On the behavior of Ising spin glasses in a uniform magnetic field

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On the behavior of Ising spin glasses in a uniform magnetic field

David Huse, Daniel Fisher

To cite this version:

David Huse, Daniel Fisher. On the behavior of Ising spin glasses in a uniform magnetic field. Journal

de Physique I, EDP Sciences, 1991, 1 (5), pp.621-625. �10.1051/jp1:1991157�. �jpa-00246356�

L

Phys.

^I 1

(1991)

621-625 MAi 1991, PAGE 621

Classification

P%ysksAbsmwts

75.50-75.10

On the behavior of Ising spin glasses in

_a

uniform magnetic

field

David A~ Huse

(I)

and Daniel S. Fhher (2,*

(1)

^{AT&T Bell}

Laboratories, Murray Hill,

NJ

07974,

U.s.A.

(2) Physics Department,

Princeton

University,

Princeton, NJ

08544,

U.s.A.

(Received

30 November 199(

accepted

14 March

1991)

Abstract. We show how the recent computer ^{simulation data}

reported by

Caracciolo ei al for the three-dimensional + J

[sing spin glass

in _a uniform

magnetic

field _are, in contrast to their

claims, quite

consistent with the

simple scaling

or

droplet theory developed by

Mcmillan, ^Flsher and Huse, and

Bray

and Moore, and do not

provide

evidence for _a de AJmeida-Thouless transition.

Caracciolo _et al

[1, 2]

^have

recently reported

results of

computer

simulations of the + J

[sing spin glass

model in _a uniform

magnetic

field _{on a}

simple

^{cubic lattice.}

They

claim that their _re-

sults _are in

good agreement

^with the exotic behavior of the

infinite-range Sherrigton-Kirkpatrick (SK)

model

[3,

4] ^and are

incompatible

with the

simpler scaling

or

droplet theory _[5-8]

of finite-

dimensional

short-range spin glasses.

Here we

dhpute

that

claim, showing

^{how the} results _re-

ported

in references

[1, 2]

are

quite

conshtent with the

scaling

or

droplet theory presented

in references

[5-8].

^In

fact,

we argue ^{that in} references

[1, 2]

^the

temperatures

^studied were too

high

and the sizes studied _were too small to address the issues on which the two theoretical

approaches differ,

I.e. the nature and extent of the low

temperature

^ordered

phase.

We also

briefly

discuss

other _recent numerical studies and what

regimes

need _to be accessed to

really

address these hsues.

The + J

Ising spin glass

model with

only nearest-neighbor

interactions _{on a}

simple

cubic lattice has been

extensively

simulated

[9, 10]

^for zero

magnetic

field. These studies indicate _a

phase

transition, although

the evidence b _not

indisputable. However,

T there is indeed a

phase

transition it

appears quite

^certain that it _occurs at a

temperature

^less than 1.3

(in

units where J

= ~l and

kB

₌

I). Ogielski

_[9] estimates

Tc(h

₌

0)

₌ ^1.175 +

0.025; however,

we feel his _error bars are

overly optimistic, especially

_on the lower

temperature

side. The simulations of references 11, 2] are all

performed

at

applied

field h

= 0.2 and thus

h/Tc(0) _/

0.16. At such _an

applied

field

(scaled by 7~(0))

the reduction of

7~(h)

from

Tc(0)

in the

Sherrigton-Kirkpatrick

model is

(Tc(0) Tc(h)) /Tc(0)

_> 0.25. Thus T there _{were a} de AJmeida-Thouless

phase

^transition line

[3,4]

ⁱⁿ the

simple

cubic +J model _one would

expect 7~(h

=

0.2) _~

0.9 if _one

simply

scales the SK

phase diagram _[11].

This indicates that _even if the

system

^{did behave} as in _mean field

theory,

(* ^{l+esent address:}

Physics Department,

Harvard

University, Cambridge,

MA

02138,

U-S-A-

622 JOURNALDEPHYSIQUEI N°5

the lowest

temperature

^{simulated in references}

[1, 2],

T _ci

0.833,

would be very near to or above We de AJmeida-Thouless line. Thus it should come as no

surprise

that their data _are all consistent with

being

in We disordered

phase.

We have

argued

elsewhere

[6,

8] ^{that the ordered}

phase

of an

Ising spin glass

with

finite-range

interactions b unstable in

equilibrium

to a small

magnetic

field for all T _<

Tc(0)

and all dimen- sionalities d. The

resulting

disordered

paramagnetic phase

has _a finite correlation

length (

that

diverges

as a power ^{of the field h for} h _- 0 and T <

Tc(0).

The relaxation time for T _<

Tc(0)

b

expected _{[6, 8]}

to

diverge

as

exp (A(~ /T)

for h _- 0 and thus

(

_{- oo,} where

#

is _a

positive exponent

^{and A is the}

amplitude

for the free energy ^barriers

[6, 8].

^Thus we conclude

[6, 8]

^that there is _no de Nmeida-Thouless line in the

equilibrium phase diagram. (Mcmillan

_[5j and

Bray

and Moore [7j ^{have made similar}

arguments

^{for d} =

3,

but

they

^did not

attempt

^{to rule out the}

possibility

of _a de AJmeida-Thouless line for

high

d.

)

For small field and T

$ ^Tc (0)

the correlation

length

and correlation time are

quite large _{[6, 8],}

so it would be very difficult to determine whether or not

they

are finite in a simulation

study.

Much of data

reported

in references

[1, 2]

are from this difficult low field

regime.

We will now examine the data and

interpretation given

in reference

[I]

in the same sequence

as

the?

_are

_presented

there:

a)

MAGNEnzATION _vs. TEMPERATURE.

Figure

la of reference

[I]

shows that the

magneti-

zation at h

= 0.2 b constant within _error bars _over the

temperature

range 1.0

$

^T

$

^{1.5. As}

argued above,

even if there h _a de AJmeida-Thouless line these

temperatures

are above it. Thus the

magnetization

_appears to be

nearly temperature independent

in this

part

^{of the}

paramagnetic phase.

This

appears

^{to be in}

disagreement

with We behavior of the SK

model,

where the magne- tization becomes

temperature independent only

below

Tc(h).

Of course, as shown in

figure

16 of

reference

[I],

the average

exchange

_energy continues to

drop

after the

magnetization

saturates,

as it must since the

specilic

heat is

positive.

Witllin the

scaling

_or

droplet theory _{[6, 8],}

the actual numerical values of the

magnetization

and energy are to a

large

extent determined

by

the details of small

length

scale fluctuations

(small droplets);

these _are not universal _so it is not clear how

one could make _a detailed

prediction

of the

temperature-dependence

of the

magnetization

and We energy.

b)

CORRELATION BETWEEN CONFIGURATIONALAND ENERGY OVERLAPS. In references

Ii, _2j they

simulated

mulIiple replicas

of the _same

sample (same Ji;'s)

^and measured the

configurational overlap

of

replicas

_a and

fl

~

q +

j _{£ alai, (I)}

I=i

where each

replica

contains N

[sing spins al.

The

"energy overlap"

qe +

[ L a;alar al ₍₂₁

lo)

was also

measured,

where the sum runs over all 3N

nearest-neighbor pairs

of

spins. They

noted that in order for both _q and qe ^to differ

substantially

from

unity,

the

droplets

that constitute the

difference between the instantaneous

spin configurations

of the two

replicas

must have _{a non-zero} average surface-to-volume ratio.

Indeed,

_we have shown

[6,

8] ^{that the difference of} q from

unity

is

predominantly

due to the small

droplets;

^similar

arguments apply

to qe. These small

droplets,

N°5 ISING SPIN GLASSES IN A MAGNETIC FIELD 623

which contain

only

a few

spins,

do have _a

large

surface-to-volume ratio and it b their

presence

^that

produces

most of the deviation of _{q and qe} from

unity.

Note that these

droplets

do not all

spend approximately

^one-half of the time

flipped,

as is

implied

^{in the discussion of}

figure

13 of reference

[2];

^{rather the} average

polarizations

of the

droplets

^will

generally

be

continuously

distributed

[8].

The

strong

correlation observed between _{q and qe} is inevitable: when _q is _near either I _or

-I,

there _are few

droplets _present

so qe must be near

I,

while when the _two

configurations

are

quite

different _{so q} b _near zero, _qe must be at a minimum. For _a

paramagnetic

state one may constrain q in the

thermodynamic

limit

by

the

conjugate field,

_e, and _{measure qe.} The

resulting

function

qe(q) should,

because of the above

constraints,

be well

approximated by

an even function of _q.

c)

NONTRIVIALITY _oF

P(q ).

For _a

general

finite-size

system P(q),

the

probability

distribution of _q, is

expected

to have nontrivial _structure as

long

as the linear

sample

^size L h less than or of

order the correlation

length (.

It h

only

for L »

f

that

P(q)

becomes very

sharply peaked

in _a

paramagnetic phase.

Thus the

P(q J's

for L < 14 and T < I shown in

figures

2 and 3 of reference

ill

are not

surprising,

since within any

theory

the correlation

length

is

expected

to be

large

for T _<

Tc(0)

and low field. lb demonstrate that

P(q)

is indeed nontrivial

(I.e.,

not a

delta-function)

for L _{- oo one} must show that

(q2) (q)2

e xso

/L~

does _not vanish for L _{- oo.}

However,

the data shown in

figure

5 of reference

ill

show that _xso

/L~ steadily

decreases with

increasing L,

and

are thus

quite

consistent with

P(q collapsing

to _a

single

delta-function

peak

in the

thermodynamic limit,

_as we

expect

occurs.

d)

ABSENCE oF SELF-AVERAGING oF

P(q).

One may consider the

probability

distribution

Pj (q)

^of _q ^for a

given

finite-size

sample

J and

compare

^{it to}

P(q),

which is the average ^of

Pj(q)

over all

possible samples

of that size. As _{a measure} of the

typical

difference between

Pj(q)

and

P(q)

the

quantity

S ₊

/ _dq ~ _{L (PJ}

(qJ (qJJ~)

^~~~

(3J

was

proposed

in reference

[ii,

where

NJ

^{is the number of}

samples

summed _over.

However,

this

measure

ofself-averaging

is

severely

flawed because of the

expectation (even

ⁱⁿ the SK

model)

that

P(q

contains a delta-function

component

^{for L} - oo. For

example,

if the

system

^is

paramagnetic

then for L »

(

_one

_expects Pj(q )for

a

given sample

J to be

peaked

at qj, with qj

varying by

of order

L-~/2

_from

sample

to

sample

when h

#

0. The width of thb

peak

in

Pj(q)

should also be of

ordqr L~~/2.

_{Thus the}

sample-to-sample

fluctuations in

Pj (q)

at fixed _q should be of order

P(q)

itself. Thin _means S will not vanish with

diverging

L for L >

(

even if

P(q) collapses

to a delta-function _as is

expected

in _a

(self-averaging) paramagnetic phase.

The

problem

is that S is _a useful _measure of

self-averaging only

when

P(q)

contains _no delta-function

component

^for L - oo. A

simpler

and

probably

more reliable _measure of

self-averaging

is

given by

the ratio of

the

sample-to-sample

variation of _xsG to its mean.

d)

ULTRAMETRICITY. Here _we

just

note that a

space

^of states

consbting

of one state is

trivhlly

ultrametric. A

paramagnetic phase clearly

must

satisfy

the test for

ultrametricity

done in reference

[ii.

Thus the data shown in

figure

4 _are

quite

conshtent with what should occur in a

paramagnetic phase.

624 JOURNAL DE PHYSIQUE I N°5

EmSTENCE _{oF A PHASE} TRANsiTioN. In _a

paramagnetic phase

with _a

large

correlation

length, (,

if _{one measures xsG} for various linear

sample

si2es

L,

_one should find that for L _<

2f

the finite-she effects are

strong

^and _xso ^{increases with}

increasing

L. For L »

(,

on the

(her

hand,

_xso should saturate and

approach

^the finite value it attains in the L

- oo limit. In

figure

5 of reference

[ii, xso(L)

at h

= 0.2 and T _ci 0.833 and T

= I is shown to increase with L as L varies from 6 to 14. Given the she of the _error bars and the small range of ^L it is _no

surprise

that the data _are

quite

consistent with _a

temperature-dependent

_power

law,

_xso

+w

L~(~).

_{The fact}

that xso h

increasing fairly rapidly

with L _means that the correlation

length

is

greater

than or of

order,

say, 7 lattice

units,

which is to be

expected, considering

the low field and

temperatures.

^At the zero-field critical

temperature

_{xso +w}

L2~Q,

with

[9, 10]

q ci -0.3.

Discussion.

Unfortunately,

it is not clear

how,

for three-dimensional

spin-glass models,

one can

numerically

access the low

temperature, large L, long

time

regime

in the ordered

phase

where there _are real

qualitative

differences between the behavior of the SK model

[3],

on the _one

hand,

and the

scaling

or

droplet theory [5-8],

on the other.

Perhaps

a careful

study

of

nonequilibrium

domain

growth [12]

^{for T} <

7~ might

shed _some

light

on thb

question.

In

fact,

_{no one} has

yet produced

_any solid

evidence for true

long-range spin-glass

order _at

positive temperature

in these

models,

even in zero

magnetic

field. Ml the

low-temperature

behavior observed b

quite

consistent with the

system being

at _or near a critical

point

with

spatial

correlations that

decay

as a

power

^{of the dbtance.}

The ordered

phase

is _more accessible

numerically

for four-dimensional

models,

as demon- strated

by Reger,

Bhatt and

Young _[13]. They

measured

P(q)

for the

Ising spin glass

with Gaussian- distributed

J;;'s (with

variance

unity)

_on four-dimensional

hypercubic

lattices with

periodic

bound- ary ^{conditions for sizes} up to L

= 6 and

temperatures

^{down to T} = 1.2. The zero-field transition

Tc(0)

_appears to occur near T

= 1.75. At T

= 1.2 _{one can see}

(Fig.

I of Ref.

[13])

^{that the width}

of the

peak

in

P(q)

normalized

by

its

position

decreases

considerably

_on

going

from L

= 4 _to

L =

6,

as is

expected

in the ordered

phase. However,

the si2es _are

quite

small and _one cannot tell from their data

whether,

for L _{- oo,}

P(q)

is

approaching simply

a delta-function _{as we}

expect

or

is instead

approaching

a distribution with _a continuous

part,

as does the SK model. It is

important

to

emphasize

that in

general P(q)

is not a

good

indicator of the

presence

or absence of many states

[14].

Another

approach

to the

question

of the existence of the de AJmeida-Thouless line is to ex-

amine contours of _{constant xsG} in the

(h, T) plane [15].

^{In the SK model such} contours go ^to

large

h for T

-

0,

while in the

droplet theory they

_go to h ₌ 0 for T _- 0 for _a

system

^with

continuously-distributed J,,'s,

^which iS a real

qualitative

difference. In reference

[lsj

such _a

study

was

attempted,

but since iI _was

mostly

^based on

high-T

series

expansions

which fail

t6

_converge well above

Tc(0)

and _was restricted to the + J model with its extensive

ground

state

entropy,

^{it does}

not

help

to resolve the

question

for d > 3.

Perhaps

a

thorough

^simulation

study

^of

xsG(h, T)

for

a model with

continuously-d

istributed

J;;'s

would be useful in

addressing

this

question.

Yet an-

other

promising approach

is advanced

algorithms

such as that

proposed by ^Swendson

and

Wang [16].

^The

potential

of these

algorithms

in _an

applied

^field has not

yet

^been

explored.

We conclude with

general

^remarks on the difficulties to be faced

by

numerical simulations. In order to

probe

the behavior of

spin glasses

in _a

magnetic

field

(or

in

zero-field)

at

long enough

dis- tance scales _to have _a

hope

for

distinguishing

between the theoretical

predictions

for the ordered

phase,

the

sample

sizes must be considerable

larger

than the critical zero-field correlation

length

[8]

(- (T)

which

diverges

as the zero-field critical

point

^is

approached

from below. Otherwise _one

N°5 [SING SPIN GLASSES IN A MAGNETIC FIELD 625

just

sees the critical behavior. Thus data _on

relatively

small

samples

in small fields near

Tc(0)

are not

adequate.

With _a fixed amount of available

computer ^time,

one should instead choose the

temperature

^{to maximize the ratio of L} to the correlation

length (- (T)

for which _{one can} still achieve

equilibrium.

This

optimum temperature

is

probably roughly I/2

of

Tc(0).

One must then

try

^to

dhtinguish between,

for

example,

a xso which

diverges

at zero field _as h~~H and one

which

diverges

at a critical field as

[h hc(T)]~~H.

^{In either} case, the relaxation time is

expected

to

diverge exponentially rapidly

as xso

diverges _{[8, 12],}

_so that

equilibration

of

sufficiently large systems

is

extremely

difficult.

Acknowledgements.

Daniel S. Fisher is

partially supported by

the

NSfI

DMR 8719523 and the A~P Sloan Foundation.

References

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