Dynamical criticality and domain-wall coupling in long-range Hamiltonians

Dynamical quantum phase transitions hold a deep connection to the underlying equilibrium physics of the quench Hamiltonian. In a recent study [J. C. Halimeh et al., arXiv:1810.07187] it has been numerically demonstrated that the appearance of anomalous cusps in the Loschmidt return rate coincides with the presence of bound domain walls in the spectrum of the quench Hamiltonian. Here we consider transverse-field Ising chains with power-law and exponentially decaying interactions, and show that by removing domain-wall coupling via a truncated Jordan-Wigner transformation onto a Kitaev chain with long-range hopping and pairing, anomalous dynamical criticality is no longer present. This indicates that bound domain walls are necessary for anomalous cusps to appear in the Loschmidt return rate. We also calculate the dynamical phase diagram of the Kitaev chain with long-range hopping and pairing, which in the case of power-law couplings is shown to exhibit rich dynamical criticality including a doubly critical dynamical phase.

Figure 1

The return rate for LRKC at three values of $\alpha =1.5$, 2.1, and 5.5 in (a), (b), and (c), respectively, is shown after a quench starting at $h_i=0$. Each curve represents a different final magnetic field $h_f$ located in the regions A, B, and C (see Fig. 2), respectively, from top to bottom. Quenches to region A display the traditional dynamical critical behavior, with singly critical regular cusps appearing when the system instantaneously crosses the equilibrium critical point $h_c^e=1;h_f=1.5$ for the upper (blue) curves. The intermediate (green) curves represent the return rate for quenches in region B, corresponding to $h_f=0.85$, 0.975, and 1.0 for (a), (b), and (c), respectively. In (a) and (b) these curves show two families of regular cusps originating from two dynamical critical momenta displayed in Fig. 2, while for $\alpha >3$ in (c), the region B is collapsed onto a critical line at $h_c^d=h_c^e$, and only a single critical momentum mode exists. Finally, the lower (gold) curves do not show any cusps since the system is quenched to the trivial phase $h_f=0.5$, i.e., to the deep ordered phase below $h_c^d$ at the given $\alpha$ values. In the case of LRTFIC, anomalous cusps appear in this region [18]. In all panels, solid thick curves represent the exact solution in the thermodynamic limit, while dotted-line curves are for a finite chain of length $N=1000$ with periodic boundary conditions; see Appendix. Insets show the time derivative of the return rate, indicating jumps at the critical times.

Figure 2

Excitation probabilities for the quenches in Fig. 1 from $h_i=0$ to final magnetic fields $h_f>h_c^e$ (region A) and $h_c^d<h_f<h_c^e$ (region B), shown in (a) and (b), respectively. (c) The phase boundaries $h_c^e\left(\alpha \right)$ and $h_c^d\left(\alpha \right)$ as a function of the range parameter $\alpha$. For quenches terminating in region A, one has the singly critical regular dynamical phase known from NNTFIC with a single critical mode $k^{*}$. When $h_f$ lies in region B, however, the return rate displays singularities corresponding to two different critical modes, as shown in (b), which is only possible for $\alpha <{\alpha }_{\times }$ (see text). This doubly critical regular phase vanishes for $\alpha \ge {\alpha }_{\times }$ since then $h_c^d=h_c^e$. Finally, the return rate is fully analytic in region C, where no dynamical critical momentum modes exist. In the inset of (c), the value of $h_c^d$ for $\alpha \gtrsim 3$ is shown to be very close, but not equal to $h_c^e=1$.

Figure 3

The final magnetic field at which one observes critical modes as a function of their momentum $k$ for different values of the range parameter $\alpha =1.6$, 2.2, 3.0, and 5.0 from bottom to top, respectively, in the $h_i=0$ case. All the curves start at $h_f=h_c^e=1$ for $k=0$ signaling the emergence of the singly critical regular dynamical phase for quenches from $h_i=0$ to $h_f>h_c^e$. For $\alpha <3$ the curves show a negative (nonanalytic) slope signaling the emergence of dynamical critical modes even for $h_f<h_c^e$ at $k>0$. At larger $k$, the $\alpha <3$ curves increase again, leading to two dynamical critical momenta for each $h_f$. Therefore, the $\alpha <3$ curves support a doubly critical regular dynamical phase and $h_c^d$ can be obtained as the minimum $h_f$ which supports critical modes, as shown by the dashed gray lines. For $\alpha >3$ the curves may still present negative curvature, and then support the doubly critical phase depending on the value of $h_i$. In the $h_i=0$ case, shown in the plot, the doubly critical phase ceases to exist for $\alpha \ge {\alpha }_{\times }\approx 3.35$, where the curves acquire positive curvature, regardless of the value of $h_i$.

Figure 4

In (a) and (b) we show the return rates for quenches from $h_i=0$ to $h_f=0.9$ and 1.5, respectively, for LRKC with exponentially decaying hopping and pairing. Cusps in the return rate only appear for $h_f>h_c^e$, where $h_c^e=1$ is the equilibrium (and dynamical) critical point for all $\lambda$. The solid lines represent the chain in the thermodynamic limit, while the dots are the finite-size system with periodic boundary conditions. In the case of exponentially decaying couplings, no additional dynamical critical phase appears in LRKC, in contrast to the case of power-law couplings; cf. Fig. 2. Indeed, the excitation probabilities in (c) do not cross the $1/2$ values for quenches to $h_f=0.9$ (solid lines with points) at any finite $k$. Only for quenches across the equilibrium phase transition $h_f>1$, which are indicated by solid lines in (c), does the excitation probability cross the $1/2$ threshold at a finite $k$ giving rise to cusps in the return rates shown in (b). Insets in (a) and (b) show the time derivative of the return rate, which shows jumps at the critical times for (b).

Figure 5

Return rate $r\left(t\right)$ at $\alpha =1.5$ for $N=50000$ compared to the solution obtained by (13) using the analytical expressions (A13) and (A14) in the thermodynamic limit.