Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Fourth- And fifth-order virial expansion of harmonically trapped fermions at unitarity

J.E. Drut, A. J. Czejdo, Y. Hou, K. J. Morrell

Published Nov 9, 2022• 1 citation
Publisher

Abstract

By generalizing our automated algebra approach from homogeneous space to harmonically trapped systems, we have calculated the fourth- and fifth-order virial coefficients of universal spin- fermions in the unitary limit, confined in an isotropic harmonic potential. We present results for said coefficients as a function of trapping frequency (or, equivalently, temperature), which compare favorably with previous Monte Carlo calculations (available only at fourth order) as well as with our previous estimates in the untrapped limit (high temperature, low frequency). In addition to the conventional coefficients, we provide estimates for the contributions from subspaces with varying polarization. We use our results for the virial coefficients, together with resummation techniques, to calculate the compressibility and spin susceptibility.

DOI
10.17615/3p5t-z465

Related papers

Fourth- and fifth-order virial expansion of harmonically trapped fermions at unitarity

Y. Hou, K. J. Morrell, A. J. Czejdo +1 more

By generalizing our automated algebra approach from homogeneous space to harmonically trapped systems, we have calculated the fourth- and fifth-order virial coefficients of universal spin-$\frac{1}{2}$ fermions in the unitary limit, confined in an isotropic harmonic potential. We present results for said coefficients as a function of trapping frequency (or, equivalently, temperature), which compare favorably with previous Monte Carlo calculations (available only at fourth order) as well as with our previous estimates in the untrapped limit (high temperature, low frequency). In addition to the conventional coefficients, we provide estimates for the contributions from subspaces with varying polarization. We use our results for the virial coefficients, together with resummation techniques, to calculate the compressibility and spin susceptibility.

arXiv:2104.14440•Physical Review Research•Jul 28, 2021•8 citations

Virial coefficients of trapped and untrapped three-component fermions with three-body forces in arbitrary spatial dimensions

A. J. Czejdo, Joaquín E. Drut, Y. Hou +2 more

Using a coarse temporal lattice approximation, we calculate the first few terms of the virial expansion of a three-species fermion system with a three-body contact interaction in $d$ spatial dimensions, both in homogeneous space and in a harmonic trapping potential of frequency $\ensuremath{\omega}$. Using the three-body problem to renormalize, we report analytic results for the change in the fourth- and fifth-order virial coefficients $\mathrm{\ensuremath{\Delta}}{b}_{4}$ and $\mathrm{\ensuremath{\Delta}}{b}_{5}$ as functions of $\mathrm{\ensuremath{\Delta}}{b}_{3}$. Additionally, we argue that in the $\ensuremath{\omega}\ensuremath{\rightarrow}0$ limit the relationship ${b}_{n}^{\text{T}}={n}^{\ensuremath{-}d/2}{b}_{n}$ holds between the trapped (T) and the homogeneous coefficients for arbitrary temperature and coupling strength (not merely in scale-invariant regimes). Finally, we point out an exact, universal (coupling- and frequency-independent) relationship between $\mathrm{\ensuremath{\Delta}}{b}_{3}^{\text{T}}$ in one dimension with three-body forces and $\mathrm{\ensuremath{\Delta}}{b}_{2}^{\text{T}}$ in two dimensions with two-body forces.

arXiv:2003.04958•Physical Review A•Jun 29, 2020•13 citations

Third- and fourth-order virial coefficients of harmonically trapped fermions in a semiclassical approximation

K. J. Morrell, Casey Berger, Joaquín E. Drut

Using a leading-order semiclassical approximation, we calculate the third- and fourth-order virial coefficients of nonrelativistic spin-1/2 fermions in a harmonic trapping potential in arbitrary spatial dimensions, and as functions of temperature, trapping frequency, and coupling strength. Our simple, analytic results for the interaction-induced changes $\mathrm{\ensuremath{\Delta}}{b}_{3}$ and $\mathrm{\ensuremath{\Delta}}{b}_{4}$ agree qualitatively, and in some regimes quantitatively, with previous numerical calculations for the unitary limit of three-dimensional Fermi gases.

arXiv:1908.00070•Physical Review A•Dec 12, 2019•7 citations

Fourth- and Fifth-Order Virial Coefficients from Weak Coupling to Unitarity

Y. Hou, Joaquín E. Drut

In the current era of precision quantum many-body physics, one of the most scrutinized systems is the unitary limit of the nonrelativistic spin-1/2 Fermi gas, due to its simplicity and relevance for atomic, condensed matter, and nuclear physics. The thermodynamics of this strongly correlated system is determined by universal functions which, at high temperatures, are governed by universal virial coefficients b_{n} that capture the effects of the n-body system on the many-body dynamics. Currently, b_{2} and b_{3} are well understood, but the situation is less clear for b_{4}, and no predictions have been made for b_{5}. To answer these open questions, we implement a nonperturbative analytic approach based on the Trotter-Suzuki factorization of the imaginary-time evolution operator, using progressively finer temporal lattice spacings. By means of these factorizations and automated algebra codes, we obtain the interaction-induced change Δb_{n} from weak coupling to unitarity. At unitarity, we find that Δb_{3}=-0.356(4) in agreement with previous results, Δb_{4}=0.062(2), which is in agreement with all previous theoretical estimates but at odds with experimental determinations, and Δb_{5}=0.078(6), which is a prediction. We show the impact of those answers on the density equation of state and Tan contact, and trace their origin back to their polarized and unpolarized components.

arXiv:2004.08685•Physical Review Letters•Jul 28, 2020•18 citations

Thermodynamics of rotating quantum matter in the virial expansion

Casey Berger, K. J. Morrell, Joaquín E. Drut

We characterize the high-temperature thermodynamics of rotating bosons and fermions in two-dimensional (2D) and three-dimensional (3D) isotropic harmonic trapping potentials. We begin by calculating analytically the conventional virial coefficients ${b}_{n}$ for all $n$ in the noninteracting case, as functions of the trapping and rotational frequencies. We also report on the virial coefficients for the angular momentum and associated moment of inertia. Using the ${b}_{n}$ coefficients, we analyze the deconfined limit (in which the angular frequency matches the trapping frequency) and derive explicitly the limiting form of the partition function, showing from the thermodynamic standpoint how both the 2D and 3D cases become effectively homogeneous 2D systems. To tackle the virial coefficients in the presence of weak interactions, we implement a coarse temporal lattice approximation and obtain virial coefficients up to third order.

arXiv:2004.02344•Physical Review A•Aug 6, 2020•5 citations

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

A. J. Czejdo, Joaquín E. Drut, Yaqi Hou +1 more

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.

arXiv:2112.07744•Condensed Matter•Jan 24, 2022•5 citations
By generalizing our automated algebra approach from homogeneous space to harmonically trapped systems, we have calculated the fourth- and fifth-order virial coefficients of universal spin- fermions in the unitary limit, confined in an isotropic harmonic potential. We present results for said coefficients as a function of trapping frequency (or, equivalently, temperature), which compare favorably with previous Monte Carlo calculations (available only at fourth order) as well as with our previous estimates in the untrapped limit (high temperature, low frequency). In addition to the conventional coefficients, we provide estimates for the contributions from subspaces with varying polarization. We use our results for the virial coefficients, together with resummation techniques, to calculate the compressibility and spin susceptibility.
By generalizing our automated algebra approach from homogeneous space to harmonically trapped systems, we have calculated the fourth- and fifth-order virial coefficients of universal spin-12\frac{1}{2}21​ fermions in the unitary limit, confined in an isotropic harmonic potential. We present results for said coefficients as a function of trapping frequency (or, equivalently, temperature), which compare favorably with previous Monte Carlo calculations (available only at fourth order) as well as with our previous estimates in the untrapped limit (high temperature, low frequency). In addition to the conventional coefficients, we provide estimates for the contributions from subspaces with varying polarization. We use our results for the virial coefficients, together with resummation techniques, to calculate the compressibility and spin susceptibility.
Using a coarse temporal lattice approximation, we calculate the first few terms of the virial expansion of a three-species fermion system with a three-body contact interaction in ddd spatial dimensions, both in homogeneous space and in a harmonic trapping potential of frequency $\ensuremath{\omega}$. Using the three-body problem to renormalize, we report analytic results for the change in the fourth- and fifth-order virial coefficients $\mathrm{\ensuremath{\Delta}}{b}_{4}$ and $\mathrm{\ensuremath{\Delta}}{b}_{5}$ as functions of $\mathrm{\ensuremath{\Delta}}{b}_{3}$. Additionally, we argue that in the $\ensuremath{\omega}\ensuremath{\rightarrow}0$ limit the relationship ${b}_{n}^{\text{T}}={n}^{\ensuremath{-}d/2}{b}_{n}$ holds between the trapped (T) and the homogeneous coefficients for arbitrary temperature and coupling strength (not merely in scale-invariant regimes). Finally, we point out an exact, universal (coupling- and frequency-independent) relationship between $\mathrm{\ensuremath{\Delta}}{b}_{3}^{\text{T}}$ in one dimension with three-body forces and $\mathrm{\ensuremath{\Delta}}{b}_{2}^{\text{T}}$ in two dimensions with two-body forces.
Using a leading-order semiclassical approximation, we calculate the third- and fourth-order virial coefficients of nonrelativistic spin-1/2 fermions in a harmonic trapping potential in arbitrary spatial dimensions, and as functions of temperature, trapping frequency, and coupling strength. Our simple, analytic results for the interaction-induced changes $\mathrm{\ensuremath{\Delta}}{b}_{3}$ and $\mathrm{\ensuremath{\Delta}}{b}_{4}$ agree qualitatively, and in some regimes quantitatively, with previous numerical calculations for the unitary limit of three-dimensional Fermi gases.
In the current era of precision quantum many-body physics, one of the most scrutinized systems is the unitary limit of the nonrelativistic spin-1/2 Fermi gas, due to its simplicity and relevance for atomic, condensed matter, and nuclear physics. The thermodynamics of this strongly correlated system is determined by universal functions which, at high temperatures, are governed by universal virial coefficients b_{n} that capture the effects of the n-body system on the many-body dynamics. Currently, b_{2} and b_{3} are well understood, but the situation is less clear for b_{4}, and no predictions have been made for b_{5}. To answer these open questions, we implement a nonperturbative analytic approach based on the Trotter-Suzuki factorization of the imaginary-time evolution operator, using progressively finer temporal lattice spacings. By means of these factorizations and automated algebra codes, we obtain the interaction-induced change Δb_{n} from weak coupling to unitarity. At unitarity, we find that Δb_{3}=-0.356(4) in agreement with previous results, Δb_{4}=0.062(2), which is in agreement with all previous theoretical estimates but at odds with experimental determinations, and Δb_{5}=0.078(6), which is a prediction. We show the impact of those answers on the density equation of state and Tan contact, and trace their origin back to their polarized and unpolarized components.
We characterize the high-temperature thermodynamics of rotating bosons and fermions in two-dimensional (2D) and three-dimensional (3D) isotropic harmonic trapping potentials. We begin by calculating analytically the conventional virial coefficients bn{b}_{n}bn​ for all nnn in the noninteracting case, as functions of the trapping and rotational frequencies. We also report on the virial coefficients for the angular momentum and associated moment of inertia. Using the bn{b}_{n}bn​ coefficients, we analyze the deconfined limit (in which the angular frequency matches the trapping frequency) and derive explicitly the limiting form of the partition function, showing from the thermodynamic standpoint how both the 2D and 3D cases become effectively homogeneous 2D systems. To tackle the virial coefficients in the presence of weak interactions, we implement a coarse temporal lattice approximation and obtain virial coefficients up to third order.
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.