Decoupled-mode dynamical scaling theory of the ferromagnetic phase transition

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Decoupled-mode dynamical scaling theory of the ferromagnetic phase transition

Richard A. Ferrell

To cite this version:

Richard A. Ferrell. Decoupled-mode dynamical scaling theory of the ferromagnetic phase transition.

Journal de Physique, 1971, 32 (1), pp.85-88. �10.1051/jphys:0197100320108500�. �jpa-00207027�

DECOUPLED-MODE DYNAMICAL SCALING THEORY

OF THE FERROMAGNETIC PHASE TRANSITION (*)

Richard A. FERRELL

University

^of

Maryland, College Park, Maryland (Reçu

^{le 14}

septembre 1970)

Résumé. ^- Le calcul du comportement critique ^{de la diffusion de} spin s’effectue _par applica-

tion simple de la formule de Kubo. Une approximation ^de découplage permet l’évaluation ^{de la} fonction de corrélation _pour le courant de spin, ^ce qui amène à des résultats identiques ^{à ceux de}

Résibois et Piette, ^obtenus par ^une autre méthode.

Abstract. ^- The dynamical scaling theory ^of phase transitions begins ^{below the} phase ^transi-

tion and extrapolates ^the frequencies, by ^{means of the} temperature continuity ^{at a finite wave} number, through the critical point ^{into the} temperature region above the transition. Applied ^to

the ferromagnet, ^this approach ^{leads to} the observed five-halves critical dispersion ^{at the Curie} point. ^{In the} paramagnetic region ^it yields the critical slowing ^{down of} spin diffusion with a critical exponent of one-third. An alternative approach beginning ^{in the} paramagnetic region ^{has been} developed from the basic statistical mechanics of spin interaction. In the present paper, ^we present

a simplified version of this approach, ^{based on the} fluctuation-dissipation theorem. The relevant form of this theorem is easily ^derived along familiar lines and involves obtaining ^an expression ^for

the spin ^{current. The} spin ^{diffusi on} coefficient is then expressed ^{in terms} of the ratio of the fluctuations in spin ^{current aud} spin density. The correlation function for the current fluctuations

can be evaluated by factoring (decoupled-mode approximation). The results are shown to be equi-

valent of those obtained by ^{the other} methods, ^{and a} simple explanation ^is given for the rise of the

scaling ^function outside of the hydrodynamic region.

1. Introduction. - The

theory of dynamical scaling [1]

of

phase

transitions is based on the observation that _any critical temperature

dependence

^{can be}

expressed

^{as a} function of the correlation

length ^K-1.

When x-1 becomes greater ^{than a}

particular

^wave-

length

^under

Consideration,

say

k-l, K

^{is to be}

replaced by

^{k. In this} way, for

example,

the temperature-

dependent

^linear

dispersion

relation for second sound in

superfluid helium,

(Ok ^oc

(T¡ - T)1’3 ^k,

^{becomes at the}

lambda

point,

^{T =}

TÂ,

the critical

dispersion

^rela-

tion wk

^oc

^k3’2. Similarly, Halperin

^and

Hohenberg [2]

have

applied

^{the same idea to the}

antiferromagnet,

with the same critical

dispersion

^relation

ensuing

^as

in

superfluid

^helium. In a further

application,

^the

present ^author

[3]

^{has noted that} úJk ^oc

k 5/2

should be

expected

^for the critical

dispersion

relation of an

isotropic ferromagnet.

All of the above treatments

begin

^{with the ordered}

phase

^{of broken} symmetry, ^at temperatures below the critical temperature, ^{and are based} upon know-

ledge (by

^{means of}

hydrodynamics

^{or a sum}

rule)

of the temperature

dependence

of the collective modes

(*) ^Research supported ⁱⁿ part by the National Science Foun- dation, ^The Office of Naval Research, and the Air Force Office of Scientific Research. Paper presented ^at the International Conference on Magnetism, Grenoble, September ^1970.

associated with the broken symmetry. ^{Such modes do}

not exist in the fluid systems where there has been

developed

^{a method referred to}

alternatively

^as

« mode-mode »

[4, 5]

^{and as}

« decoupled-mode theory » [6].

^{This method can be}

applied

^to tempera- tures above the critical

point,

^{which makes it more}

direct if this is the

region

of interest. A further advan- tage ^{is that it}

provides

^{a detailed account of how}

K-1

becomes

replaced by

^{k-l as} it becomes greater ^than

K-’.

Calculations of the « mode-mode

coupling »

type ^{have also been carried out}

by

Résibois and co-

workers

[7, 8]

^and

by Wegner [9]

^on

magnetic

systems.

The purpose of the present paper is ^to demonstrate how the same results for the

ferromagnet

can be obtained in the

decoupled-mode theory,

^{as well as to}

present ^a

simple physical interpretation

^{of the main} feature of the results.

The

decoupled-mode dynamical scaling theory

^is

most

straightforward

^{for a} system ^{in which the order} parameter ^is

conserved,

^{as is the case in the ferroma-} gnet. The relaxation of the order parameter ^{is then} studied

by considering

^{the current which is associated}

with the order parameter

by

^the

equation

^{of conti-}

nuity.

^{The form of the current is}

readily

^determined

in each individual case. A

simple physical

argument suffices for the

ferromagnet.

^{Consider two}

spins S(1)

and

S(2)

^{at two}

points

^{1 and}

2, interacting by

^the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197100320108500

86

Heisenberg

energy

S(1).S(2).

^The torque ^exerted

by S(2)

^on

S(1) clearly only depends

upon their

difference,

^{and is}

proportional

^{to the} vector cross

product

This torque ^{can be}

regarded

^{as a flow of}

angular

momentum from one of the

spins

^to the other. Now

passing

^to the continuum

picture,

^we

replace

^S

by

the

spin density

m, and the finite difference

by

^the

gradient.

^{Thus we} find for the current

density

^of

spin flowing

^{across a surface}

perpendicular

to, say, ^a cartesian axis _{« z »,}

az

^{is the} z-component ^{of the}

gradient

operator and ^C is a materials constant

depending

upon the

strength

and _range of the

Heisenberg interaction,

^{as well as}

on the

density

^of

spins.

^{It is}

easily

^verified that eq.

(1),

when substituted into the

equation

^of

continuity

is

equivalent (in fact, identical)

^{to the}

equation

^of

motion for m. Because of the

isotropy,

it suflices to

study

^{one Cartesian} component ^of

Jz,

^{which we}

write as

J3,. (In

^{order to} avoid confusion in the nota-

tion,

^{we use different}

subscripts

^for

denoting

^the

directions of

spin

and direction of

flow.)

^{In terms of}

the Cartesian components ^{of m} this becomes

Eq. (3)

^{can be}

employed

^to determine the relaxation rate of the

spin fluctuations, by

^{means of the}

spin conductivity

^{À. This} transport coefficient _expresses the

steady-state spin

^current which flows in the _pre-

sence of a

gradient

^{of external}

magnetic

^{field. It is}

given by

the Kubo formula

[10, 11 ]

where the

angular

^brackets

signify

the thermal

equili-

brium _average of the

product

^{of the} current at the

arbitrary space-time points

^{2 and 1.}

Eq. (4)

^{is a}

rigorous expression

^{for the}

conductivity

^which

is, however, impossible

^{to evaluate} without the intro- duction of some

approximations.

^{The most}

important

of these is the

decoupling approximation

^described

immediately

^{in the next} section. The remainder of Section II is devoted to

exhibiting

^{the two successful}

predictions

^{of the}

dynamical scaling theory ;

^{i. e.}

the critical five-halves

dispersion

^{relation and the one-}

third-power

temperature

dependehce

of the diffusion coefficient. Section III discusses some of the aspects of the ^more detailed results which can be obtained from the

decoupled-mode theory.

^{Here we also} present

a

general physical

argument

pertinent

^to the numerical coefficient which relates the

dynamical

^{behavior at}

k/K

^{= 0 to that at}

k/x

^{= oo.}

Finally,

^{Section IV}

constitutes ^a brief _summary.

II.

Dynamical scaling.

^- The substitution

of eq. (3)

into _eq.

(4)

^{leads to}

where we have introduced

the time-dependent

^order parameter correlation function

The last line _{of eq.}

(5)

is obtained

by

^a

decoupling approximation,

^{where we}

replace

^the average of the

product

^{ôf four}

magnetizations by

^the

product

^{of two}

averages, each of the form of _eq.

(6).

^The

decay

^rate

of a fluctuation of wave number k is

generally (for

^an

isotropic ferromagnet

^{in which there is a}

spin

^conser-

vation

law)

where the critical

slowing

^{down of the}

spin

^diffusion

can be

expressed by

The unknown exponent ^{x is to} be determined self-

consistently,

^{as well as the}

scaling

^function

Q(x).

This function is normalized in the

hydrodynamic

limit x ⁼ 0

by 0’(0)

^{= 1 and} represents ^{for other} values of its argument ^a numerical correction to the

qualitative

^{rule that x is to be}

replaced by

^{k in the}

non-hydrodynamic

^{limit of}

klic -->

^oo.

For the

qualitative

purposes of this section we can

set

0’(0)

⁼

1,

^{and use} the Ornstein-Zernike form for the

equal-time

correlation

function,

Substitution of _eq.

(7), (8),

^and

(9)

into eq.

(5) gives

The Kubo formula is

easily generalized

^to

give

^{a wave}

number

dependent conductivity, by simply including

a factor _exp

ikz21

^{in the}

integrand

^of eq.

(5). Eq. (10)

then also

applies

^{in the}

non-hydrodynamic region by replacing

^K

by k,

^so that in the two

limiting

^cases

we find

To determine the unknown critical index ^x it suffices to

study

^the

limiting

^case

k/x

^{= oo.}

Requiring

that _eq.

(11)

^{in this case} should lead ^to a diffusion coefficient in agreement ^with eq.

(8),

^and

recalling

that the wave number

dependent susceptibility x(k)

is

proportional

^{to the} correlation function

g(k),

^we

obtain

from which it follows that

Substitution of this value back into _eq.

(8) gives

ⁱⁿ

the two

limits,

^{with the aid} of eq.

(9)

In the

hydrodynamic limit,

if the critical exponent for the correlation

length

^{is v =}

2/3,

^{we have}

D(o)

^oc

(T - Tc)1’3

^{-- a critical}

slowing

^{down of}

spin

diffusion which differs

markedly

^{from the}

predic-

tion of mean field

theory.

^{In the} other limit

of klK

^{= oo}

we have the critical

dispersion

^relation

III.

Scaling

^{function. - The} discussion in the

preceding

^{section has} concentrated on the behavior of the

spin

relaxation

along

^{the two lines}

k/x

^{= 0}

and

k/x

^{= oo. This behavior has} been studied

quali- tatively, neglecting

^numerical constants. But all of the above

equations

^which have been put into ^{the form} of

proportionality relationships

^{can as} well be written

as direct

equalities, retaining

^all of the numerical factors. It is a

straightforward

^{task to} carry this out.

Retaining

^also

u(klk)

^and

writing

^eq.

(5)

^{for an}

arbitrary

^value

of k/K,

^{we arrive at an}

integral equation

identical to that solved

numerically by

^iteration

by

Résibois and Piette

[8].

The solution of this

integral equation

determines the numerical coefficients of eq.

(14)

^not

only along

^{the lines of the two}

limiting

cases k/x ⁼ 0 and _oo, but also

along

^{a line of}

arbitrary

value of

kjx.

The variation of the coefficient as a

function of

k/x

^is

expressed by Q(k/x).

The results of Résibois and Piette

[8]

^{have been}

replotted

^{in this}

form and are exhibited

by

^{the curve R-P in}

figure

^1.

It should be noted that this curve is

completely

monotonic and rises

smoothly

^from

o-(o)

^{= 1 to its}

asymptotic

^value

o-(oo)

^{= 2.3. The}

original

presenta- tion

by

Résibois and Piette

[8]

^of their results exhibits

a

minimum,

^{but it is clear from} the present alternative mode of

presentation

^that this minimum has no

phy-

sical

significance.

On the other

hand,

the considerable rise of

Q(x)

^to

its

asymptotic

^{value can be} understood on the basis of a

simple physical

argument. ^But before

explaining

this argument it may be useful to report the numerical results in a

different,

^but

mathematically

^similar

problem

in critical

phenomena.

^{This is the}

phenome-

non of

paraconductivity,

^{or the onset of}

superconduc-

FIG. 1. ^- Three different scaling ^functions dx) ^{vs. x =} k/K,

the ratio of wave number to inverse correlation length, ^illus- trating ^the qualitative validity ^of the rule of thumb concerning the rise of u(x) ^{outside the} hydrodynamic region x « 1. Curve

« L » illustrates the diffusion in a binary liquid, ^where practi- cally ^{no rise is} expected, ^{while a} strong ^{rise is} expected ^for

curves « Q » ^{and « R-P ». «} Q ^» corresponds ^{to the Fourier} transform of the _square of the Orstein-Zernike function, ^while

« R-P » exhibits the numerical results obtained by Résibois and Piette (reference 8) ^{for the} isotropic ferromagnet.

tivity

^{above the}

superconducting

transition

[12, 13, 14].

Just as in the present

problem

^of

spin conductivity,

the electrical

conductivity

^can be calculated

by

^the

Kubo

formula, using

^the

factoring approximation.

The behavior of the

conductivity

^{in the two limits}

is

easily

^{shown to be}

This can be written in more

precise

^{form as}

where

Co

^{is a} certain numerical constant and the

scaling

^{function for the}

superconductor

^{is normalized}

by u s( 0)

^{= 1. The}

integration

^{can be carried out in}

closed form

[15]

and exhibits a monotonic rise in

us(x),

^{with the}

asymptotic

^value

The exact coincidence of this value with the Résibois- Piette

[8]

^{value for the}

ferromagnet

^of

u(oo)

^{= 2.3}

is,

of _course,

nothing

^{other than a} numerical accident.

The strong ^{rise in both}

scaling

^functions

is,

however,

a

qualitative

feature which we believe to be common to

problems

^{of this} type.

The

general

^{feature is} the factor of

g(k)2

^{which has}

entered the

integrand

of eq.

(5)

^{as a result of the}

decoupling approximation.

This factor tends to

dominate in

integrals

^{of this} type, ^and

corresponds

^to

the _square of the Ornstein-Zernike correlation function

88

We now note that the correlation

length

^{enters the}

problem only through

^the

quantity

^{2 K. Thus it}

is,

as a

rough

^rule

of thumb,

^{2 x and not}

simply K

^{which we}

should

expect to be replaced

^{in the}

non-hydrodynamic limit by k

^for

example,

^{in the}

superconducting problem

where in the

hydrodynamic

^limit

Âs(O)

⁼

Co K-1,

we should substitute ^K

k j2

^{and obtain}

This would lead to the result

Os(oo)

⁼

2.0,

^{not far from}

the exact value 2.3

quoted

^above.

Similarly,

^{in the}

hydrodynamic

^{limit of the}

ferromagnet according

to eq.

(11)

^and

(13), Ào

^oc

^K-3/2. Again using

^{the rule}

of thumb K -->

k/2,

^{we find in this case}

again

^not

seriously

^{removed from the correct value}

of 2.3.

Clearly

^this

rough

^{rule of thumb can be}

expected

to

give only

^a crude indication of the trend of the

scaling

^function

Q(x).

^{But we} do believe that it is

qualitatively

^correct and that it does

explain why u(x)

has to increase as x increases. A measure of the error

in the rule of thumb is

given by

the Fourier transform of the _{square of} the Ornstein-Zernike function

where we have introduced the

scaling

^function

This function is

plotted

ⁱⁿ

figure

^{1 as curve}

« Q ».

We note that the

asymptotic

^{value is}

QQ(oo) = n

so that the rulc of thumb underestimates the rise

by

the factor

2/n,

^{or 38}

%.

^{A further test} of the rule of thumb is found in the critical

slowing

^{down in a}

binary liquid,

^{where one} of the modes in the decou-

pling approximation

^{is the} non-critical shear mode.

Thus

G(r) only

^{appears once in the}

integrand,

^{so that}

the rule of thumb would

require replacing

^x

by k,

without the factor of two. In other

words,

^{we would} expect

O’L( (0)

^{= 1.0. The actual}

scaling

^{function is}

plotted

^{as curve « L » in}

figure

^{1. The}

asymptotic

value in this case is

1.18,

^{so the rule of thumb under-} estimates the rise in this case

by

¹⁵

%, [5, 6].

These

examples

demonstrate that the rule of thumb

presented

^{here can in no} way be used to obtain

precise

numerical results for the

scaling

function. But it can

be

expected

^{to serve as a useful}

qualitative guide.

In

particular,

^it

explains

^the strong rise in the

scaling function,

^{which is the}

principal qualitative

^{feature of}

the Résibois-Piette

[8]

calculations.

IV.

Summary. -

In summary, ^we have shown how the

study

^of the fluctuations in the

spin

^current

and the use of the Kubo formula

permits

^the

theory

of

dynamical scaling

^{to be put into a}

particularly

^sim-

ple

^and transparent form. ^{We have} furthermore noted

a

rough

^{rule of thumb which}

explains qualitatively

the rise in the

scaling

^{function outside the}

hydrody-

namic

region.

^The interested reader is referred to the literature for further discussion of the

theory [16]

as well as for a survey of the

experiment

^{status of the}

subject [17, 18].

References

[1] ^FERRELL (R. A.), ^MÉNYHARD (N.), ^SCHMIDT (H.), SCHWABL (F.) ^and SZÉPFALUSY (P.), Phys. ^Rev.

Letters, 1967, 18, 891 ; Phys. Letters, 1967, 24A, 493, ^{and Ann.} Phys. (N. Y.), 1968, 47, ^565.

[2] ^HALPERIN (B. I.) and HOHENBERG (P. C.), Phys. ^Rev.

Letters, 1967, 19, 700, ^and Phys. Rev., 1969, 177,

952.

[3] ^FERRELL (R. A.), ^{« Field} Theory ^{of Phase} Transitions »,

International Symposium ^on Contemporary Phy- sics, Trieste, 1, 129, IAEA, Vienna, 1969. Uni-

versity ^of Maryland Department ^of Physics ^and Astronomy ^Technical Report ^{n° 848}

(July

1968).

See bibliography for other work in the field.

Proc. of the Eleventh Intern. Conf. on Low Tem- perature Physics, University ^{of St Andrews Prin-} ting Department, 1968, p. 27.

[4] ^SWIFT

(J.)

and KADANOFF (L. P.), ^Ann. Phys.

(N.

Y.), 1968, 50, 312; ^KADANOFF (L. P.) ^{and SWIFT} (J.), Phys. Rev., 1968, 166, ^{89. See also FIXMAN}

(M.),

J. Chem. Phys., 1967, 47, ^2808.

[5] ^KAWASAKI (K.), ^{to be} published, ^and Phys. Letters, 1969, 30A, ^325.

[6] ^FERRELL

(R.

A.), Physical ^Review Letters, 1970, 24, 1169.

[7]

RÉSIBOIS (P.)

^{and DE LEENER} (M.), Physics Letters, 1967, 25A, ^65.

[8] ^RÉSIBOIS

(P.)

and PIETTE

(C.),

Phys. ^{Rev. Letters} 1970, 24, ^514.

[9] ^WEGNER

(F.),

Zeitschrift fur Physik, 1968, 216, 433 ; 1969, 218, ^260.

[10] ^CALLEN

(H.)

and WELTON (T.), Phys. ^Rev., 1951, 83, ^{34. KUBO} (R.), ^J. Phys. ^Soc. Japan, 1957, 12, 570; Lectures in Theoretical Physics, ^{Vol. 1}

(Interscience,

^New York, 1959), ^{ch. 4.}

[11] ZWANZIG

(R.),

^{Ann. Rev.} Phys. Chem., 1965, 16, ^67.

[12] ^FERRELL (R. A.), ^{J. Low} Temp. Phys., 1969, 1, ^241.

[13] ^SCHMIDT

(H.),

^Z. Physik, 1968, 216, ^336.

[14] ^ASLAMAZOV (L. G.) and LARKIN (A. I.), Phys. Letters, 1968, 26A, ^238.

[15] ^CYROT

(M.)

and FERRELL

(R.),

^Jour. of Low Temp.

Physics (to ^be

published).

[16] ^FERRELL

(R.),

^« Decoupled-Mode

Dynamical

Scaling Theory ^{of Phase} Transitions », ^{Univ. of Md.}

Department ^of Physics ^and Astronomy ^Techni- cal Report ^{n° 71-005}

(July,

1970) and Procee-

dings ^{of a} Conference on Dynamics ^{of Critical} Phenomena, ^Fordham Univ., ^New York, ^June 1970, ^Editor J. Budnick.

[17] ^COLLINS

(M.

F.), MINKIEWICZ (V. J.), ^NATHANS

(R.),

PASSELL (L.) and SHIRANE

(G:),

Phys. Rev., 1969,

179, ^417.

[18] ALS-NIELSEN (J.), ^BNL preprint 14783 and Phys.

Rev. Letters (to ^be published).