Reentrancy and Dynamic Specific Heat of Ising Spin Glasses

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Reentrancy and Dynamic Specific Heat of Ising Spin Glasses

Mai Li, Tran Hung, Marek Cieplak

To cite this version:

Mai Li, Tran Hung, Marek Cieplak. Reentrancy and Dynamic Specific Heat of Ising Spin Glasses.

Journal de Physique I, EDP Sciences, 1996, 6 (2), pp.257-263. �10.1051/jp1:1996101�. �jpa-00247182�

Reentrancy and Dynamic Specific Heat of Ising Spin Glasses

Mai Suan Li (~>~>*)

,

Tran

Quang Hung

_(~>**) and Marek

Cieplak (~)

(~) Institute of

Physics,

Pohsh

Academy

of

Sciences,

02-668 Warsaw, Polan

(~)

Faculty

of

Engineering

and

Design Kyoto

Institute of

Technology, Sakyo-ku, Kyoto

606,

Japan

(Received

27 June 1995,

accepted

19

October1995)

PACS.7.5.10.Nr

Spin-glass

and otl~er random mortels PACS.75.10.Hk Classical

spin

mortels

Abstract. The

dynamic susceptibility

and

dynamic

specific heat of

Ising spin glasses

with

non-symmetric

distributions of the

exchange couplings

_are studied

by

the Monte Carlo method and _are

compared

to the results obtained

previously by

the local _mean field method. The two methods

yield qualitatively

different results but they still suggest lack of any reentrancy

phenomena.

One of the

interesting problems

of the

spin glass (SG) physics

is _a

question

^{of tl~e}

reentrancy:

is tl~ere _a transition from tl~e

ferromagnetic phase

to tl~e SG

phase

_on

lowering

tl~e

temperature,

T. Tl~e

reentrancy pl~enomenon

bas been observed

experimentally

in many materials

iii

but

a tl~eoretical

justification

of tl~e

pl~enomenon

_is

missing.

Tl~e exact Parisi solution of tl~e infinite range SG model [2j ^{mies the} reentrancy ^out.

Similarly,

ail theoretical studies of tl~e short _range

Ising

_or

anisotropic Heisenberg

SG'S

give

_no evidence for tl~e

pl~enomenon

_[3].

We bave

speculated

_[4j tl~at tl~e

origins

of

reentrancy migl~t

be of _a

dynamical

_{nature, i e.}

resulting

from finiteness of observation times. We bave

attempted

to

study

tl~is

by

_using

the Glauber

dynamics

_{[5j for}

Ising

SG'S witl~in tl~e local _mean field

approximation [6j.

^Tl~is

approacl~

bas

yielded

_a double

peak

structure, _{as a} function of

T,

in tl~e real part ^{of tl~e}

dynamic susceptibility.

The

positions

of tl~ese

peaks, l~owever,

_are sucl~ tl~at tl~e

reentrancy

is not

explained and, furtl~ermore, tl~ey merged

into _one

peak

in tl~e static limit.

In this _{paper, we} ask whether the

dynamical origins

of the

reentrancy

would be revealed if

a more accurate treatment of tl~e

system

is

employed. Specifically,

_we use tl~e Monte Carlo

metl~od _to determine tl~e _ac

susceptibility, x'(uJ)

_(uJ is tl~e

frequency),

of short _range

Ising

_sys-

tems witl~

asymmetric couphngs

and

again

find _no reentrancy. ^Tl~ere are,

l~owever, qualitative changes compared

to tl~e local _mean field metl~od tl~ere is _no double

peak

structure in

x'(uJ)

that bas been found in the

approximate

treatment.

Another issue of this _paper is wl~at is the temperature

dependence

of the

dynamic specific

heat

c(uJ).

^The motivation for this is _as follows. Within the local _mean field

approximation

^the

maxima in the real and

imaginary parts

_move towards

l~igher temperatures

_on

decreasing

^the

frequency

_[4j wl~ereas _an exact treatment for _six-spin clusters

I?i yields opposite

results. Our

(*)Author

for

correspondence (e-mail: msli@hier.kit.ac.jp) (**)

On leave from Hanoi Technical

University

©

Les

Éditions

_de

Physique

1996

258 JOURNAL DE

PHYSIQUE

I N°2

l/Ionte Carlo results for

c(uJ) qualitatively

_agree with those obtained for the

six-spin clusters,

i-e- similar to that found for the

susceptibility.

The

dynamic specific

heat has been measured in

glasses [8,9j

but net yet ⁱⁿ SG'S. Its

expenmental

features remain to be determined.

We consider trie Hamiltonian

7i =

~j _J~jS~Sj

H

~jS~, _il)

<ii> i

wl~ere

S,

= +1 _are tl~e

spins

located _on the cubic lattice of N sites, <

1j

> denotes summation

aven nearest

neighbors,

the

couplings _J~j

_are Gaussian distributed witl~ the _mean

Jo

and the

dispersion

J. The

magnetic

field considered here is time

dependent

and

H =

Ho

sinuJt.

(2)

TO

study dynamic _quantities

_we

employ

tl~e Monte Carlo simulation witl~ tl~e standard

l/Ietropolis updating.

We first focus _on tl~e _ac

susceptibility

witl~ tl~e eye on a

possible

reen-

trancy

at finite

frequencies.

We follow tl~e

approacl~

taken _in reference

[loi.

In order to deterniine

x(uJ)

_one sl~ould switcl~ _{on a} small

amplitude

field

given by equation (2)

and then start

monitoring

tl~e

niagnetization

^after _some time

to

wl~icl~ is chosen _so that ail transient

exponentials

_con be considered extinct. ive

study

L _x L _x L

systems

^witl~ L=10 and it is

suflicient to cl~oose to as 50000 l/Ionte Carlo steps per

spins (MCS/S).

Once _{we pass} tl~e point t =

to,

,ve average

mit)

_over

typically

2000 4000

periods

To

(To

_"

2~/uJ)

and extract tl~e real

and

imaginary parts

^of the

susceptibility tl~rough Àfjtj

=

Ho IX' sinjùJtj

~"

cosjùJtj j3j

The

periods

we have considered _are

10,

20 and 100 MCS

/S multiplied by

2~. The results of _our calculations for t.he standard SG'S

Jo

"

0)

_are

presented

in

Figure

1. We take

Ho

to be

equal

to

0.01J: smaller values of

Ho

leave tl~e result almost

unchanged.

In ail _our calculations _we choose the cubic

system

with tl~e linear size L

= 10. For

large

^uJ's tl~e maximum of

x'(uJ)

is ratl~er broad. On

decreasing

_uJ the

position, Tw,

of the maximum moves towards lower temperatiires and it

sl~arpens

_up. In tl~e de limit tl~e _maximum

acquires

a

cusp-like

_appearance and Tw

-

Tg,

wl~ere

Tg

^{is tl~e}

equilibrium

transition

temperature.

^The

absorptive susceptibility ~",

_on the otl~er

hand,

bas _a small but distinct

anomaly

around

Tw.

TO obtain the de

susceptibility

_we follow Bl~att and

Young iii] using ^10~ MCS/S

for

equilibration

and anotl~er

105 MCS/S

for averaging. Tl~e results _are

averaged

over 10

samples.

It sl~ould be noted tl~at tl~e

corresponding

maximum of tl~e static

susceptibility t(uJ

=

0)

is located _at T _m 1.2J

(more

accurate àfonte Carlo calculations

involving

tl~e finite _size

scahng

_{give an} estimate of

Tg

m J

iii _).

Tl~e results

presented

in

Figure

1 are very similar to tl~ose obtained for tl~e standard SG

Aui-~Mn~

witl~

z ₌ 0.0298

[12]

^{and for}

Euo.2Sro.sS [13j

^{and also} to wl~at

l~appens

_in tl~e local _mean field

approximation [4j.

Tl~e

question

we ask _{now is} whether

reentrancy pl~enomenon

can take

place

_on short time scales wl~en

only

_some

partial equilibration

of the

system

con be acl~ieved. TO _answer this

question

we

study

tl~e

equilibrium phase diagram

first. The results of _our calculations _ai-e

presented

in

Figure

2. Tl~e

paramagnet (PM

SG and

ferromagnet (FM)

SG

boundary

fines

are obtained from the maxima of tl~e static

susceptibility X(0) (see Fig. 3).

Tl~e SG FM transition

boundary

is obtained from tl~e

dependence

of

x(0)

_on

Jo

for various temperatures

(see Fig. 4). Clearly,

tl~ere is _no reentrancy ^{in tl~e static} regime. We _now calculate tl~e _ac

susceptibility

for _varions values of tl~e _mean,

Jo,

of tl~e

couplings.

Our Monte Carlo simulations

indicate tl~at

(see Fig. 2)

tl~e

ferromagnetically

ordered

phase

exists at low T's for

Jo

> 0.45J.

Figure

5 shows tl~e

T-dependence

of

x'

and

x"

^for

Jo

# 0.5J for 3 diiferent

frequencies.

Tl~e

08

Standard SG

(Jo=0)

X~ WTO

0 6 Ù

ÎÎI

*. *.* o 05

w~.o o'

~ 04

oz

X

0

0 1 2 3 4

kBT/J

Fig.

1. TÎ~e static and

dynamic susceptibility

of the standard SG

(Jo

=

0).

The values of

frequency

are indicated. The system ^{size L} = 10 and the results _are

averaged

_over 10

samples.

The open

squares, black

triangles,

_squares and circles

correspond

to _{uJT0 "} 0, 0.01, 0.05 and ù-1

respectively.

On

decreasing

_uJ the maxima of

x'(uJ)

moves towards the

equilibrium

value of the critical temperature

Tg m1.2J.

5.o

~To

.."

<'

- 0 10

_;." _-." ' 4

o05

w'

w---~ ..~ 0 01 ..

_;"' _o"

__; __.

PM

__,«

_ "" _o^"'

3 .;"_...

1

,

..'

"' ,"""

E~ "

,:."

44 ;.""'

«.""''

i 0 FM

SG

00

00 02 0A 06 08 10

J~/J

Fig.

2. The T Jo

phase diagram.

The solid fines

correspond

to the

equilibrium

_case

(the

_curves

are obtained from maximum of the

susceptibility). PM,

FM and SG denote the paramagnet, _spm

glass

and

ferromagnetic phases respectively.

Within indicated _error bars the SC Fil transition

boundary

is a

straight

fine. The black

triangles,

_squares and

hexagons correspond

to T,,,ax of

x'(uJ)

obtained for

uJTo " ù-1, 0.05 and 0.01

respectively (see Fig.

5 and 6

below).

similar

plots

for

Jo

" 0.7J is sl~own _m

Figure

6. In ail _{cases we} observe

only

_one

peak

in

x'(uJ)

as a function of T

(tl~e positions

^{of tl~ese}

peaks

^for _{various uJ} and

Jo

are

plotted

in

Fig. 2)

and tl~e

reentrancy pl~enomenon

does trot _anse. It appears tl~at the double

peak

structure _m

x'(uJ) predicted by

tl~e local _mean field metl~od

applied

to tl~e Glauber

dynamics

is eitl~er _a

fallacy

of tl~e

approximate

method _or it is real but _at the _same time too weak to be detected

by

260 JOURNAL DE

PHYSIQUE

I N°2

Jo

6 0

Ù

^°°

. . -.

: w +-o

.'h

ù '. +

i ~

, .,

4.0

.,

~

/

"h

; ^{.. '}

ù ^" '

ù

~

~

/

y '

o ~

/ é

"

~

0

0 1

BT là

Fig.

3. TÎ~e temperature

dependence

of the static

susceptibility x(0)

for _various values of Jo. The open squares, black

triangles,

_squares and circles correspond to Jo

= 0, 0.3J, OAJ and 0.7J respectively.

Depending

_on Jo ^the maxima of these _curves

correspond

to the PM SG and PM FM

phase

transition shown _in tl~e

phase diagram

in

Figure

2.

o 8

,o,

» h ~B~~~

1 : w --o---.o o 9

/ 1 _{mû 7}

~ ~

~°

i

ÙÎ~

: 4

: 1

,» -~ à

.° ," _'*.. '.

~ 0.4

:"'

_".

_',

," h _é

' + ..

" é

,,4, h _':

,«"' '«_ _.,

~," *, _: ;4

0 2 ~~ ', _j

"-« h à

«

O.ù

00 02 04 06 08 10

Jo/J

Fig.

4. The

dependence

of

x(0)

_on Jo for _vanous values of T. The open squares, black

triangles,

squares and circles

correspond

to

kBT/J

= 0.3, 0.5. 0.7 and 0.9. The _maxima

correspond

to the SG-FM

transition shown

in

Figure

2.

the Monte Carlo

averaging

_process. In any event, the

origms

of _any reentrancy ^{in SG} systems remain obscure.

Tl~e

specific-l~eat spectroscopy

^bas been demonstrated to be _a useful tool for

studying

relax- ation processes m

supercooled liquids [8,9j.

In _a

typical

_{experiment [8j one} immerses _a l~eater mto tl~e

liquid

and

applies

_a smusoidal current at

frequency

_uJ. Tl~e _power

dissipated

contains

a de comportent ^aiid an ac comportent at

frequency

_{uJ, rangmg} between 0.2 Hz and 6 kHz. Tl~e

ac comportent ^results m

pl~ase-sl~ifted

temperature oscillations and dermes tl~e

complex specific

l~eat.

3 0 ~o-0.5~

~~~

o o~

»..m o 05

wo~ o 1

2 0 '~

i

o zo

°~ ^{i o}

0 0

0 1 2 3 4 5

kBTlà

Fig.

5. The

dynamic susceptibility

for different

frequencies (black triangles,

_squares ^and circles denote _uJTo

= 0.01,0.05 and o-1

respectively)

and Jo=0.5. The results _are

averaged

_aven 10

samples.

5 Jo#0 7J

;~;,

_~To

: ; « . .. o o~

' : o os

« .« ».. ~ oi

k

; ',

'~ _.~ '.

j'

0

~P 1~

3 / [

/

;[

2

j

~~ i

j

i _<

0

0 2 3 4 5

kBT/J

Fig.

6. The _{same as in}

Figure

5 but for Jo

" 0.7.

Birge

and

Nagel

_[8]

pointed

eut tl~at in unfrustrated materials and standard

liquids

tl~e real part ^{of tl~e}

specific

l~eat

c'(uJ)

ares net

practically depend

on uJ and tl~e

imaginary

part is almost _zero. In the _case of

SG'S, however,

and

presumably

of other systems ^with many modes of

configurational relaxation,

the

dynamic specific

heat may be used _as another

probe

of SG

dynamics _I?i

Indeed the

dynamic specific

heat

couples

to the time evolution of

even-spin

correlations whereas the

dynamic susceptibility

is given in terms of

odd-spin

correlations.

By

working

with _a

six-spin

cluster

Cieplak

and Szamel demonstrated I?i ^that the temperature and

frequency dependence

of the

dynamic specific

heat is similar _to that of the

dynamic

262 JOURNAL DE

PHYSIQUE

I N°2

o

C'

/d

/ »'. ..,

~ ;"' m

d / ,' / ^~V'

ç~ ^f_# ,

0 1;,,' _~T~

-- 00

oo'

w ~ o 0025

".+ 005

C~~

0

0 2 3

kBT/J

Fig.

7. The real and

imaginary

parts ^{of the}

dynamic specific

heat for the 3D system

(L

=

10)

as a function of T. The solid fine

corresponds

to the

equilibrium

specific heat which is obtained

by

using 10~

MCS/S

for

equilibration

and another 10~

MCS/S

for _averaging. The black

triangles,

circles

and _squares

correspond

to _{uJTo =} 0.01, 0.025 and 0.05,

respectively.

The results _are averaged _over 10

samples.

susceptibility except

that the former is not aifected

by

_processes

coupled

to the very

longest

relaxation time in the

system.

When _one fixes

uJ and

plots

c' and c" _{as a} function of T then _one

gets

a curve witl~ _a maximum. On

decreasing

_uJ tl~e

position

of the _{maximum moves} towards

lower

temperatures

and it

sl~arpens

_up. Tl~e studies of clusters _are

illuminating

but the clusters do trot

display

_any limite critical

temperature.

^In _[4] we studied the

dynamic specific

l~eat of tl~e

system undergoing

tl~e transition to tl~e SG

phase

at T

#

0 using Glauber

dynamics

and tl~e local _mean field

approximation. Contrary

to tl~e results for clusters tl~e maxima of

c(uJ)

moves towards

l~igher

temperature on

decreasing

_uJ. In this _{paper we} reconsider tl~is

problem

by

Monte Carlo simulations.

In tl~e

experiments

on

liquids,

_an

oscillatory

l~eat _causes

temperature

oscillations. For _a theoretical

study

of SG'S it is _more convenient to define tl~e _same

dynamic specific

l~eat

by inverting

tl~e situation. We tl~us propose tl~at tl~e

temperature

^of tl~e

spin system

varies

periodically

with time:

T(t)

= T ₊

ôTsin(uJt)

,

(4)

where ôT is assumed to be small relative _to T.

The

perturbation

in T results in _a

oscillatory

evolution of the internai energy. The

oscillatory part

^{of tl~e} average

Hamiltonian,

< H >, will be denoted

by

à < H _>. When transients aie out _{one can} define c' and c" _as follows

à _< H >

IN

= ôT

jc'jùJ) sinjùJtj c"jùJ) CosjùJtj j5)

Similar to calculations of

susceptibility

we

skip

first

lo~ MCS/S

and average à < H _{> aven} 2000 4000

periods.

Tl~e

periods

we bave considered _are

20,

40 and 100

MCS/S multiplied by

2~. Our calculations bave been carried out for tl~e 3D

system

of _size L

= 10.

Averaging

of

c'(uJ)

and

c"(uJ)

is

performed

over 10

samples.

The

amplitude

ôT is taken to be

equal

to

0.01J/kB (the

results remain almost the _same for smaller

ôT).

Figure

7 shows the temperature

dependence

of tl~e

dynamic specific

heat for selected values of _uJ. The

longer

tl~e

period

of

oscillations,

tl~e closer c' resembles tl~e

equilibrium specific

heat. Tl~e maxima of c' and c" _move toward lower temperatures as uJ is decreased. It should be noted that the _same

tendency

has been observed in the

6-spin

cluster and iii experiments

on

glasses.

It

is, however, opposite

to what observed within the

dynamic

local _mean field

approximation

_[4]. The failure of the latter

approach

in

predicting

correct

positions

^of minima should be related to the fact that the _energy

ought

to be determined from

two-spiii

correlations but not from the local

magnetizations.

It would be

interesting

to compare the results _on

dynamic specific

heat with

experiments

and to

clarify

the

origins

of

reentrancy.

On the theoretical

side,

_we observe that the local _mean field

approach applied

to the

dynamics

allers

apparent advantages

over the l/Ionte Carlo method calculational

speed

and

large signal

to noise ratio but is showii here to be

quahtati,~ely

unreliable for frustrated

systems.

Acknowledgments

~ie

acknowledge partial support

^from the Polish _ageiicy I(BN

(grant

number 2P302 127

07).

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