On the minimization of Hamiltonians over pure Gaussian states

TL;DR

This paper demonstrates that minimizing a polynomial Hamiltonian over pure Gaussian states simplifies it by removing non-particle-number-preserving terms, resulting in a form interpretable as quasiparticle excitations, a process termed Beliaev's Theorem.

Contribution

It introduces Beliaev's Theorem, showing how Hamiltonians can be Wick-ordered to eliminate certain terms after minimization over Gaussian states.

Findings

Non-particle-number-preserving terms are eliminated after minimization.

Remaining Hamiltonian terms can be interpreted as quasiparticle excitations.

The process simplifies the Hamiltonian's structure for analysis.

Abstract

A Hamiltonian defined as a polynomial in creation and annihilation operators is considered. After a minimization of its expectation value over pure Gaussian states, the Hamiltonian is Wick-ordered in creation and annihillation operators adapted to the minimizing state. It is shown that this procedure eliminates from the Hamiltonian terms of degree 1 and 2 that do not preserve the particle number, and leaves only terms that can be interpreted as quasiparticles excitations. We propose to call this fact Beliaev's Theorem, since to our knowledge it was mentioned for the first time in a paper by Beliaev from 1959.