Ground States of Fermionic lattice Hamiltonians with Permutation Symmetry

TL;DR

This paper investigates the ground states of permutation-invariant fermionic lattice Hamiltonians, introducing a new family of states for efficient computation and applying them to the Fermi-Hubbard model with improved results over traditional methods.

Contribution

It constructs a novel family of fermionic states for permutation-invariant systems, enabling easier ground state calculations and surpassing generalized Hartree-Fock results in the Fermi-Hubbard model.

Findings

Constructed a parametric family of fermionic states for large systems.

Explicitly built states for one and two modes per site.

Achieved results beyond generalized Hartree-Fock theory.

Abstract

We study the ground states of lattice Hamiltonians that are invariant under permutations, in the limit where the number of lattice sites, N -> \infty. For spin systems, these are product states, a fact that follows directly from the quantum de Finetti theorem. For fermionic systems, however, the problem is very different, since mode operators acting on different sites do not commute, but anti-commute. We construct a family of fermionic states, \cal{F}, from which such ground states can be easily computed. They are characterized by few parameters whose number only depends on M, the number of modes per lattice site. We also give an explicit construction for M=1,2. In the first case, \cal{F} is contained in the set of Gaussian states, whereas in the second it is not. Inspired by that constructions, we build a set of fermionic variational wave functions, and apply it to the Fermi-Hubbard model in two spatial dimensions, obtaining results that go beyond the generalized Hartree-Fock theory.