Critical scaling near and far from Tc ; from ferromagnets to heavy fermions

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Critical scaling near and far from Tc ; from ferromagnets to heavy fermions

J. Souletie

To cite this version:

J. Souletie. Critical scaling near and far from Tc ; from ferromagnets to heavy fermions. Journal de

Physique, 1988, 49 (7), pp.1211-1217. �10.1051/jphys:019880049070121100�. �jpa-00210803�

Critical scaling ^{near and far from} Tc ; ^from ferromagnets ^to heavy

fermions

J. Souletie

Centre de Recherches sur les Très Basses Températures, CNRS, ^BP 166X, 38042 Grenoble Cedex, ^France

(Requ ^{le 4 décembre 1987,} révisé le 3 mars 1988, accepté ^{le 8 mars} 1988)

Résumé.

Nous montrons que les hypothèses qui conduisent aux lois d’échelle dans la description ^des

transitions de phase impliquent ^des singularités essentielles dans la limite Tc

^{0 et si} Tc ^0, des lois de

puissance ^{de T} qui peuvent ^{avec succès} s’appliquer à des situations concrètes : par exemple ^les fermions ^lourds.

Nous discutons aussi les effets d’un cross-over dimensionnel, de la dilution et du désordre.

Abstract.

We show that the assumptions of critical scaling imply ^essential singularities in the limit where

Tc

0 and power laws of the temperature ^for Tc ^{0 which are} appropriate ^to describe realistic physical

situations such as e.g. heavy fermions. We discuss also the effects of a dimensional cross-over as well as that of dilution and disorder.

Classification

Physics ^Abstracts

75.40

75.40D

75.10

75.10H

The two assumptions ^{of static} scaling.

We write the Gibbs potential ^{of N} interacting particles ^of moment g ^as that of n independent particles ^{of moment} 4eff :

By definition the coherence length03BE is the size of these particles ^and n ’" g - d in ^{dimension d. We}

assume similarly ^that 03BCeff - 03BEd ^Then

from which we obtain

or

where f (x) ^{is an odd} function of X. In the para-

magnetic regime the coefficients of the expansion ^of

q (M) should be analytical ⁱⁿ J/ T if J is the interaction at least within some convergence radius

j/Tc ^{which defines} Tc. ^{The idea of} scaling ^{is to}

mimick the pathology on C ^when J/T ^reaches

JIT, by lower values as a power law of the variable t :

We assume therefore that

with the effect that the consequent pathologies ^of

the different susceptibilities ^or thermodynamic quan- tities are either described or dominated by ^a power law of t. As there are only ^two assumptions (Eq. (2)

and (5)) ^{in the so-called static} scaling hypothesis [1],

there is at least one relation between any three exponents ^{which we} may define. Thus from

equation ^{3 we} have for the susceptibility y ^{and the} magnetization ^{M :}

Similarly differentiating equation (2) ^{with respect to}

T we obtain for the entropy S ^{and the} specific ^heat Cp:

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049070121100

1212

hence

Sweeping ^the J / Tc ^ratio.

Equation (5) ^{like all} associated power laws has ^at least three parameters (03BEo, T, ^and v). ^{A convenient}

way ^to eliminate one parameter ^{is to} differentiate to reach e.g. :

A plot ^of PC (T) ^{vs. T yields a straight line whose}

intercept ^with P e (T)

^{0 is Tc and with} P 6 (T) ^{= 1 is}

Tc ^{+ nJ such that nJ}

vTc (see Fig. la ^and 2a).

Depending ^{therefore on the} quantity ^{X which we}

consider (X ^{= 03BE,} C p ^{T 2, X T, M... ) we have one of a}

stack of straight ^{lines of} intercept Tc ^{from which we}

may define ^an energy which directly provides ^the corresponding exponent [2, 3]. ^{We have :}

Fig. ^1.

^{Plot of} PX(T) _ - ^{a In} T/a ^{In X vs. T} showing

how T, and the different characteristic energies ^xJ (aJ ^and gJ for X = C p T 2 ^{and X T} respectively) ^{can be}

determined from the experimental data when X - Xo(I -

Tcl7)-X’ITC ^for Tc > 0 (a), ^for Tc = 0 (b) ^{and for} Tc : 0 (c).

Fig. ^2.

(a) - ^{a In} T / a ^{In X T vs. T in the} Heisenberg ^3-d system KzCuCI4.2HzO [9, 2] ^and (b) Log (X T/C ) ^vs.

-

In (1 - T,,IT) showing how the Curie constant can be obtained from the data as the intercept ^{with the x}

^{0 axis.}

Between any such three energies ^{there is at least}

one relation which follows from the relation between the corresponding exponents : e.g. ^we deduce from

equation (10)

and the general ^{form of X is}

Once Tc ^{is known a} plot ^of log X ^vs.

log (1- TCIT) provides ^a straight line whose inter- cept ^{with the} log (1- Tcl T)

^{0 axis is} log Xo (see Fig. 2b). The value of Xo should be consistent with the high temperature paramagnetic limit which it characterizes i. e. for example x T - C =

g2 A 2S(S+ 1 )/3 k, C --., Co - interatomic distance.

Equations (12) ^or (14) ^{are not a new} hypothesis

but a consequence ^of the validity ^of equation (5).

This can also be seen by expanding e.g. X T ^{in terms}

of T- l. Then

with af = y ( y + 1 ) ... ( y ⁺ $ -1 ) T,,flf ! ^{The en-}

ergy y Tc ^can be determined at different orders in f

from the ratio ap/a p _ ^{1 of any} successive terms as in the method of high temperature expansions (see e.g.

Ref. [1]). ^If equation (5) ^{is true,} [yT, ]f ^{does not} depend ^on f, we ^have gJ

y Tc ]I ^and equation (12)

follows. On a realistic lattice, equation (5) ^can only

be perfectly ^{true at a scale} larger ^{than the} interatomic distance. On such a lattice, ^{with near} neighbour interactions, f ^{moments are} involved in the interac- tions which contribute in T- p _{and at} can be estimated

by enumerating ^{all the} corresponding graphs. ^{In this}

case [ y TC]1 ^{is the mean field Curie} temperature since to order one in T-1 we have

Theoretically ^{it is obtained} by summing ^the pair

interaction J, upon the shell of first neighbours. ^But

the goal ^of high temperature expansions ^{is to} ap-

proach ^{as much as} possible ^the [y7"c]oo limit in order to determine a renormalised value of gJ, averaged

over a large ^{number f of} spins, ^{which has a chance to}

be independent of the lattice structure and the same

for the same space and spin dimensionality. ^{If such a}

limit exists the universal value of y is gJ / Tc, ^{it is} bigger the smaller Tc ^{and it} diverges ^if Tc ^cancels.

Alternatively, gJ ^{can be viewed as the} renormalised value of the interaction in the continuum where

equation (5) would be valid at all scales. In the

figures ^{1 and 2a it} corresponds ^{to the} intercept ^{of the} tangent ^at Tc ^{to the} P xT ( T ) ^{curve with the}

y = 1 axis. The figure 3b shows that y is correlated

Fig. ^3.

(a) Correlation of the measured exponent ^value with the J/Tc ^{ratio in} LiTbxYl -xF4. ^{We take J}

TMF

xT,(x = 1). ^{The data are} from reference [11] ; (b) ^corre-

lation of the exponent ^{value with the} 1 / Te ^{ratio in}

different ferromagnetic (F) ^model systems compiled ⁱⁿ

reference [6]. The data for the 3-d spin glass ^{are from}

reference [10].

with the 7/7c ^{ratio in} different model systems ^which confirms the validity ^{of the second} scaling hypothesis (Eq. (5)). ^{It tells us} also that [YTc]oo ^{is not very}

different from [ytcl, ^whose scatter, associated ^with the different lattices used, appears ^{in the error bar}

on TMF. ^This explains why equation (5), ^{with the}

renormalised value of gJ

[yTc]. ^{measured at} 7e, ^{can account for the} susceptibility ^{measured in the}

S

1/2 Heisenberg ^3-d ferromagnet K2CuCl4.2 H20

at temperatures up ^to 4 Tc (see Fig. 2). Actually [ytcl. ^{is a} much better estimate than [yTc], to ^the TMF ^measured by experimentalists. ^{Such a result} [2, 9] ^{which is} supported by experimental (2-5) ^and

theoretical (6, 7) evidence in other Bravais and self- similar [8] ^{lattices is often obscured} by ^{an unfortu-}

nate linearization of the scaling ^{variable. This trans-}

formation of t into t’

TITC - 1 restricts the appli- cability ^{of the} scaling assumption ^{into a critical} regime (T - Tc)lTc ^{: 8 which} vanishes if Tc ^cancels.

It has dramatic consequences in all studies which like

ours are interested in the Tel T --. 0 limit of the model.

Given a set of finite energies gJ, dnJ, etc..., ^we

see in the figure 1 that the associated stack of

straight ^lines which aims Tc

⁰ separates ^different regimes ^of equation (13) :

1. the case Te:> 0 is that of traditional phase

transitions. The coherence volume diverges ^at Tc ^{with an infinite} slope ^since a In a In T -+ oo , triggering associated pathologies of y T, C p T 2 9 ... In

order to keep ^the entropy finite and positive, thermodynamics impose ^{d u ,1 which in turn re-} quires a -- 1 and

where we have used equations (8), (12) ^and (13).

The exponents ^{increase with the} j/Tc ^{ratio as} equation (12) predicts : this is illustrated figure ^3b

with classical data of the literature (Refs [6, 10]) ^and figure ^{3a with} experimental ^{data in} LiTbxYl -xF4 [11]. This latter result, by the way, which implies

that the exponent varies with dilution raises a

problem ^{which we} will rediscuss later on (see ^also Ref. [12]) ;

2. for Tc

^{0 we have a In} T/a In Cd

kT/8 (see Fig.1b) ^which implies that 03BE d ^{is an} exponential ^of

8/T. Consequently :

where we know from equation (13) ^that

and 5 is the exponent which monitors the magnetiza-

ton at Tc : ^{we have} H/Tc ^M

^{M- a where 5 = 1 +} l’ / f3 ^{follows from} equation (4). ^{In the case of a} ferromagnetic interaction we want also the magneti-

zation to be saturated at T

0 which happens ^{to be}

also the critical temperature. ^{This would then} impose

5

oo and gJ

^aJ

^{dnJ = 6} which restitutes the well-known results valid for the linear Ising ferromagnet

and

The essential singularity ^{is obtained as} the limit of

a power law whose exponent diverges

Examples of this behaviour are the anisotropic spin glass Feo.3M&>.7CI2 [3] ^{or the 2-d} Heisenberg ^fer- romagnet [20] ^{or its} realisation by 3He layers ^ad-

sorbed on graphite [21] ;

3. the case Tc ⁰ (see Fig.1c) ^{describes a coherence} length which still diverges ^{at T}

0 because J/T diverges ^{but with a finite} slope (aln/alnT=

Cst.) ^{because we are not at the} pole 7/7c ^{which is in}

the unphysical range of negative temperatures. ^We

have e.g. from equation (14)

1214

It follows from equations (12) ^and (13) ^that

dnJ

(2 - a ) Tc ^{which makes a}

2 if Tc -- ^{0. The}

entropy (Eq. (8)) ^is well behaved at all physical temperatures starting ^like

We have for the specific heat and the susceptibility

and

where a

2 and 2 - « y « - 2 follows from

equations (13) ^and (17). ^{For each a,} therefore, ^we

have two windows (y

0 and y 0) of solutions whose susceptibility ^{cancels or} diverges ^{like a} power of T depending ^{on the} antiferromagnetic ^{or ferro-} magnetic character of the interaction. Despite ^the

fact that 6 diverges ^{at T}

^{0 the true} scaling point

remains Tc ^{0 where C} and all its derivatives are

singular.

This case describes situations which pertain ^to paramagnetism ^{and true} paramagnetism ^{is obtained}

in the limit JITC -+ 0 where the interaction vanishes and we have X T --+ C, ç --+ ç 0’ ^{etc. The} figure ^4a displays typical shapes ^{of the} susceptibility ^be-

haviour. The Curie-Weiss law is obtained for

JITC = 1 ^{and J and} Tc ^{0. The} specific ^{heat is}

shown on the figure ^{4b. Notice} that, ^{as it} depends ^on

3 parameters only, ^{if we fix the} entropy ^{the total}

curve is determined by ^the specification ^{of two} points (or ^of Cp(T .ax)). ^These solutions for Tc ⁰

present many features which would permit ^to

rationalize the properties ^of nearly ferromagnetic

systems ^{and of} heavy ^fermions.

Fig. ^4.

(a) ^Different shapes ^{of the reverse} susceptibility

obtained for T. a 0 and J a 0. The dash-dotted lines (for Tc

0) mark the limit between d dc (inner curves) ^and

d

d, (when Tc >. 0) ; (b) typical shapes ^{of the} specific

heat predicted ^when Tc ^{0. The curves} correspond ^{to the}

same total entropy ^{R Ln 2.}

The figure ^{5a for} example ^{is an} adjustement ^of

c p/ T

^{A/ (T + I Tc} )3 ^to the data of CeCu6 [13].

The figure ^{5b shows that the same} picture ^{is consist-}

ent with the susceptibility ^{since the} CP/XT ^ratio

varies like (T - Tc )-2 ^{with the same} Tc’" - 6 K.

Note however that if the entropy ^is imposed ^{to be}

R In (2 S + 1 ) and the Curie constant C

92/-k 2S(S+ 1 ) the model leads in the T -+ 0 limit to a Wilson ratio

for S = 1/2 rather different from the experimental

value of 0.7.

Fig. ^5.

(a) ^The Cp/T vs. ^{T data in CeCu6} (Ref. [13]) ^can

be fitted to an expression ^A ( T - Tc)-3 ^with T, = - ^{5.8 K :}

solid line on the figure ; (b) ^the Cp/XT ratio in the same

system is consistent with an expression ( T - Tc)- 2 ^{with the}

same T ^{value as} evidenced from the linearity ^{in a}

XT/Cp ^{vs. T plot.} The correction which aligns ^the points ^at T > 2 K is deduced from the CP/T ^{vs. T plot of}

figure ^5a.

Disorder.

The plot ^of P x (T) ^is appropriate ^{to discuss a}

dimensional cross-over between a higher tempera-

ture dimensionality d1 and d2 =1= d1. ^The figure ^6a

illustrates one plausible scenario which depends ^on whether dl ’ dc. ^{In the} logic ^{of the} above discussion

we associate the lower critical dimensionality dc ^with

the case where Tc = 0 ^{and the case d} dc ^with Tc ^{« 0. The} figure 6a features a situation where

d1 > d2 which could be appropriate ^{to describe} e.g. ^a size effect : _{say when} the coherence length ^increases

over the thickness of the sample. ^The figure ^6b

features a case where d1 d2 ^{which could} apply ^to

the case of weakly coupled ^{chains or} layers. ^On decreasing ^the temperature the interchains (inter- layers) interaction when added over the increasing

volume of the coherent cluster, ^sums up ^{to a} value sufficient to couple ^{with the} other chains (layers)

and make the system ^3-d. K2CuF4 ^{is a known}

example ^{of a} layered system ^often presented ^{as an}

Fig. ^6.

(a) Description ^{of a} dimensional cross-over from

dl to d2 > d1 (d, is the lower critical dimensionality d,, ^{in this} example) ; (b) experimental data in the S = 1/2 Heisenberg ferromagnet K2CuF4 (Ref. [14]) ^which

exhibits a cross-over from 2-d to 3-d behaviour.

illustration of Heisenberg 2-d behaviour [14]. ^Indeed

the P x (T) ^{curve is in the high} temperature regime

consistent with an interpretation ^{in terms of a}

transition at Tc

^{0 but on} lowering ^the temperature it deviates to reach a finite Tc ^{with a} slope ^consistent

with the known value of the exponent ^for Ising systems ^{at 3-d : y}

^1.25 [2, 14] (see Fig. 6b).

The same framework is useful to describe the effect of dilution. A dilute ferromagnet ^at p >

p, ^can be characterized by ^its structural coherence

length 03BEs = (p - Pc)- v’. According to current pic-

tures about percolation, ^{at scales )} : 03BE s ^{there is no}

information which would permit ^to situate the system with respect ^to pc ^{and it} therefore behaves with the fractal dimensionality ^{of the} backbone : the latter is often viewed as a network of red bonds which

account for most of the size of the object, connecting

at blobs where most of the mass is sitting [15,16].

Elaborating upon this model, ^de Jongh [17] ’reasons

that the magnetic correlation length ^will first have to

develop along the red bonds and exhibit therefore a

1-d character. This will singularise Ising spins ^which

still exhibit a transition at T,

^{0 from other models}

for which dc

1. In this picture a cross-over to three dimensional behaviour as sketched in the figure ^6b

will occur at T, where 03BEid - 03BE1-d - 03BEs-03BE3d, ^{and the}

transition is reached after the exponent has reached its critical value which happens only ^{within a critical} regime Tc ^T Tx ^{which is the narrower the closer}

we are to p,. The data of reference [10] ^shown figure ^3a exhibit, ^{as the} percolation threshold is

approached, ^a correlation of the measured y (which

increases by ^{a factor} 2.5) ^{with the} JIT, ^{ratio. This}

result which agrees with ^our equation (12) ^con-

tradicts the expectation ^from theory and simulation that exponents ^should vary only by ^a discontinuous step if the class of universality ^{is modified.} Probably

the exponent experimentally reported ^{is an effective} exponent whose increase traduces the expected

increase in the curvature of the Px (T ) ^{curve on}

approaching pc. We ^are presently exploring ^this

possibility ^{which seems the} only way ^to reconciliate

theory ^and experiment ^{in this case} and in any other

case where we anticipate ^a modification of the

7/Tc ratio without a modification of the universality

class.

Finally ^{a smaller} disorder such as introduced in e.g. amorphous magnets ^{can be} approached by introducing ^{a finite} distribution of J values. At high

temperatures ^the PX (T ) ^line aims Te> ^{which is}

associated with the average interaction (J). ^The

true Tc corresponds ^to Jc W’ ^{If the class of} universality ^is conserved, ^which depends upon the realisation of some kind of Harris criterion, ^{we will}

observe a cross-over between two Px (T ) lines which

are affine in the ratio of (j)/Jc ^{as shown on the} figure ^{7a. This means an effective} exponent Yeff (T)

which starting ^{from its universal value} ’Ye at 7e ^{increases to a maximum value to match the} slope

of P x (T) ^at the inflexion point which characterizes the cross-over regime and decreases again ^to

Y eff ~ yc at temperatures larger ^{than the} width of the

distribution, ^where (J) ^{can be defined. This be-}

haviour of the exponents has been observed in several amorphous ferromagnets [18] (see Fig. 7b).

Fig. ^7.

(a) ^The Px (T ) dependence anticipated ^{in the}

presence of small disorder ^as explained ^{in the text. The} slope ^{in the} cross-over regime corresponds ^{to an increase}

of the effective exponent ; (b) ^{the effective} exponent measured in the amorphous ferromagnet Fe20Nis6B24 [16].

The y (T) (,) ^are deduced from the original ^y

(0, 0 ) by ^the correspondance y - 1

(y * - 1)T/7c ^{which ac-}

counts for the different scaling variables used (linear ^and

non linear).

Dynamics.

Parallel to the static scaling hypothesis (Eqs. (3) ^and (5)) ^the dynamic scaling hypothesis [1] ^{fixes the}

relaxation time T which it is necessary ^to wait in order to observe the system ^at equilibrium. ^The assumption ^is

T is another line of the

1216

stack displayed figure ¹ which defines another en-

ergy :

The divergence ^{of T on} approaching Tc (critical slowing down) ^{has the} shape ^{of that} of e ^{and is} displayed figure ^{7b. In} particular ^{we find}

As a consequence ^{of this} hypothesis ^{it is} impossible

to observe a system ^at equilibrium ^{within a} given

time _Tm associated with the measure below a certain

blocking temperature TB(Tm) such that T(TB) =

T m. We have

The dependence ^of TBI (J ^on Tel (J ^{for all} possible

values of Tc -- 0 ^{is shown on the} figure ^{8a where we}

have assumed that z is fixed. TB, ^{of course, is} larger

than 7e and the situation which is blocked strictly speaking ^{is a} paramagnetic situation. But a typical length scale has been frozen ^{in the} system : ^the correlation length ^at TB ^{which is}

In the spirit ^{of the} scaling equation (1) ^{we can}

consider the system ^{as an} assembly ^of NI 6d super-

paramagnetic grains ^of moment 6d which will react as such to further perturbations ^{or more} precisely ^as

a ferromagnet ^at scales 6 -- CB ^{and as a} 3-d paramag- net at larger ^scales.

Fig. ^8.

^The divergence of the relaxation time

in the different regimes (1 ^to 4) ^that equation

To(1 - Tc/T)-znJ/Tc ^can describe for Tc -- ⁰ is schematized in the figure ^{b. The} ergodicity ^{is lost at a frontier} TB ^{in the} paramagnetic regime ^where

T m the time available for an experiment. ^The dependence ^{of the} blocking temperature TB ^on the critical temperature Tc ^{is shown in} figure ^{a for two} different values of Tm (A ^and B).

Other hypotheses.

The static and dynamic scaling hypotheses ^are by ^no

means the only ways to account for the divergence ^of

a coherence length ^{at a} transition. There are in the literature other possibilities ^a number of which can

be expressed ^{under one or} the other form of the

general ^four parameters expression

where X can be X T, Cp T 2, ^{T etc. For} TF

^{0 this}

is the expression ^{that Binder and} Young [19] prom- oted to fit the dynamics ^of transition at Tc

^{0. The}

case Tp

0, ^{03C3 = 1} restitutes the exponential ^sin- gularities ^{which we found to} be the natural answer of static and dynamic scalings ^for Tc

0. For a = 1 and TF ^{finite we have the} phenomenological ^Fulcher

law and the case Q = 1/2 is that of the Kosterlitz- Thouless transition. This is now PX(T) ^{which be-}

haves as a power law :

We may in the spirit ^{of the} previous analysis

assimilate Px(T) ^{with its} tangent ^{in a} limited range of temperatures and define T,*(T) ^and J*(T) + Tc* ^{at the} intercepts ^{with the} Px(T)

^{0 and} Px(T) ^{= 1 axes} respectively. Equation (21) ^then

describes the divergence ^{of the effective} exponent

J*lTc* ^when Tc* approaches ^{a finite} TF whereas the

same effect in equation (19) resulted from the cancel- lation of Tc ^{at fixed J. This} aptitude ^{could make the} equation (25) well fitted to describe a cross-over.

Conclusions.

The PX(T ) representation helps ^{visualize the differ-}

ent regimes ^of the mathematics underlying ^the phenomenological scaling theory. ^{The classical re-}

sults are recovered together ^{with a} number of less obvious points : ^for example ^a correlation _appears between the value of the exponent ^{d v and the}

//7c ^{ratio and an essential} singularity ^{comes out as}

the natural Tc

0 limit of the model. We do not know of any previous attempt ^to associate the

Tc ⁰ regime of the mathematical expression ^with physical situations despite the fact that the Curie and Curie Weiss laws are associated with this case. The

efficiency of the model is surprisingly good ^{in the}

case of the heavy ^fermion problem ^{as we will show in}

more detail in another paper. Basic to our point ^is

the ability of the model to provide ^{a sensible account}

of the high temperature situation when the non

linear variable J/T - JI Tc is used. It is clear that the

development ^{of the} notion of critical regime ^has

been hampered by the fact that effects associated

with the linearization of the scaling variable have

been confused with real physical limitations. Once this is accepted ^{it is} possible ^to reintroduce the notion of a critical regime ^{on a} semiquantitative basis, by discussing ^{as we did the effect of a cross-}

over or that of major ^{and minor} disorder. Notice that the aptitude ^{of the model to describe the} high temperature situation is also reflected in the aptitude

which it shows to account for the Tc

^{0 limit which} corresponds ^{to the same limit t = 1 of the non}

linearized scaling ^{variable t.}

Acknowledgments.

The author acknowledges discussions with J.

Flouquet, ^J. Hamman, ^K. Matho, ^{J. J.} Prejean,

J. L. Tholence and C. Tsallis. He is indebted to G.

Deutscher who pointed ^out several similarities with his own treatment of localization effects in clusters

near the percolation ^threshold (Ref. [12]) ^{and to}

J. P. Renard who attracted his attention _upon the results in the LiTbxYl-xF4 system.

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