a. klumper- spectral properties of quantum spin chains and chalker-coddington-networks of...

8/3/2019 A. Klumper- Spectral properties of quantum spin chains and Chalker-Coddington-networks of Temperley-Lieb type

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Spectral properties of quantum spin chains and

Chalker-Coddington-networks of Temperley-Liebtype

A. Klumper

University of Wuppertal - Department of Physics

with: Aufgebauer, Brockmann, Nuding, Sedrakyan

18.06.10
http://find/http://goback/


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Contents

1 Spin models of Temperley-Lieb-TypeBiquadratic Spin-1 Quantum Chain + generalizations

Chalker-Coddington network

TL-reference model: s= 1/2 Heisenberg chain

2 Open boundary conditionsConstruction of sectors

Multiplicities ( thermodynamics for h= 0)3 Periodic boundary conditions

Construction of sectors ( twist, exponents, thermodynamics)Multiplicities

4 Summary

A. Klumper (University of Wuppertal) TL-equivalence for finite T, h 28.07.09 2 / 32


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Spin models of Temperley-Lieb-Type Biquadratic Spin-1 Quantum Chain + generalizations

Biquadratic Spin-1 Quantum Chain

Most general su(2) isotropic quantum s= 1 spin chain withnearest-neighbour interaction:

H=L

j=1 cos SjSj+1 + sin

SjSj+12

Special/integrable points: cot = 3,

1, 0

Here of particular interest:1

0

cos

sin

E = E , E : 8 + 1

su(3): 3 x 3

E = E , E : 6 + 3

E , E , E : 1 + 3 + 5

0 2 1

1 2

2 0

su(3): 3 x 3

purely biquadratic case, representation of Temperley-Lieb algebra

Parkinson 87/88; AK 89, 93; Barber, Batchelor 89; Albertini 00; Alcaraz, Malvezzi 92; Koberle, Lima-Santos 94, 96; Kulish 2003; ...



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Spin models of Temperley-Lieb type

Hamiltonian H= i=1 bi Hilbert space HN = (V)N

local interactions proportional to projectors onto su(2)-singletsbi = | |i,i+1. For V = C3

| = ,=1,0,1

, |, , = 0 0 1

0 1 01 0 0

, + = idaccidentally higher su(3) symmetry of type [3] [3]:

bi commutes with g g g g... where g := (g1

)T

for any invertible 3 3 matrix g, (if g is unitary: g= g).Generalization to dim(V) = 2s+ 1 =: n for arbitrary spin

(s= 1/2, 1, 3/2, 2...) and su(n) symmetry.A. Klumper (University of Wuppertal) TL-equivalence for finite T, h 28.07.09 4 / 32


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Temperley-Lieb relations

TLN(): (open boundary)Generators bi for i= 1,...,N 1 with relations:

b2i = bi for i= 1, 2,..., N 1

bi bi1 bi = bi

bi bj = bj bi for |ij| > 1PTLN(): (closed boundary) additional generator bN:

b2N = bN

bjbNbj = bj; bNbjbN = bN for j= 1,N 1bjbN = bNbj for j= 1,N 1


S i d l f T l Li b T Bi d i S i Q Ch i li i


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graphical notation for bi = id(i1) | |i,i+1 id(Ni1)

bi = id(i1)

id(Ni1)

Temperley-Lieb condition : bibi+1bi = bi

= = id

Temperley-Lieb condition : b2i = bi

=


S i d l f T l Li b T Ch lk C ddi t t k


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Spin models of Temperley-Lieb-Type Chalker-Coddington network

Chalker-Coddington network

3-state model with staggered su(2

|1) symmetry for spin quantum Hall

effect

2 2= (1t ) + t

2 2= (1t ) + t

A A

B B

= 1 1 +1 = 1

= diag(+1,1,+1)

= = = diag(+1,+1,+1)

boundary conditions/super-trace: closed loops evaluate to 1

Gruzberg, Ludwig, Read 1999


Spin models of Temperley Lieb Type TL reference model: s 1/2 Heisenberg chain


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Spin models of Temperley-Lieb-Type TL-reference model: s= 1/2 Heisenberg chain

Temperley-Lieb Models: Summary of Examples

i) Biquadratic chains and generalizations

V V =2sJ=0

VJ, | su(2) singlet, = dim(V)singlet of two spin-s reps quarkanti-quark singlet of su(n),n= 2s+ 1

ii) generalizing model (arbitrary anisotropy and spin)

| Uq(su(2)) singlet ; =s

l=s

(q2)l for q Rhere most important: q= 1, i su(n), su(s+ 1, s) = n, 1.

iii) reference system: XXZ-model s= 1/2 Bethe-Ansatz| | =

q1/2 |+ q1/2 |+

q1/2 +| q1/2 +|

hermitian for q

R, TL-parameter = q+ q1, interaction = 2


Spin models of Temperley Lieb Type TL reference model: s = 1/2 Heisenberg chain


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Temperley-Lieb Equivalence at T> 0?

Two different models being representations of the same TL-algebra

(same !) have same spectrum, but multiplicities may differ!

TL equivalent are:

3-state biquadratic chain and 2-state XXZ chain with = 3/2

Thermodynamics differ! Aufgebauer, AK 2010

0 1 2 3 4 5T

0

0,1

0,2

0,3

0,4

0,5

0,6

specif

ic

heat

antiferromagnetic case: S=1 and XXZ

0 1 2 3 4 5T

0

0,1

0,2

0,3

0,4

specif

ic

heat

ferromagnetic case: S=1 and XXZ


Spin models of Temperley-Lieb-Type TL-reference model: s = 1/2 Heisenberg chain


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Temperley-Lieb Equivalence and CFT?

Two different models being representations of the same TL-algebra

(same !) have same spectrum for open boundary conditions, butcentral charge and scaling dimensions may differ!

TL equivalent are:

3-state su(2

|1) invariant chain and 2-state XXZ chain with = 1/2

Scaling dimensions for = cos Heisenberg chain

x =1 /

2S2 +

1

2(1 /)m2

1

3S2 +3

4m2

with central charge c= 1.

su(2|1) quantum chain has different characteristics, in particular c= 0.


Open boundary conditions Construction of sectors


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Open boundary conditions

Decomposition of Hilbert space into irreducibles of TLN() (generic )

Equiclasses of irred. indexed

by Nand k ; 0 k N2 O(N, k)decomposition rule:

O(N, k) TLN1()

(P. Martin: Potts models and related problems in statistical mechanics)




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Reference States alias 1d-TL-reps

Bethe-ansatz reference states (by definition) are annihilated by all bis

N := { | bi = 0 : 1 i N 1} O(N, 0)

General representation isomorphic to O(N, k)

constructed from reference state on chain with N 2k sites and kmany local TL-singlets

v1 := k with N2kspans TLN-invariant subspace

multiplicity of O(N, k) in HN is equal to dim(N2k)




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p y

example k= 1 N= 4

b2

b1

b2( )b3

b2

b3 b2( )

b2b1

b3b2

orthogonal-basis:

v1 = , v2 = 2 1

b2( ) 1

( )

v3 = 2

13 2

b3 v

2 2 1v

2polynomials : P0() = 1, P1() = , Pk() = Pk1 Pk2


Open boundary conditions Multiplicities ( thermodynamics for h= 0)


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p y p ( y )

Dimension of N (sits in N1) is kernel of

map bN1 : N1 h N2

which is surjectiv!Proof: For given N2 consider the k= 1 TLN-sector

vN1()bN1 P1PN2

Dimension formula for kernel and image yields recursion relation

dim(N) = ndim(N1) dim(N2) n= dim(h)

dim(N) = PN(n) =n+n2 4N+1 nn2 4N+1

2N+1n2 4

Multiplicities for open boundaries derived earlier by double centralizer

property Kulish, Manojlovic and Nagy 2008.


Periodic boundary conditions Construction of sectors ( twist, exponents, thermodynamics)


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Temperley-Lieb Equivalence of Partition Functions

The spectra of two Hamiltonian in equivalent representations O(N, k)

are identical. Asymptotically, there are zN2k

many of suchrepresentations/sectors, with fugacity

z=

n+

n2 42

We sum over all sectors and in each one over all energies

Z(T,h= 0) =N

k=0 allEkeEk zN2k.

which gives the grand-canonical partition function of the XXZ

reference model with magnetic field

N2k = M Z(T, h= 0) = Tre(HXXZ(T ln z)M) = ZXXZ(T,T ln z)A. Klumper (University of Wuppertal) TL-equivalence for finite T, h 28.07.09 15 / 32



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Antiferromagnetic Biquadratic Spin-1 Quantum Chain:

Specific Heat and Entropy

0 1 2 3 4 5T

0

0,1

0,2

0,3

0,4

0,5

0,6

specific

heat


0 1 2 3 4 5T

0

0,5

1

entropy


(Note the thermodynamically activated behaviour with very small gap)




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Ferromagnetic Biquadratic Spin-1 Quantum Chain:

Specific Heat and Entropy

0 1 2 3 4 5T

0

0,1

0,2

0,3

0,4

spe

cific

heat


0 1 2 3 4 5T

0

0,5

1

entropy


There is residual entropy!

S(T = 0) = ln z= ln

n+

n2 42

n=3 0.9624...

and activated behaviour with noticeable gap.A. Klumper (University of Wuppertal) TL-equivalence for finite T, h 28.07.09 17 / 32



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Partition Function with external magnetic field

lattice path integral approach classical model based on TL

N

L

with periodic boundary conditions of column-to-column transfer matrix

magnetic field leads to twisted boundary conditions




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Periodic and twisted boundary conditions

Consider models like biquadratic chain with

= id

on periodically closed chain.

For k= 1 :

N2 =bNbN1 b2 N2 =

b1

(bNbN1

b2

N2

) =

= T2(N2) Take pN2 and eigenstate of T = eiPMomentum eigenvalue of translationally invariant reference state

enters!A. Klumper (University of Wuppertal) TL-equivalence for finite T, h 28.07.09 19 / 32



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Periodic and twisted boundary conditions

Consequence:

TL-equivalence of periodic biquadratic chain to twisted XXZ

Goal: Decompose the Hilbert space into PTL-representations (see

Martin, Saleur 93) isomorphic to twisted XXZ-PTLN-representations.

HN =

N1i=1

12S+i Si+1 + Si S+i+1 + (q+ q1)Szi Szi+1

+1

2

e2i S+NS

1 + e

2i SNS+1 + (q+ q

1)SzNSz1

.

bN acts as projector | | in hN h1| = ei q1/2 |+ ei q1/2 |+

PTLN()-relations fulfilled for all

C




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Bethe-Ansatz for twisted XXZ in Sz =constant:

eigenstates in PTLN-subrepresentation

| |+(N2) b2 |+ | |+(N3)

graphical notation:

N2 =

b2

b1 bNbN1 b2 | |+(N2) = (1)N ei2 | |+(N2)




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Construction for arbitrary k

p

N2k T = ei

(b1bN b2k+2)(b3b2) (b2k+1b2k)k = ()N(e2i) k graphically:

b2b3 b2k

b2k+1 b1b2k+2

sector P(N, k, ):k construct PTLN-invariant subspace.dim(P(N, k, )) Nk




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Special extremal case P(N,N/2, )

Even N

Sector k= N2 : appended reference state trivial imaginary twist-angle, i.e. magnetic field

= i ln1

2

n+

n2 4

n= dim(h)

yields thermodynamics of biquadratic quantum spin chain for arbitrary

T and h.

Sectors with k< N2 : not needed.




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Temperley-Lieb Equivalence at finite T and h

The partition functions for the biquadratic chain and the XXZ chain are

identical Z(T,h) = ZXXZ(T,h) provided

1 + 2 cosh

h

T

= 2 cosh

h

T

0 1 2 3 4 5T

0

0,1

0,2

0,3

0,4

suscept

ibility


0 1 2 3 4 5T

0

1

2

3

4

5

6

suscep

tibility





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Chalker-Coddington network at critical point

Isotropic case tA = tB corresponds to integrable, critical vertex model.

transfer matrices

T and T1 2

Two commuting families of diagonal-to-diagonal transfer matrices, T1

and T2. Logarithmic derivative yields TL-Hamiltonian.A. Klumper (University of Wuppertal) TL-equivalence for finite T, h 28.07.09 25 / 32



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Chalker-Coddington network: conformal properties

Sector k= L2 :

appended reference state trivial, real twist-angle = /3.Central charge: c() = 1 6(/)

2

1 / 0

Sectors with k< L2

= l L 2k, l= 0, 1,..., L 2k 1,

(If magnetization of state is odd: l= 1/2, 3/2,..., L 2k 1/2.)

Scaling dimension: x= 4S2 + 9(m /)2 112

, s= Sm

Log-corrections: Ex,s ECFTx,s = 2v

(1/12 x)2 s2) logL

L3,

Aufgebauer, Brockmann, Nuding, AK, Sedrakyan (2010)A. Klumper (University of Wuppertal) TL-equivalence for finite T, h 28.07.09 26 / 32

Periodic boundary conditions Multiplicities

p


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Dimension of pN

Multiplicities of scaling dimension identical to dimension of space of

reference states.

pN is kernel of the map

bN : N N2 VN V1 VN1.

Again surjectivity can be proven, dimension formula:

dim(pN) = dim(N) dim(N2).

dim(pN)(n) =

n+

n2 42

N+

nn2 4

2

N.



p


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Spectrum of T in pN

T = eiP diagonalisable in p

NEigenvalues of momentum operator P:

2

Nl, l {0, 1,...,N 1}

(1) Spectrum of T in HN(2) Spectrum of T in pN, multiplicities of k-sectors

ad (i):If N is prime: multiplicities M(l= 0) = (2

N2)N + 2, M(l= 0) = (2

N2)N .

N= 6: M(0) = 14, M(1) = 9, M(2) = 11, M(3) = 10.




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(1) X = {xi} basis of V XN =xi1 xi2 xiN

T acts on XN partition of T-orbits(p): dimension of p-periodic subspace

p|N

(p) = (N) (N) := dim(HN) = nN

(N) = n|N

(n) Nn

: Mobius function

Multiplicity of momentum eigenvalues

M2Nl;HN =

n| (l,N)

(Nn)Nn

=

n| (l,N)

nN

n| N

n

(n) Nnn

.




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(2) Formula for multiplicities of momentum eigenvalues in pN similar

to above

M2N l; p

N = n| (l,N)(Nn)

Nn

= n| (l,N)

n

Nn| Nn

(n) Nnn .where now (N) = dim(pN).


Summary

S


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Summary

direct-sum decomposition of Hilbert space(1) open boundaries :

irreducible TLN-representations

(2) periodic boundaries :

PTLN-representations depend on additional parameter

XXZ-sectors with twist, spectrum via Bethe-Ansatz(3) applications:

thermodynamics (arbitrary T, h), conformal properties

(i) multiplicities known thermodynamics(ii) nongeneric Temperley-Lieb parameter?

(iii) correlation functions?


Summary

D iti f P i t O


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Decomposition of P into Os

decomposition rule for P(N, k, ) into irreducible TLN-representations:( > 2 and 2k< N)

P(N, k, ) TLN= kl=0O(N, k l)

dim(P(N, k, )) = Nkproof by construction of orthogonal basis

polynomials:

D,k () = Pk() Pk2() ()k (e2i + e2i)

2 cos(2)

for k 2.


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