Non-congruent phase transitions in strongly interacting matter within the Quantum van der Waals model

The non-congruent liquid-gas phase transition (LGPT) in asymmetric nuclear matter is studied using the recently developed Quantum van der Waals model in the grand canonical ensemble. Different values of the electric-to-baryon charge ratio, $Q/B$, are considered. This non-congruent LGPT exhibits several features which are not present in the congruent LGPT of symmetric nuclear matter. These include a continuous phase transformation, a change in the location of the critical point, and the separation of the critical point and the endpoints. The effects which are associated with the non-congruent LGPT become negligible for the following cases: when $Q/B$ approaches its limiting values, $0.5$ or $0$, or if quantum statistical effects can be neglected. The latter situation is realized when the particle degeneracy attains large values, $g\gtrsim 10$.


I. INTRODUCTION
An infinite hypothetical system of interacting neutrons and protons in equal proportions is called symmetric nuclear matter. The known phenomenology of the nucleon-nucleon interaction suggests short range repulsion and intermediate range attraction. This yields a first-order liquidgas phase transition (LGPT) from a diluted (gaseous) to a dense (liquid) phase in symmetric nuclear matter, and, correspondingly, for a discontinuity of the particle number density as a function of pressure.
Asymmetric nuclear matter is of interest for both heavy ion-collisions and nuclear astrophysics: neutron-rich matter is present in compact stars, binary neutron star merges [30], and it is relevant for type-II supernova evolution [31,32]. Asymmetric nuclear matter exhibits a strong dependence on the isospin. As the Q/B ratio is fixed, this additional isospin degree of freedom can not be exploited by the system in pure phases. In the mixed phase, the total asymmetry is constant, while the local asymmetries in the gaseous fraction and in the liquid fraction can be different. Indeed, it is thermodynamically favorable for the total system if the liquid fraction is more symmetric than the gaseous one -this isospin distillation phenomenon has been predicted [33][34][35] long before for the exactly analogous phenomenon dubbed strangeness distillation in the first analysis of a production mechanism for possible strangelet formation in high energy heavy ion collisions and astrophysical situations. The additional isospin degree of freedom changes the energy density. This contribution becomes negligible at end points of the mixed phase. Hence, the pressure does not stay constant, but continuously changes as the system crosses the mixed phase region, while pressures of components remain equal in every point of the mixed phase [36]. Other features of the phase transitions (PTs) are modified as well. For instance the chemical potentials show similar behavior as pressure. This leads to an additional dimension in the phase diagram. Such PTs are called "Gibbs PTs", or, following the more recent terminology, "non-congruent PTs" [26]. This notion contrasts with the "Maxwell" or "congruent" PTs, where only one globally conserved charge is allowed. Asymmetric nuclear matter is therefore a role model for non-congruent PTs. These are most relevant also for the conjectured PT between a hadron gas and a quark-gluon plasma. The later may occur in the course of binary neutron stars merges and in relativistic heavy-ion collisions, where the associated distillation process was first proposed for strangeness as a signature of that PT [33][34][35].
For asymmetric nuclear matter, the order parameter therefore is not longer given by the difference between the net baryon densities of the liquid and the gaseous phases, n l − n g , but rather by the asymmetry factor, Q/B [25]. When Q/B is fixed, the system moves along the PT line in the (µ B , T )-plane. This corresponds to a continuous transformation from one pure phase to the other [20]. Only in the three special cases of asymmetry, Q/B = 0, 0.5, and 1, the phase transformation does lead to the appearance of discontinuities in thermodynamic variables, which is common for congruent PTs.
The properties of nuclear matter can be described by a variety of different models. Here we employ an extension of the classical van der Waals (vdW) model which was recently generalized to include the effects of quantum statistics, special relativity, grand canonical ensemble, and mixtures of different sized constituents. This Quantum vdW (QvdW) model has been further developed and applied to the description of symmetric nuclear matter in Refs. [37][38][39][40][41][42][43]. The QvdW approach models the repulsive interactions by the excluded-volume corrections, while the attractive interactions are modeled by a density-proportional mean field. The QvdW model describes the basic properties of nuclear matter rather well, the results are similar to the Walecka model [44]. A generalized QvdW-type formalism, based on models of real gases equations of state allows variations of the excluded-volume effects and of the attractive mean field [42].
The multi-component QvdW formalism, employed in the present work, allows for the study of systems with arbitrary numbers of different components [45,46].
The present paper studies the LGPT in asymmetric nuclear matter using the QvdW model. Section II introduces the QvdW model with separate baryonic and electric chemical poten-tials for neutrons and protons. Section III considers four special cases with congruent LGPTs, namely, the limiting cases of: 1) symmetric nuclear matter, Q/B = 0.5; 2) the extreme asymmetry, Q/B = 0; 3) an arbitrary Q/B ratio in the Boltzmann approximation; 4) an arbitrary Q/B ratio and infinite degeneracy. Section IV studies the general case of a non-congruent LGPT in nuclear matter with intermediate values of the asymmetry factor, 0 < Q/B < 0.5, and with physical values of (spin) degeneracy. Section V presents the calculation of the susceptibilities of the fluctuations of the baryonic and of the electric charges for asymmetric nuclear matter.
Section VI considers the non-congruent LGPTs within QvdW model generalized to take into account the isospin-dependencies of the attractive and repulsive parameters. A summary in Sec. VII closes the article.

II. NUCLEAR MATTER WITH TWO DIFFERENT CONSERVED CHARGES
We consider an infinite system of interacting nucleons consisting of neutrons and protons which differ only by the electric charge they carry. The total baryonic, B, and electric, Q, charges of the system in the grand canonical ensemble are regulated by the corresponding chemical potentials, µ B and µ Q . Then µ n = µ B is the chemical potential of the neutrons and µ p = µ B + µ Q is the chemical potential of the protons. The QvdW model yields the total pressure of the nucleons as [43]: Here T is the temperature, p id n , p id p are the pressures of the ideal Fermi gas of the neutrons and the protons, respectively. µ * B is the shifted baryon chemical potential due to the QvdW interactions: Here it is assumed that the repulsive excluded volume terms and the mean field attraction terms of protons and neutrons do not differ, and that their masses do not differ either. The interaction parameters, a and b, yield, respectively, the strength of the attraction and of the repulsion between the nucleons. As the interactions between all protons and neutrons here are assumed to be the same, also the shift in the chemical potential is the same for both, protons and neutrons. The densities of the baryonic and the electric charges are given by the partial derivatives of the pressure with respect to the corresponding chemical potentials, Here n id n = n id n (T, µ * B ) and n id p = n id p (T, µ * B + µ Q ) are the ideal gas densities of neutrons and protons, respectively. The pressure and the density of the ideal Fermi gas of neutrons, j = n, and protons, j = p, are given by The density of states corresponding to the momentum k is given by g j is the number of the internal quantum states -the degeneracy factor of the neutrons and the protons, which are spin 1/2 particles, therefore g n = g p = 2. The masses of both, neutrons and protons, are assumed to be equal, with m n = m p = 938 MeV.
As the values of the charges, B and Q, are conserved, the total system is also required to have the charge ratio Q/B, The thermodynamical functions at given T -and µ B -values are calculated by solving the self-consistent system of the four transcendental equations, (1),(2),(3), and (8), with respect to the four unknown quantities, µ * , µ Q , p, and n B .

III. FOUR SPECIAL CASES FOR CONGRUENT PHASE TRANSITIONS
A. Symmetric nuclear matter (g = 4) The LGPT in symmetric nuclear matter was studied within the QvdW model in Ref. [37].
The case of symmetric nuclear matter corresponds to a fixed value of the baryon-to-charge ratio, Q/B = 0.5. In this case, as follows from Eqs. (1),(3), and (8), the electric chemical potential is always zero, µ Q ≡ 0, and the multiplicities of both neutrons and protons are regulated by a single chemical potential, µ B . Hence, the system of nucleons is a single-component system.
In this case, the degeneracy factor of the nucleons is to be taken as g = 4, which includes two (isospin) charge states and two spin states.
The QvdW interactions are taken to be equal for all pairs of nucleons: a = 329 MeV fm 3 for the attractive term and b = 3.42 fm 3 for the repulsive term. These a and b values were obtained in Ref. [37] by fitting the binding energy and the saturation density of the ground state (GS; T = 0, p = 0) of symmetric nuclear matter: The position of the critical point (CP) of symmetric nuclear matter within QvdW model is [37]: The LGPT line in the (µ B , T ) coordinates as well as the LGPT region in the (n B , T ) coordinates were obtained in Ref. [37] within the QvdW model for symmetric nuclear matter.
They are presented in Fig. 1 (a) and (b), respectively, by the solid blue curves. The CP and the GS are represented by the blue star and the blue square, respectively.

B. Neutron matter (g = 2)
Another limiting case is the completely asymmetric nuclear matter consisting of neutrons only 1 . This corresponds to a zero asymmetry parameter, Q/B = 0, implying µ Q → −∞. In analogy to the symmetric nuclear matter case, the system is described by the single-component QvdW equation, but with a smaller degeneracy factor, g = 2, which counts only the two spin states of the neutron. The same values of the interaction parameters a and b are used as in the case of the nucleon-nucleon interaction in symmetric nuclear matter.
The values of the thermodynamic quantities in the nuclear GS of symmetric nuclear matter are fixed to known empirical values, see Eq. (9). These values fix parameters a and b for nucleons, as well as the value of the shifted chemical potential in the GS, µ * GS = 998 MeV 1 Note that neutron matter as discussed here does not match neutron star matter -the latter must include light and heavy nuclei, beta-equilibrium, leptons, strange hadrons, and eventually also a quark matter contribution. So the term neutron matter is used here only for a hypothetical state of neutrons only.
(µ GS = 921.5 MeV) [37]. As the degeneracy factor for pure neutron matter is twice smaller than that of symmetric nuclear matter, different GS properties for pure neutron matter are expected as compared to the symmetric nuclear matter. A straightforward calculation yields: One sees that the neutrons binding energy in the GS is positive, E GS b = 0.33 MeV > 0, thus the GS of pure neutron matter does exist in this model but neutron matter is not self-bound.
The position of the CP is determined by the following equations, which give, Mixed phase boundaries at given T < T c are derived from the Gibbs equilibrium condition: where which is independent of g-, m-, and Q/B-values [38]. The position of the CP is determined solely by the interaction parameters a and b [47]: Hence, the non-congruence of the LGPT vanishes 2 for arbitrary Q/B and g values, and the position of the CP is given by Eq. (16). Figure 1 where the critical values for the Boltzmann case are given by Eq. (16).
exhibits a CP at close to the position of the classical vdW CP (16).
The value of saturation density, n GS

IV. NON-CONGRUENT PHASE TRANSITION
The so-called non-congruent LGPTs occur for 0 < Q/B < 0.5. In this case µ Q is finite, and two conserved charges B and Q must be considered, which are tuned by the corresponding chemical potentials, µ B and µ Q . The electric-to-baryon charge ratio is kept fixed, Q/B = const.
While points (G1,L1) or (L2,G2) both have the same temperature, pressure, and chemical potentials, they differ by the values of the shifted chemical potential, µ * B , and densities of charges, n B , n Q . The requirement (8) of the constant charge ratio is imposed on the pure phases (G1, L2) only: The    The asymmetry in the gaseous fraction of the mixture is always larger then the asymmetry in the liquid fraction: this is the isospin distillation phenomenon, which is just the equivalent of the strangeness distillary considered in Refs. [33][34][35][36]. At T → 0, the infinitesimal gaseous fraction approaches the composition of the pure neutron matter, (Q/B) local → 0, while the infinitesimal liquid fraction approaches the composition of the symmetric nuclear matter, (Q/B) local → 0.5.
The system can be described in the mixed phase for an arbitrary proportion of fractions by introducing an additional parameter χ -the share of the total volume which is occupied by the liquid fraction. χ = 0 and χ = 1 correspond to the purely gaseous and to the purely liquid phase, respectively, while 0 < χ < 1 is realized for the mixed phase. The densities of the baryonic and of the electric charge for both, pure phases and the mixed phase, are given by Here n G B , n G Q and n L B , n L Q are the charge densities of the gaseous and the liquid fractions, respectively. In general, the volumes occupied by both fractions are finite. Hence, conservation laws shall be applied to the total mixture only, but not to the different components separately [33][34][35][36]. The requirement of a constant charge ratio (8) for both, pure phases and the mixed phase, reads The chemical potentials µ B and µ Q can be found at constant T and χ from equation (26) evaluated simultaneously with the Gibbs equilibrium condition,   in such a way that, for all T , ξ B = 0 at the start of the PT and ξ B = 1 at the finish of the PT. Q/B = 0.3 is used in Fig. 9. The two lines correspond to temperatures T = 15 MeV < T c and T c < T = 18.714 MeV < T TEP . The fraction of the liquid is zero in both cases at the start of the PT. The liquid fraction starts to increase monotonically when the density increases. In the case of subcritical temperatures, T < T c , the liquid fraction reaches χ = 1 at the finish of the PT. Correspondingly, the gaseous fraction reaches 1 − χ = 0, i.e., only the pure liquid phase is left. In contrast, the so-called retrograde condensation [48,49] occurs for supercritical temperatures in the narrow temperature range, T c < T < T TEP : at some value of the density, n max B , the fraction of the liquid phase reaches its maximum value, χ max , and with further increase of density, decreases rapidly. No liquid remains at the finish of the PT, and the system is in the purely gaseous phase again. Retrograde condensation is a unique feature of non-congruent PTs [50].

V. FLUCTUATIONS OF BARYONIC AND ELECTRIC CHARGES
The scaled variances of baryonic and electric charges fluctuations can be calculated as  respectively. This yields the following expressions: .
Here n N = n B − n Q is the neutrons number density. The ideal gas scaled variances of the baryon, neutron, and proton multiplicity fluctuations are given by , respectively. The ideal gas variance of particles multiplicity fluctuations is calculated as For symmetric nuclear matter, µ Q = 0, Eq. (30) reduces to the following form, where The scaled variances given by Eq. (34) satisfy the relation where q = n Q /n B is the probability of detecting a proton, i.e., the charge fluctuations are obtained from binomial folding of baryon number fluctuations. This is a well-known generic feature of Boltzmann systems [51]. In contrast, the quantum statistics does violate the relation (35).
The correlation between the baryonic and the electric charges is calculated as follows:

VI. ISOSPIN-DEPENDENT INTERACTION PARAMETERS
Throughout this work we have assumed that the QvdW interaction parameters are the same for proton-proton, proton-neutron, and neutron-neutron interactions, i.e. the isospin dependence of the N N potential was not considered. The model predicts the symmetry energy value of about J 20 MeV, which is 10-15 MeV lower than the empirical estimate [52]. In addition, the liquid-gas phase transition in pure neutron matter, predicted by the model (see Fig. 1), appears to be ruled out by chiral effective field theory [53]. An improved description of asymmetric nuclear matter within the QvdW approach can be achieved by considering isospin dependent QvdW parameters.
In Ref. [46] the multi-component QvdW model was formulated where one can specify the attractive and repulsive QvdW parameters for each pair of particle species. This formalism is applied here for asymmetric nuclear matter. We define a nn , a pp , and a pn = a np as attractive QvdW parameters for neutron-neutron, proton-proton, and neutron-proton interactions, respectively. Similarly, b nn , b pp , and b pn = b np are repulsive QvdW parameters. The assumed isospin symmetry yields b pp = b nn and a pp = a nn . Therefore, the model has four QvdW interaction parameters: a nn , a np , b nn , and b pn . The isospin-dependent multi-component QvdW equation for the pressure is Here the shifted chemical potentials µ * n and µ * p are given by The generalized equations for neutron and proton densities are [46] n n (T, If all QvdW parameters are assumed to be equal, a nn = a np , b nn = b np , Eqs. The symmetric nuclear matter, µ Q = 0, n N = n Q , reduces to a single-component system with interaction parameters a = a nn + a np 2 The values of a and b are fixed by fitting the known ground state properties of symmetric nuclear matter: a = 329 MeV fm 3 and b = 3.42 fm 3 (see Sec. III A). Therefore, the model has two free parameters: a np /a nn and b np /b nn . We fix these parameters to reproduce the constraints on the symmetry energy and its slope at normal nuclear density. We take the following values: a np /a nn = 2.5 and b np /b nn = 1.7. This yields symmetry energy J = 32 MeV and its slope L = 51 MeV which are consistent with empirical constraints [54][55][56][57][58][59][60].
The red line in Fig. 12 shows the dependence of the binding energy on n B at zero temperature for pure neutron matter within QvdW model with a np /a nn = 2.5 and b np /b nn = 1.7. The gray band represents result for pure neutron matter within chiral effective mean field model [53] with NN and 3N interactions and a renormalization-group evolution. The width of the band is mainly due to uncertainties in 3N forces. In the considered case a np /a nn = 2.5 and b np /b nn = 1.7 the minimum in the binding energy as a function of density is absent for Q/B 0.08. T T E P , a p n / a n n = 2 . 5 , b p n / b n n = 1 . 7 T T E P , a p n = a n n , b p n = b n n T c , a p n = a n n , b p n = b n n T c , a p n / a n n = 2 . 5 , b p n / b n n = 1 . n T E P , a p n / a n n = 2 . 5 , b p n / b n n = 1 . 7 n T E P , a p n = a n n , b p n = b n n n c , a p n = a n n , b p n = b n n n c , a p n / a n n = 2 . 5 , b p n / b n n = 1 . 7 X Figure  The comparison between isospin-independent and isospin-dependent parameterizations of the QvdW interactions in Fig. 13 shows that the former is appropriate for possible applications to heavy-ion collisions, which are typically characterized by Q/B 0.4.

VII. SUMMARY
The non-congruent liquid-gas phase transition in asymmetric nuclear matter with two glob- The Quantum van der Waals model with isospin-dependent interaction parameters, constrained to empirical values of the symmetry energy and its slope, has also been considered.