Microscopic theory of a dense Fermi liquid at low temperatures

Physica

52 (1971)

165-192

MICROSCOPIC

I. FORMAL

0 North-Hoiiand

Publishing

THEORY

OF A DENSE

LOW

TEMPERATURES

DEVELOPMENT

AND

University

Statistical of Texas

FERMI

QUASIPARTICLE

L. E. Center for

co.

LIQUID

AT

INTERPRETATION

REICHLt

Mechanics

at Austin,

and Thermodynamics,

Austin,

Texas

78712,

USA

and E. University

of Denver,

Received

R.

TUTTLE

Denver,

Colorado

13 October

80210,

USA

1970

Synopsis The microscopic equilibrium theory of a Fermi liquid developed and Mohling is considered in the high-density, low-temperature region. in this limit, appearing considered,

a set of polarization-type

in the

theory.

explicit

expressions

distribution are derived. to those postulated by theory

is developed

In the

diagrams

approximation for

the

grand

dominate in which potential

by Lee, Yang, It is found that,

the graphical only and

these particle

summations diagrams

are

momentum

It is found that the system obeys a set of equations Landau. A “statistical” quasiparticle interpretation

similar of the

and discussed.

1. Introduction. In recent years, semiphenomenological theories have had considerable success in explaining the experimentally observed temperature dependences of the thermodynamic and transport properties of liquid sHe. The first step toward a useful theory of the liquid state was made by Landaui) who postulated that, at low temperatures, the quasiparticle states were in one-to-one correspondence with the states of an ideal Fermi gas, and that the internal energy was a functional of the quasiparticle distribution. The entropy, when considered as a function of the quasiparticle distribution function, therefore has the same form as that of an ideal Fermi gas. The maximization of the entropy, subject to the constraints of constant energy and number of quasiparticles, then leads to a quasiparticle distribution function of the same form as that of an ideal gas. The consequences of this theory were investigated by a number of authors, * Work based, partial fulfillment

in part, on a dissertation submitted to the University of requirements for Ph. D. in Physics, 1969. 165

of Denver

in

166

L. E. REICHL

AND

E. R. TUTTLE

and it was found that the specific heat 1~2) and the transport coefficientsaT4) had temperature dependences similar to those obtained for a gas of weakly interacting fermions. At the time, these results appeared to be supported by the experimental data. One therefore expected that a microscopic theory of liquid sHe should, at sufficiently low temperatures, have a functional form similar to that postulated by Landau. It was shown that both Green function approachs) and the perturbation theory approach used by Bloch, Balian and de Dominicis6) did indeed yield such forms. On the other hand, it did not prove possible to calculate the properties of the system numerically with either of these theories, and the theories which were most successful from the calculational viewpoint7) were not clearly related to the Landau theory. In 1965, the problem was complicated further when Andersons) noted a disagreement between the theoretical predictions and the experimental specific heat curve. Whereas the theoretical deviation from linearity was proportional to T3, the experimental deviation was found to be proportional to T3 In T. Discrepancies between theory and experiment were also found in the transport coefficientss). Shortly thereafter, Doniach and Engelsbergio) showed, through a Green function model calculation, that spin fluctuations in the liquid could account for the observed Ts In T dependence of the specific heat curve. Since the work of Doniach and Engelsberg, the effects of spin and density fluctuations on the equilibrium (and transport) properties of liquid aHe have been studied by a number of authors, Amit et al. 11) have used the Green function formalism to calculate corrections to the single-particle mass operator and the specific heat which arise from the interaction of quasiparticles with spin and density fluctuations. In this calculation, the basic postulates of the Landau theory are assumed; and the coefficient of the T3 In T term in the specific heat is expressed in terms of Landau’s phenomenological parameters FL and Znis). A similar calculation has also been performed by Brenig and Mikeskars) . Emery14) has obtained the correction to the specific heat through use of a Boltzmann-type equation, in which the transition probability has been corrected to take into account screening by the spin fluctuations. He has then calculated the quasiparticle damping and, from this, found the correction term. Once again, the coefficient of this term is expressed in terms of the Landau parameters. Finally, W. Brenig et aZ.15), E. Riedell’j), and W. F. Brinkman and S. Engelsberg17) have calculated the specific-heat correction using the random-phase approximation in a system interacting through a contact potential. These are essentially perturbation theory calculations; for a strongly coupled system, such as liquid sHe, the results can only be considered as qualitative. Furthermore, the relationship of such a theory to the quasiparticle picture of the liquid is not completely clear.

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All of the past work on spin fluctuations has several features in common. It is assumed that the basic fermions (whether they are called quasiparticles or not) obey an ideal Fermi distribution. In each case, the theory is based on a model in which the coupling may be assumed be weak, and it is not possible to determine the form of the interaction between basic fermions from microscopic principles. Therefore, it is not possible to calculate the thermodynamic properties of the system numerically. Finally, the temperature range over which these theories may be expected to hold is very small ll). In this and a subsequent paper, we wish to examine the equilibrium properties of liquid sHe, and of dense Fermi liquids in general, from a point of view different from those used in the past. We shall attempt to arrive at a theory which shows clearly the relation between Landau theory and spin and density fluctuations in the liquid; and, which will allow us to obtain numerical values for thermodynamic properties of the liquid from purely microscopic considerations. We shall begin, in this paper, by deriving an equilibrium theory of a low temperature, dense, homogeneous, Fermi liquid from the exact microscopic theory which has been developed by Lee and Yang is) and extended by Mohlingl9). Lee and Yang first showed that the Ursell cluster functions of quantum statistics (and therefore the thermodynamic properties of quantum statistical many-body systems) can be expressed in terms of the unsymmetrized Boltzmann cluster functions introduced by Kahn and UhlenbecksO). The Boltzmann cluster functions can in turn be expanded in a series of integrals over coupled two-body cluster functions. The rl transformation introduced by Mohling resums the series thus obtained for the grand potential in such a way that apparently divergent low-temperature contributions are completely removed. Furthermore, the cl-transformed theory has a simple quasiparticle interpretation. The interaction between quasiparticles is determined through the reaction matrix gr(kiks 1k&4) which, in turn, can be calculated directly from the microscopic two-particle interaction. In its most general form the theory is well-behaved even for particles which interact through a potential containing an infinite repulsive core. We shall here consider the formal application of this theory to dense Fermi liquids at low temperature. In section 2, we show that the vertex functions for the primed master c-graphs in the Mohling theory can be greatly simplified, and take a form very similar to that appearing in simple perturbation theory, except that the reaction matrix gi(kiks 1K&4) replaces the perturbation theory matrix element . We then find, in analogy to the work of Hugenholtzsi), that the sums over primed master C-graphs can be approximated by sums over primed polarization C-graphs. With the introduction of the simplified vertex functions, it is possible to perform the temperature integrations appearing in the polarization C-graphs.

This is done in section 3, and the resulting expressions for the sums over polarization 0- and l-graphs are written in a form convenient for further analysis. In section 4, we obtain explicit expressions for the grand potential, the and the quasiparticle energy. We also particle momentum distribution, establish a set of relationships between these quantities and derive a particularly simple form for the entropy. Finally, we observe that it is possible to think of the liquid as a system of quasiparticles. The quasiparticles are shown to interacting “statistical” obey a set of equations similar to those postulated by Landau. In this case, however, the equations are exact and do not break down at finite temperatures. 2. Microscopic theory of a Fermi liquid in the high-density limit. As discussed in the previous section, an exact microscopic theory of low-temperature Fermi systems has been developed by Mohlingia) based on the quantummechanical cluster expansion for the grand partition function first introduced by Lee and Yang is). It has been shown that, in the perturbation theory limit, this theory is equivalent to the work of de Dominicisss). Recently, this theory was considerably simplifiedss) *, and we use the notation of the simplified version in the present work. In this section, we examine this theory in the low-temperature, high-density limit. It has been shown (TI) that the grand potential and the particle momentum distribution for a Fermi liquid can be written in the following form:

Qf(B,g,J4 = F ln [ 1 + evIlBk - ddll + QF' (8,g,Q2) - $ ~&gjndW(h> tz, k)~(~)P(tz - h) - y'(k)1 -

T [jdt

LZ'(t, t, k) - /dt ec@"(t, t, k)],

= v’(k) ;dt %‘(/I,t, A). 0

(2-l) (2.2)

In these equations, Sz is the volume of the system; g is the chemical potential; and /? is (knT)-1, where T is the absolute temperature and k~ is Boltzmann’s constant. The function wi is the quasiparticle energy oic = Wk + d(k),

(2.3)

where wlc = G%s/2m and d(K) is the Hartree-Fock single-particle interaction energy. The symbol K = (k, a) denotes both momentum and spin, and m * Here after referred

to as TI.

MICROSCOPIC

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169

is the mass of a particle in the system. The function y’(k), which is given

~‘(4 = [exp[B(dc - g)l + q-1, is the quasiparticle

momentum

by

(2.4)

distribution.

The functions

L?"(t2, tl, k)

Q'@2, h, h),

and

9”“‘(ts,

ti, k)

which appear in eqs. (2.1) and (2.2) can be expressed in terms of primed master C-graphs [cf. (TI)]. Let us now consider a low-temperature, high-density, Fermi system. We assume that the two-particle interaction has an attractive part sufficiently strong that the many-body system is bound. Because of this, we shall use the R-graph expansion of (TI), as the convergence of this expansion is expected to better than that of the simple perturbation (V-graph) expansion for system with strong interparticle interactions. In this case, the natural expansion function is the generalized primed pair function,

which is defined by

tde klk2 ’

= qt, - La)qt2 - to)

xi[ k3k41 to + qta -

to) 1

x IW2 -

k&4 1ia’

t1 klkz ’ h) B(h -

~2) - v’(k2) a(0 -

(2.6)

s2)l

and the primed pair function is given by

=g1(hblhk4) expLh(&l +4, -oh, -&,)I Ma 1to (

t1 klk2 [

+

’

C fz(Wz

kske

1

I h.ks I hh)

P

wit,

+

x

expth(wj,,

+

wk,

-

wk,

-

ok,)]

x

eWo(wk,

+

wk,

-

WI,

-

wit,)].

wi.

-

w)iE5 -

w&

)

(2.7)

The generalized primed pair function [cf. eq. (2.5)] is the vertex function in the primed master C-graphs. The functions gl(klk2 1k3k4) and fz(Wz

I k&s I k&4)

L. E. REICHL

170

are defined and discussed gi(M~

AND E. R. TUTTLE

elsewhere

249 25).

It is assumed that the functions

I k&4) and 1

c /2(hk2 kske

j h5k6 / k3k4)

p

‘d,

+

“ic,

-

mk,

-

Wk, >

are of the same order of magnitude. Nevertheless, in the low-temperature, high-density limit, it is possible to neglect the second term in eq. (2.7) when calculating the grand potential and the momentum distribution. For, if we calculate the nth-order contribution to the grand potential or the particle momentum distribution we find that the terms containing one or more powers of the function fs(kiks ]ksk$ ]k&) belong to one of four types, all of which can be shown to be small compared to those terms containing only the function gi(Kiks 1K&4). (1) Terms proportional to I’ ~‘(ks)exp[p(wi, + ohz - OJc)k, - Ok,)]. At low temperatures, the distribution functions are exponentially small unless ok, and wi, are less than g. However, in a bound Fermi system at temperatures approaching absolute zero, g is negative; (~0;~ + dcB -

wk,

-

wk,)

is therefore negative in the region in which the distribution functions are not negligibly small. Thus terms of this type vanish exponentially as T approaches zero and may be neglected. (2) Terms which give a negative contribution These terms are unphysical and are removed through particle ensemble theorys6).

to the entropy. the use of quasi-

(3) Terms which “exchange” the momenta in gl(W2 I M4). The combifunction fa(kiks / k&6 1K&I) sometimes appears in the particular nation

x

-P

H P

1 &,

+ mi, -

Wk,

-

Ok,

1 “;c, + “h, -

wk,

-

Lob,

I

’

Whenever this occurs, there is also a term in which the above expression is replaced by gi(ksk4 1Klk4.Now the function gi(kiks I k&4 is not hermitian ;

MICROSCOPIC

THEOKY

OF A DENSE FERMI

LIQUID.

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171

to be precise, gl(h'h/k3~4)

x

P [C

=

gl(k3k41klk2)

+

~~~fe(klk21k5k6/k3k4)

-

@k,

1 Wit, + ‘&, -

wk,

>

1

-P

‘&, + “ic, -

ok,

-

Wk,

(2.8)

*

That is, the terms containing fa(Krks lk5k~lh&4) can be combined with the term gr(kak4) krka) to yield gi(Kika 1K&4). In a dense Fermi system, the function gl(Kika 1k&4) appears to have a fairly sharp maximum about its diagonal elements *, and the major contribution to a given diagram therefore comes from the region in momentum space where this function is almost diagonal. Clearly, in this region, the term in eq. (2.8) which contains 3k)4 is much smaller than gl(k3k4 (klk2) and can be neglected in comparison with it. In the same approximation, of course, gl(klk2 1k3k4) can be assumed to be hermitian; i.e.,

fz(hkzlk5k61k

gi(kikz

I k&4) = gl(W4 I k&2).

(2.9)

(4) Terms which modify the vertex function gl(klk21k3k4). There also occur terms containing fz(klk2lksk6l k3k4) which have the same basic structure as terms of lower orders in that one or more of the functions gl(klkzl k3k4) appearing in the lower-order terms is replaced by a function containing several powers of gl(klk2 / k3k4) and fe(klk2 1ksk6 / k3k4) 22). Upon analysis, it is seen that such terms can be regarded as consisting of vertex modifications of the corresponding lower-order terms. If, as hypothesized, the function gl(klkz / kskd) is a good expansion function for strongly coupled systems, then the terms containing fz(klk2 1ksk6 1k3k4) are less important than lower-order terms of the same structure, and can be considered as corrections to the latter. We therefore neglect such terms in our calculations. If the term containing fz(klkz 1k5k6 1k3k4) in eq. (2.7) for the primed pair function is neglected, then the generalized primed pair function of eq. (2.5) takes on the following simple form: ‘It2 klk2 R [ k&4

- to) 1=[WI ”

to

Y'(h)lP(h - to)- v'(h)]

x gl(klk2 I k&4) exp[h(&, + cd, -

oic,-

oic,)].

(2.10)

This vertex function is identical to the vertex function appearing in “timedependent” perturbation theory except that the antisymmetrized matrix * This is based on a set of model calculations which are discussed in some detail in a succeeding paper.

L. E. REICHL

172

AND

E. R. TUTTLE

element of the potential, (K1k2 IV1 kskd, which appears there is replaced here by the function gl(klkz 1k3ka) (cf. TI). In particular, the dependence upon the temperature variables and the distribution functions is identical to that appearing in the simple perturbation theory. Similarly, if we expand the grand potential, given in eq. (2.1), in powers of the approximate primed pair function of eq. (2.10), we obtain a series whose terms are identical to those of the “time-dependent” perturbation series, except that, as expected, the matrix element (klkz 1PI k3k& is everywhere replaced by gl(klkz 1k3k4). (To show this, one first expands the grand potential in powers of the generalized primed pair function of eq. (2.5) to obtain the series of primed contracted O-graphs (cf. TI). One then replaces this pair function by its approximate form, as given in eq. (2.10), and observes that this is equivalent to the primed contracted V-graph expansion, with gl(klk2 Ik3k4) as expansion parameter. It is then obvious that this expansion is isomorphic to the simple perturbation expansion.) It is now a simple matter to determine the dominant terms in the expansion in the low-temperature, high-density limit. We first note that the function gl(klk2 Ik3k4) contains a momentum-conserving d-function and, at least for certain potentials, has a fairly sharp maximum for small momentum transfer q = ks - kl = k4 - kz. Because gl(klkzl k3k4) has these properties, and because the series of approximate expanded master graphs is isomorphic to the simple perturbation series, the conclusions drawn by Hugenholtzrr) in his study of the perturbation expansion for a low-temperature, Fermi system apply equally well to the pair-function expansion used here. In particular, for a high-density system, the dominant terms in the nth order are those which correspond to the nth-order polarization diagrams in the perturbation series. Following Hugenholtz, it is easy to show that all other &h-order terms will be multiplied by factors of order q/kF in comparison with the polarization diagram terms. The Fermi momentum kF is defined by the equation

N

k;

-=-9 f-2

3x2

(2.11)

and q is the average momentum transfer at a vertex of the diagram. Because only the region of small momentum transfer contributes to the integrals, in the high-density limit (2.12) and these other terms can be neglected in comparison with the polarization graph terms of the same order. We can then show that neglecting all but the polarization diagram terms in the sum over primed master O-graphs is

MICROSCOPIC

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OF A DENSE FEKMI LIQUlD.

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173

3

K,

s;

2rl

Fig. 1. Primed polarization O-graphs. a) One-vertex graph. b) Two-vertex graph. c) n-vertex graph. The symmetry numbers S, are shown beneath each graph.

equivalent to keeping only those master O-graphs having the structure shown in fig. 1, and setting the line factor %‘(t, s, K) equal to 6(t - s) in these graphs. The above analysis enables us to simplify greatly the exact formal expressions for szf(/?, g, Q) and given in eqs. (2.1) and (2.2). We define a primed polarization 0- or 1-graph to be a primed master 0- or 1-graph having structure shown in figs. 1 or 2, respectively, in which (a) each primed generalized pair function is given by eq. (2. IO), and (b) whenever the function s’(t, s, k) occurs as a line factor, it is replaced by 6(t - s). We then define X’(ts,

tr, k)p = C (all different

primed polarization

from which we determine the polarization-graph particle interaction energy, d(k), to be

L-graphs)k,

contribution

(2.13)

to the quasi-

174

L. E. REICHL

AND E. R. TUTTLE

Fig. 2. Primed polarization L-graphs. a) One-vertex graph. b) Two-vertex graph. c) n-vertex graph. The symmetry numbers .sn are shown beneath each graph.

LIP(A) = {that

part of Y’(t2,

tl, K)p which is independent

and tl and is multiplied

by [e(t, -

tl) -

of t2

y’(k)]}.

(2.14)

This gives us 9’&2,

Similarly, 9’W2,

t1, K)P

=

X’(i2,

t1, K)P

-

dP(k)[qt2

-

h)

-

y'(k)l.

(2.15)

it is easy to show that h,

YP

=

in this approximation.

9’(L2,

(2.16)

t1, k)P

Finally,

we define the quantity

Q&(B, g, Q2)= C (all primed

polarization

O-graphs).

(2.17)

MICROSCOPIC

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Upon replacing the quantities in eqs. (2.1) and (2.2) by their polarization graph approximations, we obtain the following expression for Qf(/3, g, Q) and :

-

F {Y’(K) In Y’(K) + [l -

+ QG(B,

Y’(K)] In [l -

Y’(K)]} (2.18)

g, Q) ;

and
= v’(h) -

y’(k) ;dt

_Y(@, t, k)p.

(2.19)

0

In eq. (2.18), the logarithmic term of eq. (2.1) has been rewritten in a more useful form. If the number of quasiparticles is equal to the number of particles, as we shall show in section 4, then the first term of this equation reduces to & in the polarization graph approximation. 3. Evaluution of the polarization graphs. As we have seen in the previous section, the grand potential, the particle momentum distribution, and the quasiparticle energy can be evaluated explicitly in the polarization graph approximation if we can obtain explicit expressions for the functions QZ$(~, g, Q), Y(tz, tl, k)p, and dp(K). The last two of these functions can be determined from the function Y’(t 2, tl, K)p. We therefore only need to evaluate OF;@, g, 52) and X’(tz, tl, k)p, and in this section we address ourselves to this problem. Let us first observe that there is only one primed polarization O-graph and one primed polarization L-graph of each order; these are shown in figs. 1 and 2, respectively. We can therefore write

QF;(B,g,Qn)= ?L=l fi -QFk,,,(B, g, Q)

(3.1)

x’(ts,

(3.2)

and ti, k)p =

% X’(tz, n=l

ti, k)p,n,

L. E. REICHL

176

AND E. R. TUTTLE

from the n-vertex primed polarwhere QFb,,(/?,g, 1;2)is th e contribution from the n-vertex ization O-graph and W(ts, tr, k)p, 1Eis the contribution primed polarization L-graph. Using the labeling conventions indicated in figs. 1 and 2, we obtain the following expressions for QF~,,(/l, g, Q) and 3-'(L2, h, k)P,n:

(3.3)

(--l)n

QF~,,(B~ g,J-4= -y-

n

c kt...kan

s 0

(3.4) and tdl 3”‘(t2,

h,

h)p,

1 =

r, ka

1 11

klk2

R [ hk2

(3.5)

tl’

taTa klk4 x’(t2,

h,

h)p,

~ (-l)n

n =

z

Sn

ka...kzn

s

@dTi . . . d 712_ 1 R [ k2h

0

x . . . x -[ where

S,

1 ”

71

;;~-;;:_,I,, -[ ;::_l~y-Jl )

= 212 if n # 2, Ss = 8 and sn = 1 if n # 2 and ss

(3.6) ZZ

2. The

generalized primed pair functions to be used in these two equations are the approximate generalized primed pair functions of eq. (2.10). The evaluation of the first-order contributions, QF~,,@, g, Q) and X’(ts, ti, kl)p, 1 proceeds somewhat differently from that of the remaining terms. We obtain Q%,,(B,

g, Q) = $

k% y’(k) y’(kz) gi(Wz 18

I hkz)

(3.7)

and s'(t2, h, kl)P,l =

-

[e(t, -

h) -

v'(h)] C v'(k2) gl(M2IM2).

(3.8)

kn

If we attempt to evaluate QFb,,(/?, g, 52) and X’(ts, ti, kl)p,n explicitly, we find that each function contains n! terms after the temperature integrations are performed. However, it is possible to consolidate these terms and to obtain relatively simple expressions for both QFb,,(B, g, i?) and S(t2, TV, k)p, n by writting the primed generalized pair function in a slightly different form and proceeding in a different way.

MICROSCOPIC

Let us introduce

THEORY

OF A DENS’E

a new function,

Q(T) = W)[l -

y'(b)1

I+(T),

FERbIf

LIQUID.

1

177

defined by

- e(-T) v’(b)} exp[T(g

-

(3.9)

U&J],

where e(T) is the Heaviside function. Then the approximate ized pair function may be written in the form

primed general-

=

Dl(h - to) R&2 - to) gl(hk2 Ik&4) eqQ$& - h - b)]

x exp[h&

+ t24,l exp[--to (4, +

When this expression (3.6), the exponential

(3.10)

for the pair function is substituted terms cancel and we obtain

(-l)n g, Q) = 7

QK+(B,

wl,)j.

n

‘dj

x kl...kan

s

I . . . dWr(t,

-

into eqs. (3.4) and

tr)

Dz(t1

-

tn)

0

D4(t2+)

X D3(tl-t2)

X ...X D2n-3

(tn-2--n-1)

D2n-2

(&-1-&a-2)

D2n-1(~n-l--tn)D2n(~n--tn-l)gl(k1k4)k2k3)gl(k3hi~ k4k5)

x

(3.11)

X ... X gl(k2n-3k2nlk2n-2k2n-1)gl(kzn-lkzlkznh)

and (-l)n 3-‘(t2,

h,

kl)p,

n =

Sn

X

&(TI-~2)

‘dTl

r,

~

&(~2-tl)

x D2nh-~-1)

kz...kzn

. . . dTn-lDl(t2

-

71)

D2(71

Do

h)

X ...X D2n-2 (~n-l-Tn-2)D2n-l(~n-l-V)

exp[(tl--t2)(g--w~,)lgl(klk4Ik2k3)gl(k3k6)k4k (3.12)

X ... X gl(k2n-3k2n I k2n-2k2n-l)gl(k2n-lk2lk2&1).

The function

-

I 0

possesses the following

simple Fourier

expansion: (3.13)

where

q(G) = (4,

- g - 2,)-l

(3.14)

and 2, = (2r +

1) xi//l.

(3.15)

Despite the apparent similarity between 9&.) and the single-particle Green functions), we must remember that, in the equilibrium theory being

L. E. REICHL

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AND E. R. TUTTLE

used here, the quasiparticle energies, wit, are average or “statistical” energies and hence are always real. When the Fourier expansion of @(T) is substituted into eqs. (3.11) and (3.12)) the temperature integrations become trivial and merely introduce Kronecker delta functions, of the form 8(r$ + rj, ok + YI), into the equations for -Q&,,(B, g, 0) and .X’(&, ti, WP, n. These Kronecker delta functions, in turn, allow us to define a new variable, ~1, such that

.51= 2xi(rsj-1 Summation expressions

-

rsj)/p = 2ni l//3 = .zsj-1 -

zzg

(all j).

(3.16)

over the Kronecker delta functions then leads to the following for LIF;,, (B, g, Q) and x’(t2, h, h)~, n:

QF;,,(B,

g, Q) = gl(h3k6

X

X R(Q)

s

(--l)n

c

n

1k4h)

;

kl...kan

t=

. . . gl(h2n-lk2

Al, k2) X(&Z

gl(hk4

1k2h)

-co

1k2nkl)

t h3, h4) *. f R(EZj h2n-1,

(3.17)

k2n)

and .X’(t2,

tl, kl)P,

n =

(--l)n n snp

ka...kzn

. .a

gl(&n-l&

C

x

gl(k3h6

k&5)

x

%,t,(Q

k1, kz) R(Q,

5

g1(W4

Ik2k3)

1=--m

I k&l)

k3, k4) a.. R(Ez,

k2n-1,

k2n)

(3.18)

;

where R(Ez, ki, &) E and

1

OQ

-iL=~, [oi, 1

gtlta(Fz)

1

O”

klJ k2) = 7 .,:_

- g-

zr][dc, -

g+

(3.19) &I

-

Zi]

expM - t2) (g - d, + .a)] [o& - g - zl][u);l - g + El - zl]

(3.20) .

It is important to note that the energy denominators which appear in the functions R(q, ki, kj) and c%?~~L,(E~, kl, k2) give rise to no singularities or irregularities in the functions X’(ts, ti, kl)p,n and QFb,,@, g, Q) $. In the previous section, we performed the 4. Thermodynamic properties. temperature integrations in the primed polarization 0- and L-graphs, and for the functions QFb,,(/?, g, 52) and obtained explicit expressions s’(tz, ti, ki)P, n. In the present section, we shall use these results to obtain explicit expressions for various thermodynamic quantities, and establish simple relationships between the total energy and the quasiparticle energy * This problem has been examined earlier in detail by De Dominicis (ref. 22) in his study of the polarization diagrams in ordinary perturbation theory.

MICROSCOPIC THEORY

and between the particle

OF A DENSE FERMI LIQUID.

and quasiparticle

momentum

I

distribution.

179

Finally,

we shall conclude this section with a discussion of the quasiparticle interpretation of the microscopic theory and its relation to the Landau theory of Fermi liquids. It is important to note that no attempt will be made to sum explicitly over all orders, n, in our expressions for various thermodynamic quantities. Although the results obtained here are in a form suitable for a study of the formal properties of these quantities, they are not in a form suitable for a numerical study. The problem of obtaining the grand potential in a form suitable for calculation is considered in a following paper. We first consider the functions QFk,,(/3, g, Q) and X’(ts, ti, K~)P,~, as given at the end of the previous section. It is possible, using standard techniques, to perform the summations indicated in eqs. (3.19) and (3.20) When this is done, and the results substituted into eqs. (3.17) and (3.18) for the functions QFb,,(/?, g, .f2) and X’(ts, ti, K~)P,~, we obtain

(-1)”

QFb,n(Ag,Q) = ---g--X

g1@3h

lk4h5)

n

c kl...kan

g1(W4

...

gl(k2n-lk2

I W3)

I h2nkl)

P(hl,

...j

P-1)

k2n)

and 2’(t2,

tl, h)P,

x

1k4h5)

gl(h3h6

where the functions P(h

(--l)n -~ @n

n =

*a*, k2n)

r, g1(W4 kz...ka,,

I k2k3)

I k2nh)

... gl(k2n-lk2

gt,t,(kl,

P(Ki, . . . . Ksn) and Ptlt,(Ki, E

X

. . . [v’(kSn)

x

**a (El +

,-

m -

CY’(h2)

-

~‘(h)][~‘(~4)

v’(hZn-l)][(EZ

Oiczn -

+

“ic,

. . . . k2,) are -

-

. . ., AZn),

(4.2)

defined

v’(k3)]

Wjcl)(Q

+

Ok,

-

Ui,)

oicz”_,)]-l

(4.3)

and ~)tJ,(h,

.*.,

k2n)

s

,=g,

x

w2-h)

-

x

exp[(t2--tl)(&-wi,

LY’(k4)

V’(kl)l

-

-

Y’(k3)]

w,--tl) -

-

..a [Y’(h2n)

-

Y’(k2n-l)]

v’(K2)]

~2)l)

x [(El - dp + wicJ(Q+"ic, - dc,) X *. * (82+ 4,”

-

&,,_,)I-1.

(4.4

E. B. I?l?fCHL ANiI B. i?. +l?Ul+TLE

180

Despite appearances, the expression for .X’(ts, tr, ki)~,~ cannot yet be separated into temperature-dependent and -independent parts because the second term in eq. (4.4) still contains a part which is independent of temperature. The summation over integers in each of the functions P(ki, . . . . ks,) and ~lt&l* * f *>kzn) can be performed by methods similar to those used in the evaluation of K(Q, ki, kj) and &t,(ei, kl, kz). After some manipulation, we obtain Q%&%

B(- 1)” 2s

g, Q) = -

n

X ...

X gl(k2n-lk2

X ...

X

2

1h2h3)

gl(hk4

gl(k3h

1k4k5)

kl...kan

1h2nkl)

y’(kl)[l

-

Y’(h2)][“‘(k4)

y’(K3)]

oi4 - wi3)

- oiE~+

[Y’(kzn) - Y’(k2n-l)][(oi1

-

x (&, - “ic, + wit, - &c,) *** (wk, - w& + co&” - ~ic*,_,)l-i.

(4.5)

In obtaining eq. (4.59, we have made use of the relationship S, = +zsn between the symmetry numbers of the O-graphs and the L-graphs. Similar operations on eq. (4.2) give us the following expression for X’(ts, ti, ki)p, n:

x’(t2,

tl, kl)P,

x

[qt2

(-1)” ~ Sn

n =

-

h)

-

[1 -

y’(h)1

i

[ [Y’(k4)

x

v’(k3)

-

x

-

d,

... x

+

[1 -

-

[Y’(k2n-2)

-

wi,

+

-

-

d,

+

. . . [Y’(k2n)

f&J

. f. (4,

y’(K4)1[9J’(kt3)

oi,

[v’(hZn)

... x (wi,

x

d(k3)]

f&, + c&, -

(&,-

(wk,

-

(dc,

y’(k2)l

wk,) (dc,

-

-

wi, -

-

d,)

Y’(k2n-I)1

d,

+ dcB, -

Y’(K5)]

dc,

+

wit, -

wit,)

v’(k2n-l)] WL

-

Y’(h2n-3)]

pie,,_,)

v’(h2n-l)Ll

-

Y’(h2n)l

4&I)

MICROSCOPIC

+

THEORY

OF A DENSE

FERMI

LIQUID.

I

181

[I - V’(K2)1[1 - Y’(k)] V’(k4) W2 - fl) (Gc, - “i, + “ic, - mi,)

K

y’(K2) v’(k)[I

-

x +

expL(t2

...

-

+

- v’(k4)l O(jl - t2)

- 0ic, + 0.4,- C&J]

h) (4

)

[ 1 - Y’(k2)l[I - Y’(h2n-1)l Y’(h2n) W2 - tl) (w&

(

- w&, + ai. - wi,)

V’(k2) Y’(k2n-l)[ 1 - Y’(h2n)l O(il - L2)

-

(wipn_1-

“i,” + WI, -

oi,)

>

x exp[(t2 - h)(d~, - w& + f&,. - &_

(4.6)

In this form, it is easy to see from eqs. (2.14) and (2.15) that the first term of eq. (4.6) contributes to d~,~(kl) while the second contributes to 2’@2,

h,

kl)P, 12.

Particle momentum distribution. The definition of the particle momentum distribution, in the polarization graph approximation, is given in eq. (2.19). Comparison of eqs. (2.15), (2.19) and (3.2) shows that it may be written as a sum of contributions from each order:

= Y’@l) -

:

<@l)>P,,,

(4.7)

n=l

where <@l)>P,

% = y’(kl) {d=V,

t, kl)~, 12

(4.8)

0

is the contribution to from the Izth-order polarization L-graph. function Y(s, t, k 1) P,~ is just the temperature-dependent part of y’(s,

The

t, kl)P, n.

From eq. (3.8) we see that identically zero so that <+l)>P,l

= 0.

the first-order

contribution,

Z’(/I, t, kl)p,l

is

(4.9)

L. E. REICHL

182

AND

E. R. TUTTLE

In higher orders, U’@, t, k 1) p, n is given by the second term of eq. (4.6). When this is substituted into eq. (4.8) and the indicated temperature integration is performed, we obtain


(--l)n Y'(h) --y-

=

x

z

gl(hk41k2h3)

gl(k3hik4k5)

...gl(hz-l~2~hnh)

kz...kan

x

{

x

... x

+

(dcg

...

[1

-

-

wit, t

[Y’(h2n)

+

[l -

d&3)1

9J'(k2)1[1dc,

-

Y’(k2n-1)1[1

-

v’(kZ)][v’(h4)

Cl -v’(k2+1)1

~‘(k2~)[1

Each of the exponential

-

wk,

+

Y’(k5)l wit, -

c&c,)

exp[B(~;,-o;,+w;~-~~~)l]

9”(h3)] -

mi,,+wk,-

(dc, -

-

(d&

x (dc,,_*-

Y’(~4)[~‘(b)

o&)2

a.. [Y’(k2n-2)

dczn +

wk, -

-

Y’(k2n-3)]

&)2

-exPIB(~iC1-~ic,+~ica,-~ta,_l)ll

f&J.. . (wi,,_,-

. c4 1oJ

&,+Gczn-2-

d&,-J I

*

terms in eq. (4.10) can be rewritten if we make use

of the identity ~‘(W[l

--v’(k2)1[1 =

--y’(h21-1)l

[l -v’(kl)]d(k2)

y’(ksj-i)[l

The final form of
X

C

wk, +

&,,-&,,_,)I

(4.11)

--Y’(kzj)].

l)n

___

gl(k1k4

-

n is thus

(-

p, n = Y’(h)

y’(h23) exp[B(&,

Sn

Ik&s) gi(ksk6 Ik&5) . . . gi(ksn-rks I ksnki)

ka...kan

[1

X

v’(k2)lP

-

(4,

I

-

-

dc, + [v’(k6) -

x(

dc,

+

**. +

-

oi,

+

v’(k3)ly’(k4) dcp -

dcy

v’(k.511 a*. [v’(ksn)

“He -

wi,)

*. . (4,

- Y’(k2n-1)l

- dc, + OjGan - Oic,“_,)

[1 - ~‘(k2)l[~‘(k4)

x ... x

(O;Czn_l -

dc,,

+

dc,

-

[v’(ksn-2) - Y’(ksn-s)][l ((GC,,_,-

w&)2

-

(pi,“_,

-v’(ksn-i)]

wit*, + (&,,_, -

y’(k3)I -

dan

+

v’(ksn)

tic,“_,)

wi4 -

-

ok,)

MICROSCOPIC

-

[l -

x

c

THEORY

OF A DENSE

FERMI

LIQUID.

I

183

V’(ki)] F

gl(klk41k2k3)gl(K3K6jk4K5)...g1(K2n-1k21k2nk1)

kz...kzn

Y'(~Z)Y'(K3)[1 - v'(k4)] &,

-

Wjcd +

";c,-

WJ x

-

... x

Cl,;,)’

y'(&)l

**.[Y'(h2?&)- y'(~2?&-1)l

[Y'('h-2) - v'(h2n-3)] v'(h2n-l)[1 - Y'(h2n)] x ... x

(&,,_,

-

&,,

+ &,_,

- &,,_,)

.

(4.12)

The particle momentum distribution (vz(&)> can now be found by substituting eq. (4.12) into eq. (4.7). It is easy to see that the momentum distribution is not a step function, even at zero temperature. In particular at zero temperature, the first term in eq. (4.12) represents the &h-order contribution to the depletion of states below the Fermi level; and the second term, the &h-order contribution to the occupation of states above the Fermi level. Obviously, neither of these terms is, in general, zero and the particle momentum distribution in the interacting system is not, therefore, equal to the free-particle Fermi distribution. We also note that the momentum distribution depends on the temperature only through the quasiparticle distribution function Y’(K) and the quasiparticle energy wk. We shall now show that, even through the particle and quasiparticle momentum distributions differ ((n(k)> - Y’(K) # 0), the total number quasiparticles is equal to the total number of particles. The number of particles

in the system is clearly

=

c



2

Y'(h) -

kl

m =

kl

c n=l

c P,,. kl

(4.13)

We begin by summing <~~(KI))P, n as given by eq. (4.12)) over the momentum ki. In the second term of this summation, we let ksg ++ ks+i (i = 1, 2, . . . . a). If we then observe that, in the polarization graph approximation the function gl(Klk2 1K3k4) is hermitian, i.e., g1(M21h3~4)

=

gl(~3ww2)>

(4.14)

L. E. REICHL AND E. R. TUTTLE

184

it is easy to see that Z


=

(all 12).

0

kl Thus, eq. (4.13) becomes

simply

z y’(k),

=

(4.15)

k

and we conclude that the number of particles in the system exactly the number of quasiparticles.

equals

Quasiparticle energy and grand potential. It is now a simple matter to find the quasiparticle energy, ok. From eqs. (2.3), (2.15), (3.8) and (4.6) we immediately obtain

“ic,

=

WkI +

: n=l

=

ok,

-

+

5

(-:; _

n=2

dP,n(h)

Iz Y’(h)

g1(hk2

IW2)

?z Sn

2 gl(hk4 ks...ksn

1k2k3)

gl(k3h

/ k4h)

... gl(k2n-lk2

1 k2nh)

y’(k3)l dc, - oi, + “i, - 4,) (4, - d. + Ok, - 4,) [l - v’(k2)1[~‘(h) -

x

...

[y’(bn) -

Y’(hn-l)]

v’(h)[l - v’(k4)][v’(h)- y'(kdl

x

x

(~~,-wjG,+wjG,-oiE,)(w~,-wic,+oic,-~iG,) ...

x

[y’(bn) -

Y’(kzn-l)]

[y’(h) - y'(h)1 - *** x

1..

x

[~‘(k2n-2) - d(kZn-a)]

Y'(k2n-l)Ll

-

~‘(k2n)I

We observe that eq. (4.16) is actually a very complicated integral equation, as both the distribution function v’(k) and the vertex function, gl(klk2 1k3k4) contain ok. Also the function wi given above is real, not complex.

MICROSCOPIC

In a similar

THEORY

OF A DENSE FERMI

way, the grand potential,

LIQUID.

I

Q/(b, g, Q), can be obtained

185

from

eqs. (2.18)) (3.1), (3.7) and (4.5) : Q/(/K g, Q) = &a9 -

+

-

B T y’(h) Ok

$ (v’(K) In y’(k) + [I -

!E QG,,(B,

y’(k)])

g, Q)

12=1

-

y’(k)] In [l -

=

&a9

p F v'(~)wc

-

$

+

c Y'@l)Y'(KZ) g1(hkzlhk2) 2 klka

{v'(k) In

y’(k) + [l -

Y’(K)] ln[l - v’(K)]}

B

X . . . X (w;c, -

OiE, + o&,

-

&*,_,)I.

(4.17)

Neither CO&nor .Qnf(/?, g, Q) as given by eqs. (4.16) and (4.19) is in a form which is useful for numerical calculation. However, as we shall now show, eqs. (4.16) and (4.17) can be used to establish useful relationships between Inf(/?, g, a), ok, Y’(K), and the total energy, . Let us begin by considering the variation of Qf(b, g, Sz) with respect to the distribution function, Y’(K).’ We find that

O” v='P,

WJf) -

b’ h)

=

Bd(4 + mzl

n)

svl(kl)

Upon taking the functional derivative d(kl) explicitly, we obtain $

v%,,) sv’(h)

(4.18)

.

of QF~,,(~,

= B z y’(b) gl(hkz I klkz) =

-PAP,

g, Q) with respect

l(h)>

to

(4.19)

t The vertex function, gl(klkz 1kakq), depends on the quasiparticle energy wi and therefore, through eq. (4.15) on v’(k). However, in performing the functional differentiation indicated in eq. (4.17) this dependence has been suppressed because the contributions appearing from this dependence are of higher order (cf. section 2) than those we are keeping in the polarization graph approximation. It has been shown in ref. 14 that if all terms in G/(/3, g, 8) are kept, additional terms in A(k) will cancel these additional contributions and Qf(p, g, ~2) will still be a stationary function of y’(k).

L. E. REICHL

186

AND

E. R. TUTTLE

and W-%,,)

=-/q_

TX gl(W4

av’(kl)

‘12

X . . . X gl(hn-h _

B (--1P ~ &a

C

D>(k,

Ikmh)

D< (h,

gl(kh

Ihh)

Ikzn-lb)

. . . . kzn) and D,(kl, . . . . bn) = -

x ... x

+ (d-

gl(bh

I hW

. . .> hn)

gl(W5

Ik&s)

ka...kan

X . . . x gl(hnh

where D,(kl,

I k&s)

kz...kan

D> (h,

(4.20)

. . ., km),

. . . . kzn) are defined as:

v’(b) [v’(k) - v'(h)1 (wi,wit, + "i, - d,)(&, - 4. + 4, - dc,) i

[v'(h)-v'(km-I)] (dc- d, + QAizn - d*,_J [1- v'(h)lv'(k4)[v'(ks) - v'(b)1 Q&i,+f&,-d,)(Q&,--wjc,+"ic, -&,) [v'(kiw) - v'(hn-I)]

x ...x

[v'(h) - v'(b)1 + ... + (ct.&,"_, - da, + 4, - dcl)(~~*,_l - &*, + wk,- dc,) x ...x

Lv'(km-2) - v'(hn-$I[1 - v'(hs--l)lv'(kw.)

D<(kl, . . ..kzn) =

x ...x -

P - v'(kz)Ib'(h) - v'(h)1 dc, + "ic,wi,)(&,&,+ "i,- d,) i (Gc,-

[v'(h)- v'(hn--1)l (wk,- wit, + dcZ"- Gc2"_,)

v'(ks)[l - v'(k4)][v'(h) - v'(kJl - "ia+ 4, - dc,) (d,- 4, + "k,- wkl)(oi,

x ...x

[v'(hs) -v'(bn--l)l (d- 4, + Gczn- tic,"_,)

- ...-

[v'(ka) - v'(h)1 (LOiGzn_l - Or&"+ "ic, - 4c1)(~~,,_l - 4," + 4, - d*)

x ...x

[v’(kzn-2) - v’(kzn-3)]~‘(kzn-1)[1

-

v'(b)1

MICROSCOPIC

THEORY

OF A DENSE

FERMI

It can be shown that D,(Ki, . . . . ksn) and D,(ki, make use of this, and the fact that gi(Kiks/k& see

LIQUID.

I

187

. . . . k24 are equal. If we is hermitian, it is easy to

that

wJ%J = --B&, n(h).

(4.21)

By’(h)

Upon substitution

WJf)

of eqs. (4.19) and (4.21) into eq. (4.18)‘we now find that (4.22)

()

-zzz.

Sv’ (k)

Therefore, the grand potential is a stationary function with respect to changes in the quasiparticle distribution v’(k). It can be shown that the grand potential is also stationary with respect to changes in the particle momentum distribution. It is now easy to determine the total energy, , of the system. From thermodynamics we know that the total energy is related to the grand potential through the equation = g -

& (Qf)g,n*

(4.23)

Because the chemical potential, g, appears in the grand potential only through the quasiparticle distribution function, v’(k), eq. (4.23) can be rewritten in the form

(Qfkn - 5

(E> = g(N) - +

>

(4.24)

L7,Q

where the subscript v’ in the second term indicates that the distribution function v’(k) is to be held constant in performing the differentiation. By eq. (4.22) the last term of eq. (4.24) is zero, leaving = However,

g(N) - $

(Qf)y,,n.

(4.25)

from eqs. (3.7) and (4.5) it is easy to see that (4.26)

as that the total energy of the system is given by

=

r, v’(k) wk k

-

$ f&(/3,

g, ~2)

= IZ v’(k) w/c - ; C v’(h) v’(h) gl(Mz I k&z) + k

klka

L. E. REICHL

188

+

&

C-1)”

g

n=2 x

-

gl(~1~4lk2k3)gl(k3k6~~4~5)...gl(k2n-l~2~~2~~1)

v'(k2)][y'(k4)- v'(k3)]...[+(kZn) - Y'(k2n-l)]

x [(wit,- "ic, + X . . . X (d,

E. R. TUTTLE

kl...kn*

sn

Y’(kl)[l

c

AND

-

"i,-

“k,

+

oi,)(d,-WiE, da”

-

+

"i, - dc,)

(4.27)

dc*,_,)l-1.

It is interesting to note that depends upon the temperature only through the distribution function Y’(K). It is now possible to establish a particularly simple relationship between the energy and the quasiparticle energy ojE. If we take the variational derivative of with respect to y’(k), we immediately find [cf. eq. (4.21)] that

6 =wk+d(k) 6v’(kl)

(4.28)

=o,&.

Thus, if we add one quasiparticle of momentum k to the system (i%‘(k) = l), then the total energy of the system increases by an amount o.B~.This is just the energy of the added quasiparticle. Entropy and specific heat. From thermodynamics, the entropy can be shown to be related to the grand potential by the equation



(4.29)

which with the aid of eq. (4.22),may be written -

h

=

in the form (4.30)

Qf - Pg + @.

If we substitute eqs. (4.17) and (4.27) for equation, we find that = -kg

x {v’(k) In v’(k) + [l -

Qf(p,

v’(k)] ln[l

k

g, Q) and into

- v’(k)]}.

this (4.31)

This exactly the form which would be obtained for a gas of noninteracting particles with distribution function v’(k). The heat capacity at constant volume, Cn, can be obtained from either the entropy [eq. (4.31)] or the energy [eq. (4.27)]:

CI-J= __ hi

-/I+$ (),N,,, = B&( $$) ,Q

3 v’(k)

ok

-

QF;, B

1(N),Q .

(4.32)

MICROSCOPIC

If this result rather

THEORY

is rewritten

than the variables

OF A DENSE

in terms

FERMI

LIQUID.

of the more natural

(p, , 9) the specific

1

189

variables

(B, g> 4

heat becomes

(4.33) The chemical potential g is determined from eq. (4.15)) which relates the number of particles to the distribution function v’(k). This point will be discussed in more detail in a subsequent paper where we obtain an explicit expression for C,. However, it is sufficient to note here that the heat capacity can be found if the function QnFb@?, g, Sz) is sufficiently well defined. 5. Conclusions. Let us now collect and examine more carefully some of the results obtained so far in this section. The results of most interest are th e following :

(2.4)

(a) y’(k) = [exp[B(d - s)l + 11-l;

(4.15)

(b) = z v’(k); (c) Q&k g, Q) = @g(N) - C {y’(k)In v’(k) + 11

d(k)] ln[l

-

v’(k)]}

k -

VJf) (4 --&,(k)

(e)

B E y'(k) ak k

=

+

Q&(/t

(4.17)

g, Q);

(4.22)

0;

(4.27)

= F y’(k) NE>

(4.28)

(f) By’(k)=d (g)

= -kg

T {v’(k) In v’(k) + [l -

d(k)] ln[l

-

y’(k)lI.

(4.3 1)

It is important to remember that all of the above relationships have been deduced directly from microscopic theory. On the other hand, they can also be used as the basis for a semiphenomenological theory. At first glance, the equations given above appear to be exactly the equations of Landau’s theory of a Fermi liquidi). However, in the Landau theory one deals with dynamical quasiparticles. These quasiparticles have finite life times, and the quasiparticle energies therefore have an imaginary

190

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E. R. TUTTLE

part. As a result, the Landau theory holds only for liquids at low temperatures (where collision frequencies between quasiparticles are small) and only for quasiparticle states in the neighborhood of the Fermi surface. Under these conditions one can assume that the dynamical quasiparticles have a fairly well-defined momentum; i.e., they will occupy a given state for a relatively long time. In contrast to this, the microscopic theory we are using is an equilibrium theory, and all the quantities appearing above are “thermally” or “statistically” averaged functions. In equilibrium, because of detailed balance, as many quasiparticles are scattered into a particular state as are scattered out of it. We are no longer looking at a dynamical quasiparticle in a particular state but at a statistical quasiparticle in a particular state. Thus the statistical quasiparticle energy is a real function, and the concept of quasiparticle life times is not relevant. Furthermore, the basic relationships given above [cf. eqs. (a)-(g)] hold f or all states of the system and for all temperatures. (While we have derived eqs. (a)-(g) for all orders only in the “polarization graph” approximation, they have been shown to be correct through third order when all terms in the primed master polarization graphs are includedsJ> 26). They have also been established to all orders in the perturbation theory limit 337 27)) and are probably correct in general.) From eqs. (a)-(g) and the discussion given above, we can now view a dense Fermi liquid as composed of a gas of interacting “statistical” quasiparticles, whose momentum distribution, Y’(K), has the same functional form as the free Fermi distribution, with the quasiparticle energy oh replacing the singleparticle energy ok of the free Fermi case. The total number of these quasiparticles is equal to the number of particles in the system, as shown by eq. (b). The entropy (S) has the same form as the entropy of a free Fermi gas [eq. (g)]. Thus, the states of the quasiparticles system are in one-to-one correspondence with the states of a free Fermi gas. The form of the grand potential [eqifc)] has been discussed earlier in section 2. As noted there, the first three terms in eq. (c) have analogs in the free Fermi-gas system, while the term !X&!?, g, Q) contains most of the interaction. Furthermore, the grand potential is stationary with respect to changes in the distribution function y’(k) [eq. (d)], in agreement with the conclusions reached by Bloch and de Dominicis6). Finally, the quasiparticle energy wit is simply the variational derivative of the energy with respect to the distribution function v’(k), as one would expect. We should note, however, that this is the energy of a “statistical” quasiparticle with momentum k, and is not equal to the “dynamical” quasiparticle energy, as determined from the poles of the single-particle Green function. It is, none the less, well defined, and, in the perturbationtheory limit, the prescription which we have used to determine wit is in agreement with that used by de Dominiciszs).

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191

In this paper, we have derived expressions for the thermodynamic properties of a dense Fermi liquid from an exact microscopic theory of manybody systems. As observed above, the results have forms very similar to those postulated by Landau, and this similarity has enabled us to develop a quasiparticle picture of such systems. We have seen that the liquid behaves as a gas of interacting quasiparticles whose states are in one-to-one correspondence with the states of an ideal gas. Furthermore, these quasiparticles have a basically “statistical” nature; one either may consider them to have an infinite life time or may interpret all results obtained as relating to quasithermal averages ; e.g., d(k) is the average number of “dynamical” particles in the state K. This absence of all time dependence on the microscopic level arises because we consider only the averaged occupation numbers of the quasiparticle states and are not concerned with the transition probabilities between the states. forms of the distribution function, In this theory, the “Landau-like” entropy, etc., hold at all temperatures, as opposed to the usual theories involving “dynamical” quasiparticles, which are limited to very low temperatures where only those states close to the Fermi surface are important. Similar functional forms have been obtained by Bloch and de Dominicis in the perturbation-theory limit; we have shown that many of the features of their work can be extended into the region of high densities, low temperatures, and strong interactions, where perturbation theory is no longer applicable. The authors would like to thank the National Acknowledgements. Aeronautics and Space Administration for its support of this research through N.A.S.A. Grant NsG-618; and, also, the National Science Foundation for its support of this research. One author (L.E.R.) would like to thank the Universite Libre de Bruxelles, Chimie Physique II for its hospitality during the summer of 1970.

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