Out-of-equilibrium phase diagram of long-range superconductors

Within the ultimate goal of classifying universality in quantum many-body dynamics, understanding the relation between out-of-equilibrium and equilibrium criticality is a crucial objective. Models with power-law interactions exhibit rich well-understood critical behavior in equilibrium, but the out-of-equilibrium picture has remained incomplete, despite recent experimental progress. We construct the rich dynamical phase diagram of free-fermionic chains with power-law hopping and pairing and provide analytic and numerical evidence showing a direct connection between nonanalyticities of the return rate and zero crossings of the string order parameter. Our results may explain the experimental observation of so-called accidental dynamical vortices, which appear for quenches within the same topological phase of the Haldane model, as reported by Fläschner et al. [Nat. Phys. 14, 265 (2018)]. Our work is readily applicable to modern ultracold-atom experiments, not least because state-of-the-art quantum gas microscopes can now reliably measure the string order parameter, which, as we show, can serve as an indicator of dynamical criticality.

Figure 1

Dispersion relation ${\omega }_k$ (10) together with the Fourier transforms of the hopping amplitude $J_k$ (6) and pairing amplitude ${\mathrm{\Delta }}_k$ (7) for quasimomenta $k$ within the first Brillouin zone. The dispersion relation is calculated with the following long-range hopping and pairing exponents $\alpha$ and $\beta$ and chemical potential $h$, listed from top to bottom of the left end of each solid curve: $(\alpha ,\beta ,h)=(2.8,1.7,1.4)$ in purple, $(\alpha ,\beta ,h)=(2.3,1.2,1.4)$ in violet, $(\alpha ,\beta ,h)=(3,1.9,0.4)$ in green, $(\alpha ,\beta ,h)=(1.8,1.5,0.4)$ in blue.

Figure 2

The four different forms that function (31) can take for quenches from $h_i=0$. In form (i), we see that there exists only the singly critical phase, where quenches to $h_f>h_c^e=1$ generate a unique critical momentum mode that gives rise to critical times in the return rate. In form (ii), we see again the singly critical phase, but also the doubly critical phase, which occurs for quenches within the topologically nontrivial phase and allows for two critical momentum modes. In form (iii), the singly critical phase emerges for quenches just above $h_c^e$ and also for large quenches. Quenches in between lead to a triply critical phase with three critical momentum modes. Finally, form (iv) exhibits the richest dynamical criticality, in that again the doubly critical phase emerges for quenches within the topologically nontrivial phase, the triply critical phase appears for quenches in a region above $h_c^e$, but for quenches above that region only the singly critical phase exists.

Figure 3

Phase diagram of the model in the $(\alpha ,\beta )$ plane. In the red-colored region the doubly critical phase exists, while in the white region $c_1>0$ and there is no doubly critical phase.

Figure 4

The value $h_c^d$ which delimits the appearance of the doubly critical phase for a quench starting at $h_i=0$ is shown as a function of $\beta$ for different values of $\alpha$.

Figure 5

Time evolution of the squared longitudinal magnetization of the TFIC (25) and the norm of the string order parameter of its Jordan-Wigner mapping for a quench from $h_i=0$ to $h_f=3$. The result for the magnetization is simulated using TDVP [66, 67, 68].

Figure 6

The time evolution of the return rate $r\left(t\right)$ and SOP norm $\mathcal{M}\left(t\right)$ for the Kitaev chain with long-range hopping and pairing with decay exponents $\alpha$ and $\beta$, respectively, for three different values of $\alpha \in \{1.75,2.25,3.25\}$, each at three different values of $\beta \in \{1.50,2.50,3.50\}$. These results are for quenches from $h_i=0$ to $h_f>h_c^e$, i.e., in the singly critical dynamical phase. The SOP is calculated for $N=4000$ sites while the return rate $r\left(t\right)$ for $N=10000$ sites, where these system sizes are chosen such that the corresponding quantity reaches convergence. The second time-derivative (curvature) of the return rate, $\ddot{r}\left(t\right)$, is also provided in order to highlight cusps that may not appear clearly in $r\left(t\right)$. As can be seen, the cusps of $r\left(t\right)$ and the zeros of $\mathcal{M}\left(t\right)$ exhibit the same periodicity.

Figure 7

Return rate and SOP norm for a quench from $h_i=0$ into the the doubly critical phase with $h_f=1.075h_c^d<h_c^e$, for two choices of $(\alpha ,\beta )$. In both panels the top quadrant shows the SOP norm and the middle quadrant the return rate. The bottom quadrant shows the curvature $\ddot{r}\left(t\right)$ of the return rate in order to identify the critical times, which are further indicated by the vertical dashed lines. The critical times exhibit two distinct periodicities, distinguished by the black and gray dashed lines, respectively, arising from the two distinct critical momenta of this quench. The system sizes were chosen as in Fig. 6.

Figure 8

Return rate and SOP norm for a quench from $h_i=0$ into the the triply critical phase with $h_f=1.05>h_c^e$, for two choices of $(\alpha ,\beta )$. The quadrants are arranged as in Fig. 7. The vertical lines indicate the critical times with three distinct periodicities (black dashed, dark gray dashed, light gray dot-dashed) arising from the three critical momenta of this quench. The system sizes were chosen as in Fig. 6.

Figure 9

Comparison between the return rate of the TFIC (25) and that of the LRKC (1) in the limit $\alpha ,\beta \to \infty$ for a quench from $h_i=0$ to $h_f=3$ (left panel) and a quench from $h_i\to \infty$ to $h_f=0.5$ (right panel). The return rates of the TFIC are computed using TDVP.