Critical Exponents of Quark Matter

I investigate the ferromagnetic phase transition inside strong quark matter (SQM) with one gluon exchange interaction between strong quarks. I use a variational method and the Landau-Fermi liquid theory and obtain the thermodynamics quantities of SQM. In the low temperature limit, the equation of state (EOS) and critical exponents for the second-order phase transition (ferromagnetic phase transition) in SQM are analytically calculated. The results are in agreement with the Ginzberg-Landau theory.


Exchange and Direct Contributions to the Energy Density
The Landau-Fermi liquid interaction function is related to the Lorentz invariant matrix element via , the polarized states have greater energy than the un-polarized states [24,25].The low temperature domain characteristic is Fermi energy of system.If the Fermi energy of system is greater than thermal energy, then we can use the low temperature limit.In SQM, the order of Fermi energy is greater than 20 MeV [25]; and low temperature means T MeV.Therefore, the ferromagnetic phase can appear in the non-relativistic region [25].Therefore only strange quarks are involved and the results of the present paper are applicable to SQM (The reason will be explained in Section II).The resulting scattering matrix elements in the non-relativistic region automatically have spin-dependent terms and we do not insert the spin-spin interaction manually [26].I use a variational approach [27][28][29][30][31][32][33][34][35] to obtain the equation of state of the system.By varying the free energy with respect to p (the polarization parameter) and the effective mass at various densities and temperatures, we can minimize the free energy for a given density and temperature.

, ,
, (antiparallel spins), we have the flip interaction.So the exchange energy density for the flip and non-flip interactions can be written as  correspond to In the above equations, where the  and p are the density of the spin up and spin down electrons and the polarization parameter, re- is the Fermi distribution functions.In the non-relativistic region, we can use the approximation and then we have [23] 2 Using Equation ( 5), if we have      then the Lorentz invariant matrix elements vanish, which means that in the non-relativistic region the spin flip contribution to the energy density vanishes.The spin non-flip exchange   and kinetic energy density in the non-relativistic case at zero temperature is In the ultra-relativistic region , the exchange energy density is proportional to and the kinetic energy is also proportional to the un-polarized state is energetically favorable and we will write all equations in the non-relativistic limit.

Equation of State at Low Temperature
To obtain the equation of state at low temperature Fermi , I use the variational method with the following approximation to the single particle energy in the Fermi distribution function [2,27,35]: Using the above approximation, I use the redefined kinetic energy instead of the sum of kinetic and potential energy in the distribution function, and the kinetic energy in the other parts of the equations remains unchanged [2,27,33,35].For a fully polarized state p = 1 and an unpolarized state p = 0, I can describe the system with a single effective mass and chemical potential.For partially polarized states , we must use separate effective masses and chemical potentials for the spin up and spin down states.If we use the following relation, (where is the Lerch Phi function), we can write the kinetic energy as: To obtain the kinetic energy at low temperature, we use the relations Here, ± refers to spin up and down states.If we use the low temperature expansion of the chemical potential, then we have: In Equation (11), kin is the kinetic energy density of the system.Similar to the above, the exchange energy and entropy density of the system become Here,  , kin 0  , and ex 0  are the effective mass of the electrons, the non-relativistic kinetic and exchange energies at zero temperature, respectively.Using the results of Equation ( 12), it follows that the free energy density is m By minimizing the free energy, one can find  as a function of p: For p = 0 and non-interacting systems, Equation ( 14) simply yields  .We must notice here that if we set 0   (non-interacting system), then p = 0.At zero temperature, minimization of the free energy becomes simpler, and one obtains 0 0 0.
The resulting equation from Equation ( 15) is This result is very similar to the well-known results of the spontaneous magnetization of an imperfect Fermi gas [21].But in our calculation, the interaction part of the Hamiltonian is not independent of the spin alignment, and this dependence changes the right side of Equation (16).

Critical Exponents
Similar to previous work on a Fermi gas [36], solving Equation ( 16) yields Figure 1.Also the results for nonzero temperature are similar to the results of [36].Here I am interested in the density and temperature dependence of the polarization parameter and the other thermodynamic quantities.At zero temperature, we can expand the magnetic susceptibility As π , the first term in Equation ( 17) becomes zero.So The magnetization is proportional to the polarization  Expanding  near  , we find    does not depend on  , so we have 0  .

Results and Discussion
I use the variational approach to obtain the critical exponents of quark matter.The method is based on the minimization of the free energy (corresponding to maximum entropy at equilibrium).Using this method, Fermi systems, such as an electron gas or quark matter, can be in a ferromagnetic phase for a specific value of the density and temperature.For quark matter, this can be happen at low temperature and high density.For SQM in chemical equilibrium, the density of the u, d, and s quarks can be calculated from the weak interactions between quarks [25].The up and down quarks are in the ultra-relativistic region (because of their small mass) and only the s quarks can be in the ferromagnetic phase [25].Therefore, all equations are for strange quarks.This phase transition is second order [36] and one can calculate the critical exponents of this phase transition.The results are the same as for the Landau mean field theory.

Figure 1 .
Figure 1.The polarization as a function of the Fermi momentum and coupling constant.

I
would like to thank the Research Council of the University of Tehran and the Institute for Research and Planning in Higher Education for financial support under contract No. 138-569.