Accelerating the convergence of exact diagonalization with the transcorrelated method: Quantum gas in one dimension with contact interactions

TL;DR

This paper introduces a transcorrelated method that exactly handles the wave function cusp in 1D quantum gases with contact interactions, significantly accelerating the convergence of exact diagonalization calculations.

Contribution

The authors develop a non-hermitian transcorrelated Hamiltonian approach that improves wave function smoothness and convergence rates in diagonalization of contact-interacting quantum gases.

Findings

Error scales as M^{-3} with the transcorrelated method

Conventional methods scale as M^{-1} or M^{-2}

Transcorrelated approach enhances convergence efficiency

Abstract

Exact diagonalization expansions of Bose or Fermi gases with contact interactions converge very slowly due to a non-analytic cusp in the wave function. Here we develop a transcorrelated approach where the cusp is treated exactly and folded into the many-body Hamiltonian with a similarity transformation that removes the leading order singularity. The resulting transcorrelated Hamiltonian is not hermitian but can be treated numerically with a standard projection approach. The smoothness of the wave function improves by at least one order and thus the convergence rate for the ground state energy improves. By numerical investigation of a one-dimensional gas of spin-$\frac{1}{2}$ fermions we find the error in the transcorrelated energy to scale as $M^{-3}$ with a single-particle basis of $M$ plane waves compared to $M^{-1}$ for the expansion of the original Hamiltonian and $M^{-2}$ using conventional lattice renormalization.