Probing the role of long-range interactions in the dynamics of a long-range Kitaev chain

We study the role of long-range interactions (more precisely, the long-range superconducting gap term) on the nonequilibrium dynamics considering a long-range $p$-wave superconducting chain in which the superconducting term decays with distance between two sites in a power-law fashion characterized by an exponent $\alpha$. We show that the Kibble-Zurek scaling exponent, dictating the power-law decay of the defect density in the final state reached following a slow (in comparison to the time scale associated with the minimum gap in the spectrum of the Hamiltonian) quenching of the chemical potential $\mu$ across a quantum critical point, depends nontrivially on the exponent $\alpha$ as long as $\alpha <2$; on the other hand, for $\alpha >2$, we find that the exponent saturates to the corresponding well-known value of $1/2$ expected for the short-range model. Furthermore, studying the dynamical quantum phase transitions manifested in the nonanalyticities in the rate function of the return possibility $I\left(t\right)$ in subsequent temporal evolution following a sudden change in $\mu$, we show the existence of a new region; in this region, we find three instants of cusp singularities in $I\left(t\right)$ associated with a single sector of Fisher zeros. Notably, the width of this region shrinks as $\alpha$ increases and vanishes in the limit $\alpha \to 2$, indicating that this special region is an artifact of the long-range nature of the Hamiltonian.

Figure 1

The variation of the defect density $n_d$ with inverse quenching rate ${\tau }_Q$. Left: The scaling exponent (determined by the slope of the curve) depends on the interaction range $\alpha$ for $\alpha <2$. The solid lines show the theoretically predicted values $1/(2\alpha -2)$, and the symbols are from the numerical solution of the differential equations (9) using the Runge-Kutta method with $L=10000$, ${\mu }_i=-10$, and ${\mu }_f=0$. Right: The scaling exponent no longer changes with the range of interaction for $\alpha \ge 2$ but rather gets saturated to the short-range value $1/2$. In the marginal case $\alpha =2$, the exponent reaches the short-range limit of $1/2$ only in the asymptotic limit.

Figure 2

Left: Plot of the transition probability (27) as a function of $k$ for different values of ${\mu }_i$ with ${\mu }_f=1$; the long-range interaction parameter $\alpha$ is fixed to $\alpha =1.3$. We find two distinct regimes: (i) The transition probability $p_k$ crosses the value 1/2 for three values of $k$ (i.e., yielding three values of $k_{*}$) for ${\mu }_i=-3.2$ and once for ${\mu }_i=-6.5$, while for the other two values it crosses $p_k=1/2$ once and touches at another value of $k$. Right: Phase boundary separating the three-crossing region (A) from the single-crossing regions (B) is shown in the $\alpha -{\mu }_i$ plane. On the phase boundary $p_k$ crosses (and touches) the $p_k=1/2$ line twice, as shown in the left panel. It is noteworthy that the width of region A shrinks as $\alpha$ increase and vanishes as $\alpha \to 2$.

Figure 3

Left: The temporal evolution of ${\nu }_D\left(t\right)$ as a function of $t$ corresponding to region A shown in the right panel of Fig. 2. We find that corresponding to $n=0$, there are three instants, $t_0^{1*}$, $t_0^{2*}$, and $t_0^{3*}$, at which ${\nu }_D\left(t\right)$ makes discontinuous changes by a factor of unity. However, whether the DTOP jumps or drops is determined by the slope of the $p_k$ vs $k$ curve at $k_{*}$ (i.e., the crossing points) as depicted in the left panel of the Fig. 2. Right: ${\nu }_D$ vs $t$ for region B in the right panel of Fig. 2; we have only one instant of nonanalyticity corresponding to each $n$, and we show jumps in the DTOP corresponding to $n=0$ and $n=1$. Notably, in this case there is no drop in ${\nu }_D\left(t\right)$ at DQPTs.