Signatures of topology in quantum quench dynamics and their interrelation

Motivated by recent experimental progress in the study of quantum systems far from equilibrium, we investigate the relation between several dynamical signatures of topology in the coherent time evolution after a quantum quench. Specifically, we study the conditions for the appearance of entanglement spectrum crossings, dynamical quantum phase transitions, and dynamical Chern numbers. For noninteracting models, we show that in general there is no direct relation between these three quantities. Instead, we relate the presence of level crossings in the entanglement spectrum to localized boundary modes that may not be of topological origin in the conventional sense. We exemplify our findings with explicit simulations of one-dimensional two-banded models. Finally, we investigate how interactions influence the presence of entanglement spectrum crossings and dynamical quantum phase transitions, by means of time-dependent density matrix renormalization group simulations.

Figure 1

Time evolution of the entanglement spectrum eigenvalues (solid lines) and of the rate function $f\left(t\right)$ (dashed red line) for the quench protocol: $\vec{d}\left(k\right)=(J_x,0,0)$ and $\overrightarrow{d^{\prime }}\left(k\right)=(J_{x}cosk,J_{x}sink,0)$. Here $L=1000$ sites and $J_x=1$. In this case, the ESCs and the DQPTs, signaled by the cusps in $f\left(t\right)$, appear at the same instants in time, while the DCN is finite ($\pm 1$).

Figure 2

Time evolution of the largest entanglement spectrum eigenvalues (solid lines) and of the rate function $f\left(t\right)$ (dashed red line) for the quench protocol: $\vec{d}\left(k\right)=(J_x,0,J_{z}cosk)$ and $\overrightarrow{d^{\prime }}\left(k\right)=(J_{x}cosk,J_{x}sink,J_z)$. Here $L=1000$, $J_x=1$, and $J_z=1$. In this case, the presence of DQPTs is not accompanied by ESCs, while the DCN is finite ($\pm 1$).

Figure 3

For the quench protocol $\vec{d}\left(k\right)=J(\beta ,0,\alpha )$ to $\overrightarrow{d^{\prime }}\left(k\right)=J(cosk,sink,\alpha )$, the parameters $\alpha$ and $\beta$ determine the existence of four regions where DQPTs and/or ESCs appear or do not appear. The existence of ESCs extends to the blue line as well (and thus for any $\alpha =0$), which corresponds to the case of DCN quantized to one.

Figure 4

Dynamics of various signatures for the quench protocol: $\vec{d}\left(k\right)=J(\beta ,0,\alpha )$ and $\overrightarrow{d^{\prime }}\left(k\right)=J(cosk,sink,\alpha )$ with $\alpha =\beta =0.5$ and $J=1$. (a) Time evolution of the entanglement spectrum ${\lambda }_m\left(t\right)$ (for $m=1,...,6$, solid lines), and of the rate function $f\left(t\right)$ (dashed red line). (b) Time evolution of the eigenvalues $h_n\left(t\right)$ of the parent Hamiltonian ${\widehat{H}}_P\left(t\right)$ with open boundary conditions. A pair of zero-energy modes appears when the entanglement spectrum is degenerate. (c) Time evolution of the Zak phase $\mathcal{Z}\left(t\right)$ of the parent Hamiltonian $h_{\mathrm{P}}(k,t)$ (solid blue line). The Zak phase does not equal $\pi$ (marked by the red horizonal line) at the times of the ESCs (marked by vertical dashed black lines, in green the entanglement spectrum for clarity), implying the absence of inversion as well as other possible protecting symmetries in the AZ sense.

Figure 5

Time evolution of the four largest eigenvalues ${\lambda }_m\left(t\right)$ of the entanglement spectrum for the quench protocol: $\vec{d}\left(k\right)=J(\beta ,0,\alpha )$ and $\overrightarrow{d^{\prime }}\left(k\right)=J(cosk,sink,\alpha )$ with $\alpha =\beta =0.5$ and different values of the Hubbard interaction term: $U=0.5J$ (a) and $U=J$ (b). (c) The entanglement spectrum at a fixed time $t^{*}$ [see the black arrow in panel (a)] for which we have crossings between the two largest eigenvalues at $U=0.5J$. Data obtained with DMRG for a chain of $L=96$ unit cells with OBC and $J=1$, using time step $dt=0.0025$ (in units $1/J$). The truncation error is smaller than ${10}^{-12}$ at all times.

Figure 6

The rate function $f\left(t\right)$ for the quantum quench $\vec{d}\left(k\right)=J(\beta ,0,\alpha )$ and $\overrightarrow{d^{\prime }}\left(k\right)=J(cosk,sink,\alpha )$ with $\alpha =\beta =0.5$ for different values of the Hubbard interaction term $U$. Data obtained with DMRG for a chain of $L=96$ unit cells with OBC and $J=1$, using time step $dt=0.001$. The truncation error is smaller than ${10}^{-12}$ at all times.

Figure 7

Time evolution of the rate function $f\left(t\right)$ after a quench in the interacting SSH model from finite (large) $U$ to $U^{\prime }=0$. Black line: $f\left(t\right)$ from Eq. (5), valid for PBC, for a quench from $U=10J$ to $U^{\prime }=0$. Red, blue, green, and violet lines show $f\left(t\right)$ from DMRG simulations with OBC, for a quenches to $U^{\prime }=0$ starting from $U=5J,10J,20J$, and $100J$, respectively. In the plot $J=1$, $L=1056$, the time step $dt=0.0025$ (in units $1/J$). The truncation error is smaller than ${10}^{-12}$ at all times.

Figure 8

Time evolution of the six largest eigenvalues ${\lambda }_m\left(t\right)$ of the entanglement spectrum for a quench in the interacting SSH model from $U=5J$ to $U^{\prime }=0$ with $J=1$. Data obtained with DMRG on a chain of $L=1056$ unit cells with OBC, with time step $dt=0.0025$ (in units $1/J$). The truncation error is smaller than ${10}^{-12}$ at all times.

Figure 9

Dynamics of various signatures for the quench protocol: $\vec{d}\left(k\right)=J(0,0,\alpha )$ and $\overrightarrow{d^{\prime }}\left(k\right)=J(cosk,sink,\alpha )$ with $\alpha =0.5$ and $J=1$. (a) Time evolution of the entanglement spectrum ${\lambda }_m\left(t\right)$ (for $m=1,...,4$, solid lines) and of the rate function $f\left(t\right)$ (dashed red line). (b) Time evolution of the Zak phase $\mathcal{Z}\left(t\right)$ of the parent Hamiltonian $h_{\mathrm{P}}(k,t)$ (solid blue line). The Zak phase equals $\pi$ (marked by the red horizonal line) at the times of the ESCs (marked by vertical dashed black lines; in green is the entanglement spectrum for clarity).

Figure 10

(a) Density profile $|{\varphi }_{L/R}{\left(j\right)|}^2\equiv |\langle j,A|{\varphi }_{L/R}\rangle {|}^2+{\left|\langle j,B|{\varphi }_{L/R}\rangle \right|}^2$ of the two zero-energy modes $|{\varphi }_L\rangle$ and $|{\varphi }_R\rangle$ of the parent Hamiltonian (A19) with OBC. (b) Evolution of ${\vec{n}}_{\mathrm{P}}(k,t_1^{*})$ as a function of $k$ in the Brillouin zone.

Figure 11

Dynamics of various signatures for the quench protocol: $\vec{d}\left(k\right)=J(\beta ,0,\alpha )$ and $\overrightarrow{d^{\prime }}\left(k\right)=J(\delta +cosk,sink,\alpha )$ with $\beta =\alpha =0.5$, $\delta =0.2$, and $J=1$.