# Slow Dynamics and Kohlrausch Relaxation in Isolated Disordered Many-Body Systems

## TL;DR

This paper demonstrates that Kohlrausch relaxation law is a universal feature in isolated disordered many-body quantum systems, persisting in the thermodynamic limit and characterized by a peak in relaxation time with disorder strength.

## Contribution

It reveals the generic occurrence of stretched exponential relaxation in closed quantum systems and introduces a simple non-Hermitian model to analytically describe single-spin dynamics.

## Key findings

- Kohlrausch law applies broadly to disordered quantum systems.
- Relaxation time peaks at intermediate disorder levels.
- Non-Hermitian model accurately captures single-spin auto-correlation dynamics.

## Abstract

The Kohlrausch(-Williams-Watts) law of stretched exponential relaxation has been observed for more than a century and a half in diverse complex classical systems. Here we show that this law describes relaxation quite generically in closed (executing Schr\"{o}dinger dynamics), interacting disordered many-body systems across a range of system sizes using interaction range and disorder strength as primary tuning parameters. This we observe for both time-independent and periodically driven (Floquet) systems. Finite-size analysis indicates the persistence of this non-thermal relaxation regime in the thermodynamic limit thus defining a distinct dynamical regime. This regime exhibits a peak in the time-scale of the perceptible relaxation, upon crossing over from weak to strong disorder. We provide a simple picture of this behavior, which naturally accounts for its general occurrence. Formation of spin-glass -- one of the possible mechanisms for stretched relaxation appears incidental to the occurrence of Kohlrausch law in our context. Finally, we provide a simple non-Hermitian Hamiltonian formulation for the dynamics of a single spin embedded in the disordered chain. This provides an analytical formula that captures not only the Kohlrausch relaxation of the disorder averaged auto-correlation but also captures the largely diverse dynamics of an arbitrary target spin in the system. Our work hence also provides a concrete quantification of the ``pre-thermal slowness" in many-body disordered system.