Tricritical behavior of the massive chiral Gross-Neveu model

The phase diagram of the massive chiral Gross-Neveu model (the $1+1$-dimensional Nambu-Jona-Lasinio model at large $N$) is investigated in the vicinity of the tricritical point. Using the derivative expansion, the grand canonical potential is cast into the form of a Ginzburg-Landau effective action. Minimization of this action by variational and numerical methods reveals both 1st and 2nd order phase transitions to a chiral crystal phase, separated by a tricritical line. These findings are contrasted to the massive Gross-Neveu model with discrete chiral symmetry where only 2nd order transitions have been observed.

Figure 1

Phase diagram of the massless ${\mathrm{NJL}}_2$ model in the $(\mu ,T)$ plane according to Ref. 14. In the chiral crystal phase, the condensate has helical symmetry, see Eq. (3).

Figure 2

Phase boundary at $\gamma =0.0005$ according to the variational calculation based on the ansatz (22), separating the homogeneous phase from the crystal phase. A 1st order line to the left of the tricritical point ${\mathrm{P}}_{\mathrm{t}}$ goes over into a 2nd order line to the right of ${\mathrm{P}}_{\mathrm{t}}$. The points labeled (a, b, c) refer to the effective potentials shown in Fig. 3.

Figure 3

Evolution of effective potential when crossing the phase boundary in Fig. 2 at three different temperatures, illustrating the difference between 1st and 2nd order transitions. A minimum to the left of $M_{\max }$ refers to the crystal phase, a minimum to the right of $M_{\max }$ to the homogeneous phase. Figures 3a, 3b, 3c correspond to the points a, b, c marked in Fig. 2.

Figure 4

Two-dimensional phase boundary in $(\gamma ,\mu ,T)$ space, separating a massive Fermi gas from the chiral crystal, according to the variational calculation. Notice that only the vicinity of the tricritical point is shown.

Figure 5

(a) Projection of the 3d phase boundary of Fig. 4 onto the $(\mu ,T)$ plane. (b) Same plot, replacing the variable $T$ by $\tau =\sqrt{T_{\mathrm{c}}-T}$. (c) Rescaling both axes by a common factor ${\gamma }^{-1/3}$ removes the $\gamma$ dependence.

Figure 6

Perturbative phase boundary in the rescaled variables $(\nu ,\sigma )$ defined in Eq. (43), computed from the parametric representation (50).

Figure 7

Ratio $R=A/B$ of the main axes of the ellipse traced out by $\Phi$ in the complex plane, right at the perturbative phase boundary.

Figure 8

Sample periodic orbits of a classical charged particle in an electromagnetic field, equivalent to solutions of the Euler-Lagrange equations for the rescaled GL action. The dot at $\phi \approx 0.78$ corresponds to the time independent solution right at the tricritical point $(\nu ,\sigma )=(2,99,1.54)$. The curves are obtained for the parameters $(\nu ,\sigma )=(2.93,1.62)$, (2.58, 2.15), (2.36, 2.57), (2.20, 2.93), (2.09, 3.25), (1.99, 3.54) from inside out. Since the shape of the trajectories is reflection symmetric only the lower half is shown.

Figure 9

“Universal” phase boundary in rescaled variables obtained from perturbative, analytical (2nd order) and nonperturbative, numerical (1st order) calculations.

Figure 10

Like Fig. 4, but result from numerical minimization of the GL effective action. The whole surface has been reconstructed from the single curve in Fig. 9 by using scaling. The curves $\gamma =\mathrm{const}$ would all have the same shape in a $(\mu ,\tau )$ plot, but a different scale. The second set of curves drawn onto the surface connects points related by scaling. The narrow gap between the phase boundary and the $T$ axis is an artefact of the limited range of $\sigma$ when determining the 1st order curve in Fig. 9.