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
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
Motivation• What’s a spin chain?• Use a modular tensor category/quantum group!
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
The Heisenberg model
1! 1 = 11! ! = !
! ! ! = 1 + !
Fusion rules for SU(2)3 SU(2) spins
12! 1
2= 0 + 1singlet
triplet
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
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)
The golden chain
! ! ! !! !
The golden chain
! ! ! !! !
! ! ! ! ! !
!
!
1
. . .
1! ! = !
The golden chain
! ! ! !! !
! ! ! ! ! !
!
!
1
. . .
1! ! = !
Hilbert space: |x1, x2, x3, . . .!
! ! ! ! !
! . . .x1 x2 x3
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.
The golden chainWe want to construct a local Hamiltonian .H =
!
i
Hi
The golden chainWe want to construct a local Hamiltonian .H =
!
i
Hi
! !
x1 x2 x3
The golden chainWe want to construct a local Hamiltonian .H =
!
i
Hi
! !
x1 x2 x3
! !
x1 x3
x̃2
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
"
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
"
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:
The golden chainHi = Fi !1
i FiLocal Hamiltonian:
Hi = !!
!!2 !!3/2
!!3/2 !!1
"
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)
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
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
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
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 = !
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
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
Topological symmetryen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
!
!!
Topological symmetryen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
2irrelevant operatorsrelevant operators
!
!!
Relevant perturbations
! !!
! !!
Topological symmetryen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
2irrelevant operatorsrelevant operators
!
!!
Relevant perturbations
! !!
! !!
Topological symmetryen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
2irrelevant operatorsrelevant operators
!
!!
staggered
Relevant perturbations
! !!
! !!
Topological symmetryen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
2irrelevant operatorsrelevant operators
!
!!
staggered
prohibited bytranslational symmetry
Relevant perturbations
! !!
! !!
Topological symmetryen
ergy
E
(K)
momentum K0 !I
!
!!
!!!
2irrelevant operatorsrelevant operators
!
!!
staggered
prohibited bytranslational symmetry
!
no flux -flux!
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!
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!
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!
!!!
!!!
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.
Topological stability for all kMinimal models have a coset description:
Mk = su(2)1!su(2)k!1su(2)k
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
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!
Majumdar-Ghosh chain
! ! !!!
! ! !!!
!
! !
!
!
!
!!
!!
a) ! ! !!!! ! ! !!b)
!
! !
! !!!
!
!
!
! ! !
! !
!
!c)
Consider a (competing) three anyon-fusion term➠ neither translational nor topological symmetry are broken
SU(2)3 anyons
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
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.)
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)
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)
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. ! , "
su(2)5spin-1 chain phase diagram
C. Gils et. al., in preparation
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
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
Nordita program
August 17 - September 11, 2009:
Quantum Hall Physics - Novel systems and applications