Quasiparticle origin of dynamical quantum phase transitions

Considering nonintegrable quantum Ising chains with exponentially decaying interactions, we present matrix product state results that establish a connection between low-energy quasiparticle excitations and the kind of nonanalyticities in the Loschmidt return rate. When domain walls in the spectrum of the quench Hamiltonian are energetically favored to be bound rather than freely propagating, anomalous cusps appear in the return rate regardless of the initial state. In the nearest-neighbor limit, domain walls are always freely propagating and anomalous cusps never appear. As a consequence, our work illustrates that models in the same equilibrium universality class can still exhibit fundamentally distinct out-of-equilibrium criticality. Our results are accessible to current ultracold-atom and ion-trap experiments.

Figure 1

Dynamical phase diagrams of (1) in the $(\lambda ,h_{\mathrm{f}})$ plane with an initial transverse-field strength of (a) $h_{\mathrm{i}}=0$ and (b) $h_{\mathrm{i}}\to \infty$. The different regions and transitions are explained in the text.

Figure 2

The energy gap in the topologically nontrivial (dashed lines) and topologically trivial (solid lines) sectors for (1) $\lambda =0.4$ [black (dark)] and $\lambda =0.8$ [yellow (light)] as a function of transverse-field strength $h$. The dotted line indicates the edge of the two-domain-wall continuum: if we find an excitation below this continuum, this corresponds to a stable local excitation. Both gaps approach zero at the equilibrium critical point. There is no topological sector in the paramagnetic phase, so we cannot define a crossover there.

Figure 3

Loschmidt return rates after quenching a fully $z$-polarized state with $\widehat{H}\left(h_{\mathrm{f}}\right)$ for $\lambda =0.4$ at (a) $h_{\mathrm{f}}=2J$, (b) $h_{\mathrm{f}}=3.65J$, and (c) $h_{\mathrm{f}}=4.5J$. Dotted gray lines indicate sections of the branch cut above the return rate (solid blue line), and cusps form when they intersect, whereas the dashed orange line is the order parameter. Here, two phases appear separately as (a) the ADP for small quenches $h_{\mathrm{f}}<h_{\mathrm{c}}^{\mathrm{d}}<h_{\mathrm{cross}}$ and (b) the RDP for large quenches $h_{\mathrm{f}}>h_{\mathrm{cross}}>h_{\mathrm{c}}^{\mathrm{d}}$, but the return rate also exhibits (c) both cusps for intermediate quenches $h_{\mathrm{f}}\in (h_{\mathrm{c}}^{\mathrm{d}},h_{\mathrm{cross}})$. Note that for $\lambda =0.4$ and quenches from $h_{\mathrm{i}}=0$, we have $h_{\mathrm{c}}^{\mathrm{d}}<h_{\mathrm{cross}}$ [see Fig. 1]. See Fig. 4 where, at $\lambda =0.8$ and quenches from $h_{\mathrm{i}}=0$, we have $h_{\mathrm{c}}^{\mathrm{d}}>h_{\mathrm{cross}}$.

Figure 4

Same as Fig. 3, but for $\lambda =0.8$ with (a) $h_{\mathrm{f}}=1.35J$, (b) $h_{\mathrm{f}}=1.65J$, and (c) $h_{\mathrm{f}}=3J$. The return rate shows one of three distinct phases: (a) the ADP for small quenches $h_{\mathrm{f}}<h_{\mathrm{cross}}<h_{\mathrm{c}}^{\mathrm{d}}$, (b) the TDP for intermediate quenches $h_{\mathrm{f}}\in (h_{\mathrm{cross}},h_{\mathrm{c}}^{\mathrm{d}})$, and (c) the RDP for large quenches $h_{\mathrm{f}}>h_{\mathrm{c}}^{\mathrm{d}}>h_{\mathrm{cross}}$. Note that for $\lambda =0.8$ and quenches from $h_{\mathrm{i}}=0$, we have $h_{\mathrm{c}}^{\mathrm{d}}>h_{\mathrm{cross}}$ [see Fig. 1]. See Fig. 3 where, at $\lambda =0.4$ and quenches from $h_{\mathrm{i}}=0$, we have $h_{\mathrm{c}}^{\mathrm{d}}<h_{\mathrm{cross}}$.

Figure 5

Same as Fig. 3, but starting from a fully $x$-polarized state ($h_{\mathrm{i}}\to \infty$), and while showing the transverse magnetization ${\mathcal{M}}_x\left(t\right)$ rather than the order parameter, as the latter is always zero here due to both the initial state and quenching Hamiltonian being $Z_2$ symmetric. (a) The return rate shows the TDP for $h_{\mathrm{f}}=6J>h_{\mathrm{c}}^{\mathrm{e}}$, (b) the RDP for $h_{\mathrm{f}}=4.5J\in (h_{\mathrm{cross}},h_{\mathrm{c}}^{\mathrm{e}})$, and (c) both regular and anomalous cusps for $h_{\mathrm{f}}=2J<h_{\mathrm{cross}}$.

Figure 6

Same as Fig. 5, but for $\lambda =0.8$ with (a) $h_{\mathrm{f}}=3J>h_{\mathrm{c}}^{\mathrm{e}}$, (b) $h_{\mathrm{f}}=2J\in (h_{\mathrm{cross}},h_{\mathrm{c}}^{\mathrm{e}})$, and (c) $h_{\mathrm{f}}=J<h_{\mathrm{cross}}$. The dynamical behavior is qualitatively the same as at other values $\lambda$ for this quench.

Figure 7

Loschmidt return rates after quenching a fully $x$-polarized state with $\widehat{H}\left(h_{\mathrm{f}}\right)$ for (a)–(c) $\lambda =2$ and (d)–(f) $\lambda =3$. Dotted lines indicate sections of the branch cut above the return rate (solid blue line), and cusps form when they intersect. The transverse magnetization is represented by a dashed orange line. Just as in the main text, we see three distinct dynamical phases: (a), (d) the TDP for $h_{\mathrm{f}}>h_{\mathrm{c}}^{\mathrm{e}}$; (b), (c) the RDP for $h_{\mathrm{f}}\in (h_{\mathrm{cross}},h_{\mathrm{c}}^{\mathrm{e}})$; and (c), (f) a coexistence region of the ADP and RDP for $h_{\mathrm{f}}<h_{\mathrm{cross}}$.

Figure 8

Different values of the maximum bond dimension $D_{\max }$ were used in our simulations, indicating convergence at $D_{\max }=350$ or lower with a time step of $\delta t=0.001/J$.