# Time Dependent Variational Principle for Tree Tensor Networks

## TL;DR

This paper extends the Time Dependent Variational Principle (TDVP) to general loop-free tensor networks, enabling efficient simulation of quantum systems with complex Hamiltonians, including long-range interactions.

## Contribution

It generalizes TDVP to any finite loop-free tensor network, allowing for broader applications in quantum many-body simulations.

## Key findings

- TDVP can be applied to Fork Tensor Product States for multi-orbital Anderson models.
- Enables accounting for off-diagonal hybridizations in the bath.
- Applicable to systems with spin-orbit coupling and lattice distortions.

## Abstract

We present a generalization of the Time Dependent Variational Principle (TDVP) to any finite sized loop-free tensor network. The major advantage of TDVP is that it can be employed as long as a representation of the Hamiltonian in the same tensor network structure that encodes the state is available. Often, such a representation can be found also for long-range terms in the Hamiltonian. As an application we use TDVP for the Fork Tensor Product States tensor network for multi-orbital Anderson impurity models. We demonstrate that TDVP allows to account for off-diagonal hybridizations in the bath which are relevant when spin-orbit coupling effects are important, or when distortions of the crystal lattice are present.