Critical behavior of a charged Bose gas

Journalof Low TemperaturePhysics. Vat. 15, Nos. 5/6, 1974

Critical Behavior of a Charged Bose Gas R. F. Bishop* Department o f Theoretical Physics. The Schuster Laboratory, Manchester University. Manchester, England ( R e c e i v e d N o v e m b e r 26, 1973)

A fully self-consistent Hartree-Fock theory, using the Coulomb interaction screened by the polarization insertions calculated in the self.consistent randomphase approximation, is applied to the d-dimensional, dense, charged Bose gas at temperatures close to the transition temperature To. The quasiparticle energy spectrum is calculated and shown to behave at Tc like e(k) = Ak* for small k, and tr is calculated as a function of the dimensionality d. The change in transition temperature from that of an ideal gas at the same density, and of the chemical potential are shown to be given by (To - Tco)fFco m Xr~d-2)/3 and Vc ~ Yr~2/3, where rs is the ratio of the interparticle spacing to the Bohr radius. Approximate expressions are given for the coefficients X and Y. The critical exponents are calculated, and the system is shown to obey exact scaling. 1. INTRODUCTION The charged Bose gas was first investigated theoretically (at zero temperature) by Foidy, 1 who showed that the Coulomb interaction becomes relatively easy to handle in the high-density limit, just as in the case of the charged Fermi gas. 2 In this limit a small parameter r, appears in the problem, equal to the ratio of the interparticle spacing to the Bohr radius. Using a Bogulyubov transformation, 3 Foldy evaluated the leading order expression for the ground-state energy per particle at zero temperature in this limit, and found it to be proportional to r7 3/,. Other authors *-9 have extended the zero-temperature formalism to verify these results and to obtain the next few terms in the series expansion in (fractional) powers of r,. Similar methods have also been employed to study the charged Bose gas in the high-temperature limit. 2,t° The very interesting region of intermediate temperatures has only begun to be studied recently, tl-t3 One knows that an ideal Bose gas undergoes * T h i s w o r k w a s c o m p l e t e d w h i l e the a u t h o r h e l d a n S R C P o s t d o c t o r a l R e s e a r c h F e l l o w s h i p .

601 O 1974 Plenum Publishing Corporation, 227 West 17th Street, New York, N,Y. 1001I. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical. photocopying, microfilming, recording, or otherwise, without written permission of the publisher.

602

R.F. Bishop

a phase transition at a critical temperature T~0 below which the system contains a condensate of particles macroscopically occupying a single state, and it is of considerable theoretical interest to investigate the effects of interparticle interactions on this transition. Choosing the interaction to be a pure Coulomb repulsion provides the theorist with a nontrivial, but analytically simple example of an interacting boson assembly. The model serves as a classic example of a system with long-range interaction, and one may hope that some of its properties may be shared by systems interacting via more complicated interactions containing a long-range part. In particular, we shall explicitly demonstrate in our calculation those features of the system encountered in the transition region that arise from the overlap of the lowmomentum singularity in the interaction with the corresponding singularity in the boson distribution function at small values of the momentum. The charged Bose gas has also some interest as a model of several real physical systems. For example, many years ago Ginzburg 1. and Schafroth 15 realized that an ideal charged Bose gas (i.e., where the Coulomb interaction is ignored) displays a Meissner effect at temperatures below T~o, and could thus serve as a model for superconductors. It has since been shown that the inclusion of the Coulomb interaction does not destroy the Meissner effect, t6 although the electrodynamics of the system becomes essentially local in the high-density limit ~7 as compared with the noniocal electrodynamics of a BCS superconductor.IS More recently it has also been suggested that the charged Bose gas below its condensation temperature might be important in the study o"fsome systems of astrophysical interest, 19,20 such as, for example, white dwarf stars, which are presumed to consist largely of ionized helium. Also, in recent years an enormous amount of work has been expended on the general theory of phase transitions, and in particular on the investigation of the critical indices 2t'22 that describe the singular behavior of the various thermodynamic functions in the vicinity of the transition. These investigations have now received added impetus from the very recent work of Wilson 23 and its extension and application by other authors. In this context the charged Bose gas provides a simple theoretical model of a system with long-range repulsive interactions that exhibits critical behavior, and it can be used as a testing ground for various theoretical approaches. Thus, for example, the methods of Wilson 2a have been applied to this system by Ma. 12 In the present work the charged Bose gas is examined in the highdensity limit at temperatures equal to and above the critical point, and the whole formalism is carried through with the dimensionality d of the system as an arbitrary (and continuous) variable. A fully self-consistent set of approximations is made, and th~/se presumably become exact as the density becomes infinite. The formalism employed is that of finite-temperature

Critical Behavior of a Charged Bose Gas

603

("imaginary-time") many-body theory, and attention is focused on the oneand two-body Green's functions, which are used to derive the thermodynamic functions. Within the set of approximations made, the critical behavior of the system is identified explicitly, by calculating all of the critical exponents. It is demonstrated that the system obeys exact scaling, and all of the usual scaling relations are shown to be satisfied. In the next section the system and its parameters are defined, and the various thermodynamic functions are set up in a manner convenient to the problem. The many-body theory formalism used is described in Section 3, and the necessary Green's functions are defined and related to the thermodynamics of the system. The critical exponents are defined in terms of the limiting behavior of these Green's functions. The self-consistent approximation that allows the problem to become tractable is made in terms of the same Green's functions, and is described in Section 4. It comprises, basically, the random phase approximation for the Coulomb potential, together with the Hartree-Fock approximation for the resulting screened interaction. Both of these approximations are made fully self-consistent, which is vital to any treatment of the critical region. It is shown explicitly that the resulting set of approximations is both self-consistent and consistent with the various exact relations for the thermodynamic functions of the system obtained from the Green's functions. The resulting set of coupled integral equations is examined in the highdensity limit and at the critical temperature T~, in Section 5. It is shown that the quasiparticle spectrum takes a particular form, which can be obtained from the solution to a single integral equation. The form of the spectrum at low momentum is shown to be e(k) ~ k ~, where ~ < 2: and the exponent a is solved for explicitly as a function of the dimensionality d. The change in transition temperature Tc - T¢ofrom the ideal Bose gas is investigated in Section 6, and it is shown that at high density T~ - Too = Arts d-2)la to leading order in the parameter rs. There is seen to be no phase transition for d ___2, just as for the ideal gas. The chemical potential/z~ at the critical.point is shown to behave as a function of density like la~ = Br~/s for all d, at high density. The numerical constants A and B are related to the form of the quasiparticle spectrum for all values of the momentum, and approximate expressions are given from the knowledge of the low-momentum form only, found in the previous section. The treatment of the self-consistent set of equations is extended in Section 7 to temperatures T > T~, where it is shown explicitly that the critical behavior of the system near the transition region can be found exactly from a knowledge of the quasiparticle spectrum at low momentum values only. Thus, all of the critical exponents are identified as explicit functions of d and the exponent a found in Section 5. It is demonstrated that the system in

604

R . F . Bishop

this set of approximations obeys exact scaling, and all of the usual scaling relations are verified. A brief summary of the results is given in Section 8, and some integrals needed in the work are evaluated in the appendix. 2. T H E R M O D Y N A M I C S AND NOTATION The system under consideration is taken to be a large number N of identical spinless charged bosons, of charge e and mass m, together with a uniform background of opposite charge to ensure that the whole system is electrically neutral, The entire system is constrained to occupy a volume f / i n d dimensions. It has become common in recent years in the context of phase transitions to treat the number of spatial dimensions d which the system is constrained to occupy as a continuous variable. For example, Wilson 23 has developed an analysis of phase transitions in which the quantity e = 4 - d is treated as a perturbation parameter. For generality, and to compare with such work, we shall also treat d as a continuous parameter. In such case an infinitesimal volume element in d dimensions is given by (2~)-d dex = ( 2 n ) - l K d _ l x a - I d x s i n a-2 0dO;

0 < 0 < n

(la)

or

(2r0-d dec = Kax d- 1 dx

(lb)

where K d = 2 l-en-a/2/F(½d)

(2)

where in Eq. (lb) the integration over the angular variable 0 has been performed, and it is necessary that d > 1 for the integral to exist. We define a parameter r s = ro/ao, where r o is the average interparticle spacing, and ao = hZ/me z = 2/e 2 is the Bohr radius.* The high-densitY criterion is then quantified by requiring rs I. The supposed solution for ~(k) of Eq. (48) is now substituted into Eq. (45), together with the derived expressii3n for D*(q). After the potential has been replaced by its analytic form from Eq. (6), and after e 2 has been replaced using Eq. (50), we easily find

MJ2

~o(t)= t2 + - - v ,2~-~-p

f d~s ~[n(

:-o~o-~/21s + tl) - n(v~7~/2s)]

I x

Sa-l

+

Vc2# - 1 - 3 p s d - 2 e

The exponential functions in the distribution functions are again expanded to leading order in powers of v~, to give ~o(t )

tz

.+__zJ2 My:-'

_3o~ dasF

1 J(2~)"L~(s + t)

s2~-a sjj : ' - , + ,,,2 o -

1 ]

1 - 3p

It is apparent at this point that the ansatz of Eq. (48) does indeed solve the original set of coupled integral equations in the high-density limit (vC--, 0), if we choose v = ~(2~ - 1)

(52)

and if ~o(t) satisfies the integral equation

j 2 ~ ddsr. = : +

1

J (2,e'Lq,(s + t)

I]

S 2¢r-d

+

(53)

616

R.F.

Bishop

This is still a rather complicated nonlinear integral equation (although admittedly much simpler than the original coupled equations), and it seems unlikely that an exact analytic solution can be found. We shall not inquire whether Eq. (53) actually has a solution, but assuming that a unique solution exists, it still remains to verify the limiting behavior of Eq. (49). After the substitution s = t x the limit t --* 0 is taken in Eq. (53), using the limiting form of Eq. (49). It is seen that the limiting form is validas long as a < 2, and we obtain an implicit solution for a, M(d, ¢) = N(d, a) = (2n)-d f ddx[ll + XI-" -- X-~]X 2°-d

(54)

It is seen that the constant J has disappeared from this equation, and can only be obtained from the detailed solution to Eq. (53). The restriction a < 2 is physically reasonable, since for a > 2 the critical behavior of the charged Bose gas would be the same as for the ideal gas. The integral N(d, a) is evaluated in the appendix as Eq. (AI5), where it is shown that it remains finite only under the conditions d > max (1, a) and 0 < a < 2, both of which are already built into our theory. The implicit equation for a of Eq. (54), using Eqs. (AI2) and (AI5), reduces to -

-

-

d>l,

)r(½d ÷ ½a) = r(½

0" 2

(63)

in the same range of validity as Eq. (55). Replacing the integrals P and M by their values from the appendix, and substituting Eq. (63) into Eq. (58), gives the result that the leading order change in the transition temperature in powers of rs is, for 2 < d < 4, -

-

~o

,~

r(1 + ½d)((½d)a(a - ½)( ×

FzM L ' (r(i1½d) ¥ d)J

]

2 ~j2]

cosec

(64)

where we have substituted for vc from Eq. (50). The constants Za and M are given by Eqs. (7) and (Al2), respectively, and only J remains unknown until a more detailed calculation is performed. Since a is determined by Eq. (55), it is easy to see that every factor in Eq. (64) is positive definite in the range 2 < d < 4 with the exception of the factor (d - 1 - a). From Fig. 3, one observes that d-l-oX0,

dX2.87

This gives T~ < T~o for 4 > d > 2.87, and T~ > T~o for 2 < d < 2.87, and in particular T~ < T~o for d -- 3. There does not appear to be any obvious physical reason why T~ - T,o should change sign at d = 2.87, and this is perhaps just an indication that the approximation of replacing ¢(x) by J x ~ in Eq. (60) istoo drastic, and hence that the numerical factor in Eq. (64) should not be taken too seriously.

R. F. Bishop

620

The value of the chemical potential at the critical temperature It~ can also be evaluated using Eq. (30), f ddq , , V(q) Pc = ~,*(k = O) = j (-~)dn~ql 1 _ V(q)O*(q)

where the integral is evaluated at T = T~. Changing integration variable q ~ s = viPfl~/2q and using Eqs. (6), (48), (50), and (51), we obtain after the

limit v¢ ~ 0 is taken as before, that to leading order in re, #¢ is given by ~ dds

j2

fld~

1

- - v 2" .~ M (2n) a

S 2t~-d

~O(S) S 2~'- 1 +

1

(65)

Using Eqs. (50) and (52), we see that to leading order in powers of r~, I1~ ~

(66)

r2~/3

rs~O

independent of d for 2 < d < 4. Again the numerical proportionality constant must be evaluated from Eq. (65), which requires an exact knowledge of the function tp(s). If we again make the approximation of replacing ~o(s)by its asymptotic form from Eq. (49), we obtain •

~c ~lc ,~ Kd

j

v 2p

fO~

S¢- 1

ds s2 ~- ~ + 1

and the integral can be performed using Eq. (62), since a > I, to give fldAc ~ g d ~ v 2 p

~z

2a--1

cosec 2 ~ tr --l

Replacing vc in favor of r~ from Eq. (50) gives finally .., ~ K a t rid 2/a 2 1is flc~c ~2a-----~ c°sec2-a-Z-T Z a M - t l 2 F(l"+~d) ~

(67)

where the factors K~, Za and M are known from Eqs. (2), (6), and (A12), respectively, and only J requires a more detailed calculation. 7. T H E CRITICAL INDICES

From the basic set of coupled equations derived in Section 4 we are now in a position to derive the critical indices defined in Eqs. (25) and (26). Two of them have in fact been essentially derived already. Thus, from Eqs. (24), (32), (48), and (4.9), we have G(k, 0) ~-, k-~ k~O

Critical Behavior of a Cl~u~ed Bose Gas

621

which immediately identifies the exponent r/from Eq. (25) as q = 2 - tr

(68)

Similarly from Eqs. (51) and (26), the exponent ~ can also be identified as = d - 2a

(69)

We also observe that G(k, 0) and D*(k, 0) take their asymptotic forms for small k only for k max(1,~)

(All)

and repeated use of the duplication formula (A5) gives the final expression

M(d,~) =

(4=)_,/2 r2(½d

-

~)r(~

-

r 2 ( ½ a ) r ( d - a)

½d)

'

½d < a < d;

d > 1 (AI2)

The IntegralS(d, ~) The integral N(d, o) is defined in Eq. (54) as N =- N(d, o) = *Note that the power

of sin 0 is rectified in our Eq. (A8).

f

ddx[I 1 + x]-" -

X-o]X

2o-d

is erroneouslygivenas the negativeofthe correctresult. This error

Critt~l Behavior of a Charged Bose Gas

633

After the simple change of integration variable, x ~ y = - 1 - x in the first term this becomes

f

N =

d'y[ll + yl 2 " - d - y 2 , - , ] y - , ,

= (2~)- t K d_ ,

dO sin d- 2 0

dy

x yd-~'-l[(1 + 2ycos 0 + y2}~-dn _ y2~-,] = (2~)- tK,_ 1

I2 d0sin ' - 2 0 Io° d x

x x - t-~[(1 + 2x cos 0 + x2) ~-"/2 - I]

(A13)

where the substitution y = x-1 has been made. The radial integration can now be performed as above for the evaluation of M(d, a ) , using a simple extension of the relation given from Ref. 33 as Eq. (A8). This integral is most readily evaluated by converting it into a contour integral around the branch cuts. The second term in the square brackets ofEq. (AI3) then merely ensures that the contribution from the infinitesimal circle about the branch point at the origin vanishes. Hence, we find d x x - 1 - ~ [ ( 1 + 2x cos 0 +

x2) a-d/2

-

1]

= 2` / 2 - ` - 1/2F(½d - tr + ½ ) B ( - t r , d - a)(sin 0) 1/2-'/2+" •x

--,/2°1/2-'/2+~t""¢1/2 w-,o 0),

0 < a < d;

a < 2

(AI4)

and substitution into Eq. (A13) yields N = ( 2 n ) - t K , _ 12` / 2 - ` - t/2F(~ - a + ½)B(-tr, d - a) X

fO

.,.ID (c.;~ 13~d/2 + a - 3 / 2 D 1/2 - , / 2 + al,~.rt © ~ u t , ~,alLt ~ 9 ~t ,12 - 1/2 %,,,,,..,0t,/

The angular integration can now be performed as above, to give N = ½K a_ l r'(½d - a + ½ ) B ( - t r , d - a)

x

r(½d - ½)r(o)

d > 1;

• r(½d + ½o)r(~r)r(½d - ½o + ½)r(~ - ½~)'

a>0

This expression is simplified by repeated use of the duplication formula (A5) to give the final result, r ( - ½~)r(~)r(½d - ½~)

N(d, o ) = (4~) -d/z r(½d - o)r(½d + ½a)r(½o)"

d>max(1,a);

0