Abstract
The virial expansion provides a non-perturbative view into the thermodynamics of quantum many-body systems in dilute regimes. While powerful, the expansion is challenging as calculating its coefficients at each order n requires analyzing (if not solving) the quantum n-body problem. In this work, we present a comprehensive review of automated algebra methods, which we developed to calculate high-order virial coefficients. The methods are computational but non-stochastic, thus avoiding statistical effects; they are also for the most part analytic, not numerical, and amenable to massively parallel computer architectures. We show formalism and results for coefficients characterizing the thermodynamics (pressure, density, energy, static susceptibilities) of homogeneous and harmonically trapped systems and explain how to generalize them to other observables such as the momentum distribution, Tan contact, and the structure factor.
Highlights
IntroductionThe virial expansion (VE) (see, e.g., Reference [2] for an introduction to both the classical and quantum cases and Reference [3] for a comprehensive review) aims to tackle the finite-temperature quantum many-body problem by breaking it down into contributions from subspaces of the full Fock space corresponding to a fixed (and small) particle number
Quantum many-body systems are notoriously difficult to compute
The virial expansion (VE) is an expansion of the quantum many-body thermodynamics in powers of the fugacity z that is capable of non-perturbatively characterizing such many-body systems in dilute, high-temperature regimes
Summary
The virial expansion (VE) (see, e.g., Reference [2] for an introduction to both the classical and quantum cases and Reference [3] for a comprehensive review) aims to tackle the finite-temperature quantum many-body problem by breaking it down into contributions from subspaces of the full Fock space corresponding to a fixed (and small) particle number In this sense, the VE is effectively an expansion around a dilute limit in which the interparticle distance is much p larger than every other scale in the system, in particular the thermal wavelength λ T = 2πβ, where β = 1/T is the inverse temperature and we have used units such that h = k B = m = 1. We are getting ahead of ourselves; let us start from the beginning
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