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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.

The properties of Fermi systems have been investigated in several works [1-20]. One of the important cases is the study of the magnetic properties of an electron gas [3-20]. Spontaneous magnetization may appear at different densities for different temperatures and the polarization of the system is a function of the density and temperature. By assuming a spin-spin interaction inside the system, we can study its magnetic properties. For gaseous systems, statistical methods for an imperfect Fermi gas show that the system can be in its ferromagnetic phase [21-23]. All relations are written in the non-relativistic and low temperature limit, because in the ultra-relativistic region, 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 [

The Landau-Fermi liquid interaction function is related to the Lorentz invariant matrix element via

If (parallel spins), we have the spin non-flip interaction and if (antiparallel spins), we have the flip interaction. So the exchange energy density for the flip and non-flip interactions can be written as

In the above equations, correspond to

where the and p are the density of the spin up and spin down electrons and the polarization parameter, respectively. is the Fermi distribution functions. In the non-relativistic region, we can use the approximation and then we have [

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

[

To obtain the equation of state at low temperature, 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), is the kinetic energy density of the system. Similar to the above, the exchange energy and entropy density of the system become

Here, , , and 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

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 (non-interacting system), then p = 0. At zero temperature, minimization of the free energy becomes simpler, and one obtains

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 [

Similar to previous work on a Fermi gas [

As, the first term in Equation (17) becomes zero. So

The magnetization is proportional to the polarization (p). The relation between and the magnetization and magnetic field (J) is

At we have

Using (18) and (19), we find that. At non-zero temperature and, we have

Expanding near, we find

So. As, we have. Comparing with Equation (21), we find that. It can be seen that the heat capacity does not depend on, so we have.

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 [

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.