Topological classification of dynamical phase transitions

We study the nonequilibrium time evolution of a variety of one-dimensional (1D) and two-dimensional (2D) systems (including SSH model, Kitaev-chain, Haldane model, $p+ip$ superconductor, etc.) following a sudden quench. We prove analytically that topology-changing quenches are always followed by nonanalytical temporal behavior of return rates (logarithm of the Loschmidt echo), referred to as dynamical phase transitions (DPTs) in the literature. Similarly to edge states in topological insulators, DPTs can be classified as being topologically protected or not. In 1D systems the number of topologically protected nonequilibrium time scales are determined by the difference between the initial and final winding numbers, while in 2D systems no such relation exists for the Chern numbers. The singularities of dynamical free energy in the 2D case are qualitatively different from those of the 1D case; the cusps appear only in the first time derivative.

Figure 1

Illustration for the existence of perpendicular vectors if the quench connects domains with different topological numbers. (a) Topological insulators in AIII symmetry class. As $k$ goes through the Brillouin zone ${\mathbf{d}}_k$ draws a closed loop. For any parametrization of these loops there will be at least $2\Delta \nu$ wave numbers for which ${\mathbf{d}}^0\perp {\mathbf{d}}^1$ if the winding number of the two vector fields differ by $\Delta \nu$. (b) Superconductors in the BDI symmetry class. In the $k\in [0,\pi ]$ domain (solid line) there will be at least $\Delta \nu$ perpendicular $\mathbf{d}$ vectors in a quench characterized by $\Delta \nu$. (c) Superconductors in the D symmetry class. The vectors $\widehat{\mathbf{d}}$ are no longer confined to a plane, but the occurrence of perpendicular vectors is still ensured when the topology of ${\widehat{\mathbf{d}}}^1$ and ${\widehat{\mathbf{d}}}^0$ are different.

Figure 2

Fisher lines $z_n\left(k\right)$ (for $n=0..3$) and DPTs in the generalized SSH model in a quench ${\nu }_0=1\to {\nu }_1=-2$. (a) The lines of Fisher zeros are doubly degenerate due to the $(k,-k)$ symmetry; they sweep through the real axis $\left|\Delta \nu \right|=3$ times. (b) Dynamical phase transitions appear where the Fisher zeros cross the imaginary axis. The grid lines show the DPTs corresponding to the first four Fisher lines.

Figure 3

Quench in the Haldane model from $Q_0=0$ to $Q_1=1$. (a) Fisher zeros corresponding to $n=0$ in Eq. (4). The normalized density of the Fisher zeros $\rho \left(Re\right\{z\},Im\{z\left\}\right)$ is shown as the darkness of the area. (Inset) The density diverges on the imaginary axis at the boundary of the Fisher area. (b) Cusplike singularities in the first derivative of the dynamical free energy. The shaded areas emphasize the regions where the Fisher zeros cross the time axis.