Inhomogeneous Chiral Condensates Research Articles

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

We investigate the time evolution of the quark condensate toward a chiral symmetry broken phase in hot and dense quark matter using a field-theoretic quark model with nonlocal chiral-invariant four-fermion coupling. By purposely selecting a parameter set in which inhomogeneous phases are energetically disfavored, we nonetheless observe the emergence of metastable patterned configurations that appear to persist for remarkably long timescales. These findings suggest that even when not fully stable, inhomogeneous phases may play a significant role in the dynamics of chiral symmetry breaking and restoration. To gain deeper insight into these phenomena, we also analyze the impact of the dimensionality of coordinate space on both the formation and stability of inhomogeneous chiral condensates.

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 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.

The phase diagram of the (1 + 1)-dimensional Gross–Neveu model is reanalyzed for (non-)zero chemical potential and (non-)zero temperature within the mean-field approximation. By investigating the momentum dependence of the bosonic two-point function, the well-known second-order phase transition from the symmetric phase to the so-called inhomogeneous phase is detected. In the latter phase the chiral condensate is periodically varying in space and translational invariance is broken. This work is a proof of concept study that confirms that it is possible to correctly localize second-order phase transition lines between phases without condensation and phases of spatially inhomogeneous condensation via a stability analysis of the homogeneous phase. To complement other works relying on this technique, the stability analysis is explained in detail and its limitations and successes are discussed in context of the Gross–Neveu model. Additionally, we present explicit results for the bosonic wave-function renormalization in the mean-field approximation, which is extracted analytically from the bosonic two-point function. We find regions—a so-called moat regime—where the wave function renormalization is negative accompanying the inhomogeneous phase as expected.

In this review, we discuss the physical characteristics of the magnetic dual chiral density wave (MDCDW) phase of dense quark matter and argue why it is a promising candidate for the interior matter phase of neutron stars. The MDCDW condensate occurs in the presence of a magnetic field. It is a single-modulated chiral density wave characterized by two dynamically generated parameters: the fermion quasiparticle mass m and the condensate spatial modulation q. The lowest-Landau-level quasiparticle modes in the MDCDW system are asymmetric about the zero energy, a fact that leads to the topological properties and anomalous electric transport exhibited by this phase. The topology makes the MDCDW phase robust against thermal phonon fluctuations, and as such, it does not display the Landau–Peierls instability, a staple feature of single-modulated inhomogeneous chiral condensates in three dimensions. The topology is also reflected in the presence of the electromagnetic chiral anomaly in the effective action and in the formation of hybridized propagating modes known as axion-polaritons. Taking into account that one of the axion-polaritons of this quark phase is gapped, we argue how incident γ-ray photons can be converted into gapped axion-polaritons in the interior of a magnetar star in the MDCDW phase leading the star to collapse, a phenomenon that can serve to explain the so-called missing pulsar problem in the galactic center.

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 propose a new chirality-imbalance phenomenon arising in baryonic/high density matters under a magnetic field. A locally chiral-imbalanced (parity-odd) domain can be created due to the electromagnetically induced U(1)A anomaly in high-density matters. The proposed local-chiral imbalance generically possesses a close relationship to a spatial distribution of an inhomogeneous chiral (pion)-vector current coupled to the magnetic field, in which the inhomogeneity is irrespective to the pion-domain wall configuration formed by neutral pion. To demonstrate such a nontrivial correlation, we take the skyrmion crystal approach to model baryonic/high density matters. Remarkably enough, we find the chirality-imbalance distribution can have a periodicity in a high density region, and that it looks specially propagating (dubbed ‘chiral-imbalance density wave’), when the inhomogeneous chiral condensate develops to take a similar periodic distribution, so-called a chiral density wave. This implies the emergence of a nontrivial density wave for the explicitly broken U(1)A current simultaneously with the chiral density wave for the spontaneously broken chiral-flavor current. We further find that the topological phase transition in the skyrmion crystal model (between skyrmion and half-skyrmion phases) undergoes the deformation of the chiral-imbalance density wave in shape and periodicity. The emergence of this chiral-imbalance density wave could give a crucial contribution to studies on the chiral phase transition, as well as the nuclear matter structure, in compact stars under a magnetic field.

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 investigate the influence of rotation on the dynamical chiral symmetry breaking in strongly interacting matter. We develop a self-consistent Bogoliubov-de Gennes-like theoretical framework to study the inhomogeneous chiral condensate and the possible chiral vortex state in rotating finite-size matter in four-fermion interacting theories. We show that for sufficiently rapid rotation in $2+1$ dimensions, the ground state can be a chiral vortex state, a type of topological defect in analogy to superfluids and superconductors. The vortex state exhibits pion condensation, providing a new mechanism to realize pseudoscalar condensation in strongly interacting matter.

We explore magnetic field effects on the nuclear matter based on the skrymion crystal approach for the first time. It is found that the magnetic effect plays the role of a catalyzer for the topological phase transition (topological deformation for the skyrmion crystal configuration from the skrymion phase to half-skyrmion phase). Furthermore, we observe that in the presence of the magnetic field, the inhomogeneous chiral condensate persists both in the skyrmion and half-skyrmion phases. Explicitly, as the strength of magnetic field gets larger, the inhomogeneous chiral condensate in the skyrmion phase tends to be drastically localized, while in the half-skyrmion phase the inhomogeneity configuration is hardly affected. It also turns out that a large magnetic effect in a low density region distorts the baryon shape to an elliptic form but the crystal structure is intact. However, in a high density region, the crystal structure is strongly effected by the strong magnetic field. A possible correlation between the chiral inhomogeneity and the deformation of the skrymion configuration is also addressed. The results obtained in this paper might be realized in the deep interior of compact stars.

We study how an external magnetic field modifies the chiral phase structure of QCD, in particular the phases characterized by inhomogeneous chiral condensates. The magnetic field can be systematically incorporated into a generalized Ginzburg-Landau framework, and it turns out to induce a model independent universal coupling between the magnetic field and the axial isospin current. The resulting effect is found to be drastic especially in the chiral limit; no matter how small the magnetic intensity is, the tricritical Lifshitz point is totally washed out, and the real kink crystal is replaced by a magnetically induced chiral spiral. The current quark mass, on the other hand, has an opposite effect, protecting the chiral critical point from the magnetically induced chiral spiral. But once the magnetic intensity exceeds a critical value, the critical point no longer exists. We draw a semiquantitative conclusion that the critical point disappears for $\sqrt{eB}\geq 50$ MeV.

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 study the dynamics of inhomogeneous scalar and pseudoscalar chiral order parameters within the framework of the time-dependent Ginzburg-Landau equations. We utilize a nonlocal chiral quark model to obtain the phase diagram of the model as function of temperature and baryon chemical potential and study the formation of metastable spatial domains of matter where the order parameters acquire a spatial modulation in the course their dynamical evolution. We found that, before reaching the expected equilibrium homogeneous state, both scalar and pseudoscalar chiral condensates go through long-lived metastable inhomogeneous structures. For different initial configurations of the order parameters, the lifetimes of the inhomogeneous structures are compared to timescales in a relativistic heavy-ion collision.

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.

We study the dynamics of the formation of inhomogeneous chirally broken phases in the final stages of a heavy-ion collision, with particular interest on the time scales involved in the formation process. The study is conducted within the framework of a Ginzburg-Landau time evolution, driven by a free energy functional motivated by the Nambu–Jona-Lasinio model. Expansion of the medium is modeled by one-dimensional Bjorken flow and its effect on the formation of inhomogeneous condensates is investigated. We also use a free energy functional from a nonlocal Nambu–Jona-Lasinio model which predicts metastable phases that lead to long-lived inhomogeneous condensates before reaching an equilibrium phase with homogeneous condensates.

Abstract We study the possibility of two types of inhomogeneous phases in core of neutron stars: one is the Coulomb crystal, which is known as quark-hadron pasta structures, and another one is chiral crystal. In the Coulomb crystal, the inhomogeneous phase appears as the result of the balance between the surface tension and the Coulomb interaction. In chiral crystal, we study the inhomogeneous chiral condensate, which has spatial modulation. In the simple model in 1+1 dimensions, this condensate has the same feature with the FFLO state, which is well known in the condensed matter physics.

The two-flavor quark-meson model is used as a low-energy effective model for QCD to study inhomogeneous chiral condensates at finite %temperature $T$ baryon chemical potential $\mu_B$. The parameters of the model are determined by matching the meson and quark masses, and the pion decay constant to their physical values using the on-shell and modified minimal subtraction schemes. Using a chiral-density wave ansatz for the inhomogeneity, we calculate the effective potential in the mean-field approximation and the result is completely analytic. The size of the inhomogeneous phase depends sensitively on the pion mass and whether one includes the vacuum fluctuations or not. Finally, we briefly discuss the mean-field phase diagram.

We calculate dispersion relations for $\bar{D}(0^-)$, $\bar{D}^*(1^-)$, $\bar{D}_0^*(0^+)$ and $\bar{D}_1(1^+)$ mesons in the Dual Chiral Density Wave (DCDW) where the chiral symmetry is spontaneously broken by inhomogeneous chiral condensate. Employing the Bloch's theorem, we show that all the modes have opposite group velocity to the momentum in the low-momentum region and the energy is minimized at non-zero momentum. Furthermore, the magnitude of momentum which realizes the minimum energy is equal to the wave number of the density wave.

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.