Schwinger-Keldysh field theory for operator Rényi entropy and entanglement growth in non-interacting systems with sub-ballistic transports

Abstract

The notion of operator growth in quantum systems furnishes a bridge between transport and the generation of entanglement between different parts of the system under quantum dynamics. We define a measure of operator growth in terms of subsystem operator Rényi entropy, which provides a state-independent measure of operator growth, unlike entanglement entropies, and the usual measures of operator growth like out-of-time-order correlators. We show that the subsystem operator Rényi entropy encodes both spatial and temporal information, and thus can directly connect to transport for a local operator related to a conserved quantity. We construct a unified Schwinger-Keldysh (SK) field theory formalism for the time evolution of operator Rényi entropy and entanglement entropies of initial pure states. We use the SK field theory to obtain the operator Rényi and state entanglement entropies in terms of infinite-temperature and vacuum Keldysh Green's functions, respectively, for non-interacting systems. We apply the method to explore the connection between operator and entanglement growth, and transport in non-interacting systems with quasiperiodic and random disorder, like the one- and two-dimensional Aubry-André models and the two-dimensional Anderson model. In particular, we show that the growth of subsystem operator Rényi entropy and state von Neumann and Rényi entanglement entropies can capture both ballistic and sub-ballistic transport behaviors, like diffusive and anomalous diffusive transport, as well as localization in these systems.

Figures (18)

• Figure 1: Schematic of the division of the system into two subsystems, e.g., in one dimension, for the calculation of the operator Rényi entropy and state entanglement entropies. A system of length $L$ with periodic boundary condition, is partitioned into two halves $A$ and $B$. The operator $\mathcal{O}$ is placed at a distance $\sim L/4$ from the boundaries between $A$ and $B$.
• Figure 2: The Schwinger-Keldysh closed-time contour for an arbitrary initial density matrix $\rho_0$, extending from $t_0$ to $+\infty$ and returning to $t_0$. The two branches represent the forward ($+$) and backward ($-$) time evolutions, respectively.
• Figure 3: Growth of operator Rényi entropy for quasiperiodic 1D Aubry-André model in the ballistic metallic phase ($V=0.5$), at the sub-diffusive/nearly diffusive critical point ($V=1.0$), and in the localized insulating phase ($V>1$) for system size $L = 576$.
• Figure 4: Growth of (a) operator Rényi entropy $S_{op}^{(2)}$, (b) von Neumann entanglement entropy $S^{(1)}$, and (c) second Rényi entanglement entropy $S^{(2)}$ for 1D Aubry-André Model at the critical point $V_C = 1$ for different system sizes $L=128-576$. The time evolutions of $S^{(1)}$ and $S^{(2)}$ have been computed starting from the Néel state.
• Figure 5: Scaling of saturation times, (a) $t_{op}$, (b) $t_{vN}$, (c) $t_{RE}$, associated with the growth of the operator Rényi entropy and state entanglement entropies, as a function of system size $L$ in the metallic phase and at the critical point, $V=0.0-1.0$ [legend in (a)] for the 1D Aubry-André model. The exponents for the power-law growth of the time scales with $L$ are shown in Table \ref{['tab:AA_1D_SAT']}.