Quadratic band touching with long-range interactions in and out of equilibrium

Motivated by recent advances in cold atomic systems, we study the equilibrium and quench properties of two-dimensional fermions with quadratic band touching at the Fermi level, in the presence of infinitely long-range interactions. Unlike when only short-range interactions are present, both nematic and quantum anomalous Hall (QAH) states appear at weak interactions, separated by a narrow coexistence region, whose boundaries mark second- and third-order quantum phase transitions. After an interaction quench, the QAH order exhibits three distinct regions: persistent or damped oscillations and exponential decay to zero. In contrast, the nematic order always reaches a nonzero stationary value through power-law damped oscillations, due to the interplay between the symmetry of the interaction and the specific topology of the quadratic band touching.

Figure 1

The weak-coupling phase diagram of Eqs. (1) and (2) for $g_x\ge g_y$ in the weak-coupling limit is shown in the left panel. For $g_y>g_x$, $g_x$ should be replaced by $g_y$. The blue dashed/green dashed-dotted lines denote a second/topological third-order transition, respectively. Right panel: Nematic (red) and QAH (blue) order parameters are shown for $\rho g_z=\frac{1}{2}$ in the coexisting region. The red dashed line denotes the nematic gap without QAH, which is only a local minimum of the ground-state energy compared to the coexisting solution.

Figure 2

The time evolution of the QAH (blue dashed) and nematic (red solid) order parameters are shown, starting from $\mathrm{\Delta }=0.05W$ or $m_x=0.05W$, corresponding to an initial coupling $\rho g_z=0.543$ or $\rho g_x=0.819$, respectively, and $\rho W=1$. After the quench, $\rho g_{z,x}=2$ (top panel), 0.7 (middle panel), and 0.1 (bottom panel). For the latter, the steady-state value is indicated by a black dashed-dotted from Eq. (8). The tiny oscillation superimposed on the temporal decay arises from a finite cutoff, which is needed to make $S^{x,z}(t=0)$ finite.

Figure 3

The steady-state behavior of the order parameter is plotted for the pure QAH (blue) and nematic (red) state, starting from either $\mathrm{\Delta }=0.05W$ or $m_x=0.05W$ in the respective channel. The green dots denote the three qualitatively different regions of the QAH order parameter. The dashed lines are the equilibrium order parameters from the solution of Eqs. (4). The inset zooms into the weak-coupling limit.