Peierls instability for systems with several Fermi surfaces: an example from the chiral Gross-Neveu model

Abstract

As is well known, the chiral Gross-Neveu model at finite density can be solved semi-classically with the help of the chiral spiral mean field. The fermion spectrum has a single gap right at the Fermi energy, a reflection of the Peierls instability. Here, we divide the N fermion flavors up into two subsets to which we attribute two different densities. The Hartree-Fock ground state of such a system can again be found analytically, using as mean field the ``twisted kink crystal" of Basar and Dunne. Its spectrum displays two gaps with lower edges coinciding with the two Fermi energies. This solution is favored over the homogeneous one, providing us with an explicit example of a multiple Peierls instability.

Figures (11)

• Figure 1: Examples of fermion dispersion relation of massless (left) and massive (right) GN models at finite density. Shaded stripes: band gaps. Common potential parameters L15L16: $\kappa=0.8,A=1$, 1st Brillouin zone between $\pm \pi/2{\bf K}=\pm 0.7872$. Band edges: $\pm 0.6,\pm1$ (left plot), $\pm0.2428,\pm0.6472,\pm1.0290$ (right plot).
• Figure 2: Examples of fermion dispersion relations for basic twisted kink potential $\hat{\Delta}$L7L8. Shaded stripes: band gaps. Parameters: $\kappa=0.5$, $\theta/\theta_{\rm max}=1/4, 1/2, 3/4$ from left to right ($\theta_{\rm max} =4 {\bf K}'$). 1st Brillouin zone between $\pm \pi/2L=\pm 0.4457,\pm 0.6212,\pm 0.4457$ from left to right. Band edges: Continuum bands always bounded by $\pm 1$. The bound band in between has the edges $[0.5048,0.8480]$, $[-0.3333,0.3333]$, $[-0.8480,-0.5048]$ from left to right.
• Figure 3: Energy density vs. fermion density from two-component HF calculation based on the twisted kink crystal. Densities: $\rho_1=0$ (type $A$ fermions, fraction $1-\nu$), $\rho_2= \rho/\nu$ (type $B$ fermions, fraction $\nu$) with $\rho$ on the horizontal axis. Three values of $\nu$ are shown. On this scale, results are practically indistinguishable from those based on a homogeneous mean field, see Fig. \ref{['fig11']}, Appendix B.
• Figure 4: Zooming in onto the small $\rho$ region of Fig. \ref{['fig3']}. Fat curves: Details of curves labeled $\nu=1/4$ (left plot) and $\nu=1/2$ (right plot). The thin curves are from the homogeneous calculation, Appendix B. The circles correspond to the critical point where a first order phase transition occurs. These points are joined by a straight (dashed) line to the vacuum point at $\rho=0$, the mixed phase. The full calculation is lower in energy and does not show any phase transition.
• Figure 5: Like Fig. \ref{['fig4']}, but for $\nu=3/4$ (left plot) and $\nu=1$ (right plot). The $\nu=1$ curve is the result from the chiral spiral, not shown in Fig. \ref{['fig3']}. The short dotted lines tangential to the full curves at $\rho=0$ (also drawn in Fig. \ref{['fig3']}) show that the slope is consistent with the mass of a single twisted kink.