# Gaussian time dependent variational principle for the Bose-Hubbard model

## TL;DR

This paper extends Gaussian time-dependent variational principle methods to the Bose-Hubbard model, providing improved ground state approximations, excitation spectra, and response functions beyond traditional mean-field theories.

## Contribution

It introduces a Gaussian TDVP approach for the Bose-Hubbard model, surpassing mean-field approximations and benchmarking against established methods like DMRG.

## Key findings

- Accurate ground state approximations in 1d, 2d, 3d
- Identification of Goldstone, doublon, and continuum excitations
- Computed linear response functions relevant to experiments

## Abstract

We systematically extend Bogoliubov theory beyond the mean field approximation of the Bose-Hubbard model in the superfluid phase. Our approach is based on the time dependent variational principle applied to the family of all Gaussian states (i.e. Gaussian TDVP). First, we find the best ground state approximation within our variational class using imaginary time evolution in 1d, 2d and 3d. We benchmark our results by comparing to Bogoliubov theory and DMRG in 1d. Second, we compute the approximate 1- and 2-particle excitation spectrum as eigenvalues of the linearized projected equations of motion (linearized TDVP). We find the gapless Goldstone mode, a continuum of 2-particle excitations and a doublon mode. We discuss the relation of the gap between Goldstone mode and 2-particle continuum to the excitation energy of the Higgs mode. Third, we compute linear response functions for perturbations describing density variation and lattice modulation and discuss their relations to experiment. Our methods can be applied to any perturbations that are linear or quadratic in creation/annihilation operators. Finally, we provide a comprehensive overview how our results are related to well-known methods, such as traditional Bogoliubov theory and random phase approximation.