Competing phases of XXZ spin chain with frustration

Akira Furusaki (RIKEN)

July 10, 2011Symposium on Theoretical and

Mathematical Physics, The Euler International Mathematical Institute

Collaborators:Shunsuke Furukawa (U. Tokyo)Toshiya Hikihara (Gunma U.)Shigeki Onoda (RIKEN)Masahiro Sato (Aoyama Gakuin U.)

Thanks: Sergei Lukyanov (Rutgers)

Outline

1. Introduction: frustrated spin-1/2 J1-J2 XXZ chain

2. XXZ chain (J2=0): review of bosonization approach

3. Phase diagram of J1-J2 XXZ spin chain

a. J1>0 (antiferromagnetic)

b. J2<0 (ferromagnetic)

J1

J2> 0(AF)

If J2 is antiferromagnetic, spins are frustrated regardless of the sign of J1.

J1-J2 spin chain is the simplest spin model with frustration.

frustrated spin-1/2 J1-J2 chain

1,2

x x y y z zn j j n j j n j j n

n j

H J S S S S S S

Materials: Quasi-1D cuprates (multi-ferroics)

RbRb22CuCu22MoMo33OO1212

LiCuVOLiCuVO44 LiCuLiCu22OO22 NaCuNaCu22OO22

Cu(3dx2-y2)

O(2px) J1 J2

PbCuSOPbCuSO44(OH)(OH)22

b

a

c

Cu2+ spin S=1/2

Quasi-1D spin-1/2 frustrated magnets with ferro J1

J1 <0

J2 >0

Cu

Ob

a

c

0-1-2-3-4

LiCuVO4LiCu2O2Li2ZrCuO4

Enderle et al.Europhys.Lett.,2005

Masuda et al.PRL,2004; PRB,2005

Drechsler et al.PRL,2007

CuO2 chain: edge-sharing network

ferromagnetic J1

(Kanamori-Goodenough rule)

Multiferroicity

Observation of chiral ordering through electric polarization P

Seki et al.,PRL, 2008(LiCu2O2)

Model

J1

J2 (>0, antiferro)

x

yeasy-plane anisotropy

-4 4

Ferro AntiferroSpin spiral

0

finite chirality

Classical ground state

Frustration occurs when J2>0, irrespective of the sign of J1.

Frustrated spin-1/2 J1-J2 XXZ chain

The phase diagram is symmetric. J1>0 and J1<0 are equivalent under

pitch angle (applies only in the classical case)

Quantum case S=1/2

Antiferromagnetic case (J1>0,J2>0) is well understood.

- Singlet dimer order is stabilized (J2/J1>0.24).

Haldane, PRB1982Nomura & Okamoto, J.Phys.A 1994White & Affleck, PRB 1996Eggert, PRB 1996

- Vector chiral ordered phase (quantum remnant of the spiral phase) is found for small Nersesyan,Gogolin,& Essler, PRL 1998

Hikihara,Kaburagi,& Kawamura, PRB 2001

Classical spiral (chiral) order is destroyed by strong quantum fluctuations in 1D.

1 2 0.8, 0.2J J

Previous study of spin-1/2 J1-J2 chain Ground-state phase diagram for AF-J1 case

Two decoupled J2 chains

Majumdar-Ghosh line

T. Hikihara, M. Kaburagi, and H. Kawamura, PRB (2001), etc. K. Okamoto and K. Nomura, Phys. Lett. A (1992).

J1 chain

Quantum case S=1/2

Antiferromagnetic case (J1>0,J2>0) is well understood.

- Singlet dimer order is stabilized (J2/J1>0.24).

Haldane, PRB1982Nomura & Okamoto, J.Phys.A 1994White & Affleck, PRB 1996Eggert, PRB 1996

- Vector chiral ordered phase (quantum remnant of the spiral phase) is found for small Nersesyan,Gogolin,& Essler, PRL 1998

Hikihara,Kaburagi,& Kawamura, PRB 2001

Classical spiral (chiral) order is destroyed by strong quantum fluctuations in 1D.

The ferromagnetic-J1 case (J1<0,J2>0) is less understood.

1 2 0.8, 0.2J J

Goal: to determine the ground-state phase diagram

Our strategy

• perturbative RG analysis around J1=0 or J2=0.

XXZ spin chain: exactly solvable low-energy effective theory (bosonization)

• numerical methods

density matrix renormalization group (DMRG) time evolving block decimation for infinite system (iTEBD)

XXZ spin chain: brief review mostly standard textbook material,

plus some relatively new developments

XXZ spin chain

• Exactly solvable: Bethe ansatz gapless phase

1 1 1x x y y z zj j j j j jH S S S S S S

1 1

-1 １

antiferromagneticIsing order

ferromagneticIsing order

Tomonaga-Luttingerliquid

energy gap energy gap

gapless excitations

power-law correlations

LRO LRO

• Effective field theory: bosonization

2 2

eff

1cos 16

2 x x

vH dx K

K

1cos : 1 at 0, at 1

2 2K K

K

11 11 cos

2K

sin

2 1v

Luther, Peschel

, yx y i x y )(),( xx : bosonic field

cos 16 is

relevant for

irrelevant for

1

1

For 1 marginally irrelevant for 1

( )0 1

1

( 1) sin 4 ( )

1( ) ( 1) sin 4 ( )

i l ll

z ll x

S e b b l

S l a l

2 21

2 x x

vH dx K

K

rA

rSS

rA

rASS

rzz

rz

xr

xxr

x

)1(

4

1

1)1(

1220

/1100

In the critical phase

The cosine term is irrelevant in the low-energy limit

: Gaussian model

)()(),( yxiyx y )(),( xx : bosonic field

: exactly determined by Bethe ansatz

not directly obtainedfrom Bethe ansatz

,v

1 0 0 1 0 1, , , , , ,...z x xA A A a b b

Spin operators

1 1

1

2K

T. Hikihara & AF (1998)

22

0

2 2 sinh1exp

sinh cosh 14 1 2 28 1t

x

tdtA e

t t t

rA

rSS

rA

rASS

rzz

rz

xr

xxr

x

)1(

4

1

1)1(

1220

/1100

1

22

0

sinh 2 12 22 2 1exp

sinh cosh 14 1 2 2t

z

tdtA e

t t t

Lukyanov &Zamolodchikov (1997)

Lukyanov (1998)

more recently,Maillet et al.

zA

exact

numerics

from S. Lukyanov, arXiv:cond-mat/9809254

dimer correlation

(staggered) dimer correlation: as important as the spin correlations

NN bond (energy) operators

zl

zl

zelllle SSlOSSSSlO 111 )(,

4

1)(

0 1

2 2

( ) 1 cos 4 ( )

( ) ( )

l

e l

x l x l

O l c c x

c x c x

21,, lxz l( )

unknown: we have determined numerically using DMRG

0 , , ,...c c c

[cf. Eggert-Affleck (1992)]

scaling dimension = 1/2 at AF Heisenberg point

1c

known (can be obtained from energy density etc.)

Analytic results for uniform components

Uniform part of dimer operators = energy density in uniform chain

We can evaluate the coefficients of the uniform comp. from the exact results of the energy density

2210

2210

1

sin

cos

4

1

2

cos

sin2

1

IIc

IIc

z

0 22

01

1coshsinh

cosh

1coshsinh

sinh

tt

ttdtI

tt

tdtI

sin14

cos1sin

sin116

122sin

222

222

zc

c

14

1

18

cos

22

22

zc

c

Dimer operators in finite open chain

Dimer order induced at open boundaries penetrates into bulk decaying algebraically

Dirichlet b.c. for boson field : 0)1()0( LOpen boundary condition

mode expansion

nnn

n

nnn

n

n

xqix

n

xq

L

xx

10

10

cos)(

sin1

)1()(

222

2

2

2110

)12(

1

)1(12

)12(

1)(

lfc

Lc

lfcclO

l

e

)1(2sin

)1(2)(

L

xLxf

DMRG results

Calculate the local dimer operator for a finite open chain using DMRG

fit the data to the form obtained by bosonization to determine

)(lOe

1c

excellent agreement between DMRG data and bosonization forms

Numerics (DMRG)

Staggered part of the dimer operators

211

222

2

20S

)12(

1

)12(

1

)1(12)()(

lfc

lfc

LcclOlO

l

ee

)1(2sin

)1(2)(

L

xLxf

zl

zl

zelllle SSlOSSSSlO 111 )(,

4

1)(

coefficient1c

Exact formulas for are not known.1c

Hikihara, AF & Lukyanov, unpublished

• Effective field theory

2 2

eff

1cos 16

2 x x

vH dx K

K

222

1 2 214sin

4 1 1 2 2

Lukyanov & ZamolodchikovNPB (1997)

1st order perturbation in gives the leading boundary contribution tofree energy of semi-infinite (or finite) spin chains for 1 2 1

cos 16 0 for Dirichlet b.c., 0 0x

Boundary specific heat: 2

21

1 1 3 2 1b

TC

v v

Boundary susceptibility:

2 3 1

22 3

2

1 3 2 2 ' 1 2

4 2 2 1b

T

v v

• Boundary energy of open XXZ chain

:),( LmE

L spins

1

)()()(),( 2

10

L

mmmLLmE

2

212

111 2

)2()2()0()(

m

mhm

1

11

22

11

sin2

3

1

h

arccos1

1

13

2

AF & T. Hikihara, PRB 69, 094429 (2004)

lowest energy of a finite open chain with totzS mL

J1-J2 spin chain with antiferromagnetic J1

Ground-state phase diagram for AF-J1 case

Majumdar-Ghosh line

J1 chainTwo decoupled J2 chains

dimer phase 1 1 2 0j j j jS S S S

1 0 1 1 cos 4 ( )l

j j lS S c c x

2 2

eff

1cos 16

2 x x

vH dx K

K

J2>0 changes and scaling dimension of cos 16 .

If is relevant and ,

then is pinned at .

0 cos 16 x 0 or 4.

cos 4 0

dimer LRO

If is relevant and ,

then is pinned at .

0 cos 16 x 4 or 4.

sin 4 0

Neel LRO

Haldane ‘82

White & Affleck ’96……..

Ground-state phase diagram for AF-J1 case

Majumdar-Ghosh line

J1 chainTwo decoupled J2 chains

Perturbation around J1=0Two decoupled J2 chains

2 2

eff1,2

1cos 16

2 x x

vH dx K

K

( ), 0 1

, 1

( 1) sin 4 ( )

1( ) ( 1) sin 4 ( )

i l ll

z ll x

S e b b l

S l a l

1 1, 1, 1 2, 1 2h.c. cos 8 cos 8 sin 8j j j xJ S S S g g

dimer order vector chiral order 1 2 1 2

1 1,

2 2

When relevant → sin 8 0, 0d

dx

Characteristics of the vector chiral stateCharacteristics of the vector chiral state

power-law decay, incommensurate

2 sin 8d

gdx

Nersesyan-Gogolin-Essler (1998)

Vector chiral order Vector chiral order

A quantum counterpart of the classical helical stateA quantum counterpart of the classical helical state

SxSx & SxSy spin correlation

Opposite sign

Vector chiral phaseVector chiral phase

no net spin current flow

02 )2(2

)1(1 ll JJ

(1)1 ~ sin 8

z

l l ls s

dx

dss zlll

~2)2(

qrrssqrrss Kyr

xKxr

x sin~ cos~ 41

04

1

0

(1) (2), , or ,

p-type nematic Andreev-Grishchuk (1984)

J1-J2 spin chain withferromagnetic J1

Phase diagram & chiral order parameter

ferromagnetic J1 antiferromagnetic J1

The vector chiral order phase is large in the ferromagnetic J1 caseand extends up to the vicinity of the isotropic case 1.

Perturbation around J1=0Two decoupled J2 chains

2 2

eff1,2

1cos 16

2 x x

vH dx K

K

( ), 0 1

, 1

( 1) sin 4 ( )

1( ) ( 1) sin 4 ( )

i l ll

z ll x

S e b b l

S l a l

1 1, 1, 1 2, 1 2h.c. cos 8 cos 8 sin 8j j j xJ S S S g g

dimer order vector chiral order 1 2 1 2

1 1,

2 2

2 2

1 1, 1, 1 2, 1z z zj j j x xJ S S S J

2 1JK K Kv

dimension

1

1 : 2 2g K K

1

2 : 1 2g K

0 0

11 at 0, at 1

2K K

Phase diagram & chiral order parameter

ferromagnetic J1 antiferromagnetic J1

1 sin 8d

Jdx

1 ~ sin 8 ,z

l ls s 2 ~

z

l l

ds s

dx

Ground-state phase diagram for Ferro-J1 case

Two decoupled J2 chainsJ1 chain

LiCu2O2 LiCuVO4

PbCuSO4(OH)2 NaCu2O2 Rb2Cu2Mo3O12

Li2ZrCuO4

Sine-Gordon model for spin-1/2 J1-J2 XXZ chain with ferromagnetic coupling J1

We begin with the J2=0 limit. J1 chain

Ferromagnetic

Easy-plane

Effective Hamiltonian (sine-Gordon model)

TL-liquid (free-boson) part irrelevant perturbation

TL-liquid parameter velocity

Ferromagnetic SU(2) Heisenberg

2 2

eff

1cos 16

2 x x

vH dx K

K

1cos , 0

2

1

2K

1

sin

2 1v J

Spin and dimer operators

If the cosine term becomes relevant, then

Neel order

dimer order

J1 chainBKT-type RG equation

11 sin 4

jzj xS

0 11 sin 4ji

jS e b b

1 1 1 cos 4j

j j j jS S S S c

cos 16

Exact coupling constant in the J1 chain (J2=0)

It vanishes and changes its sign at i.e.,

Relation between and excitation gapsof finite-size systems from perturbation theory for cosine term

estimated bynumerical diagonalization

Exact value is known in J1 chain

S. Lukyanov, Nucl. Phys. B (1998).

“Dimer” gap “Neel” gap

We can check the position of =0 from numerical-diagonalization result.

J2=0

This relation is stable against perturbations conserving symmetries.

Generally the exact value of is not known in the presence of such perturbations (J2).

However, the position of =0 is determined by the equation which can be numerically evaluated.

J2 perturbation makes the term relevant .Neel and dimer phases are expected to emerge.

Neel order

dimer orderGaussian phase transition point (c=1)

Phase diagram and Neel/dimer order parameters Ground-state phase diagram of easy-plane anisotropic J1-J2 chain

J1 chain

Curves of =0

Irrelevant relevant

dimer<0

Neel>0

dimer<0

Neel>0

Furukawa, Sato & AFPRB 81, 094410 (2010)

Direct calculation of order parameters from iTEBD method

>0<0 <0

XY component of dimer Z component of dimer

Neel operator (Z component of spin)

Neel phase The emergence of the Neel phase is against our intuition:ferromagnetic & easy-plane anisotropy

Spin correlation functions in the Neel phase

Short-range behavior is different from that of the standard Neel order.

1 0J 1.

Dimer phase

FM-J1 case AF-J1 case

dimer phase in the AF-J1 region

dimer phase in the FM-J1 region

Neel

On the XY line (=0)

“triplet” dimer “singlet” dimer J1-J2 XY chain with FM J1 J1-J2 XY chain with AF J1

rotation at every even site

Dimer order parameter

Different dimer order

dimer

1 2 3

dimer

1 2 0S S

zoom

weak dimer order

string order parameterdimer order parameter

1

2 1 2 2 1 2 2 1 21

expk

j j l l k kl j

S S i S S S S

2 1,2 : dimerized bondj j

1 2 3

long-range string order

2 2 1The string op is short-ranged for .j jS S

Sato, Furukawa, Onoda & AFMod. Phys. Lett. 25, 901 (2011)

Summaryferromagnetic J1 antiferromagnetic J1

2 1J JSato, Furukawa, Onoda & AFMod. Phys. Lett. 25, 901 (2011)

Furukawa, Sato & AFPRB 81, 094410 (2010)

Construction of ground-state wave function of J1-J2 chain

Neel order!

projection to single-spin space

a trimer state in every triangle

multi-magnon instability

Phase diagram in magnetic field (h>0, J1<0, J2>0, )

Vector-chiral phase

Antiferro-triatic Antiferro-nematic

k (2)

k (1)

k (2)

SDW2

SDW2SDW3

SDW3

Hikihara, Kecke, Momoi & AF PRB 78, 144404 (2008);Sudan et al. PRB 80, 140402 (2009)

Nematic

Nematic (IC)

1

1

J1-J2 Heisenberg spin chain in magnetic field

J1<0 J1>0J2>0

Okunishi & Tonegawa (2003);McCulloch et al. (2008);Okunishi (2008);Hikihara, Momoi, AF, Kawamura (2010)