Chiral spirals from noncontinuous chiral symmetry: The Gross-Neveu model results

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.

Figure 1

The Jacobi elliptic functions, $\mathrm{sn}(\xi |\lambda )$, $\mathrm{cn}(\xi |\lambda )$, and $\mathrm{dn}(\xi |\lambda )$ at $\lambda =0.6$. $\mathbf{K}(\lambda )$ is the quarter period.

Figure 2

The Jacobi complete elliptic function of the first kind $\mathbf{K}(\lambda )$ (normalized by $\pi /2$). The period grows logarithmically as $\lambda$ approaches 1, which corresponds to the dilute limit of the fermion density.

Figure 3

The dispersion relation between the energy $\tilde{\omega }$ and quasimomentum $\tilde{Q}$ (normalized by $\mathcal{A}$). $\lambda$ is chosen to be 0.6. The gap is opened at the quasimomentum $\pi /2\mathbf{K}$ ($\eta =0$), which should be assigned as ${\tilde{p}}_F$. The energy at the band edge is ${\tilde{\omega }}_F=\sqrt{{\lambda }_1}$ (${\tilde{\omega }}_F^{\prime }=1$) for the first (second) energy branch. The plot is symmetric with respect to $\tilde{\omega }\to -\tilde{\omega }$, and the Dirac sea also has an energy gap of the same size.

Figure 4

Left: The “elliptic” chiral spirals at $\lambda =0.9$. We plot the amplitude-free parts of condensates, ${\mathcal{N}}_s⟨\overline{\psi }\psi ⟩\equiv ⟨\overline{\psi }\psi ⟩\times G/N\mathcal{A}=-\mathcal{M}$ and ${\mathcal{N}}_p⟨\overline{\psi }\mathrm{i}{\gamma }_5\psi ⟩\equiv ⟨\overline{\psi }\mathrm{i}{\gamma }_5\psi ⟩\times 2\pi /N{\mathbf{K}}^{\prime }\mathcal{A}={\partial }_{\xi }\mathcal{M}$. Right: The plots at $\lambda =0.9$ for the (normalized) scalar (S), pseudoscalar (PS), and fermion number density (n). For the fermion number density, we divide by $N{p}_F/\pi$. The fermion number is stuck at the location of the domain wall where the scalar density passes zero.

Figure 5

The quasimomentum-energy dispersion. Left: For the GN model. The spectra contain the gapped region in the Fermi and Dirac sea. The size of the gap decreases like $\sim M_0\times M_0/p_F$ as $p_F$ becomes large. Right: For the ${\mathrm{NJL}}_2$-type models. The gaps open only at the Fermi points. The size of the gap is known to be $\sim M_0$, independently of the value of $p_F$.