Geometrical quench and dynamical quantum phase transition in the $\alpha -T_3$ lattice

We investigate quantum quenches and the Loschmidt echo in the two-dimensional, three-band $\alpha -T_3$ model, a close descendant of the dice lattice. By adding a chemical potential to the central site, the integral of the Berry curvature of the bands in different valleys is continously tunable by the ratio of the hopping integrals between the sublattices. By investigating one and two filled bands, we find that dynamical quantum phase transition (DQPT), i.e., nonanalytical temporal behavior in the rate function of the return amplitude, occurs for a certain range of parameters, independent of the band filling. By focusing on the effective low-energy description of the model, we find that DQPTs happen not only in the time derivative of the rate function, which is a common feature in two-dimensional models, but also in the rate function itself. This feature is not related to the change of topological properties of the system during the quench but rather follows from population inversion for all momenta. This is accompanied by the appearance of dynamical vortices in the time-momentum space of the Pancharatnam geometric phase. The positions of the vortices form an infinite vortex ladder, i.e., a macroscopic phase structure, which allows us to identify the dynamical phases that are separated by the DQPT.

Figure 1

Left panel: The $\alpha -T_3$ lattice can be considered as two interpenetrating the honeycomb lattices. There is no hopping between the two rim sites, labeled ($A,B$); however, they are both connected to the hub site, labeled ($H$), with hopping amplitudes $\gamma$, ${\gamma }^{\prime }$. Their ratio $\alpha ={\gamma }^{\prime }/\gamma$ serves as a tuning parameter that distinguishes different lattice stuctures. When $\alpha =0$ we recover the familiar honeycomb lattice and with $\alpha =1$ the dice lattice. Right panel: The dispersion relation near a Dirac point with finite $\mu$ chemical potential on the hub site.

Figure 2

The plots of the real part of the universal rate functions for quenches that open a gap $({\mu }_0=0\to \mu \ne 0)$. Left panel: The first derivative of the rate function when the quench parameters are chosen so that they do not satisfy inequality (11): ${\alpha }_0=0.1\to \alpha =1.8$. Right panel: The rate function with parameter, so that they satisfy inequality (11): ${\alpha }_0=0.1\to \alpha =5$. On both plots the red arrows indicate the kinks that results from the nonanalytic behavior.

Figure 3

The real part of the universal rate function and the Pancharatnam geometrical phase for quenches that open a gap $({\mu }_0=0\to \mu \ne 0)$. Top row: The plot of the rate functions with parameters; top left: ${\alpha }_0=0.1\to \alpha =5$; and top right: ${\alpha }_0=0.1\to \alpha =3.2$. Bottom row: The contourplots of the Pancharatnam geometric phases. The dashed lines indicate the exact moment of the DQPTs (gray dashed lines on the top row and white dashed lines on the bottom row), we can observe that the vortices are created pairwise and with different circulation during the time interval of the plateaus. With fixed ${\alpha }_0$ and decreasing $\alpha$ the intervals become larger and they inevitably merge together for parameters that does not satisfy (11). For better visibility some vortices are highlighted with white circles in the bottom right panel.

Figure 4

The ${\alpha }_0-\alpha$ parameter space, where the colors depict the different regions whether DQPTs cannot occur at all (white), DQPTs occur in the first derivative of the rate function only (blue) and DQPTs occur in the rate function itself (orange). The left panel shows the regions during a gapless to gapped quench scenario ${\mu }_0=0\to \mu \ne 0$ and the right panel corresponds to quenches within the same gapped phase ${\mu }_0=\mu$.