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A microscopic theory of the lambda transition, II
ANNALS
OF PHYSICS
147. 244-266
A Microscopic
(1983)
Theory
of the Lambda
TADASHI Insrim fir Auf der Morgenstelle
Theoretische 14, 7400
II
TOYODA
Physik Xbingen
Received
Transition,
der Unicersiriii Tiibingen. 1. Federal Republic of German)
October
1. 1982
The previously proposed finite temperature field theory of the lambda transition based on the Schwinger functional method is investigated further. A systematic method for calculating the higher-order loop terms is presented by introducing the one-loop Green’s functions, which are found to be a natural finite temperature extension of the Beliaev-Hugenholtz-PinesGavoret-Nozieres zero-temperature Green’s functions. The application of the finite temperature loop expansion to the dynamical properties is presented by calculating the retarded density correlation functions at the one-loop level. The result gives a microscopic basis for the form of the dynamical structure factor recently proposed by Woods and Svensson. From a general point of view. without using any approximations or model interactions, Goldstone’s theorem for the lambda transition at finite temperature IS presented.
1.
INTRODUCTION
In the previous paper [ 1 ] (hereafter referred to as I) the present author proposed a new microscopic theory of the lambda transition based on the finite temperature Schwinger functional method together with the finite temperature loop expansion. It is the purpose of the present paper to extend the formalism to give an explicit way of calculating higher-order loop terms and also to show how to use the finite temperature loop expansion in treating various correlation functions which are closely connected to the dynamical properties of the system. In addition to these rather prac tical purposes, the present paper is also intended to clarify some basic general aspects of the formalism such as a tinite temperature extension of Goldstone’s theorem for the lambda transition. Since 1941, when Landau [2] proposed an energy spectrum for elementary excitations on the superfluid state of liquid “He, which has played an essential role in understanding properties of superfluidity, the theoretical derivation of the spectrum has been one of the central problems in the microscopic theory of the lambda transition. In 1959, Hugengoltz and Pines [3] proved rigorously that there is a gapless excitation energy spectrum in the superfluid phase of liquid 4He at zero temperature. which later proved to be a special case of Goldstone’s theorem (41. From the form of the grand potential obtained in I, the energy spectrum of an elementary excitation was identified and it was shown that the energy spectrum is gapless within the approximation. However, the treatment was based on the one-loop approximation and 244 0003.4916/83 CopyrIght All rights
$7.50
Q 1983 by Academic Press. Inc. of reproduction in any form reserved.
THEORY
OF
LAMBDA
TRANSITION.
II
245
therefore it is desirable to investigate the properties of such an excitation energy spectrum more rigorously without depending on any approximations. In the present paper we give for the first time the proof of the Hugenholtz-Pines-Goldstone theorem for the lambda transition at finite temperature based on the finite temperature Schwinger functional method. One of the essential features of our formalism is the explicit introduction of a symmetry breaking external field variable into the Hamiltonian. In 1 the order parameter of the lambda transition was defined as
Y= -(@,;.
( . . . ),, E Tr(p”, ...I.
where p^, is the density operator for the grand canonical
(1.1)
ensemble and has the form (1.2)
If the hamiltonian I? commutes with the number operator fi = ,( dx’?$, where ‘? and p are the second quantized field operators for “He atoms, then the order-parameter defined above is always zero unless the thermodynamical limit has been taken. In order to treat such a limit rigorously it may be desirable to elaborate on the algebraic approach [ 5 1. On the other hand, as has been shown in I, such a difficulty in defining the nonvanishing order-parameter can also be solved by introducing a symmetrybreaking external field into the hamiltonian together with the functional Legendre transformation that eliminates the external field variable in favor of the orderparameter. The physical meaning of this procedure can be illustrated intuitively as follows: In the real system there are always small symmetry-breaking perturbations. Near the critical point certain responses of the system become singular and some symmetry-breaking perturbations are enormously amplified to become the orderparameter and thus the phase-transition is triggered. Once such an order-parameter emerges, it will be continuously amplified by the singular responses of the system and eventually becomes a macroscopic quantity. During the process any detailed information about the symmetry-breaking perturbations that have triggered the whole process will be lost and only the fact that there was such a symmetry-breaking perturbation can remain, Of course, the order-parameter cannot continue to grow indefinitely. Once the magnitude of the order-parameter reaches its equilibrium value. i.e., the value that gives the minimum of the grand potential, growth must stop. Thus. it is necessary to obtain the grand potential as a functional of the order-parameter by eliminating the external field using the Legendre transformation, as has been shown in 1.
In addition to the Legendre transformation the finite temperature loop expansion makes the calculation of the grand potential possible, in principle, up to the desired order. However, in I only the one-loop contribution was treated and a practical method of calculating higher-order loop terms was not presented. In the present paper we give such a perturbational calculation method introducing new finite temperature
246
TADASHI TOYODA
Green’s functions, which are found to be a natural finite temperature extension of the zero-temperature Green’s functions previously introduced by Beliaev [6 ], Hugenholtz and Pines [3] and Gavoret and Nozieres [71. In the perturbational calculation of higher-order loop terms, the Legendre transformation plays an essential role in eliminating a certain class of reducible diagrams. This will be shown explicitly at the two-loop level. In I the entire treatment was restricted to the equilibrium case. However. the dynamical property of the lambda transition has recently attracted much attention ]S] and it is desirable to extend the theory to the dynamical case. This is of great interest because the conventional theory for the dynamical property of the lambda transition is essentially phenomenological 191. For example, one of the most successful theories of such a dynamical property, developed by Ferrell and his collaborators (IO], can be linked to the microscopic theory through the relation between the superfluid velocity and the phase of the order-parameter. However. as has been pointed out by Anderson [ 111, there has been no acceptable proof for such a relation. That is, despite the great successof the conventional phenomenological theory of the dynamical critical phenomena of the lambda transition, a fundamental understanding of the phenomena still remains to be found. In the present paper we show how the finite temperature loop expansion can be used to calculate the density correlation functions, which are closely related to the dynamical property of the system. The present paper is restricted to the basic formulation of the method and a further attempt to investigate the dynamical critical phenomena based on the present formalism will be the subject of another paper. Within the one-loop approximation, we give a microscopic basis for the Ansatz for the dynamical structure factor recently proposed by Woods and Svensson ] 12]. The paper is organized as follows: In Section 2 the theory is formulated in a general way and Goldstone’s theorem for the lambda transition at finite temperature is presented. In Section 3 the one-loop Green’s functions are introduced. In Section 4 the method for calculating higher-order loop contributions to the effective potential is shown by presenting a calculation of the two-loop terms. In Section 5 the application of the present formalism to the dynamical properties is shown by calculating the retarded density correlation functions. In Section 6 discussions and concluding remarks are given.
2. GENERAL
FORMALISM
AND GOLDSTONE'S
THEOREM
FOR T#O
In this section we discuss some general aspects of the formalism which are independent of model hamiltonians or approximations and give a proof of the Hugenholtz-Pines-Goldstone theorem for finite temperatures. We start with the functional F[J] defined in (3.10) of 1 as F(J] = In E-J],
(2.1)
THEORY
OF LAMBDA
where .?lJ] is the grand partition
function fi=
f?,,,
TRANSITION,
241
II
of the system with the hamiltonian + I?,,,.
(2.2)
The first term fiSYm commutes wit.. the number operator 6 and the second term is the coupling between the system and the symmetry-breaking external field
ci,,,= 1.dx(J”@ +J!P’\. In (2.1) and in the following has been made, i.e., J = J(x, functional F/J 1 defined by Green’s functions defined as
discussion, we assume that Wick’s rotation defined in I r), where r is the temperature variable defined in I. The (2. I ) is the generating functional for the temperature [ 13 ] p
GY.“‘(P
(2.3)
,..“, py: 9, T...,4,rr) = . bJ*(p,)
WfqJ)
‘.. 6J*(p,y)
Wq,)
... W%,)
(2.4)
and
where J(p) and J*(p) are the Fourier transforms and J”(x, r) with the notation
In the following
discussions
of the external field variables J(x. r)
we also use the notation (2.7)
As discussed defined as
in I, the change of variables from J and J* to ‘Y and Y*, which are
Y(p)=-
WJI dJ*( P)’
UI*(p)r-
dF[J] ~J(P; ’
(2.8)
can be made by the functional Legendre transformation introducing the new functional r( Y\.
riyl +F[Jl =j. ” {J*(P)
U’(P) + J(P) Y*(P))>
(2.9)
248
TADASHI
TOYODA
where Y(p) is the Fourier transform of !P(x, r) defined in (4.1) of I. As in I, here again E’[JJ and r( Y] are the abbreviated expressions for FIJ, J* 1 and r[ Y, ‘P* 1. respectively. Now we define the vertex functions
and
These vertex functions define the functional Taylor expansion of the effective potential r( ul]. If there is no spontaneously broken symmetry, only the G(,‘.““‘s and r(h’*N”~ are nonvanishing. On the other hand, if there is a spontaneously broken symmetry, we cannot always have such simplicity. However, in the conventional method the effective potential is usually constructed using the vertex functions calculated for the symmetric theory. Then, if the effective potential has a minimum for the nonzero value of the order-parameter, it is concluded that a spontaneously broken symmetry takes place. It should be remarked here that if we want to investigate physical properties of the system, which are directly related to the vertex functions, such as dynamical properties, we must find vertex functions for the system with a spontaneously broken symmetry. This can be done, for example, by expanding the effective potential obtained in the symmetric theory about the nor-zero value of the order-parameter, which minimizes the effective potential, and then identifying the coeflicients as the vertex functions [ 13 1. Before discussing Goldstone’s theorem let us first derive some relations that connect Green’s functions and vertex functions. Such relations are of practical importance because they give explicit prescriptions for calculating vertex functions in terms of Green’s functions, which can, in turn, be calculated by the conventional pertur bational method. Using (2.8) and (2.9) it is straightforward to show
@I Yl = J*(p) ___ WP)
and
w Yl pp*(p)
(2.12)
= J(p).
Various relations connecting the Gi’*““s and ry.““s can be obtained functional derivatives of (2.8) and (2.12) [ 141. For example, we obtain
p)(p; k) rp2)(;k,p) zyO)(p,
k;) I-j’*“(k;p)
G;‘,“(q;p) G$‘).“(;p, q) G:2-“(p, q:) G;‘.“(p; q)
by taking
1 (2.13)
THEORY
OF LAMBDA
TRANSITION,
11
249
where we have defined (2.14)
Repeating similar procedures, it is also straightforward to obtain the relations that connect the Gy-“j’s and l-y.“” s. The procedure is technically the same as that of the well-known #4-theory, which can be found in the literature ] 14 ] and therefore need not be repeated here. If the system is spatially uniform, i.e., translationally invariant, the Green’s functions and the vertex functions can be written as G:‘%v)
= J(P, 4) G,,(p)+
G:O*“(; q,p) = J(P, -4)
C,,(p),
G:‘~“‘(qd
Gz,(p).
T”“YPi J
y)(:
= 4~. 9) =&A
-4)
(2.15)
9) r,,(P).
q. p) = qp, -9) rlz(p).
and ~:‘*“‘(q.P:) = &I, -9) l-?,(P), Then (2.13) becomes
r,,(p) T,,(P) TV, ~&J)
(2.16)
As discussedin Section 1. the emergenceof the order-parameter is triggered by the symmetry-breaking external field of small magnitude. Therefore, it is interesting to see the direct relation between the variation of the symmetry-breaking external field variable J and the variation of the order-parameter y/ Such a relation can be obtained easily from (2.12) [ 141. For the uniform system we find (2.17)
where &f(p) and 6Y(p) are the variations of the external field variable J(p) and the order-parameter Y(p), respectively. If we have nonvanishing 6Y in the limit of 6J = 0, then becauseof (2.17) we must have (2.18)
250
TADASHITOYODA
The meaning of this condition becomes more transparent, if we write the Green’s functions in terms of the vertex functions using (2.16) as 1 G,,(P) G,?(P) IGo, GAPI I = ir,,(P)Tzz(P)-T,z(P)TzI(P)t
I-??(P)
-r,,t
-f:,(p)
T,,(p)
P)
(2.19) I .
That is, if there is a spontaneously broken symmetry, the above Green’s functions have a singularity at p = 0 (i.e., p = 0 and CU,,= 0). This is just Goldstone’s theorem (41 for the lambda transition at finite temperatures.
3. ONE-LOOP
GREEN'S
FUNCTIONS
In I we obtained the logarithm of the grand partition function in the form
where A is a dimensionless parameter which indicates the order of the finite temperature loop expansion and will be set to one at the end of the calculation. The functionals SIP,,, J], S.,(I,V], S,,[w] and S,,]w] are given as
- an . dx( (p; I+v)?+ cc.), !
(3.3)
(3.4) and
Smlwl = -a, f dx Iv14.
(3.5)
THEORY
In the above expressions
OF
LAMBDA
we have introduced
TRANSITION,
251
II
the notation
which will be used in the rest of this paper. The symbols used in (3.2) are same as those of I. Namely, m is the mass of a ‘He atom, ,LQis the bare chemical potential, and CI,,is the bare coupling constant. The fields (D,, and 9,:: satisfy the Euler equations
1 L-i,+$-v*+.u,I(Po-2a,/cp&,4,=J
and
h2
iT+mv2+po Following
I
fQ~-2cz,/(40/~(Do*=J*.
I, we introduce the finite temperature FlJl = ~“l%3Jl
+ @,[%I
(3.8)
loop expansion of F[Jj as
+ ~*Mol
+ W),
(3.9)
where ‘pU and c,Y;~have the functional dependent e on J and J* through the Euler equations (3.7) and (3.8). In order to find the F,,[c,J,,]‘s we expand the second term on the RHS of (3.1) in powers of A:
+Aln
fPv/*Q’v/
~~~{~S,,[~J+is,?[yll.‘xexp{~,~~jl(3.l0) !I
In the rest of this section, we shall show a systematic the above expansion. First we define
way to evaluate each term in
(3.11) in where the form ( . . . } is any functional
of I+Jand I,V*. Second, we rewrite
.S.,[ ~1, given by (3.3),
where we have defined (3.13)
252
TADASHI
TOYODA
‘The expression for S,d] w] gives the corresponding Green’s functions for the evaluation of the functional expectation value (3.11) in the well-known form of Wick’s theorem. For simplicity and clarity in the following we assume that q~,, is uniform. i.e.. independent of x, and define rWe introduce the Fourier transforms
for I,Y(X) and v*(x)
y(x) = j eip+ y(p) P
and
y*(x)
as
= I_ emin’ y”(p),
(3.15)
“P
where we have used the notation notation
defined by (2.7) and have also introduced
px=p Then, S, [ w] can also be written
(3.14)
u,ql;.
the
(3.16)
* X-coo,~.
as
where we have defined ? E(P)=~P?-~,+4a,,lCul?.
(3.18)
The inverse of the 2 x 2 matrix in (3.17) gives the one-loop Green’s functions G(p) = -
icu, - E(p) -2r*
(3.19)
with
G(P)=
i% + E(P) D(P) ’
K(p) = g$
(3.20)
and D(p) = Wf + E(p)’ - 4 151’.
(3.21)
It should be noted here that in the limit of It]’ --f 0, the Green’s function G(p) reduces to the ideal boson Green’s function, i.e. [ 15 1,
,&
G(P)=
.
-1
1w, - (h’/2m)
pz f/f”’
(3.22)
253
THEORY OF LAMBDATRANSITION.
The generating functional for the one-loop Green’s functions can be written as (3.23) where and
W(P)/
= w+(P).
MC-P)l.
(3.24)
4) G(P),
(3.25)
The one-loop Green’s functions can be obtained from Z , [MI: 6 dM*(P)
6 -Z,IM 8Wq)
= (V(P)
11-O
v*(s)).,
=&P,
= (V*(P) w*(q)), = J(P, --4) K(P).
(3.26)
= (w(p)wtq)),
K*(p).
(3.27)
1 Z,IMI
(3.28) I, +I1
11 +o -EM*
ii
6 -Z.,[Ml dM*(q)
= &P.
-9)
I,~.,,
These one-loop Green’s functions and the formula ( ... If(p) ... y/*(q) .‘. ),, = 1 . . . bMl(p)
. .. &
...
give Wick’s theorem for evaluating the functional expectation value defined in (3.11)
1’61. 4. EFFECTIVE
POTENTIAL
AT THE TWO-LOOP LEVEI
The one-loop Green’s functions and Wick’s theorem presented in Section 3 make it possible to calculate systematically the higher-order contributions to the effective potential in the finite temperature loop expansion. In order to illustrate the method for calculating the higher-order terms, in this section we show explicitly the calculation of the effective potential at the two-loop level with the Legendre transformation. The two-loop contribution to F[J] defined by (3.9) can be written using (3.10) and (3.11) as
Fzl%l= (SR?lIUI)f+wmlv12)I.
(4.1)
It is convenient to introduce the Feynman diagrams shown in Fig. 1 for the calculation of the expectation value ( . . ),, in (4.1). In the following discussion we
254
TADASHI
FIG.
1.
TOYODA
(a)
(b)
Cc)
Cd)
k)
(f)
(a) C(p).
(b) K(p).
(c) K*(p).
(d) 0~
-2a,,V,,.
(e) o* s p2a~,C”?.
assume that qO is uniform. Using the diagrams we find the following (S,,), and i(Si,), in terms of the one-loop Green’s functions,
(f)
- UII,
expressions
of
(~,*Ivl).4 = n, + 2172,
(4.2)
n, = -aoP/P’RI_K(p)K*(q).
(4.3)
with
(4.4) and
:&,[v]*)~=
~{A,+2A2}+c.c.]+2A3+4A~+Ao
(4.5)
with
A,
= 02Pvj- ?’ K(P) G(q) G(P + 91, P 4
(4.8)
A, = I4*PY I’ I’ G(p) K(q) K*(p + 9). ‘P-9
(4.9)
THEORY
OF LAMBDA
(a)
TRANSITION.
(b)
(@) FIG.
2.
II,,
(b)l7,.
Cd)
Cc)
(0 (a)
255
II
(9) (c) A,,
(d) AZ (e) A,.
(f) A,.
(g) A,.
where 0 = --2a,,p0, as defined in Fig. 1. The diagrams corresponding to n,, II?, A,, AZ, n 3 and A, are shown in Fig. 2. We shall see later in this section that the term /i,, will be exactly cancelled in the final form of the effective potential by the functional Legendre transformation. Although the frequency sums in the above expressions are straightforward to carry out [ 171, we shall not present them here but leave them to the next paper, because the results are lengthy and require further investigation. In order to obtain the effective potential r using Fz[p,,] obtained above, first we introduce the expansion
r[Y]=r,[Y]+~r,[Y]+~*r*[Y]+0(~“~
(4.10)
Yx)
(4.11)
and = Y”(X) + AY,(x)
+ A2Y2(x) + 0(/l’).
We found in I that (4.12)
v’,(x) = -%(X). We can also show that, as shown form
[T+Yx)~Yy,(x)l=-I
in the Appendix, the one-loop term Y,(x)
cipx [J;,(-P>A-P)I [ Fiii P
EF$i],
has the
(4.13)
256
TADASHI
TOYODA
where 7, (k) and &z(k) are -%Y = 4k
f,(x)
=
ji
0)
28* j G(P) + ej K(p) (9 1 P P
eik.‘J
(4.14)
(k),
and
J;,(k)= W- 0) j2ejpG(P)+ 8*i, K*(p) ( , f2(x) = !I eik”A(k).
(4.15)
At the beginning of this section we obtained the two-loop term FZ, which is a functional of qO. Now using (4.1 l), (4.12) and (4.13) we can eliminate q0 in favor of Y,, and y/, . Thus we can obtain Tz [ !P] from F,, F, , and F, using (4.12) and (4.13). The effective potential r can be expressed in terms of the F,'s using (2.9) and (3.9). We find
r[Y’l =-F,[v,,Jl
-~F,b,l
-~2F2[~,J
+W”)
+ J~dx{J*Y++Y*}.
(4.16)
The first term F,, has been treated in I and has been found to be F,[rp,,JI The last expression q-q
=S[(o,,J]=S/-Y,J]
-12S,,[Y,]
+ O(A’).
(4.17)
in (4.17) can be obtained by using (3.4) of I, i.e., = SIP, -1Y,
+ O(l’>]
= S[&]
+ /z2S,,,[ Y,] + O(l’).
(4.18)
From (4.16) and (4.17) we find
z-,/Y] =-S(-Y,J]
+jd.Y{J*Y+JY*}.
(4.19)
The second term in (4.16) can also be obtained similarly,
F,Ivol=F,[-PI
+~J‘V,(x) u’,(x)+f,(x) VY4Jdx+O(~2)~
(4.20)
where we have used (A.8) and (A.9). Then from (4.16), (4.17), and (4.20) we find
I-,[Y]
=-F,(-!P].
(4.2 1)
THEORY
OF LAMBDA
TRANSITION.
II
257
The third term in (4.16) gives simply F2l%l = F,l-YYI Substituting
+ o(n).
(4.22)
(4.22). (4.20). and (4.17) into (4.16). we obtain f[Y]=-S(-Y,J]i
/'(J*Y+JY*)ds-AF,/-Y/I
- /m,[-Y]
+ O(/i3).
(4.23)
from which we find r,l Yl = -F,l-Y]
+ S.,(Y, 1- j. {f,(x) Y,(x) +./-2(x) Y/~(x)} dx.
(4.24)
Using (4.13), (3.17), and (3.19) we can readily find that the second and third terms on the RHS of (4.24) have the same structure and can obtain
I - ( If,(x) ‘u,(x) $-f,(x)
F-(x)\ dx (4.25)
If we introduce the explicit forms ofy, andy? given by (4.14) and (4.15) into (4.25), we can immediately find that (4.25) is equivalent to A, in (4.5). Finally, from (4.24) and (4.25) we obtain fz[Y]=-Fz[-Y]+Ao(q70=:-Y).
(4.26)
Thus the two-loop contribution to the effective potential I- can be obtained by calculating F,[cp,], omitting the term ,4,. Then we replace (p,, with -Y. In order to obtain the final result we must carry out the renormalization using the conditions
1181 (4.27) and (4.28)
TADASHITOYODA
258
To do so, we will encounter integrals similar to (6.14) of I, which may be evaluated by Mellin-transformation. For the two-loop calculation in addition to the above renormalization conditions the well-known #‘-theory also suggests the renormalization of the scale of r(*‘, which is defined in Section 2, as [ 191 rgyfk, pu,A) = Z(a, pu,A) r’2’(k, pu,A).
(4.29)
where Z’j2 is the field renormalization constant, rr’ is the renormalized vertex constant, and /i is the large momentum cutoff introduced in I. At present the explicit calculation of the renormalization at the two-loop level for the present theory has not been carried out.
5. RETARDED
DENSITY
CORRELATION
FUNCTIONS
In the framework of the linear responsetheory, dynamical properties of the system in the vicinity of the equilibrium state can be described by various correlation functions [20]. In this section we show how the finite temperature loop expansion method can be used to calculate such correlation functions. Although it is possibleto formulate the present theory for general correlation functions, here we discuss the retarded density correlation function as a specific example because of its obvious significance. The retarded real-time density correlation function is defined as ]2 1] D&f,
x’t’) EE-i([n;,(xt),
ii,,(x’t’)])c
a(t - t’).
(5.1)
where the notation ( . .. )o has been defined in Section 1 and
&(xf) = !iqxt) !P,,(xt)- (a:, QG
(5.2)
and (5.3) The subscript H meansthe Heisenberg picture. In this section, instead of the retarded real-time correlation function we consider the Wick-rotated retarded correlation function, which will be called simply the retarded correlation function here, defined as 1211 c/(x,x’) = -(T,[ii(x) fi(x’)])G, (5.4) where we have used the notation x = (x, r) and z(x) is the Wick-rotated ;,(xf). The symbol T,[ ... ] means “take the r-ordered product.” In the rest of this section we assumethat the system is spatially uniform and introduce the Fourier transform of P(x, x’), Q(x,
x’)
=
eiP(X-X’) 1
P
&IQ),
(5.5)
THEORY
OF
LAMBDA
TRANSITION,
where we have used p = (p, on) defined in (2.6). Then this ‘r(p) representation qp)=L
.cc &,’ 271 J -x
259
II
has the Lehmann
A (P3 w’) icu,-co”
(5.6)
whose spectral function A also gives the Lehmann representation of the real-time retarded correlation function defined by (5.1) [21]. The ensemble expectation value for the r-ordered product of the field operators, such as (5.4) can be calculated as the functional expectation value defined by
where S., , S,, , and S,, have been defined in Section 3 and A is the dimensionless parameter indicating the order of the finite temperature loop expansion. Using (5.7) and (5.4), we can write the retarded correlation function i;/’ (.x, x’) as -2(x,-K’)=
((D*(-~)v)(x)v7*(~~‘)v)(x’))s-(l(P(~~)Iz)s(I~(”‘)I~)s.
(5.8)
Then changing the field variables for the functional integration in (5.7) from q~to i,~ defined by ]22] V=%+V
(5.9)
we can rewrite (5.8) as 2(x,x’)
= !z,(x, x’) + Q,*(x, x’).
(5.10)
where
(5.11)
In the calculation of the expectation value ( ... )s given by (5.7), the expansion of the RHS of (5.7) in powers of A gives the finite temperature loop expansion. Because only the expectation value ( ... ).? defined by (3.11) appears in the expansion, we can use the one-loop Green’s functions introduced in Section 3 to make a systematic perturbational calculation. For the lowest order in the loop expansion we have ( . . . lg . . . )s =
( *. . ly * *. ) 1 $- o(n).
(5.13)
260
TADASHI
Then, for the lowest-order forms
TOYODA
loop terms the retarded correlation
functions
have the
Y, = Yj’)
+ U(A)
(5.14)
p,, = up
+ U(A),
(5.15)
{-v:, + E(k)
E(k + P) + 4a; I v7,,1’}
and
where
QYYP)
= I,
D(k) D(k + P)
’
(5.17)
with k = (k, v,)
(5.18)
and
62;;‘(p) = -21~,12{E(P)-2ao/~,,12~ D(P) where we have assumedq0 is uniform and have used (3.19), (3.20). (3.21), (3.25), (3.26), and (3.27). Expressions (5.17) and (5.20) contain the bare parameters ,u” and a, and the classical field pO. As shown in Section 5 of I, at the lowest order in the loop expansion we can replace ,q, and a, by the renormalized physical parameters p and a, respectively. The classical field q0 can also be replaced by -Y at the lowest order, as shown in Section 4. After the renormalization and the Legendre transformation, instead of (5.17) and (5.20) we have {-vi +ER(k)ER(k + p) + 4a j Yl’} D”(k) DR(k t-p)
(5.21)
and c2P*(p; !q E
- 2 1Y12{ER(p) - 2a 1Y12} DR(p)
(5.22)
where 2
ER(p)=&p2-,u+4a
(5.23)
TRANSITION,11
261
DR(p) E co:,+ ER(p)2 - 4u2 ~!q4.
(5.24)
THEORY OF LAMBDA
and
Finally, the lowest-order loop approximation for the retarded correlation function C/(x, x’) has,the form eip(-y-~yGqyp;
C/(x,x’)=!
Y) + u;,(p;
Y)} + O(A).
(5.25)
P
The two correlation functions Qp(p; !?) and U:,(p; ul) have very different properties. The first, 9:) goes smoothly over to the correlation function of the normal bose gas in the !?+ 0 limit. This can be seenby making use of (3.22). On the other hand, the second, g:,, is proportional to 1y]‘, which may be identified as the condensate density, and vanishes in Y -+ 0 limit. Thus, the result (5.25) shows that the retarded density correlation function CZ(x, x’) can be split into the “normal part,” pip, and the “superfluid part, ” g”p,. It should be remarked that below T., the normal part also dependson the order-parameter. Although the present calculation is based on the one-loop approximation, higher-order contributions can be calculated systematically by making use of the one-loop Green’s functions and the Feynman diagrams given in Section 4. Recently Yamada [23] made an analysis of the responsefunctions of liquid ‘He II based on the two-fluid model and showed that the Ansatz on the dynamical structure factor made by Woods and Svensson [ 121 can be explained on the basis of the twofluid model. The present result supports, at least qualitatively, Yamada’s analysis from a microscopic point of view.
6. DISCUSSION AND CONCLUDING
REMARKS
In Section 2 we formulated the finite temperature Schwinger functional formalism for the grand canonical ensemblein a general way and proved the Hugenholtz-PinesGoldstone theorem for finite temperatures. To the author’s knowledge this is first time this has been done. A remark on the Hugenholtz-Pines-Goldstone theorem for finite temperatures must be given here. The fact that Green’s functions have a singularity at p = 0 does not necessarily mean that the energy spectrum of the corresponding boson is gapless.This can only be seenwhen the chemical potential of that boson is known. For elementary excitations we expect that the chemical potential is zero. However, as shown in Section 5, the one-loop Green’s function G(p), which can be identified as the Green’s function of an elementary excitation boson with a gapless energy spectrum and zero chemical potential below the lambda point, can also be identified as the temperature Green’s function of a boson with nonzero chemical potential above the lambda point. In other words the Green’s function obtained in Section 3 contains the energy spectrum of a boson which can be identified as a Landau type of elementary excitation below the lambda point and can also be identified as a normal
262
TADASHI
TOYODA
4He boson above the lambda point. This indicates that in order to identify the thermal quasi-particle energy spectrum a careful treatment of the corresponding chemical potential is essential. The one-loop Green’s functions introduced in Section 3 make it possible to calculate higher-order loop terms systematically in the form of Wick’s theorem. It is obvious that these one-loop Green’s functions are a natural finite temperature extension of the zero-temperature Green’s functions introduced by Beliaev [6 ], Hugenholtz and Pines [3], and Gavoret and Nozieres [7]. One of the essential features of our one-loop Green’s functions is that they contain the order-parameter of the lambda transition in a consistent way. As we discuss later in this section, this point is crucial for the explanation of the result of Woods and Svensson for the dynamical structure factor. As shown in Section 5, these one-loop Green’s functions can be used to calculate the density correlation functions within the finite temperature loop expansion scheme. Since the density correlation functions are directly related to the real-time Green’s functions, it is also possible to treat dynamic critical phenomena within the framework of the present formalism. The general algorithm for calculating the density correlation functions up to the L-loop level is as follows: First, calculate the effective potential up to L-loop terms using the one-loop Green’s functions defined in Section 3, which contain the bare parameters ruOand a, and also the “classical field” (Do. Then making the Legendre transformation and renormalizing, we obtain the grand potential with the renormalized parameters and the orderparameter at the L-loop level. From the grand potential we calculate the equilibrium value of the order parameter and also the chemical potential as a function of the other relevant variables, as shown in I at the one-loop level. Then substituting this chemical potential expression and the equilibrium value of the order-parameter into the one-loop Green’s functions, we can construct the perturbation calculation for the density correlation functions up to the L-loop level. It should be noted that the entire calculation is controlled by the finite temperature loop expansion. In Section 5 we showed this procedure at the one-loop level. Also in Section 5, analyzing the response functions, we gave a microscopic basis for the Ansatz for the dynamical structure factor recently proposed by Woods and Svensson [ 121. Recently Griffin and Talbot [24] also attempted to explain the Woods-Svensson result for the dynamical structure factor, especially the normal part, on the basis of the Bogoliubov approximation for the single-particle spectral densities and by taking the energy spectrum of the excitations as known. Therefore, their approach is not really microscopic but has the advantage of being much closer to the experiments in several aspects. Although our one-loop approximation is very similar there are two essential differences. Our to the Bogoliubov approximation, microscopically derived energy spectrum contains the order-parameter of the lambda transition in a consistent manner, and also the finite temperature loop expansion makes it possible to calculate higher-order corrections systematically, while such a calculation is not possible in the Bogoliubov approximation. The fact that our oneloop Green’s functions contain the order-parameter in a consistent way makes it possible to explain the smooth transition of the normal part of the dynamical
THEORY
OF LAMBDA
TRANSITION,
263
11
structure factor from below the lambda point to above the lambda point in the Woods-Svensson result, which has not been explained in the Griffm-Talbot theory. Since we have taken a purely microscopic approach, at present our result is rather qualitative and it is desirable to treat higher-order loop terms as well as to extend the formalism to more realistic potentials. In concluding this paper we may state that the present formalism, which can be seen as a natural finite temperature extension of the previous zero-temperature theories, makes it possible to treat the lambda transition in a unified microscopic theory, and it can also be used to treat dynamic critical phenomena of the system within the framework. At present, the results obtained are rather qualitative but it has at least clarified some fundamental aspects of the lambda transition, which have not been explained by any previous theories.
APPENDIX
The one-loop contribution to the order-parameter, obtained from (3.9) and (2.8), which give
Y,(x)
and Y:(x),
can be
where we have used the fact that F,[c+o,j has no explicit J dependence. A similar expression for Y*(x) can also be obtained and we can write them as CA.2)
with (A.31 and
(A.4 1
264
TADASHI
TOYODA
From (3.9) and (3.10) we have
where S, [ I,V] actually contains (pO, i.e.,
&
S,[wl = -4%P,*
w*(x)
w@) - 2o,rp,V*(x)
v*(-u)*
(A.61
v(x) - 2u, cp,*w(x) V(-~).
(A.7)
and m
6
Thus we immediately f,(x)
s, [WI = -4% (Dov*(x) get = -4aoG(w*(x)
v(X)),~ - 2aovo(v*(x)
w*(x))..,
b4.8)
v(x)).,
d-u))..,.
(A.9)
and f,(x) For the following I’, (4 and f2M
= -4a,v,(v*tx)
discussion,
- 2a,cp,*(v@)
it is convenient to introduce the Fourier transforms
f,(x) = ji e’P”.7,?;,(p),
of
(A. 10)
where
and f>:,(k) = W, 0) In the above expressions
i
281 G(P) + d* 1 K*(P)). P
(A.12)
P
we have used %= -2a,fp,.
The functional derivatives of p. and (o$ with regard to J and J* can be obtained from the Euler equations (3.7) and (3.8), from which we can readily find
THEORY
OF LAMBDA
TRANSITION,
265
II
The 2 x 2 matrix operator on the LHS has already appeared in (3.12) and its inverse operator gives the one-loop Green’s functions as shown in (3.19). Therefore we have
g,,(x,x’) g,,(x,x’) g*,(-x3 x’) Introducing
(A.14)
g&2-~‘)
(A.14) and (A.lO)
into
(A.2),
we obtain
G(P) K*(P) 1W:(x).Y,(x)]= - I_e-i’JxI.&PXJ,(-P)l [K(p) G(-P)1’
(A.15)
“P
which gives the one-loop contribution
to the order-parameter
defined in (4.11).
ACKNOWLEDGMENTS I would like to thank Professors N. M. Hugenholtz (Groningen), Y. Kuroda (Nagoya), Y. Nagaoka (Kyoto). S. Nakajima (Tokyo), T. Nishiyama (Osaka), T. Tsuneto (Kyoto), T. Usui (Nagoya). and K. Yamada (Nagoya) for useful discussions and comments. I would also like to thank Professor K. Wildermuth for his warm hospitality at the Institut fiir Theoretische Physik der Universitiit Tiibingen. and Dr. L. Rikus for carefully reading the manuscript.
REFERENCES I. 2. 3. 4. 5. 6. I. 8. 9. 10.
I I. 12. 13. 14. IS. 16. 17.
T. TOYoDA, Ann. Ph.~s. (N.Y.) 141 (1982). 154; Phys. Lelf. 87A (1981). 91. L. D. LANDAU, J. Phys. (U.S.S.R.) 5 (1941). 71; 11 (1947). 91. N. M. HUC~ENHOLTZ AND D. PINES. Php. Rev. 116 (1959), 489. J. GOLDSTONE, NUOUO Cimenlo 19 (1961), 154. H. ARAKI AND E. J. WOOIJS, J. Math. Phlx 4 (1963), 637; N. M. HUGENHOLTZ. i?f “Advances in Solid State Physics” (0. Madelung, Ed.), pp. 641-646, Pergamon Vieweg. 1972. S. T. BELIAEV, Soviet Phys. JETP 7( 1958). 289. J. GAVOR~~T AND P. NOZI&ES. Ann. PhJjs. (N.Y.) 28 (1964). 349. “Dynamical Critical Phenomena and Related Topics.” See, for example, C. P. ENZ (Ed.), Proceedings, Geneva, 1979, Lecture Notes in Physics No. 104. Springer-Verlag, Berlin. 1979. See, for example, P. C. HOGENBEKG AND B. I. HALPERIN. Rev. Mod. Php. 49 (1977), 435. R. A. FEIIRELL. N. MENYHARD. H. SCHMIDT, F. SCHWABL. AND P. SZBPFALUSY. Ann. Phys. (N.Y.) 47 (1968). 565; R. A. FERRELL AND J. K. BHATTACHARJEE. Phys. Rec. B 24 (198 I). 507 1: J. K. BHATTACHAFUUEE AND R. A. FERRELL. Phys. Ret!. B 25 (1982). 216. P. W. ANDERSON, Rep. Mod. Phys. 38 (1966). 298. A. D. B. WOODS AND E. C. SVENSSON. Phw. Rec. Left. 41 (1978), 974. See. for example. D. A~r-r. “Field Theory, the Renormalization Group. and Critical Phenomena,” McGraw-Hill, New York, 1978. G. JONA-LASINIO. Nuouo Cimento 34 (1964). 1790. See. for example, A. ISIHARA. “Statistical Physics.” Chap. 15. Academic Press, New York, 1971. See, for example. E. BRBZIN. et al., in “Phase Transitions and Critical Phenomena” (C. Domb and M. S. Green. Eds.). Vol. 6. pp. 125-247, Academic Press. New York. 1976. See, for example, A. L. FEWER AND J. D. WALECKA, “Quantum Theory of Many-Particle Systems” Chap. 7. McGraw-Hill. New York, 1971.
266
TADASHI
TOYODA
18. Equations (4.27) and (4.28) correct Eqs. (5.7). (5.8). and (5.9) in I. Also, the first equation of (5.5) in I should read -a, + a: f::“:@,,) + O(ai) = --a. 19. See, for example, J. ILIOPOULOS et al., Rev. Mod. Phw. 47 (1975). 165. 20. R. KUBO, J. Phvs. Sot. Japan 11 (1957); see also Chap. 13 of Ref. (15 1. 21. See, for example, Chap. 9 of Ref. [ 171. 22. Section 3 of I. 23. K. YAMADA. Progr. Theoret. Phys. 63 (1980), 715. 24. A. GRIFFIN AND E. TALBOT, Phys. Rev. B 24 (1981). 5075: A. GRIFFIN. Phvs. Rec. B 19 (1979), 5946.
OF PHYSICS
147. 244-266
A Microscopic
(1983)
Theory
of the Lambda
TADASHI Insrim fir Auf der Morgenstelle
Theoretische 14, 7400
II
TOYODA
Physik Xbingen
Received
Transition,
der Unicersiriii Tiibingen. 1. Federal Republic of German)
October
1. 1982
The previously proposed finite temperature field theory of the lambda transition based on the Schwinger functional method is investigated further. A systematic method for calculating the higher-order loop terms is presented by introducing the one-loop Green’s functions, which are found to be a natural finite temperature extension of the Beliaev-Hugenholtz-PinesGavoret-Nozieres zero-temperature Green’s functions. The application of the finite temperature loop expansion to the dynamical properties is presented by calculating the retarded density correlation functions at the one-loop level. The result gives a microscopic basis for the form of the dynamical structure factor recently proposed by Woods and Svensson. From a general point of view. without using any approximations or model interactions, Goldstone’s theorem for the lambda transition at finite temperature IS presented.
1.
INTRODUCTION
In the previous paper [ 1 ] (hereafter referred to as I) the present author proposed a new microscopic theory of the lambda transition based on the finite temperature Schwinger functional method together with the finite temperature loop expansion. It is the purpose of the present paper to extend the formalism to give an explicit way of calculating higher-order loop terms and also to show how to use the finite temperature loop expansion in treating various correlation functions which are closely connected to the dynamical properties of the system. In addition to these rather prac tical purposes, the present paper is also intended to clarify some basic general aspects of the formalism such as a tinite temperature extension of Goldstone’s theorem for the lambda transition. Since 1941, when Landau [2] proposed an energy spectrum for elementary excitations on the superfluid state of liquid “He, which has played an essential role in understanding properties of superfluidity, the theoretical derivation of the spectrum has been one of the central problems in the microscopic theory of the lambda transition. In 1959, Hugengoltz and Pines [3] proved rigorously that there is a gapless excitation energy spectrum in the superfluid phase of liquid 4He at zero temperature. which later proved to be a special case of Goldstone’s theorem (41. From the form of the grand potential obtained in I, the energy spectrum of an elementary excitation was identified and it was shown that the energy spectrum is gapless within the approximation. However, the treatment was based on the one-loop approximation and 244 0003.4916/83 CopyrIght All rights
$7.50
Q 1983 by Academic Press. Inc. of reproduction in any form reserved.
THEORY
OF
LAMBDA
TRANSITION.
II
245
therefore it is desirable to investigate the properties of such an excitation energy spectrum more rigorously without depending on any approximations. In the present paper we give for the first time the proof of the Hugenholtz-Pines-Goldstone theorem for the lambda transition at finite temperature based on the finite temperature Schwinger functional method. One of the essential features of our formalism is the explicit introduction of a symmetry breaking external field variable into the Hamiltonian. In 1 the order parameter of the lambda transition was defined as
Y= -(@,;.
( . . . ),, E Tr(p”, ...I.
where p^, is the density operator for the grand canonical
(1.1)
ensemble and has the form (1.2)
If the hamiltonian I? commutes with the number operator fi = ,( dx’?$, where ‘? and p are the second quantized field operators for “He atoms, then the order-parameter defined above is always zero unless the thermodynamical limit has been taken. In order to treat such a limit rigorously it may be desirable to elaborate on the algebraic approach [ 5 1. On the other hand, as has been shown in I, such a difficulty in defining the nonvanishing order-parameter can also be solved by introducing a symmetrybreaking external field into the hamiltonian together with the functional Legendre transformation that eliminates the external field variable in favor of the orderparameter. The physical meaning of this procedure can be illustrated intuitively as follows: In the real system there are always small symmetry-breaking perturbations. Near the critical point certain responses of the system become singular and some symmetry-breaking perturbations are enormously amplified to become the orderparameter and thus the phase-transition is triggered. Once such an order-parameter emerges, it will be continuously amplified by the singular responses of the system and eventually becomes a macroscopic quantity. During the process any detailed information about the symmetry-breaking perturbations that have triggered the whole process will be lost and only the fact that there was such a symmetry-breaking perturbation can remain, Of course, the order-parameter cannot continue to grow indefinitely. Once the magnitude of the order-parameter reaches its equilibrium value. i.e., the value that gives the minimum of the grand potential, growth must stop. Thus. it is necessary to obtain the grand potential as a functional of the order-parameter by eliminating the external field using the Legendre transformation, as has been shown in 1.
In addition to the Legendre transformation the finite temperature loop expansion makes the calculation of the grand potential possible, in principle, up to the desired order. However, in I only the one-loop contribution was treated and a practical method of calculating higher-order loop terms was not presented. In the present paper we give such a perturbational calculation method introducing new finite temperature
246
TADASHI TOYODA
Green’s functions, which are found to be a natural finite temperature extension of the zero-temperature Green’s functions previously introduced by Beliaev [6 ], Hugenholtz and Pines [3] and Gavoret and Nozieres [71. In the perturbational calculation of higher-order loop terms, the Legendre transformation plays an essential role in eliminating a certain class of reducible diagrams. This will be shown explicitly at the two-loop level. In I the entire treatment was restricted to the equilibrium case. However. the dynamical property of the lambda transition has recently attracted much attention ]S] and it is desirable to extend the theory to the dynamical case. This is of great interest because the conventional theory for the dynamical property of the lambda transition is essentially phenomenological 191. For example, one of the most successful theories of such a dynamical property, developed by Ferrell and his collaborators (IO], can be linked to the microscopic theory through the relation between the superfluid velocity and the phase of the order-parameter. However. as has been pointed out by Anderson [ 111, there has been no acceptable proof for such a relation. That is, despite the great successof the conventional phenomenological theory of the dynamical critical phenomena of the lambda transition, a fundamental understanding of the phenomena still remains to be found. In the present paper we show how the finite temperature loop expansion can be used to calculate the density correlation functions, which are closely related to the dynamical property of the system. The present paper is restricted to the basic formulation of the method and a further attempt to investigate the dynamical critical phenomena based on the present formalism will be the subject of another paper. Within the one-loop approximation, we give a microscopic basis for the Ansatz for the dynamical structure factor recently proposed by Woods and Svensson ] 12]. The paper is organized as follows: In Section 2 the theory is formulated in a general way and Goldstone’s theorem for the lambda transition at finite temperature is presented. In Section 3 the one-loop Green’s functions are introduced. In Section 4 the method for calculating higher-order loop contributions to the effective potential is shown by presenting a calculation of the two-loop terms. In Section 5 the application of the present formalism to the dynamical properties is shown by calculating the retarded density correlation functions. In Section 6 discussions and concluding remarks are given.
2. GENERAL
FORMALISM
AND GOLDSTONE'S
THEOREM
FOR T#O
In this section we discuss some general aspects of the formalism which are independent of model hamiltonians or approximations and give a proof of the Hugenholtz-Pines-Goldstone theorem for finite temperatures. We start with the functional F[J] defined in (3.10) of 1 as F(J] = In E-J],
(2.1)
THEORY
OF LAMBDA
where .?lJ] is the grand partition
function fi=
f?,,,
TRANSITION,
241
II
of the system with the hamiltonian + I?,,,.
(2.2)
The first term fiSYm commutes wit.. the number operator 6 and the second term is the coupling between the system and the symmetry-breaking external field
ci,,,= 1.dx(J”@ +J!P’\. In (2.1) and in the following has been made, i.e., J = J(x, functional F/J 1 defined by Green’s functions defined as
discussion, we assume that Wick’s rotation defined in I r), where r is the temperature variable defined in I. The (2. I ) is the generating functional for the temperature [ 13 ] p
GY.“‘(P
(2.3)
,..“, py: 9, T...,4,rr) = . bJ*(p,)
WfqJ)
‘.. 6J*(p,y)
Wq,)
... W%,)
(2.4)
and
where J(p) and J*(p) are the Fourier transforms and J”(x, r) with the notation
In the following
discussions
of the external field variables J(x. r)
we also use the notation (2.7)
As discussed defined as
in I, the change of variables from J and J* to ‘Y and Y*, which are
Y(p)=-
WJI dJ*( P)’
UI*(p)r-
dF[J] ~J(P; ’
(2.8)
can be made by the functional Legendre transformation introducing the new functional r( Y\.
riyl +F[Jl =j. ” {J*(P)
U’(P) + J(P) Y*(P))>
(2.9)
248
TADASHI
TOYODA
where Y(p) is the Fourier transform of !P(x, r) defined in (4.1) of I. As in I, here again E’[JJ and r( Y] are the abbreviated expressions for FIJ, J* 1 and r[ Y, ‘P* 1. respectively. Now we define the vertex functions
and
These vertex functions define the functional Taylor expansion of the effective potential r( ul]. If there is no spontaneously broken symmetry, only the G(,‘.““‘s and r(h’*N”~ are nonvanishing. On the other hand, if there is a spontaneously broken symmetry, we cannot always have such simplicity. However, in the conventional method the effective potential is usually constructed using the vertex functions calculated for the symmetric theory. Then, if the effective potential has a minimum for the nonzero value of the order-parameter, it is concluded that a spontaneously broken symmetry takes place. It should be remarked here that if we want to investigate physical properties of the system, which are directly related to the vertex functions, such as dynamical properties, we must find vertex functions for the system with a spontaneously broken symmetry. This can be done, for example, by expanding the effective potential obtained in the symmetric theory about the nor-zero value of the order-parameter, which minimizes the effective potential, and then identifying the coeflicients as the vertex functions [ 13 1. Before discussing Goldstone’s theorem let us first derive some relations that connect Green’s functions and vertex functions. Such relations are of practical importance because they give explicit prescriptions for calculating vertex functions in terms of Green’s functions, which can, in turn, be calculated by the conventional pertur bational method. Using (2.8) and (2.9) it is straightforward to show
@I Yl = J*(p) ___ WP)
and
w Yl pp*(p)
(2.12)
= J(p).
Various relations connecting the Gi’*““s and ry.““s can be obtained functional derivatives of (2.8) and (2.12) [ 141. For example, we obtain
p)(p; k) rp2)(;k,p) zyO)(p,
k;) I-j’*“(k;p)
G;‘,“(q;p) G$‘).“(;p, q) G:2-“(p, q:) G;‘.“(p; q)
by taking
1 (2.13)
THEORY
OF LAMBDA
TRANSITION,
11
249
where we have defined (2.14)
Repeating similar procedures, it is also straightforward to obtain the relations that connect the Gy-“j’s and l-y.“” s. The procedure is technically the same as that of the well-known #4-theory, which can be found in the literature ] 14 ] and therefore need not be repeated here. If the system is spatially uniform, i.e., translationally invariant, the Green’s functions and the vertex functions can be written as G:‘%v)
= J(P, 4) G,,(p)+
G:O*“(; q,p) = J(P, -4)
C,,(p),
G:‘~“‘(qd
Gz,(p).
T”“YPi J
y)(:
= 4~. 9) =&A
-4)
(2.15)
9) r,,(P).
q. p) = qp, -9) rlz(p).
and ~:‘*“‘(q.P:) = &I, -9) l-?,(P), Then (2.13) becomes
r,,(p) T,,(P) TV, ~&J)
(2.16)
As discussedin Section 1. the emergenceof the order-parameter is triggered by the symmetry-breaking external field of small magnitude. Therefore, it is interesting to see the direct relation between the variation of the symmetry-breaking external field variable J and the variation of the order-parameter y/ Such a relation can be obtained easily from (2.12) [ 141. For the uniform system we find (2.17)
where &f(p) and 6Y(p) are the variations of the external field variable J(p) and the order-parameter Y(p), respectively. If we have nonvanishing 6Y in the limit of 6J = 0, then becauseof (2.17) we must have (2.18)
250
TADASHITOYODA
The meaning of this condition becomes more transparent, if we write the Green’s functions in terms of the vertex functions using (2.16) as 1 G,,(P) G,?(P) IGo, GAPI I = ir,,(P)Tzz(P)-T,z(P)TzI(P)t
I-??(P)
-r,,t
-f:,(p)
T,,(p)
P)
(2.19) I .
That is, if there is a spontaneously broken symmetry, the above Green’s functions have a singularity at p = 0 (i.e., p = 0 and CU,,= 0). This is just Goldstone’s theorem (41 for the lambda transition at finite temperatures.
3. ONE-LOOP
GREEN'S
FUNCTIONS
In I we obtained the logarithm of the grand partition function in the form
where A is a dimensionless parameter which indicates the order of the finite temperature loop expansion and will be set to one at the end of the calculation. The functionals SIP,,, J], S.,(I,V], S,,[w] and S,,]w] are given as
- an . dx( (p; I+v)?+ cc.), !
(3.3)
(3.4) and
Smlwl = -a, f dx Iv14.
(3.5)
THEORY
In the above expressions
OF
LAMBDA
we have introduced
TRANSITION,
251
II
the notation
which will be used in the rest of this paper. The symbols used in (3.2) are same as those of I. Namely, m is the mass of a ‘He atom, ,LQis the bare chemical potential, and CI,,is the bare coupling constant. The fields (D,, and 9,:: satisfy the Euler equations
1 L-i,+$-v*+.u,I(Po-2a,/cp&,4,=J
and
h2
iT+mv2+po Following
I
fQ~-2cz,/(40/~(Do*=J*.
I, we introduce the finite temperature FlJl = ~“l%3Jl
+ @,[%I
(3.8)
loop expansion of F[Jj as
+ ~*Mol
+ W),
(3.9)
where ‘pU and c,Y;~have the functional dependent e on J and J* through the Euler equations (3.7) and (3.8). In order to find the F,,[c,J,,]‘s we expand the second term on the RHS of (3.1) in powers of A:
+Aln
fPv/*Q’v/
~~~{~S,,[~J+is,?[yll.‘xexp{~,~~jl(3.l0) !I
In the rest of this section, we shall show a systematic the above expansion. First we define
way to evaluate each term in
(3.11) in where the form ( . . . } is any functional
of I+Jand I,V*. Second, we rewrite
.S.,[ ~1, given by (3.3),
where we have defined (3.13)
252
TADASHI
TOYODA
‘The expression for S,d] w] gives the corresponding Green’s functions for the evaluation of the functional expectation value (3.11) in the well-known form of Wick’s theorem. For simplicity and clarity in the following we assume that q~,, is uniform. i.e.. independent of x, and define rWe introduce the Fourier transforms
for I,Y(X) and v*(x)
y(x) = j eip+ y(p) P
and
y*(x)
as
= I_ emin’ y”(p),
(3.15)
“P
where we have used the notation notation
defined by (2.7) and have also introduced
px=p Then, S, [ w] can also be written
(3.14)
u,ql;.
the
(3.16)
* X-coo,~.
as
where we have defined ? E(P)=~P?-~,+4a,,lCul?.
(3.18)
The inverse of the 2 x 2 matrix in (3.17) gives the one-loop Green’s functions G(p) = -
icu, - E(p) -2r*
(3.19)
with
G(P)=
i% + E(P) D(P) ’
K(p) = g$
(3.20)
and D(p) = Wf + E(p)’ - 4 151’.
(3.21)
It should be noted here that in the limit of It]’ --f 0, the Green’s function G(p) reduces to the ideal boson Green’s function, i.e. [ 15 1,
,&
G(P)=
.
-1
1w, - (h’/2m)
pz f/f”’
(3.22)
253
THEORY OF LAMBDATRANSITION.
The generating functional for the one-loop Green’s functions can be written as (3.23) where and
W(P)/
= w+(P).
MC-P)l.
(3.24)
4) G(P),
(3.25)
The one-loop Green’s functions can be obtained from Z , [MI: 6 dM*(P)
6 -Z,IM 8Wq)
= (V(P)
11-O
v*(s)).,
=&P,
= (V*(P) w*(q)), = J(P, --4) K(P).
(3.26)
= (w(p)wtq)),
K*(p).
(3.27)
1 Z,IMI
(3.28) I, +I1
11 +o -EM*
ii
6 -Z.,[Ml dM*(q)
= &P.
-9)
I,~.,,
These one-loop Green’s functions and the formula ( ... If(p) ... y/*(q) .‘. ),, = 1 . . . bMl(p)
. .. &
...
give Wick’s theorem for evaluating the functional expectation value defined in (3.11)
1’61. 4. EFFECTIVE
POTENTIAL
AT THE TWO-LOOP LEVEI
The one-loop Green’s functions and Wick’s theorem presented in Section 3 make it possible to calculate systematically the higher-order contributions to the effective potential in the finite temperature loop expansion. In order to illustrate the method for calculating the higher-order terms, in this section we show explicitly the calculation of the effective potential at the two-loop level with the Legendre transformation. The two-loop contribution to F[J] defined by (3.9) can be written using (3.10) and (3.11) as
Fzl%l= (SR?lIUI)f+wmlv12)I.
(4.1)
It is convenient to introduce the Feynman diagrams shown in Fig. 1 for the calculation of the expectation value ( . . ),, in (4.1). In the following discussion we
254
TADASHI
FIG.
1.
TOYODA
(a)
(b)
Cc)
Cd)
k)
(f)
(a) C(p).
(b) K(p).
(c) K*(p).
(d) 0~
-2a,,V,,.
(e) o* s p2a~,C”?.
assume that qO is uniform. Using the diagrams we find the following (S,,), and i(Si,), in terms of the one-loop Green’s functions,
(f)
- UII,
expressions
of
(~,*Ivl).4 = n, + 2172,
(4.2)
n, = -aoP/P’RI_K(p)K*(q).
(4.3)
with
(4.4) and
:&,[v]*)~=
~{A,+2A2}+c.c.]+2A3+4A~+Ao
(4.5)
with
A,
= 02Pvj- ?’ K(P) G(q) G(P + 91, P 4
(4.8)
A, = I4*PY I’ I’ G(p) K(q) K*(p + 9). ‘P-9
(4.9)
THEORY
OF LAMBDA
(a)
TRANSITION.
(b)
(@) FIG.
2.
II,,
(b)l7,.
Cd)
Cc)
(0 (a)
255
II
(9) (c) A,,
(d) AZ (e) A,.
(f) A,.
(g) A,.
where 0 = --2a,,p0, as defined in Fig. 1. The diagrams corresponding to n,, II?, A,, AZ, n 3 and A, are shown in Fig. 2. We shall see later in this section that the term /i,, will be exactly cancelled in the final form of the effective potential by the functional Legendre transformation. Although the frequency sums in the above expressions are straightforward to carry out [ 171, we shall not present them here but leave them to the next paper, because the results are lengthy and require further investigation. In order to obtain the effective potential r using Fz[p,,] obtained above, first we introduce the expansion
r[Y]=r,[Y]+~r,[Y]+~*r*[Y]+0(~“~
(4.10)
Yx)
(4.11)
and = Y”(X) + AY,(x)
+ A2Y2(x) + 0(/l’).
We found in I that (4.12)
v’,(x) = -%(X). We can also show that, as shown form
[T+Yx)~Yy,(x)l=-I
in the Appendix, the one-loop term Y,(x)
cipx [J;,(-P>A-P)I [ Fiii P
EF$i],
has the
(4.13)
256
TADASHI
TOYODA
where 7, (k) and &z(k) are -%Y = 4k
f,(x)
=
ji
0)
28* j G(P) + ej K(p) (9 1 P P
eik.‘J
(4.14)
(k),
and
J;,(k)= W- 0) j2ejpG(P)+ 8*i, K*(p) ( , f2(x) = !I eik”A(k).
(4.15)
At the beginning of this section we obtained the two-loop term FZ, which is a functional of qO. Now using (4.1 l), (4.12) and (4.13) we can eliminate q0 in favor of Y,, and y/, . Thus we can obtain Tz [ !P] from F,, F, , and F, using (4.12) and (4.13). The effective potential r can be expressed in terms of the F,'s using (2.9) and (3.9). We find
r[Y’l =-F,[v,,Jl
-~F,b,l
-~2F2[~,J
+W”)
+ J~dx{J*Y++Y*}.
(4.16)
The first term F,, has been treated in I and has been found to be F,[rp,,JI The last expression q-q
=S[(o,,J]=S/-Y,J]
-12S,,[Y,]
+ O(A’).
(4.17)
in (4.17) can be obtained by using (3.4) of I, i.e., = SIP, -1Y,
+ O(l’>]
= S[&]
+ /z2S,,,[ Y,] + O(l’).
(4.18)
From (4.16) and (4.17) we find
z-,/Y] =-S(-Y,J]
+jd.Y{J*Y+JY*}.
(4.19)
The second term in (4.16) can also be obtained similarly,
F,Ivol=F,[-PI
+~J‘V,(x) u’,(x)+f,(x) VY4Jdx+O(~2)~
(4.20)
where we have used (A.8) and (A.9). Then from (4.16), (4.17), and (4.20) we find
I-,[Y]
=-F,(-!P].
(4.2 1)
THEORY
OF LAMBDA
TRANSITION.
II
257
The third term in (4.16) gives simply F2l%l = F,l-YYI Substituting
+ o(n).
(4.22)
(4.22). (4.20). and (4.17) into (4.16). we obtain f[Y]=-S(-Y,J]i
/'(J*Y+JY*)ds-AF,/-Y/I
- /m,[-Y]
+ O(/i3).
(4.23)
from which we find r,l Yl = -F,l-Y]
+ S.,(Y, 1- j. {f,(x) Y,(x) +./-2(x) Y/~(x)} dx.
(4.24)
Using (4.13), (3.17), and (3.19) we can readily find that the second and third terms on the RHS of (4.24) have the same structure and can obtain
I - ( If,(x) ‘u,(x) $-f,(x)
F-(x)\ dx (4.25)
If we introduce the explicit forms ofy, andy? given by (4.14) and (4.15) into (4.25), we can immediately find that (4.25) is equivalent to A, in (4.5). Finally, from (4.24) and (4.25) we obtain fz[Y]=-Fz[-Y]+Ao(q70=:-Y).
(4.26)
Thus the two-loop contribution to the effective potential I- can be obtained by calculating F,[cp,], omitting the term ,4,. Then we replace (p,, with -Y. In order to obtain the final result we must carry out the renormalization using the conditions
1181 (4.27) and (4.28)
TADASHITOYODA
258
To do so, we will encounter integrals similar to (6.14) of I, which may be evaluated by Mellin-transformation. For the two-loop calculation in addition to the above renormalization conditions the well-known #‘-theory also suggests the renormalization of the scale of r(*‘, which is defined in Section 2, as [ 191 rgyfk, pu,A) = Z(a, pu,A) r’2’(k, pu,A).
(4.29)
where Z’j2 is the field renormalization constant, rr’ is the renormalized vertex constant, and /i is the large momentum cutoff introduced in I. At present the explicit calculation of the renormalization at the two-loop level for the present theory has not been carried out.
5. RETARDED
DENSITY
CORRELATION
FUNCTIONS
In the framework of the linear responsetheory, dynamical properties of the system in the vicinity of the equilibrium state can be described by various correlation functions [20]. In this section we show how the finite temperature loop expansion method can be used to calculate such correlation functions. Although it is possibleto formulate the present theory for general correlation functions, here we discuss the retarded density correlation function as a specific example because of its obvious significance. The retarded real-time density correlation function is defined as ]2 1] D&f,
x’t’) EE-i([n;,(xt),
ii,,(x’t’)])c
a(t - t’).
(5.1)
where the notation ( . .. )o has been defined in Section 1 and
&(xf) = !iqxt) !P,,(xt)- (a:, QG
(5.2)
and (5.3) The subscript H meansthe Heisenberg picture. In this section, instead of the retarded real-time correlation function we consider the Wick-rotated retarded correlation function, which will be called simply the retarded correlation function here, defined as 1211 c/(x,x’) = -(T,[ii(x) fi(x’)])G, (5.4) where we have used the notation x = (x, r) and z(x) is the Wick-rotated ;,(xf). The symbol T,[ ... ] means “take the r-ordered product.” In the rest of this section we assumethat the system is spatially uniform and introduce the Fourier transform of P(x, x’), Q(x,
x’)
=
eiP(X-X’) 1
P
&IQ),
(5.5)
THEORY
OF
LAMBDA
TRANSITION,
where we have used p = (p, on) defined in (2.6). Then this ‘r(p) representation qp)=L
.cc &,’ 271 J -x
259
II
has the Lehmann
A (P3 w’) icu,-co”
(5.6)
whose spectral function A also gives the Lehmann representation of the real-time retarded correlation function defined by (5.1) [21]. The ensemble expectation value for the r-ordered product of the field operators, such as (5.4) can be calculated as the functional expectation value defined by
where S., , S,, , and S,, have been defined in Section 3 and A is the dimensionless parameter indicating the order of the finite temperature loop expansion. Using (5.7) and (5.4), we can write the retarded correlation function i;/’ (.x, x’) as -2(x,-K’)=
((D*(-~)v)(x)v7*(~~‘)v)(x’))s-(l(P(~~)Iz)s(I~(”‘)I~)s.
(5.8)
Then changing the field variables for the functional integration in (5.7) from q~to i,~ defined by ]22] V=%+V
(5.9)
we can rewrite (5.8) as 2(x,x’)
= !z,(x, x’) + Q,*(x, x’).
(5.10)
where
(5.11)
In the calculation of the expectation value ( ... )s given by (5.7), the expansion of the RHS of (5.7) in powers of A gives the finite temperature loop expansion. Because only the expectation value ( ... ).? defined by (3.11) appears in the expansion, we can use the one-loop Green’s functions introduced in Section 3 to make a systematic perturbational calculation. For the lowest order in the loop expansion we have ( . . . lg . . . )s =
( *. . ly * *. ) 1 $- o(n).
(5.13)
260
TADASHI
Then, for the lowest-order forms
TOYODA
loop terms the retarded correlation
functions
have the
Y, = Yj’)
+ U(A)
(5.14)
p,, = up
+ U(A),
(5.15)
{-v:, + E(k)
E(k + P) + 4a; I v7,,1’}
and
where
QYYP)
= I,
D(k) D(k + P)
’
(5.17)
with k = (k, v,)
(5.18)
and
62;;‘(p) = -21~,12{E(P)-2ao/~,,12~ D(P) where we have assumedq0 is uniform and have used (3.19), (3.20). (3.21), (3.25), (3.26), and (3.27). Expressions (5.17) and (5.20) contain the bare parameters ,u” and a, and the classical field pO. As shown in Section 5 of I, at the lowest order in the loop expansion we can replace ,q, and a, by the renormalized physical parameters p and a, respectively. The classical field q0 can also be replaced by -Y at the lowest order, as shown in Section 4. After the renormalization and the Legendre transformation, instead of (5.17) and (5.20) we have {-vi +ER(k)ER(k + p) + 4a j Yl’} D”(k) DR(k t-p)
(5.21)
and c2P*(p; !q E
- 2 1Y12{ER(p) - 2a 1Y12} DR(p)
(5.22)
where 2
ER(p)=&p2-,u+4a
(5.23)
TRANSITION,11
261
DR(p) E co:,+ ER(p)2 - 4u2 ~!q4.
(5.24)
THEORY OF LAMBDA
and
Finally, the lowest-order loop approximation for the retarded correlation function C/(x, x’) has,the form eip(-y-~yGqyp;
C/(x,x’)=!
Y) + u;,(p;
Y)} + O(A).
(5.25)
P
The two correlation functions Qp(p; !?) and U:,(p; ul) have very different properties. The first, 9:) goes smoothly over to the correlation function of the normal bose gas in the !?+ 0 limit. This can be seenby making use of (3.22). On the other hand, the second, g:,, is proportional to 1y]‘, which may be identified as the condensate density, and vanishes in Y -+ 0 limit. Thus, the result (5.25) shows that the retarded density correlation function CZ(x, x’) can be split into the “normal part,” pip, and the “superfluid part, ” g”p,. It should be remarked that below T., the normal part also dependson the order-parameter. Although the present calculation is based on the one-loop approximation, higher-order contributions can be calculated systematically by making use of the one-loop Green’s functions and the Feynman diagrams given in Section 4. Recently Yamada [23] made an analysis of the responsefunctions of liquid ‘He II based on the two-fluid model and showed that the Ansatz on the dynamical structure factor made by Woods and Svensson [ 121 can be explained on the basis of the twofluid model. The present result supports, at least qualitatively, Yamada’s analysis from a microscopic point of view.
6. DISCUSSION AND CONCLUDING
REMARKS
In Section 2 we formulated the finite temperature Schwinger functional formalism for the grand canonical ensemblein a general way and proved the Hugenholtz-PinesGoldstone theorem for finite temperatures. To the author’s knowledge this is first time this has been done. A remark on the Hugenholtz-Pines-Goldstone theorem for finite temperatures must be given here. The fact that Green’s functions have a singularity at p = 0 does not necessarily mean that the energy spectrum of the corresponding boson is gapless.This can only be seenwhen the chemical potential of that boson is known. For elementary excitations we expect that the chemical potential is zero. However, as shown in Section 5, the one-loop Green’s function G(p), which can be identified as the Green’s function of an elementary excitation boson with a gapless energy spectrum and zero chemical potential below the lambda point, can also be identified as the temperature Green’s function of a boson with nonzero chemical potential above the lambda point. In other words the Green’s function obtained in Section 3 contains the energy spectrum of a boson which can be identified as a Landau type of elementary excitation below the lambda point and can also be identified as a normal
262
TADASHI
TOYODA
4He boson above the lambda point. This indicates that in order to identify the thermal quasi-particle energy spectrum a careful treatment of the corresponding chemical potential is essential. The one-loop Green’s functions introduced in Section 3 make it possible to calculate higher-order loop terms systematically in the form of Wick’s theorem. It is obvious that these one-loop Green’s functions are a natural finite temperature extension of the zero-temperature Green’s functions introduced by Beliaev [6 ], Hugenholtz and Pines [3], and Gavoret and Nozieres [7]. One of the essential features of our one-loop Green’s functions is that they contain the order-parameter of the lambda transition in a consistent way. As we discuss later in this section, this point is crucial for the explanation of the result of Woods and Svensson for the dynamical structure factor. As shown in Section 5, these one-loop Green’s functions can be used to calculate the density correlation functions within the finite temperature loop expansion scheme. Since the density correlation functions are directly related to the real-time Green’s functions, it is also possible to treat dynamic critical phenomena within the framework of the present formalism. The general algorithm for calculating the density correlation functions up to the L-loop level is as follows: First, calculate the effective potential up to L-loop terms using the one-loop Green’s functions defined in Section 3, which contain the bare parameters ruOand a, and also the “classical field” (Do. Then making the Legendre transformation and renormalizing, we obtain the grand potential with the renormalized parameters and the orderparameter at the L-loop level. From the grand potential we calculate the equilibrium value of the order parameter and also the chemical potential as a function of the other relevant variables, as shown in I at the one-loop level. Then substituting this chemical potential expression and the equilibrium value of the order-parameter into the one-loop Green’s functions, we can construct the perturbation calculation for the density correlation functions up to the L-loop level. It should be noted that the entire calculation is controlled by the finite temperature loop expansion. In Section 5 we showed this procedure at the one-loop level. Also in Section 5, analyzing the response functions, we gave a microscopic basis for the Ansatz for the dynamical structure factor recently proposed by Woods and Svensson [ 121. Recently Griffin and Talbot [24] also attempted to explain the Woods-Svensson result for the dynamical structure factor, especially the normal part, on the basis of the Bogoliubov approximation for the single-particle spectral densities and by taking the energy spectrum of the excitations as known. Therefore, their approach is not really microscopic but has the advantage of being much closer to the experiments in several aspects. Although our one-loop approximation is very similar there are two essential differences. Our to the Bogoliubov approximation, microscopically derived energy spectrum contains the order-parameter of the lambda transition in a consistent manner, and also the finite temperature loop expansion makes it possible to calculate higher-order corrections systematically, while such a calculation is not possible in the Bogoliubov approximation. The fact that our oneloop Green’s functions contain the order-parameter in a consistent way makes it possible to explain the smooth transition of the normal part of the dynamical
THEORY
OF LAMBDA
TRANSITION,
263
11
structure factor from below the lambda point to above the lambda point in the Woods-Svensson result, which has not been explained in the Griffm-Talbot theory. Since we have taken a purely microscopic approach, at present our result is rather qualitative and it is desirable to treat higher-order loop terms as well as to extend the formalism to more realistic potentials. In concluding this paper we may state that the present formalism, which can be seen as a natural finite temperature extension of the previous zero-temperature theories, makes it possible to treat the lambda transition in a unified microscopic theory, and it can also be used to treat dynamic critical phenomena of the system within the framework. At present, the results obtained are rather qualitative but it has at least clarified some fundamental aspects of the lambda transition, which have not been explained by any previous theories.
APPENDIX
The one-loop contribution to the order-parameter, obtained from (3.9) and (2.8), which give
Y,(x)
and Y:(x),
can be
where we have used the fact that F,[c+o,j has no explicit J dependence. A similar expression for Y*(x) can also be obtained and we can write them as CA.2)
with (A.31 and
(A.4 1
264
TADASHI
TOYODA
From (3.9) and (3.10) we have
where S, [ I,V] actually contains (pO, i.e.,
&
S,[wl = -4%P,*
w*(x)
w@) - 2o,rp,V*(x)
v*(-u)*
(A.61
v(x) - 2u, cp,*w(x) V(-~).
(A.7)
and m
6
Thus we immediately f,(x)
s, [WI = -4% (Dov*(x) get = -4aoG(w*(x)
v(X)),~ - 2aovo(v*(x)
w*(x))..,
b4.8)
v(x)).,
d-u))..,.
(A.9)
and f,(x) For the following I’, (4 and f2M
= -4a,v,(v*tx)
discussion,
- 2a,cp,*(v@)
it is convenient to introduce the Fourier transforms
f,(x) = ji e’P”.7,?;,(p),
of
(A. 10)
where
and f>:,(k) = W, 0) In the above expressions
i
281 G(P) + d* 1 K*(P)). P
(A.12)
P
we have used %= -2a,fp,.
The functional derivatives of p. and (o$ with regard to J and J* can be obtained from the Euler equations (3.7) and (3.8), from which we can readily find
THEORY
OF LAMBDA
TRANSITION,
265
II
The 2 x 2 matrix operator on the LHS has already appeared in (3.12) and its inverse operator gives the one-loop Green’s functions as shown in (3.19). Therefore we have
g,,(x,x’) g,,(x,x’) g*,(-x3 x’) Introducing
(A.14)
g&2-~‘)
(A.14) and (A.lO)
into
(A.2),
we obtain
G(P) K*(P) 1W:(x).Y,(x)]= - I_e-i’JxI.&PXJ,(-P)l [K(p) G(-P)1’
(A.15)
“P
which gives the one-loop contribution
to the order-parameter
defined in (4.11).
ACKNOWLEDGMENTS I would like to thank Professors N. M. Hugenholtz (Groningen), Y. Kuroda (Nagoya), Y. Nagaoka (Kyoto). S. Nakajima (Tokyo), T. Nishiyama (Osaka), T. Tsuneto (Kyoto), T. Usui (Nagoya). and K. Yamada (Nagoya) for useful discussions and comments. I would also like to thank Professor K. Wildermuth for his warm hospitality at the Institut fiir Theoretische Physik der Universitiit Tiibingen. and Dr. L. Rikus for carefully reading the manuscript.
REFERENCES I. 2. 3. 4. 5. 6. I. 8. 9. 10.
I I. 12. 13. 14. IS. 16. 17.
T. TOYoDA, Ann. Ph.~s. (N.Y.) 141 (1982). 154; Phys. Lelf. 87A (1981). 91. L. D. LANDAU, J. Phys. (U.S.S.R.) 5 (1941). 71; 11 (1947). 91. N. M. HUC~ENHOLTZ AND D. PINES. Php. Rev. 116 (1959), 489. J. GOLDSTONE, NUOUO Cimenlo 19 (1961), 154. H. ARAKI AND E. J. WOOIJS, J. Math. Phlx 4 (1963), 637; N. M. HUGENHOLTZ. i?f “Advances in Solid State Physics” (0. Madelung, Ed.), pp. 641-646, Pergamon Vieweg. 1972. S. T. BELIAEV, Soviet Phys. JETP 7( 1958). 289. J. GAVOR~~T AND P. NOZI&ES. Ann. PhJjs. (N.Y.) 28 (1964). 349. “Dynamical Critical Phenomena and Related Topics.” See, for example, C. P. ENZ (Ed.), Proceedings, Geneva, 1979, Lecture Notes in Physics No. 104. Springer-Verlag, Berlin. 1979. See, for example, P. C. HOGENBEKG AND B. I. HALPERIN. Rev. Mod. Php. 49 (1977), 435. R. A. FEIIRELL. N. MENYHARD. H. SCHMIDT, F. SCHWABL. AND P. SZBPFALUSY. Ann. Phys. (N.Y.) 47 (1968). 565; R. A. FERRELL AND J. K. BHATTACHARJEE. Phys. Rec. B 24 (198 I). 507 1: J. K. BHATTACHAFUUEE AND R. A. FERRELL. Phys. Ret!. B 25 (1982). 216. P. W. ANDERSON, Rep. Mod. Phys. 38 (1966). 298. A. D. B. WOODS AND E. C. SVENSSON. Phw. Rec. Left. 41 (1978), 974. See. for example. D. A~r-r. “Field Theory, the Renormalization Group. and Critical Phenomena,” McGraw-Hill, New York, 1978. G. JONA-LASINIO. Nuouo Cimento 34 (1964). 1790. See. for example, A. ISIHARA. “Statistical Physics.” Chap. 15. Academic Press, New York, 1971. See, for example. E. BRBZIN. et al., in “Phase Transitions and Critical Phenomena” (C. Domb and M. S. Green. Eds.). Vol. 6. pp. 125-247, Academic Press. New York. 1976. See, for example, A. L. FEWER AND J. D. WALECKA, “Quantum Theory of Many-Particle Systems” Chap. 7. McGraw-Hill. New York, 1971.
266
TADASHI
TOYODA
18. Equations (4.27) and (4.28) correct Eqs. (5.7). (5.8). and (5.9) in I. Also, the first equation of (5.5) in I should read -a, + a: f::“:@,,) + O(ai) = --a. 19. See, for example, J. ILIOPOULOS et al., Rev. Mod. Phw. 47 (1975). 165. 20. R. KUBO, J. Phvs. Sot. Japan 11 (1957); see also Chap. 13 of Ref. (15 1. 21. See, for example, Chap. 9 of Ref. [ 171. 22. Section 3 of I. 23. K. YAMADA. Progr. Theoret. Phys. 63 (1980), 715. 24. A. GRIFFIN AND E. TALBOT, Phys. Rev. B 24 (1981). 5075: A. GRIFFIN. Phvs. Rec. B 19 (1979), 5946.