# Entanglement spectrum of the degenerative ground state of Heisenberg ladders in a time-dependent magnetic field

## TL;DR

This paper explores the relationship between entanglement and subsystem Hamiltonians in Heisenberg ladders under time-dependent magnetic fields, revealing conditions where entanglement spectra are nontrivial despite proportionality to subsystem Hamiltonians.

## Contribution

It extends perturbation theory analysis to degenerate ground states of Heisenberg ladders, showing nontrivial entanglement spectra can occur even when Hamiltonians are proportional.

## Key findings

- Entanglement Hamiltonian proportional to subsystem Hamiltonian under certain conditions.
- Degenerate ground states can have nontrivial entanglement spectra.
- First-order perturbation theory reveals complex entanglement structures.

## Abstract

We investigate of the relationship between the entanglement and subsystem Hamiltonians in the perturbative regime of strong coupling between subsystems. One of the two conditions that guarantees the proportionality between these Hamiltonians obtained by using the nondegenerate perturbation theory within the first order is that the unperturbed ground state has a trivial entanglement Hamiltonian. Furthermore, we study the entanglement Hamiltonian of the Heisenberg ladders in a time-dependent magnetic field using the degenerate perturbation theory, where couplings between legs are considered as a perturbation. In this case, when the ground state is two-fold degenerate, and the entanglement Hamiltonian is proportional to the Hamiltonian of a chain within first-order perturbation theory, even then also the unperturbed ground state has a nontrivial entanglement spectrum.