“phase transitions” at finite temperature in finite systems

Volume 196, number 3 PHYSICS LETTERS B 8 October 1987

" P H A S E T R A N S I T I O N S " AT F I N I T E T E M P E R A T U R E IN F I N I T E SYSTEMS

E.D. DAVIS Department of Physics, University of the Witwatersrand, Johannesburg 2001, South Africa

and

H.G. MILLER Theoretical Physics Division, National Research Institute for Mathematical Sciences, Council for Scientific and Industrial Research, Pretoria 0001, South Africa

Received 12 March 1987

Indications of phase transitions (present in the thermodynamic limit) are seen in the solution for a finite number of particles of an exactly solvable model. The gross structure of the implied exact "'phase diagram" is reproduced by self-consistent mean- field approximations, but not, in general, the precise location of the "phase" boundaries.

At finite temperatures, realistic calculations [ 1-3 ] in nuclear systems, even in a variational scheme, are difficult to perform. Shape transitions as well as the loss o f superfluidity have been observed and the critical temperatures at which these phenomena occur have been determined in finite-temperature mean- field calculations [4,5]. However, it is by no means clear how meaningful these predictions are. In finite systems, the thermal fluctuations about equilibrium values are expected to be large [6], particularly in the region where a phase transition takes place. For example, studies o f their effect on an order parameter have concluded that the superconducting to normal phase transition is effectively washed out [ 7,8 ]. Furthermore, na'fvely, one expects the results found at T = 0 to remain valid, at least, for small temperatures. However, the limit T--.0 is not necessarily continuous [ 9,10].

In this paper we consider the phase transitions predicted by the finite-temperature Har t ree-Fock (FTHF), Bardeen-Cooper-Schrieffer (FTBCS) and Har t ree-Fock-Bogol iubov (FTHFB) approximations [4] when applied to the exactly solvable Agassi model [11,12]. In contrast to earlier works (which looked at fluctuations in approximate quantities), we study the fluctuations around an exact ensemble

average. In spite of the fact that the fluctuations are large, we see evidence for phase transitions in finite systems which agrees with the gross phase structure predicted by finite-temperature mean-field calculations.

The Agassi model consists o f N interacting fer- mions that occupy two levels, each of degeneracy/2 (12 even); the hamiltonian is

t

r n , m ' > 0

V ,t t ---~ ~ Co,,,C~m,C_o,,~,C_a,,~ ,

rrl tr~"

(1)

where a labels the levels, m the states within a level and corn creates a fermion in the single-particle state la in ) . We take a to be + 1 / - I for the upper/lower level, and m to have the range rn = + 1, __+ 2, ..., + £2/2. In what follows we assume Vand g to be positive and the number o f particles to be equal to g2.

In applying FTHFB to the Agassi model, one can assume that each quasi-particle state has the same thermal occupation probability. (This is true o f many similar models, for example the model studied in ref.

0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

277


[13] when the cranking frequency ~ = 0 , which is the limit appropriate to our work). It is always possible to decompose the HFB transformation into three simpler transformations via the Bloch-Messiah theorem [ 14], and we make the natural assumption [ 15 ] that these transformations have the same form at T~:0 as at T = 0 [ l 1,12]. Because the occupation probabilities are all the same, the consequences o f this decomposit ion are formally the same as in zero temperature HFB - viz. ensemble averages depend only on the first two transformations.

The solutions of interest of the corresponding FTHFB equations (obtained by variation o f the grand potential functional ~) are as follows:

(1) A "spherical" F T H F solution. This solution is always present, but is stable (that is, corresponds to a minimum of ¢) only if

tanh x < 1/Z , (2)

and

tanh x < USa, (3)

where x satisfies

1 + (g/e) t a n h x = 4 t x , (4)

and %= ( I 2 - 1 ) V/e, ,S'a=,S'+ V/E= (.(2-- 1 )g/e + V/e and t= kT/e, k being Boltzmann's constant.

(2) A "deformed" FTHF solution. This exists as long as the inequality in eq. (2) is not satisfied and is stable, provided Z > Z'~.

(3) A FTBCS superconducting solution. This exists as long as the inequality in eq. (3) is not satisfied and is stable, provided ,S~>Z.

Not surprisingly, these results are very. similar to those found at T = 0 [ 12]. Again, there is no full HFB solution. At T = 0 the deformed HF and BCS solutions are formally similar. This property persists at finite temperature. We can therefore conclude that, in the Agassi model, the effect of temperature on pairing is the same as its effect on deformation.

The phase boundaries when t = 1.5 and -(2=20 (of the phase diagram implied by these results) are given by curves A, B and C in fig. 1. The boundaries of the spherical HF phase are obtained by combining eq. (4) with eqs. (2) and (3), respectively, and solving for the critical value o f Vas a function ofg. Observe that temperature has no effect on the deformed HF to BCS transition line (in contrast to other model

7

8

5

3

• ;J •

2 " "'''% •

o I 2 3.

Q

e

i T

5

i •

I" I

K

6 7 [

Fig. 1. Approximate and "exact" phase diagrams for the Agassi model when t= 1.5, -(2=20. Curves A, B and C represent the spherical-to-superconducting, spherical-to-deformed and superconducting-to-deformcd phase boundaries, respectively, calculated in the appropriate finite-temperature mean-field approximation. The points correspond to the maxima which occur in an exact calculation of the specific heat, Cv.

studies [15]. On the other hand, the size of the spherical HF region increases with increasing 7: In the absence o f the pairing interaction, the value o f z at which the spherical to deformed HF transition occurs is Z~. = coth(1/4t). This result is consistent with the T = 0 limit. (As T-~0, Zc-~l). Even when the pairing interaction is present, it can be used to esti- mate where the spherical to deformed HF transition is. In the same way, the value of Z at which the spherical HF to BCS transition occurs is also approx- imately equal to coth(l /4t) . These estimates improve with increasing ,(2.

All of these transitions may be classified as being continuous (or of second order). It has been shown [ 15,16] that, for quasi-spin models similar to the one considered here, FTHFB provides an exact descrip- tion in the thermodynamic limit. This means that, in this limit, the Agassi model does experience these continuous phase transitions.

It can be shown quite generally [ 17 ] that, if H is the hamiltonian of a system, then the specific heat Cv (in units o f k) is given by

C~ = (kT) 2 ( < I F > - ( < I t > ) 2 ) , (5)

278


where ( ) denotes the canonical ensemble average. Since the specific heat vanishes in the limit T~0 , it is a direct measure of the extent of thermal fluctuations. It is well known [18] that phase transitions are characterised by an increase in the extent of fluctuations. Exact studies in large but finite systems, which, in the thermodynamic limit, experience a continuous phase transition, have shown C~ peaks (smoothly) in the region where this transition occurs [ 19]. Thus, by studying Cv, we can simultaneously extract information about the magnitude of thermal fluctuations and see whether phase transitions, which would occur in the thermodynamic limit, can still be felt in small finite systems.

To calculate Cv we have taken the ensemble over which the average is performed to be the collective subspace associated with the Agassi model [ 11,20], since this subspace determines the structure in Cv. Fig. 2 contains some examples of C,. as a function of X and Z when t= 1.5. The specific heat does indeed exhibit peak structure. To establish how it may be interpreted consistently as the remnant of the phase

L / X I

0.6

O4

0.2 _~ . . . . . ~ _ 2 4 e -£

', )

/ / \ . . . .

//'~.

o 2 4 6 x

0.8

0.6

10. Er/£,

0.4

02

00

Fig. 2. The specific heat per particle, C,/.Q, (in units of k) for different values of the interaction strengths x and e when t = 1.5.

transitions in the thermodynamic limit, it is suffi- cient to determine the loci of these peaks in the X-Z plane. These are given by the dots in fig. 1.

Clearly, the regions I, II and III are to be identified with the spherical, superconducting and deformed phases respectively. The region IV can be associated with a deformed-superconducting or hybrid phase which does not appear to be describable by FTHFB. However, it may also be perceived as just a transi- tional region linking essentially the superconducting and deformed regions, for, as #2 increases (or the thermodynamic limit is approached), the width of this region decreases (fig. 2b). With this in mind, it is possible to associate the peaks in C,. with transitions found in the thermodynamic limit.

The behaviour of C, demonstrates that the phase transitions predicted by FTHFB (or other methods [5]) are relevant in finite systems. In fact, a result of the discussion in the previous paragraph was the "exact" phase diagram (given by the dots in fig. 1) for the Agassi model when g2=20 and t= 1.5. Com- parison of the exact and approximate phase diagrams shows that, although the location of the approximate superconducting-to-deformed transition is essentially correct, the size of the approximate spherical phase at this temperature is grossly over- estimated. At other temperatures, the findings are similar, and only as T ~ 0 can the agreement between the approximate and exact phase diagrams be considered reasonable everywhere. It must therefore be concluded that, in general, FTHFB does not predict critical interaction strengths, or equivalently, critical temperatures reliably.

Finally, we comment on the magnitude of the exactly calculated fluctuations away from the phase boundaries. Typically, they are significant in the spherical region (fig. 2a) but negligible in the superconducting and deformed regions (fig. 2b). Note that this trend occurs at a fixed temperature. It is in line with the finding in the model study of ref. [ 15] that the convergence to the thermodynamic limit of the exact results, as the particle number was increased, was slowest in the spherical (or normal) phase.

This trend has implications for the spherical-to- deformed and spherical-to-superconducting transitions. When thermal fluctuations are significant, the average value of any particular parameter can be quite different from that predicted by the HFB

279


a p p r o x i m a t i o n [6] . In par t icular , o rde r pa rame te r s

like the pa i r ing gap - which wi th in the mean- f i e ld

desc r ip t ion are au toma t i ca l l y zero in the spher ica l

phase - could in a m o r e e labora te t r e a t m e n t be sig-

n i f icant ly d i f ferent f rom zero. In fact, exact ly this is

seen in the results o f refs. [7 ,8] . Taken in isolat ion,

it impl ies that the t rans i t ions f rom spher ica l - to-

d e f o r m e d and sphe r i ca l - to - supe rconduc t ing are

washed out. Howeve r , ou r work ind ica tes that a dif-

ferent ( and pragmat ica l ly a d v a n t a g e o u s ) v i e w p o i n t

should be adopted : the t rans i t ions do occur , but they

co r re spond to a progress ion f rom a region in which

a stat ic self-consis tent m e a n field by i tse l f is useful

to a region in which it is not.

We are grateful for use o f the c o m p u t e r faci l i t ies

o f the W i t s - C S I R Schon land Research Cen t r e for

Nuc lea r Sciences, and thank R.M. Quick and J.P.

Vary for s t imula t ing discussions. O n e o f us ( E . D . D . )

is par t icular ly indeb ted to W.D. Heiss for m a n y use-

ful c o m m e n t s and a careful reading o f the manuscr ip t .

ReJerences

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“phase transitions” at finite temperature in finite systems

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