Space-time duality between quantum chaos and non-Hermitian boundary effect

Quantum chaos in Hermitian systems concerns the sensitivity of long-time dynamical evolution to initial conditions. The skin effect discovered recently in non-Hermitian systems reveals the sensitivity to the spatial boundary condition even deep in the bulk. In this Letter, we show that these two seemingly different phenomena can be unified through the space-time duality. The intuition is that the space-time duality maps unitary dynamics to nonunitary dynamics and exchanges the temporal direction and spatial direction. Therefore, the space-time duality can establish the connection between the sensitivity to the initial condition in the temporal direction and the sensitivity to the boundary condition in the spatial direction. Here, we demonstrate this connection by studying the space-time duality of the out-of-time-ordered correlator in a concrete chaotic Hermitian model. We show that the out-of-time-ordered correlator is mapped to a special two-point correlator of a non-Hermitian system in the dual picture. For comparison, we show that the sensitivity disappears when the non-Hermiticity is removed in the dual picture.

Figure 1

Schematic of the space-time duality of a quantum circuit. (a), (b) The space-time duality between two-qubit quantum circuits $\widehat{u}$ (left) and $\widehat{v}$ (right) with $v_{i_1,j_1}^{i_2,j_2}=u_{i_1,i_2}^{j_1,j_2}$. The green and blue boxes respectively represent two-qubit gates $e^{i{J}_z{\widehat{\sigma }}_1^z{\widehat{\sigma }}_2^z}$ and $e^{i{\tilde{J}}_z{\widehat{\sigma }}_1^z{\widehat{\sigma }}_2^z}$. The yellow and orange boxes respectively represent single-qubit gates $e^{i{J}_x{\widehat{\sigma }}_x}$ and $e^{i{\tilde{J}}_x{\widehat{\sigma }}_x}$, with the relation between $J_z$ and ${\tilde{J}}_z$, and relation between $J_x$ and ${\tilde{J}}_x$ given by Eq. (3). (c), (d) The space-time duality between two general operators $\widehat{U}$ and $\widehat{V}$, with $L_{\tilde{r}}=L_t$ and $L_{\tilde{t}}=L_r$.

Figure 2

(a) Correlation function $\mathcal{F}\left(\varphi \right)$ defined in Eq. (8). The solid line represents the temporal direction. Different branches of forward and backward evolutions in the double Keldysh contour are distinguished by different colors, and each part has a length $L_t$. (b) The space-time dual of $\mathcal{F}\left(\varphi \right)$. The solid line now represents the spatial direction with $L_{\tilde{r}}=4L_t$. Squares and triangles are respectively labeled as $e^{\pm i\varphi \widehat{W}}$ and $\widehat{O}$ acting on the double Keldysh contour in (a) or their dual circuit acting on different spatial points on the spatial contour in (b). The distance between the triangle and square is $L_t$ in both cases.

Figure 3

(a1), (a2) $\mathcal{F}\left(\varphi \right)$ as a function of $\varphi$ for different $L_t$; (b1), (b2) $\mathcal{F}\left(\varphi \right)$ as a function of $L_t$ for different $\varphi$; (c1), (c2) ${\partial }^2\mathcal{F}\left(\varphi \right)/\partial {\varphi }^2{|}_{\varphi =0}$ as a function of $L_t$ for different $L_r$. $L_r=8$ for (a1), (a2) and (b1), (b2). The top row plots the function given by Eq. (8), or equivalently Eq. (9), with $\widehat{O}={\widehat{\sigma }}_1^z$ and $\widehat{W}={\sum }_{r=1}^{L_r}{\widehat{\sigma }}_r^z$. We choose parameters $\{J_x,J_z,h\}=\{1,1,0.5\}$. The bottom row plots a modified Eq. (9) which eliminates the non-Hermiticity (see text for details).

Figure 4

The duality between the light cone in (a) unitary dynamics and in (b) nonunitary dynamics. The timelike regime [shaded area in (a)] is translated into a spacelike regime [shaded area in (b)] after space-time duality. The correlation is nonzero in the shaded regimes.