Dynamical Quantum Phase Transitions: Role of Topological Nodes in Wave Function Overlaps

A sudden quantum quench of a Bloch band from one topological phase toward another has been shown to exhibit an intimate connection with the notion of a dynamical quantum phase transition (DQPT), where the returning probability of the quenched state to the initial state—i.e., the Loschmidt echo—vanishes at critical times $\{t^{*}\}$. Analytical results to date are limited to two-band models, leaving the exact relation between topology and DQPT unclear. In this Letter, we show that, for a general multiband system, a robust DQPT relies on the existence of nodes (i.e., zeros) in the wave function overlap between the initial band and the postquench energy eigenstates. These nodes are topologically protected if the two participating wave functions have distinctive topological indices. We demonstrate these ideas in detail for both one and two spatial dimensions using a three-band generalized Hofstadter model. We also discuss possible experimental observations.

Figure 1

Geometric representation of the DQPT condition $G(t^{*})=\sum _n{z}_n(t^{*})=0$. $\{z_n(t^{*})\}$ must form a closed polygon in the complex plane and would hence satisfy a generalized triangle inequality $|z_n|\le \sum _{m\ne n}|z_m|$. Wave function normalization $⟨\mathrm{\Psi }|\mathrm{\Psi }⟩=\sum _n|z_n|=1$ then leads to $|z_n|\le \frac{1}{2}$.

Figure 2

Plot of ${\psi }_{\mathrm{Max}\mathrm{Min}}$ [Eq. (9)] as functions of the postquench $t$. The prequench state is prepared by filling one of the three bands ${\psi }^{(1,2,3)}$ of the generalized Hofstadter model Eq. (11) with parameters $t_i=3$ and $m_i=2.8$. Postquench $H(\mathit{k})$ has fixed $m_f=3$ and a varying $t_f$, sweeping it through six topological phases labeled by its three Chern numbers (ordered from the lower to the higher band). The prequench Hamiltonian is in phase 4. A robust DQPT can be identified by ${\psi }_{\mathrm{Max}\mathrm{Min}}=0$ [Eq. (10)]. Note that ${\psi }_{\mathrm{Max}\mathrm{Min}}$ changes between zero and nonzero only at phase boundaries, verifying robust DQPT as a feature of topological phases insensitive to parameter tuning. See the SM [43] for a detailed account of all 18 types of quenches shown here.