Based on the generalized Ginzburg-Landau theory, chiral phase transition is discussed in the presence of magnetic field. Considering the chiral density wave we show chiral anomaly gives rise to an inhomogeneous chiral phase for nonzero quark-number chemical potential. Novel Lifshitz point appears on the vanishing chemical potential line, which may be directly explored by the lattice QCD simulation.Comment: 4pages,2figure

We study the phase diagram of the Nambu-Jona-Lasinio model in the external magnetic field within the mean-field approximation, taking into account the inhomogeneous chiral condensate. It is shown that there appears a new type of the chiral condensate, endowed with two features of real kink crystal and dual chiral density wave, in the magnetic field. We also show that there are first order phase transitions between different inhomogeneous phases in the presence of magnetic field.

First, we develop the counting rule for Nambu-Goldstone bosons to the system including two-time derivative terms. In this case, the type-II Nambu-Goldstone bosons may appear along with the massive Nambu-Goldstone ones. The number of the bosons is not reduced in contrast to the system without two-time derivative terms. We also investigate the reduction of the degrees of freedom from the perspective of the Dirac-Bergmann theory of constraints and reproduce the counting rule for Nambu-Goldstone bosons without Lorentz invariance. Then, we construct the generic Higgs model and study a Higgs phenomenon with these Nambu-Goldstone bosons on the basis of the Dirac-Bergmann theory. We show that the gauge fields in this system absorb all of the Nambu-Goldstone bosons such as the type-I, type-II and massive ones.

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The inhomogeneous chiral phase is discussed in QCD at finite temperature and/or density. We study the phase diagram on the density-temperature plane by taking into account the effect of the current mass by a variational method. It is demonstrated that our framework well describes the inhomogeneous phase over the whole phase region. Keywords: quark matter, QCD phase diagram,chiral symmetry,inhomogeneous condensate, variational methodRecently the inhomogeneous chiral phases have been actively studied in the QCD phase diagram; where the chiral condensates (order parameter) have a spatial dependence. Inhomogeneous order parameter has been studied as well in condensed matter physics; e.g., spin density wave [1,2], charge density wave [3], Fulde-Ferrell-Larkin-Ovchinikov superconductor [4,5] and so on. We here consider the similar subject about chiral transition within QCD. It has been shown in [6,7] that there are exact solutions of inhomogeneous chiral condensates in 1+1 dimensions and inhomogeneous phases actually develops up to a critical temperature. It has been shown that analytic solutions are no longer real functions but complex functions such as D(z) = ψ ψ + i ψ iγ 5 τ 3 ψ = ∆(x)e iθ(x) , and represented by the Jacobi elliptic functions in the two dimensional NJL model in the large N c limit. In [8], it has been shown that one dimensional inhomogeneous structure can be embedded in 1+3 dimensions. He showed that the energy spectrum of quarks in the 1+1 dimensional systems can be generalized to 1+3 dimensions by operating the Lorentz boost in the perpendicular direction to the spatial modulation. He then applied such a method for the real kink crystal (RKC), which has a spatial dependence in the amplitude of D(z); D(z) = ∆(z), and has shown that the inhomogeneous chiral phases may appear in the low temperature and/or moderate density region before chiral transition in the QCD phase diagram in 1+3 dimensions. Similar procedure can be applied for another type of condensations. Actually, prior to [8], [9] have suggested that dual chiral density wave (DCDW), which has a spatial dependence in the phase of D(z); D(z) = ∆e iθ(z) , also appears in the QCD phase diagram within the 1+3 dimensional NJL model.