Toward an Automated-Algebra Framework for High Orders in the Virial Expansion of Quantum Matter

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.

Quatrièmes coefficients d’amas et du viriel d’un gaz unitaire de fermions pour un rapport de masse quelconque

“…the 1 + 3 case being deduced from the 3 + 1 case by exchanging the two spin states thus changing the mass ratio α = m ↑ /m ↓ to its inverse 1/α. For α = 1, conjecture ( 5) is in agreement with the quantum Monte Carlo calculation of reference [42] down to the minimal accessible value of ω, ω ≈ 1; it is also in agreement with a recent, more powerful numerical calculation down to values of ω 1 [43,44]. After using relation (3), it is also in agreement with the same powerful numerical calculation performed directly in the spatially homogeneous case of a quantization box [45].…”

“…le cas 1 + 3 se déduisant du cas 3 + 1 par échange des deux états de spin donc changement du rapport de masse α = m ↑ /m ↓ en son inverse 1/α. Pour α = 1, la conjecture (5) est en accord avec le calcul de Monte-Carlo quantique de la référence [42] jusqu'à la valeur minimale de ω accessible, ω ≈ 1 ; elle est en accord avec un calcul numérique récent plus performant jusqu'à des valeurs de ω 1 [43,44]. Après utilisation de la relation (3), elle est aussi en accord avec le même calcul numérique performant effectué directement dans le cas spatialement homogène d'une boîte de quantification [45].…”

Thermodynamics of spin-1/2 fermions on coarse temporal lattices using automated algebra

“…The bold-diagrammatic Monte Carlo (BDMC) results of [17] appear as pink empty squares, the auxiliary-field quantum Monte Carlo (AFQMC) data of [18] are shown as orange empty squares and the self-consistent results of [19] are given by the green empty squares. The vertical grey line around β ⁢ μ = 2.4 shows the location of the superfluid phase transition bounded by values corresponding to a critical temperature of 0.152 [11,20] and 0.171 [18,21]. Finally, the QTCE results are shown as light- and dark-shaded blue bands, correspondingly marking the effects of renormalizing with the second- and fifth-order VE.…”

“…As a specific example pursued by our group, automated algebra has successfully pushed the boundaries of the quantum virial expansion (VE) for spin-1/2 Fermi gases in a wide range of settings (e.g. [8][9][10]11] for a recent review), managing to obtain precise estimates of up to the fifth-order coefficient. Encouraged by the success of automated algebra for the VE, here we explore a different route that is not a priori restricted to low-density regimes.…”