A RESULTS RELATED TO L_{F}

# Degenerate limit thermodynamics beyond leading order for models of dense matter

## Abstract

Analytical formulas for next-to-leading order temperature corrections to the thermal state variables of interacting nucleons in bulk matter are derived in the degenerate limit. The formalism developed is applicable to a wide class of non-relativistic and relativistic models of hot and dense matter currently used in nuclear physics and astrophysics (supernovae, proto-neutron stars and neutron star mergers) as well as in condensed matter physics. We consider the general case of arbitrary dimensionality of momentum space and an arbitrary degree of relativity (for relativistic mean-field theoretical models). For non-relativistic zero-range interactions, knowledge of the Landau effective mass suffices to compute next-to-leading order effects, but in the case of finite-range interactions, momentum derivatives of the Landau effective mass function up to second order are required. Numerical computations are performed to compare results from our analytical formulas with the exact results for zero- and finite-range potential and relativistic mean-field theoretical models. In all cases, inclusion of next-to-leading order temperature effects substantially extends the ranges of partial degeneracy for which the analytical treatment remains valid.

###### keywords:
Hot and dense matter, thermal effects, potential and field-theoretical models.
1\fntext

[fncc]c.constantinou@fz-juelich.de

\fntext

[fnbm]bm956810@ohio.edu

\fntext

[fnmp]prakash@ohio.edu

\fntext

[fnjl]james.lattimer@stonybrook.edu

## 1 Introduction

Homogeneous bulk matter comprised of fermions is commonly encountered in astrophysics, condensed matter physics, and nuclear physics. For extreme degenerate to near-degenerate conditions which prevail when the temperature is small compared to the Fermi temperature, Landau’s Fermi Liquid Theory (FLT) has been a useful guide to describe the thermodynamic and transport properties of the system (see, e.g., (1) and references therein). The equation of state (EOS) of dense matter in cold and catalyzed neutron stars, for example, is dominated by the zero-temperature properties (which predominantly determine the structure of and neutrino interactions within the star) while finite-temperature corrections (important for the cooling of neutron stars) are adequately given by the degenerate limit expressions from FLT. The leading order FLT corrections to the energy density and pressure are quadratic in the temperature; corrections to the entropy and specific heats are linear in the temperature. However, matter in supernovae and proto-neutron stars (2); (3), especially in situations in which collapse to a black hole occurs, may reach temperatures exceeding the Fermi temperature, in which case the finite-temperature contributions extend beyond those given by the FLT. In neutron star mergers, it is likely that in some cases a hyper-massive neutron star, or HMNS, is formed: the merged remnant mass exceeds the cold maximum mass. The metastable support is provided by rotation, including differential rotation, and thermal effects. The timescale over which collapse to a black hole eventually occurs, potentially observable in gravitational wave signatures, will therefore be sensitive to thermal effects (4). In this contribution, we derive analytical formulas for next-to-leading order temperature effects in the state variables of interacting nucleons in both the non-relativistic and relativistic limits for a variety of nuclear interaction models.

For non-relativistic models with zero-range interactions, knowledge of the Landau effective mass is sufficient to satisfy thermodynamic identities. However, in the general case of finite-range interactions, momentum derivatives of the Landau effective mass function up to second order are required. We compare results from the analytical expressions to exact numerical calculations for zero- and finite-range potential models as well as for relativistic mean-field theoretical models. The analytic next-to-leading order expressions lead to an improvement of the leading order results of FLT, as demonstrated by the wider ranges of degeneracy and temperature for which they remain valid. In addition, we derive relations in a form that are independent of the dimensionality of the momentum space under consideration. Therefore, although our discussion focuses on examples from dense matter physics, which are three-dimensional systems in momentum space, the expressions derived can have a wider application to certain problems in condensed matter physics in which the momentum space is two-dimensional.

The paper is organized as follows. In Sec. 2, the formalism to calculate next-to-leading order corrections to the results of FLT in D-dimensions is developed. Analytical formulas appropriate for 3-dimensions are given in Sec. 3, whereas Sec. 4 contains results for 2-dimensions. The formalism is applied to zero- and finite-range potential models and a relativistic field-theoretical model in Sec. 5. Numerical results for these models are presented in Sec. 6 where the extent to which the next-to-leading order corrections improve the FLT results are demonstrated. Section 7 presents a summary and conclusions. Useful formulas for the evaluation of the thermal properties are provided in Appendices A, B, and C.

## 2 General Considerations

For a generic Hamiltonian density where and are the number and kinetic energy densities respectively, the single-particle potential is obtained from a functional differentiation of with respect to , and can contain terms depending on as well as the momentum :

 U(n,p)=δHδn=U(n)+R(p), (1)

where denotes contributions that depend on only. Note that above may also be -dependent but we will suppress this for notational simplicity.

The study of the thermodynamic properties of a fermion system involves integrals of the form

 I=∫∞0dp g(p)f(p),f(p)=[1+exp(ϵ−μT)]−1, (2)

where is the temperature, is the chemical potential, and the single-particle spectrum

 ϵ=p22m+U(n,p). (3)

The structure of the function is determined by the state variable under consideration. In general, integrals involving the Fermi function do not admit analytical solutions and thus require numerical treatment. In the low-temperature limit, however, when the degeneracy parameter

 η=μ−ϵ(p=0)T (4)

is large, these integrals can be approximately evaluated employing the Sommerfeld expansion (see, e.g., (5)) by transforming Eq. (2) to

 I=∫∞0dyϕ(y)1+exp(y−η)\lx@stackrelη≫1⟶∫η0ϕ(y) dy+π26dϕdy∣∣∣y=η+7π4360d3ϕdy3∣∣∣y=η+…. (5)

with the identification , and the substitution

 y≡ϵ−U(n)T=p22mT+R(p)T, (6)

from which it follows that

 dydp=pMTandϕ(y)=MTpg(p), (7)

where the Landau effective mass function

 M(p)=m[1+mpdR(p)dp]−1. (8)

This function is implicitly temperature-dependent and its relation to the Landau effective mass is

 M(p=pF;T=0)=m∗, (9)

where is the Fermi momentum. From the relations in Eq. (7)

 dϕdy = −T2M2gp3[1−p(g′g+M′M)] (10)
 d3ϕdy3 = −15T4M4gp7[1−p(g′g+53M′M) (11) + Extra open brace or missing close brace − p315(g′′′g+7M′2g′M2g+6M′g′′Mg+4M′′gMg + M′3M3+M′′′M+4M′M′′M2)],

where the primes denote differentiation with respect to .

For a system in dimensions having internal degrees of freedom, the number density is given by

 n=CD∫dp pD−1 fpwithCD=γ(2πℏ)D DπD/2(D/2)!. (12)

The combination of Eqs. (5),(10), and (11) with results in

 n = CDD{pDμ+π26DpD−4μM2μT2(D−2+pμM′μMμ) (13) + Missing or unrecognized delimiter for \left + (pμM′μMμ)3+(7D−18)(pμM′μMμ)2 + (6D2−40D+59)pμM′μMμ+p3μM′′′μMμ + 4 p3μM′μM′′μM2μ+(4D−11)p2μM′′μMμ]+…},

where the subscript denotes quantities evaluated at , i.e.,

 ϵ=μ=p2μ2m+R(pμ)+U(n). (14)

For particles in a volume , the number density at and at finite is the same. Equating the result in Eq. (13) to its counterpart , and perturbatively inverting we get

 pμ=pF[1−π26m∗2T2p4F(D−2+pFM′Fm∗)+…] (15)

with

 pF=(nDCD)1/DandM′F=dMdp∣∣∣pF. (16)

As our main goal here is to derive the next-to-leading order correction in temperature for the entropy density , it suffices to truncate the series expansion of to . We will show below that higher-order terms do not contribute at this level of approximation where we may also neglect the temperature dependence of and its derivatives. The result in Eq. (15) helps us to work only with quantities defined on the Fermi surface as done in Landau’s Fermi-Liquid theory (1); (6); (7). The entropy density is formally given by

 s=−CD∫dp pD−1[fplnfp−(1−fp)ln(1−fp)]. (17)

Integrating this expression twice by parts we obtain

 s=1T{τm(12+1D)+n(U−μ)+CD∫dp pD−1fp[R(p)+pDdR(p)dp]} (18)

where

 τ=CD∫dp pD+1fp (19)

is the kinetic energy density. With the aid of Eq. (14) for the chemical potential, Eq. (18) can be written as

 s=1T{τm(12+1D)−np2μ2m+CD∫dp pD−1fp[R(p)−R(pμ)+pDdR(p)dp]} (20)

from which we identify the functions

 g1s(p) = pD+1m(12+1D)−pD−1p2μ2m (21) g2s(p) = pD−1[R(p)−R(pμ)+pDdR(p)dp] (22)

to be used in the Sommerfeld expansion. For both of these functions, the first term on the right-hand side of Eq. (5) involving an integral vanishes yielding

 s = π23CDpD−2μMμT+7π490CDpD−6μM3μT3[(D−4)(D−2) (23) + p2μM′2μM2μ+p2μM′′μMμ+(3D−7)pμM′μMμ].

Use of Eqs. (15) and (16) in the above result delivers the working expression for in terms of quantities defined on the Fermi surface:

 s ≃ π23CDpD−2Fm∗T+π445CDpD−6Fm∗3T3[(D−9)(D−2) (24) + 72p2FM′2Fm∗2+72p2FM′′Fm∗+(16D−39)2pFM′Fm∗],

where the term is the well known result from FLT. We note that a large number of cancellations occur in obtaining Eqs. (23) and (24) despite the complexity of of Eqs. (10) and (11). For a system composed of different kinds of particles the total entropy density is a sum of the contributions from the individual species where, in Eq. (24), the Fermi momentum, the effective mass, and its derivatives all carry a particle-species index .

Equation (24) forms the basis from which other properties of the system can be derived. For example, the entropy per particle is the simple ratio , whereas the thermal energy, pressure, and chemical potential can be obtained through the application of the appropriate Maxwell relations (5):

 Eth=∫T dS,Pth=−n2∫dSdndTandμth=−∫dsdndT (25)

[for a multiple-species system, ].

The specific heats at constant volume and pressure are given by the standard thermodynamics expressions (5)

 CV = T∂S∂n∣∣∣n=∂Eth∂T∣∣∣n (26) CP = T∂S∂n∣∣∣P=CV+Tn2(∂Pth∂T∣∣n)2∂P∂n∣∣T. (27)

We note that the formalism above has not considered effects, for example, from single particle-hole excitations, or from collective and pairing correlations near the Fermi surface (1); (8); (9). Contributions from these sources must be added to those considered here when appropriate.

## 3 Results for D=3

For a single-species system of spin particles in 3 dimensions [for which ], the entropy density becomes

 s=pFm∗T3ℏ3−2π215ℏ3m∗3T3p3F(1−LF), (28)

where

 LF≡712p2FM′2Fm∗2+712p2FM′′Fm∗+34pFM′Fm∗. (29)

We stress that, in general,

 dM(p)dp∣∣∣pF=M′F≠m∗′=dM(pF)dpF (30)

as can contain both and (via ). In terms of the level-density parameter (where is the Fermi temperature), Eq. (28) can be written as

 s=2anT−165π2a3nT3(1−LF). (31)

The quantity arises from nontrivial momentum dependencies in the single-particle potential. For free gases (where ), and for systems having only contact interactions where (such as Skyrme models), .

Equation (31) in conjunction with Eqs. (25)-(27) leads to

 S = 2aT−165π2a3T3(1−LF),Eth=aT2−125π2a3T4(1−LF) (32) Eth = aT2−125π2a3T4(1−LF) (33) Pth = 23anQT2−85π2a3nQT4(1−LF+n2QdLFdn) (34) μth = −a(1−2Q3)T2+45π2a3T4[(1−LF)(1−2Q)−ndLFdn] (35) CV = 2aT−485π2a3T3(1−LF)andCP=CV+169a2Q2T3dP0dn, (36)

where

 Q=1−3n2m∗dm∗dn. (37)

In the derivation of Eq. (36) we have assumed that the zero-temperature pressure is such that . This condition will not be met in situations where is relatively flat as in the vicinity of a critical point. When this is the case, we must use Eq. (27) for , with

 (∂Pth∂T∣∣∣n)2 = (43anQT)2[1−485π2a2T2(1−LF+n2QdLFdn)] (38) ∂P∂n∣∣∣T = dP0dn+23aQT2(1−2Q3+ndQdn) (39) − 58π2a3QT4[(1−2Q+ndQdn)(1−LF+n2QdLFdn) − ndLFdn(1−2Q+n2Q2dQdn)+n22Qd2LFdn2].

Similar considerations as with Eq. (36) hold for the ratio of the specific heats

 CPCV=1+89aQ2T2dP0dn. (40)

Other quantities of interest in astrophysical applications include the thermal index

 Γth=1+PthnEth=1+2Q3−45π2a2nT2dLFdn (41)

 ΓS=CPCVnP∂P∂n∣∣∣T=nP0[dP0dn+23aQT2(1+2Q3+ndQdn−nP0dP0dn)] (42)

where, in addition to Eqs. (34),(39) and (40), the approximation

 1P≃1P0(1−PthP0) (43)

was used. In its native variables , is given by

 ΓS=nP∂P∂n∣∣∣S=nP0+nQS26a[dP0dn+QS26a(1+23Q+nQdQdn)]. (44)

To arrive to Eq. (44) one begins by inverting Eq. (32) for the small parameter

 aT=S2+S35π2(1−LF) (45)

which is then employed in the expression for the thermal pressure with the results

 Pth = nQ6aS2+nQ30π2aS4(1−LF−3n2QdLFdn) (46) ∂Pth∂n∣∣∣S = Q6aS2(1+23Q+nQdQdn) (47) + Q30π2aS4[(1+23Q+nQdQdn)(1−LF) − 2ndLFdn(1+32Q)−3n22Qd2LFdn2].

Finally, the result is truncated to in both the numerator as well as the denominator. We refrain from invoking approximation (43) as for nuclear systems, can cross 0 at low densities. This is not a problem in the variables because the degenerate approximation breaks down at sufficiently low density regardless of . In the variables , however, for small values of the entropy the system remains degenerate irrespective of the density, and thus division by zero is avoided (as could happen if Eq. (43) is used).

We point out that the adiabatic index is related to the squared speed of sound according to

 (csc)2=ΓSPh+mn, (48)

where is the enthalpy density.

## 4 Results for D=2

In condensed matter physics, 2-dimensional systems are of much interest. In the current framework, the entropy density is

 s=π23C2m∗T+7π490C2m∗3T3p4F(p2FM′2Fm∗2+p2FM′′Fm∗−pFM′Fm∗) (49)

with . A noteworthy feature of this result is that the term receives contributions only from the derivatives of the effective mass function with respect to at the Fermi surface. Thus, it is absent not only for free gases but also for systems with contact interactions where the -dependence of implies that .
In terms of the level density parameter , and

 pF = (2nC2)1/2,Q=1−nm∗dm∗dn (50) LF = p2FM′2Fm∗2+p2FM′′Fm∗−pFM′Fm∗ (51)

 S = 43aT+5645π2a3T3LF,Eth=23aT2+1415π2a3T4LF (52) Pth = 23anQT2+1415π2a3nQT4(LF−n3QdLFdn) (53) μth = −23aT2(1−Q)−1445π2a3T4[LF(1−3Q)+ndLFdn] (54) CV = 43aT+5615π2a3T3LFandCP=CV+169a2Q2T3dP0dn. (55)

The above results do not include the effects of collective excitations near the Fermi surface or of non-analytic contributions. As pointed out in Ref. (9), 2-dimensional Fermi systems in condensed matter physics (even with contact interactions) have contributions to the entropy from interactions separate from those due to the collective modes. These contributions arise from non-analytic corrections to the real part of the self-energy.

## 5 Application to Models

In what follows, we compare the analytical results from the leading order corrections to Landau Fermi-liquid theory to the results of exact numerical calculations of the thermal state variables. These comparisons are made using models that are widely used in nuclear and neutron star phenomenology. In the category of non-relativistic potential models, we begin with the model, referred to as MDI(A), that reproduces the empirical properties of isospin symmetric and asymmetric bulk nuclear matter (10), optical model fits to nucleon-nucleus scattering data (11), heavy-ion flow data in the energy range 0.5-2 GeV/A (12), and the largest well-measured neutron star mass of 2 (13); (14). This model, which is based on Refs. (15); (16), incorporates finite range interactions through a Yukawa-type, finite-range force, is contrasted with a conventional zero-range Skyrme model known as SkO (17). Both models predict nearly identical zero-temperature properties at all densities and proton fractions, including the neutron star maximum mass, but differ in their predictions for heavy-ion flow data (18). To provide a contrast, we also investigate a relativistic mean-field theoretical (MFT) model (10) which yields zero-temperature properties similar to those of the two non-relativistic models chosen here. For all three models, we consider nucleonic matter in its pure neutron-matter (PNM, with ) and symmetric nuclear matter (SNM, with ) configurations.

### 5.1 Finite-range potential models

For the MDI(A) model (16); (10), the momentum-dependent part of the single-particle potential is given by

 R(p) = 2Cγn02(2πℏ)3∫d3p′ fp′11+(→p−→p′Λ)2 (56) \lx@stackrelT=0⟶ Missing or unrecognized delimiter for \left + (Λ2+p2F−p2)4Λpln[Λ2+(p+pF)2Λ2+(p−pF)2]}. (57)

For the coefficients , , and we use the values 0.16 fm, -23.06 MeV, -128.9 MeV and 420.9 MeV, respectively. Explicit expressions for the derivatives of and their connection with and are provided in Appendix A. The MDI Hamiltonian density is shown in Appendix B. For details of the exact numerical calculations, see Ref. (10).

### 5.2 Zero-range Skyrme models

Zero-range Skyrme models belong to that subset of the case for which . This is because, for these models, the momentum-dependent part of the potential has the form

 R=β(n)p2 (58)

( is a density dependent factor) which renders the generalized effective mass to be independent:

 M=m1+mpdRdp=11+2mβ(n), (59)

and therefore its derivatives . Hence as well. Consequently, the results in Sec. 3 for Skyrme models simplify considerably. Results to be shown here are for the SKO model (17), the exact numerical calculations for which are described in Ref. (10).

### 5.3 Relativistic models

The single-particle energy spectrum of relativistic mean-field theoretical models (19) obtained from the nucleon equation of motion has the structure

 ϵ=E∗+U(n),E∗=[p2+M∗2(n,T)]1/2. (60)

The single-particle potential is the result of vector meson exchanges whereas the Dirac effective mass arises from scalar meson interactions. The implementation of the above equations in the Sommerfeld expansion is made possible by the identification

 y=E∗T,dydp=pE∗Tandϕ(y)=E∗Tpg(p). (61)

The calculation of , and proceeds as in the non-relativistic case with the replacement [cf. Eq. (8)]. In particular, for we have

 pμ=pF[1−π26E∗2FT2p4F(D−2+pFE∗′FE∗F)] (62)

where

 E∗F=(p2F+M∗2)1/2andE∗′F=dE∗dp∣∣∣pF=pFE∗F. (63)

The simple dependence of on the momentum in Eq. (60) leads to the correspondingly straightforward expression (63) for which, as we will show soon hereafter, results in an elementary form for and by extension the whole set of the MFT thermodynamics can be written in an uncomplicated manner.

Substituting Eq. (63) into Eq. (62) yields

 pμ=pF[1−π26E∗2FT2p4F(D−2+p2FE∗2F)]. (64)

The twice-by-parts integration of Eq. (17) for the entropy density in the relativistic context gives

 s=CDT∫dp pD−1fp(E∗+pDdE∗dp−E∗μ) (65)

where one observes the analogy with the integral term of Eq. (20). Using

 gs(p)=pD−1(E∗+pDdE∗dp−E∗μ) (66)

we proceed as before to get the entropy density in terms of as

 s=π23CDpD−2μE∗μT+7π290CD(D−2)(D−4)pD−6μE∗3μT3[1+3(D−4)p2μE∗2μ] (67)

which, with the aid of Eq. (62), becomes

 s = π23CDpD−2FE∗FT+π445CD(D−2)(D−9)pD−6FE∗3FT3 (68) × [1+112(D−9)p2FE∗2F−52(D−2)(D−9)p4FE∗4F].

In the derivation of the last equation the weak temperature of in the degenerate limit has been ignored (but not of ). Combining Eq. (68) with Eqs. (25)-(27) in , and using the definitions [here, the Fermi temperature

 a=π22E∗Fp2F,q=M∗2E∗2F(1−3nM∗dM∗dn)andLF=1112p2FE∗2F−512p4FE∗4F, (69)

we obtain

 S = 2aT−165π2a3T3(1−LF),Eth=aT2−125π2a3T4(1−LF) (70) Pth = 13anT2(1+q)−45π2a3nT4[1−LF+q(1−LF3−109p4FE∗4F)] (71) μth = −23aT2(1−q2)−45π2a3T4q(1−LF3−109p4FE∗4F) (72) CV = 2aT−485π2a3T3(1−LF)andCP=CV+49a2T3(1+q)2dP0dn. (73)

As in the nonrelativistic case, when conditions are such that is small, one must use derivatives of the pressure with respect to and that include thermal contributions to in the calculation of . Explicitly,

 (∂Pth∂T∣∣∣n)2 = 49a2n2T2(1+q)2 (74) × {1−485π2a2T21+q[1−LF+q(1−LF3−109p4FE∗4F)]} ∂P