Oct. 28, 2005 KIAS 1

One-dimensional approach tofrustrated magnets

Akira Furusaki (RIKEN)

Collaborators: Oleg Starykh (Utah)

Leon Balents (UCSB)PRB 72, 094416 (2005)

Oct. 28, 2005 KIAS 2

Phases of antiferromagnets Neel ordered phase

Valence Bond Solid

Spin liquid (RVB)

Tomonaga-Luttinger liquid (d=1)

2

1

algebraic correlations

No local order parameter

dimer LRObroken translational symmetry

staggered magnetization

gapless magnons, S=1

gapped magnons, S=1

deconfined massive spinons, S=1/2

deconfined massless spinons, S=1/2

Oct. 28, 2005 KIAS 3

Frustrated antiferromagnets: a route to exotic spin liquids ?

(classical) antiferromagnets on frustrated lattices haveextensive ground-state degeneracy

Lifting degeneracy by quantum fluctuations exotic phases ?

cf. fractional quantum Hall liquids Landau levels with extensive degeneracy + interactions Laughlin states, anyons,..

fractionalized spin liquids?

Oct. 28, 2005 KIAS 4

Geometric frustration

?

spins on a triangle interactingvia antiferromagnetic exchange

2-fold degeneracy

Lattices with triangles geometric frustration

2dTriangular lattice -(ET)2X, Cs2CuCl4….

Kagome lattice

3dPyrochlore lattice spinel oxides,….

Oct. 28, 2005 KIAS 5

Pyrochlore: most frustrated 3d latticeCorner-sharing 3d lattice oftetrahedra (4 spins in each)

On every tetrahedron:

Extensive degeneracy = # tetrahedra

Oct. 28, 2005 KIAS 6

2-dimensional pyrochlore = checkerboard lattice

1Jcorner-sharing geometry,but and are not equivalent2J

Y

X

J1

J2

.N.N.N

''2.N.N

'1 rrrr SSJSSJH

S=1/2 antiferromagnetic Heisenberg model

rS

on and

Oct. 28, 2005 KIAS 7

related 2d lattice models

J1-J2 model

Shastry-Sutherland model

J1: N.N. AF exchangeJ2: N.N.N. AF exchange

Oct. 28, 2005 KIAS 8

J1-J2 model

2d J1-J2 model

LRO:21 JJ ,

:21 JJ ,0&0, LRO

finite-temperature Ising transition

12 2JJ Maximal frustration

2J1J

J

Anisotropic model

Starykh & Balents, PRL (2004)

ColumnarDimer phase

Classical phase boundary J2=J1/2

J2/J

J1/J

Oct. 28, 2005 KIAS 9

Shastry-Sutherland model [Physica 108B, 1069 (1981)]

dimer plaquette singlet Neel

Koga and Kawakami, PRL 84, 4461 (2000)

21 / JJ

Oct. 28, 2005 KIAS 10

Ground-state phase diagram ofthe checkerboard model (2d pyrochlore)

J1/J2

?0

Neel (square lattice of J1 bonds)Plaquette phaseChainsJ2>>J1

1

(classicallydecoupled)

= 4-spin singlet

earlier results

?

Oct. 28, 2005 KIAS 11

J １>>J2 B. Canals [PRB 65, 184408 (2002)]

spin wave analysis J(q)

12 5.0 JJ

)1,1( Eigenvector of lowest eigenstate

12 JJ 12 2JJ

Oct. 28, 2005 KIAS 12

J １～ J2 Exact diagonalization [Fouet et al., PRB 67, 054411 (2003)]

J1=J2, up to N=36 (N=40)

ground state S=0: 2-fold degeneracy (N>>1)

many low-lying excited states in the S=0 sector

valence bond crystal(plaquette singlet)

Oct. 28, 2005 KIAS 13

Series expansion [Brenig and Grzeschik, PRB 69, 064420 (2004)]

triplet dispersion

Oct. 28, 2005 KIAS 14

Quadrumer boson approximation

2

42

2

31

2

43211

144332211

2SSSSSSSS

J

SSSSSSSSJ

10 2JE

11 JE

1,1,0 24131234 SSS

1,1,1 24131234 SSS

2232

10S

2

10

2

10

2

10

0

T

T

T

100

TTTTTTSS

hard-core bosons

Oct. 28, 2005 KIAS 15

rewrite the Hamiltonian in terms of the T bosons discard quartic and higher-order terms

quadratic approximation diagonalization by Bogoliubov transformation dispersion of triplet bosons

211211 coscos

3

4kkJJJJkE

(1) unstable when 21 5

8JJ bosons condense at 0k

Antiferromagnetic LRO: Neel state

(2) unstable when 21 11

8JJ

bosons condense at ,k

Magnetic LRO

Oct. 28, 2005 KIAS 16

Comparison with 1/S

Tchernyshyov, Starykh, Abanov & Moessner, PRB 68, 144422 (2003)

(a)

(b) (c) (d) (e)

Oct. 28, 2005 KIAS 17

J １ <<J2

decoupled spin chains

AF coupling is frustrated.

sliding Luttinger liquids ?

Starykh, Singh & Levin, PRL 88, 167203 (2002)

Oct. 28, 2005 KIAS 18

A single Heisenberg chain (S=1/2)

jjj SSJH 1

Low-energy theory

SU(2)1 WZW theory

ggggggxdi

ggxdS

Tr12

Tr8

1 32

)2(SUg

xNxJaS jj

1/

xaSSSS jjjjj 211 1

jax

staggered dimerization

gTr~

giN Tr~

Oct. 28, 2005 KIAS 19

Quasi LRO (algebraic correlations)

2

1~)0(x

JxJ aa

x

x1

~0

x

NxN aa 1~0

dominant correlations

staggered magnetization

staggered dimerization

Neel order

Dimer order (VBS)

uniform magnetization is subdominant

Oct. 28, 2005 KIAS 20

Weak perturbation: J1

xNxJamnS mhn

mhh ,, 1/,2/1

yNayJxNaxJJV mvym

nvnm

mhxn

mh ,,,

,,1 1212

xNaxJa

mnSmnS

mhxn

mh

hh

,, 12

,2/1,2/1

Inter-chain interaction

vh JJ

If we keep only the term, V is not relevant sliding Luttinger liquid

However, the and terms are dangerously irrelevant!NJ

NN

Oct. 28, 2005 KIAS 21

2nd order perturbation in V

Operator Product Expansion

222

,0,,

v

xxNxJ

abb

xa

yNayJxNaxJJV mvym

nvnm

mhxn

mh ,,,

,,1 1212

uniform magnetization staggered magnetization = staggered dimerization

nm

nvmhnm manad

v

aJSV

,,,2

2212 ,,1

3

relevant operator allowed by symmetry

Mean-field analysis

mmh na 1,

nnv ma 1,

crossed-dimer state

Oct. 28, 2005 KIAS 22

comparison with exact diagonalization (36 spins)

lkjilkji SSSSSSSSklijD

;

Sindzingre, Fouet, Lhuillier, PRB 66, 174424 (2002)

dimer-dimer correlations

Oct. 28, 2005 KIAS 23

Global phase diagram scenario I: direct transition between crossed-dimer and p

laquette VBS

21 / JJNeelPlaquetteCrossed Dimer ? ?

? = 1st order transition or intermediate coexistence phase

Continuous transitions are forbidden by Landau-Ginzburg-Wilson theory.

Oct. 28, 2005 KIAS 24

scenario II: crossed-dimer and plaquette VBS via an intermediate ordered phase

? ?Crossed Dimer Neel* Plaquette NeelO(3)21 / JJ

? = 1st order transition or intermediate coexistence phase

Oct. 28, 2005 KIAS 25

application to 3d pyrochlore

3d crossed-dimer phase for ?21 JJ

1J2J

Oct. 28, 2005 KIAS 26

Summary Global phase diagram of the AF Heisenberg model on

the 2d-pyrochlore (checkerboard lattice)

1d-approach can give some useful hints for understanding 2d and 3d frustrated magnets.

21 / JJNeelPlaquetteCrossed Dimer ? ?

PRB 72, 094416 (2005)

Oct. 28, 2005 KIAS 27

appendixLR JJJ

uniform part of magnetization

,,4

10,,0,,

22xxN

vyNxJxNxJ abcabc

xy

baRy

bx

aR

,,4

10,,0,,

22xxN

vyNxJxNxJ abcabc

xy

baLy

bx

aL