# Supersymmetric Many-Body Systems from Partial Symmetries: Integrability, Localization and Scrambling

## TL;DR

This paper constructs supersymmetric many-body quantum systems using partial symmetries, demonstrating their integrability, localization, and non-thermalizing behavior through analytical and numerical analysis.

## Contribution

It introduces a novel framework using symmetric inverse semigroups to realize supersymmetry in many-body systems, revealing new phases protected by the Witten index.

## Key findings

- Systems are supersymmetric with a finite Witten index.
- Hamiltonians are integrable with mixed product and entangled spectra.
- Disorder induces many-body localization, preventing thermalization.

## Abstract

Partial symmetries are described by generalized group structures known as symmetric inverse semigroups. We use the algebras arising from these structures to realize supersymmetry in (0+1) dimensions and to build many-body quantum systems on a chain. This construction consists in associating appropriate supercharges to chain sites, in analogy to what is done in spin chains. For simple enough choices of supercharges, we show that the resulting states have a finite non-zero Witten index, which is invariant under perturbations, therefore defining supersymmetric phases of matter protected by the index. The Hamiltonians we obtain are integrable and display a spectrum containing both product and entangled states. By introducing disorder and studying the out-of-time-ordered correlators (OTOC), we find that these systems are in the many-body localized phase and do not thermalize. Finally, we reformulate a theorem relating the growth of the second Renyi entropy to the OTOC on a thermal state in terms of partial symmetries.