Quantum Quench in $\mathcal{P}\mathcal{T}$-Symmetric Luttinger Liquid

A Luttinger liquid (LL) describes low energy excitations of many interacting one dimensional systems, and exhibits universal response both in and out of equilibrium. We analyze its behavior in the non-Hermitian realm after quantum quenching to a $\mathcal{PT}$-symmetric LL by focusing on the fermionic single particle density matrix. For short times, we demonstrate the emergence of unique phenomena, characteristic to non-Hermitian systems, that correlations propagate faster than the conventional maximal speed, known as the Lieb-Robinson bound. These emergent supersonic modes travel with velocities that are multiples of the conventional light cone velocity. This behavior is argued to be generic for correlators in non-Hermitian systems. In the long time limit, we find typical LL behavior, extending the LL universality to the nonequilibrium, non-Hermitian case. Our analytical results are benchmarked numerically and indicate that the dispersal of quantum information is much faster in non-Hermitian systems.

Figure 1

The single particle density matrix is plotted from Eq. (10) for $g_2/v=0.8$ (blue) and 0.95 (red) line with $x=100\alpha$. The $n=1$ light cone from the sum in $G_r(x,t)$ corresponds to the conventional light cone with $2\tilde{v}$ velocity, while the first two supersonic features are denoted by $n=2$ and 3 with $4\tilde{v}$ and $6\tilde{v}$ velocity, respectively.

Figure 2

The complex many-body energy spectrum, $E_n$ of Eq. (13) for $N=14$ at half filling is plotted for $J_z=-0.3J$ (red dots) and the noninteracting, Hermitian case with $J_z=0$ (blue squares) for comparison from exact diagonalization using periodic boundary conditions. During the time evolution with $e^{-itH}$, states with the largest imaginary part contribute the most, which is the upper flat part of the spectrum. Out of these, states with increasing $\mathrm{Re}{E}_n$ have less influence due to their decreasing overlap with the initial state. Due to the normalization in Eq. (11), the imaginary offset of energies drops out from the expectation values.

Figure 3

Contour plot of the density correlator, ${\chi }_{nn}(x,t)/{\chi }_{nn}(x,0)$, where denominator cancels the initial spatial correlation in the ground state, and all features result from the non-Hermitian quench dynamics with $J_z/J=-0.3$. The white dashed lines denote the $n=1$, 2, and 3 modes by using the sound velocity, $\tilde{v}=1.02J$ without any fitting.