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Microscopic theory of a dense Fermi liquid at low temperatures
Physica
52 (1971) 193-204 o North-Holland
MICROSCOPIC
THEORY AT LOW
Publish&
Co.
OF A DENSE
FERMI
LIQUID
TEMPERATURES
II. SPIN AND DENSITY
FLUCTUATIONS
L. E. REICHLt Center for Statistical Mechanics and Thermodynamics. University of Texas at Austin, Austin, Texas 78712, USA
and E. R. TUTTLE University
of Denver,
Denver, Colorado 80210, USA
Received 13 October 1970
synopsis The theory of a dense low-temperature Fermi liquid which was developed in an earlier paper is applied to the study of spin and density fluctuations. Under certain conditions, the grand potential takes on a particularly simple form, which contains terms characteristic of spin and density fluctuations as well as terms of the type usually obtained in the Landau theory. A simple model calculation indicates that the approximations made in obtaining the grand potential are probably justified for a system similar to liquid sHe. All quantities appearing in the theory are directly calculable from the microscopic properties of constituent particles in the liquid.
1. Introduction. In a previous paper, we developed a microscopic theory of a dense low-temperature Fermi liquidr) * and showed that a self-consistent picture of the system as a gas of interacting, stable, quasiparticles could be deduced from the theory. In that work, an expression for the grand potential was derived which contained only quantities directly calculable from the two-particle interaction; this result was obtained by considering only those diagrams (primed polarization O-graphs) of greatest importance for this particular system. The results of recent experimental and theoretical work on liquid sHe, indicate that spin fluctuations should have an important effect on the grand potentials). We therefore wish to examine the results obtained in RTI to see that such effects have indeed been included. t Work based, in part, on a dissertation submitted to the University of Denver in partial fulfilment of requirements for Ph. D. in Physics, 1969. * Hereafter referred to as RTI. 193
L.
194
REICHL AND E. R. TUTTLE
I?.
In section 2, we show that if certain assumptions are made about the quasiparticle interaction in a dense Fermi system, the primed polarization O-graphs may be summed explicitly. The resulting expression for the grand potential then contains a term of exactly the form previously obtained by Brinkman and Engelsbergs) and shown by them to contribute a T3 In T dependence to the specific heat. In section 3, we discuss briefly the results of a rough model calculation of the quasiparticle interaction. It is found that, on the whole, the approximations made in section 2 appear to be justified for a system similar to liquid sHe. 2. Summation over polarization O-graphs. In a previous paper (RTI), we made a formal study of the properties of a dense Fermi liquid at very low temperatures, and obtained explicit expressions for the grand potential, the momentum distribution, and the quasiparticle energy. We then saw that the general form of these quantities enabled us to evolve a self-consistent quasiparticle picture of the system. In this paper, we concentrate upon the grand potential. This function can, as we have seen, be written in the form
Qf (B,g*Q)P -
z
=
/% T
y’(k)
-
B z
In y’(k) + [l -
{v’(k)
+ Q&Q*
y’(k)
Wk
y’(k)] ln[l - Y’(K)]} (1.2.18)
g> Q),
where QG(BJ
g, Q) = -,
QG,,(B>
(1.3.1)
g, Q)
from the &h-order polarization and QFb?,?,(P, g, Q) is the contribution O-graph. As was shown in RTI, these graphs can be explicitly evaluated giving 52%
I(/% g, Q)
=
f
k5 Y’(h) 1a
y’(h)
g1(W2
(1.3.7)
I k&2)
and QG,,(B>
(--lP g, Q) = 7
A
x
gl(k3&
1h4k5)
x
R(EZ ) kl,
c
:
kl...kzn
. . . gl@27&-lk2
K2) R(EZ,
K3,
K4)
g1(hk4
I k2k3)
l=--(x) 1K2&1) . . * +z,
h-1,
Kzn),
(1.3.17)
the function R( .Q, kt, k~) has been defined in eq. (1.3.19), and the vertex function gi(Kiks )k&4 is defined and discussed in the appendix.
MICROSCOPIC
In this section,
THEORY
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195
of paramagnetic
systems, and use the properties of the vertex function to obtain an approximation to the sum over polarization O-graphs indicated above. We first observe that the vertex function gi(kika 1k&d) conserves momentum, this can be indicated explicitly by writing gi(kiks 1k&4) in the form g1(W2
I k3k4)
=
L 1 - (2x)3 52
2
6(3)(h
+
k2 -
k3 -
k4) g”l(h2,
K12 I h34),
(2.1)
where the relative momentum kis and the center of mass momentum Kis are defined as kis = &(ka - ki) and Kis = kl + kz, respectively. If we now introduce the momentum transfer q = kl -
k2 = k3 -
k4 = . . . = k2n-l - kgn
and use the momentum conserving 6 functions to perform the summations over even-numbered momenta in eq. (1.3.17), then QFb,,(@, g, Q) becomes (-1)”
QF&,(B, g, q =
D12=-m *
(2x)3
7
-
?z
01; k3-q,
[
x
&(kl,
x
gl(k3,~3;k5-q,u6Ik3-q,~4;k5,(T5)
X
... x
gl(kzn-1,
dkl-q,a2;
~2n-1;
h-q,
rc
$
c
k3,as)
~2 I k2n-1
kl, ~1)
- q, u2n;
(2.2)
where
vb(kl - q) - &(kd
R,,u, (EZ,q, kl) = EZ +
&(kl - q) - dl(kd *
In a homogeneous, paramagnetic system, both the quasiparticle distribution function and the quasiparticle energy are independent of spin in the absence of a magnetic field, and the quasiparticle energy depends only on the magnitude of the momentum. As a result, the only quantities in eq. (2.2) which depend upon the spin are the reduced vertex functions g”i(ki, ~1; k3 -
q,
04 I kl
-
q,
02; k3> ~3).
If the two-particle interaction is central and spin independent, then spin dependence of this function is of a particularly simple form : g’l(h2,
Kl2
I h34)
&
=
-,
(2L + 1)
the
1 @12’$34)
~12
L
x
[&*b,&r,b,
-
(-
1)L~L-104~~*631 hL(h2,
K12
lk34).
(2.4)
L. E. REICHL
196
AND
E. R. TUTTLE
The function g”i,~(kis, Kis I&) can be determined from the two-particle wavefunction, as discussed in the appendix. In this case, the spin summations indicated in eq. (2.2) are easily performed, giving
x R(Ez, q, k3) . .. Wz >q, km-l) [gs(h, k3 - q I kl - q, k3)
x gs(ks,kc,- q Ik3 - q, kci).. . g&n-l,
- q (kzn-1 -
q, kl)
k3- q Ih - q, k3)ga(ks,kci- q 1k3 - q, kii)
+ %A X
kl
. . . x gab-l,
h - q Ikm-1 - q, kl),
(2.5)
where g,(ki, kz
x
Ihr h) = 2
-&l,L(~lPi
j.
(2~5+ 1112- (- 1Fl h(bW
Kl2jk34)
(2.6)
and g,(ki, kz I h, h)
= - &-g,,(ZL
+ l)(+h(k2'~34)
(2.7) In general, the function gi,~(kis, Ki2 I k34) is extremely complicated. However, it is possible to obtain some idea of its general behavior from a model calculation for a relatively simple potential function which roughly approximates that between two sHe atoms (cf. section 3). We shall find that g(a)s(kl, kz - q 1kl - q, k2) is fairly sharply peaked about q = 0, indicating that the long-range part of the quasiparticle interactions is relatively important. This property, when combined with the behavior of the function R(EI, q, k) effectively restricts the momenta kl, k3, . . . . kzn-l to the neighborhood of the Fermi surface at low temperature. Furthermore, we assume that glajs(kl, k2 - q) kl - q, kz) is a slowly varying function of ,. ” cos 81s = kl.k2. If this is so, it is convenient Legendre series of the form g(a)&,
k2 -
q I k1-
to expand
q, k2) =
&@“(h,
gtajs(kl, k2 -
k2, q) h(cos
In this expansion, we should note that each coefficient,
q, kz) in a
q (kl -
yg”(ki,
f312).
(2.8)
ks, q) should
be peaked about zero momentum waves [r,~(Kis, g(,),(ki,
KisIks4)
transfer
and depends upon all the partial
[cf. eqs. (2.6) and (2.7)].
ks - q I kl
-
Thus, if
q, kz)
varies slowly as a function of cos 012, the first term in eq. (2.8) dominates the series, and we can neglect the remaining terms. The expansion of eq. (2.8) is effectively the same as that made by Landau, and the coefficients y(L&‘*(kl,k2, q) can be easily related to the Landau phenomenological parameters FL and 2~4). It is of interest to note that Landau’s assumption of the importance of the first term (L = 0) of this series is at least partially justified by our model calculation (cf. section 3). We also observe that it is possible, at least in principle, to calculate the coefficients y$“(kl, k2, q) directly from the microscopic He-He potential function. We now substitute eq. (2.8) into eq. (2.5), retaining only the L = 0 term; this yields the following result for QFb,,(/l, g, ii?):
(--lP :
%%(Ba g,Q) = --yX
{[2#(kF,
c [x(q’,El)]”
l=-ca
It
@I”
+
p
3[2Y?‘@F,
Cm3
(2.9)
where x(q, El) =
PJ3 sz
v’(k - q) - v’(k) ?
EZ+ w’(k - q) -
(2.10)
w’(k)
and the momenta kt, kj in the Legendre coefficient have been restricted to the Fermi surface. If desired, of course, higher terms in eq. (2.8) can be retained; these, however, are expected to be small, and merely complicate the result. If we now combine eqs. (1.3. l), (1.4.17), and (2.9), and perform the sum over n indicated in eq. (1.3. l), we finally obtain the following expression for the grand potential:
szf(j9,g, Q) = rBg
B z
Y’(k)
Wk +
-
i +ti,
5
{[2~r’(kFj
-
i ,-,
5
{[ln(l
+
3 [ln(l
+
;
&@‘@F,
& 1
+
y’(h)
@I2
+t’(~F,
q)
x(q,
- [1 - v’(k)] ln[l - y'(k)13
y’(k2)
+
g1(h
k2 I h,
3[2d?(kF,
!I’) X(!L
~2)) -
AZ)
!d12)[X(%
6~))
2y?‘(kF,
-
?‘t’(kF,
!I) x(Q,
a2
!I’) X(%
Q)I).
El)]
(2.11)
L. E. REICHL
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The first three terms in this expression have already been discussed (cf. RTI). The next two are of the type usually appearing in the theory of a Fermi liquid, and contribute terms proportional to T, T3, etc. to the specific heat, in agreement with the usual theory of a Fermi liquid. The last term, however, is of exactly the same form as that found by Brinkman and Engelsbergs) in their study of spin and density fluctuations. Thus at low temperatures, where the momenta are, to good approximation, restricted to the Fermi surface, this term will yield a T3 In T contribution to the specific heat, in addition to other terms of the expected type. This becomes especially clear if we note that, in the vicinity of the Fermi sphere, the quasiparticle energy is given approximately by 4c=
!Vk2 ~2m* -
(2.12)
44,
where m* is the density-dependent quasiparticle effective mass, and u(p) is a density-dependent effective potential. At low temperatures, the chemical potential, g, is approximately equal to co&, so that the effective potential z+(p) cancels out of the distribution functions and energy denominators of x(q, EZ), and the temperature dependence of the last term becomes identical to that of the expression obtained by Brinkman and Engelsberg. In this case, however, the expression (2.11) has been derived directly from an exact microscopic theory of a Fermi liquid and contains only quantities which can be calculated from the interaction between particles in the liquid. 3. Model calculation. of gl(klkz 1k3k4). In arriving at eq. (2.1 l), we have made a number of assumptions about the behavior of the vertex function gl(klkz 1k3k4). In this section, we shall look at the results of a simple model calculation in an attempt to find out to what extent these assumptions are justified. We shall be particularly interested in the possible application of the theory to liquid sHe. As we have observed in the appendix, the function gl(klk2 / k3k4) can be found analytically if the radial wave equation can be solved analytically. One of the simplest potential functions for which this can be done, and which contains at least the gross features of the sHe-aHe interaction is the following: -fWo V(r) = I -vo I O
Olr
b < r,
i.e., a finite repulsive core surrounded by a constant attractive region. The explicit expression for gl,L(klz, K12 1k34) obtained for this potential has been given elsewheres), and will therefore not be repeated here. The parameters a, b, WO and Ire were chosen to approximate the aHe interatomic potential as closely as possible and calculations, to first order
MICROSCOPIC
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in gi(Kika 1Klkz), were made of the average energy per particle, the chemical potential, and the equilibrium density. It was found that the long-range part of the interaction could not be neglected, an additional contribution proportional to r-6 was added as a perturbation in the region Y > b5). With this approximation, the binding energy per particle was found to be approximately 1 K, the particle density approximately 0.9 pexp, and the effective mass approximately 1.5 maHe. These results are as good as can be expected considering that all higher-order terms in these quantities have been neglected. It was found that the results were extremely insensitive to variations of b (the square well diameter) and c, the coefficient of the r-6 term, provided the overall strength of the attraction was kept constant in these variations by simultaneously varying I/O. A similar insensitivity was found to variations of the repulsive core diameter, provided the repulsive strength Woas was kept constant. From these observations one can conclude that the detailed structure of the interaction is comparatively unimportant, as long as the gross features of the potential are included and the overall strengths of the attractive and repulsive parts of the potential are preserved. This observation is borne out to some extent by the work of OstgaardG), who obtains similar numerical results using a very different method. Complete second-order calculations have not been carried out, but in preliminary calculations, the second-order terms appear to be smaller than those of first order indicating that the function gi(Kiks/K&) is indeed a reasonable expansion function. Detailed calculations with this model were made of the function gi,L(Kis, Kis 1K34) to investigate the relative magnitudes of the diagonal and off-diagonal terms. It was found that g”i,~(kis, Kis I&) has a maximum when k34 = k12 and that it drops off fairly rapidly as one departs from the diagonal. This is particularly noticeable for the higher partial waves, but even for L = 0 and L = 1, one typically finds that g”i,(e,l)(klz, Klz 1k12/2) m &cl, (0, ~(klz, Klz /klz). One therefore can conclude the processes with small momentum transfer (klz m k34) dominate the behavior of the system. Upon restricting our consideration to the diagonal terms, we find that for the potential we are considering, S-wave contribution to gl(klkz j klkz) is repulsive, while all higher partial-wave contributions are attractive. Furthermore, the S- and P-wave contributions are of about the same magnitude in the region of consideration; higher-order contributions fall off rapidly as a function of the angular momentum. Because of this cancellation of the S- and P-wave terms, the function gl(klkz 1klkz) is dominated by cl, z(klz, K12 1klz). In contrast to this, in the calculation of ga(klkz /klkz) the S- and P-wave terms have the same sign and reinforce each other. We can conclude from this that the overall signs of the quantities -$$(kF, q) x(g, Ed) and -2yt(kF, q) x(q’, ~1) will be agreement with the sign obtained experi-
L. E. REICHL
200
AND
E. R. TUTTLE
mentally for the Landau parameters 20 and Fad). However, our calculations are not conclusive enough at this point to make a numerical estimate of these quantities. Finally, in order to obtain some idea of the relative importance of the coefficients yp’*(K~, q) in the Legendre expansion [cf. eq. (2.8)] for the functions g”(,,,(kiks 1klk~) we can set the magnitudes of ki and kz equal to the Fermi momentum and plot g’(a)s(klk2/klkz) as a function of cos 012, where 131sis the angle between kl and k2. We find that g(,),(klkz 1klkz) is a slowly varying function of cos 1312. From this we conclude that, in the Legendre (a)s(kF, q) is definitely the most expansion given by eq. (5.8) the coefficient yO important. The above conclusions have been verified only for the particular model described above. However, the insensitivity of the results to the fine details of the two-body interaction makes it reasonable to assume that the qualitative behavior of the function gl(klk2 )k3k4) will not be greatly changed if a more accurate approximation to the actual sHesHe interaction is used. It therefore appears that the approximations made in obtaining eq. (2.11) should be relatively model independent. 4. Discussion. In this paper, we have shown that the existence of spin and density fluctuations is consistent with the statistical quasiparticle theory of a dense Fermi liquid derived in RTI. From a study of the vertex function, gl(klk2 I k3k4), we have been able to obtain a simple approximation to the grand potential. We find that the grand potential contains terms of the Landau type which give contributions to the specific heat proportional to T, T3 etc., and that it contains a term which is of the same form as that considered by Brinkman and Engelsberg in their model calculation of the effects of spin and density fluctuations on low-temperature Fermi systems. In our case, however, all quantities are directly calculable from the microscopic properties of the constituent atoms. Although detailed calculations have not yet been carried out using this theory, rough preliminary calculations indicate that various contributions to the thermodynamic properties of liquid sHe do have the correct signs and are probably of the same order of magnitude as values obtained experimentally for the liquid. In previous work on spin and density fluctuations it has been customary to divide the entropy into “Landau” type and “non-Landau” type contributions. The “Landau” type contribution has the form
= -kg
F {d(k) In(k) + [l -
where kg is Boltzmann’s momentum distribution.
v’(k)] ln[l
-
v’(k)]},
(4.1)
constant, and v’(k) is the dynamical quasiparticle This term represents the entropy obtained if the
MICROSCOPIC
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OF A DENSE
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LIQUID.
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201
states of the interacting systems are in one-to-one correspondence with the states of a free Fermi gas; at extremely low tempertures this term dominates the entropy. It is found that the contribution associated with spin and density fluctuations is contained in the “non-Landau” terms of the entropy. In the present work (cf. RTI), we have found that the entropy has exactly the functional form given in eq. (3.1) but with the statistical quasiparticle momentum distribution. In this case, the quasiparticle energy is chosen differently and only in the zero-temperature limit it is equal to the quasiparticle energy determined from the poles of the single-particle Green function. Nevertheless, the contributions of spin and density fluctuations to the thermodynamic properties of the system are easily identifiable, as we saw in section 2. Thus the functional form of the entropy given in eq. (4.1) is in no way incompatible with the existence of such fluctuations, provided the quasiparticle energy is chosen appropriately. Recently, it has been shown by JonesT) that the structure of the vertex function, gi(kik2 (k3K4) is not unique; i.e., there exists a family of different possible vertex functions which can appear in the primed master graphs. While this arbitrariness in the vertex function does not change the formal structure of the results in this paper, it could affect numerical calculations. Thus, before undertaking detailed calculations, it will be necessary to investigate the properties of each possible vertex function to determine which is most suitable for liquid sHe. It is hoped that in the near future this can be done and that complete numerical calculations of liquid sHe can be undertaken and compared with experiment. Acknowledgements. The authors would like to thank the National 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 UniversitC Libre de Bruxelles, Chimie Physique II for its hospitality during the summer of 1970.
APPENDIX
In this appendix, we define the vertex function gi(kiks /k&4) which is used as an expansion parameter in the theory developed here. Of necessity, the exposition given here is greatly abbreviated; more details can be found in the original papers59 8). The function gi(Kiks 1k&4) can be written in terms of the matrix elements between free-particle states of an operator
er:
g1(M21k3K4)
=
-<*
+
<~4~31Glw2>*,
(A.1)
L. E. REICHL
202
where, if As is the two-particle
AND E. R. TUTTLE
hamiltonian,
(A4 The operator 8’ acts on the states to the right to give the sum of the quasiparticle energies characteristic of those states, and P denotes Cauchy principal part. The operator Gi is clearly not hermitean due to its dependence on 8’. Furthermore, it is easy to see from eq. (A.2) that gi(kiksjk&) is well defined for all potentials and does not necessarily become large as the potential V increases. We therefore expect that an expansion in powers of gi(Kiks 1K&a) will converge more rapidly than the perturbation expansion, particularly for strong interactions. It has been shown that there is an easier way to determine gi(Kiks 1K&4) than that of taking the matrix elements of the operator ei directly. In fact, in this alternative approach, the function gi(Kiks lksk4) can be evaluated analytically if the two-body wavefunctions for the interaction chosen can be found analytically. For a spin-independent, two-body, central potential the function gi(Kiks 1K&Q) can be written in the following form:
gl(hkzl k3k4) =
dt3)(kl +
k2 -
k3 -
k4)
[
where h(M21k3k4j
=
&
-,
(2L +
(24 3 1
-sz
g"l(hk21~3~4)>
(A.3)
1) &2'i34)
L
’
1 ~ kiz
&(kis,
K12lk34)C&,o,&,o,
-
(-l)L
~~,d~,cd.
(A4
In this equation, k12 is the relative momentum, kl2 = +(kz - kl) ; Klz is the total momentum, Kis = kl + k 2 ; klz = IklzJ ; and (~1 is the .z component of spin of particle 1. We observe from eq. (A.3) that the function gl(klkz 1k3k4) conserves momentum; this property of the function has been used in performing some of the momentum integrations in section 2. Also, the entire spin dependence of gl(klkz 1k3k4) is contained in the term in the second line of eq. (A.4), i.e., gl,L(kls, Kl2) k34) is spin independent. Finally, gl,L(klp, Kl2 / k34) can be written in terms of the reaction matrix : h,L(km
K12
I h4) = cm2&L(h) ca
2@k12 +-
xm
s
dko
lAtL)I ko> cm2 &(ko)
0
1 Ok,
+
Ok,
-
WO
(A.5)
MICROSCOPIC
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OF A DENSE
FERMI
where dL(k) is the phase shift for the Lth partial w. = @Kf,/4m
LIQUID.
II
203
wave, and
+ @k~/m.
(If the two-body potential being considered possesses bound states, then another term must be added. However, as we are primarily interested in the term has been application of the theory to liquid sHe this additional omitted.) The reaction matrix is defined in terms of the two-body radial wavefunction through the following equation :
f
s
drFL(kr)
U(r) (rl koL>,
(Ad
0
where U(r) is the reduced potential, U(r) = mV(r)/hs and FL(kr) is the rationalized spherical Bessel function, defined as FL(~) = pit(p). The normalization of the radial wavefunction is chosen such that its asymptotic form is
zoo
sin
koR -
$
1.
+ BL(ko)
The phase shift dL(k) can be determined A matrix through the equation
from the diagonal
elements
(k IA(h)/ k) = tan SL(k).
of the
(A7
We thus see that the function gl(klkz I k3k4) is, at least in principle, calculable for any given two-body interaction. The function fz(klk2 I kskc )k 3k 4) can also be written in terms of the A matrix. It is related to gl(klkz )k3k4) through the equation
gl(hkz I k&4) = fl(hkz I k&4) + C fz(Wz I k&s I k&4) kske
1
1 Wk, + mfi* -
Uk, -
Oka
oh1 + mi, -
ok,
-
mk,
where the first and second terms of this equation correspond, through eqs. (A.3) and (A.4), to the first and second terms, respectively, of eq. (A.5). Calculations of t~,~(k~s, Klz 1k34) have been made for several different potential functions 5,s). The results of one such calculation, which provides some justification for the approximations made in this paper, are discussed in section 3.
204
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II
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9)
Reichl, L. E. and Tuttle, E. R., Physica 52 (1971) 165. For a brief summary of this work, we refer to the introduction in RTI. Brinkman, W. F. and Engelsberg, S., Phys. Rev. 169 (1968) 417. Wheatley, J. C., in Quantum Fluids, edited by D. F. Brewer, North-Holland Co. (Amsterdam, 1966). Tuttle, E. R., thesis, Univ. of Colorado, 1964 (unpublished). Ostgaard, E., Phys. Rev. 170 (1968) 257. Jones, R. W., thesis, Univ. of Colorado, 1970 (unpublished). Mohling, F., Phys. Rev. 122 (1961) 1062; 124 (1961) 583. Mohling, F., Phys. Rev. 128 (1962) 1365.
Publ.
52 (1971) 193-204 o North-Holland
MICROSCOPIC
THEORY AT LOW
Publish&
Co.
OF A DENSE
FERMI
LIQUID
TEMPERATURES
II. SPIN AND DENSITY
FLUCTUATIONS
L. E. REICHLt Center for Statistical Mechanics and Thermodynamics. University of Texas at Austin, Austin, Texas 78712, USA
and E. R. TUTTLE University
of Denver,
Denver, Colorado 80210, USA
Received 13 October 1970
synopsis The theory of a dense low-temperature Fermi liquid which was developed in an earlier paper is applied to the study of spin and density fluctuations. Under certain conditions, the grand potential takes on a particularly simple form, which contains terms characteristic of spin and density fluctuations as well as terms of the type usually obtained in the Landau theory. A simple model calculation indicates that the approximations made in obtaining the grand potential are probably justified for a system similar to liquid sHe. All quantities appearing in the theory are directly calculable from the microscopic properties of constituent particles in the liquid.
1. Introduction. In a previous paper, we developed a microscopic theory of a dense low-temperature Fermi liquidr) * and showed that a self-consistent picture of the system as a gas of interacting, stable, quasiparticles could be deduced from the theory. In that work, an expression for the grand potential was derived which contained only quantities directly calculable from the two-particle interaction; this result was obtained by considering only those diagrams (primed polarization O-graphs) of greatest importance for this particular system. The results of recent experimental and theoretical work on liquid sHe, indicate that spin fluctuations should have an important effect on the grand potentials). We therefore wish to examine the results obtained in RTI to see that such effects have indeed been included. t Work based, in part, on a dissertation submitted to the University of Denver in partial fulfilment of requirements for Ph. D. in Physics, 1969. * Hereafter referred to as RTI. 193
L.
194
REICHL AND E. R. TUTTLE
I?.
In section 2, we show that if certain assumptions are made about the quasiparticle interaction in a dense Fermi system, the primed polarization O-graphs may be summed explicitly. The resulting expression for the grand potential then contains a term of exactly the form previously obtained by Brinkman and Engelsbergs) and shown by them to contribute a T3 In T dependence to the specific heat. In section 3, we discuss briefly the results of a rough model calculation of the quasiparticle interaction. It is found that, on the whole, the approximations made in section 2 appear to be justified for a system similar to liquid sHe. 2. Summation over polarization O-graphs. In a previous paper (RTI), we made a formal study of the properties of a dense Fermi liquid at very low temperatures, and obtained explicit expressions for the grand potential, the momentum distribution, and the quasiparticle energy. We then saw that the general form of these quantities enabled us to evolve a self-consistent quasiparticle picture of the system. In this paper, we concentrate upon the grand potential. This function can, as we have seen, be written in the form
Qf (B,g*Q)P -
z
=
/% T
y’(k)
-
B z
In y’(k) + [l -
{v’(k)
+ Q&Q*
y’(k)
Wk
y’(k)] ln[l - Y’(K)]} (1.2.18)
g> Q),
where QG(BJ
g, Q) = -,
QG,,(B>
(1.3.1)
g, Q)
from the &h-order polarization and QFb?,?,(P, g, Q) is the contribution O-graph. As was shown in RTI, these graphs can be explicitly evaluated giving 52%
I(/% g, Q)
=
f
k5 Y’(h) 1a
y’(h)
g1(W2
(1.3.7)
I k&2)
and QG,,(B>
(--lP g, Q) = 7
A
x
gl(k3&
1h4k5)
x
R(EZ ) kl,
c
:
kl...kzn
. . . gl@27&-lk2
K2) R(EZ,
K3,
K4)
g1(hk4
I k2k3)
l=--(x) 1K2&1) . . * +z,
h-1,
Kzn),
(1.3.17)
the function R( .Q, kt, k~) has been defined in eq. (1.3.19), and the vertex function gi(Kiks )k&4 is defined and discussed in the appendix.
MICROSCOPIC
In this section,
THEORY
we restrict
OF A DENSE
ourselves
FERMI
LIQUID.
to the consideration
II
195
of paramagnetic
systems, and use the properties of the vertex function to obtain an approximation to the sum over polarization O-graphs indicated above. We first observe that the vertex function gi(kika 1k&d) conserves momentum, this can be indicated explicitly by writing gi(kiks 1k&4) in the form g1(W2
I k3k4)
=
L 1 - (2x)3 52
2
6(3)(h
+
k2 -
k3 -
k4) g”l(h2,
K12 I h34),
(2.1)
where the relative momentum kis and the center of mass momentum Kis are defined as kis = &(ka - ki) and Kis = kl + kz, respectively. If we now introduce the momentum transfer q = kl -
k2 = k3 -
k4 = . . . = k2n-l - kgn
and use the momentum conserving 6 functions to perform the summations over even-numbered momenta in eq. (1.3.17), then QFb,,(@, g, Q) becomes (-1)”
QF&,(B, g, q =
D12=-m *
(2x)3
7
-
?z
01; k3-q,
[
x
&(kl,
x
gl(k3,~3;k5-q,u6Ik3-q,~4;k5,(T5)
X
... x
gl(kzn-1,
dkl-q,a2;
~2n-1;
h-q,
rc
$
c
k3,as)
~2 I k2n-1
kl, ~1)
- q, u2n;
(2.2)
where
vb(kl - q) - &(kd
R,,u, (EZ,q, kl) = EZ +
&(kl - q) - dl(kd *
In a homogeneous, paramagnetic system, both the quasiparticle distribution function and the quasiparticle energy are independent of spin in the absence of a magnetic field, and the quasiparticle energy depends only on the magnitude of the momentum. As a result, the only quantities in eq. (2.2) which depend upon the spin are the reduced vertex functions g”i(ki, ~1; k3 -
q,
04 I kl
-
q,
02; k3> ~3).
If the two-particle interaction is central and spin independent, then spin dependence of this function is of a particularly simple form : g’l(h2,
Kl2
I h34)
&
=
-,
(2L + 1)
the
1 @12’$34)
~12
L
x
[&*b,&r,b,
-
(-
1)L~L-104~~*631 hL(h2,
K12
lk34).
(2.4)
L. E. REICHL
196
AND
E. R. TUTTLE
The function g”i,~(kis, Kis I&) can be determined from the two-particle wavefunction, as discussed in the appendix. In this case, the spin summations indicated in eq. (2.2) are easily performed, giving
x R(Ez, q, k3) . .. Wz >q, km-l) [gs(h, k3 - q I kl - q, k3)
x gs(ks,kc,- q Ik3 - q, kci).. . g&n-l,
- q (kzn-1 -
q, kl)
k3- q Ih - q, k3)ga(ks,kci- q 1k3 - q, kii)
+ %A X
kl
. . . x gab-l,
h - q Ikm-1 - q, kl),
(2.5)
where g,(ki, kz
x
Ihr h) = 2
-&l,L(~lPi
j.
(2~5+ 1112- (- 1Fl h(bW
Kl2jk34)
(2.6)
and g,(ki, kz I h, h)
= - &-g,,(ZL
+ l)(+h(k2'~34)
(2.7) In general, the function gi,~(kis, Ki2 I k34) is extremely complicated. However, it is possible to obtain some idea of its general behavior from a model calculation for a relatively simple potential function which roughly approximates that between two sHe atoms (cf. section 3). We shall find that g(a)s(kl, kz - q 1kl - q, k2) is fairly sharply peaked about q = 0, indicating that the long-range part of the quasiparticle interactions is relatively important. This property, when combined with the behavior of the function R(EI, q, k) effectively restricts the momenta kl, k3, . . . . kzn-l to the neighborhood of the Fermi surface at low temperature. Furthermore, we assume that glajs(kl, k2 - q) kl - q, kz) is a slowly varying function of ,. ” cos 81s = kl.k2. If this is so, it is convenient Legendre series of the form g(a)&,
k2 -
q I k1-
to expand
q, k2) =
&@“(h,
gtajs(kl, k2 -
k2, q) h(cos
In this expansion, we should note that each coefficient,
q, kz) in a
q (kl -
yg”(ki,
f312).
(2.8)
ks, q) should
be peaked about zero momentum waves [r,~(Kis, g(,),(ki,
KisIks4)
transfer
and depends upon all the partial
[cf. eqs. (2.6) and (2.7)].
ks - q I kl
-
Thus, if
q, kz)
varies slowly as a function of cos 012, the first term in eq. (2.8) dominates the series, and we can neglect the remaining terms. The expansion of eq. (2.8) is effectively the same as that made by Landau, and the coefficients y(L&‘*(kl,k2, q) can be easily related to the Landau phenomenological parameters FL and 2~4). It is of interest to note that Landau’s assumption of the importance of the first term (L = 0) of this series is at least partially justified by our model calculation (cf. section 3). We also observe that it is possible, at least in principle, to calculate the coefficients y$“(kl, k2, q) directly from the microscopic He-He potential function. We now substitute eq. (2.8) into eq. (2.5), retaining only the L = 0 term; this yields the following result for QFb,,(/l, g, ii?):
(--lP :
%%(Ba g,Q) = --yX
{[2#(kF,
c [x(q’,El)]”
l=-ca
It
@I”
+
p
3[2Y?‘@F,
Cm3
(2.9)
where x(q, El) =
PJ3 sz
v’(k - q) - v’(k) ?
EZ+ w’(k - q) -
(2.10)
w’(k)
and the momenta kt, kj in the Legendre coefficient have been restricted to the Fermi surface. If desired, of course, higher terms in eq. (2.8) can be retained; these, however, are expected to be small, and merely complicate the result. If we now combine eqs. (1.3. l), (1.4.17), and (2.9), and perform the sum over n indicated in eq. (1.3. l), we finally obtain the following expression for the grand potential:
szf(j9,g, Q) = rBg
B z
Y’(k)
Wk +
-
i +ti,
5
{[2~r’(kFj
-
i ,-,
5
{[ln(l
+
3 [ln(l
+
;
&@‘@F,
& 1
+
y’(h)
@I2
+t’(~F,
q)
x(q,
- [1 - v’(k)] ln[l - y'(k)13
y’(k2)
+
g1(h
k2 I h,
3[2d?(kF,
!I’) X(!L
~2)) -
AZ)
!d12)[X(%
6~))
2y?‘(kF,
-
?‘t’(kF,
!I) x(Q,
a2
!I’) X(%
Q)I).
El)]
(2.11)
L. E. REICHL
198
AND
E. R. TUTTLE
The first three terms in this expression have already been discussed (cf. RTI). The next two are of the type usually appearing in the theory of a Fermi liquid, and contribute terms proportional to T, T3, etc. to the specific heat, in agreement with the usual theory of a Fermi liquid. The last term, however, is of exactly the same form as that found by Brinkman and Engelsbergs) in their study of spin and density fluctuations. Thus at low temperatures, where the momenta are, to good approximation, restricted to the Fermi surface, this term will yield a T3 In T contribution to the specific heat, in addition to other terms of the expected type. This becomes especially clear if we note that, in the vicinity of the Fermi sphere, the quasiparticle energy is given approximately by 4c=
!Vk2 ~2m* -
(2.12)
44,
where m* is the density-dependent quasiparticle effective mass, and u(p) is a density-dependent effective potential. At low temperatures, the chemical potential, g, is approximately equal to co&, so that the effective potential z+(p) cancels out of the distribution functions and energy denominators of x(q, EZ), and the temperature dependence of the last term becomes identical to that of the expression obtained by Brinkman and Engelsberg. In this case, however, the expression (2.11) has been derived directly from an exact microscopic theory of a Fermi liquid and contains only quantities which can be calculated from the interaction between particles in the liquid. 3. Model calculation. of gl(klkz 1k3k4). In arriving at eq. (2.1 l), we have made a number of assumptions about the behavior of the vertex function gl(klkz 1k3k4). In this section, we shall look at the results of a simple model calculation in an attempt to find out to what extent these assumptions are justified. We shall be particularly interested in the possible application of the theory to liquid sHe. As we have observed in the appendix, the function gl(klk2 / k3k4) can be found analytically if the radial wave equation can be solved analytically. One of the simplest potential functions for which this can be done, and which contains at least the gross features of the sHe-aHe interaction is the following: -fWo V(r) = I -vo I O
Olr
b < r,
i.e., a finite repulsive core surrounded by a constant attractive region. The explicit expression for gl,L(klz, K12 1k34) obtained for this potential has been given elsewheres), and will therefore not be repeated here. The parameters a, b, WO and Ire were chosen to approximate the aHe interatomic potential as closely as possible and calculations, to first order
MICROSCOPIC
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199
in gi(Kika 1Klkz), were made of the average energy per particle, the chemical potential, and the equilibrium density. It was found that the long-range part of the interaction could not be neglected, an additional contribution proportional to r-6 was added as a perturbation in the region Y > b5). With this approximation, the binding energy per particle was found to be approximately 1 K, the particle density approximately 0.9 pexp, and the effective mass approximately 1.5 maHe. These results are as good as can be expected considering that all higher-order terms in these quantities have been neglected. It was found that the results were extremely insensitive to variations of b (the square well diameter) and c, the coefficient of the r-6 term, provided the overall strength of the attraction was kept constant in these variations by simultaneously varying I/O. A similar insensitivity was found to variations of the repulsive core diameter, provided the repulsive strength Woas was kept constant. From these observations one can conclude that the detailed structure of the interaction is comparatively unimportant, as long as the gross features of the potential are included and the overall strengths of the attractive and repulsive parts of the potential are preserved. This observation is borne out to some extent by the work of OstgaardG), who obtains similar numerical results using a very different method. Complete second-order calculations have not been carried out, but in preliminary calculations, the second-order terms appear to be smaller than those of first order indicating that the function gi(Kiks/K&) is indeed a reasonable expansion function. Detailed calculations with this model were made of the function gi,L(Kis, Kis 1K34) to investigate the relative magnitudes of the diagonal and off-diagonal terms. It was found that g”i,~(kis, Kis I&) has a maximum when k34 = k12 and that it drops off fairly rapidly as one departs from the diagonal. This is particularly noticeable for the higher partial waves, but even for L = 0 and L = 1, one typically finds that g”i,(e,l)(klz, Klz 1k12/2) m &cl, (0, ~(klz, Klz /klz). One therefore can conclude the processes with small momentum transfer (klz m k34) dominate the behavior of the system. Upon restricting our consideration to the diagonal terms, we find that for the potential we are considering, S-wave contribution to gl(klkz j klkz) is repulsive, while all higher partial-wave contributions are attractive. Furthermore, the S- and P-wave contributions are of about the same magnitude in the region of consideration; higher-order contributions fall off rapidly as a function of the angular momentum. Because of this cancellation of the S- and P-wave terms, the function gl(klkz 1klkz) is dominated by cl, z(klz, K12 1klz). In contrast to this, in the calculation of ga(klkz /klkz) the S- and P-wave terms have the same sign and reinforce each other. We can conclude from this that the overall signs of the quantities -$$(kF, q) x(g, Ed) and -2yt(kF, q) x(q’, ~1) will be agreement with the sign obtained experi-
L. E. REICHL
200
AND
E. R. TUTTLE
mentally for the Landau parameters 20 and Fad). However, our calculations are not conclusive enough at this point to make a numerical estimate of these quantities. Finally, in order to obtain some idea of the relative importance of the coefficients yp’*(K~, q) in the Legendre expansion [cf. eq. (2.8)] for the functions g”(,,,(kiks 1klk~) we can set the magnitudes of ki and kz equal to the Fermi momentum and plot g’(a)s(klk2/klkz) as a function of cos 012, where 131sis the angle between kl and k2. We find that g(,),(klkz 1klkz) is a slowly varying function of cos 1312. From this we conclude that, in the Legendre (a)s(kF, q) is definitely the most expansion given by eq. (5.8) the coefficient yO important. The above conclusions have been verified only for the particular model described above. However, the insensitivity of the results to the fine details of the two-body interaction makes it reasonable to assume that the qualitative behavior of the function gl(klk2 )k3k4) will not be greatly changed if a more accurate approximation to the actual sHesHe interaction is used. It therefore appears that the approximations made in obtaining eq. (2.11) should be relatively model independent. 4. Discussion. In this paper, we have shown that the existence of spin and density fluctuations is consistent with the statistical quasiparticle theory of a dense Fermi liquid derived in RTI. From a study of the vertex function, gl(klk2 I k3k4), we have been able to obtain a simple approximation to the grand potential. We find that the grand potential contains terms of the Landau type which give contributions to the specific heat proportional to T, T3 etc., and that it contains a term which is of the same form as that considered by Brinkman and Engelsberg in their model calculation of the effects of spin and density fluctuations on low-temperature Fermi systems. In our case, however, all quantities are directly calculable from the microscopic properties of the constituent atoms. Although detailed calculations have not yet been carried out using this theory, rough preliminary calculations indicate that various contributions to the thermodynamic properties of liquid sHe do have the correct signs and are probably of the same order of magnitude as values obtained experimentally for the liquid. In previous work on spin and density fluctuations it has been customary to divide the entropy into “Landau” type and “non-Landau” type contributions. The “Landau” type contribution has the form
= -kg
F {d(k) In(k) + [l -
where kg is Boltzmann’s momentum distribution.
v’(k)] ln[l
-
v’(k)]},
(4.1)
constant, and v’(k) is the dynamical quasiparticle This term represents the entropy obtained if the
MICROSCOPIC
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OF A DENSE
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LIQUID.
II
201
states of the interacting systems are in one-to-one correspondence with the states of a free Fermi gas; at extremely low tempertures this term dominates the entropy. It is found that the contribution associated with spin and density fluctuations is contained in the “non-Landau” terms of the entropy. In the present work (cf. RTI), we have found that the entropy has exactly the functional form given in eq. (3.1) but with the statistical quasiparticle momentum distribution. In this case, the quasiparticle energy is chosen differently and only in the zero-temperature limit it is equal to the quasiparticle energy determined from the poles of the single-particle Green function. Nevertheless, the contributions of spin and density fluctuations to the thermodynamic properties of the system are easily identifiable, as we saw in section 2. Thus the functional form of the entropy given in eq. (4.1) is in no way incompatible with the existence of such fluctuations, provided the quasiparticle energy is chosen appropriately. Recently, it has been shown by JonesT) that the structure of the vertex function, gi(kik2 (k3K4) is not unique; i.e., there exists a family of different possible vertex functions which can appear in the primed master graphs. While this arbitrariness in the vertex function does not change the formal structure of the results in this paper, it could affect numerical calculations. Thus, before undertaking detailed calculations, it will be necessary to investigate the properties of each possible vertex function to determine which is most suitable for liquid sHe. It is hoped that in the near future this can be done and that complete numerical calculations of liquid sHe can be undertaken and compared with experiment. Acknowledgements. The authors would like to thank the National 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 UniversitC Libre de Bruxelles, Chimie Physique II for its hospitality during the summer of 1970.
APPENDIX
In this appendix, we define the vertex function gi(kiks /k&4) which is used as an expansion parameter in the theory developed here. Of necessity, the exposition given here is greatly abbreviated; more details can be found in the original papers59 8). The function gi(Kiks 1k&4) can be written in terms of the matrix elements between free-particle states of an operator
er:
g1(M21k3K4)
=
-<
+
<~4~31Glw2>*,
(A.1)
L. E. REICHL
202
where, if As is the two-particle
AND E. R. TUTTLE
hamiltonian,
(A4 The operator 8’ acts on the states to the right to give the sum of the quasiparticle energies characteristic of those states, and P denotes Cauchy principal part. The operator Gi is clearly not hermitean due to its dependence on 8’. Furthermore, it is easy to see from eq. (A.2) that gi(kiksjk&) is well defined for all potentials and does not necessarily become large as the potential V increases. We therefore expect that an expansion in powers of gi(Kiks 1K&a) will converge more rapidly than the perturbation expansion, particularly for strong interactions. It has been shown that there is an easier way to determine gi(Kiks 1K&4) than that of taking the matrix elements of the operator ei directly. In fact, in this alternative approach, the function gi(Kiks lksk4) can be evaluated analytically if the two-body wavefunctions for the interaction chosen can be found analytically. For a spin-independent, two-body, central potential the function gi(Kiks 1K&Q) can be written in the following form:
gl(hkzl k3k4) =
dt3)(kl +
k2 -
k3 -
k4)
[
where h(M21k3k4j
=
&
-,
(2L +
(24 3 1
-sz
g"l(hk21~3~4)>
(A.3)
1) &2'i34)
L
’
1 ~ kiz
&(kis,
K12lk34)C&,o,&,o,
-
(-l)L
~~,d~,cd.
(A4
In this equation, k12 is the relative momentum, kl2 = +(kz - kl) ; Klz is the total momentum, Kis = kl + k 2 ; klz = IklzJ ; and (~1 is the .z component of spin of particle 1. We observe from eq. (A.3) that the function gl(klkz 1k3k4) conserves momentum; this property of the function has been used in performing some of the momentum integrations in section 2. Also, the entire spin dependence of gl(klkz 1k3k4) is contained in the term in the second line of eq. (A.4), i.e., gl,L(kls, Kl2) k34) is spin independent. Finally, gl,L(klp, Kl2 / k34) can be written in terms of the reaction matrix
K12
I h4) =
2@k12 +-
xm
s
dko
lAtL)I ko> cm2 &(ko)
0
1 Ok,
+
Ok,
-
WO
(A.5)
MICROSCOPIC
THEORY
OF A DENSE
FERMI
where dL(k) is the phase shift for the Lth partial w. = @Kf,/4m
LIQUID.
II
203
wave, and
+ @k~/m.
(If the two-body potential being considered possesses bound states, then another term must be added. However, as we are primarily interested in the term has been application of the theory to liquid sHe this additional omitted.) The reaction matrix is defined in terms of the two-body radial wavefunction
f
s
drFL(kr)
U(r) (rl koL>,
(Ad
0
where U(r) is the reduced potential, U(r) = mV(r)/hs and FL(kr) is the rationalized spherical Bessel function, defined as FL(~) = pit(p). The normalization of the radial wavefunction is chosen such that its asymptotic form is
zoo
sin
koR -
$
1.
+ BL(ko)
The phase shift dL(k) can be determined A matrix through the equation
from the diagonal
elements
(k IA(h)/ k) = tan SL(k).
of the
(A7
We thus see that the function gl(klkz I k3k4) is, at least in principle, calculable for any given two-body interaction. The function fz(klk2 I kskc )k 3k 4) can also be written in terms of the A matrix. It is related to gl(klkz )k3k4) through the equation
gl(hkz I k&4) = fl(hkz I k&4) + C fz(Wz I k&s I k&4) kske
1
1 Wk, + mfi* -
Uk, -
Oka
oh1 + mi, -
ok,
-
mk,
where the first and second terms of this equation correspond, through eqs. (A.3) and (A.4), to the first and second terms, respectively, of eq. (A.5). Calculations of t~,~(k~s, Klz 1k34) have been made for several different potential functions 5,s). The results of one such calculation, which provides some justification for the approximations made in this paper, are discussed in section 3.
204
MICROSCOPIC
THEORY
OF A DENSE FERMI LIQUID.
II
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9)
Reichl, L. E. and Tuttle, E. R., Physica 52 (1971) 165. For a brief summary of this work, we refer to the introduction in RTI. Brinkman, W. F. and Engelsberg, S., Phys. Rev. 169 (1968) 417. Wheatley, J. C., in Quantum Fluids, edited by D. F. Brewer, North-Holland Co. (Amsterdam, 1966). Tuttle, E. R., thesis, Univ. of Colorado, 1964 (unpublished). Ostgaard, E., Phys. Rev. 170 (1968) 257. Jones, R. W., thesis, Univ. of Colorado, 1970 (unpublished). Mohling, F., Phys. Rev. 122 (1961) 1062; 124 (1961) 583. Mohling, F., Phys. Rev. 128 (1962) 1365.
Publ.