# Strong Disorder Renormalization for the dynamics of Many-Body-Localized systems : iterative elimination of the fastest degree of freedom via the Floquet expansion

## TL;DR

This paper reformulates the Strong Disorder Renormalization approach for many-body localized systems using Floquet theory, enabling a systematic iterative elimination of the fastest degrees of freedom to analyze their dynamics.

## Contribution

It introduces a Floquet-based high-frequency expansion framework to derive renormalization rules for many-body localized systems, extending previous RG methods.

## Key findings

- Derivation of effective Floquet Hamiltonian using high-frequency expansion.
- Establishment of a flow equivalent to RSRG-X for eigenstate construction.
- Application of the framework to the random-transverse-field XXZ chain in MBL phase.

## Abstract

The Vosk-Altman Strong Disorder Renormalization for the unitary dynamics of various random quantum spin chains is reformulated as follows : the local degree of freedom characterized by the highest eigenfrequency $\Omega$ can be considered as a high-frequency-Floquet-periodic-driving for the neighboring slower degrees of freedom. Then the two first orders of the high-frequency expansion for the effective Floquet Hamiltonian can be used to generate the emergent Local Integrals of Motion (LIOMs) and to derive the renormalization rules for the effective dynamics of the remaining degrees of freedom. The flow for this effective Floquet Hamiltonian is equivalent to the RSRG-X procedure to construct the whole set of eigenstates that generalizes the Fisher RSRG procedure constructing the ground state. This general framework is applied to the random-transverse-field XXZ spin chain in its Many-Body-Localized phase, in order to derive the renormalization rules associated to the elimination of the biggest transverse field and to the elimination of the biggest coupling respectively.