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Transition temperature of a dense charged bose gas
ANNALS
OF PHYSICS:
64,
Transition
1-20 (1971)
Temperature
of a Dense Charged L. FETTER+
ALEXANDER Institute of Theoretical Stanford University,
Bose Gas*
Physics, Department Stanford, California
of Physics 94305
Received July 21, 1970
The linear response of a normal dense charged Bose gas to an applied scalar potential is studied in the random-phase approximation. Although the plasma frequency remains constant, the asymptotic screening around an impurity changes character at the transition temperature from exponential to algebraic. A self-consistent Hartree-Fock theory with the static screened interparticle potential yields the shift in the transition temperature (T, -~. TCu)/TCO w -0.026 rlg’3, where TCU is the transition temperature of an ideal Bose gas with density n and particle mass M, r8 = (e2MM’r2)(3,‘4an)1~3, and e is the boson charge.
1. INTRODUCTIOX The low-temperature behavior of an ideal gasdependscrucially on its quantum statistics. For example, a perfect Bose gas undergoes a phase transition at a critical temperature TCo,where a macroscopic condensate first starts to appear. In a perfect Fermi gas, however, the exclusion principle forbids such condensation. and the associatedphase diagram has no singularities. The striking difference between bosons and fermions persists in the presence of interactions, asseenin the contrast between the physical systemsHe3and He4. This distinction may also be studied with theoretical models, such as a dense charged gas placed in a rigid uniform background to ensure charge neutrality. Although the high- and low-temperature limits have been studied extensively [l-3], the very interesting topic of the phase transition in the charged Bose system appears untouched. As a first step in such a study, we here examine the behavior as the phase transition is approached from above. For definiteness, we consider the linear responseto an external scalar potential (Section 2). The resulting screening of a static impurity clearly foreshadows the approaching phase transition * Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, U. S. Air Force, under AFOSR Contract No. F44620-68-C-0075. + Alfred P. Sloan Research Fellow.
1 0 1971 by Academic Press, Inc.
2
FETTER
(Section 3). The corresponding screened interparticle potential then enables us to determine the critical temperature with a self-consistent Hartree-Fock theory (Section 4). 2. LINEAR RESPONSE TO AN EXTERNAL
POTENTIAL
Consider a gas of noninteracting spinless bosons with mass M per particle. At a density n and temperature T, the system has two characteristic energies: The thermal energy kBT and the quantum-mechanical zero-point energy ti2n2/3/M associated with localization within a volume n-l. Quantum effects predominate below the critical temperature Tco, which is evidently of order A2n2/3/MkB. More precisely, a detailed calculation yields [4] 277752 [* Tco = Mk,
213
I
M 3.31 g
,
(1)
B
where <(#) M 2.612 is the Riemann zeta function. The corresponding qualitative analysis of an interacting gas is more complicated owing to the characteristic energies associated with the strength and range of the interparticle potential. In the special case of a pure Coulomb gas, however, the range is infinite, and the only new quantity with the dimensions of an energy is e2n1J3,where e is the charge per particle, At high density (e2n1i3< fi2n2/3jM), the interparticle spacing r. E (3/4m) l/3 is much smaller than the Bohr radius a, = fi2/Me2 for the bosons, and it is conventional to introduce the small parameter r, = ro/uo < 1. We here treat only the region near the phase transition, so that T = Tco and e2n1J3< k,T, The linear response to an external potential v&x, t) is governed by the exact density-density correlation function for the unperturbed but interacting medium. Thus if n(x, t) is the Heisenberg operator for the particle density and 5(x, t) = n(x, t) - (n(x, t)) is the deviation from the mean value, then the induced density 6(n(x, t)) is given by [5] 6(n(x, t)) = 1 d3x’ dt’ IIR(x, 1; x’, t’) ey&x’,
f’),
(2)
where JP is the retarded commutator P(x,
t; x’, t’) = -if?1([8(x,
t), 6(x’, t’)]) B(t - t’),
(3)
and the angular brackets denote a grand canonical ensemble average at temperature T = (kg&l and chemical potential p, (4)
DENSE
BOSE
3
GAS
In a uniform equilibrium system, translational invariance in space and time guarantees that flR takes the form ZP(x - x’, t - t’). With the Fourier transform 2~(~,
t)i = (2+4
j d3q dw ei(q.xuw*)S(n(q,
w)i
(5)
and similar equations for v&q, w) and IIR(q. ~1, Eq. (21 becomes a simple product Wz(q, 0~)) = nYq,
w> eq,,(q, ~1.
(6)
Thus the problem is reduced to that of finding nR(q, w), which can be interpreted as the exact retarded polarization of the medium. The function flR is simply related to a corresponding time-ordered polarization 17. In the absence of Bose condensation (the more general case is treated in Ref. [6]), IT satisfies a simple Dyson’s equation. Consequently, nR can be expressed exactly in terms of the retarded proper polarization .IT*R(q, w), nR(q, w) = 1 -
n*R(q, w) I/,(q) n*yq,
Z).
(7)
Here V,,(q) is the Fourier transform of the Coulomb potential, modified at q = 0 to account for the uniform neutralizing background j4rre2/q2 ~ObI) = [O
q f 0, q = 0.
(8)
Unfortunately, Lr *R itself contains contributions of all orders in perturbation theory. Thus it is necessary to introduce approximations, and we make the simplest possible one, in which n*R is replaced by that for a perfect Bose gas noR above its transition temperature. This approach is frequently known as the randomphase approximation [7, 81. The function flOR is readily determined with standard techniques [5] to be
where eke = ?iek2/2M and nko = {exp[j?(E,O - p)] - 1)-l
(10)
is the usual Bose-Einstein distribution function. As a first application of Eq. (7), we show that a charged Bose gas has a long-
4
FETTER
wavelength collective model (plasma oscillation) at all temperatures above its transition temperature. The real part of Eq. (9) can be rewritten exactly as Re LroR(q, w) = 2c,OP 12%
(2~)~ (ho - fi2k *$M),
- (6,“)”
(11)
where P denotes the Cauchy principal value. An expansion for small q yields Re noR(q, OJ) = &
With the observation n = (2?~)-~
and the definition
I
d3k nko
(13)
of the mean-square velocity (u2> = n-l(2~)-~
j d3k nso(hk/M)2,
(14)
Eq. (12) becomes Re noR(q, OJ) w &
4Yv2) [ 1 + -;u;i-].
(15)
A system has a collective mode with frequency on and damping constant yq when the exact polarization LrR(q, w) has pole at w = wq - iy, . For small damping, Eq. (7) shows that wq is determined by the approximate equation 1 = V,(q) Re 17*R(q, wa). In the random-phase
approximation, this relation is easily inverted to give D 2 + 92(u2), WQ = WPZ
(16)
(17)
where wvz = (4rrne2/M)1/2
(18)
is the plasma frequency. Note that wDI is constant for a gas at fixed volume, and that the temperature and chemical potential affect only the correction term proportional to q2. The damping of plasma oscillations is determined by Im D*R. Although this quantity can be evaluated in closed form in the random-phase approximation [9], it is not especially relevant here and will not be considered further.
DENSE
BOSE
GAS
3. SCREENING OF A STATIC IMPURITY We now turn to the screening of a static point charge, where ‘P&q, Co)= 4xzeq-“2X6(W).
(19)
In the sameapproximation asthat usedabove, the induced charge density becomes
W(q, w)i = 7
27r6(w)
= Z27iqw)
n”R(q, QJ) 1 - V,(q) n”vl. WI
(20)
Vu(s)fioR(% 0) 1-
C’,(q)ITOYq, 0) ’
which is real becauseIm I?OR(q,0) = 0. Unlike the collective plasma oscillations, however, even the leading contribution to Eq. (20) dependsstrongly on T and p. The first task is the determination of 170R(q,0), which we shall henceforth denote F,,(c/). A simple transformation of Eq. (20) yields
It is convenient to introduce the thermal wavelength A = (2nk2/MkBT)‘~2
(22)
and the dimensionlessvariables s = q/l, t = kf, and N = -p/kBT ;-: 0. (Recall that p < 0 for a perfect Bose gas above its transition temperature.) In this way, Eq. (21) becomes t
exp(,r :-
f”/&-
In -1
1
2r + s ( ?f-s
1,
(23)
where the symmetry of the integrand allows us to extend the lower limit to - CC. This function is similar to one studied by Schafroth [IO]; it is evaluated in Appendix A. where we show that (2‘9 Here g, is a dimensionlessfunction that remains finite as CLand q/l tend to zero.
6
FETTER
Equation (20) can now be used to study the spatial form of the induced charge density p&x) around an impurity. The inverse Fourier transform gives PdX)
= ze j &
V,(q) F,(q) f+a.x 1 _ Ir,(q) Fo(q)
= Ze
eia’x q2 ~(4)
=-- ze
s
-$$
1
O”
x 27s s 0
q dq sin qx
(25)
4~e2Fo(d q2 - 4re2Fo(q)
’
This expression has several interesting features. First, the total screening charge is (26)
= -Ze, where the last equality follows because -‘Ocq
= ‘) = ~Ajlk
BT
jrn0 exp(cu +$/477)
- 1
(27)
is positive definite. Thus the impurity is wholly screened at large distances. Second, Eqs. (24) and (25) may be combined to yield v3{27r-’ arctan[s/4(cY7r)1’2] + s&r, CX)}, psc(x) =- -&& jmdss sin(sx/A) s3 + vS{27r-l arctan[s/4((1/77)1/2] + sy(s, a)}
(28)
0
where v3 = 8r3e2/AkBT
(29)
is a small parameter for a high-density gas near Tco. The substitution s = vf shows that CJImakes a negligible contribution to p&x) as v -+ 0, and we therefore obtain the final form (30)
where &c(z) = &
s sin szv32+ arctan(s/q) s,” ds s3 + v327r-l arctan(s/q)
(31)
and 7 is a second small parameter vj E 4(m)1/2.
(32)
DENSE
BOSE
I
GAS
For an ideal Bose gas [l 11, 7 = 35(9( T - TOO)/TcO< 1.
(33)
It is evident that S,,(z) vanishes identically for v := 0, which reflects the physical observation that the screening vanishes in a perfect gas. The other simple limiting case occurs at Tro, when a! = 0 and 27r-l arctan (s/v) == 1 for all s.
Since the length x enters only through the parameter vx/Ll, the characteristic screening distance is (1( TCo)/v( TCo) * 0. ~OU,$~. Equation (34) can be expressed in terms of sine and cosine integrals; a straightforward analysis then gives the following limiting forms:
where y = 0.577 is Euler’s constant. Note that S,,(z) at TCu vanishes for both large and small z. (It also must vanish at least once for an intermediate value.) Furthermore, the asymptotic tail of p&x) at TCo is algebraic rather than exponential. Both these features are in marked contrast with the Thomas-Fermi or Debye-Hiickel theory, where the corresponding screening function is a pure exponential [5]. We now turn to the more general question of the screening function &(;) for arbitrary small values of v and 7. Since the integrand of Eq. (31) is even in s. the equation may be rewritten arctan(s/q) &c(z)= 5 ?‘I, d.Yseiyzv327r1 $- 1~32~iarctan(siq) ’ ~3
(36)
which allows us to deform the integration contour into the upper half-plane. The asymptotic behavior for large z is then determined by the singularity of the integrand that is nearest the real axis. The singularities in the upper half-plane are of two types: (1) A branch cut running along the imaginary axis from s = in to s = ice, and (2) a pair of simple poles. The detailed analysis can be simplified by
8
FETTER
using the variable 7 = s/v, so that the relevant poles are the roots of the transcendental equation 73 + (V/T)” 2rr-1 arctan 7 = 0
(37)
for Im T > 0. (a) Relatively
far from the transition
temperature
Tcor~‘3 Q T -
(V Q -q or
T,‘),
these two poles are on the imaginary axis at s1 = i(2v3/q)1iz and Xl’ = iq[l - 2 exp(-q3.rrjv3)]. The first pole clearly dominates the asymptotic behavior. Hence &(z) cc exp(- j s1 [ z), and the screening length is of order A[(T - Tco)/rsTco]‘12.
(b) As v/q increases, the two poles approach
each other, coalescing at Tco)]. The screening function is now proportional to exp( - 1sZ 1z), so that the screening length initially decreases with increasing v for fixed 7. (c) When V/T becomes still larger, the poles move symmetrically into the first and second quadrants. For the particular value v/v = 1.07 [viz., when Tcorl’3 = 1.89 (T - Tco)], the poles reach the line s = iq. In this case, the poles and branch cut both contribute to the long-range part of S.&z). (d) If v/q > 1.07, the branch cut at iq dominates the asymptotic form, which is now proportional to exp(-qz). In this region the screening length is L’I/T and increases like (T - Tea)-’ as T---f Tc O.In the limit T = Tco, the screening function changes from exponential to algebraic and takes the form 5&(z). -z-~ [see Eq. (35b)].
s3 = 0.89qi when v/q = 0.92 [viz., when Tcars“’ = 1.62 (T -
The above analysis demonstrates that the long-range screening near Tco is quite complicated, depending sensitively on the relative values of the small parameters rit3 and I - T/Tc O. This behavior arises from the approaching phase transition; it therefore differs greatly from that of either a Boltzmann or Fermi gas. A slightly different quantity of interest is the static screened interaction V(q) between two particles in the medium. Since 1 - V,(q)fT*R(q, 0) acts as the static dielectric constant, we may use the same approximation as in Eq. (20) to write
V/(4)=
V,(q) 1-
V,(q) mJR(q, 0) *
An inverse Fourier transform yields the corresponding V(x) = ezx-‘S(x/A),
(38)
screened spatial potential (39)
9
DENSE BOSE GAS
where sdsin sz s3-7 v32.rr-1arctanm
’
(40)
The detailed analysis is quite similar to that for S&z) and will not be given here. Unlike the induced screening charge, however, we see that S(z) == 1 for v 2 0, which is also evident from physical considerations.
4. SELF-CONSISTENT
HARTREE-FOCK
THEORY FOR T > Tc
In the previous sections, the chemical potential has been taken to be that for an ideal Bose gas, vanishing at the unperturbed transition temperature TCo. This description cannot be fully self-consistent, becausep itself is altered by the presence of interactions. To include such an effect, we now attempt an improved (HartreeFock) theory, which allows us to determine the shift in the transition temperature to leading order in the interaction. For a weak short-range potential, the Hartree-Fock equations take Ihe intuitive form [12] n = (2~r-~
I
d3k n, ,
(41)
n,<= [exp P(Q - CL)- 11-l,
(42)
cl<= E,,.o +
(43)
(2x)-3
! d39[vo(0) t V,,(k - q)] II,, ,
where V&k - q) is the Fourier transform of the interparticle potential and the two terms in the integral in Eq. (43) represent the direct and exchange interactions, respectively. The self-consistency is now explicit, becausethe excitation spectrum cl{ both determines and is determined by the distribution function n,. . In their present form, the coupled Hartree-Fock equations can be expanded at long wavelengths to find the transition temperature of a weakly interacting neutral Bose gas [13-l 51. For a charged system, however, they must be modified in two distinct ways. First, the direct interaction V,(O) vanishes identically owing to the overall charge neutrality. Second, the bare potential V’*(y) ==:4re2/9” is unbounded at long wavelengths. Although this behavior does not invariably cause problems, it leads to serious difficulties at T, , where the Bose-Einstein distribution function II,, also develops a long-wavelength singularity. For this reason, screening plays an essential role in the phase transition of a dense charged Bose gas, and we shall replace V,,(q) by the static screenedinteraction
V(q) = 1 - V,(q) V,(q) F(q) ’
(44)
10
FETTER
where F(q) is also determined self-consistently, d3k
nk+,, - nk
Fcq) = s (2x)3
tk+q -
Thus we obtain the modified Hartree-Fock
lk -
equations
n = (271)~~ d3k nk , s
(46)
ek = cko i- (27r)-3 - d3q V(k - q) n, ,
(47)
J
with nK given by Eq. (42). It is clear that an exact solution of these coupled equations is out of the question, and we merely seek the lowest-order correction to the properties of the noninteracting system. An assembly of bosons remains normal so long as nk is bounded for all k. For an ideal gas, nko is largest at k = 0, so that the long-wavelength (k/l < 1) behavior of eke - y fixes the unperturbed critical temperature TCo. We shall assume that a similar description applies to an interacting charged Bose gas and write the single-particle spectrum in the approximate form (again for kA < 1) ck m co + ti2k2/2M* + 8~ .
(48)
Here 8~ is an additional shift in the spectrum that cannot be represented with an effective mass M*. As shown below, Sr, vanishes as k + 0, and the critical temperature TC of a dense charged Bose gas at fixed n is determined by the condition EO
In particular,
-p=O
at
Tc.
(49)
Eqs. (46) and (47) then reduce to d3k I n m (27?
exp[(ti2k2/2M* d3q
Ek = ‘Ice +
I
1 + &,J/k,T,] Vo(k
(2~)~ 1 -
-
- 1 ’
q)
V,(k - q) F(k - q)
1 x
(50)
exp[(h2q2/2M* + &,)/k,T,]
- 1 ’
(51)
where
F(q)
1 exp[(fz2k2/2M* + 6+)/kBTc] - 1 (52)
DENSE
BOSE
II
GAS
It is evident from Eq. (51) that the corrections to the free-particle spectrum vanish as rs and hence v3 tend to 0. Since we want only the leading contribution to Tc - TCO, it is then permissible to simplify Eqs. (51) and (52) by neglecting 8~ entirely and setting M* = M. As a result, F(q) reduces to that for a perfect Bose gaswithp-zOandT= T,, -2r”/qfl,4k,Tc,
F(q) * while the excitation EL e .FI; ” f
spectrum
sd3q
(53)
becomes 1
4ne2 (2~)~ q2 + (8rr3e”/qAc4kBTe)
exp[@(k
+ q)2/2MkBT,,]
Here II, = (2m52/MkBTc)1~2 is the thermal wavelength at TC , We once again introduce the dimensionless variables s = qil, , t = k/l, yP3 == 8n3e2/A,kBTc , so that the shift in the spectrum can be written
Ek- Eli0 k,Tc
(54)
-. 1 ’
, and
1 s Ye3 - __-d3s --, (2n)3 s3 + vc3 exp[(s + t)2/4.rrl - 1 2.rr2 J = ___
(55)
The right side of Eq. (55) is formally of order vC3, but a direct expansion in powers of vC3 leads to divergent integrals, thus confirming the importance of screening. Instead, it is preferable to use the expansion [16] (56) to rewrite
the distribution
1 exp[(s + t)2/4rr] -
function I = (s$ty
---
1 2 4v
+ gl [(s+ t)? 8$jn+ (s--_____ ’ -t t)2+ Win 1
(57)
For real s and / t / < 2x ~‘2, the fractions in brackets may be expanded in a Taylor series: resumming the terms for r = 0 with Eq. (56), we find 1 .-__ exp[(s + t)2/4x]?
=
4x (s -+
ty
4rr --T .P
! n=l)-(2s.tit’)[ + 46.t)”[ _
1 exp(s2/4r)
+4$
($2
- 1
1 (3 - 8772in)2 + Ig,2inj3
(s2
+
;n2jn)l
-
1
+ ($ + igin) II + O(f3).
12
FETTER
When this expression is substituted into Eq. (55), we readily identify the various contributions to the long-wavelength spectrum [cf. Eq. (48)], (59)
M -M* - * = 8vc3 g I& 8.5 z&
[ (9 - Lin)”
-
+
(9 + iwifl~
d3s
2vc3 =
; y> 1% [ (S2-
S3
s
1
(243
s3 :
Ye3
[ (s +
;m2jn)3
+ (s2+ Ig,& (60)
11’
t)” --.
1 s2
(61)
I
The constant q,/kBTc may be evaluated to leading order in vc by rewriting the integrand
+ -;;2
3
I
~
(z)
s 1 s3 + Yc3 [ exp(s2/47r) -
1
--.
47T s2 I
(62)
The first term is of order vc2, as shown by the substitution s = v&. In contrast, the second term is of order yc3 because it remains well defined if the denominator s3 + ~3 is replaced by s3. Consequently, Eq. (62) becomes [17]
(63)
=
2vc2 3.rr26’
which yields the chemical potential at Tc [cf. Eq. (49)], PL(Tc)w 0.77r?‘kBTC .
A similar analysis shows that M/M*
- 1 is of order vc3, which will turn out to be
DENSE
negligible in the present calculation. be evaluated explicitly,
BOSE
13
GAS
We are left with the term &;,. ) which will now
Here I is a definite integral, 1 -I- y 1 -y
! -2, 1
(65)
and we have used the substitution s = r/u in obtaining the last line of Eq. (64). This integral is studied in Appendix B, where the following limiting forms are derived:
(66b) I(x) -
-
8 3Tr -\/‘3 x2
The function 47r&JkBTCvC3 = (~A,/v,)~ f(kA,/v,) is shown in Fig. I; since it changes sign near the point k = v,/n, , we confirm the previous assertion that SE cannot be represented with a single effective mass. The self-consistent value of Tc can now be found explicitly from Eqs. (50) and (641, WL” =
* ---d”t -__ (2n)3 J exp((4r))l
1 t’[(M/M*)
where we again use the substitution t = k& the corresponding expression with I+ = 0,
+
__f(t/vr)]l
--
I ’
(67)
. It is convenient to add and subtract
(68)
+I& C
exp{(47r)’
t?[(M/M*)1
+
Z(t/+)]l
~- 1 ~- exp(rz/4n) 1 --- --- 1 I ’
14
FETTER
FIG. 1. Additional
contribution
to the long-wavelength
excitation spectrum.
The first term can be evaluated exactly, and the second can be rewritten with the further substitution t = vex,
vc2x2 r&3 = 5 (;) + $ j,” d.x[exp{(4z)-’ vC2x2[(M/M*) + W)]} - 1 -
vc2x2
exp(vc2x2/47r) - 1
1
(69)
The integral clearly remains well-defined if vC + 0 in the integrand. M/M* = 1 + 0(vC3), we obtain the approximate result
nAe3 w5(;)+ 7,
Since (70)
where
J = s;dx[ 1:1(x)- l1 is a dimensionless
number. A numerical integration J es 0.036.
gives (72)
DENSE
BOSE
GAS
Equation (70) is easily solved for Tc , and a simple rearrangement the desired expression, Tc - Tco 4Jv, :a - ___ TCO 37m)
c=a-0.026r:‘3.
I5
[cf. Eq. (l)] yields
(73)
Note the nonanalytic dependence (cc ezj3) on the coupling constant, which arises solely from the term SE in the excitation spectrum. As a corollary, the slope of the phase transition curve in the T-rs plane diverges to - cc for rs 4 0.
5.
DISCUSSION
The present calculation relies on a self-consistent description of a dense charged Bose gas in the region 0 < T - T, < T, . In particular, we have determined the screening of a charged impurity and the shape of the phase transition curve for rs << 1. This analysis represents only a first step toward a complete description, and several questions remain unanswered: An expansion of the Helmholtz free energy for small positive values of (T - T,)/ Tc would allow a calculation of the specific heat just above Tc . Another interesting topic is the diamagnetism, which may be studied with the theory of linear response.Since a densecharged Bose gas exhibits a Meissner effect near T = 0 [18], it seemslikely that the magnetic susceptibility would exhibit anomalies as T + T,+. A more difficult problem is the extension of the theory through the phase transition into the condensed phase. The Hartree-Fock description neglects fluctuations entirely, but such effects are important in the immediate vicinity of r,. . ln addition, the usual Bogolyubov approximation [19] breaks down as T--t T,> . as seen in its prediction of multivalued thermodynamic functions in that region [ 14, 201. For these reasons, any theory that is valid on both sides of the phase transition will doubtless require a wholly new approach. Recently, Chernikova [21] has made considerable progress in studying the phase transition of a neutral Bose gas, and similar techniques may prove useful for the charged system.
APPENDIX
A
The static screeningfunction for an ideal Bosegasis given by the exact expression [seeEq. (23)] F”(4) = -
(Al)
where (42)
FETTER
16
We first note thatf(x,
a) may be rewritten as a contour integral [IO],
f(x, 4 = j, 4 exp(ol Zy2)_ 11%($g)Y
(A31
where the integrand is rendered single valued by a cut from -x to x and C runs along the real axis passing above the branch cut (see Fig. 2). This representation is easily verified, because a simple analysis shows that Y+x log (------) Y--x
= In I$-$/
- k-03(x2 - y’)
for y along C. Hence the imaginary part of Eq. (A3) cancels identically, and the real part reproduces Eq. (A2). In addition to the branch cut, the integrand of Eq. (A3) has an infinite sequence of simple poles at y2 = --01 + 2mri, where n = 0, &l, f2,.... Deform the contour to C’, consisting of the real axis for 1y / > x plus a semicircle in the upper half-plane of radius just larger than x (see Fig. 2). If x < (a2 + 47r2)lj4, the integrals along C and C’ differ only by the contribution of the simple pole at y = idi2, which gives the expression
f(x,4 = nilog( $I$ z) + j,, dyexp(a +4’y2, _ 1log($-&$). (A9 The first term can be rewritten 27r arctan(x/o11j2); since ) y ) > x all along C’, the second term can be expanded in a Taylor series, and we find
f(x,4 = 2rarctan (s) + 2 z fi = 2rr arctan ($$)
+ 2ni go fi
j,, 4 exp(ol cy2,_ 1 G,p-j(a),
y PLANE
FIG. 2. Integration
contours in y plane.
DENSE
BOSE
17
GAS
where (A7) Here G,(a) is a single-valued analytic function of cr for all values of u [22], and the contour c” is shown in Fig. 3. If Re u is temporarily taken positivfe, the contour C” may be deformed to the positive real axis plus a small circle about the point t --I. G,(a)
:- p’-oy-l
+ &
fq’ dt
-0
tO-l
exp(cY.+ t) -- 1
(1 _~ pin)
(Re CT‘. 0).
(A8)
t PLANE
FIG.
3. Integration
contour
in t plane.
This last integral is essentially the Bose-Einstein function [23] (A91 and we therefore obtain G,(a) = eino[zo-l - T-I sin ror( a) 6,( ,I)].
(A IO)
In addition, F,(a) has the expansion [23] % F,(n) = T( 1 - cr) W-l + c (- ,u)lf i(U,,; ‘I) . ,1=0
(All)
A combination of Eqs. (AIO) and (Al 1) with the identity r(a) r( 1 - a) -= 7-rcsc 77u yields m GA4
59s/w1-2
=
-
j-(;‘:
u)
I”
(-cl)”
<(a I1 !
--
nl
(A12)
18
FETTER
Although this result has only been derived for Re CT> 0, each side is a singlevalued analytic function of u, and we use the principle of analytic continuation to extend it throughout the o plane. The function f(x, CX)is now determined explicitly by Eqs. (A6) and (A12). In particular, X-lf(x, CY)= 27i-x-i arctan@/&?) + 27r112&a) (A13) for small x and (II, which verifies the previous assertion about the remainder function ~JIappearing in Eq. (24). APPENDIX
B
The function I(X) in Eq. (64) is defined by the integral
031) It obeys a simple functional
equation, which can be derived as follows: Consider
c=- 2x3 m 79 I 0 =- 2x3 * 3-2 s 0
032)
where the last line is obtained with the substitution (B2) may be combined to give x-31(x-l)
+ x31(x) = f
jm dy [%
y-l = y’, Equations (Bl) and
ln / s
/ - 2 z: z -!3 1.
(B3)
0
The first integral is precisely one [24], and the second can be evaluated with the change of variables y = xz [17], ~-~Z(x-l)
+ x3Z(x) = I - $
jw dz xz$;;-l
(B4) = 1 - (8,3sD\/j)(x
+ x-l).
This functional relation is very useful, because it relates I for 1 < x < 03 to I for 0 -C x -C 1. In particular, we see that I(1) = &El - 16 &/9r]
w 0.0099.
(B5)
DENSE
BOSE
GAS
I9
The limiting behavior for small x can be derived by dividing the integration two regions y -C 1 and y > 1, I(x)
= I,(x)
-1 12(x).
into
(B~J)
where
(B8) and I, has been rewritten integrand of II ,
2x3 -- __ 7T2
a1
with the substitution
J’ -
_I’--~.
the
dy
.! 0 y3(.y3
-t Y”>
The last term is of order x2 as x -+ 0, and the remaining evaluated exactly to give
The other contribution
We first rearrange
definite integral can be
can be expanded directly for small x,
I2(-x) = $-
j1 dy [ y2 in I*
/ - 2~,] -~
0(x3)
0
(BI 1)
2 -- & (1 - In 2), which yields the final form (B12) The corresponding J%. @4),
asymptotic I(-4 -
behavior
for large x follows
-s/37?- d’3 x’)
(x -
a).
immediately
from (B13)
20
FETTER
ACKNOWLEDGMENT I am grateful to Dr. G. E. Walkerfor computingthe curve in Fig. 1 andthe numericalvalue
in Eq. (72).
REFERENCES 1. M. GELL-MANN AND K. A. BRUECKNER, Phys. Rev. 106 (1957), 364. E. W. MONTROLL AND J. C. WARD, Phys. FIuids 1 (1958), 55. L. L. FOLDY, Phys. Rev. 12.4 (1961), 649. 4. F. LONDON, “Supertluids,” Vol. II, p. 40, Dover Publications, New York, 1964. 5. A. L. FETTER AND J. D. WALECKA, “Quantum Theory of Many-Particle Systems,” Chaps. 5 and 9, McGraw-Hill, New York, 1971. 6. N. M. HUGENHOLTZ AND D. PINES, Phys. Rev. 116 (1959), 489. 7. J. HUBBARD, Proc. Roy. Sot. (London), Ser. A 243 (1957), 336. 8. P. NOZIBRES AND D. PINES, Nuovo Cimento 9 (1958), 470. 9. R. D. PUFF AND J. S. TENN, Phys. Rev. A 1 (1970), 125. 10. M. R. SCHAFROTH, Phys. Rev. 100 (1955), 463. 11. Ref. [4], p. 46. 12. Ref. [5], Section 27. 13. M. LUBAN, Phys. Rev. 128 (1962), 965. 14. V. K. WONG, Ph.D. Thesis, University of California, Berkeley, 1966, unpublished. 15. Ref. [5], Section 28. 16. E. C. TITCHMARSH, “The Theory of Functions,” 2nd ed., p. 113, Oxford University Press, Oxford, 1960. 17. H. B. DWIGHT, “Tables of Integrals and Other Mathematical Data,” 4th ed., p. 213, 856.08, Macmillan, New York, 1968. 18. A. L. FETTER, Ann. Phys. (New York) 60 (1970), 464. 19. N. N. BOGOLYUBOV, J. Phys. (U.S.S.R.) 11 (1947), 23. 20. K. HUANG, C. N. YANG, AND J. M. LUTTINGER, Phys. Rev. 105 (1957), 776. 21. D. M. CHERNIKOVA, Zh. Eskp. Teor. Fiz. 57 (1969), 2125 [English translation in Soo. Phys. JETP 30 (1970), 11541. 22. E. T. WHI~~AKER AND G. N. WATSON, “A Course of Modern Analysis,” 4th ed., p. 266, Cambridge University Press, Cambridge, 1962. 23. Ref. [4], p. 203. 24. Ref. [17], p. 243, 864.12. 2. 3.
OF PHYSICS:
64,
Transition
1-20 (1971)
Temperature
of a Dense Charged L. FETTER+
ALEXANDER Institute of Theoretical Stanford University,
Bose Gas*
Physics, Department Stanford, California
of Physics 94305
Received July 21, 1970
The linear response of a normal dense charged Bose gas to an applied scalar potential is studied in the random-phase approximation. Although the plasma frequency remains constant, the asymptotic screening around an impurity changes character at the transition temperature from exponential to algebraic. A self-consistent Hartree-Fock theory with the static screened interparticle potential yields the shift in the transition temperature (T, -~. TCu)/TCO w -0.026 rlg’3, where TCU is the transition temperature of an ideal Bose gas with density n and particle mass M, r8 = (e2MM’r2)(3,‘4an)1~3, and e is the boson charge.
1. INTRODUCTIOX The low-temperature behavior of an ideal gasdependscrucially on its quantum statistics. For example, a perfect Bose gas undergoes a phase transition at a critical temperature TCo,where a macroscopic condensate first starts to appear. In a perfect Fermi gas, however, the exclusion principle forbids such condensation. and the associatedphase diagram has no singularities. The striking difference between bosons and fermions persists in the presence of interactions, asseenin the contrast between the physical systemsHe3and He4. This distinction may also be studied with theoretical models, such as a dense charged gas placed in a rigid uniform background to ensure charge neutrality. Although the high- and low-temperature limits have been studied extensively [l-3], the very interesting topic of the phase transition in the charged Bose system appears untouched. As a first step in such a study, we here examine the behavior as the phase transition is approached from above. For definiteness, we consider the linear responseto an external scalar potential (Section 2). The resulting screening of a static impurity clearly foreshadows the approaching phase transition * Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, U. S. Air Force, under AFOSR Contract No. F44620-68-C-0075. + Alfred P. Sloan Research Fellow.
1 0 1971 by Academic Press, Inc.
2
FETTER
(Section 3). The corresponding screened interparticle potential then enables us to determine the critical temperature with a self-consistent Hartree-Fock theory (Section 4). 2. LINEAR RESPONSE TO AN EXTERNAL
POTENTIAL
Consider a gas of noninteracting spinless bosons with mass M per particle. At a density n and temperature T, the system has two characteristic energies: The thermal energy kBT and the quantum-mechanical zero-point energy ti2n2/3/M associated with localization within a volume n-l. Quantum effects predominate below the critical temperature Tco, which is evidently of order A2n2/3/MkB. More precisely, a detailed calculation yields [4] 277752 [* Tco = Mk,
213
I
M 3.31 g
,
(1)
B
where <(#) M 2.612 is the Riemann zeta function. The corresponding qualitative analysis of an interacting gas is more complicated owing to the characteristic energies associated with the strength and range of the interparticle potential. In the special case of a pure Coulomb gas, however, the range is infinite, and the only new quantity with the dimensions of an energy is e2n1J3,where e is the charge per particle, At high density (e2n1i3< fi2n2/3jM), the interparticle spacing r. E (3/4m) l/3 is much smaller than the Bohr radius a, = fi2/Me2 for the bosons, and it is conventional to introduce the small parameter r, = ro/uo < 1. We here treat only the region near the phase transition, so that T = Tco and e2n1J3< k,T, The linear response to an external potential v&x, t) is governed by the exact density-density correlation function for the unperturbed but interacting medium. Thus if n(x, t) is the Heisenberg operator for the particle density and 5(x, t) = n(x, t) - (n(x, t)) is the deviation from the mean value, then the induced density 6(n(x, t)) is given by [5] 6(n(x, t)) = 1 d3x’ dt’ IIR(x, 1; x’, t’) ey&x’,
f’),
(2)
where JP is the retarded commutator P(x,
t; x’, t’) = -if?1([8(x,
t), 6(x’, t’)]) B(t - t’),
(3)
and the angular brackets denote a grand canonical ensemble average at temperature T = (kg&l and chemical potential p, (4)
DENSE
BOSE
3
GAS
In a uniform equilibrium system, translational invariance in space and time guarantees that flR takes the form ZP(x - x’, t - t’). With the Fourier transform 2~(~,
t)i = (2+4
j d3q dw ei(q.xuw*)S(n(q,
w)i
(5)
and similar equations for v&q, w) and IIR(q. ~1, Eq. (21 becomes a simple product Wz(q, 0~)) = nYq,
w> eq,,(q, ~1.
(6)
Thus the problem is reduced to that of finding nR(q, w), which can be interpreted as the exact retarded polarization of the medium. The function flR is simply related to a corresponding time-ordered polarization 17. In the absence of Bose condensation (the more general case is treated in Ref. [6]), IT satisfies a simple Dyson’s equation. Consequently, nR can be expressed exactly in terms of the retarded proper polarization .IT*R(q, w), nR(q, w) = 1 -
n*R(q, w) I/,(q) n*yq,
Z).
(7)
Here V,,(q) is the Fourier transform of the Coulomb potential, modified at q = 0 to account for the uniform neutralizing background j4rre2/q2 ~ObI) = [O
q f 0, q = 0.
(8)
Unfortunately, Lr *R itself contains contributions of all orders in perturbation theory. Thus it is necessary to introduce approximations, and we make the simplest possible one, in which n*R is replaced by that for a perfect Bose gas noR above its transition temperature. This approach is frequently known as the randomphase approximation [7, 81. The function flOR is readily determined with standard techniques [5] to be
where eke = ?iek2/2M and nko = {exp[j?(E,O - p)] - 1)-l
(10)
is the usual Bose-Einstein distribution function. As a first application of Eq. (7), we show that a charged Bose gas has a long-
4
FETTER
wavelength collective model (plasma oscillation) at all temperatures above its transition temperature. The real part of Eq. (9) can be rewritten exactly as Re LroR(q, w) = 2c,OP 12%
(2~)~ (ho - fi2k *$M),
- (6,“)”
(11)
where P denotes the Cauchy principal value. An expansion for small q yields Re noR(q, OJ) = &
With the observation n = (2?~)-~
and the definition
I
d3k nko
(13)
of the mean-square velocity (u2> = n-l(2~)-~
j d3k nso(hk/M)2,
(14)
Eq. (12) becomes Re noR(q, OJ) w &
4Yv2) [ 1 + -;u;i-].
(15)
A system has a collective mode with frequency on and damping constant yq when the exact polarization LrR(q, w) has pole at w = wq - iy, . For small damping, Eq. (7) shows that wq is determined by the approximate equation 1 = V,(q) Re 17*R(q, wa). In the random-phase
approximation, this relation is easily inverted to give D 2 + 92(u2), WQ = WPZ
(16)
(17)
where wvz = (4rrne2/M)1/2
(18)
is the plasma frequency. Note that wDI is constant for a gas at fixed volume, and that the temperature and chemical potential affect only the correction term proportional to q2. The damping of plasma oscillations is determined by Im D*R. Although this quantity can be evaluated in closed form in the random-phase approximation [9], it is not especially relevant here and will not be considered further.
DENSE
BOSE
GAS
3. SCREENING OF A STATIC IMPURITY We now turn to the screening of a static point charge, where ‘P&q, Co)= 4xzeq-“2X6(W).
(19)
In the sameapproximation asthat usedabove, the induced charge density becomes
W(q, w)i = 7
27r6(w)
= Z27iqw)
n”R(q, QJ) 1 - V,(q) n”vl. WI
(20)
Vu(s)fioR(% 0) 1-
C’,(q)ITOYq, 0) ’
which is real becauseIm I?OR(q,0) = 0. Unlike the collective plasma oscillations, however, even the leading contribution to Eq. (20) dependsstrongly on T and p. The first task is the determination of 170R(q,0), which we shall henceforth denote F,,(c/). A simple transformation of Eq. (20) yields
It is convenient to introduce the thermal wavelength A = (2nk2/MkBT)‘~2
(22)
and the dimensionlessvariables s = q/l, t = kf, and N = -p/kBT ;-: 0. (Recall that p < 0 for a perfect Bose gas above its transition temperature.) In this way, Eq. (21) becomes t
exp(,r :-
f”/&-
In -1
1
2r + s ( ?f-s
1,
(23)
where the symmetry of the integrand allows us to extend the lower limit to - CC. This function is similar to one studied by Schafroth [IO]; it is evaluated in Appendix A. where we show that (2‘9 Here g, is a dimensionlessfunction that remains finite as CLand q/l tend to zero.
6
FETTER
Equation (20) can now be used to study the spatial form of the induced charge density p&x) around an impurity. The inverse Fourier transform gives PdX)
= ze j &
V,(q) F,(q) f+a.x 1 _ Ir,(q) Fo(q)
= Ze
eia’x q2 ~(4)
=-- ze
s
-$$
1
O”
x 27s s 0
q dq sin qx
(25)
4~e2Fo(d q2 - 4re2Fo(q)
’
This expression has several interesting features. First, the total screening charge is (26)
= -Ze, where the last equality follows because -‘Ocq
= ‘) = ~Ajlk
BT
jrn0 exp(cu +$/477)
- 1
(27)
is positive definite. Thus the impurity is wholly screened at large distances. Second, Eqs. (24) and (25) may be combined to yield v3{27r-’ arctan[s/4(cY7r)1’2] + s&r, CX)}, psc(x) =- -&& jmdss sin(sx/A) s3 + vS{27r-l arctan[s/4((1/77)1/2] + sy(s, a)}
(28)
0
where v3 = 8r3e2/AkBT
(29)
is a small parameter for a high-density gas near Tco. The substitution s = vf shows that CJImakes a negligible contribution to p&x) as v -+ 0, and we therefore obtain the final form (30)
where &c(z) = &
s sin szv32+ arctan(s/q) s,” ds s3 + v327r-l arctan(s/q)
(31)
and 7 is a second small parameter vj E 4(m)1/2.
(32)
DENSE
BOSE
I
GAS
For an ideal Bose gas [l 11, 7 = 35(9( T - TOO)/TcO< 1.
(33)
It is evident that S,,(z) vanishes identically for v := 0, which reflects the physical observation that the screening vanishes in a perfect gas. The other simple limiting case occurs at Tro, when a! = 0 and 27r-l arctan (s/v) == 1 for all s.
Since the length x enters only through the parameter vx/Ll, the characteristic screening distance is (1( TCo)/v( TCo) * 0. ~OU,$~. Equation (34) can be expressed in terms of sine and cosine integrals; a straightforward analysis then gives the following limiting forms:
where y = 0.577 is Euler’s constant. Note that S,,(z) at TCu vanishes for both large and small z. (It also must vanish at least once for an intermediate value.) Furthermore, the asymptotic tail of p&x) at TCo is algebraic rather than exponential. Both these features are in marked contrast with the Thomas-Fermi or Debye-Hiickel theory, where the corresponding screening function is a pure exponential [5]. We now turn to the more general question of the screening function &(;) for arbitrary small values of v and 7. Since the integrand of Eq. (31) is even in s. the equation may be rewritten arctan(s/q) &c(z)= 5 ?‘I, d.Yseiyzv327r1 $- 1~32~iarctan(siq) ’ ~3
(36)
which allows us to deform the integration contour into the upper half-plane. The asymptotic behavior for large z is then determined by the singularity of the integrand that is nearest the real axis. The singularities in the upper half-plane are of two types: (1) A branch cut running along the imaginary axis from s = in to s = ice, and (2) a pair of simple poles. The detailed analysis can be simplified by
8
FETTER
using the variable 7 = s/v, so that the relevant poles are the roots of the transcendental equation 73 + (V/T)” 2rr-1 arctan 7 = 0
(37)
for Im T > 0. (a) Relatively
far from the transition
temperature
Tcor~‘3 Q T -
(V Q -q or
T,‘),
these two poles are on the imaginary axis at s1 = i(2v3/q)1iz and Xl’ = iq[l - 2 exp(-q3.rrjv3)]. The first pole clearly dominates the asymptotic behavior. Hence &(z) cc exp(- j s1 [ z), and the screening length is of order A[(T - Tco)/rsTco]‘12.
(b) As v/q increases, the two poles approach
each other, coalescing at Tco)]. The screening function is now proportional to exp( - 1sZ 1z), so that the screening length initially decreases with increasing v for fixed 7. (c) When V/T becomes still larger, the poles move symmetrically into the first and second quadrants. For the particular value v/v = 1.07 [viz., when Tcorl’3 = 1.89 (T - Tco)], the poles reach the line s = iq. In this case, the poles and branch cut both contribute to the long-range part of S.&z). (d) If v/q > 1.07, the branch cut at iq dominates the asymptotic form, which is now proportional to exp(-qz). In this region the screening length is L’I/T and increases like (T - Tea)-’ as T---f Tc O.In the limit T = Tco, the screening function changes from exponential to algebraic and takes the form 5&(z). -z-~ [see Eq. (35b)].
s3 = 0.89qi when v/q = 0.92 [viz., when Tcars“’ = 1.62 (T -
The above analysis demonstrates that the long-range screening near Tco is quite complicated, depending sensitively on the relative values of the small parameters rit3 and I - T/Tc O. This behavior arises from the approaching phase transition; it therefore differs greatly from that of either a Boltzmann or Fermi gas. A slightly different quantity of interest is the static screened interaction V(q) between two particles in the medium. Since 1 - V,(q)fT*R(q, 0) acts as the static dielectric constant, we may use the same approximation as in Eq. (20) to write
V/(4)=
V,(q) 1-
V,(q) mJR(q, 0) *
An inverse Fourier transform yields the corresponding V(x) = ezx-‘S(x/A),
(38)
screened spatial potential (39)
9
DENSE BOSE GAS
where sdsin sz s3-7 v32.rr-1arctanm
’
(40)
The detailed analysis is quite similar to that for S&z) and will not be given here. Unlike the induced screening charge, however, we see that S(z) == 1 for v 2 0, which is also evident from physical considerations.
4. SELF-CONSISTENT
HARTREE-FOCK
THEORY FOR T > Tc
In the previous sections, the chemical potential has been taken to be that for an ideal Bose gas, vanishing at the unperturbed transition temperature TCo. This description cannot be fully self-consistent, becausep itself is altered by the presence of interactions. To include such an effect, we now attempt an improved (HartreeFock) theory, which allows us to determine the shift in the transition temperature to leading order in the interaction. For a weak short-range potential, the Hartree-Fock equations take Ihe intuitive form [12] n = (2~r-~
I
d3k n, ,
(41)
n,<= [exp P(Q - CL)- 11-l,
(42)
cl<= E,,.o +
(43)
(2x)-3
! d39[vo(0) t V,,(k - q)] II,, ,
where V&k - q) is the Fourier transform of the interparticle potential and the two terms in the integral in Eq. (43) represent the direct and exchange interactions, respectively. The self-consistency is now explicit, becausethe excitation spectrum cl{ both determines and is determined by the distribution function n,. . In their present form, the coupled Hartree-Fock equations can be expanded at long wavelengths to find the transition temperature of a weakly interacting neutral Bose gas [13-l 51. For a charged system, however, they must be modified in two distinct ways. First, the direct interaction V,(O) vanishes identically owing to the overall charge neutrality. Second, the bare potential V’*(y) ==:4re2/9” is unbounded at long wavelengths. Although this behavior does not invariably cause problems, it leads to serious difficulties at T, , where the Bose-Einstein distribution function II,, also develops a long-wavelength singularity. For this reason, screening plays an essential role in the phase transition of a dense charged Bose gas, and we shall replace V,,(q) by the static screenedinteraction
V(q) = 1 - V,(q) V,(q) F(q) ’
(44)
10
FETTER
where F(q) is also determined self-consistently, d3k
nk+,, - nk
Fcq) = s (2x)3
tk+q -
Thus we obtain the modified Hartree-Fock
lk -
equations
n = (271)~~ d3k nk , s
(46)
ek = cko i- (27r)-3 - d3q V(k - q) n, ,
(47)
J
with nK given by Eq. (42). It is clear that an exact solution of these coupled equations is out of the question, and we merely seek the lowest-order correction to the properties of the noninteracting system. An assembly of bosons remains normal so long as nk is bounded for all k. For an ideal gas, nko is largest at k = 0, so that the long-wavelength (k/l < 1) behavior of eke - y fixes the unperturbed critical temperature TCo. We shall assume that a similar description applies to an interacting charged Bose gas and write the single-particle spectrum in the approximate form (again for kA < 1) ck m co + ti2k2/2M* + 8~ .
(48)
Here 8~ is an additional shift in the spectrum that cannot be represented with an effective mass M*. As shown below, Sr, vanishes as k + 0, and the critical temperature TC of a dense charged Bose gas at fixed n is determined by the condition EO
In particular,
-p=O
at
Tc.
(49)
Eqs. (46) and (47) then reduce to d3k I n m (27?
exp[(ti2k2/2M* d3q
Ek = ‘Ice +
I
1 + &,J/k,T,] Vo(k
(2~)~ 1 -
-
- 1 ’
q)
V,(k - q) F(k - q)
1 x
(50)
exp[(h2q2/2M* + &,)/k,T,]
- 1 ’
(51)
where
F(q)
1 exp[(fz2k2/2M* + 6+)/kBTc] - 1 (52)
DENSE
BOSE
II
GAS
It is evident from Eq. (51) that the corrections to the free-particle spectrum vanish as rs and hence v3 tend to 0. Since we want only the leading contribution to Tc - TCO, it is then permissible to simplify Eqs. (51) and (52) by neglecting 8~ entirely and setting M* = M. As a result, F(q) reduces to that for a perfect Bose gaswithp-zOandT= T,, -2r”/qfl,4k,Tc,
F(q) * while the excitation EL e .FI; ” f
spectrum
sd3q
(53)
becomes 1
4ne2 (2~)~ q2 + (8rr3e”/qAc4kBTe)
exp[@(k
+ q)2/2MkBT,,]
Here II, = (2m52/MkBTc)1~2 is the thermal wavelength at TC , We once again introduce the dimensionless variables s = qil, , t = k/l, yP3 == 8n3e2/A,kBTc , so that the shift in the spectrum can be written
Ek- Eli0 k,Tc
(54)
-. 1 ’
, and
1 s Ye3 - __-d3s --, (2n)3 s3 + vc3 exp[(s + t)2/4.rrl - 1 2.rr2 J = ___
(55)
The right side of Eq. (55) is formally of order vC3, but a direct expansion in powers of vC3 leads to divergent integrals, thus confirming the importance of screening. Instead, it is preferable to use the expansion [16] (56) to rewrite
the distribution
1 exp[(s + t)2/4rr] -
function I = (s$ty
---
1 2 4v
+ gl [(s+ t)? 8$jn+ (s--_____ ’ -t t)2+ Win 1
(57)
For real s and / t / < 2x ~‘2, the fractions in brackets may be expanded in a Taylor series: resumming the terms for r = 0 with Eq. (56), we find 1 .-__ exp[(s + t)2/4x]?
=
4x (s -+
ty
4rr --T .P
! n=l)-(2s.tit’)[ + 46.t)”[ _
1 exp(s2/4r)
+4$
($2
- 1
1 (3 - 8772in)2 + Ig,2inj3
(s2
+
;n2jn)l
-
1
+ ($ + igin) II + O(f3).
12
FETTER
When this expression is substituted into Eq. (55), we readily identify the various contributions to the long-wavelength spectrum [cf. Eq. (48)], (59)
M -M* - * = 8vc3 g I& 8.5 z&
[ (9 - Lin)”
-
+
(9 + iwifl~
d3s
2vc3 =
; y> 1% [ (S2-
S3
s
1
(243
s3 :
Ye3
[ (s +
;m2jn)3
+ (s2+ Ig,& (60)
11’
t)” --.
1 s2
(61)
I
The constant q,/kBTc may be evaluated to leading order in vc by rewriting the integrand
+ -;;2
3
I
~
(z)
s 1 s3 + Yc3 [ exp(s2/47r) -
1
--.
47T s2 I
(62)
The first term is of order vc2, as shown by the substitution s = v&. In contrast, the second term is of order yc3 because it remains well defined if the denominator s3 + ~3 is replaced by s3. Consequently, Eq. (62) becomes [17]
(63)
=
2vc2 3.rr26’
which yields the chemical potential at Tc [cf. Eq. (49)], PL(Tc)w 0.77r?‘kBTC .
A similar analysis shows that M/M*
- 1 is of order vc3, which will turn out to be
DENSE
negligible in the present calculation. be evaluated explicitly,
BOSE
13
GAS
We are left with the term &;,. ) which will now
Here I is a definite integral, 1 -I- y 1 -y
! -2, 1
(65)
and we have used the substitution s = r/u in obtaining the last line of Eq. (64). This integral is studied in Appendix B, where the following limiting forms are derived:
(66b) I(x) -
-
8 3Tr -\/‘3 x2
The function 47r&JkBTCvC3 = (~A,/v,)~ f(kA,/v,) is shown in Fig. I; since it changes sign near the point k = v,/n, , we confirm the previous assertion that SE cannot be represented with a single effective mass. The self-consistent value of Tc can now be found explicitly from Eqs. (50) and (641, WL” =
* ---d”t -__ (2n)3 J exp((4r))l
1 t’[(M/M*)
where we again use the substitution t = k& the corresponding expression with I+ = 0,
+
__f(t/vr)]l
--
I ’
(67)
. It is convenient to add and subtract
(68)
+I& C
exp{(47r)’
t?[(M/M*)1
+
Z(t/+)]l
~- 1 ~- exp(rz/4n) 1 --- --- 1 I ’
14
FETTER
FIG. 1. Additional
contribution
to the long-wavelength
excitation spectrum.
The first term can be evaluated exactly, and the second can be rewritten with the further substitution t = vex,
vc2x2 r&3 = 5 (;) + $ j,” d.x[exp{(4z)-’ vC2x2[(M/M*) + W)]} - 1 -
vc2x2
exp(vc2x2/47r) - 1
1
(69)
The integral clearly remains well-defined if vC + 0 in the integrand. M/M* = 1 + 0(vC3), we obtain the approximate result
nAe3 w5(;)+ 7,
Since (70)
where
J = s;dx[ 1:1(x)- l1 is a dimensionless
number. A numerical integration J es 0.036.
gives (72)
DENSE
BOSE
GAS
Equation (70) is easily solved for Tc , and a simple rearrangement the desired expression, Tc - Tco 4Jv, :a - ___ TCO 37m)
c=a-0.026r:‘3.
I5
[cf. Eq. (l)] yields
(73)
Note the nonanalytic dependence (cc ezj3) on the coupling constant, which arises solely from the term SE in the excitation spectrum. As a corollary, the slope of the phase transition curve in the T-rs plane diverges to - cc for rs 4 0.
5.
DISCUSSION
The present calculation relies on a self-consistent description of a dense charged Bose gas in the region 0 < T - T, < T, . In particular, we have determined the screening of a charged impurity and the shape of the phase transition curve for rs << 1. This analysis represents only a first step toward a complete description, and several questions remain unanswered: An expansion of the Helmholtz free energy for small positive values of (T - T,)/ Tc would allow a calculation of the specific heat just above Tc . Another interesting topic is the diamagnetism, which may be studied with the theory of linear response.Since a densecharged Bose gas exhibits a Meissner effect near T = 0 [18], it seemslikely that the magnetic susceptibility would exhibit anomalies as T + T,+. A more difficult problem is the extension of the theory through the phase transition into the condensed phase. The Hartree-Fock description neglects fluctuations entirely, but such effects are important in the immediate vicinity of r,. . ln addition, the usual Bogolyubov approximation [19] breaks down as T--t T,> . as seen in its prediction of multivalued thermodynamic functions in that region [ 14, 201. For these reasons, any theory that is valid on both sides of the phase transition will doubtless require a wholly new approach. Recently, Chernikova [21] has made considerable progress in studying the phase transition of a neutral Bose gas, and similar techniques may prove useful for the charged system.
APPENDIX
A
The static screeningfunction for an ideal Bosegasis given by the exact expression [seeEq. (23)] F”(4) = -
(Al)
where (42)
FETTER
16
We first note thatf(x,
a) may be rewritten as a contour integral [IO],
f(x, 4 = j, 4 exp(ol Zy2)_ 11%($g)Y
(A31
where the integrand is rendered single valued by a cut from -x to x and C runs along the real axis passing above the branch cut (see Fig. 2). This representation is easily verified, because a simple analysis shows that Y+x log (------) Y--x
= In I$-$/
- k-03(x2 - y’)
for y along C. Hence the imaginary part of Eq. (A3) cancels identically, and the real part reproduces Eq. (A2). In addition to the branch cut, the integrand of Eq. (A3) has an infinite sequence of simple poles at y2 = --01 + 2mri, where n = 0, &l, f2,.... Deform the contour to C’, consisting of the real axis for 1y / > x plus a semicircle in the upper half-plane of radius just larger than x (see Fig. 2). If x < (a2 + 47r2)lj4, the integrals along C and C’ differ only by the contribution of the simple pole at y = idi2, which gives the expression
f(x,4 = nilog( $I$ z) + j,, dyexp(a +4’y2, _ 1log($-&$). (A9 The first term can be rewritten 27r arctan(x/o11j2); since ) y ) > x all along C’, the second term can be expanded in a Taylor series, and we find
f(x,4 = 2rarctan (s) + 2 z fi = 2rr arctan ($$)
+ 2ni go fi
j,, 4 exp(ol cy2,_ 1 G,p-j(a),
y PLANE
FIG. 2. Integration
contours in y plane.
DENSE
BOSE
17
GAS
where (A7) Here G,(a) is a single-valued analytic function of cr for all values of u [22], and the contour c” is shown in Fig. 3. If Re u is temporarily taken positivfe, the contour C” may be deformed to the positive real axis plus a small circle about the point t --I. G,(a)
:- p’-oy-l
+ &
fq’ dt
-0
tO-l
exp(cY.+ t) -- 1
(1 _~ pin)
(Re CT‘. 0).
(A8)
t PLANE
FIG.
3. Integration
contour
in t plane.
This last integral is essentially the Bose-Einstein function [23] (A91 and we therefore obtain G,(a) = eino[zo-l - T-I sin ror( a) 6,( ,I)].
(A IO)
In addition, F,(a) has the expansion [23] % F,(n) = T( 1 - cr) W-l + c (- ,u)lf i(U,,; ‘I) . ,1=0
(All)
A combination of Eqs. (AIO) and (Al 1) with the identity r(a) r( 1 - a) -= 7-rcsc 77u yields m GA4
59s/w1-2
=
-
j-(;‘:
u)
I”
(-cl)”
<(a I1 !
--
nl
(A12)
18
FETTER
Although this result has only been derived for Re CT> 0, each side is a singlevalued analytic function of u, and we use the principle of analytic continuation to extend it throughout the o plane. The function f(x, CX)is now determined explicitly by Eqs. (A6) and (A12). In particular, X-lf(x, CY)= 27i-x-i arctan@/&?) + 27r112&a) (A13) for small x and (II, which verifies the previous assertion about the remainder function ~JIappearing in Eq. (24). APPENDIX
B
The function I(X) in Eq. (64) is defined by the integral
031) It obeys a simple functional
equation, which can be derived as follows: Consider
c=- 2x3 m 79 I 0 =- 2x3 * 3-2 s 0
032)
where the last line is obtained with the substitution (B2) may be combined to give x-31(x-l)
+ x31(x) = f
jm dy [%
y-l = y’, Equations (Bl) and
ln / s
/ - 2 z: z -!3 1.
(B3)
0
The first integral is precisely one [24], and the second can be evaluated with the change of variables y = xz [17], ~-~Z(x-l)
+ x3Z(x) = I - $
jw dz xz$;;-l
(B4) = 1 - (8,3sD\/j)(x
+ x-l).
This functional relation is very useful, because it relates I for 1 < x < 03 to I for 0 -C x -C 1. In particular, we see that I(1) = &El - 16 &/9r]
w 0.0099.
(B5)
DENSE
BOSE
GAS
I9
The limiting behavior for small x can be derived by dividing the integration two regions y -C 1 and y > 1, I(x)
= I,(x)
-1 12(x).
into
(B~J)
where
(B8) and I, has been rewritten integrand of II ,
2x3 -- __ 7T2
a1
with the substitution
J’ -
_I’--~.
the
dy
.! 0 y3(.y3
-t Y”>
The last term is of order x2 as x -+ 0, and the remaining evaluated exactly to give
The other contribution
We first rearrange
definite integral can be
can be expanded directly for small x,
I2(-x) = $-
j1 dy [ y2 in I*
/ - 2~,] -~
0(x3)
0
(BI 1)
2 -- & (1 - In 2), which yields the final form (B12) The corresponding J%. @4),
asymptotic I(-4 -
behavior
for large x follows
-s/37?- d’3 x’)
(x -
a).
immediately
from (B13)
20
FETTER
ACKNOWLEDGMENT I am grateful to Dr. G. E. Walkerfor computingthe curve in Fig. 1 andthe numericalvalue
in Eq. (72).
REFERENCES 1. M. GELL-MANN AND K. A. BRUECKNER, Phys. Rev. 106 (1957), 364. E. W. MONTROLL AND J. C. WARD, Phys. FIuids 1 (1958), 55. L. L. FOLDY, Phys. Rev. 12.4 (1961), 649. 4. F. LONDON, “Supertluids,” Vol. II, p. 40, Dover Publications, New York, 1964. 5. A. L. FETTER AND J. D. WALECKA, “Quantum Theory of Many-Particle Systems,” Chaps. 5 and 9, McGraw-Hill, New York, 1971. 6. N. M. HUGENHOLTZ AND D. PINES, Phys. Rev. 116 (1959), 489. 7. J. HUBBARD, Proc. Roy. Sot. (London), Ser. A 243 (1957), 336. 8. P. NOZIBRES AND D. PINES, Nuovo Cimento 9 (1958), 470. 9. R. D. PUFF AND J. S. TENN, Phys. Rev. A 1 (1970), 125. 10. M. R. SCHAFROTH, Phys. Rev. 100 (1955), 463. 11. Ref. [4], p. 46. 12. Ref. [5], Section 27. 13. M. LUBAN, Phys. Rev. 128 (1962), 965. 14. V. K. WONG, Ph.D. Thesis, University of California, Berkeley, 1966, unpublished. 15. Ref. [5], Section 28. 16. E. C. TITCHMARSH, “The Theory of Functions,” 2nd ed., p. 113, Oxford University Press, Oxford, 1960. 17. H. B. DWIGHT, “Tables of Integrals and Other Mathematical Data,” 4th ed., p. 213, 856.08, Macmillan, New York, 1968. 18. A. L. FETTER, Ann. Phys. (New York) 60 (1970), 464. 19. N. N. BOGOLYUBOV, J. Phys. (U.S.S.R.) 11 (1947), 23. 20. K. HUANG, C. N. YANG, AND J. M. LUTTINGER, Phys. Rev. 105 (1957), 776. 21. D. M. CHERNIKOVA, Zh. Eskp. Teor. Fiz. 57 (1969), 2125 [English translation in Soo. Phys. JETP 30 (1970), 11541. 22. E. T. WHI~~AKER AND G. N. WATSON, “A Course of Modern Analysis,” 4th ed., p. 266, Cambridge University Press, Cambridge, 1962. 23. Ref. [4], p. 203. 24. Ref. [17], p. 243, 864.12. 2. 3.