integrable models and applications florence, 15-20 september 2003 g. morandi f. ortolani e....

Integrable Models and Applications Florence, 15-20 September 2003

G. Morandi

F. Ortolani

E. Ercolessi

C. Degli Esposti Boschi

F. Anfuso

S. Pasini

P. Pieri (Univ. Camerino)

L. Campos Venuti (Univ. Stuttgart)

Spin-1 Chains: Critical Properties and CFT

Spin-1 Chains: Critical Properties and CFT

Condensed Matter Theory Group in BolognaCondensed Matter Theory Group in Bologna

M. Roncaglia


• 1D arrangement of spins with interactions

• Exact solution for S=½ Heisenberg model (Bethe, 1931)

• From spin ½ to fermions (Jordan-Wigner, 1928)

• Quantum fluctuations prevent from LRO even at T=0

• Algebraic decay of correlation functions

• Gapless excitations carrying spin ½


S=½ XXZ model

][111

zi

zi

yi

yi

xi

xi

iH

FM AFM-1 0 1

BKTFDSD

221

dd2

1 xvv

xS

Gaussian model

Continuum limit:

= compactified boson

cos


Spin-1 models

N

ijiji SSSSJH

1

2 ])([

b

• b=1 BA integrable (Lai-Sutherland, 1974)

• b=1 BA integrable (Babudjian-Takhtajan, 1982)

SU(3)k=1SU(3)k=1

SU(2)k=2SU(2)k=2

WZNWmodel

Effective 3 free Majorana fermions, c=3/2

• b=1/3 Only GS is known, short ranged (AKLT)

000000Typical configurations

k

k

jlljString

SSiSO

1

1explim

|| jk

String orderparameter 0


Spin-1 Heisenberg model

• Integer-spin chains are gapped, with finite correlation length (Haldane’s conjecture,1983)

• Verified numerically and experimentally

• Mapping onto an O(3) nonlinear sigma model

• Is the Haldane state destroyed by anisotropies?

• Non vanishing string order 0String

O

No Neel order 0)1(lim kj

kjNeel

SSO || jk


N

i

z

i

z

i

z

i

y

i

y

i

x

i

x

i SSSSSSSJH1

2

111 ][ )(

Inclusion of anisotropies (D)

D

• Lower symmetry: from SU(2) to U(1) x Z2

i

zi

zTOT

SSQuantum number conserved

=Ising-like D = single ion

=spin1/2

• Bosonization on a two-leg ladder (Schulz, 1986)

S=1


Phase diagram

Large-D: unique GS, gapped 0

StringNeelOO

Neel: double GS, gapped 0, z

StringzNeel

OO

Haldane: unique GS, gapped 0;0

StringNeelOO

XY1: gapless with spin 1 excitations

XY2: gapless with spin 2 excitationsFrom W. Chen et al., PRB 67, 104401 (2003)


Critical phases First order

c=1/2

• Haldane-Neel (c=1/2)c=1

• Haldane-Large D + XY1 + XY2 (c=1)

c=3/2 ?

• Tricritical point SU(2)k=2 WZNW (c=3/2)


Haldane-Large D

• Classically, we have a planar phase for D>

jij

jj

jj

jjje

s

l

s

ls nznΩΩS ˆ;ˆ1ˆ)1(ˆ;ˆ

2

2

2

• Ansatz: inclusion of small fluctuations along z-axis

• Path integral approach with coherent states

Integration over the field l

Continuum limit treating (x, as a slow variable


Gaussian model (c=1)

g

vv

xS x1

;1

dd2

1 22

)1(2)1(21 DsvDs

g

Compactification radiuswhere

• The vertex operators Vmn have conformal dimensions

Ζ

nmKn

K

md

mn,

42

2

• The total magnetization coincides with the winding number m

K=g

• The mass generation term is )4cos( K =dual field


Finite size effects

• CFT on a finite size chain of length L

200

6L

cveL

E

• The excited states are related to the dimensions

)(200

rrdL

vEEmnmn

• label the secondary states ),( rr

• The levels Emn are calculated numerically

• Two parameters (v,K) to be fixed

• Only K determines the universality class BKTFDSD

1/2 1 2

K


Numerical calculations

• Exact diagonalization small systems

• Density matrix renormalization group “DMRG” (White,1992)

Usually works best for gapped systems, but very accurate if combined with CFT

High accuracy approx. method (<0.0001% on GS energy)

Multi-target algorithm that converges on the first low-lying states

Every calculation is done within a given sector of the total magnetization along z-axis

Reference: “Density Matrix Renormalization”, Peschel et al. Eds. (Springer, 1998)


Along the Haldane-Large D line (D=0.99)

m=1,n=0

m=0,n=0secondary

m=2,n=0

m=0, n=1

v(th) = 2.45, K(th) = 1.285v = 2.58, K = 1.328


D v(num) K(num) v(th) K(th)

(0.5, 0.65) 2.197 1.580 2.07 1.52

(1, 0.99) 2.588 1.328 2.45 1.285

(2.59, 2.3) 3.70 0.85 3.43 0.91

(3.2, 2.9) 4.45 0.526 3.77 0.83

Tricritical point ?

Conjecture: K=1/2 (SD) instead of K=1 (FD)

Towards the tricritical point


XY2 phase

• Perturbative approach for large negative D

i

zi

SDH20 ||

No 0’s

zi

ziiiii SSSSSSH

i111

1

2

1 Spin 1 Spin 1/2

zj

Sjj

SS

jj

SS

jj

SS

jj

SS

zj

2j

2j

2zj

21zj

21

00

][111

zi

zi

yi

yi

xi

xi

eff

iJH

1||4|;|/1 DDJ whereEffectiveS=1/2 XXZ

2nd order


eff = 0

K(num) = 0.998

Free Dirac


Open problems

• Mass generation away from the H-D critical line is described by a sine-Gordon theory. What is the string-order parameter in the continuum?

• What is the nature of the transition between the two critical phases XY1 and XY2?

• Tricritical point?


Conclusions

Reference:C. Degli Esposti Boschi, E. Ercolessi, F. Ortolani and M. Roncaglia, “On c=1 Critical Phases in Anisotropic Spin-1 Chains”, to appear in EPJ B, cond-mat/0307396.

• A combined use of both analytical and numerical calculations gives interesting quantitative results for the study of critical phases.

• We have found that the c=1 phases in the S=1 -D model are described by a pure Gaussian theory (without any orbifold)

• There are strong indications that the tricritical point at large D and does not belong to the universality class described by a SU(2)k=2 WZNW theory.

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