Self-Consistent Crystalline Condensate in Chiral Gross-Neveu and Bogoliubov–de Gennes Systems

We derive a new exact self-consistent crystalline condensate in the ($1+1$)-dimensional chiral Gross-Neveu model. This also yields a new exact crystalline solution for the one dimensional Bogoliubov–de Gennes equations and the Eilenberger equation of semiclassical superconductivity. We show that the functional gap equation can be reduced to a solvable nonlinear equation and discuss implications for the temperature-chemical potential phase diagram.

Figure 1

Band spectrum of the BdG Hamiltonian (4) for the complex crystal condensate (13), as a function of the twist parameter $\theta$. The dashed line denotes the infinite period limit [in this case $\nu \to 0$], which is Shei’s solution (11). At $\theta =2\mathbf{K}$, we recover the symmetric spectrum of the real kink crystal, relevant to the ${\mathrm{GN}}_2$ system.

Figure 2

The amplitude $M(x)$ [solid line] and phase $\chi (x)$ [dashed line] of $\Delta =M{e}^{i\chi }$ in (13), over several periods. The amplitude is periodic while the phase rotates by an angle $2\phi$ each period.

Figure 3

Crystalline complex kink (13) plotted as a function of $x$ over one period. The cross section denotes $\Delta (x)=S(x)-iP(x)$, and indicates a net rotation through the twist parameter $2\phi$ over one period.

Figure 4

Phase diagram of the chiral ${\mathrm{NJL}}_2$ model, from a Ginzburg-Landau analysis based on a crystalline condensate of the form in (13). Along the $T$ axis (below $T_c$) the condensate is of the form of Shei’s twisted kink (11), while on the $T=T_c$ line, the condensate reduces to the form of the LOFF chiral spiral [12]. The condensate is depicted as in Fig. 3.