We develop a method that uses truncation-order-dependent reexpansions constrained by generic strongcoupling information to extrapolate perturbation series to the nonperturbative regime.The method is first benchmarked against a zero-dimensional model field theory and then applied to the dilute Fermi gas in one and three dimensions.Overall, our method significantly outperforms Padé and Borel extrapolations in these examples.The results for the ground-state energy of the three-dimensional Fermi gas are robust with respect to changes in the form of the reexpansion and compare well with quantum Monte Carlo simulations throughout the BCS regime and beyond.We develop a method that uses truncation-order-dependent reexpansions constrained by generic strongcoupling information to extrapolate perturbation series to the nonperturbative regime.The method is first benchmarked against a zero-dimensional model field theory and then applied to the dilute Fermi gas in one and three dimensions.Overall, our method significantly outperforms Padé and Borel extrapolations in these examples.The results for the ground-state energy of the three-dimensional Fermi gas are robust with respect to changes in the form of the reexpansion and compare well with quantum Monte Carlo simulations throughout the BCS regime and beyond.

Jan 1, 20208 citations

Constrained Extrapolation Problem and Order‐Dependent Mappings

Corbinian Wellenhofer, Daniel R. Phillips, A. Schwenk

The problem of extrapolating the perturbation series for the dilute Fermi gas in three dimensions to the unitary limit of infinite scattering length and into the BEC region is considered, using the available strong‐coupling information to constrain the extrapolation problem. In this constrained extrapolation problem (CEP), the goal is to find classes of approximants that give well‐converged results already for low perturbative truncation orders. First, it is shown that the standard Padé and Borel methods are too restrictive to give satisfactory results for this CEP. A generalization of Borel extrapolation is given by the so‐called maximum‐entropy (MaxEnt) extrapolation method. However, it is shown that MaxEnt requires extensive elaborations to be applicable to the dilute Fermi gas and is, thus, not practical for the CEP in this case. Instead, order‐dependent‐mapping extrapolation (ODME) as a simple, practical, and general method for the CEP is proposed. It is found that the ODME approximants for the ground‐state energy of the dilute Fermi gas are robust with respect to changes of the mapping choice and agree with results from quantum Monte Carlo simulations within uncertainties.The problem of extrapolating the perturbation series for the dilute Fermi gas in three dimensions to the unitary limit of infinite scattering length and into the BEC region is considered, using the available strong‐coupling information to constrain the extrapolation problem. In this constrained extrapolation problem (CEP), the goal is to find classes of approximants that give well‐converged results already for low perturbative truncation orders. First, it is shown that the standard Padé and Borel methods are too restrictive to give satisfactory results for this CEP. A generalization of Borel extrapolation is given by the so‐called maximum‐entropy (MaxEnt) extrapolation method. However, it is shown that MaxEnt requires extensive elaborations to be applicable to the dilute Fermi gas and is, thus, not practical for the CEP in this case. Instead, order‐dependent‐mapping extrapolation (ODME) as a simple, practical, and general method for the CEP is proposed. It is found that the ODME approximants for the ground‐state energy of the dilute Fermi gas are robust with respect to changes of the mapping choice and agree with results from quantum Monte Carlo simulations within uncertainties.

arXiv:2011.02105physica status solidi (b)Mar 23, 20213 citations

Effective field theory for dilute Fermi systems at fourth order

Corbinian Wellenhofer, C. Drischler, A. Schwenk

We discuss high-order calculations in perturbative effective field theory for fermions at low energy scales. The Fermi-momentum or ${k}_{\mathrm{F}}{a}_{s}$ expansion for the ground-state energy of the dilute Fermi gas is calculated to fourth order, both in cutoff regularization and in dimensional regularization. For the case of spin one-half fermions we find from a Bayesian analysis that the expansion is well converged at this order for $|{k}_{\mathrm{F}}{a}_{s}|\ensuremath{\lesssim}0.5$. Furthermore, we show that Pad\'e-Borel resummations can improve the convergence for $|{k}_{\mathrm{F}}{a}_{s}|\ensuremath{\lesssim}1$. Our results provide important constraints for nonperturbative calculations of ultracold atoms and dilute neutron matter.We discuss high-order calculations in perturbative effective field theory for fermions at low energy scales. The Fermi-momentum or ${k}_{\mathrm{F}}{a}_{s}$ expansion for the ground-state energy of the dilute Fermi gas is calculated to fourth order, both in cutoff regularization and in dimensional regularization. For the case of spin one-half fermions we find from a Bayesian analysis that the expansion is well converged at this order for $|{k}_{\mathrm{F}}{a}_{s}|\ensuremath{\lesssim}0.5$. Furthermore, we show that Pad\'e-Borel resummations can improve the convergence for $|{k}_{\mathrm{F}}{a}_{s}|\ensuremath{\lesssim}1$. Our results provide important constraints for nonperturbative calculations of ultracold atoms and dilute neutron matter.

arXiv:2102.05966Physical Review CJul 20, 202114 citations

Lowest order constrained variational approximation for ultracold Bose-Fermi mixtures

Cordioli, Pietro

In this thesis, we investigate ultracold Bose-Fermi mixtures in both two and three dimensions using a non-perturbative approach based on the Lowest Order Constrained Variational (LOCV) method. This variational technique, originally developed in the context of nuclear matter, is applied here to dilute quantum gases at zero temperature, allowing for an accurate treatment of two-body correlations beyond perturbative limits. We begin by analysing Bose-Fermi mixtures near a broad Fano-Feshbach resonance, which enables precise tuning of the interspecies interaction strength. Particular attention is devoted to the low-dimensional case, where quantum fluctuations are enhanced and experimental realizations have become increasingly accessible. While the three-dimensional problem serves as a useful benchmark, the two-dimensional Bose-Fermi mixture presents unique challenges. Notably, no prior non-perturbative studies exist on its mechanical stability. A central goal of this work is to determine the conditions under which the 2D Bose-Fermi mixture remains mechanically stable as the boson-fermion attraction is increased. Our results demonstrate that the LOCV approach captures the relevant physics from weak to moderate coupling, offering a complementary perspective to perturbative methods or alternative non-perturbative approaches such as the T-matrix formalism. In this coupling range, we identify the critical strength of boson-boson repulsion necessary to prevent phase separation or collapse, thereby providing the first theoretical variational prediction of a stability phase diagram in this two dimensional regime. Overall, this work fills a gap in the literature and provides a useful reference point for future theoretical and experimental studies of strongly interacting Bose-Fermi systems in reduced dimensionality.In this thesis, we investigate ultracold Bose-Fermi mixtures in both two and three dimensions using a non-perturbative approach based on the Lowest Order Constrained Variational (LOCV) method. This variational technique, originally developed in the context of nuclear matter, is applied here to dilute quantum gases at zero temperature, allowing for an accurate treatment of two-body correlations beyond perturbative limits. We begin by analysing Bose-Fermi mixtures near a broad Fano-Feshbach resonance, which enables precise tuning of the interspecies interaction strength. Particular attention is devoted to the low-dimensional case, where quantum fluctuations are enhanced and experimental realizations have become increasingly accessible. While the three-dimensional problem serves as a useful benchmark, the two-dimensional Bose-Fermi mixture presents unique challenges. Notably, no prior non-perturbative studies exist on its mechanical stability. A central goal of this work is to determine the conditions under which the 2D Bose-Fermi mixture remains mechanically stable as the boson-fermion attraction is increased. Our results demonstrate that the LOCV approach captures the relevant physics from weak to moderate coupling, offering a complementary perspective to perturbative methods or alternative non-perturbative approaches such as the T-matrix formalism. In this coupling range, we identify the critical strength of boson-boson repulsion necessary to prevent phase separation or collapse, thereby providing the first theoretical variational prediction of a stability phase diagram in this two dimensional regime. Overall, this work fills a gap in the literature and provides a useful reference point for future theoretical and experimental studies of strongly interacting Bose-Fermi systems in reduced dimensionality.

Jul 24, 20250 citations

The Dilute Fermi Gas via Bogoliubov Theory

Marco Falconi, Emanuela L. Giacomelli, Christian Hainzl +1 more

We study the ground state properties of interacting Fermi gases in the dilute regime, in three dimensions. We compute the ground state energy of the system, for positive interaction potentials. We recover a well-known expression for the ground state energy at second order in the particle density, which depends on the interaction potential only via its scattering length. The first proof of this result has been given by Lieb, Seiringer and Solovej (Phys Rev A 71:053605, 2005). In this paper, we give a new derivation of this formula, using a different method; it is inspired by Bogoliubov theory, and it makes use of the almost-bosonic nature of the low-energy excitations of the systems. With respect to previous work, our result applies to a more regular class of interaction potentials, but it comes with improved error estimates on the ground state energy asymptotics in the density.We study the ground state properties of interacting Fermi gases in the dilute regime, in three dimensions. We compute the ground state energy of the system, for positive interaction potentials. We recover a well-known expression for the ground state energy at second order in the particle density, which depends on the interaction potential only via its scattering length. The first proof of this result has been given by Lieb, Seiringer and Solovej (Phys Rev A 71:053605, 2005). In this paper, we give a new derivation of this formula, using a different method; it is inspired by Bogoliubov theory, and it makes use of the almost-bosonic nature of the low-energy excitations of the systems. With respect to previous work, our result applies to a more regular class of interaction potentials, but it comes with improved error estimates on the ground state energy asymptotics in the density.

arXiv:2006.00491Annales Henri PoincaréApr 16, 202126 citations

Simulations of strongly-interacting fermion systems

Evgeni Burovski

A generic scenario of a crossover between weak-coupling and strong-coupling limits of a many-body quantum system is studied. Powerful numerical methods of investigating the strongly correlated regime are developed and applied to answering a number of important questions of the strongly correlated fermions physics. The internal structure of an exciton in an insulating medium is studied throughout the crossover between the (analytically solvable) strong- and weak-coupling limits. In the intermediate regime, accurate results for the exciton ground state are obtained and used to establish the limits of applicability of the corresponding limiting cases for the first time. A novel combination of the diagrammatic determinant Monte Carlo and the worm algorithm is developed and used for investigating the unitarity regime of the BEC-BCS crossover via simulating the dilute Fermi-Hubbard model. Superfluid phase transition and thermodynamic properties of the normal state are characterized for both lattice fermions and continuum unitary Fermi gas. The relation to current experiments with trapped ultracold quantum gases is discussed.A generic scenario of a crossover between weak-coupling and strong-coupling limits of a many-body quantum system is studied. Powerful numerical methods of investigating the strongly correlated regime are developed and applied to answering a number of important questions of the strongly correlated fermions physics. The internal structure of an exciton in an insulating medium is studied throughout the crossover between the (analytically solvable) strong- and weak-coupling limits. In the intermediate regime, accurate results for the exciton ground state are obtained and used to establish the limits of applicability of the corresponding limiting cases for the first time. A novel combination of the diagrammatic determinant Monte Carlo and the worm algorithm is developed and used for investigating the unitarity regime of the BEC-BCS crossover via simulating the dilute Fermi-Hubbard model. Superfluid phase transition and thermodynamic properties of the normal state are characterized for both lattice fermions and continuum unitary Fermi gas. The relation to current experiments with trapped ultracold quantum gases is discussed.

Jan 1, 20070 citations

We discuss high-order calculations in perturbative effective field theory for fermions at low energy scales. The Fermi-momentum or

{k}_{\mathrm{F}}{a}_{s}

expansion for the ground-state energy of the dilute Fermi gas is calculated to fourth order, both in cutoff regularization and in dimensional regularization. For the case of spin one-half fermions we find from a Bayesian analysis that the expansion is well converged at this order for $|{k}_{\mathrm{F}}{a}_{s}|\ensuremath{\lesssim}0.5$. Furthermore, we show that Pad\'e-Borel resummations can improve the convergence for $|{k}_{\mathrm{F}}{a}_{s}|\ensuremath{\lesssim}1$. Our results provide important constraints for nonperturbative calculations of ultracold atoms and dilute neutron matter.