Stability of dynamical quantum phase transitions in quenched topological insulators: From multiband to disordered systems

Dynamical quantum phase transitions (DQPTs) represent a counterpart in nonequilibrium quantum time evolution of thermal phase transitions at equilibrium, where real time becomes analogous to a control parameter such as temperature. In quenched quantum systems, recently the occurrence of DQPTs has been demonstrated, both with theory and experiment, to be intimately connected to changes of topological properties. Here, we contribute to broadening the systematic understanding of this relation between topology and DQPTs to multiorbital and disordered systems. Specifically, we provide a detailed ergodicity analysis to derive criteria for DQPTs in all spatial dimensions and construct basic counterexamples to the occurrence of DQPTs in multiband topological insulator models. As a numerical case study illustrating our results, we report on microscopic simulations of the quench dynamics in the Harper-Hofstadter model. Furthermore, going gradually from multiband to disordered systems, we approach random disorder by increasing the (super)unit cell within which random perturbations are switched on adiabatically. This leads to an intriguing order of limits problem which we address by extensive numerical calculations on quenched one-dimensional topological insulators and superconductors with disorder.

Figure 1

(a) Interpretation of the Pancharatnam geometrical phase on the Bloch sphere as half of the surface area enclosed by the trajectory up to time $t$ and the geodesic curve leading back to the initial wave function. (b) Time evolution snapshots of the geometrical phase for the three-band Hofstadter model after a quench. Phase vortices are circled in red, and the marked areas show the admissible region according to the criterion (5) (cross-hatched) and the complement of the exclusion (6) (dotted). (c) Corresponding rate function and its derivative. Cusps of $g^{\prime }\left(t\right)$ hallmark DQPTs, i.e., the (dis)appearance of Fisher zeros and phase vortex pairs.

Figure 2

Pancharatnam geometrical phase and rate function for the three-band Hofstadter model as in Fig. 1, but for a $(k$-independent) initial state with complex random entries. The lack of momentum symmetry of the geometrical phase is expected (see main text). Note that the dynamical criterion for Fisher zeros (dotted areas) only depends on the spectrum of $H\left(k\right)$ and thus agrees with Fig. 1.

Figure 3

Pancharatnam geometrical phase for the disordered Kitaev chain with period $\ell$ and increasing disorder strength $\mathrm{\Delta }{\mu }_{\text{max}}$. Each row corresponds to a fixed disorder strength (starting from zero disorder in the top row), and each column to a fixed supercell size $\ell$. The dashed vertical lines mark the critical momentum $k_c$ of the ordered system $\left(\mathrm{\Delta }{\mu }_{\text{max}}=0\right)$.

Figure 4

Rate function and its derivative around the first critical time point $t_c$, for the disordered Kitaev chain with random disorder realizations in the supercell representation.

Figure 5

Histogram and corresponding variance for random disorder realizations $\left(\mathrm{\Delta }{\mu }_{\text{max}}=\frac{1}{2}\right)$ of the rate function $g^{\ell }\left(t\right)$ evaluated at $t=1$. The variance exhibits a $\sim 1/\ell$ scaling, with $\ell$ the supercell size.