Fermi surface topology in the case of spontaneously broken rotational symmetry

The relation between the broken rotational symmetry of a system and the topology of its Fermi surface is studied for the two-dimensional system with the quasiparticle interaction $f(\mathbf{p}, \mathbf{p}')$ having a sharp peak at $|\mathbf{p}-\mathbf{p}'|=q_0$. It is shown that, in the case of attraction and $q_0=2p_{\mathrm{F}}$ the first instability manifesting itself with the growth of the interaction strength is the Pomeranchuk instability. This instability appearing in the $L=2$ channel gives rise to a second order phase transition to a nematic phase. The Monte Carlo calculations demonstrate that this transition is followed by a sequence of the first and second order phase transitions corresponding to the changes in the symmetry and topology of the Fermi surface. In the case of repulsion and small values of $q_0$, the first transition is a topological transition to a state with the spontaneously broken rotational symmetry, namely, corresponding to the nucleation of $L\simeq \pi(p_{\mathrm{F}}/q_0-1)$ small hole pockets at the distance $p_{\mathrm{F}}-q_0$ from the center and the deformation of the outer Fermi surface with the characteristic multipole number equal to $L$. At $q_0\to0$, when the model under study transforms to the two-dimensional Nozières model, the multipole number characterizing the spontaneous deformation is $L\to\infty$, whereas the infinitely folded Fermi curve acquires the Hausdorff dimension $D=2$ which corresponds to the state with the fermion condensate.