Flow towards diagonalization for Many-Body-Localization models : adaptation of the Toda matrix differential flow to random quantum spin chains

TL;DR

This paper explores the adaptation of the Toda matrix differential flow to random quantum spin chains, providing a new continuous method for diagonalizing many-body Hamiltonians relevant to Many-Body Localization.

Contribution

It introduces a novel adaptation of the Toda differential flow for many-body Hamiltonians, specifically applied to the random XXZ chain with potential insights into Many-Body Localization.

Findings

The flow coincides with the Toda flow for free-fermion chains.

It offers an alternative to discrete flows for studying localization.

Potential to compare with existing discrete flow methods.

Abstract

The iterative methods to diagonalize matrices and many-body Hamiltonians can be reformulated as flows of Hamiltonians towards diagonalization driven by unitary transformations that preserve the spectrum. After a comparative overview of the various types of discrete flows (Jacobi, QR-algorithm) and differential flows (Toda, Wegner, White) that have been introduced in the past, we focus on the random XXZ chain with random fields in order to determine the best closed flow within a given subspace of running Hamiltonians. For the special case of the free-fermion random XX chain with random fields, the flow coincides with the Toda differential flow for tridiagonal matrices which is related to the classical integrable Toda chain and which can be seen as the continuous analog of the discrete QR-algorithm. For the random XXZ chain with random fields that displays a Many-Body-Localization transition, the present differential flow should be an interesting alternative to compare with the discrete flow that has been proposed recently to study the Many-Body-Localization properties in a model of interacting fermions (L. Rademaker and M. Ortuno, Phys. Rev. Lett. 116, 010404 (2016)).