Nuclear Physics A262 (1976) 527 - 538; @ North-Holland Publishing Co., Amsterdam

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LANDAU THEORY OF RELATIVISTIC FERMI LIQUIDS +

GORDON BAYM and SIU A. CHIN

Department of Physics, University of Illinois, Urbana, Illinois 61801

Received 2 January 1976

Abstract: A relativistic extension of the Landau Fermi liquid theory, applicable to the study of high density matter, is developed. Consequences of Lorentz invariance in the theory are explored. The formalism is illustrated by a study of relativistic Fermi systems weakly interacting via scalar and vector meson exchange. Second order exchange energies for both massless scalar and massless vector interactions are calculated in terms of Landau parameters on the Fermi surface. Zero sound and “color-plasma oscillations” are studied in quark matter with SU(3) color gluon coupling.

1. Introduction and basic definitions

The Landau theory of Fermi liquids ‘3 2, has enjoyed considerable success as a framework for understanding the properties of non-relativistic systems of inter- acting fermions, ranging from low temperature liquid 3He [ref. 3)] and electrons in metals 4), to nuclear matter ‘). [For a recent review see ref. 6).] Extension of the theory to relativistic Fermi liquids can be expected to prove of similar utility in describing the dense matter found in compact astrophysical objects, such as for example, possible quark matter in neutron stars 7), and relativistic electron gases.

It is our purpose in this paper to indicate the general form of such an extension of the theory, and, as an illustrative application of the theory, to discuss in this framework relativistic matter interacting via scalar and vector meson fields in lowest order perturbation theory. As we shall see much of the non-relativistic theory can be taken over to the relativistic problem with usually small, but occasionally inter- esting, modifications. We shall not provide microscopic derivations of any of the results of the Landau theory, but leave this problem as an interesting theoretical challenge. Let us begin by stating the basic definitions of the relativistic Landau theory.

As in the non-relativistic Landau theory the low-lying excited states of a relativistic Fermi liquid can be described in terms of interacting quasiparticle excitations, plus possible collective modes. Let us denote the (smoothed) quasiparticle distribution function, as usual, by nPO, where p is the quasiparticle momentum, and o is the spin index. The particle density n is then equal to the quasiparticle density

n = s

dmpo, (1)

+ Research supported in part by US National Science Foundation Grants GP 40395, DMR75-22241, and MPS74-22148.

527

528 G. BAYM AND S. A. CHIN

where dr denotes cgd3p/(2n)3. Also, the total momentum density of the system is

9 I= dzpnpa. f

(2)

The energy per unit volume of a low-lying state can be regarded as a functional E(n,,) of the distribution function; the first variation of E with respect to nPV defines the quasiparticle energies apa :

6E = s

dze&z,,. (3)

The entropy per unit volume of a macroscopic state has the usual form (with Bol~a~n’s constant set equal to unity)

(4)

From minimizing E- TS at fixed temperature T and number y1 one finds the usual result for the distribution function for a state in thermal equilibrium

y1 pa = (exp ~-ts~~-~)/~~~l)-‘, (5)

where p is the chemical potential. At T = 0, a quasiparticle whose momentum is pf, the Fermi momentum, has energy p; for a system of particles with degeneracy g (= 2 for spin-+), pf = (6n%/g)*. Quasiparticle states with momenta close to the Fermi surface are long-lived, with a width CC (sp - p)‘.

The quasiparticle energy spa is itself a functional of +a< ; the first variation of of s,, defines the Landau Fermi liquid intemction~~~.~,~,:

8sp, = j

dzlf,,, p,a.8np,a,. (6)

Since fPe,pPa, is a second variation of E it is s~metric under interchange of its arg~ents.

We define, as usual, the Landau parameters.fs by

(7)

where 0 is the angle between p and p’, both taken on the Fermi surface, and the integration is over all directions of p. The density of states at the Fermi surface is given by

@)

then the d~ensionless Landau par~eters are F; = .~(O)~. The spin antisy~et~c parameters fi and e may be defined similarly.

RELATIVISTIC FERMI LIQUIDS 529

Up to this point the equations we have written down are identical to those in the non-relativistic Landau theory. The differences between the relativistic and non- relativistic theory appear when we consider the consequences of Lorentz invariance. We turn now to studying the behavior of the quasiparticle energies and interactions under Lorentz transformations.

2. Lorentz invariance

Consider a system in volume Vof total energy 6 and total momentum P. Suppose that we observe the system from a frame moving with velocity - v, and ask how the quasiparticle energies and momenta in the new frame are related to those in the old frame. From Lorentz invariance we have

%J) = [&(O) + P(O) . V-JIG

P(v) = P(0) - a(8 . P(O))(l - y) + b(O)vy, (9)

where E(v) and P(v) are the energy and momentum as observed from the moving frame, and B(0) and P(0) are the energy and momentum observed from the original frame; y = (1 -v’)-*, 6 is the unit vector along v, and we set c = 1.

Imagine now that we add, in the original frame, a particle (or equivalently a quasi- particle) of momentum p to the system; the momentum of the system then becomes P(O)+p, while the energy becomes &‘(O)+aJO). (We suppress the spin index here.) In the new frame, the momentum increases, according to (9) by

@ = p - v*(v* . p)( 1 - y) + &,(o)vy, (10)

while the energy increases by

a# = (a,(O) + P - 4% (11)

Eqs. (10) and (11) are the transformation laws for quasiparticle momentum and energy. These laws are identical to those for free particles, and imply that sp(v)’ -p2 = $-p2. However &i-p2 is not in general independent of the momentum p, since .@v) is a functional of the “moving” distribution n@,,(v), and thus, in the presence of interactions, does not equal .+O), which is a functional of the original distribution function n,,,(v = 0). The two distribution functions are related by

n+) = n,(O), (12)

where jj is given in terms of p by (10). By expanding the transformation laws (10) and (11) to first order in v we can

immediately derive the relativistic generalization of the Landau effective mass relation for excitations above the ground state:

(13)

530 G. BAYM AND S. A. CHIN

The quantity ~(1 -I-$Ff) plays the role of the effective mass at the Fermi surface ;

also_ ~~&~~~)~~ = z’r, the Fermi velocity. To derive (13) we write, to first order in o,

Ed’ E&O)+- i dZIf,,~(n,~(u)- n,,(O)); 04) J

this term represents the change in sl due to the change in the distribution function. To first order, n,,(u) = B,,_~~,&I) = np,(O>--ap,r . V,,+(O). Then using E@(U) = ap(0)-t-P * v, and ~~(0) = ~~(0) -zp(0)v * Vp,(O)+ . . ., we find that

sp(0)Vpap(O) = pi- s

dz’j&s,V,,n,,(O). (15)

Since in the ground state pzII’ = @(ll--.Q), eq. (15) implies

E~$,,@P + P s

dz’@,, - ,u) j”,,$ . fi%,r/c3p = p.

Setting p = pf and using (7) we find that

( > ag Pf SP,2 f,” ---. ‘f = ap Pf = ji 2nZ 3

W)

Eq. (13) follows directly from this relation, together with (8). The transformation properties offppj are more complex than in the non-relativistic

theory, where~~~,(~) in the moving frame is related to f,,,(O) in the original frame

by f,,~(r) = fp-mz+nrv (0), with IYL the particle bare mass. [This relation does not imply that fdepends only on p-p’.] To find the relativistic transformation proper- ties we first consider how the distribution function in the moving frame varies when the distribution function in the original frame is varied. As n,,(O) is varied, the momentum p in the moving frame corresponding to a given momentum p in the original frame will, according to eq. (lo), vary by

SP = u$e,(O) = uy s

dz’&,,(O)Sn,,(O); .

(18)

such a variation does not occur in the non-relativistic theory. Thus from (12) we have

&z,(u) + sp * Viinp(u) = 6n,(O), (19)

where the variations in np and n, are at fixed p and p respectively. The first variation of both sides of eq. (1 l), at fixed p, is therefore

RELATIVISTIC FERMI LIQUIDS 531

If we use p’, related to p’ by eq. (10) (with primes), as an integration variable, then +

d? = dz’(1 + u . Vpsp,(0))~. (21)

Using (18) and (19) and identifying coefficients of dr’bn,(O) on both sides of (20), we find the (complicated) relation between the functions f in the two frames ; in the limit of small u this relation is

f&(u) = [l -u . (Vpsp(0)+ Vpsp(0))]fpp(O)+ s

dr”j--,,,(O)v . V,,,np,,(0)f,,,,.(O), (22)

where j and j’ are related respectively to p and p’ by eq. (10). The terms on the right proportional to v are new features of the relativistic theory. Eq. (22) can also be derived by varying both sides of (15) with respect to n,.(O).

In the limit of weak interactions between the quasiparticles we can neglect the final term on the right side of (22), as well as the dependence off,,, on the distribution function. Then eq. (22) reduces to

Vp(EpOfpOpp’) + Vp’(EpOd-pOpp’) = 0, (23)

where E; = (p2 +m’)*, and j&, denotes the lowest order quasiparticle interaction. By contrast, in the non-relativistic theory&. is a function only of p - p’ in the weak interaction limit.

We consider next the transformation properties of the energy density. To do this we first construct the stress tensor Tfly of the system by taking moments of the Landau kinetic equation for the distribution function

2 4VPEP.VrnP-VrEp.Vpnp = 2 ( > collisions

Multiplying both sides by sp, summing over all p, and using conservation of quasi- particle energy in collisions, we have

s drs$ +V; s

dze,V,s,n, = 0. (25)

From (3) the first term is the time derivative of E = Too. Eq. (25) is thus the con- servation law a,Top = 0, from which we identify

Toi = s d7qaEp/api)np. (26)

Taking moments of (24) with respect to pi we find the conservation law aMTip = 0

+ Note that d3p/.sP is not invariant under a Lorentz transformation. since VP&, # p/c) in the presence of interactiona.

532 G. BAYM AND S. A. CHIN

where ~‘0 = gi =

s dqn,,, (27)

[In deriving (28) one uses fdzaPVin, = V,E,] The total stress tensor is thus of the form

T”v =: @y_gflv E- ( Jd%J~ (29)

where @“ is the metric tensor (go0 = - l),

POi = TOi, f+iO = TiO, (30)

Too = dzs n P P’

$” =

s d~~i(~&~/~p~)~~. (31)

Under a Lorentz transformation Z@’ itself transforms as a tensor. We demonstrate this explicitly for the (0,O) component. Viewing the system from a frame moving with velocity -u we have r;i”“O(v) = jd?sii(P)ni;(u). Now let p, related to fi by (lo), be the new variable of integration. Then n@(u) = n,(O), cl? = dz(1 +v . VPsP(0))y, and Q(S) = (~~(0) + p . v)y ; hence

PO(v) = y”[ PO(O) + v,(P’(O) + P(0)) f v,u,F(O)], (32)

which is the correct tensor transformation law. The transformations of the other components follow similarly by use of eqs. (10) and (11).

In order then that P transform as a tensor, E- fdze,n, must transform as a scalar. Going to the moving frame, and equating to zero the term in ~(~)-~d~&~(~)~~(~) of first order in u, we have -fd~iie~(u)n,(O) = 0. But since under a variation of u,

at fixed p,

&,(I+ = s

drlf,,,bn,, (u) = - f

dt’&sP,~ * V,,n,,(O),

we see, on integrating by parts that quite generally

(33)

s dzdz’n,n,~V,,(e,, fppt) = 0. (34)

multiplying both sides of eq. (15) by n,(O), integrating over p and using eq. (34) we find that, as expected, To’ = T”.

In the limit of weak interactions,

zp = E;+ s d+f,opapv, (35)

RELATIVISTIC FERMI LIQUIDS 533

(36)

Then the fact that E- jdzaP,n, is a scalar implies that the interaction energy in this limit, ~~dzdz’j$n,nP1, also transforms as a scalar.

3. Fermi liquid properties

In this section we summarize briefly the forms of simple physical properties of the system, such as the compressibility, first sound velocity and specific heat.

The compressibility of the Fermi liquid may be derived as in the non-relativistic theory by differentiating the relation ,u = .aq with respect to ~1, taking into account the dependence of ep on the distribution function. This gives

Eq. (17) then implies the useful relation

(38)

The relativistic first sound velocity is given, as usual, by

ap ap ap an n ap +_=_--=--; aE ap an aE p an (39)

eq. (38) thus yields

lp2 l+F”, c2=-L

lpf2 1

_=__ 3~~ l+$F”, 3~’

1+ (40)

The specific heat of a relativistic Fermi liquid at low temperature may be derived as in the non-relativistic theory 1, ‘, 6), and one finds the same form

cv = +&(0)7Y (41)

4. Weakly interacting systems

As an illustrative application of the theory developed in the preceding sections, we turn now to a study of relativistic, high density matter interacting via a scalar field 4 and a vector meson field A, [refs. ‘- ‘“)I, in lowest order perturbation theory. This study is relevant for weakly interacting Fermi systems such as the relativistic electron gas, and quark matter as may possibly occur in neutron stars. To keep the discussion general, we begin by considering both the scalar and vector mesons to

534 G. BAYM AND S. A. CHIN

be massive. Their couplings to spin-i fermions are described by a Lagrangian

9 = 9; + 9; + 9; + ig,&%A, +g$+#, (42)

where A?$, 9’; and 9’: are the free Lagrangian densities of a Dirac field 9 with mass M, the vector field A, with mass m, and the scalar field C/I with mass rc respect- ively.

The Landau Fermi liquid interactionf,,, P,sP is related to the two particle forward t scattering amphtude via

f MM

pa,p’o’ = F Eo - &pa, p’o”

P P’ (43)

where APO,P,oJ is the usual Lorentz invariant matrix element I’). The spin symmetric part ~ff~~,~.~, is given by

f;p, = t c $F ; ~po,p’o” ad p

(44)

To second order in the coupling constants, A’po,p,a, consists of the usual direct and exchange amplitudes, which may be evaluated easily by the conventional Feyuman rules. For example, in the case of vector interaction, the direct contribution is just

where p . p’ = - E$$, fp . p’. Similarly, the exchange contribution is given by

For the scalar case, we have

(46)

(47)

(48)

ThenS =~,diT+fsv,ex+fSs,dir+~,ex. One may check explicitly that the results (45)- (48) indeed satisfy the relation (23).

t As in the non-relativistic theory, the forward scattering amplitude here is defined by first letting the three-momentum transfer go to zero and then letting the energy transfer.go to zero. These two limits are not interchangeable in a many-particle system, except in lowest order.

RELATIVISTIC FERMI LIQUIDS 535

If one wishes, one may now directly use E,di,, f”,,,,, &ur and x,,, in eqs. (35) and (36) and obtain, to second order, the single particle excitation spectrum, as well as the energy density of the system. For the direct contributions, this yields

&p = EGO+ $n2- $5 $M[xq-ln(x+q)], P

E = E”+ i $n2- 2 $A44[xq-h(x+q)]2,

(49)

(50)

where E” denotes the energy density of a free Fermi gas of density n, x = p,lM,

and q z (1 +x2)*. In eqs. (49) and (50), (M3/27c2)[xy--ln(x+r])] = Sdz’(M/$)n$. The direct use of eqs. (35) and (36) for the exchange contributions requires evaluation of integrals over the Fermi sphere. Within the present framwork, however, the energy density and quasiparticle spectrum near the Fermi surface may be computed directly in terms of Landau parameters on the Fermi surface, by use of the relation (38), as follows. The single particle excitation spectrum near the Fermi surface is

of the form

sp = P+@-P,). (51)

The chemical potential p can be found from the Landau parameters f; and f; by integrating eq. (38) with respect to the density, while vr is given in terms of ,u and f; by eq. (17). The energy density can then be obtained in terms of p by integration of the relation ,U = dE/dn.

We illustrate this method of calculation by computing the second order exchange energy of the relativistic electron gas. In this case, g”, = 4rre2, and we have from(46)

f," -$f,s = - 6 ; j+ d(cos 0) [“;;(?!I-;;; ‘)I(1 - cos 0) = $ (I _ 2~2,$), f -1 f f

(52)

where sr = (p: + M’)*. [Note thatf; andfs, and indeed all the Landau parameters, are separately divergent except in the extreme relativistic limit, A4 -+ 0; however, the combination f;--if; is always finite.] For convenience, we rewrite eq. (38) as

(53)

To calculate p to second order, it is sufficient to let ,u = ‘Ed in the right side of eq. (53). With the constant of integration adjusted so that @r = 0) = M, the integration of eq. (54) yields

p2= t$+ $M”[xq-3ln(x+q)]

536 G. BAYM AND S. A. CHIN

or

p = Ef+ ; f ~M’[X~-3 ln(x+r/)]. f

Thus to second order, the exchange energy density is given by

Ey = s

1 da[p--s,] = sg,2M4 x- iln (x+$ 1

1 --

= (27r)4gv 2~4~x4-~~~~-ln(~+~)]2~.

(54)

This expression for the relativistic electron exchange energy density agrees with the previous calculation by Zapolsky I’) and Akhiezer and Peletminskii ’ 3), who arrived at this result by a direct evaluation of the double integral in eq. (36). For pf B- M,

Fvx = &ppf4132rr4. Similarly, we obtain the exchange energy for the case of a massless scalar interaction

Quark matter, weakly interacting via exchange of massless vector color gluons, is expected to have a zero sound collective mode in which the distribution function n,(rt) is independent of the spin and internal degrees of freedom of the quarks; i.e., in this mode, the total density oscillates and the matter remains locally color neutral, with no local spin fluctuations. For simplicity we assmne all flavors (u, d, s, . . .)

of quarks present to have the same density. The quark-gluon interaction Lagrangian is

zint = ~~~a~~~=i~~~~i~~, (57)

where eai is the quark field of color a and flavor i, the “; are the gluon fields (CI = 1, . . . X), &, are the SU(3) generators and g, is the color coupling constant. To second order in g,, the spin, color and flavor symmetric Landau parameter_& is given by the exchange term [cf. eq. (46)“J only; the direct term vanishes because the color symmetric combination ( N tra) does not couple to the gluons. Thus,

(58)

where k is the number of flavors. In the limit of massless quarks [as used in the MIT bag model ‘3 to a first approximation]

(59)

thus to order g,“, only$ is non-vanishing. Since to the same order Fo = $/3rc2 >, 0, one finds from solution of the kinetic equation (24) that the system has a (collisionless)

~ELAT~ISTI~ FERMI LIQUIDS 537

zero sound mode ‘TV)], o = coq, of velocity

c0 = 2+(1+2exp[-2/F:]), (60)

where, from eq. (38) and (17),

v, = pJJL = 1 -$P,, (61)

to order t g,“. By Contras t, the (collision dominated) first sound velocity, from eq. (40), equals ,,/i to this order. We note that in this massless limit, both the velocity of zero and first sound are independent of the density of the system.

In addition to this zero sound mode, and the usual color and spin symmetric electromagnetic plasma oscillations, quark matter has a spin and flavor symmetric “color plasma excitation” of frequency given, to order gs, by v

~0: = g,2n/6p (62)

(or more generally by xig$ni/6& near zero wave number. In this mode one has linear combinations of motions in which two color components counteroscillate while the third remains at rest. We note also that the exchange energy of relativistic quark matter is ~jg~~~~8~4 per unit volume, to order gz.

For general fermion mass M, the Landau parameter (58) evaluated on the Fermi surface diverges in the forward direction as (1 - cos 19)~ ‘, where 8 is the angle between p and p’. In the limit M -+ 0 this divergence disappears. However when we consider higher order corrections to (58) due to polarization effects of the medium on the exchanged gluons, we find a reappearance of this divergence even for M = 0. The polarization corrections differ for exchange of transverse and longitudinal gluon degrees of freedom. By direct calculation we find that for quark mass M,

(63)

where q = p-p’, q” = E;--I$; the longitudinal and transverse polarization opera- tors are given in terms of the (irreducible) current-current correlation functions by

where $; = R~i,$aiyC&,i, TZ, = (0, fi) and 12 .g = 0, and in (64) there is no sum over a. When the If-functions are neglected, eq. (63) reduces to the previous result (58).

In the limit 141 +,O, with q” fured, no0 + ~j(g~~i~~)q2/(qo)2 to order gf; the lon~tudinal term thus exhibits the plasma oscillation (62). Similarly in this limit

JI, 4 Cig,Zn,/6/+ an d one finds as well a transverse plasma oscillation of the same frequency (62) at zero wave number. For quasiparticle interactions on the Fermi surface q” --) 0; then for small q, LZoo(q, 0) + -~g~~,.&~,/3~~. Due to this screening

+ A massless two component plasma possesses a similar zero sound mode in which the two components oscillate in phase to preserve local charge neutrality.

t* An analogous mode is discussed in ref. 15); see also ref. 7).

538 G. BAYM AND S. A. CHIN

the longitudinal contribution to (63) is finite as q -+ 0. On the other hand IIT(q, q" = 0) -+ -4nq’x, as q -+ 0, where x = (g,“/48n3)~iln(2&Mi) is the effective “magnetic” susceptibility of the matter. [For Mi = 0, the Mi in the loga- rithm is replaced by 141.1 This lack of static transverse screening means that the transverse term in (63) remains divergent as p -+ p’ on the Fermi surface t. As a consequence, in computing the zero sound mode to higher order in g,“, one cannot directly take, for use in the kinetic equation, the q" -+ 0 limit of the Landa.uparam- eters, but rather one must retain the dependence of the Landau parameters on the frequency and wave vector of the zero sound mode in solving the kinetic equation. The elucidation of this problem is beyond the scope of this paper.

One of the authors (G.B.) would like to thank the Aspen Center for Physics, where this work was begun, for its hospitality during the summer of 1975.

References

1) L. D. Landau, ZhETF (USSR) 30 (1956) 1058 [JETP (Sov. Phys.) 3 (1957) 9201; ZhETF (USSR) 32 (1957) 59 [JETP (Sov. Phys.) 5 (1957) 1011

2) A. A. Abrikosov and I. M. Khalatnikov, Rep. Progr. Phys. 22 (1959) 329 3) J. C. Wheatley, Prog. Low Temp. Phys. vol. 6 (North-Holland; Amsterdam 1970) p. 77 4) P. M. Platzman and P. A. Wolff, Solid State Phys., suppl. 13 (Academic Press, NY, 1973) 5) A. B. Migdal, Nuclear theory: the quasiparticle method (Benjamin, New York, 1968) 6) G. Baym and C. J. Pethick, inPhysics of liquid and solid helium, vol. 2, ed. K. H. Bennemann and

J. B. Ketterson (Wiley-Interscience. in press) 7) J. C. Collins and M. J. Perry, Phys. Rev. Lett. 34 (1975) 1353 8) Ya. B. Zel’dovich, ZhETF (USSR) 41 (1961) 1609 [JETP (Sov. Phys.) 14 (1962) 11431 9) J. D. Walecka, Ann. of Phys. 83 (1974) 491

10) G. Marx, Nucl. Phys. 1 (1956) 660 11) J. J. Sakurai, Advanced quantum mechanics (Addison-Wesley, Reading, 1967) 12) H. S. Zapolsky, Cornell University LNS Rept. (unpublished, Sept. 1960);

E. E. Salpeter, Ap J. 134 (1961.) 669 . 13) I. A. Akhiezer and S. V. Peletminskii, ZhETF (USSR) 38 (1960) 1829 [JETP (Sov. Phys.) 11 (1960)

13161 14) A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn and V. F. Weisskopf, Phys. Rev. D9 (1974) 3471 15) M. B. Kislinger and P. D. Morley, EFI preprint 75-8 (University of Chicago)

+ It would be useful to know whether this lack of static transverse screening affects Collins and Perry’s argument 7, concerning the absence of the Yang-Mills infrared problem in a many-particle system.