Physics Letters A 165 ( 1992 ) 96-99

North-Holland PHYSICS LETTERS A

Scaling function near a fluctuation-induced tricritical point

M.L. Lyra Departamento de Fisica, Universidade Federal de Alagoas, Maceid, AL 57061, Brazil

and

M.D. Coutinho-Filho Departamento de Fisica, Universidade Federal de Pernambuco, Recife, PE 50739, Brazil

Received 16 September 199 1; revised manuscript received 2 March 1992; accepted for publication 4 March I992 Communicated by A.R. Bishop

A free energy functional containing a critical n-vector field coupled to a non-critical scalar one, which commonly appears in the

spin-phonon problem and near the ferromagnetic transition of the Hubbard model, is used within a field theoretical and renor-

malization group framework to obtain, to first order in t= 4-d, the universal scaling function of the equation of state near a

fluctuation-induced tricritical point.

1. Introduction

It has been reported by many authors [ 1 ] that the nature of a phase transition, as predicted by mean field approaches, can be modified when fluctuations are taken into account. In the spin-phonon problem, for example, where the critical spin fluctuations are coupled to non-critical site and volume fluctuations, constraints imposed on the volume fluctuations can dramatically change the characteristics of the tran- sition [ 2 1. If the specific heat critical exponent (Y is positive in the unconstrained system, a tricritical point emerges with no renormalization of the critical exponents [ 3 1. On the other hand, if (Y < 0, a fluc- tuation-induced tricritical point (FITP), with ex- ponents renormalized a la Fisher [ 41, takes place. The constraint induces an effective long range in- teraction whose sign plays a fundamental role on the characterization of the nature of the transition [ 5 1. In the O(n) field theory with cubic anisotropy [ 6 ] a FITP with cubic exponents arises for n<4. Re- cently [ 71, a FITP with renormalized Heisenberg critical exponents was predicted to occur near the ferromagnetic transition of the Hubbard model in three dimensions when constraints are imposed on

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the uniform charge fluctuations. These FITPs rep- resent a special class of tricritical points due to the fact that they do not appear in the mean field theory and also do not obey the e6 field theory commonly used to study the conventional tricritical points [ 81. Recently [ 91 , #I fluctuation-induced first-order tran- sition was also predicted to occur in unconventional superconductors (high-T, oxides and heavy-fermion systems).

In this work we will determine, to first order in E =4-d, the universal form of the equation of state near a FITP with renormalized 0 (n) critical expo- nents. We will use a free energy functional contain- ing an 0 (n) critical field coupled to a constrained non-critical scalar one such as appears in problems like the spin-phonon and the Hubbard model. In section 2 we report, to leading order in e, the renor- malization group results for the tricritical exponents and a generic equation of state near the critical points. In section 3 we use scaling arguments to write the equation of state near the FITP in a universal form and compare the scaling function with that for a standard 0 (n ) critical point.

*’ For standard type superconductors, see ref. [ lo].

Elsevier Science Publishers B.V.

Volume 165, number 2 PHYSICS LETTERS A llMay1992

2. Renormalization group results to leading order in 4

The free energy functional we study is composed of a g4 field theory coupled to a scalar field as follows,

MS,, %) = t c (rs +4*)S;S-, 4

+trc c e$l-g+&, c C,SIC.S,-k . (1) q+o k,q#O

In ( 1) the zero momentum component of the sca- lar field fluctuations pqEo is avoided [ 111. This functional appears in the constrained spin-phonon problem with A,, real, and in the ferromagnetic tran- sition of the Hubbard model under constrained charge field with A,, pure imaginary. The constraint can be interpreted as a choice of a particular path in a higher parameter space containing a hidden vari- able used to maintain constant the uniform com- ponent of I. As the scalar field appears only in a qua- dratic fashion it can be integrated out, and the partition function can be written as

Z=ew( -PO) j g[S,l exp[ -PF(S,) I , (2)

where F(S,) is the generalized 0( n ) @4 functional,

BF(Sq) = t 1 (rs +4*)S,*S-,

+A c sq, ~sJ&~~-q,--42--43 -1, c w2_, 2 41.42.43 b-0

(3)

with A, =A&/2r, and S: = & &‘sq_k. A generalized renormalization group program with dimensional regularization, containing diagrams with two dis- tinct momenta conservation on the interaction ver- tices, gives the fixed points with respective critical exponents governing the flux diagram in the renor- malized coupling constants plane (g,, gC) [ 71. Four fixed points (FPs) can be observed in the flux dia- gram: a trivial Gaussian FP, which is the most un- stable one; a spherical FP (renormalized Gaussian); an O(n) FP (n= 1 (Ising), n=2 (XY), n=3 (Hei- senberg ) ); and a renormalized 0 (n ) FP. For n c 4 the flux lines crossing the mean field line of insta- bility emerge from the O(n) FP and thus it is a FITP.

For n>4 the FITP is the renormalized O(n) with tricritical exponents given by

@,=l, (y*=3- 2 =- 40 +O(eO) (Y (4-n)c ’

B n+8 pt= Cy = (4-n)E ___ +o(tO) ) (4)

where the exponents are defined as in ref. [ 71. No- tice that (Y, and & are, to leading order in e, of 0 ( t - ’ ) , reinforcing the concept of a FITP.

The equation of state can be obtained through H=/3dF/aMI f, where t is the renormalized reduced temperature, M is the renormalized thermodynamic average of the critical field after the symmetry break- ing, and F is the renormalized Helmholtz free en- ergy. The free energy can be calculated exactly to or- der e by summing over all one-loop renormalized vertex functions at zero external momenta. The equation of state can be generally put in the form

H=tM+~gsM3+(n-l)K(fg,)+K(fg,-2g,), (5)

with

K(g)=+gM(t+$gM*) ln(t+fgM’) . (6)

Eq. (5) shows that the coupling parameter g, ap- pears explicitly only in the contribution due to lon- gitudinal fluctuation modes. Nevertheless, at the re- normalized 0 (n) FP, the effect of this coupling is also felt on the transversal modes as the fixed point value of g, changes.

3. Universal scaling function

At the renormalized O(n) fixed point, the cou- pling parameters are given by

24 * 6(4-n) t g:=n(n+8)e, g,- n(n+8) *

In the critical domain the equation of state obeys the Widom scaling form [ 121, valid near the fixed point over the line of zero auxiliary tricritical field

181,

H=M*“f( t/M”&) , (8)

in which the function f (x) is regular around zero and thus describes the two regions above and below T,.

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Volume 165, number 2 PHYSICS LETTERS A 1 I May 1992

The critical exponents & and pR are, to order E,

&=6=3+e,

/$&L 1, n+2

1-Q 2 4(n+8)e.

Defining y=gSM2, the scaling variable x=

tv- ‘12*, and choosing the following normalization

conditions [ 13 1,

H=MsR, at x=0 , (10)

and

H=O, atx=-1, (11)

the equation of state takes the form (8), wheref(x) is

f(x)=l+x+efi(x) 2

with

h(x)= & (

t (n-1)(x+1) ln(x+l)

+ i (n+2)[x+j(n+2)] ln[x+f(n+2)]

-q (x+1) h[$(n+2)]

+(n+2)xln(+n) . >

(12)

(13)

We now examine this scaling function together with the correspondent one for an 0 (n) FP [ 13 1. For n=4 these two scaling functions coincide as the re- normalized 0 ( n ) and 0 (n ) FPs are the same to O(e). For n<4 (fig. 1) the O(n) FP is the FITP, and for n > 4 the FITP is the renormalized 0 (n ) FP. A general feature of figs. 1 and 2 is that in the or- dered region (x<O) the scaling function near the FITP is very close to but above that corresponding to the most stable FP, whereas in the symmetric re- gion (x> 0) the former departs from the latter more significantly but from below. These two regimes can also be obtained by looking, to order t, to the sign of the specific heat exponent: (Y<O (n>4); a>0 ( n < 4). Calculations to 0 ( e2 ) and higher order sug- gest that this condition is more adequate. In this way, forn=3andd=3 (t=l),wherea=-0.11 [14],a

12.c -.

9 0 -I

, ,I’ i /

t, , ’ ’ >,‘I / /”

Fig. 1. Universal scaling function to first order in t, and n=2:

(a) for an O(n) critical point (solid line) and (b) for a renor-

malized O(n) critical point (dashed line). In this case the

O( n =2) critical point is a fluctuation-induced tricritical point.

Go ! , ,,/_’ , / -4 3 -?

I 0 0 0 2.C 4 0 6.C 8.0 10 !I

Fig. 2. Same as in fig. I for n = 6. In this case (n > 4) the renor-

malized 0( n = 6) critical point is a fluctuation-induced tricriti- cal point.

renormalized Heisenberg FITP emerges, in contrast

to the prediction to first order in t. The results reported in this paper can be improved

by calculation with higher-order E expansion and by observing that the functionf(x) must satisfy the an- alyticity condition laid down by Grifftths for large x

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Volume 165, number 2 PHYSICS LETTERS A 11 May 1992

[ 15 1. These will be subjects of further work.

Acknowledgement

This work was partially supported by CNPq, CAPES and FINEP (Brazilian Research Agencies).

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