# Time dynamics with matrix product states: Many-body localization transition of large systems revisited

## TL;DR

This paper compares two matrix product state algorithms, tDMRG and TDVP, for simulating many-body localization transitions, revealing that their effectiveness varies with the system's phase and size.

## Contribution

It provides a comparative analysis of tDMRG and TDVP algorithms across different phases of many-body localized systems, highlighting their respective strengths and limitations.

## Key findings

- tDMRG performs better in localized and crossover regimes.
- TDVP outperforms tDMRG in delocalized regimes.
- Previous estimates of critical disorder strength are challenged.

## Abstract

We compare accuracy of two prime time evolution algorithms involving Matrix Product States - tDMRG (time-dependent density matrix renormalization group) and TDVP (time-dependent variational principle). The latter is supposed to be superior within a limited and fixed auxiliary space dimension. Surprisingly, we find that the performance of algorithms depends on the model considered. In particular, many-body localized systems as well as the crossover regions between localized and delocalized phases are better described by tDMRG, contrary to the delocalized regime where TDVP indeed outperforms tDMRG in terms of accuracy and reliability. As an example, we study many-body localization transition in a large size Heisenberg chain. We discuss drawbacks of previous estimates [Phys. Rev. B 98, 174202 (2018)] of the critical disorder strength for large systems.