Dynamics of entanglement after exceptional quantum quench

We investigate a quantum quench from a critical to an exceptional point. The initial state, prepared in the ground state of a critical Hermitian system, is time evolved with a non-Hermitian SSH model, tuned to its exceptional point. The single particle density matrix exhibits supersonic modes and multiple light cones, characteristic of non-Hermitian time evolution. These propagate with integer multiples of the original Fermi velocity. In the long time limit, the fermionic Green's function decays spatially as $1/x^2$, in sharp contrast to the usual $1/x$ decay of noninteracting fermions. The entanglement entropy is understood as if all these supersonic modes arise from independent quasiparticles (though they do not), traveling with the corresponding supersonic light cone velocity. The entropy production rate decreases with time and develops plateaus during the time evolution, signaling the distinct velocities in the propagation of nonlocal quantum correlations. At late times, the entanglement entropy saturates to a finite value, satisfying a volume law.

Figure 1

Schematics of the non-Hermitian SSH model after the quench. The small/big circles denote the $A/B$ sublattices, subject to balanced ($h$) gain (red)/loss (blue), respectively. The thick solid/thin dashed lines stand for the alternating hoppings, $J+\frac{h}{2}$ and $J-\frac{h}{2}$. The distance between two adjacent $A\mathrm{s}$ (or $B\mathrm{s}$) is $a$.

Figure 2

Time dependence of the absolute value of the off-diagonal Green's function is plotted for $x/a=1000$ with $h/J=1$ (blue) and 0.5 (red). The other, $AA, BA$, and $BB$ components are indistinguishable from $AB$ on this scale. Each peak at integer $x/2v_{F}t$ is identified as a supersonic mode. The inset depicts the time averaged off-diagonal Green's function in the steady state for $h=0.1$ (black) and 0.01 (green), the diagonal components vanish for $x\ne 0$. The spatial decay changes from $1/x$ to $1/x^2$ with increasing $x$, the crossover occurs at $x\sim J/h$.

Figure 3

The time evolution of the entanglement entropy is shown for several subsystem sizes. The initial linear growth in time, the gradual slope changes, and the saturation for late time, obeying a volume law, is apparent.

Figure 4

The time evolution of the entropy production rate (gray line) is shown for a typical quench; the moving average (red dashed line) is also depicted to clearly visualize the plateaus. The vertical black dashed lines indicate the expected plateau positions from the velocity of the supersonic modes for several $2v_{F}t/L=1/m$ with $m=1$...7.

Figure 5

The post-quench time dependence of diagonal (a) and off-diagonal (b) Green's matrix elements governed by the Lindblad equation. Both matrix elements exhibit exponential decay after transients. No signal of supersonic modes can be observed.