Symmetry-breaking Fermi surface deformations from central interactions in two dimensions

We present a mean-field theory of the Pomeranchuk instability in two dimensions, starting from a generic central interaction potential described in terms of a few microscopic parameters. For a significant range of parameters, the instability is found to be pre-empted by a first-order quantum phase transition. We provide the ground-state phase diagram in terms of our generic parameters.

Figure 1

Shape of Fermi surface of 2D continuum system, before (thinner line) and after (thicker line) a Pomeranchuk instability. Several possible symmetries $\left(l=2,3,4,5\right)$ are shown for the Pomeranchuk order.

Figure 2

(Color online) Values of the amplitude on the Fermi surface of the deformation potential, $\Lambda$, that minimize the free energy (solid lines), plotted as functions of the coupling strength in units of its “critical” value, $V/V_{\text{crit}}$. (a) $V^{\prime }{k}_F/V_{\text{crit}}=-1$ and $V^{\left(n\right)}{k}_F^n/V_{\text{crit}}\approx 0$ for all $n=2,3,\dots$ [Eq. (27)] (b) $V^{\prime }{k}_F/V_{\text{crit}}=1$, $V^2{k}_F^2/V_{\text{crit}}=-2+1/50$, $V^3{k}_F^3/V_{\text{crit}}=-1$, and $V^{\left(n\right)}{k}_F^n/V_{\text{crit}}\approx 0$ for $n\ge 4$ [Eq. (30)]. The dashed line in panel (b) indicates an additional stationary point, but it is a maximum, not a minimum. The region where the transition takes place has been blown out in the inset. The dotted lines indicate the critical coupling of Eq. (23) [panels (a) and (b)] and the lower bound in Eq. (32) [panel (b)]. The structure of the free energy in the different parameter regions has been sketched for illustration.

Figure 3

(Color online) Phase diagram for small symmetry-breaking deformations of the Fermi surface with a given value of the angular momentum quantum number $l$. In the symmetric state, the Fermi surface is circular. In the broken-symmetry state, it has one of the configurations of Fig. 1. In the shaded region one of the two states is stable, and the other metastable, so the transition takes place through a first-order jump. The parameter $V^{‴}{k}_F^3/V_{\text{crit}}=-1/20$, which controls the width of this region [Eq. (32)].