Color Ferromagnetism in Quark Matter
Abstract
We show a possibility that there exists a color ferromagnetic state in quark matter, in which a color magnetic field is spontaneously generated. The state arises between the hadronic state and the color superconducting state when the density of quarks is varied. Although the state ( Savvidy state ) has been known to involve unstable modes of gluons, we show that the modes compose a quantum Hall state to stabilize the ferromagnetic state. We also show that the order of the phase transition between the state and the quark gluon plasma is of the first order.
pacs:
PACS 12.38.t, 12.38.MN, 24.85.+p, 73.43.fQuark Matter, Color Superconductivity, Quantum Hall States
Quark matter possesses several phase structures when its temperature and densiy are varied; the hadronic phase, the quark gluon plasma phase, and the color superconducting phase. Especially, recent progress[1] in the color superconducting phase has been paid much attention. The superconducting phase is realized as dynamical effects of quark matter, namely, the condensation of Cooper pairs of quarks due to instability of Fermi surface of quark gas against attractive forces. On the other hand, hadronic phase is realized as dynamical effects of gluons, namely, the condensation of color magnetic monopoles[2]. SU(3) gluons become effectively Abelian gluons and color magnetic monopoles at large distance[3], and the monopoles condense to form a dual superconductor. Therefore, both of these phases are characterized as superconducting states, i.e. electric and magnetic superconductors.
In this letter we point out a possible existence of a color ferromagnetic state[4] in which the color magnetic field is spontaneously generated. It is very intriguing that a quantum Hall state[5] of some gluons, which has been known previously to be unstable modes[6], is formed to stabilize the ferromagnetic state. Such a quantum Hall state carrying color charges is shown to be possible only when quark matter is present. The quark matter is shown to have lower energy density in the ferromagnetic phase than that in the color superconducting state when the chemical potential is less than approximately . On the other hand, when the chemical potential is larger than , the energy density in the superconducting phase is lower than that in the ferromagnetic state. Hence, the ferromagnetc phase is expected to arise between the hadronic phase and the superconducting phase. The actual value of is a fairly important quantity characterizing the typical energy scale of the state although it is not yet determined in this paper. Here we quote a typical energy scale in QCD, GeV simply for indicating a order of the value. Furthermore, we show that the ferromagnetic phase of quark matter may arise at chemical potentials larger than where is the radius of the qurak matter produced in heavy ion collisions.
Let us first explain our strategy. We consider mainly the SU(2) gauge theory with massless quarks of two flavours and make a comment on the case of SU(3) gauge theory. We are concerned with the chemical potential around MeV so that the coupling is not small. But we assume that the results obtained with the one loop approximation are physically correct. Then, we use the one loop effective potential of color magnetic fields calculated previously [4]. Similarly, quarks in the magnetic field are assumed to be almost free particles, which interact with the spontaneously generated color magnetic field. Thus, important quantities in this paper are the one loop effective potential of the color magnetic fields and the energy density of quarks in the field. By comparing the energy density of the quarks in the color magnetic field with that of the quarks in the BCS state, we find that a ferromagnetic phase is realized energetically for chemical potential less than .
As is well known[4, 6], the effective potential of the constant color magnetic field has the minimum at nonzero magnetic field and also has an imaginaly part at the minimum:
The presence of the imaginary part of the potential leads to excitation of unstable modes around the minimum . This is similar to the case that when we expand a potential of a scalar field around the local maximum, , i.e. wrong vacuum, unstable modes are present with their energies such as . They are excited and eventually lead to the stable vaccum with condensation of a constant unstable mode with . In the gauge theory similar condensations of gluon unstable modes are expected to arise.
In order to explain the unstable modes in the gauge theory, we decompose the gauge fields such that denote color components. Then, we may suppose that the field is the ’electromagnetic’ field of the U(1) gauge symmetry and is the charged vector field, where indices
(1)  
with , where we have omitted a gauge term . We can derive using the Lagangian that the energy of the charged vector field under the magnetic field is given by with a gauge choice, and , where we have taken the spatial direction of being along axis. ( the integer ) denote contributions of spin components of ( Landau levels ) and denotes momentum in the direction parallel to the magnetic field. We note that in each Landau level, there are degenerate states in momentum with the degeneracy given by per unit area; where represents Harmonic oscillation of the nth order with its center in .
It turns out from the spectrum that the states with parallel magnetic moment () in the lowest Landau level ( ) are unstable when . Therefore, we expect from the lesson in the scalar field that the unstable modes are excited and lead eventually to true stable vacuum with the condensation of the unstable modes with . Actualy, there have been several attempts[7] to find the true vacuum by making the condensation of the unstable modes. The difficulty to find the true vacuum is that since the unstable modes are in the lowest Landau level so that the condensation of the modes gives rise to nonuniform state; their wave functions are localized in the coordinate . Furthermore, it was difficult to see whether or not any unstable modes dispappear in the condesed state. In the final section of the paper, we will see that the excitation of the modes leads to a uniform quantum Hall state ( QHS ) without any unstable modes. Most effecient way to see the uniformness of the QHS is using ChernSimons gauge theory[8, 5] in dimensions.
First, we estimate energies of quarks in the magnetic field and compare energy density of quarks in the field with that of quarks in BCS state. First of all, we note that the energy of the quark in the color magnetic field is given by , where and is a momentum parallel to the magnetic field and takes a value of , representing spin contributions. Remember that since we are considering SU(2) gauge theory with massless quarks of two flavours, we have two types of quarks whose color charges are positive and negative, respectively for each flavour. Both of the quarks have the identical energy. There are many degenerate states specified by momentum in each Landau level. The degeneracy is given by per unit area, where the factor comes from positive and negative color charged quarks with each flavour.
The number density of the quarks is estimated such that
Then, we have proved numerically by equating both densities, that for any , where the ratio goes smoothly to zero ( one ) as goes to zero ( infinity ). In general, from the dimensional analysis the ratio is a function only of the variable, . Therefore, we find that the energy density of quarks in the magnetic field with any strength is lower than that of free quarks.
We have estimated numerically how the energy density becomes lower with increasing the Fermi energy,
(2) 
for , , , and , respectively. We have also found numerically that the above quantity goes to zero such as with increasing.
We should compare these values with those of the energy decrease when the BCS state is realized. The energy decrease may be estimated as follows. That is, only the quarks in the vicinity of the Fermi surface whose width may be given by a gap energy , gain energy by making Cooper pairs. Thus the decrease of the energy density is given such as . Normalizing it by , we find . Thus, the decrease of the energy ratio in the BCS state is slower than that of the energy ratio in the ferromagnetic state when the Fermi energy increases. It implies that the energy gain of quarks in the BCS state is much larger than that in the ferromagnetic state for sufficiently large chemical potential. ( The one loop effective potential energy of the magnetic field is much smaller than those of the quarks so that we have ignored the energy in the discussion. ) Hence, the BCS state is realized for such a high density of quarks. On the other hand, for sufficiently small chemical potential the energy gain of quarks in the ferromagnetic state is larger than that in the BCS state. Actually, the energy gain [9] in the BCS state is at most about percents of since has been estimated at most as MeV when the Fermi energy is about GeV. On the other hand, the energy gain in the presence of the magnetic field becomes percents when the Fermi energy is equal to . Although we have not yet determined the value of , we expect that the value of is order of several handred MeV, e.g. MeV. Hence, we find that the phase boundary between the ferromagnetic phase and the superconducting phase is present around the chemical potential of quarks being about MeV. We are addressed later with the minimum chemical potential needed for the realization of the ferromagnetic phase.
Up to now, we have considered the energy densities of the quarks and the gauge fields at zero temperature. It is easy to calculate the free energy at finite temperature of the quarks and the gluons in the magnetic field. In the case we neglect the contributions of unstable modes; the modes condense to form a quantum Hall state in which any unstable modes are absent.
We have numerically estimated the free energy and found that the color magnetic field becomes large with the chemical potential. This is owing to the magnetic moment generated by the quarks,
We proceed to show briefly that the unstable modes compose a quantum Hall state and that a stable ferromagnetic state is realized. We note that the unstable modes are states in the lowest Landau level with the energy given by ; . The denenerate states in the Landau level may be specified by a momentum whose wave functions behave such as with must be excited and we must take average over in a state with the unstable modes excited. Along this strategy ones have tried[7] to find variationally a stable ground state. However, the resultant state has not been uniform in space and not been shown to have no unstable modes although it is stable in the variational parameters. . In order to form spatially uniform state, various modes in
In order to see a uniform QHS composed of the unstable modes we use ChernSimon gauge theory. Note that the unstable modes with can be described effectively by a dimentional scalar field coupled with the magnetic field ( ): , with , where denotes the coherent length of the magnetic field, namely, its extention in the direction of the field and the index runs from to . This Lagrangian can be obtained by taking only the unstable modes with from eq(1); we have ignored the other modes coupled with the unstable modes. The second term represents anomalous magnetic moment which leads to the instability of the state . The term can be regarded as a negative mass so that the model is similar to the model of the scalar field with the double well potential. We see that the particles of interact with each other through a delta function potential. Thus, we expect that the excitations of the unstable modes may eventually form a QHS similar to the case of electrons interacting repulsively with each others under magnetic field.
In order to describe QHSs, we rewrite the Lagrangian by using ChernSimons gauge field [8],
(3) 
with antisymmetric tensor with , where the statistical factor should be taken as integer to keep the equality of the system described by to that of . Namely, in two dimentional space the statistics of particles can be changed by attaching a ficticious flux to the particles. The introduction of the field is to attach the flux to the particles of . When we take integer, the statistics of the particles does not change. In our case the bosons of become new bosons of . The equivalence of this lagrangian to the original one has been shown [8] in a operator formalism although the equivalence had been known in the path integral formalism using the world lines of the particles.
It is well known that QHSs can be described by this type of ChernSimons gauge theory even in the mean field approximation [5]. Equations of motion are given by
(4)  
(5)  
(6) 
where we have taken .
Using these equations, we now explain how the ferromagnetic state is stabilized by making a QHS. The essence is that in the QHS the ficticious flux is used to cancel on average with the real magnetic flux, i.e. . Consequently, the field does not feel any gauge field and the lagrangian eq(3) is reduced to one representing an usual double well potential. Hence we find a uniform solution, : is obtained by solving the above equations which are reduced to
We should point out that since the condensate of possesses a color charge ( ), the charge must be supplied from somewhere in color neutral system. Without the supplier of the color charge, the condensation can not arise so that the quantum Hall state is not realized. Thus, the color ferromagnetic state is not stabilized. Quark matter is a supplier of the color charge. Hence, the stable ferromagnetic state is possible in dense quark matter; some color charges of quarks are transmitted to the condensate. Then, the color charge density of the quark matter with the radius should be larger than that of the color condensate, . Thus, it follows that the chemical potential, , should be larger than in order for the ferromagnetic phase to arise in the quark matter. Since it is a value nessesary for the realization of the phase, a critical value separating the two phases, hadronic phase and ferromagnetic phase is larger than it.
Up to now, we have discussed the ferromagnetic state of the SU(2) gauge theory with massless quarks of two flavours. The orientation in color space of the magnetic field generated spontaneously can be taken arbitrary in the case of the SU(2) gauge theory. On the other hand, in the SU(3) gauge theory the different choice of the orientation leads to different physical unstable modes[7]; three unstable modes are present in general but one of the modes vanishes in a specific orientation. The orientation can be determined in the quark matter by minimizing the energy density of the quarks; the energy depends on the orientation of the magnetic field. Namely, when we take ( ), we impose the color neutrality of the quark matter and minimize its energy. Then, we can find the orientation of the magnetic field. The detail will be published in near future.
Finally, we show that a real observable magnetic field is produced by quarks rotating around the color magnetic field. If the number of the color positive charged quarks is the same as that of the color negative charged quarks, the total real magnetic moment produced by the quarks vanishes. But the number of the color positive charged quarks and that of the color negative charged quarks is different in the color neutral system due to the gluon condensation with the color charges in the QHS. Therefore, the real magnetic moment produced by, for example, up quarks does not vanish. Taking being several and the radius of quark matter being several fm, we can show that the real magnetic field with strength Gauss is produced in the color ferromagnetic phase of the quark matter, which may be generated by heavy ion collisions.
We would like to express thanks to Profs. T. Kunihiro, T. Hatsuda and M. Asakawa for useful discussions. One of the authors ( A. I. ) also expressess thanks to the member of theory group in KEK for their hospitality.
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