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Functional integrals for condensed bose systems
Physica 98A 154-168 @ North-Holland
FUNCTIONAL
INTEGRALS
Publishing Co.
FOR CONDENSED
Masakazu Department
BOSE
SYSTEMS
ICHIYANAGI
of Applied Physics, Osaka University, Suita, Osaka, 565, Japan
Received 17 February 1970
We express an exact formulation for an interacting Bose system as a functional integration over complex functions. We propose a modification of the generating functional which leads to different limiting processes. The present article rests upon Bogoliubov’s description on the explanation of the existence of the order-parameter in terms of quasiaverages arising from a symmetry breaking field. Discussions to determine the order-parameter function are also given. We indicate that the modified functional integral method is relevant to the theory of superfluidity. The derivation of Landau-Ginzburg type equation for the order-parameter is given.
1. Introduction
That the superfluidity of liquid helium should be described a macroscopic wavefunction was emphasized by Onsager’) and Penrose*). However, we do not know how to derive an equation for the macroscopic wavefunction, which is corresponding to the Landau-Ginzburg equation for a superconductivity. Though, the so-called Gross-Pitaevskii theory3) does lead to a set of equations very similar to Landau-Ginzburg theory, this theory in its simple form makes no distinction between the superfluid density and the condensate density. The latter density is close to 10% at absolute zero temperature, whereas the superfluid part is exactly 100% of the total density. It is an essential step toward the superfluid theory to identify the superfluid velocity as being equal to (h/n)VS, S being the phase of the condensate wavefunction4). This definition of the superfluid velocity is general and possibly rigorous. However, it is not so obvious to see how to break up the total density into superfluid and normal densities. In this article, we are going to discuss the microscopic background of that ansatz for the phase of the condensate wavefunction. An essential aspect of the theory presented here is the consistent treatment of the condensate and the order-parameter. The point of view adopted is based on Bogoliubov’s description of the condensate in terms of quasi-averages arising from an infinitesimal, symmetry breaking external field5). Here we try to generalize Bogoliubov’s description to include the order-parameter. This is done through 154
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an introduction of the e-term in the Lagrangian density, which modifies a symmetry breaking field in a rather essential way to show how to derive the wave equation for the order-parameter. The techniques of functional integral method prove themselves very useful in these investigations. We have used them in order to be able to derive the equation for the order-parameter of an interacting Bose system. Some of the arguments used in such a derivation are found to be related to those used by Casher, Lurie and Revsen6), among others. However, the present author believes some point may be new, in particular the treatment of the super-fluid order-parameter. In effect the idea we develop below amounts to assuming a kind of infinitesimal symmetrybreaking fields which are in relation with the order-parameter of a Bose system. It would be mysterious to observe that the order-parameter, which is first introduced in this theory as a parameter, behaves as if it were the dynamical variable of the system. The reader will not fail to notice a close parallel with ferromagnetism. The symmetry-breaking field is introduced to lift the degeneracy of the states of the system with respect to the phase of the Bose fields. Therefore, it would be expected to prove that the outcome in the presence of the symmetry-breaking field is not depend on the magnitude of the order-parameter. To prove this may be quite troublesome and reserves a separate article. However, we may note that the order-parameter simply manifest itself as a convention by which we describe the condensation process. In order to make the article reasonably self-contained, we have presented technical details associated with the basic concepts such as the gauge invariance and the idea of asymptotic field for a homogeneous Bose system’). The introduction of the E-term in the Lagrangian replaces the external fields J and J* by J-E@ and J*- &D*, @ and @* being the parameters to be specified as the order-parameters. E is an infinitesimal. Stand in view of the above fact, we rest upon Bogoliubov’s description of the Bose system, if we expect that the spontaneous breakdown of symmetry can be effected by introducing an external field proportional to the order-parameter. The plan of this article is as follows: The next section is devoted itself to give a preliminary discussion on the method of the functional integral. The occurrence of the E-term is discussed. In section 3, we will discuss some consequences concerning the gauge invariance of the formulation. The physical meaning of the parameter in the e-term is studied in section 4. In section 5, by generalizing the discussions in the preceding sections to an inhomogeneous Bose system, we are going to develop a theory for the orderparameter rather than the condensate wavefunction. By a single train of argument akin to the one for a homogeneous Bose system, we will show that
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M. ICHIYANAGI
it is possible to derive the macroscopic equation that the order-parameter should obey. In section 6, some concluding remarks are given.
2. Functional-integral
method
In this section we address ourselves to develop a theory for Bose-Einstein condensation of an interacting Bose system. Our theory in this article rests upon the explanation of the existence of the condensate in terms of quasiaverage’) arising from a symmetry-breaking field. We formulate our theory in terms of the functional integration, which proves itself useful in the investigation of thermal properties of an interacting Bose system and allows the formulation for the condensate in an inhomogeneous case. Our model is given by the Lagrangian
a4,0*1=-I(
dx dr [ 4*(x7) ;
a
- +*(rr)(V’+
/A)+(xT)
(2.1) where p is the chemical potential of the Bose system of the particle mass m = f, g the coupling constant, and 7 the imaginary time, 0 < T < /3 (= l/k,T, ka being the Boltzmann constant). 4(x7) is a complex scalar field and +*(x7) is a complex conjugate of &x7). Then, following Casher et a1.6) we define the generating functional
2L.t J*l = j- [d41[W*l expM4,
(2.2)
+*I + W* + J*4L
where the symbol J [d+][d&*] denotes functions &(x7) and 4*(x7); and
an integration
over the space of the
dTJ(xT)&*(XT). It is easy to see that the stationary
(2.3 conditions
are
((653& 4*l/wJ*w))
+ J(m) = 0,
(2.4a)
+*l/wx7)))
+ J*w) = 0,
(2.4b)
((63d5
where ((Q)) denotes the ensemble average of the quantity
Q;
(((2)) = (ZM J*l)-’ 1 kMl[W*lQ ewW’e[4,4*1+ W* + J*#J).
(2.5)
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We may recall that solutions of the’ stationary conditions correspond to the most probable path and the main contribution to the functional integral comes from the domain near to that path. As was shown by Wiegel’), the solutions in the limit J(xT) = J*(xT) = 0 are independent of T in the zeroth order approximation. The next point to be noted is that the functional integral of a type introduced above is invariant under the change of variables: j- [d4lbW*l~(4,4*1=
1 [d4lkM*lF[4
+ @3,4*+ @*I,
(2.6)
where @ = @(xr) is an arbitrary function. The acceptance of (2.6) leaves purposely vague an interesting possibility of interpreting the apparent action of the machinery of Bogoliubov’). Therefore, in the functional-integral formulation of the Bose-Einstein condensation, there is no way to know which one of those stationary solutions should be picked out. In what follows, we will show how to select a special one from among infinitely many stationary solutions. To do that, we introduce
the generating functional
W, J*; @I = / kMl[d+*l ewWI4, &*I + J+* + J*4 - ~14 - @p13, (2.7) where the limit E + 0 as V (the volume of our system) 4 ~4is understood above expression. Then, the new stationary conditions are (all+,
4*1/84*(x7))
+ J(XT) = l(+(rT) - @(XT)),
@2[4, ‘#‘*l/~d’(Xd + J*(XT) = d$‘*(xT) - @*(XT)),
in the (2.8a) (2.8b)
where t
(Q) = WJ, J*; @I)-’ x I [WlW4*1 exp{%4+4*1+ W* + J*4 - 44 - Q/3.
(2.9)
It is obvious that the above conditions are not invariant under the change of variables as expected. Eqs. (2.8) clearly show the reason why we have introduced the e-term in the Lagrangian. To see this, let us for a moment consider an ideal Bose system, for which (2.8) read (PO+ E)@(Xr)) = -(J - e@),
(2.1Oa)
t In this article, four kinds of ensemble average are used. The reader may not fail to distinguish form among them; (( )) is defined by (2.3, ( ) by (2.9), ( )I by (4.8), and ( )2 by (5.5), respectively. It should be noted that in the limit of E + 0 ( ) may or may not be equal to (( )), since the latter has many solutions.
1.58
M. ICHIYANAGI
(po + E)((b*(xT)) = -(J”
- CD*),
(2. lob)
where p. is the chemical potential for the system. Here, in (2.10), we have assumed that the parameters @ and @* are constants, independent of x and T. Then, it is easy to see that the average values of I and +*(x7) have non-zero limits; lii
(c#J(XT)) = @.
(2.11)
It has been shown that in the ideal Bose system the nonvanishing limit of (4(x7)) cannot obtained from the mean field theory alone?. Thus, we see that the E-term lets the value of the condensate (or the order-parameter) to be ascertained, that is to say, the e-term selects a special value of the condensate from among many possible solutions of the mean-field theory, when the parameter @ is given.
3. Gauge invariance
and phase of the condensate
In this section we exploit the consequences of gauge invariance in order to see how the e-term selects a special solution from among many stationary solutions. We first note that our Lagrangian is invariant under constant phase transformations; +(X7) +
de&T),
4*(X7)
+= e-“4*(X7),
(3.1)
where 0 is a real constant phase. Then, by making use of (2.2), we see that (3.1) changes Z[J, J*] into Z[e-“J, e”J*]. Therefore, in order that Z[J, J*] is well-defined, we must have Z[e-“J, e’“J*] = Z[J, J*], implying that Z[J, J*] depends
(3.2) on J and J* only through the combination
J - J*.
A question then arises as to how the phase of the condensate is fixed by the infinitesimal external field. The Bogoliubov procedure shows that the phase of the condensate depends on the direction in the complex J-plane followed in the limiting procedure 1JI + 0. It is not unphysical to expect that Z[J, J*; CD] also depends on J and J* only through the combination J - J *. In that case, we see that under (3.1) Z[J, J*; @] -+ Z[e-‘“J, e”J*; em”@] = Z[J, J*; em”@].
Therefore,
from
(2.8), we see that the stationary
solutions
(3.3) move around
FUNCTIONAL
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according to the change of phase of &CT). To see this from a rather different point of view, we assume that @(XT)= @(const.). We now raise an interesting point which is found from the stationary conditions. From (2.8), we get (+*(x7))(=/8&*(x~)) = J*(xT)(&xT))
- (Nx7))(=/Mx7)) - J(x7)(d*(xT))
+ l{@*(#+))
- %#J*(x~)>1,
(3.4)
where L!?= 6p[4,4*]. From the point of view of the gauge invariance, the two sides of (3.4) have different characters from each other. This fact indicates that we have
J(xT)(+*(xT))- J*(xT)(‘#+d) ‘d@*@‘(d) Then, when we take a variation J*(xT), we get @(XT)) =
l
- @‘@*(XT))).
(3.5)
of the both side of (3.5) with respect
j dy d+#‘*(XT)&‘T’))@
- (+(xT)&‘T’))@*),
to
(3.6)
where (. * -) denotes the ensemble average (2.9) when J = J* = 0. Equation (3.6) is a key point to determine (+(x7)) in terms of @ and @*. We are now at the position to do that problem. We write &XT) in terms of the asymptotic field”‘); (b(XT)
=
&, * X(X7) + * . .,
4*(X7) = 2,
(3.7) ’ X*(X7)
+
* . *.
where z+ is the renormalization constant of the asymptotic field x(x7) and the dots denote the higher order combinations in the asymptotic fields. Then, by making use of (3.7) in (3.6) and after simple manipulation, we arrive at
(&XT)) = 2,
’ @,
(p + M),
(3.8)
when we take the limit of E + 0. Eq. (3.8) corresponds to the eq. (5.4) of the previous paper’@). It should be noted that (3.8) shows that the phase of the condensate (+(X7)) is fixed by the quantity @ in the e-term. If 2, < 1, the expectation value (c#J(xT)) is not equal to the parameter CD.This result is expected in (2.8); that is to say, if we have (c$(xT)) = @, the e-term does not play any role to determine the stationary solutions. An ideal Bose system is purposely an exceptional case which brings the expectation value (+(x7)) to the parameter.
M. ICHIYANAGI
160
4. Stationary phase and renormalization
constant
So far the parameter COwas not specified in any physical sense. In this section, however, we treat an example in which the parameter @ is thought of as a stationary phase’). By making use of (2.6) and (2.7) we get
Z[J, J*; ‘PI = exp{Y[@, @*I + J@* + J*@}
[d4l[W*l expW[9 + @,4* + @*I I - .3[@, @*I+ .&#I* + J*d, - ~141’). x
(4.1)
The action in this model is given by W[J, J*; @] = -p-’
In Z[J, J*; @]
= -p-‘{2[@,
CD*]+ J@* + J*@}+
W,[J, J*; @I,
(4.2)
where W’[J, J*; @I = -p-’
Z,[J, J*; @I =
In Z,[J, J*; @],
[d+][d+*]
(4.3)
exp{5!?[+ + CD,+* + @*I
- U[@, @*I + Jc$* + J*c#J - E[c#J[“}.
(4.4)
Let us introduce the effective action r[(+), (c$*); E] which is obtained from the action W[J, J*; @] by a Legendre transformation; r[G$), M*); cl= WJ, J*; @I - S-‘(J*(4) + J(+*)).
(4.5)
In order to evaluate r[(~$), (4”); e], J(m) and J*(m) must be eliminated. To do this, let us choose the parameters to satisfy the following relations12) @[-F + g]@]2]+ J(w) @*[-CL + g]@]2]+ J*(m)
= 0, = 0.
(4.6)
Then, from the definition of (+(x7)) we have @(x7)) = @ + (&(r7))1,
(4.7)
(+(YT)), = -f?SW,[J,
(4.8)
J”; @l/SJ*(m).
Here, in (4.8) J(m) and J*(m) are eliminated in favor of @ and @*. Thus, (4(x7)), is a complicated functional of CDand @*. The following notations are
FUNCTIONAL
now introduced
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by writing .%‘[+.+ @, c$* + @*I in the form:
.z[f#J + CD,4* + @*I = Lz[@, CD*1- J”#I - .@* + L?‘[+, d*; @, @*I + $*G-‘4
+ +I*,
(4.9)
where C8= (f*),
(4.10)
6* = (4, +*I,
--++y+V2+X,
2.2
G=
(4.11) $-•+p+V2+&
z2
i
1.
Here in (4.11) Z, and X2 are defined by 2, . (c#J)(xT)= g I dy d7’ ‘(‘$$r;;$;;)) =g
dy d7, s(]+(x7)]2+*(x’)) I
x2 * @(XT))
(+(y~‘)) (+*(y71))
w*(Ya
= g /
dy dr’ s(J’(X’)‘2’*(xT))
= g
dy dT1 ~(td’(d*d’(XT))
%+(YT’))
I
3
(4.12)
(&T’))
(,qyyTt))
s@*(YT’)>
(4.13)
Then, by making use of (4.6) and (4.4), we obtain Z,[J, J*; @I = To proceed, z2i.t
J*;
I
[d&][d&*] exp{&$*G-‘4 + Y[4, +*; CD,@*I}.
it is needed to introduce @I
=
ZdJ,
J*;
(4.14)
the object (4.15)
@l/Zd@l,
where &[@I = 1 [d+][d+*] exp{:&*G-‘4). The quantity Z,,[@] is the partition propagator G. Then, we have Z,,[cP] =
fl (1 - e&k)-‘,
function
(4.16) for free fields described
by the
(4.17)
k
where w& is the phonon frequency, which is the eigenvalue of G. Equation (4.15) shows us that only linked terms need be considered when the pertur-
162
M. ICHIYANAGI
bation scheme is derived from it directly. The appearance of Z,,[@] in (4.15) indicates that the propagator in this scheme is G. It is noted that G depends on @ and @* by its definition, and correspondingly wk may depend on @ and @*. Eq. (4.8) can be rewritten in the form ilW,[J
J”; @] a@ aw,[J, ;@ aJ*+
J*; @]
a@*
(4.18)
By making use of (4.14)-(4.17) and (4.18), we have
aW,[J, J*; @I a@*
(4.19)
where W,[J, J*; CD]= -p-’
In &[J, J*; @I.
(4.20)
In (4.20) J and J* must be eliminated in favor of 0 and @*. Then, in the first order in the interaction described by Y[&, c$*; @, @*I, from (4.7) and (4.19) we obtain the approximate expression for (I); (+(x7)) = @ + (first term of (4.19)) (4.21)
= Q/1(1+ Al(mc(@l*)], where A is a numerical factor given by A
= p
c
k
efi”k . (1 - epwk)-2,
k
and m = iis the particle mass and c the sound velocity of the system given by (4.23)
c = (gl@l’). To derive (4.21), we have used the fact that the chemical potential reads
(4.24)
CL= gl@12. The result tells us that the renormalization
constant 2, is approximately
given
by 2, = (1 + A/mcl@l*)-‘. 5. The order-parameter
What has Bose-Einstein
(4.25)
of a Bose system
condensation
treated in the previous sections to do
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with superfluidity? The connection between Bose-Einstein condensation and superfluidity is as follows*). We consider the superfluid from the point of view of a macroscopically occupied quantum state with its wave function of the form P(x) = exp[iS(x)]R(x), S and R being real functions. We identify the superfluid velocity as being equal to 2hVS, indicating that the superfluid circulation will be quantized in units of h/m, (m = i). In usual theories4*“), the superfluid velocity is defined in terms of the phase of the condensate wavefunction, whose normalization gives the density of condensed particles. To be on the safe side we have confidence in treating the phase of the condensate as an essential object. We must note however that no one knows the wave equation that the macroscopic wavefunction (denoted the superfluid orderparameter) should obey. One should recall that in section 2 we have seen that the phase of the condensate is fixed by the parameter in a unified way. Thus, we are interested in particular in the generalization to a nonhomogeneous Bose system having a macroscopic wavefunction below a certain transition temperature, It is a realistic application of the present theory potential in section 2 to derive an equation for the macroscopic wavefunction ?P(x~). Let us first introduce the object W[J; !P] = -p-l
In
I
[d4][d4*]
exp{Z[4 + F, 4* + !P*]
- Z[ F, ?Py*]+ A$* + J*+ - +$1’}, We now want to evaluate this at the particular
(5.1)
values of J and J*, where
J(XT) + 6 W*/@P’*(x7) = 0, J*(xT) + SW*/&P(x7) = 0.
(5.2)
In (5.2), W2 is given by w2
=
WJ;
~llr=-sw*,su*,J*=-SW2,6’Y,
which is the functional, integro-differential equation. By noticing that W[J, J*; P], (4.2), is given by
(5.3) W2 is the solution of it.
W[J, J*; P] = Y[W, ?P*]+ J?P* + J*!P + W2, we have
(5.4)
M. ICHIYANAGI
164
It is easy to prove the relation SWzIW(X7) = (63[4
+ V, f$* + Vy*]/S~(XT)),,
(5.5)
which is derived from (5.1) by taking the functional differentiation with respect to +(x7). In (5.5), (* . *)2 means the ensemble average which is taken with respect to the action (5.1). Then, from (5.5) and (5.2) we obtain (621+ + q, 4* + ~*l/s&(xT)), = 0,
(5.6)
(MT))* f 0,
(5.7)
and
where we have put J = .I* = 0 in the sense of quasi-average. Here as a mild consolution check, let us verify that this expression has the right form such as the Gross-Pitaevskii equation, if we disregard the quantities (+(x7))* as a small correction to the order-parameter. However, it is not the outcome of (5.6), since such an equation can be found in the stationary phase approximation’). To obtain higher order corrections to that, we may employ a diagramatic method, for the functional formulation presented has a consistent perturbation scheme6). Eq. (5.6) shows that how the wave equation of I is governed by the dynamics of the system; that is to say, the macroscopic wavefunctions W(x7) is chosen weakly independent to the Bose system whose Lagrangian is 6p[&, +*I, but once the macroscopic wavefunction is introduced into the Lagrangian, its dynamics is strongly correlated to the one of the given system. It is now worth while to note that the conditions (5.2) are written in the form J(XT) + @z[&, +*]/ad’*(XT)) = 0, (5.8) J*(XT)
+ @z[+,
4*1/s&(X7))
= 0,
which resemble to the conditions (2.4). It is noted that the ensemble averages in (5.8) depend implicitly on the order-parameter ~J~(xT).Therefore we can verify the reason why we have introduced the modified ensemble which has the symmetry breaking fields in its action, (5.1). To do this, we must prove (see, Appendix) 6wz/@+?XT) = @z[+, 4*]/a&(Xr)).
(5.9)
The essential aspect of the present theory is that the phase of the condensate (+(x7)) is fixed in the action of the macroscopic order-parameter ly(x~). This ensures us from identifying the superfluid velocity with the gradient in the phase of condensate. It must now be argued whether the absolute value of !&XT) may be irrelevant or not, though the phase of ly(x~) is of essential
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importance. Let us consider a case of ferromagnetic system below its transition temperature. In that system, once an external magnetic field is switched on, the system has its own magnitude of magnetization which is unsuspectingly independent of the strength of the external field. From this we learn the possible situation which may be encountered in a theory of superfluidity. However, Schick and Zilselp give an example of an ideal Bose gas which contradicts the above statement, showing that the particle distribution function has an infinitely sharp peak without any fluctuation. This shows explicitly that the system changes abruptly if we introduce the symmetry breaking field.
6. Concluding
remarks
By employing the method of functional integration, we have treated the condensate wavefunction and the order-parameter of an interacting Bose system. This method enables us to make distinction between the condensate wavefunction (&(x7)) and the order-parameter W(XT). From the analysis we have seen how it is that the order-parameter in an explicit form fixes the condensate wavefunction. We have treated the inhomogeneous Bose system. This theory rests upon Bogoliubov’s description of the equilibrium state5), and especially on the explanation of the existence of the condensate in terms of quasi-average arising from an infinitesimal external field. We have introduced the e-term in the generating functional. We have argued that the possible symmetry breaking in the phase of the condensate can be effectuated by introducing that e-term. In other words, the e-term selects a special one from among many solutions from the condensate, when the order-parameter is specified. This result clearly, but not unexpectedly, shows us that the superfluid velocity, which is essentially related to the phase of the order-parameter, can be defined in terms of the phase of the condensate wavefunction. In case of a homogeneous Bose system at absolute zero, the condensate wavefunction is simply proportional to the order-parameter. The coefficient of the proportion is the renormalization constant of the asymptotic fields of the system. This result suggests that it may be the order-parameter introduced whose normalization is equal to the density of superfluid part. It is important to note that the e-term has another operational significance. To see this, let us note that the canonical partition function for an ideal Bose system is given by &[J, J*; E] = exp[JO,* + Jzcr] exp - 2 [esCk*-~a-‘)- I]-‘)Jk(*], c
k
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M. ICHIYANAGI
Jk =
I
dx exp(ik - x)J(x),
(6.2)
where (Y is the order-parameter.
Then, we have
P > PC(P)
vi
Zo[J,
.r*; E] =. exp{-~Id3~~,1,1:}exp(los*+I:r:) ,
((j3)
P < PC(P)
where 2, = l& exp(@)
(6.4)
and pc(P)
= m
1 I
d3k
,&_
1,
where is the critical density. From (6.3), we see that a Bose-Einstein condensation manifests itself, if the density p is higher than p&3). We have been interested in this article in the way how to derive an equation for the order-parameter. Explicit discussion was given when the orderparameter is introduced in the e-term. Eq. (5.6) stands for the equation for the order-parameter, which shows how the order-parameter, which is introduced externally, is governed by the dynamics of the system. The discussion, especially on the properties of Wz, (5.3), is meant as a possibility because the actual evaluation must be so complicated that it deserves a separate article.
Appendix A
Proof of eq. (5.9) The equation for W, is W, = -p-’
In 1 [d&][d+*] exp{Z[& + !P, c$* + V*]
64.1)
FUNCTIONAL
INTEGRALS
Now, due to a functional 0= =
I
167
BOSE SYSTEMS
integral by parts we have
&
[d+l[W*l
FOR CONDENSED
exd- * *I + !P, d* + P*]/S&(XT) - a W,/PP(xr)}
[d+][d~*]{&Y[~
exp[- . -1. (A.3
Thus, we get -=SW2 W(XT)
(cwd
(A.3)
+ q, gJ* + ly*l/84J(xT))2.
Here, in (A.2) the exponential easy to get
factor of (A.l) is suppressed
in short. It is also
SW2 = (cw[c#J+ 9, +* + P*]/s?P(xT))2 - cw[?P, v*]/iw(xT) 8w(XT)
~
-
I
dy dT’{&T’))#2W2/6~(XT)6~(yT’)
+ (~*(yT’))2s2W~/S~(XT)lj~*(YT’)}.
(A.4)
Then, by using (A.3) in (A.4), we arrive at the equation 0 = %?[@, q*]/6q(XT) +
I
dy dT’{(~(J’T’))~82w2/~~(XT)8?@T’)
+ (~*(YT’))$ WJS!P(XT)S?P*(YT’)}.
(A.3
Eq. (A.2) can be written in the form
0 = I ~d~l[d4*1~6=%~,4*]/6d’(XT) - 6w2/6p(XT)} xexp[f$+d-q,(b*++*where the abbreviation equation shows us that 6w2/6p(=)
= (=i+,
!P*],
in the exponential
(A.@ factor
+*l/&f+T)),
where the ensemble average is given by (2.9).
References
t) L. Onsager,Nuovo Cim. Suppl. 6
(1949) 249.
is self-explanatory.
This
(A.7)
168
M. ICHIYANAGI
2) 0. Penrose, Phil. Mag. 42 (1951) 1373. See also, 0. Penrose and L. Onsager, Phys. Rev. 104 (1956) 576. 3) L.P. Pitaevskii, Soviet Phys.-JETP 13 (l%l) 451. E.P. Gross, J. Math. Phys. 4 (1963) 195. 4) P.C. Hohenberg and P.C. Martin, Ann. Phys. 34 (1965) 291. 5) J.N. Bogoliubov, Physica 26 (1960) Sl. 6) A. Casher, D. Lurie and M. Revzen, J. Math. Phys. 9 (1%8) 1312. 7) H. Umezawa, Acta Phys. Hung. 19 (1%5) 9. M. Ichiyanagi and M. Ohya, J. Phys. Sot. Japan 41 (1976) 1870. 8) F.W. Wiegel, Phys. Reports 16C (1975) 57. 9) M. Schick and P.R. Zilsel, Phys. Rev. 188 (1969) 522. 10) M. Ichiyanagi, J. Phys. Sot. Japan 38 (1975) 1586. 11) P.W. Anderson, Rev. Mod. Phys. 38 (1966) 298. 12) For an ideal Bose system, see the paper by J.D. Gunton and M.J. Buckingham, 166 (1968) 152. 13) B.D. Josephson, Phys. Letters 21 (1966) 608.
Phys.
Rev.
FUNCTIONAL
INTEGRALS
Publishing Co.
FOR CONDENSED
Masakazu Department
BOSE
SYSTEMS
ICHIYANAGI
of Applied Physics, Osaka University, Suita, Osaka, 565, Japan
Received 17 February 1970
We express an exact formulation for an interacting Bose system as a functional integration over complex functions. We propose a modification of the generating functional which leads to different limiting processes. The present article rests upon Bogoliubov’s description on the explanation of the existence of the order-parameter in terms of quasiaverages arising from a symmetry breaking field. Discussions to determine the order-parameter function are also given. We indicate that the modified functional integral method is relevant to the theory of superfluidity. The derivation of Landau-Ginzburg type equation for the order-parameter is given.
1. Introduction
That the superfluidity of liquid helium should be described a macroscopic wavefunction was emphasized by Onsager’) and Penrose*). However, we do not know how to derive an equation for the macroscopic wavefunction, which is corresponding to the Landau-Ginzburg equation for a superconductivity. Though, the so-called Gross-Pitaevskii theory3) does lead to a set of equations very similar to Landau-Ginzburg theory, this theory in its simple form makes no distinction between the superfluid density and the condensate density. The latter density is close to 10% at absolute zero temperature, whereas the superfluid part is exactly 100% of the total density. It is an essential step toward the superfluid theory to identify the superfluid velocity as being equal to (h/n)VS, S being the phase of the condensate wavefunction4). This definition of the superfluid velocity is general and possibly rigorous. However, it is not so obvious to see how to break up the total density into superfluid and normal densities. In this article, we are going to discuss the microscopic background of that ansatz for the phase of the condensate wavefunction. An essential aspect of the theory presented here is the consistent treatment of the condensate and the order-parameter. The point of view adopted is based on Bogoliubov’s description of the condensate in terms of quasi-averages arising from an infinitesimal, symmetry breaking external field5). Here we try to generalize Bogoliubov’s description to include the order-parameter. This is done through 154
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155
an introduction of the e-term in the Lagrangian density, which modifies a symmetry breaking field in a rather essential way to show how to derive the wave equation for the order-parameter. The techniques of functional integral method prove themselves very useful in these investigations. We have used them in order to be able to derive the equation for the order-parameter of an interacting Bose system. Some of the arguments used in such a derivation are found to be related to those used by Casher, Lurie and Revsen6), among others. However, the present author believes some point may be new, in particular the treatment of the super-fluid order-parameter. In effect the idea we develop below amounts to assuming a kind of infinitesimal symmetrybreaking fields which are in relation with the order-parameter of a Bose system. It would be mysterious to observe that the order-parameter, which is first introduced in this theory as a parameter, behaves as if it were the dynamical variable of the system. The reader will not fail to notice a close parallel with ferromagnetism. The symmetry-breaking field is introduced to lift the degeneracy of the states of the system with respect to the phase of the Bose fields. Therefore, it would be expected to prove that the outcome in the presence of the symmetry-breaking field is not depend on the magnitude of the order-parameter. To prove this may be quite troublesome and reserves a separate article. However, we may note that the order-parameter simply manifest itself as a convention by which we describe the condensation process. In order to make the article reasonably self-contained, we have presented technical details associated with the basic concepts such as the gauge invariance and the idea of asymptotic field for a homogeneous Bose system’). The introduction of the E-term in the Lagrangian replaces the external fields J and J* by J-E@ and J*- &D*, @ and @* being the parameters to be specified as the order-parameters. E is an infinitesimal. Stand in view of the above fact, we rest upon Bogoliubov’s description of the Bose system, if we expect that the spontaneous breakdown of symmetry can be effected by introducing an external field proportional to the order-parameter. The plan of this article is as follows: The next section is devoted itself to give a preliminary discussion on the method of the functional integral. The occurrence of the E-term is discussed. In section 3, we will discuss some consequences concerning the gauge invariance of the formulation. The physical meaning of the parameter in the e-term is studied in section 4. In section 5, by generalizing the discussions in the preceding sections to an inhomogeneous Bose system, we are going to develop a theory for the orderparameter rather than the condensate wavefunction. By a single train of argument akin to the one for a homogeneous Bose system, we will show that
156
M. ICHIYANAGI
it is possible to derive the macroscopic equation that the order-parameter should obey. In section 6, some concluding remarks are given.
2. Functional-integral
method
In this section we address ourselves to develop a theory for Bose-Einstein condensation of an interacting Bose system. Our theory in this article rests upon the explanation of the existence of the condensate in terms of quasiaverage’) arising from a symmetry-breaking field. We formulate our theory in terms of the functional integration, which proves itself useful in the investigation of thermal properties of an interacting Bose system and allows the formulation for the condensate in an inhomogeneous case. Our model is given by the Lagrangian
a4,0*1=-I(
dx dr [ 4*(x7) ;
a
- +*(rr)(V’+
/A)+(xT)
(2.1) where p is the chemical potential of the Bose system of the particle mass m = f, g the coupling constant, and 7 the imaginary time, 0 < T < /3 (= l/k,T, ka being the Boltzmann constant). 4(x7) is a complex scalar field and +*(x7) is a complex conjugate of &x7). Then, following Casher et a1.6) we define the generating functional
2L.t J*l = j- [d41[W*l expM4,
(2.2)
+*I + W* + J*4L
where the symbol J [d+][d&*] denotes functions &(x7) and 4*(x7); and
an integration
over the space of the
dTJ(xT)&*(XT). It is easy to see that the stationary
(2.3 conditions
are
((653& 4*l/wJ*w))
+ J(m) = 0,
(2.4a)
+*l/wx7)))
+ J*w) = 0,
(2.4b)
((63d5
where ((Q)) denotes the ensemble average of the quantity
Q;
(((2)) = (ZM J*l)-’ 1 kMl[W*lQ ewW’e[4,4*1+ W* + J*#J).
(2.5)
FUNCTIONAL
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We may recall that solutions of the’ stationary conditions correspond to the most probable path and the main contribution to the functional integral comes from the domain near to that path. As was shown by Wiegel’), the solutions in the limit J(xT) = J*(xT) = 0 are independent of T in the zeroth order approximation. The next point to be noted is that the functional integral of a type introduced above is invariant under the change of variables: j- [d4lbW*l~(4,4*1=
1 [d4lkM*lF[4
+ @3,4*+ @*I,
(2.6)
where @ = @(xr) is an arbitrary function. The acceptance of (2.6) leaves purposely vague an interesting possibility of interpreting the apparent action of the machinery of Bogoliubov’). Therefore, in the functional-integral formulation of the Bose-Einstein condensation, there is no way to know which one of those stationary solutions should be picked out. In what follows, we will show how to select a special one from among infinitely many stationary solutions. To do that, we introduce
the generating functional
W, J*; @I = / kMl[d+*l ewWI4, &*I + J+* + J*4 - ~14 - @p13, (2.7) where the limit E + 0 as V (the volume of our system) 4 ~4is understood above expression. Then, the new stationary conditions are (all+,
4*1/84*(x7))
+ J(XT) = l(+(rT) - @(XT)),
@2[4, ‘#‘*l/~d’(Xd + J*(XT) = d$‘*(xT) - @*(XT)),
in the (2.8a) (2.8b)
where t
(Q) = WJ, J*; @I)-’ x I [WlW4*1 exp{%4+4*1+ W* + J*4 - 44 - Q/3.
(2.9)
It is obvious that the above conditions are not invariant under the change of variables as expected. Eqs. (2.8) clearly show the reason why we have introduced the e-term in the Lagrangian. To see this, let us for a moment consider an ideal Bose system, for which (2.8) read (PO+ E)@(Xr)) = -(J - e@),
(2.1Oa)
t In this article, four kinds of ensemble average are used. The reader may not fail to distinguish form among them; (( )) is defined by (2.3, ( ) by (2.9), ( )I by (4.8), and ( )2 by (5.5), respectively. It should be noted that in the limit of E + 0 ( ) may or may not be equal to (( )), since the latter has many solutions.
1.58
M. ICHIYANAGI
(po + E)((b*(xT)) = -(J”
- CD*),
(2. lob)
where p. is the chemical potential for the system. Here, in (2.10), we have assumed that the parameters @ and @* are constants, independent of x and T. Then, it is easy to see that the average values of I and +*(x7) have non-zero limits; lii
(c#J(XT)) = @.
(2.11)
It has been shown that in the ideal Bose system the nonvanishing limit of (4(x7)) cannot obtained from the mean field theory alone?. Thus, we see that the E-term lets the value of the condensate (or the order-parameter) to be ascertained, that is to say, the e-term selects a special value of the condensate from among many possible solutions of the mean-field theory, when the parameter @ is given.
3. Gauge invariance
and phase of the condensate
In this section we exploit the consequences of gauge invariance in order to see how the e-term selects a special solution from among many stationary solutions. We first note that our Lagrangian is invariant under constant phase transformations; +(X7) +
de&T),
4*(X7)
+= e-“4*(X7),
(3.1)
where 0 is a real constant phase. Then, by making use of (2.2), we see that (3.1) changes Z[J, J*] into Z[e-“J, e”J*]. Therefore, in order that Z[J, J*] is well-defined, we must have Z[e-“J, e’“J*] = Z[J, J*], implying that Z[J, J*] depends
(3.2) on J and J* only through the combination
J - J*.
A question then arises as to how the phase of the condensate is fixed by the infinitesimal external field. The Bogoliubov procedure shows that the phase of the condensate depends on the direction in the complex J-plane followed in the limiting procedure 1JI + 0. It is not unphysical to expect that Z[J, J*; CD] also depends on J and J* only through the combination J - J *. In that case, we see that under (3.1) Z[J, J*; @] -+ Z[e-‘“J, e”J*; em”@] = Z[J, J*; em”@].
Therefore,
from
(2.8), we see that the stationary
solutions
(3.3) move around
FUNCTIONAL
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according to the change of phase of &CT). To see this from a rather different point of view, we assume that @(XT)= @(const.). We now raise an interesting point which is found from the stationary conditions. From (2.8), we get (+*(x7))(=/8&*(x~)) = J*(xT)(&xT))
- (Nx7))(=/Mx7)) - J(x7)(d*(xT))
+ l{@*(#+))
- %#J*(x~)>1,
(3.4)
where L!?= 6p[4,4*]. From the point of view of the gauge invariance, the two sides of (3.4) have different characters from each other. This fact indicates that we have
J(xT)(+*(xT))- J*(xT)(‘#+d) ‘d@*@‘(d) Then, when we take a variation J*(xT), we get @(XT)) =
l
- @‘@*(XT))).
(3.5)
of the both side of (3.5) with respect
j dy d+#‘*(XT)&‘T’))@
- (+(xT)&‘T’))@*),
to
(3.6)
where (. * -) denotes the ensemble average (2.9) when J = J* = 0. Equation (3.6) is a key point to determine (+(x7)) in terms of @ and @*. We are now at the position to do that problem. We write &XT) in terms of the asymptotic field”‘); (b(XT)
=
&, * X(X7) + * . .,
4*(X7) = 2,
(3.7) ’ X*(X7)
+
* . *.
where z+ is the renormalization constant of the asymptotic field x(x7) and the dots denote the higher order combinations in the asymptotic fields. Then, by making use of (3.7) in (3.6) and after simple manipulation, we arrive at
(&XT)) = 2,
’ @,
(p + M),
(3.8)
when we take the limit of E + 0. Eq. (3.8) corresponds to the eq. (5.4) of the previous paper’@). It should be noted that (3.8) shows that the phase of the condensate (+(X7)) is fixed by the quantity @ in the e-term. If 2, < 1, the expectation value (c#J(xT)) is not equal to the parameter CD.This result is expected in (2.8); that is to say, if we have (c$(xT)) = @, the e-term does not play any role to determine the stationary solutions. An ideal Bose system is purposely an exceptional case which brings the expectation value (+(x7)) to the parameter.
M. ICHIYANAGI
160
4. Stationary phase and renormalization
constant
So far the parameter COwas not specified in any physical sense. In this section, however, we treat an example in which the parameter @ is thought of as a stationary phase’). By making use of (2.6) and (2.7) we get
Z[J, J*; ‘PI = exp{Y[@, @*I + J@* + J*@}
[d4l[W*l expW[9 + @,4* + @*I I - .3[@, @*I+ .&#I* + J*d, - ~141’). x
(4.1)
The action in this model is given by W[J, J*; @] = -p-’
In Z[J, J*; @]
= -p-‘{2[@,
CD*]+ J@* + J*@}+
W,[J, J*; @I,
(4.2)
where W’[J, J*; @I = -p-’
Z,[J, J*; @I =
In Z,[J, J*; @],
[d+][d+*]
(4.3)
exp{5!?[+ + CD,+* + @*I
- U[@, @*I + Jc$* + J*c#J - E[c#J[“}.
(4.4)
Let us introduce the effective action r[(+), (c$*); E] which is obtained from the action W[J, J*; @] by a Legendre transformation; r[G$), M*); cl= WJ, J*; @I - S-‘(J*(4) + J(+*)).
(4.5)
In order to evaluate r[(~$), (4”); e], J(m) and J*(m) must be eliminated. To do this, let us choose the parameters to satisfy the following relations12) @[-F + g]@]2]+ J(w) @*[-CL + g]@]2]+ J*(m)
= 0, = 0.
(4.6)
Then, from the definition of (+(x7)) we have @(x7)) = @ + (&(r7))1,
(4.7)
(+(YT)), = -f?SW,[J,
(4.8)
J”; @l/SJ*(m).
Here, in (4.8) J(m) and J*(m) are eliminated in favor of @ and @*. Thus, (4(x7)), is a complicated functional of CDand @*. The following notations are
FUNCTIONAL
now introduced
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161
by writing .%‘[+.+ @, c$* + @*I in the form:
.z[f#J + CD,4* + @*I = Lz[@, CD*1- J”#I - .@* + L?‘[+, d*; @, @*I + $*G-‘4
+ +I*,
(4.9)
where C8= (f*),
(4.10)
6* = (4, +*I,
--++y+V2+X,
2.2
G=
(4.11) $-•+p+V2+&
z2
i
1.
Here in (4.11) Z, and X2 are defined by 2, . (c#J)(xT)= g I dy d7’ ‘(‘$$r;;$;;)) =g
dy d7, s(]+(x7)]2+*(x’)) I
x2 * @(XT))
(+(y~‘)) (+*(y71))
w*(Ya
= g /
dy dr’ s(J’(X’)‘2’*(xT))
= g
dy dT1 ~(td’(d*d’(XT))
%+(YT’))
I
3
(4.12)
(&T’))
(,qyyTt))
s@*(YT’)>
(4.13)
Then, by making use of (4.6) and (4.4), we obtain Z,[J, J*; @I = To proceed, z2i.t
J*;
I
[d&][d&*] exp{&$*G-‘4 + Y[4, +*; CD,@*I}.
it is needed to introduce @I
=
ZdJ,
J*;
(4.14)
the object (4.15)
@l/Zd@l,
where &[@I = 1 [d+][d+*] exp{:&*G-‘4). The quantity Z,,[@] is the partition propagator G. Then, we have Z,,[cP] =
fl (1 - e&k)-‘,
function
(4.16) for free fields described
by the
(4.17)
k
where w& is the phonon frequency, which is the eigenvalue of G. Equation (4.15) shows us that only linked terms need be considered when the pertur-
162
M. ICHIYANAGI
bation scheme is derived from it directly. The appearance of Z,,[@] in (4.15) indicates that the propagator in this scheme is G. It is noted that G depends on @ and @* by its definition, and correspondingly wk may depend on @ and @*. Eq. (4.8) can be rewritten in the form ilW,[J
J”; @] a@ aw,[J, ;@ aJ*+
J*; @]
a@*
(4.18)
By making use of (4.14)-(4.17) and (4.18), we have
aW,[J, J*; @I a@*
(4.19)
where W,[J, J*; CD]= -p-’
In &[J, J*; @I.
(4.20)
In (4.20) J and J* must be eliminated in favor of 0 and @*. Then, in the first order in the interaction described by Y[&, c$*; @, @*I, from (4.7) and (4.19) we obtain the approximate expression for (I); (+(x7)) = @ + (first term of (4.19)) (4.21)
= Q/1(1+ Al(mc(@l*)], where A is a numerical factor given by A
= p
c
k
efi”k . (1 - epwk)-2,
k
and m = iis the particle mass and c the sound velocity of the system given by (4.23)
c = (gl@l’). To derive (4.21), we have used the fact that the chemical potential reads
(4.24)
CL= gl@12. The result tells us that the renormalization
constant 2, is approximately
given
by 2, = (1 + A/mcl@l*)-‘. 5. The order-parameter
What has Bose-Einstein
(4.25)
of a Bose system
condensation
treated in the previous sections to do
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163
with superfluidity? The connection between Bose-Einstein condensation and superfluidity is as follows*). We consider the superfluid from the point of view of a macroscopically occupied quantum state with its wave function of the form P(x) = exp[iS(x)]R(x), S and R being real functions. We identify the superfluid velocity as being equal to 2hVS, indicating that the superfluid circulation will be quantized in units of h/m, (m = i). In usual theories4*“), the superfluid velocity is defined in terms of the phase of the condensate wavefunction, whose normalization gives the density of condensed particles. To be on the safe side we have confidence in treating the phase of the condensate as an essential object. We must note however that no one knows the wave equation that the macroscopic wavefunction (denoted the superfluid orderparameter) should obey. One should recall that in section 2 we have seen that the phase of the condensate is fixed by the parameter in a unified way. Thus, we are interested in particular in the generalization to a nonhomogeneous Bose system having a macroscopic wavefunction below a certain transition temperature, It is a realistic application of the present theory potential in section 2 to derive an equation for the macroscopic wavefunction ?P(x~). Let us first introduce the object W[J; !P] = -p-l
In
I
[d4][d4*]
exp{Z[4 + F, 4* + !P*]
- Z[ F, ?Py*]+ A$* + J*+ - +$1’}, We now want to evaluate this at the particular
(5.1)
values of J and J*, where
J(XT) + 6 W*/@P’*(x7) = 0, J*(xT) + SW*/&P(x7) = 0.
(5.2)
In (5.2), W2 is given by w2
=
WJ;
~llr=-sw*,su*,J*=-SW2,6’Y,
which is the functional, integro-differential equation. By noticing that W[J, J*; P], (4.2), is given by
(5.3) W2 is the solution of it.
W[J, J*; P] = Y[W, ?P*]+ J?P* + J*!P + W2, we have
(5.4)
M. ICHIYANAGI
164
It is easy to prove the relation SWzIW(X7) = (63[4
+ V, f$* + Vy*]/S~(XT)),,
(5.5)
which is derived from (5.1) by taking the functional differentiation with respect to +(x7). In (5.5), (* . *)2 means the ensemble average which is taken with respect to the action (5.1). Then, from (5.5) and (5.2) we obtain (621+ + q, 4* + ~*l/s&(xT)), = 0,
(5.6)
(MT))* f 0,
(5.7)
and
where we have put J = .I* = 0 in the sense of quasi-average. Here as a mild consolution check, let us verify that this expression has the right form such as the Gross-Pitaevskii equation, if we disregard the quantities (+(x7))* as a small correction to the order-parameter. However, it is not the outcome of (5.6), since such an equation can be found in the stationary phase approximation’). To obtain higher order corrections to that, we may employ a diagramatic method, for the functional formulation presented has a consistent perturbation scheme6). Eq. (5.6) shows that how the wave equation of I is governed by the dynamics of the system; that is to say, the macroscopic wavefunctions W(x7) is chosen weakly independent to the Bose system whose Lagrangian is 6p[&, +*I, but once the macroscopic wavefunction is introduced into the Lagrangian, its dynamics is strongly correlated to the one of the given system. It is now worth while to note that the conditions (5.2) are written in the form J(XT) + @z[&, +*]/ad’*(XT)) = 0, (5.8) J*(XT)
+ @z[+,
4*1/s&(X7))
= 0,
which resemble to the conditions (2.4). It is noted that the ensemble averages in (5.8) depend implicitly on the order-parameter ~J~(xT).Therefore we can verify the reason why we have introduced the modified ensemble which has the symmetry breaking fields in its action, (5.1). To do this, we must prove (see, Appendix) 6wz/@+?XT) = @z[+, 4*]/a&(Xr)).
(5.9)
The essential aspect of the present theory is that the phase of the condensate (+(x7)) is fixed in the action of the macroscopic order-parameter ly(x~). This ensures us from identifying the superfluid velocity with the gradient in the phase of condensate. It must now be argued whether the absolute value of !&XT) may be irrelevant or not, though the phase of ly(x~) is of essential
FUNCTIONAL
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165
importance. Let us consider a case of ferromagnetic system below its transition temperature. In that system, once an external magnetic field is switched on, the system has its own magnitude of magnetization which is unsuspectingly independent of the strength of the external field. From this we learn the possible situation which may be encountered in a theory of superfluidity. However, Schick and Zilselp give an example of an ideal Bose gas which contradicts the above statement, showing that the particle distribution function has an infinitely sharp peak without any fluctuation. This shows explicitly that the system changes abruptly if we introduce the symmetry breaking field.
6. Concluding
remarks
By employing the method of functional integration, we have treated the condensate wavefunction and the order-parameter of an interacting Bose system. This method enables us to make distinction between the condensate wavefunction (&(x7)) and the order-parameter W(XT). From the analysis we have seen how it is that the order-parameter in an explicit form fixes the condensate wavefunction. We have treated the inhomogeneous Bose system. This theory rests upon Bogoliubov’s description of the equilibrium state5), and especially on the explanation of the existence of the condensate in terms of quasi-average arising from an infinitesimal external field. We have introduced the e-term in the generating functional. We have argued that the possible symmetry breaking in the phase of the condensate can be effectuated by introducing that e-term. In other words, the e-term selects a special one from among many solutions from the condensate, when the order-parameter is specified. This result clearly, but not unexpectedly, shows us that the superfluid velocity, which is essentially related to the phase of the order-parameter, can be defined in terms of the phase of the condensate wavefunction. In case of a homogeneous Bose system at absolute zero, the condensate wavefunction is simply proportional to the order-parameter. The coefficient of the proportion is the renormalization constant of the asymptotic fields of the system. This result suggests that it may be the order-parameter introduced whose normalization is equal to the density of superfluid part. It is important to note that the e-term has another operational significance. To see this, let us note that the canonical partition function for an ideal Bose system is given by &[J, J*; E] = exp[JO,* + Jzcr] exp - 2 [esCk*-~a-‘)- I]-‘)Jk(*], c
k
166
M. ICHIYANAGI
Jk =
I
dx exp(ik - x)J(x),
(6.2)
where (Y is the order-parameter.
Then, we have
P > PC(P)
vi
Zo[J,
.r*; E] =. exp{-~Id3~~,1,1:}exp(los*+I:r:) ,
((j3)
P < PC(P)
where 2, = l& exp(@)
(6.4)
and pc(P)
= m
1 I
d3k
,&_
1,
where is the critical density. From (6.3), we see that a Bose-Einstein condensation manifests itself, if the density p is higher than p&3). We have been interested in this article in the way how to derive an equation for the order-parameter. Explicit discussion was given when the orderparameter is introduced in the e-term. Eq. (5.6) stands for the equation for the order-parameter, which shows how the order-parameter, which is introduced externally, is governed by the dynamics of the system. The discussion, especially on the properties of Wz, (5.3), is meant as a possibility because the actual evaluation must be so complicated that it deserves a separate article.
Appendix A
Proof of eq. (5.9) The equation for W, is W, = -p-’
In 1 [d&][d+*] exp{Z[& + !P, c$* + V*]
64.1)
FUNCTIONAL
INTEGRALS
Now, due to a functional 0= =
I
167
BOSE SYSTEMS
integral by parts we have
&
[d+l[W*l
FOR CONDENSED
exd- * *I + !P, d* + P*]/S&(XT) - a W,/PP(xr)}
[d+][d~*]{&Y[~
exp[- . -1. (A.3
Thus, we get -=SW2 W(XT)
(cwd
(A.3)
+ q, gJ* + ly*l/84J(xT))2.
Here, in (A.2) the exponential easy to get
factor of (A.l) is suppressed
in short. It is also
SW2 = (cw[c#J+ 9, +* + P*]/s?P(xT))2 - cw[?P, v*]/iw(xT) 8w(XT)
~
-
I
dy dT’{&T’))#2W2/6~(XT)6~(yT’)
+ (~*(yT’))2s2W~/S~(XT)lj~*(YT’)}.
(A.4)
Then, by using (A.3) in (A.4), we arrive at the equation 0 = %?[@, q*]/6q(XT) +
I
dy dT’{(~(J’T’))~82w2/~~(XT)8?@T’)
+ (~*(YT’))$ WJS!P(XT)S?P*(YT’)}.
(A.3
Eq. (A.2) can be written in the form
0 = I ~d~l[d4*1~6=%~,4*]/6d’(XT) - 6w2/6p(XT)} xexp[f$+d-q,(b*++*where the abbreviation equation shows us that 6w2/6p(=)
= (=i+,
!P*],
in the exponential
(A.@ factor
+*l/&f+T)),
where the ensemble average is given by (2.9).
References
t) L. Onsager,Nuovo Cim. Suppl. 6
(1949) 249.
is self-explanatory.
This
(A.7)
168
M. ICHIYANAGI
2) 0. Penrose, Phil. Mag. 42 (1951) 1373. See also, 0. Penrose and L. Onsager, Phys. Rev. 104 (1956) 576. 3) L.P. Pitaevskii, Soviet Phys.-JETP 13 (l%l) 451. E.P. Gross, J. Math. Phys. 4 (1963) 195. 4) P.C. Hohenberg and P.C. Martin, Ann. Phys. 34 (1965) 291. 5) J.N. Bogoliubov, Physica 26 (1960) Sl. 6) A. Casher, D. Lurie and M. Revzen, J. Math. Phys. 9 (1%8) 1312. 7) H. Umezawa, Acta Phys. Hung. 19 (1%5) 9. M. Ichiyanagi and M. Ohya, J. Phys. Sot. Japan 41 (1976) 1870. 8) F.W. Wiegel, Phys. Reports 16C (1975) 57. 9) M. Schick and P.R. Zilsel, Phys. Rev. 188 (1969) 522. 10) M. Ichiyanagi, J. Phys. Sot. Japan 38 (1975) 1586. 11) P.W. Anderson, Rev. Mod. Phys. 38 (1966) 298. 12) For an ideal Bose system, see the paper by J.D. Gunton and M.J. Buckingham, 166 (1968) 152. 13) B.D. Josephson, Phys. Letters 21 (1966) 608.
Phys.
Rev.