Dynamical phase transitions, time-integrated observables, and geometry of states

We show that there exist dynamical phase transitions (DPTs), as defined by Heyl et al. [Phys. Rev. Lett. 110, 135704 (2013)], in the transverse-field Ising model (TFIM) away from the static quantum critical points. We study a class of special states associated with singularities in the generating functions of time-integrated observables as found by Hickey et al. [Phys. Rev. B 87, 184303 (2013)]. Studying the dynamics of these special states under the evolution of the TFIM Hamiltonian, we find temporal nonanalyticities in the initial-state return probability associated with dynamical phase transitions. By calculating the Berry phase and Chern number we show the set of special states have interesting geometric features similar to those associated with static quantum critical points.

Figure 1

In the left panel is the FCS phase diagram of the TFIM, regions I and II are the dynamically ordered and disordered regimes, respectively [26]. The counting field is labeled $s$ and $\lambda$ is the transverse magnetic field strength. We consider “quenches” from points $(\lambda ,s)\to (\lambda ,0)$. The right panel shows the large deviation function associated with the return probability of this protocol. Dynamical nonanalyticities are found when the “quench” crosses the FCS critical line, analogous to the effects of a quantum quench across a static quantum critical point.

Figure 2

(a) The parameter manifold $M^2$ is topologically equivalent to the $S^2$ sphere as the $k=\pi$ mode is $\phi$ independent and the $k=0$ mode is (at most) only dependent on $\phi$ up to a gauge transformation. (b) Examining the manifold at each point in the FCS phase diagram we find that at $k_{\lambda }$ the manifold is $\phi$ independent. The $s$ states below $\left|\lambda \right|=1$ are found to be completely $\phi$ independent on the $s=0$ line. However above the $\left|\lambda \right|=1$ line the manifold once again becomes topologically equivalent to the $S^2$ sphere at $s=0$.

Figure 3

The top panel shows slices of the Berry phase density at various $\lambda$ values. No extremum or singular features are visible in the vicinity of the FCS critical line, see Fig. 1. However, the derivative of this density $d\tilde{\beta }/ds$ has extremum located at the FCS critical line; this is shown in the bottom panel. For $\left|\lambda \right|>1$ no such extremum are present; this is due to a lack of any FCS critical points in this parameter regime.

Figure 4

The Chern number $C$ associated with the manifold of $|s_{k,t}\rangle$ states has a “kink” at the FCS critical line; this is shown for various $\lambda$ slices in the top panel. The derivative of the Chern number displays divergences at the FCS critical line; these features are usually associated with quantum criticality but now mark the FCS critical line, bottom panel.