The interweaving chiral spirals (ICS), that is defined as superposition of differently oriented chiral spirals, is important for qualitative understandings of the intermediate quark density region as well as quantitative estimates of the Quarkyonic region. We discuss how to construct the ICS, taking the (2+1) dimensional Fermi system as an example. We postulate that the presence of the ICS would delay the occurrence of the chiral restoration as well as deconfinement phase transition, by tempering the growth of quark fluctuations.
We elaborate how to construct interweaving chiral spirals in (2+1) dimensions, defined as a superposition of chiral spirals oriented in different directions. We divide a two-dimensional Fermi sea into distinct wedges, characterized by the opening angle 2Theta and depth Q ~ pF, where pF is the Fermi momentum. In each wedge, the energy is lowered by forming a single chiral spiral. The optimal values for Theta and Q are chosen by balancing this gain in energy versus the cost of deforming the Fermi surface (which dominates at large Theta) and patch-patch interactions (dominant at small Theta). Usi…
The nuclear matter, deconfined quark matter, and Quarkyonic matter in low temperature region are classified based on the 1/Nc expansion. The chiral symmetry in the Quarkyonic matter is investigated by taking into account condensations of chiral particle-hole pairs. It is argued that the chiral symmetry and parity are locally violated by the formation of chiral spirals, < psibar exp(2 i mu z gamma^0 gamma^z) psi >. An extension to multiple chiral spirals is also briefly discussed.
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 b…
We discuss the formation of quarkyonic chiral spirals in the presence of a magnetic field. The explicit breaking of the rotational symmetry by the external magnetic field gives rise to an additional chiral spiral that varies along the field direction and rotates in the chiral space between pion and magnetic moment components.
We investigate the sign problem of the fermion determinant at finite baryon density in (1+1) dimensions, in which the ground state in the chiral limit should be free from the sign problem by forming a chiral spiral. To confirm it, we evaluate the fermion determinant in the continuum theory at the one-loop level and find that the determinant becomes real as expected. The conventional lattice formulation to implement a chemical potential is, however, not compatible with the spiral transformation. We discuss an alternative of the finite-density formulation and numerically verify the chiral spiral…
We study the modification of the chiral phase structure of QCD due to an external magnetic field. We first demonstrate how the effect of magnetic field can systematically be incorporated into a generalized Ginzburg-Landau framework. We then analyze the phase structure in the vicinity of the chiral critical point. In the chiral limit, the effect is found to be so drastic that it totally washes the tricritical point out of the phase diagram, bringing the continent for the chiral spiral. This is the case no matter how small is the intensity of the magnetic field. On the other hand, the current qu…
We argue that the presence of a very strong magnetic field in the chirally broken phase induces inhomogeneous expectation values, of a spiral nature along the magnetic field axis, for the currents of charge and chirality, when there is finite baryon density or an imbalance between left and right chiralities. This "chiral magnetic spiral" is a gapless excitation transporting the currents of (i) charge (at finite chirality), and (ii) chirality (at finite baryon density) along the direction of the magnetic field. In both cases it also induces in the transverse directions oscillating currents of c…
We investigate the effect of a magnetic field on the vacuum of low-temperature QCD at large-N_c in the presence of a chiral chemical potential, using the holographic Sakai-Sugimoto model. Above some critical chemical potential we find an instability, which triggers a decay of the homogeneous vacuum to a non-homogeneous configuration with a spiral form, which we construct explicitly. We find that this decay is suppressed for sufficiently large magnetic field, and determine the critical strength. We find that the new vacuum does not exhibit the chiral magnetic effect. This is related to the way …
The phase diagram of the two-dimensional Nambu-Jona-Lasinio (or chiral Gross-Neveu) model is characterized by an order parameter in the form of a chiral spiral. Its radius vanishes at a critical temperature, its period depends only on the chemical potential. We generalize these findings to chirally imbalanced systems by including a chiral chemical potential $μ_5$. The relationship between the present, static approach and a previous, time dependent one is traced back to a half-local symmetry which the NJL$_2$ model shares with massless Dirac fermions, but which has been neglected so far. The st…
Vacuum properties of quantum chromodynamics in strong magnetic and finite electric fields are investigated. We show that when a uniform electric field is instantaneously applied in the parallel direction to a strong magnetic field, it induces temporal oscillation of the chiral and pion condensates. This is a temporal analog to the chiral spiral. The oscillation originates with the propagation of the collective mode, which is protected by the axial anomaly and thus nondissipative.
We study the ground state of baryonic/axial matter at zero temperature chiral-symmetry broken phase under a large magnetic field, in the framework of holographic QCD by Sakai-Sugimoto. Our study is motivated by a recent proposal of chiral magnetic spiral phase that has been argued to be favored against previously studied phase of homogeneous distribution of axial/baryonic currents in terms of meson super-currents dictated by triangle anomalies in QCD. Our results provide an existence proof of chiral magnetic spiral in strong coupling regime via holography, at least for large axial chemical pot…
We explore the inhomogeneous QCD phases at finite density and temperature using a (3+1)-dimensional Nambu-Jona-Lasinio (NJL) model in the large Nc limit with an additional attractive tensor-tensor interaction channel. For single modulated solutions, the problem reduces to solving the gap equation of a Chiral Gross-Neveu (NJL2) theory, whose minimum solution is a periodic array of chirally twisted kinks. At zero temperature, the minimum solution reduces to the quarkyonic chiral spiral found in quarkyonic matter. The connection between the quarkyonic matter and our (3+1)-dimensional NJL model is…
We argue that in cold, dense quark matter, in the limit of a large number of colors the ground state is unstable with respect to creation of a complicated Quarkyonic Chiral Spiral (QCS) state, in which both chiral and translational symmetries are spontaneously broken. The entire Fermi surface is covered with patches of QCSs, whose number increases as the quark density does. The low energy excitations are gapless, given by Wess-Zumino-Novikov-Witten model plus transverse kinetic terms localized about different patches.
In the framework of chiral perturbation theory we investigate whether, in the presence of nucleons, a many-particle system may lower its energy density by changing its mesonic vacuum from a homogeneous to a spiral configuration. Numerical values for the critical particle number densities for the spiral configuration to occur are given. They lie in the region of nuclear matter density where the applicability of chiral perturbation theory should be checked carefully.
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 t…
For a scalar theory with a global $O(N)$ symmetry, when $N=2$ a spatially inhomogeneous condensate arises when the term in the Lagrangian with two spatial derivatives has a negative coefficient. If the condensate for such a chiral spiral includes only one mode, characterized by a momentum $k_0 \hat{z}$, then in perturbation theory at nonzero temperature the propagator for the static mode has a double pole when $\vec{k}^2 = k_0^2$. We conjecture that since chiral spirals spontaneously break both global and spacetime symmetries, that such double poles are a universal property of their static tra…
We consider the formation of chiral density waves in Quarkyonic matter, which is a phase where cold, dense quarks experience confining forces. We model confinement following Gribov and Zwanziger, taking the gluon propagator, in Coulomb gauge and momentum space, as 1/(p^2)^2. We assume that the number of colors, N, is large, and that the quark chemical potential, mu, is much larger than renormalization mass scale, Lambda_QCD. To leading order in 1/N and Lambda_QCD/mu, a gauge theory with Nf flavors of massless quarks in 3+1 dimensions naturally reduces to a gauge theory in 1+1 dimensions, with …
In the limit of a large number of flavors, the ground state of the chiral Gross-Neveu model at finite fermion density exhibits a spatially periodic mean field in the form of the chiral spiral, thereby breaking translational invariance. Here we show that the ground state of the same model at finite fermion current density gives rise to a mean field which is periodic in time, a temporal chiral spiral. Since the current density is the same as axial charge density in two dimensions and axial charge is conserved, this may serve as an example of a time crystal. More specifically, as mean field theor…
We give an elementary derivation of the chiral magnetic effect based on a strong magnetic field lowest-Landau-level projection in conjunction with the well-known axial anomalies in two- and four-dimensional space-time. The argument is general, based on a Schur decomposition of the Dirac operator. In the dimensionally reduced theory, the chiral magnetic effect is directly related to the relativistic form of the Peierls instability, leading to a spiral form of the condensate, the chiral magnetic spiral. We then discuss the competition between spin projection, due to a strong magnetic field, and …
A mini-review of non-uniform phases in quark matter is presented, with particular attention to the pion condensation, also known as chiral density waves or chiral spirals. The phase diagram of strongly-interacting matter may involve such a phase, placed on the quarkyonic island between the baryonic phase and the chirally-restored quark-gluon plasma.
We investigate the phase portrait of the (1+1)-dimensional massless two-flavored NJL$_2$ model containing a quark number chemical potential $μ$ and an isospin chemical potential $μ_I$ in the limit of a large number of colors $N_c\to\infty$. Particular attention is paid to the question to what extent the inclusion of an isospin asymmetry affects chiral condensates to have a spatial inhomogeneity in the form of the so-called chiral density waves (CDW) (chiral spirals). It is shown that at zero temperature and comparatively small values of $μ$, i.e. at $μ<μ_c\approx 0.68M_0$ ($M_0$ is the dynamic…
We derived an analytic structure of the quark mass correction to chiral separation effect (CSE) in small mass regime. We confirmed this structure by a D3/D7 holographic model study in a finite density, finite magnetic field background. The quark mass correction to CSE can be related to correlators of pseudo-scalar condensate, quark number density and quark condensate in static limit. We found scaling relations of these correlators with spatial momentum in the small momentum regime. They characterize medium responses to electric field, inhomogeneous quark mass and chiral shift. Beyond the small…
