eddy ardonne · 2017. 6. 6. · the golden chain τ τ τ τ τ τ τ τ τ τ τ τ τ τ 1. . . 1...

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