Inhomogeneous Polyakov loop induced by inhomogeneous chiral condensates

The Nambu-Goldstone modes on the exotic chiral condensed phase with chiral and tensor-type quark-antiquark condensates are investigated by using the two-point vertex functions. It is shown that one of the Nambu-Goldstone modes appears as a result of meson mixing. As is well known, another method to find the Nambu-Goldstone modes is given by the use of the algebraic commutation relations between broken generators and massless modes obtained through the spontaneous symmetry breaking. This method is adopted to the cases of the chiral symmetry breakings due to the tensor-type condensate and the inhomogeneous chiral condensate. The result obtained by the use of the meson two-point vertex functions is obviously reproduced in the case of the tensor-type condensate. Furthermore, we investigate the general rules for determining the broken symmetries and the Nambu-Goldstone modes algebraically. As examples, the symmetry breaking pattern and the Nambu-Goldstone modes due to the tensor-type condensate or the inhomogeneous chiral condensate are shown by adopting the general rules developed in this paper in the algebraic method. Published by the American Physical Society 2025

It is known that, in the chiral limit, spatially inhomogeneous chiral condensate occurs in the Nambu-Jona-Lasinio (NJL) model at finite density within a mean-field approximation. We study here how an introduction of current quark mass affects the ground state with the spatially inhomogeneous chiral condensate. Numerical calculations show that, even if the current quark mass is introduced, the spatially inhomogeneous chiral condensate can take place. In order to obtain the ground state, the thermodynamic potential is calculated with a mean-field approximation. The influence of finite current mass on the thermodynamic potential consists of following two parts. One is a part coming from the field energy of the condensate, which favors inhomogeneous chiral condensate. The other is a part coming from the Dirac sea and the Fermi sea, which favors homogeneous chiral condensate. We also find that when the spatially inhomogeneous chiral condensate occurs, the baryon number density becomes spatially inhomogeneous.

We investigate the effect of strange quark degrees of freedom on the formation of inhomogeneous chiral condensates in a three-flavor Nambu--Jona-Lasinio model in mean-field approximation. A Ginzburg-Landau study complemented by a stability analysis allow us to determine in a general way the location of the critical and Lifshitz points, together with the phase boundary where the (partially) chirally restored phase becomes unstable against developing inhomogeneities, without resorting to specific assumptions on the shape of the chiral condensate. We discuss the resulting phase structure and study the influence of the bare strange-quark mass $m_s$ and the axial anomaly on the size and location of the inhomogeneous phase compared to the first-order transition associated with homogeneous matter. We find that, as a consequence of the axial anomaly, critical and Lifshitz point split. For realistic strange-quark masses the effect is however very small and becomes sizeable only for small values of $m_s$.

We study the phase structure of the two-flavor Nambu--Jona-Lasinio (NJL) model in the chiral limit, extending a previous study of the competition of an inhomogeneous chiral phase and a two-flavor color-superconducting (2SC) phase [1, 2]. There, an analytic expression for the dispersion relations for quasiparticle excitations in the presence of both a particular inhomogeneous chiral condensate, the so-called chiral density wave (CDW), and a homogeneous 2SC condensate was found. In this work we show how to determine the dispersion relations for arbitrary modulations of the chiral condensate in the presence of a homogeneous 2SC condensate, if the dispersion relations in the absence of color superconductivity are known. In our calculations, we employ two different Ans\"atze for the inhomogeneous chiral condensate, the CDW as well as the real-kink crystal (RKC). Depending on the value of the diquark coupling we find a region of the phase diagram where the inhomogeneous chiral and the 2SC condensates coexist, confirming results of Refs. [1, 2]. Decreasing the diquark coupling favors the inhomogeneous phase over the coexistence phase. On the other hand, increasing the diquark coupling leads to a larger 2SC phase, while the inhomogeneous chiral and the coexistence phases become smaller. In agreement with previous studies the RKC Ansatz is energetically preferred over the CDW Ansatz. Both Ans\"atze lead to a qualitatively similar phase diagram, however the coexistence phase is smaller for the RKC Ansatz.

We update a previous study of the effects of vector interactions in the Nambu--Jona-Lasinio model on the formation of inhomogeneous chiral symmetry breaking condensates. In particular, by properly considering a spatially modulated vector mean-field associated with the quark number density of the system we show that, as the value of the vector coupling increases, a chiral density wave modulation can become thermodynamically favored over a real sinusoidal modulation. This behavior is found both via a Ginzburg-Landau analysis close to the Lifshitz point, as well as with a full numerical diagonalization of the mean-field Dirac Hamiltonian at vanishing temperature.

We investigate inhomogeneous chiral condensation under rotation considering finite size effects and boundary conditions in the holographic QCD model. The rotational suppression effect determined by $\Omega r$ is confirmed in the holographic model which is not influenced by the boundary conditions. For chiral condensation at the center, it is found that under Neumann boundary condition the finite size exhibits two opposite effects, i.e., catalysis at high temperatures and inverse catalysis at low temperatures. In contrast, under Dirichlet boundary condition, the effect of finite size on condensation is inverse catalysis, and small size induces a phase transition from inhomogeneous to homogeneous phase. The temperature-angular velocity phase diagrams of QCD are obtained for different boundary conditions and sizes, and it is found that the critical temperature decreases with angular velocity.

We present the detailed properties of a self-consistent crystalline chiral condensate in the massless chiral Gross-Neveu model. We show that a suitable ansatz for the Gorkov resolvent reduces the functional gap equation, for the inhomogeneous condensate, to a nonlinear Schroedinger equation, which is exactly soluble. The general crystalline solution includes as special cases all previously known real and complex condensate solutions to the gap equation. Furthermore, the associated Bogoliubov-de Gennes equation is also soluble with this inhomogeneous chiral condensate, and the exact spectral properties are derived. We find an all-orders expansion of the Ginzburg-Landau effective Lagrangian and show how the gap equation is solved order by order.

It is shown that the inhomogeneous chiral condensate in the Gross-Neveu (GN) model takes the chiral spiral form, even though the thermodynamic functional depends only on the chiral scalar density. It is the inhomogeneity of the chiral scalar condensate that drives the spatial modulations of the pseudoscalar one. The result has broader implications once we start to think of fundamental theories behind the effective models. In particular, some effective interactions---which may be omitted for descriptions of the homogeneous phases---can be dynamically enhanced due to the spatial modulations of the large mean fields. Implications for the four-dimensional counterparts of the GN model are discussed. In a quark matter context, proper forms of the effective models for the inhomogeneous phases are speculated, through considerations on the Fermi-Dirac sea coupling.

We investigate the dilepton production rates from annihilation processes of charged pion pairs with modified pion dispersion relations in the inhomogeneous chiral condensed phase. We assume a dual chiral density wave as an inhomogeneous chiral condensate, and obtain the dispersion relations of the Nambu-Goldstone modes in the inhomogeneous chiral condensed phase. We use a low energy effective Lagrangian based on the O(4) symmetry which is expanded by the order parameter up to the sixth order. The obtained dispersion relations are anisotropic and quadratic for the momentum. We evaluate the electron-positron production rates by charged pion-pair annihilations as functions of an invariant mass using the obtained dispersion relations. Basically, the production rate in the inhomogeneous chiral condensed phase has a steeper overall slope with respect to an invariant mass than that in the homogeneous chiral condensed phase. Therefore, the production rate may be enhanced when the invariant mass is around twice the pion mass. Published by the American Physical Society 2024

We investigate the possibility of spatially inhomogeneous chiral and Cooper, or superconducting, pairing in the (1+1)-dimensional model by Chodos et al. [Phys. Rev. D 61, 045011 (2000)] generalized to continuous chiral invariance. The consideration is performed at nonzero values of temperature T and quark number chemical potential μ. In the framework of the Fulde–Ferrel inhomogeneity ansatz for chiral and Cooper condensates, we show that if G1>G2, where G1 and G2 are the coupling constants in the quark–antiquark and diquark channels, then in the (μ, T)-phase diagram the superconducting phase is suppressed by spatially inhomogeneous chiral spiral phase with broken chiral symmetry. In contrast, in the above mentioned original Chodos et al. model, where only the opportunity for homogeneous condensates is taken into account, the superconducting phase is realized at sufficiently high values of μ at arbitrary values of G2>0, including the interval 0<G2<G1.

We present an extensive study on inhomogeneous chiral condensates in QCD at finite density in the chiral limit using a generalized Ginzburg-Landau (GL) approach. Performing analyses on higher harmonics of one-dimensionally (1D) modulated condensates, we numerically confirm the previous claim that the solitonic chiral condensate characterized by Jacobi's elliptic function is the most favorable structure in 1D modulations. We then investigate the possibility of realization of several multidimensional modulations within the same framework. We also study the phase structure far away from the tricritical point by extending the GL functional expanded up to the eighth order in the order parameter and its spatial derivative. On the same basis, we explore a new regime in the extended GL parameter space and find that the Lifshitz point is the point where five critical lines meet at once. In particular, the existence of an intriguing triple point is demonstrated, and its trajectory consists of one of those critical lines.

Nambu--Jona-Lasinio-type models have been used extensively to study the dynamics of the theory of the strong interaction at finite temperature and quark chemical potential on a phenomenological level. In addition to these studies, which are often performed under the assumption that the ground state of the theory is homogeneous, searches for the existence of crystalline phases associated with inhomogeneous ground states have attracted a lot of interest in recent years. In this work, we study the Polyakov-loop extended Nambu--Jona-Lasinio model and find that the existence of a crystalline phase is stable against a variation of the parametrization of the underlying Polyakov loop potential. To this end, we adopt two prominent parametrizations. Moreover, we observe that the existence of a quarkyonic phase depends crucially on the parametrization, in particular in the regime of the phase diagram where inhomogeneous chiral condensation is favored.

Using the Dirac-mode expansion method, which keeps the gauge invariance, we analyze the Polyakov loop in terms of the Dirac modes in SU(3) quenched lattice QCD in both confined and deconfined phases. First, to investigate the direct correspondence between confinement and chiral symmetry breaking, we remove low-lying Dirac-modes from the confined vacuum generated by lattice QCD. In this system without low-lying Dirac modes, while the chiral condensate $\langle \bar{q} q\rangle$ is extremely reduced, we find that the Polyakov loop is almost zero and $Z_3$-center symmetry is unbroken, which indicates quark confinement. We also investigate the removal of ultraviolet (UV) Dirac-modes, and find that the Polyakov loop is almost zero. Second, we deal with the deconfined phase above $T_c$, and find that the behaviors of the Polyakov loop and $Z_3$-symmetry are not changed without low-lying or UV Dirac-modes. Finally, we develop a new method to remove low-lying Dirac modes from the Polyakov loop for a larger lattice of $12^3 \times 4$ at finite temperature, and find almost the same results. These results suggest that each eigenmode has the information of confinement, i.e., the "seed" of confinement is distributed in a wider region of the Dirac eigenmodes unlike chiral symmetry breaking, and there is no direct correspondence between confinement and chiral symmetry breaking through Dirac-eigenmodes.

The quark-meson model is often used as an effective low-energy model for QCD to study the chiral transition at finite temperature $T$, baryon chemical potential $\mu_B$, and isospin chemical potential $\mu_I$. The parameters of the model are determined by matching the meson and quark masses, as well as the pion decay constant to their physical values using the on-shell and modified minimal subtraction schemes. In this paper, we study the possibility of different phases at zero temperature. In particular, we investigate the competition between an inhomogeneous chiral condensate and a pion condensate. For the inhomogeneity, we use a chiral-density wave ansatz. For a sigma mass of $600$ MeV, we find that an inhomogeneous chiral condensate exist only for pion masses below approximately 37 MeV. We also show that due to our parameter fixing, the onset of pion condensation takes place exactly at $\mu_I={1\over2}m_{\pi}$ in accordance with exact results.

The Nambu--Jona-Lasinio model with a Polyakov loop is extended to the finite isospin chemical potential case, which is characterized by the simultaneous coupling of the pion condensate, chiral condensate, and Polyakov loop. The pion condensate, chiral condensate, and Polyakov loop as functions of temperature and isospin chemical potential are investigated by minimizing the thermodynamic potential of the system. The resulting $(T,{\ensuremath{\mu}}_{I})$ phase diagram is studied with emphasis on the critical point and Polyakov loop dynamics. The tricritical point for the pion superfluidity phase transition is confirmed, and the phase transition for isospin symmetry restoration in the high isospin chemical potential region perfectly coincides with the crossover phase transition for the Polyakov loop. These results agree with the lattice QCD data.