Measuring the distance between quantum many-body wave functions

TL;DR

This paper investigates how the distance between quantum many-body wave functions evolves under chaotic dynamics, revealing its relation to operator scrambling, and demonstrates its effectiveness in characterizing quantum chaos and thermalization.

Contribution

It introduces a new measure of wave function distance based on reduced density matrices, linking it to operator scrambling and demonstrating its utility in analyzing chaos and thermalization.

Findings

Distance $d^2(t)$ grows rapidly in non-local models with power-law interactions.

In local models, $d^2(t)$ does not exhibit exponential growth.

Long-time $d^2(t)$ saturates and decays exponentially with system size, consistent with eigenstate thermalization.

Abstract

We study the distance of two wave functions under chaotic time evolution. The two initial states are differed only by a local perturbation. To be entitled "chaos" the distance should have a rapid growth afterwards. Instead of focusing on the entire wave function, we measure the distance $d^2(t)$ by investigating the difference of two reduced density matrices of the subsystem $A$ that is spatially separated from the local perturbation. This distance $d^2(t)$ grows with time and eventually saturates to a small constant. We interpret the distance growth in terms of operator scrambling picture, which relates $d^2(t)$ to the square of commutator $C(t)$ (out-of-time-order correlator) and shows that both these quantities measure the area of the operator wave front in subsystem $A$. Among various one-dimensional spin-$\frac{1}{2}$ models, we numerically show that the models with non-local power-law interaction can have an exponentially growing regime in $d^2(t)$ when the local perturbation and subsystem $A$ are well separated. This regime is absent in the spin-$\frac{1}{2}$ chain with local interaction only. After sufficiently long time evolution, $d^2(t)$ relaxes to a small constant, which decays exponentially as we increase the system size and is consistent with eigenstate thermalization hypothesis. Based on these results, we demonstrate that $d^2(t)$ is a useful quantity to characterize both quantum chaos and quantum thermalization in many-body wave functions.