Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1...
Transcript of Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1...
![Page 1: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/1.jpg)
Topological stability of q-deformedquantum spin chains
Collaborators:S. Trebst, A. Feiguin, C. Gils, D. Huse, A. Ludwig, M. Troyer, Z. Wang
Eddy Ardonne
Nordita
ERLDQS, Firenze, 090908
![Page 2: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/2.jpg)
Motivation• What’s a spin chain?• Use a modular tensor category/quantum group!
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Motivation• What’s a spin chain?• Use a modular tensor category/quantum group!
Outline• The golden (fibonacci) chain• Topological stability of criticality• Perturbing the chain• Generalizations
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The Heisenberg model
1! 1 = 11! ! = !
! ! ! = 1 + !
Fusion rules for SU(2)3 SU(2) spins
12! 1
2= 0 + 1singlet
triplet
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The Heisenberg model
1! 1 = 11! ! = !
! ! ! = 1 + !
Fusion rules for SU(2)3 SU(2) spins
12! 1
2= 0 + 1singlet
triplet
= J!
!ij"
!Jij2! !Si
2! !Sj
2
H = J!
!ij"
!Si · !Sj
H = J!
!ij"
"
ij
0
!Jij = !Si + !Sj
Heisenberg Hamiltonian
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The Heisenberg model
1! 1 = 11! ! = !
! ! ! = 1 + !
Fusion rules for SU(2)3 SU(2) spins
12! 1
2= 0 + 1singlet
triplet
= J!
!ij"
!Jij2! !Si
2! !Sj
2
H = J!
!ij"
!Si · !Sj
H = J!
!ij"
"
ij
0
!Jij = !Si + !Sj
Heisenberg Hamiltonian
H = J!
!ij"
"
ij
1
Heisenberg Hamiltonianfor (Fibonacci) anyons
“antiferromagnet” favors“ferromagnet” favors
1! (J < 0)
(J > 0)
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The golden chain
! ! ! !! !
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The golden chain
! ! ! !! !
! ! ! ! ! !
!
!
1
. . .
1! ! = !
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The golden chain
! ! ! !! !
! ! ! ! ! !
!
!
1
. . .
1! ! = !
Hilbert space: |x1, x2, x3, . . .!
! ! ! ! !
! . . .x1 x2 x3
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The golden chain
! ! ! !! !
! ! ! ! ! !
!
!
1
. . .
1! ! = !
Hilbert space: |x1, x2, x3, . . .!
! ! ! ! !
! . . .x1 x2 x3
dimL = FL+1 ! !L
! =1 +
!5
2= 1.618 . . .
Hilbert space has no natural decomposition as tensor product of single-site states.
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The golden chainWe want to construct a local Hamiltonian .H =
!
i
Hi
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The golden chainWe want to construct a local Hamiltonian .H =
!
i
Hi
! !
x1 x2 x3
![Page 13: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/13.jpg)
The golden chainWe want to construct a local Hamiltonian .H =
!
i
Hi
! !
x1 x2 x3
! !
x1 x3
x̃2
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The golden chainWe want to construct a local Hamiltonian .H =
!
i
Hi
! !
x1 x2 x3
! !
x1 x3
x̃2=
!
x̃2
F x2x̃2
F-matrix
F =!
!!1 !!1/2
!!1/2 !!!1
"
![Page 15: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/15.jpg)
The golden chainWe want to construct a local Hamiltonian .H =
!
i
Hi
SU(2) spins
12! 1
2! 1
2
12! 1
2! 1
2
Clebsch-Gordan coefficient
6-J symbolE. Wigner 1940
! !
x1 x2 x3
! !
x1 x3
x̃2=
!
x̃2
F x2x̃2
F-matrix
F =!
!!1 !!1/2
!!1/2 !!!1
"
![Page 16: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/16.jpg)
The golden chainWe want to construct a local Hamiltonian .H =
!
i
Hi
SU(2) spins
12! 1
2! 1
2
12! 1
2! 1
2
Clebsch-Gordan coefficient
6-J symbolE. Wigner 1940
! !
x1 x2 x3
! !
x1 x3
x̃2=
!
x̃2
F x2x̃2
F-matrix
F =!
!!1 !!1/2
!!1/2 !!!1
"
Hi = Fi !1i FiLocal Hamiltonian:
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The golden chainHi = Fi !1
i FiLocal Hamiltonian:
Hi = !!
!!2 !!3/2
!!3/2 !!1
"
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The golden chainHi = Fi !1
i FiLocal Hamiltonian:
Hi = !!
!!2 !!3/2
!!3/2 !!1
"
Explicit form:
off-diagonal matrix element
Hi = !P1!1 ! !!2P!1! ! !!1P!!!
!!!3/2 (|"1"" #""" | + h.c.)
A. Feiguin et. al, PRL (2007)
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conformal field theory description
CriticalityFinite-size gap
!(L) ! (1/L) z=1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1inverse system size 1/L
0
0.1
0.2
0.3
0.4
0.5
0.6
ener
gy g
ap Δ
even length chainodd length chain
Lanczos
DMRG
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conformal field theory description
CriticalityFinite-size gap
!(L) ! (1/L) z=1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1inverse system size 1/L
0
0.1
0.2
0.3
0.4
0.5
0.6
ener
gy g
ap Δ
even length chainodd length chain
Lanczos
DMRG
Entanglement entropy
SPBC(L) ! c
3log L
central chargec = 7/10
20 40 50 60 80 100 120 160 200 24030system size L
0 0
0.5 0.5
1 1
1.5 1.5
entro
py S(L)
periodic boundary conditionsopen boundary conditions
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Energy spectra
0 2 4 6 8 10 12 14 16 18momentum K [2π/L]
0 0
1 1
2 2
3 3
4 4
5 5
6 6
resc
aled
ene
rgy
E(K)
L = 36
3/40
7/8
1/5
6/5
3
0
Neveu-Schwarz sector
Ramond sector
1/5 + 1
1/5 + 21/5 + 2
0 + 26/5 + 1
3/40 + 1
3/40 + 2
7/8 + 13/40 + 2
3/40 + 33/40 + 3
7/8 + 2
1/5 + 3
6/5 + 20 + 3
primary fieldsdescendants
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Energy spectra
0 2 4 6 8 10 12 14 16 18momentum K [2π/L]
0 0
1 1
2 2
3 3
4 4
5 5
6 6
resc
aled
ene
rgy
E(K)
L = 36
3/40
7/8
1/5
6/5
3
0
Neveu-Schwarz sector
Ramond sector
1/5 + 1
1/5 + 21/5 + 2
0 + 26/5 + 1
3/40 + 1
3/40 + 2
7/8 + 13/40 + 2
3/40 + 33/40 + 3
7/8 + 2
1/5 + 3
6/5 + 20 + 3
primary fieldsdescendants
primary fields
scaling dimensions
I ! !! !!! ! !!
1/100 3/5 3/2 3/80 7/16! "# $ ! "# $
K = 0 K = !
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Mapping to Temperly-LiebThe operators form a representation of the Temperly-Lieb algebra:
X2i = dXi XiXi±1Xi = Xi Xi, Xj= 0 when |i! j| " 2
is the isotopy parameterd = !
Xi = !!Hi
![Page 24: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/24.jpg)
Mapping to Temperly-Lieb
Get: Restricted solid on solid model (RSOS)
AFM interactions: minimal model
FM interactions: parafermions
Mk
Zk c =2(k ! 1)k + 2
c = 1! 6(k + 1)(k + 2)
The operators form a representation of the Temperly-Lieb algebra:
X2i = dXi XiXi±1Xi = Xi Xi, Xj= 0 when |i! j| " 2
is the isotopy parameterd = !
Xi = !!Hi
![Page 25: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/25.jpg)
Topological symmetryen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
!
!!
![Page 26: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/26.jpg)
Topological symmetryen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
2irrelevant operatorsrelevant operators
!
!!
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Relevant perturbations
! !!
! !!
Topological symmetryen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
2irrelevant operatorsrelevant operators
!
!!
![Page 28: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/28.jpg)
Relevant perturbations
! !!
! !!
Topological symmetryen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
2irrelevant operatorsrelevant operators
!
!!
staggered
![Page 29: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/29.jpg)
Relevant perturbations
! !!
! !!
Topological symmetryen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
2irrelevant operatorsrelevant operators
!
!!
staggered
prohibited bytranslational symmetry
![Page 30: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/30.jpg)
Relevant perturbations
! !!
! !!
Topological symmetryen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
2irrelevant operatorsrelevant operators
!
!!
staggered
prohibited bytranslational symmetry
!
no flux -flux!
![Page 31: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/31.jpg)
Relevant perturbations
! !!
! !!
Topological symmetryen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
2irrelevant operatorsrelevant operators
!
!!
staggered Symmetry operator!x!
1, . . . , x!L|Y |x1, . . . , xL"
=L!
i=1
"F
x!i+1
!xi!
#x!i
xi+1
[H,Y ] = 0
with eigenvaluesS!!flux = ! Sno flux = !!!1
prohibited bytranslational symmetry
!
no flux -flux!
![Page 32: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/32.jpg)
Relevant perturbations
! !!
! !!
Topological symmetryen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
2irrelevant operatorsrelevant operators
!
!!
staggered Symmetry operator!x!
1, . . . , x!L|Y |x1, . . . , xL"
=L!
i=1
"F
x!i+1
!xi!
#x!i
xi+1
[H,Y ] = 0
with eigenvaluesS!!flux = ! Sno flux = !!!1
prohibited bytranslational symmetry
!
no flux -flux!
![Page 33: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/33.jpg)
Relevant perturbations
! !!
! !!
Topological symmetryen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
2irrelevant operatorsrelevant operators
!
!!
staggered Symmetry operator!x!
1, . . . , x!L|Y |x1, . . . , xL"
=L!
i=1
"F
x!i+1
!xi!
#x!i
xi+1
[H,Y ] = 0
with eigenvaluesS!!flux = ! Sno flux = !!!1
prohibited bytranslational symmetry
prohibited bytopological symmetry
!
no flux -flux!
![Page 34: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/34.jpg)
!!!
!!!
Topological stabilityen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
2irrelevant operatorsrelevant operators
!
!!
prohibited bytranslational symmetry
prohibited bytopological symmetry
The criticality of the chain is protected by additional topological symmetry.
Is this special to ?SU(2)3
Local perturbations do not gap the system.
![Page 35: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/35.jpg)
Topological stability for all kMinimal models have a coset description:
Mk = su(2)1!su(2)k!1su(2)k
![Page 36: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/36.jpg)
Topological stability for all kMinimal models have a coset description:
Mk = su(2)1!su(2)k!1su(2)k
The coset field inherit the topological sector via the character decomposition:
!(1)! !(k!1)
j2=
!j1
Bj1j2
!(k)j1
" = j1 ! j2 mod 1
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Topological stability for all kMinimal models have a coset description:
Mk = su(2)1!su(2)k!1su(2)k
The coset field inherit the topological sector via the character decomposition:
!(1)! !(k!1)
j2=
!j1
Bj1j2
!(k)j1
" = j1 ! j2 mod 1
The predictions have been checked numerically, and both the AFM and FM spin-1/2 chains are stable!
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Majumdar-Ghosh chain
! ! !!!
! ! !!!
!
! !
!
!
!
!!
!!
a) ! ! !!!! ! ! !!b)
!
! !
! !!!
!
!
!
! ! !
! !
!
!c)
Consider a (competing) three anyon-fusion term➠ neither translational nor topological symmetry are broken
SU(2)3 anyons
![Page 39: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/39.jpg)
Majumdar-Ghosh chain
! ! !!!
! ! !!!
!
! !
!
!
!
!!
!!
a) ! ! !!!! ! ! !!b)
!
! !
! !!!
!
!
!
! ! !
! !
!
!c)
Consider a (competing) three anyon-fusion term➠ neither translational nor topological symmetry are broken
SU(2) spins
HMG = J!
i
!T 2i!1,i,i+1
= J!
i
!Si!Si+1 +
J
2
!
i
!Si!Si+2
SU(2)3 anyons
![Page 40: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/40.jpg)
Majumdar-Ghosh chain
! ! !!!
! ! !!!
!
! !
!
!
!
!!
!!
a) ! ! !!!! ! ! !!b)
!
! !
! !!!
!
!
!
! ! !
! !
!
!c)
Consider a (competing) three anyon-fusion term➠ neither translational nor topological symmetry are broken
SU(2) spins
HMG = J!
i
!T 2i!1,i,i+1
= J!
i
!Si!Si+1 +
J
2
!
i
!Si!Si+2
SU(2)3 anyons
Hi = P!1!1 + P1!1! + P!!!1 + P1!!! + 2!!2P!!!! +!!1 (P!1!! + P!!1! ) ! !!2
!|""1"" #"1"" | + h.c.
"+
!!5/2 (|"1""" #"""" | + |""1"" #"""" |+h.c.)
![Page 41: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/41.jpg)
Phase diagram
tricritical Ising
3-state Potts
c = 7/10
c = 4/5
Majumdar-Ghosh
Z4-phase
tan ! = "/23-state Potts, S3-symmetry
tetracritical Isingc = 4/5
c = 4/5
exactsolution
exactsolution
incommensurate phase
! ! !!!
! ! !!!
!
! !
!
!
!
!!
!!
a) ! ! !!!! ! ! !!b)
!
! !
! !!!
!
!
!
! ! !
! !
!
!c)
J3 = sin !
J1 = cos !
S. Trebst et. al., PRL (2008)
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S3-symmetric point
0 0.1 0.2 0.3 0.4 0.5momentum K [2π/L]
0 0
1 1
2 2
3 3
4 4
5 5
6 6
resc
aled
ene
rgy
E(K)
2/15
4/5
4/3 4/3
2+2/15
2+4/5
2+4/3
2+2/15
2+4/5
2+4/3
4+2/15 4+2/15
4+4/5
1+2/15
1+4/5
1+4/3
0
1+2/15
1+4/5
1+4/3
3+2/15
3+4/5
3+4/3
3+2/15
3+4/5
3+4/3
2+2/15
2+4/5
2+4/3
4+2/15
4+4/5
2+0
4+0
2+2/15
2+4/5
2+4/3
4+2/15
4+4/54+4/5
4+4/3 4+4/3
3+03+2/15 3+2/15
3+0
3+4/5
3+4/3
3+4/5
4+4/3
5+05+2/15 5+2/15
5+0
L = 36primary fields descendants
2/15
I
IIIII
Conformal field theory: parafermions with central charge .c = 4/5
! = 0.176"
! ! !!!
! ! !!!
!
! !
!
!
!
!!
!!
a) ! ! !!!! ! ! !!b)
!
! !
! !!!
!
!
!
! ! !
! !
!
!c)
![Page 43: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/43.jpg)
spin-1 chainsu(2)5Take the even sector of : su(2)5 1 ! "
!! ! = 1 + " !! " = ! + " " ! " = 1 + ! + "
Use the particles as the building block of the chain, and define the hamiltonian:
!
H = sin !P! ! cos !P"
P! , P" project on the channel. ! , "
![Page 44: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/44.jpg)
su(2)5spin-1 chain phase diagram
C. Gils et. al., in preparation
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c=80/71 super cft spectrum
0 1 2 3Momentum kx
0
0.5
1
1.5
2
2.5
Resc
aled
ene
rgy
E(k x)
c=81/70 CFTL=18
θ=1.1872π
3/350
8/35
3/7
29/2809/56
27/40
73/56
269/2801+3/351+8/35
1+3/7
16/7
9/5
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Conclusions• Topological quantum spin chains have been
studied analytically and numerically
• They support different phases: (topologically) protected criticality, gapped phases etc.
• Many aspects remain open:• 2-d systems• different types of disorder (some progress)• higher genus surfaces
![Page 47: Eddy Ardonne · 2017. 6. 6. · The golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1 × τ = τ Hilbert space: |x 1, x 2, x 3, . . .! τ τ τ τ τ τ x 1 x 2 x 3.](https://reader035.fdocuments.us/reader035/viewer/2022071007/5fc477a8975d0206135e93c7/html5/thumbnails/47.jpg)
Nordita program
August 17 - September 11, 2009:
Quantum Hall Physics - Novel systems and applications