, that diagonalizes the fusion rules. ni,j 6= 0 ) hφi hφj = 0 mod 1 n1,1 = 1 The so-called ‘diagonal invariant’ always exist: Zcft = X
j
|χφj|2
The most general way to combine the chiral halves: Zcft = X
i,j
ni,jχφiχ∗
φj
ni,j ∈ Z≥0 Presence of the vacuum: Invariance under T: Invariance under S: the matrix (ni,j) commutes with the modular S-matrix, that diagonalizes the fusion rules. ni,j 6= 0 ) hφi hφj = 0 mod 1 n1,1 = 1 The so-called ‘diagonal invariant’ always exist: Zcft = X
j
|χφj|2 Finding all invariants is, in general, a hard task, but progress has been made (minimal models, su(2)k, su(3)k, parafermions...)
Cappelli et al., Gepner et al., ...
The Ising cft has three sectors: h1 = 0 , hσ = 1/16 , h1 = 1/2 χ1, χσ, χψ
The Ising cft has three sectors: h1 = 0 , hσ = 1/16 , h1 = 1/2 The Ising2 cft has nine sectors: χ(1,1) , χ(1,σ) , χ(1,ψ) , χ(σ,1) , χ(σ,σ) , χ(σ,ψ) , χ(ψ,1) , χ(ψ,σ) , χ(ψ,ψ) χ1, χσ, χψ
The Ising cft has three sectors: h1 = 0 , hσ = 1/16 , h1 = 1/2 The Ising2 cft has nine sectors: χ(1,1) , χ(1,σ) , χ(1,ψ) , χ(σ,1) , χ(σ,σ) , χ(σ,ψ) , χ(ψ,1) , χ(ψ,σ) , χ(ψ,ψ) Apart from the diagonal invariant, one also finds a block diagonal invariant: Z = |χ(1,1) + χ(ψ,ψ)|2 + |χ(1,ψ) + χ(ψ,1)|2 + 2|χ(σ,σ)|2 χ1, χσ, χψ
The Ising cft has three sectors: h1 = 0 , hσ = 1/16 , h1 = 1/2 The Ising2 cft has nine sectors: χ(1,1) , χ(1,σ) , χ(1,ψ) , χ(σ,1) , χ(σ,σ) , χ(σ,ψ) , χ(ψ,1) , χ(ψ,σ) , χ(ψ,ψ) Apart from the diagonal invariant, one also finds a block diagonal invariant: Z = |χ(1,1) + χ(ψ,ψ)|2 + |χ(1,ψ) + χ(ψ,1)|2 + 2|χ(σ,σ)|2 One sees identification of sectors: (1, 1) ∼ (ψ, ψ) (1, ψ) ∼ (ψ, 1) Confined sectors: (1, σ) , (ψ, σ) , (σ, 1) , (σ, ψ) Split sector: (σ, σ) χ1, χσ, χψ
The Ising cft has three sectors: h1 = 0 , hσ = 1/16 , h1 = 1/2 The Ising2 cft has nine sectors: χ(1,1) , χ(1,σ) , χ(1,ψ) , χ(σ,1) , χ(σ,σ) , χ(σ,ψ) , χ(ψ,1) , χ(ψ,σ) , χ(ψ,ψ) Apart from the diagonal invariant, one also finds a block diagonal invariant: Z = |χ(1,1) + χ(ψ,ψ)|2 + |χ(1,ψ) + χ(ψ,1)|2 + 2|χ(σ,σ)|2 One sees identification of sectors: (1, 1) ∼ (ψ, ψ) (1, ψ) ∼ (ψ, 1) Confined sectors: (1, σ) , (ψ, σ) , (σ, 1) , (σ, ψ) Split sector: (σ, σ) The resulting invariant describes the u(1)4 cft, and the construction amounts to the orbifold construction. χ1, χσ, χψ
In some cases, one can permute some of the labels of the primary fields (anyons), without changing the fusion rules. Let π be such a permutation: nπ(a),π(b),π(c) = na,b,c Sπ(a),π(b) = Sa,b
In some cases, one can permute some of the labels of the primary fields (anyons), without changing the fusion rules. Let π be such a permutation: nπ(a),π(b),π(c) = na,b,c Sπ(a),π(b) = Sa,b If π preserves the scaling dimension (mod 1), then one can construct new invariants from block-diagonal ones: Z = X
a
|χa|2 − → Zπ = X
a
χaχ∗
π(a)
In some cases, one can permute some of the labels of the primary fields (anyons), without changing the fusion rules. Let π be such a permutation: nπ(a),π(b),π(c) = na,b,c Sπ(a),π(b) = Sa,b Zπ = |χ(1,1)|2 + |χ(σ,σ)|2 + |χ(ψ,ψ)|2 + χ(1,σ)χ∗
(σ,1) + χ(σ,1)χ∗ (1,σ)+
χ(1,ψ)χ∗
(ψ,1) + χ(ψ,1)χ∗ (1,ψ) + χ(ψ,σ)χ∗ (σ,ψ) + χ(σ,ψ)χ∗ (ψ,σ)
Example in the Ising2 theory, starting from the diagonal invariant: If π preserves the scaling dimension (mod 1), then one can construct new invariants from block-diagonal ones: Z = X
a
|χa|2 − → Zπ = X
a
χaχ∗
π(a)
In some cases, one can permute some of the labels of the primary fields (anyons), without changing the fusion rules. Let π be such a permutation: nπ(a),π(b),π(c) = na,b,c Sπ(a),π(b) = Sa,b Zπ = |χ(1,1)|2 + |χ(σ,σ)|2 + |χ(ψ,ψ)|2 + χ(1,σ)χ∗
(σ,1) + χ(σ,1)χ∗ (1,σ)+
χ(1,ψ)χ∗
(ψ,1) + χ(ψ,1)χ∗ (1,ψ) + χ(ψ,σ)χ∗ (σ,ψ) + χ(σ,ψ)χ∗ (ψ,σ)
Example in the Ising2 theory, starting from the diagonal invariant: In this case, we have , but that’s not generic. Zπ = Z If π preserves the scaling dimension (mod 1), then one can construct new invariants from block-diagonal ones: Z = X
a
|χa|2 − → Zπ = X
a
χaχ∗
π(a)
HTFI =
L−1
X
i=0
σz
i + σx i σx i+1
σz
j
1 2 L − 1 i i + 1 spin-1/2 σx
i σx i+1
Note: we use periodic boundary conditions: crucial for our purposes! σα
j+L ≡ σα j
Symmetry: P = Y
i
σz
i
HTFI =
L−1
X
i=0
σz
i + σx i σx i+1
σz
j
1 2 L − 1 i i + 1 spin-1/2 σx
i σx i+1
Note: we use periodic boundary conditions: crucial for our purposes! σα
j+L ≡ σα j
Define fermionic levels (Jordan-Wigner): |⇤ = |0⇤ |⇥⇤ = |1⇤ σz
i = 1 − 2c† ici
ci = ⇣Y
j<i
σz
i
⌘ σ+
i
c†
i =
⇣Y
j<i
σz
i
⌘ σ−
i
Y
i
σz
i = (−1)F
Lieb, Schultz, Mattis, (1961) Pfeuty, (1970)
Symmetry: P = Y
i
σz
i
HTFI =
L−1
X
i=0
σz
i + σx i σx i+1
=
L−1
X
j=0
(2c†
jcj − 1)+ L−2
X
j=0
(cj − c†
j)(cj+1 + c† j+1)+
− (−1)F (cL−1 − c†
L−1)(c0 + c† 0)
1 2 L − 1 i i + 1
HTFI =
L−1
X
i=0
σz
i + σx i σx i+1
=
L−1
X
j=0
(2c†
jcj − 1)+ L−2
X
j=0
(cj − c†
j)(cj+1 + c† j+1)+
− (−1)F (cL−1 − c†
L−1)(c0 + c† 0)
1 2 L − 1 i i + 1 The parity of the number of fermions is conserved! The fermion boundary conditions depend on the symmetry sector: For F even: anti-periodic boundary conditions For F odd: periodic boundary conditions
HTFI =
L−1
X
i=0
⌅z
i + ⌅x i ⌅x i+1
= X
k
⇥k
kk − 1/2
r 2 − 2 cos 2⇤k L
2 L − 1 i i + 1 Momenta k: half integer for even F integer for odd F Solution: go to k-space, and perform a diagonalize a 2x2 matrix (or, in general 2 L x 2 L if couplings are disordered).
HTFI =
L−1
X
i=0
⌅z
i + ⌅x i ⌅x i+1
= X
k
⇥k
kk − 1/2
r 2 − 2 cos 2⇤k L
2 L − 1 i i + 1 Momenta k: half integer for even F integer for odd F Solution: go to k-space, and perform a diagonalize a 2x2 matrix (or, in general 2 L x 2 L if couplings are disordered). Conformal field theory: spectrum is described in the following way: ✏i = E0L + 2⇡v L
12 + hl + hr + nl + nr
Conformal field theory: ‘towers’ of states with:
p 2
p
3 p 2
2 p k 1 2 3 4 5 6 E
Transverse Field Ising model, L=16
∆E = 1 ∆k = 2π L
Conformal field theory: ‘towers’ of states with:
p 2
p
3 p 2
2 p k 1 2 3 4 5 6 E
Transverse Field Ising model, L=16
∆E = 1 ∆k = 2π L
Conformal field theory: ‘towers’ of states with:
p 2
p
3 p 2
2 p k 1 2 3 4 5 6 E
Transverse Field Ising model, L=16
∆E = 1 ∆k = 2π L
Conformal field theory: ‘towers’ of states with:
p 2
p
3 p 2
2 p k 1 2 3 4 5 6 E
Transverse Field Ising model, L=16
∆E = 1 ∆k = 2π L
Conformal field theory: ‘towers’ of states with:
p 2
p
3 p 2
2 p k 1 2 3 4 5 6 E
Transverse Field Ising model, L=16
∆E = 1 ∆k = 2π L Primary fields: 1, σ, ψ
Conformal field theory: ‘towers’ of states with:
p 2
p
3 p 2
2 p k 1 2 3 4 5 6 E
Transverse Field Ising model, L=16
∆E = 1 ∆k = 2π L Primary fields: 1, σ, ψ Ising conformal field theory: h1 = 0 hσ = 1 16 hψ = 1 2
Conformal field theory: ‘towers’ of states with:
p 2
p
3 p 2
2 p k 1 2 3 4 5 6 E
Transverse Field Ising model, L=16
∆E = 1 ∆k = 2π L Primary fields: 1, σ, ψ Ising conformal field theory: h1 = 0 hσ = 1 16 hψ = 1 2
Relation between symmetry sector, boundary conditions for the fermions and cft sectors (primaries):
boundary condition fields 1 A 1, ψ −1 P σ
1 2 L − 1 i i + 1 3 Spectrum is the ‘product’ of two spectra of the TFI model with L/2 H(2)
TFI = L−1
X
i=0
σz
i + σx i σx i+2
1 2 L − 1 i i + 1 3 Spectrum is the ‘product’ of two spectra of the TFI model with L/2 H(2)
TFI = L−1
X
i=0
σz
i + σx i σx i+2
Both the number of fermions on the even and the odd sites is conserved modulo two Pe = Y
i,even
σz
i = (−1)Fe
Po = Y
i,odd
σz
i = (−1)Fo
Symmetries:
Relation between symmetry sector, boundary conditions for the fermions and cft sectors (primaries): HTFI H(2)
TFI
(Pe, Po) (BCe, BCo) fields (1, 1) (A, A) (1, 1), (1, ψ), (ψ, 1), (ψ, ψ) (1, −1) (A, P) (1, σ), (ψ, σ) (−1, 1) (P, A) (σ, 1), (σ, ψ) (−1, −1) (P, P) (σ, σ)
boundary condition fields 1 A 1, ψ −1 P σ
We now change our model, by adding a ‘boundary term’, that changes the boundary condition of one chain, depending on the symmetry sector of the
1 2 L − 1 i i + 1 3 H(2)
TFI = L−1
X
i=0
σz
i + σx i σx i+2
HBoundary =
L−2σx 0+
L−1σx 1
What is this new model ? H2
TFI + HBoundary
Relation between symmetry sector, boundary conditions for the fermions and cft sectors (primaries): H(2)
TFI
(Pe, Po) (BCe, BCo) fields (1, 1) (A, A) (1, 1), (1, ψ), (ψ, 1), (ψ, ψ) (1, −1) (A, P) (1, σ), (ψ, σ) (−1, 1) (P, A) (σ, 1), (σ, ψ) (−1, −1) (P, P) (σ, σ) H2
TFI + HBoundary
(Pe, Po) (BCe, BCo) fields (1, 1) (A, A) (1, 1), (1, ψ), (ψ, 1), (ψ, ψ) (1, −1) (P, P) (σ, σ) (−1, 1) (P, P) (σ, σ) (−1, −1) (A, A) (1, 1), (1, ψ), (ψ, 1), (ψ, ψ) The new model has u(1)4 critical behaviour, i.e., the other modular invariant in the Ising2 theory. This is the critical behaviour of the XY chain!
One can explicitly relate the TFI2 to the XY chain: ⇣L−1 X
i=0
σz
i + σx i σx i+2
⌘ +
L−2σx 0 +
L−1σx 1 =
X
i
τ x
i τ x i+1 + τ y i τ y i+1
H(2)
TFI + HBoundary = HXY
One can explicitly relate the TFI2 to the XY chain: ⇣L−1 X
i=0
σz
i + σx i σx i+2
⌘ +
L−2σx 0 +
L−1σx 1 =
X
i
τ x
i τ x i+1 + τ y i τ y i+1
H(2)
TFI + HBoundary = HXY
τ z
2j = σy 2jσy 2j+1
τ x
2j =
Y
i≤2j
σx
i
τ z
2j+1 = σx 2jσx 2j+1
τ x
2j+1 =
Y
i≥2j+1
σy
i
One uses a modified version of the transformation for open chains (used, f.i., by D. Fisher, but dating back to the 70’s:
One can explicitly relate the TFI2 to the XY chain: ⇣L−1 X
i=0
σz
i + σx i σx i+2
⌘ +
L−2σx 0 +
L−1σx 1 =
X
i
τ x
i τ x i+1 + τ y i τ y i+1
H(2)
TFI + HBoundary = HXY
τ z
2j = σy 2jσy 2j+1
τ x
2j =
Y
i≤2j
σx
i
τ z
2j+1 = σx 2jσx 2j+1
τ x
2j+1 =
Y
i≥2j+1
σy
i
One uses a modified version of the transformation for open chains (used, f.i., by D. Fisher, but dating back to the 70’s: So, by changing boundary conditions, one can change spin chains, such that
Let’s consider the product of n Ising models. Condensing the bosons gives the following modular invariant: Isingn − → so(n)1
Let’s consider the product of n Ising models. Condensing the bosons gives the following modular invariant: Isingn − → so(n)1 In particular, so(3)1 = su(2)2
Let’s consider the product of n Ising models. Condensing the bosons gives the following modular invariant: Isingn − → so(n)1 In particular, so(3)1 = su(2)2 Chains with su(2)2 critical points are known: in S=1 chains (BA solvable), or with long-range interactions (Greiter et al., Quella et al., Sierra et al., Tu)
Let’s consider the product of n Ising models. Condensing the bosons gives the following modular invariant: Isingn − → so(n)1 In particular, so(3)1 = su(2)2 Chains with su(2)2 critical points are known: in S=1 chains (BA solvable), or with long-range interactions (Greiter et al., Quella et al., Sierra et al., Tu) We start with three decoupled Ising chains, and add the appropriate ‘boundary’ term. After a spin-transformation, we obtain Hsu(2)2 = X
i
τ x
i τ x i+1 + τ y i τ z i+1τ y i+2
Let’s consider the product of n Ising models. Condensing the bosons gives the following modular invariant: Isingn − → so(n)1 In particular, so(3)1 = su(2)2 Chains with su(2)2 critical points are known: in S=1 chains (BA solvable), or with long-range interactions (Greiter et al., Quella et al., Sierra et al., Tu) We start with three decoupled Ising chains, and add the appropriate ‘boundary’ term. After a spin-transformation, we obtain Hsu(2)2 = X
i
τ x
i τ x i+1 + τ y i τ z i+1τ y i+2
Generalization to arbitrary so(n)1 critical chains is straightforward
This hamiltonian can be solved by Jordan-Wigner, in terms of a single
Hsu(2)2 = X
i
gxτ x
i τ x i+1 + gyτ y i τ z i+1τ y i+2
This hamiltonian can be solved by Jordan-Wigner, in terms of a single
Hsu(2)2 = X
i
gxτ x
i τ x i+1 + gyτ y i τ z i+1τ y i+2
π/6 π/3 π/2 K 1 2 3 Energy 1 - sector σ - sector ψ - sector
Spectrum of the so(3)1 chain - L=36
5 2 2 4 1 1 1 2 2 6 6 6 6 1 1 5 5 2 2 4 4 5 5 5 4 4 4 6 6 6 6 6 6 6 6 12 12
Hsu(2)2 = X
k
✏k
kk − 1/2
q 2 + 2 cos
To go beyond condensation transitions, we consider the 3-state Potts chain (compare: Fendley & Qi’s talks). Z = 1 ω ω2 X = 1 1 1 ω = e2πi/3 X3 = 1 Z3 = 1 XZ = ωZX H3−Potts = − X
i
XiX†
i+1 + Zi + h.c.
To go beyond condensation transitions, we consider the 3-state Potts chain (compare: Fendley & Qi’s talks). The 3-state Potts chain at it’s critical point: Z3 parafermion CFT, with c=4/5. Field content: Scaling dimensions: {1, ψ1, ψ2, τ0, τ1, τ2} h1 = 0 , hψ1,2 = 2/3 , hτ0 = 2/5 , hτ1,2 = 1/15 Z = 1 ω ω2 X = 1 1 1 ω = e2πi/3 X3 = 1 Z3 = 1 XZ = ωZX H3−Potts = − X
i
XiX†
i+1 + Zi + h.c.
The product of two Z3 parafermion theories: no bosons or fermions! But, there are 16 modular invariants, giving two different partition functions (not obtainable as orbifolds):
The 3-state Potts chain at it’s critical point: Z3 parafermion CFT, with c=4/5. Field content: Scaling dimensions: {1, ψ1, ψ2, τ0, τ1, τ2} h1 = 0 , hψ1,2 = 2/3 , hτ0 = 2/5 , hτ1,2 = 1/15
The product of two Z3 parafermion theories: no bosons or fermions! But, there are 16 modular invariants, giving two different partition functions (not obtainable as orbifolds):
The 3-state Potts chain at it’s critical point: Z3 parafermion CFT, with c=4/5. Field content: Scaling dimensions: {1, ψ1, ψ2, τ0, τ1, τ2} h1 = 0 , hψ1,2 = 2/3 , hτ0 = 2/5 , hτ1,2 = 1/15
hl + hr ‘degenaracy’ 1 2/15 4 4/15 4 4/5 2 14/15 4 4/3 4 22/15 8 8/5 1 32/15 4 8/3 4
ZDiagonal
hl + hr ‘degenaracy’ 1 4/15 4 4/5 2 14/15 4 17/5 8 4/3 4 22/15 8 8/5 1 8/3 4
ZPermutation
The Potts chain realizing the (Z3 parafermion)2 theory is simply: H(2)
3−Potts = −
X
i
XiX†
i+2 + Zi + h.c.
The Potts chain realizing the (Z3 parafermion)2 theory is simply: H(2)
3−Potts = −
X
i
XiX†
i+2 + Zi + h.c.
The appropriate chain for the other invariant reads H(2,perm)
3−Potts = −
⇣X
i
XiX†
i+2 + Zi + h.c.
⌘ − Y
i,even
Z†
i − 1
L−1X1 −
Y
i,odd
Zi − 1
L−2X0 + h.c.
= − ⇣X
i
˜ Xi ˜ X†
i+1 + ˜
Zi ˜ Zi+1 + h.c. ⌘
The Potts chain realizing the (Z3 parafermion)2 theory is simply: H(2)
3−Potts = −
X
i
XiX†
i+2 + Zi + h.c.
The appropriate chain for the other invariant reads H(2,perm)
3−Potts = −
⇣X
i
XiX†
i+1 + ZiZi+1 + h.c.
⌘
The Potts chain realizing the (Z3 parafermion)2 theory is simply: H(2)
3−Potts = −
X
i
XiX†
i+2 + Zi + h.c.
The appropriate chain for the other invariant reads H(2,perm)
3−Potts = −
⇣X
i
XiX†
i+1 + ZiZi+1 + h.c.
⌘ This model is closely related to the ‘quantum torus chain’ Qin et al. HQTC = ⇣X
i
cos(θ)XiX†
i+1 + sin(θ)ZiZ† i+1 + h.c.
⌘
The Potts chain realizing the (Z3 parafermion)2 theory is simply: H(2)
3−Potts = −
X
i
XiX†
i+2 + Zi + h.c.
The appropriate chain for the other invariant reads H(2,perm)
3−Potts = −
⇣X
i
XiX†
i+1 + ZiZi+1 + h.c.
⌘ This model is closely related to the ‘quantum torus chain’ Qin et al. HQTC = ⇣X
i
cos(θ)XiX†
i+1 + sin(θ)ZiZ† i+1 + h.c.
⌘ By coupling three 3-state Potts chains, one can again condense a boson: H(3,cond)
3−Potts = −
⇣X
i
XiX†
i+1 + ZiZi+1Zi+2 + h.c.
⌘
We can construct interesting spin-chains in analogy with 2d topological condensation transitions as well as modular invariance Construction works for Jordan-Wigner solvable models, ‘BA’ solvable models, and non-integrable models. Latter category (non discussed here): S=1 Blume-Capel model, giving a N=1 susy cft, (A,E) exceptional modular invariant. Open questions: Can we do this without coupling several chains together (4-state Potts?) How general is this method? Can we learn something about the modular invariant partition functions? Study of the phase diagrams of the new models is underway
We can construct interesting spin-chains in analogy with 2d topological condensation transitions as well as modular invariance Construction works for Jordan-Wigner solvable models, ‘BA’ solvable models, and non-integrable models. Latter category (non discussed here): S=1 Blume-Capel model, giving a N=1 susy cft, (A,E) exceptional modular invariant. Open questions: Can we do this without coupling several chains together (4-state Potts?) How general is this method? Can we learn something about the modular invariant partition functions? Study of the phase diagrams of the new models is underway