Critical behavior of the antiferromagnetic Heisenberg model on a stacked triangular lattice

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Critical behavior of the antiferromagnetic Heisenberg model on a stacked triangular lattice

T. Bhattacharya, A. Billoire, R. Lacaze, Th. Jolicoeur

To cite this version:

T. Bhattacharya, A. Billoire, R. Lacaze, Th. Jolicoeur. Critical behavior of the antiferromagnetic

Heisenberg model on a stacked triangular lattice. Journal de Physique I, EDP Sciences, 1994, 4 (2),

pp.181-186. �10.1051/jp1:1994131�. �jpa-00246896�

Classification Physics Abstracts

75.10J

Short Communication

Critical behavior of the antiferromagnetic Heisenberg

mortel

_{on a}

stacked triangular lattice

T.

Bhattacharya (*),

A.

Billoire,

R. Lacaze

(*")

^{and Th. Jolicoeur}

(**)

Service de Physique

Théorique

de Saclayj Laboratoire de la Direction des Sciences de la Matière du CEAj 91191 Gif _sur Yvette Cedexj France

(Received

14 May1993j received in final form 24 September1993, accepted 2 December

1993)

Abstract. We estimate, using _a large-scale Monte Carlo simulation, trie critical exponents of the antiferromagnetic Heisenberg mortel

on a stacked triangular lattice. We obtain the following

estimates:

~/v

= 2.011 + 0-014j _v = 0.585 ~ 0.009. These results contradict

a perturbative

2 + _e Renormalization Group calculation that points to Wilson-Fisher

O(4)

behaviour. While these results _may be coherent with 4 _e results from Landau.Ginzburg analysisj they show trie

existence of _an unexpectedly nch structure of trie Renormalization Group flow _{as a} function of trie dimensionality and trie number of components of the order parameter.

1. Introduction.

There is at present a

satisfactory understanding

of the critical behaviour of

physical

_systems

where the rotation symmetry _group

O(N)

is broken clown to

O(N -1)

at low temperatures.

Several theoretical tools _are available _to estimate the critical exponents ^and there is

good

agreement between these estimates. The Wilson-Fisher fixed

point

which describes the critical

physics

_can be

smoothly

followed between _two and four dimensions: for N > 3 the 2 _{+ e} and 4 _e renormalization _group

expansion

_merge in a continuous _manner.

The situation is much _more

complicated

when the rotation symmetry is

fully

broken in the

low-temperature phase.

A

prominent example

is found in the so-called

helimagnetic

systems where

Heisenberg

_{spins are} in _a

spiral

arrangement ^{below the critical} temperature. It is _an

interesting question,

^both

theoretically

^and

experimentally,

to know the

corresponding

univer-

saIity

class. A

widely

studied

prototypical

^mortel is the

antiferromagnetic Heisenberg

mortel

on a stacked

triangular lattice,

which is

simple

and

display

commensurate

helimagnetic

or-

(*) Present address: MS 8285, Group T-8j Los Alamos National Laboratory, NM 87544.

(** ^{Chercheur CNRS}

182 JOURNAL DE PHYSIQUE I N°2

der below _a transition

point

T~. There is little _consensus in the literature

regarding

critical

phenomena

associated with this mortel [1].

This

topic

has been

investigated by

_use of _a D

= 4 _e renormalization group calculation [2]. ^The

corresponding Ginzburg-Landau theory

for _a

N-component

vector model involves two

N-component

bosonic fields. It is found that for N

large enough

the transition is second order and not

governed by

^the

Heisenberg O(2N)

Wilson-Fisher fixed

point

but

by

_a diiferent fixed

point

which is also non-trivial for D _< 4. For smaller

N,

this _new fixed point

disappears

and there is _no stable fixed

point

which is _an indication for _a fluctuation-induced first-order

transition. The

dividing

^universal line between second-order and first-order behaviour is found to be

N~(D)

₌ 21.8 23.4e +

O(e~).

The

rapid

variation of N~ leaves _us rather uncertain about the fate of the _case D

= 3.

Clearly

_more information is needed about the

N~(D)

line _in the

(number

of

components-dimension)-plane.

A D

= 2 _{+ e} renormalization _group

study

has been

performed

for _a system ^of

Heisenberg spins

_[3]

by

_use of _a non-linear

sigma

mortel defined _{on a}

homogeneous non-symmetric

coset

space

OI 3)

_x

O(2) /O(2).

It _was found that _near two dimensions the system

undergoes

_a second order transition which is

governed by

the

O(4)

usual Wilson-Fisher fixed

point.

In fact the symmetry

O(3)

x

O(2) /O(2)

is

dynarnically enlarged

at the critical point to

O(3)

_x

O(3) /O(3)

and

O(3)

_x

O(3)

is

O(4).

This mismatch between the

expansions

_near four dimensions and _near two dimensions is

quite

unusual and does _not

happen

in the well-studied

O(N)

_-

O(N

1)

critical

phenomena.

^As _a consequence, the D

= 3 _case remains elusive and _a direct

study

in three dimensions is called for.

Some

preliminary

Monte Carlo

(MC)

simulations have shown evidence [4-7] ^for a continuous transition in the _case of trie

Heisenberg antiferromagnet

_{on a} stacked

triangular

lattice. The exponents ^found are not

compatible

with those of the

O(4)

_vector mortel in three dimensions.

We dia _a

large-scale

simulation with much better statistics than previous attempts. As _a consequence we are able _to

pin

down the transition temperature in _a very

precise

_manner and to obtain reliable estimates of critical exponents.

We thus focus _on the classical _spm model defined

by

the classical

Heisenberg

Hamiltoman:

H =

~j

J~~ $

S~. (1)

<v>

The

exchange

interaction J~~ is nonzero (J~~ = 1 in what

follows)

betweeu

nearest-neighbors

of _a stacked

triangular

lattice and the

spins

_are

three-component

unit vectors. The classical

ground

state is found

by minimizing

the Fourier transform

J(Q)

of the

couplings

_J~~. The spins

adopt

a

planar

arrangement _{on a} three-sublattice _structure with relative

angles

121~

degrees.

2. Trie simulation.

We made two successive sets of Monte Carlo simulations of this

model, using slightly

_asym-

metric lattices

(with periodic boundary conditions)

of

shape L~Lz,

_{where L is the} linear size inside the

planes,

and Lz

=

2/3L

the

stacking

size.

The first set of simulations _{was run on a} CM-2 8K

massively parallel

computer. We made

runs on lattices of size 12~8

(with

_a total of _{61~l~ l~l~l~} Monte Carlo _sweeps of the

lattice),

24~16

(with

_a total of S00 1~00

sweeps)

and 48~32

(with

_a total of1 570 000

sweeps).

In the first _{case we} used the Heat-Bath

algorithm,

in the later two _{cases we} used _an

Hybrid

Overrelaxation

algorithm

_[9] where each Heat-Bath _{sweep is} followed

by

_{an energy conserving}

sweep. The _{next set} of simulations _{was run on} CRAY vector computers on lattices of size

18~12, 24~16,

31~~21~

,

36~24 _and 48~32 _{with 4 to 8}

x 106

Hybrid

Overrelaxation _sweeps. Both simulations concentrate in the immediate vicinity ^of the transition.

Let _us note

by

Sa the total

spin

_per site for sublattice _a

(a

_E

Il, 2,3]),

_we measured the

magnetization:

ML (fl)

₌

~

_{< [Sa[ >,}

₍₂₎

the

susceptibility:

XL(à)

_"

@ _i~

<

~i

_{> <}

_i~ai

_{>~)> 131}

and the fourth-order cumulant:

BL(fl)

_" 1-

~j ~() ^~. (4)

~

~

<

a

>

Dur strategy to extract the critical exponents was first _{to estimate} the value of the critical temperature fl~, as the

point

^where

BL (fl)

is L

independent

^and then to estimate the expouents from the

following

set of

equations:

xL(flc)

m~

L~/~, (S)

ML(flc)

m~

L~%~i (6)

ôBL(fl) (7)

~

L~/~

ôfl

~_~

~~

~~~

(8)

~

_{~~ ~~}

~~~

~_~~

~ ~

The

extrapolation,

from the

fl

value used for the simulation, to fl~ ^is done

using

the

Ferrenberg-Swendsen tecllnique

_[11~]. This

technique

is invaluable _to

extrapolate

in _a

neigh-

borhood of size

+~

1/L~/~ (We

understand

that,

when used

blindly

outside such _a

tight

_range,

it may

give

_wrong

results).

An alternative strategy ^{would be} to evaluate expouents from the finite size

scaling

of the maxima of _XL

(fl), ML(fl)i ùBL(fl)

and

ù

In

ML(fl).

It would how-

ever require to

perform

simulations at four

fl values,

close to the four

(L dependent) points

where these

quantities

reach their

maxima,

otherwise _accuracy would be lost. The statistical

analysis

is done with respect to 21~

bins, using

first-order bios-corrected

jackknife (see

_e-g- Ref.

[Il]

and Refs.

therein).

The first _21~ % of each _{run is} discarded for thermalization.

From simulations of lattices of

increasing

sizes Li < L2 _<

converging

estimators of fl~

are obtained

by solving

the equations:

B~,~i(fl)

=

B~,(fli (9)

The results _can be found _in the second column of table I. Within _our statistical _accuracy ail estimates _are

compatible,

there is neither any clear lattice size

dependence,

_{nor any}

discrepancy

between the two sets of _runs. Dur final estimate is

(the quoted

_error is

deliberately

_on the safe

side)

fl~ = 1.1~443 + 0.1~l~1~2. (11~)

We _use this value to compute XL

(flc), ML (flc), ùBL (fl)

and

]

In

ML (fl)

in order to estimate

~/v, fl Iv

and _v Results _can be found in table I. We quote

separately

the estimated "direct"

184 JOURNAL DE PHYSIQUE I N°2

Table I. Critical temperature ^aud critical expouents

estimates, usiug equatious (9,S,6,

₇₎

and

(8) respectively

First two fines _are CM-2

data,

uext fines trie CRAY data. Trie first columu

gives

trie linear sizes of trie lattices used. Tlle two numbers iuside

pareutllesis

_are tlle estimated statistical _errons, direct aud

iuduced,

_ou tlle last

digits

of tlle number _ou tlleir left.

v v

2.009

48-24 1.04462

(21)

2.033

(18) (14)

1.998

(09) (25)

0.590

(20) (04)

0.576

(10)

(04)

1.04452 2.010 2.011 0.605

(02)

30-24 1.04388 (36) 1.994 (36) (11) 2.021 (13) (16) 0.578 (35)

(03)

0.583

(19) (03)

36-30 1.04408

(37)

1.988 (55) (16) 2.020

(18)

(21) 0.605

(55) (12)

0.607

(30) (04)

42-36 1.04485 (25) 2.083

(68) (14)

1.984 (20)

(31)

0.528

(48) (06)

0.548

(28) (06)

48-42 1.04427

(30)

1.986 (GI) (18) 2.004 (22) (39) 0.639

(69) (05)

0.577

(27)

(03)

2.011

statistical _errors

computed

from the

dispersion

of the results from the 21~

bins,

and the _errors induced

by

the uncertainty on the determination offl~,

computed

_as the

(absolute

value of

the)

diiference between the values obtained

using

our best estimate of fl~ and the ^values

using

the _one standard deviation estimate from

equation

_(11~). The last line in table I

gives

the results of linear lits of the CRAY data for

In(xL(flc) ), In(ML(flc)), In(]BL(fl) P"Pc

^and

In(]

In

ML(fl) P"Pc respectively.

The results of the lits _are stable

against omitting

the smallest lattice data. The

x~

is

equal

to

I.S, 3.7,

1.9 and 4.S

respectively (we

expect

x~

m

4).

Such values _mean that linear lits

give good represeutation

of the data for L > 18

(with

the current

accuracy).

Our final numbers _{are ~}

Iv

= 2.011 + 0.1~14, ^D

2fl/v

= 2.011 + 0.019, _v

= 0.SBS + l~.009.

We observe that

hyperscaling

is verified within _errors, and _~/ is very small _{([~/[ m}

o.01).

The value for _v is

clearly

not

compatible

with the value for the

O(4)

fixed point [13]

(v

+~

o.75).

Published Monte Carlo results for _{v are} o.53 + o.02 [4], ^{0.53 + o.03} [Si, ^O.SS + o.03 [6] and o.59 + o.02 [7]. ^{Our results} are in

good

_agreement with the most recent results

land

with the result ù-Si + l~.02 from _a model of commensurate

Heisenberg helimagnet

_[8]) The methods of

analysis

_are

quite

^{diiferent. Reference} _{[7] uses} data taken in _a wide _region around the transition point and

adjust

the values of the critical exponents

using

the old fashioned "data

collapsing"

method. This method has the

disadvantage

to

give

much

weight

to

points

with

large

^{value of}

(fl fl~)L~/~

^We extract the exponents

directly

from data

reweighted

to fl~. ^We have also _a much

higher

statistics: _{we use} 8 x 106 heat-bath + energy-conserving sweeps whereas [7] uses

6-20 times 20000 sweeps of _a less eflicient

algorithm.

Dur estimated statistical _{error on} the determination of fl~ ^is one order of

magnitude

smallet than the _one m reference [7].

3. Conclusion.

Let _us first summarize the

knowledge

obtained from

perturbative

^{RG studies} _{as a} function of the

dimensionality

^and the number of components ^{of the order} parameter.

i)

There is _a universal fine

Nc(D) separating

a first-order

region

from _a second-order

region

near D

= 4 _e. Its

slope

_is known from RG studies [2] ^as well _as the critical value Nc for

D = 4.

ii) Large-N

^studies [12] ^{indicate that the stable fixed}

point

found above

Nc (D) persists smoothly

in the

region

N

= co and 2 _< D _< 4.

iii)

Smoothness

along

the D

= 2 vertical axis _cari be shown

by studying

the nonlinear

sigma

model suited to the N-vector model: it is built _on the

homogeneous

_space

O(N)

_x

O(2) /O(N 2)

x

O(2)

_[14]. Due finds _a

single

^stable fixed point for ail values of N > 3. In the

longe-N

limit the exponents from this

sigma

model _are the _{same as} those of the linear model.

If _we believe in the

perturbative

RG

results,

then

necessarily

^the universal fine

Nc(D)

_cari

only

intersect the horizontal axis N

= 3 between D

= 2 and D

= 4. The

simplest hypothesis

is then that the

plane (N,D)

is divided in _two

regions by

the fine

Nc(D).

This fine _may intersect the N

= 3 axis at _a critical dimension

Dc.

Dur Monte Carlo results thus

imply

that Dc is between three and four dimensions since _we observe _a continuons transition at D

= 3.

A

possibility

would be that the critical behaviour _we observe is described

by

the fixed

point

found in the

neighborhood

of D

= 4.

However _one bas _to note that trie _sigma model

approach

shows that this fixed

point

is

O(4)

for N

= 3. This is

clearly

ruled ont

by

_our data. Trie

simplest

_scenario

suggested by

1-iii above

is wrong. It may be that trie whole

sigma

model

approach

breaks down since

perturbative

treatment of _a

sigma

model

neglects global

aspects. ^{Previous studies indeed} bave

performed,

as

usual, only perturbation theory

for

spin-wave

excitations [3]. ^Due may invoke trie

peculiar

vortices present ^{in the model} [16] ^since

Hi (SO(3))

=

22

that form

loops

in D

= 3 _as

responsible

of the breakdown of the sigma model. However vortices _are also excluded from trie

perturbative

calculations _near D

= 4 and thus it would be diilicult in this _{case to} believe _in what is found

perturbatively.

It may be also that the fixed

point

found in 2 _{+ e}

expansion

is destabilized

by

some operators

containing high

_powers of

gradients

[15].

We have thus obtained the critical behaviour of _a magret ^with a canted

ground-state.

This calls for _a

deeper understanding

of the Renormalization

Group

behaviour of canted systems since there _{is no} obvious

relationship

between the different

perturbative approaches,

_contrary

to the _case of collinear vector magrets. ^We note for the future that it would be

interesting

to obtain the critical behaviour of the XY canted

systemà,

much _more relevant to the

experimental

situation.

Acknowledgments.

We would like to thank

Guy Decaudain,

David

Lloyd

Owen,

Gyan

Bhanot and Roch Bour-

bonnais of TMC for their support. We aise thank H. T.

Diep,

A. P.

Young

and D. P.

Belanger

for various information about this

subject.

Note added in

prooE

Results

compatible

with our's have been obtained

by

D. Loison [17].

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