arXiv subjects

J. E. Drut

Publications and source records attributed to J. E. Drut.

At least 19 recordsLinked to original sources

arXiv2023

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

Recent advances in automated algebra for dilute Fermi gases in the virial expansion, where coarse temporal lattices were found advantageous, motivate the study of more general computational schemes that could be applied to arbitrary densities, beyond the dilute limit where the virial expansion is physically reasonable. We propose here such an approach by developing what we call the Quantum Thermodynamics Computational Engine (QTCE). In QTCE, the imaginary-time direction is discretized and the interaction is accounted for via a quantum cumulant expansion, where the coefficients are expressed in terms of noninteracting expectation values. The aim of QTCE is to enable the systematic resolution of interaction effects at fixed temporal discretization, as in lattice Monte Carlo calculations, but here in an algebraic rather than numerical fashion. Using this approach, in combination with numerical integration techniques (both known and alternative ones proposed here), we explore the thermodynamics of spin-1/2 fermions, focusing on the unitary limit in 3 spatial dimensions, but also exploring the effects of continuously varying the spatial dimension below 3. We find that, remarkably, extremely coarse temporal lattices, when suitably renormalized using known results from the virial expansion, yield stable partial sums for QTCE's cumulant expansion which are qualitatively and quantitatively correct in wide regions, compared with known experimental results.

cond-mat.quant-gas↗

arXiv2021

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

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-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). We use our estimates of the virial expansion, together with resummation techniques, to calculate the compressibility and spin susceptibility.

cond-mat.quant-gas↗

arXiv2020

Leading- and next-to-leading order semiclassical approximation to the first seven virial coefficients of spin-1/2 fermions across spatial dimensions

Following up on recent calculations, we investigate the leading- and next-to-leading order semiclassical approximation to the virial coefficients of a two-species fermion system with a contact interaction. Using the analytic result for the second-order virial coefficient as a renormalization condition, we derive expressions for up to the seventh-order virial coefficient $Δb_7$. Our results at leading order, though approximate, furnish simple analytic formulas that relate $Δb_n$ to $Δb_2$ for arbitrary dimension, providing a glimpse into the behavior of the virial expansion across dimensions and coupling strengths. As an application, we calculate the pressure and Tan's contact of the 2D attractive Fermi gas and examine the radius of convergence of the virial expansion as a function of the coupling strength.

cond-mat.quant-gas↗

arXiv2020

The virial expansion of attractively interacting Fermi gases in 1D, 2D, and 3D, up to fifth order

The virial expansion characterizes the high-temperature approach to the quantum-classical crossover in any quantum many-body system. Here, we calculate the virial coefficients up to the fifth-order of Fermi gases in 1D, 2D, and 3D, with attractive contact interactions, as relevant for a variety of applications in atomic and nuclear physics. To that end, we discretize the imaginary-time direction and calculate the relevant canonical partition functions. In coarse discretizations, we obtain analytic results featuring relationships between the interaction-induced changes $Δb_3$, $Δb_4$, and $Δb_5$ as functions of $Δb_2$, the latter being exactly known in many cases by virtue of the Beth-Uhlenbeck formula. Using automated-algebra methods, we push our calculations to progressively finer discretizations and extrapolate to the continuous-time limit. We find excellent agreement for $Δb_3$ with previous calculations in all dimensions and we formulate predictions for $Δb_4$ and $Δb_5$ in 1D and 2D. We also provide, for a range of couplings,the subspace contributions $Δb_{31}$, $Δb_{22}$, $Δb_{41}$, and $Δb_{32}$, which determine the equation of state and static response of polarized systems at high temperature. As a performance check, we compare the density equation of state and Tan contact with quantum Monte Carlo calculations, diagrammatic approaches, and experimental data where available. Finally, we apply Padé and Padé-Borel resummation methods to extend the usefulness of the virial coefficients to approach and in some cases go beyond the unit-fugacity point.

cond-mat.quant-gas↗

arXiv2020

The fourth- and fifth-order virial coefficients from weak-coupling to unitarity

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 temperature, 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. Implementing these factorizations and automated algebra codes, we obtain the interaction-induced change $Δb_n$ from weak coupling to unitarity. At unitarity, we find: $Δb_3 = -0.356(4)$, in agreement with previous results; $Δb_4 = 0.062(2)$, 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 track their origin back to their polarized and unpolarized components.

cond-mat.quant-gas↗

arXiv2020

Thermodynamics of rotating quantum matter in the virial expansion

We characterize the high-temperature thermodynamics of rotating bosons and fermions in two- (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 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.

cond-mat.quant-gas↗

arXiv2020

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

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 as well as in a harmonic trapping potential of frequency $ω$. Using the three-body problem to renormalize, we report analytic results for the change in the fourth- and fifth-order virial coefficients $Δb_4$ and $Δb_5$ as functions of $Δb_3$. Additionally, we argue that in the $ω\to 0$ limit the relationship $b_n^\text{T} = n^{-d/2} b_n$ holds between the trapped (T) and 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 $Δb_3^\text{T}$ in 1D with three-body forces and $Δb_2^\text{T}$ in 2D with two-body forces.

cond-mat.quant-gas↗

arXiv2020

Thermodynamics and static response of anomalous 1D fermions via quantum Monte Carlo in the worldline representation

A system of three-species fermions in one spatial dimension (1D) with a contact three-body interaction is known to display a scale anomaly. This anomaly is identical to that of a two-dimensional (2D) system of two-species fermions. The exact relation between those two systems, however, is limited to the two-particle sector of the 2D case and the three-particle sector of the 1D case. Here, we implement a non-perturbative Monte Carlo approach, based on the worldline representation, to calculate the thermodynamics and static response of three-species fermions in 1D, thus tackling the many-body sector of the problem. We determine the energy, density, and pressure equations of state, and the compressibility and magnetic susceptibility for a wide range of temperatures and coupling strengths. We compare our results with the third-order virial expansion.

cond-mat.quant-gas↗

arXiv2019

Spectral density, Levinson's theorem, and the extra term in the second virial coefficient for 1D delta-function potential

In contrast with the 3D result, the Beth-Uhlenbeck (BU) formula in 1D contains an extra -1/2 term. The origin of this -1/2 term is explained using a spectral density approach. To be explicit, a delta-function potential is used to show that the correction term arises from a pole of the density of states at zero energy. The spectral density method shows that this term is actually an artifact of the non-normalizability of the scattering states and an infrared cutoff regularization scheme has to be used to get the correct result in 1D. The formal derivation of the BU formula would miss this term since it ignores the effects of the boundary terms. While the result is shown for the delta-function potential, the method and result are valid for more general potentials. Additionally, the 1D Levinson's theorem can be extracted from the spectral density method using the asymptotic form of general potentials. The importance of the result lies in the fact that all these correction terms in 1D have a universal source: a pole at zero energy. Similar calculations using quantum field theoretical approaches (without explicit infrared cutoff regularization schemes) also show the same subtleties with the correction term originating from the zero energy scattering states (appendix A).

cond-mat.quant-gas↗

arXiv2019

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

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 $Δb_3$ and $Δb_4$ agree qualitatively, and in some regimes quantitatively, with previous numerical calculations for the unitary limit of three-dimensional Fermi gases.

cond-mat.quant-gas↗

arXiv2018

Fermi-Fermi crossover in the ground state of 1D few-body systems with anomalous three-body interactions

In one spatial dimension, quantum systems with an attractive three-body contact interaction exhibit a scale anomaly. In this work, we examine the few-body sector for up to six particles. We study those systems with a self-consistent, non-perturbative, iterative method, in the subspace of zero total momentum. Exploiting the structure of the contact interaction, the method reduces the complexity of obtaining the wavefunction by three powers of the dimension of the Hilbert space. We present results on the energy, and momentum and spatial structure, as well as Tan's contact. We find a Fermi-Fermi crossover interpolating between large, weakly bound trimers and compact, deeply bound trimers: at weak coupling, the behavior is captured by degenerate perturbation theory; at strong coupling, the system is governed by an effective theory of heavy trimers (plus free particles in the case of asymmetric systems). Additionally, we find that there is no trimer-trimer attraction and therefore no six-body bound state.

cond-mat.quant-gas↗

arXiv2018

Virial coefficients of 1D and 2D Fermi gases by stochastic methods and a semiclassical lattice approximation

We map out the interaction effects on the first six virial coefficients of one-dimensional Fermi gases with zero-range attractive and repulsive interactions, and the first four virial coefficients of the two-dimensional analog with attractive interactions. To that end, we use two non-perturbative stochastic methods: projection by complex stochastic quantization, which allows us to determine high-order coefficients at weak coupling and estimate the radius of convergence of the virial expansion; and a path-integral representation of the virial coefficients. To complement our numerical calculations, we present leading-order results in a semiclassical lattice approximation, which we find to be surprisingly close to the expected answers.

cond-mat.quant-gas↗

arXiv2017

Advances in non-relativistic matter via complex Langevin approaches

The recent progress in understanding the mathematics of complex stochastic quantization, as well as its application to quantum chromodynamics in situations that have a complex phase problem (e.g. finite quark density, real time), has opened up an intriguing possibility for non-relativistic many-body physics which has so far remained largely unexplored. In this brief contribution, I review a few specific examples of advances in the characterization of the thermodynamics of non-relativistic matter in a variety of one-dimensional cases affected by the sign problem: repulsive interactions, finite polarization, finite mass imbalance, and projection to finite systems to obtain virial coefficients.

cond-mat.quant-gas↗

arXiv2017

Equation of state of non-relativistic matter from automated perturbation theory and complex Langevin

We calculate the pressure and density of polarized non-relativistic systems of two-component fermions coupled via a contact interaction at finite temperature. For the unpolarized one-dimensional system with an attractive interaction, we perform a third-order lattice perturbation theory calculation and assess its convergence by comparing with hybrid Monte Carlo. In that regime, we also demonstrate agreement with real Langevin. For the repulsive unpolarized one-dimensional system, where there is a so-called complex phase problem, we present lattice perturbation theory as well as complex Langevin calculations. For our studies, we employ a Hubbard-Stratonovich transformation to decouple the interaction and automate the application of Wick's theorem for perturbative calculations, which generates the diagrammatic expansion at any order. We find excellent agreement between the results from our perturbative calculations and stochastic studies in the weakly interacting regime. In addition, we show predictions for the strong coupling regime as well as for the polarized one-dimensional system. Finally, we show a first estimate for the equation of state in three dimensions where we focus on the polarized unitary Fermi gas.

cond-mat.quant-gas↗

arXiv2017

Particle Projection Using a Complex Langevin Method

Using complex stochastic quantization, we implement a particle-number projection technique on the partition function of spin-1/2 fermions at finite temperature on the lattice. We discuss the method, its application towards obtaining the thermal properties of finite Fermi systems in three spatial dimensions, and results for the first five virial coefficients of one-dimensional, attractively interacting fermions.

hep-lat↗

arXiv2017

Dynamics of entanglement entropy of interacting fermions in a 1D driven harmonic trap

Following up on a recent analysis of two cold atoms in a time-dependent harmonic trap in one dimension, we explore the entanglement entropy of two and three fermions in the same situation when driven through a parametric resonance. We find that the presence of such a resonance in the two-particle system leaves a clear imprint on the entanglement entropy. We show how the signal is modified by attractive and repulsive contact interactions, and how it remains present for the three-particle system. Additionaly, we extend the work of recent experiments to demonstrate how restricting observation to a limited subsystem gives rise to locally thermal behavior.

cond-mat.quant-gas↗

arXiv2017

Thermodynamics of one-dimensional SU(4) and SU(6) fermions with attractive interactions

Motivated by advances in the manipulation and detection of ultracold atoms with multiple internal degrees of freedom, we present a finite-temperature lattice Monte Carlo calculation of the density and pressure equations of state, as well as Tan's contact, of attractively interacting SU(4)- and SU(6)-symmetric fermion systems in one spatial dimension. We also furnish a nonperturbative proof of a universal relation whereby quantities computable in the SU(2) case completely determine the virial coefficients of the SU($N_f$) case. These one-dimensional systems are appealing because they can be experimentally realized in highly constrained traps and because of the dominant role played by correlations. The latter are typically nonperturbative and are crucial for understanding ground states and quantum phase transitions. While quantum fluctuations are typically overpowered by thermal ones in one and two dimensions at any finite temperature, we find that quantum effects do leave their imprint in thermodynamic quantities. Our calculations show that the additional degrees of freedom, relative to the SU(2) case, provide a dramatic enhancement of the density and pressure (in units of their noninteracting counterparts) in a wide region around vanishing $βμ$, where $β$ is the inverse temperature and $μ$ the chemical potential. As shown recently in experiments, the thermodynamics we explore here can be measured in a controlled and precise fashion in highly constrained traps and optical lattices. Our results are a prediction for such experiments in one dimension with atoms of high nuclear spin.

cond-mat.quant-gas↗

arXiv2016

Ground-state energy, density profiles, and momentum distribution of attractively interacting 1D Fermi gases with hard-wall boundaries: a Monte Carlo study

Motivated by the realization of hard-wall boundary conditions in experiments with ultracold atoms, we investigate the ground-state properties of spin-1/2 fermions with attractive interactions in a one-dimensional box. We use lattice Monte Carlo methods to determine essential quantities like the energy, which we compute as a function of coupling strength and particle number in the regime from few to many particles. Many-fermion systems bound by hard walls display non-trivial density profiles characterized by so-called Friedel oscillations (which are similar to those observed in harmonic traps). In non-interacting systems, the characteristic length scale of the oscillations is set by (2 kF)^(-1), where kF is the Fermi momentum, while repulsive interactions tend to generate Wigner-crystal oscillations of period (4 kF)^(-1). Based on the non-interacting result, we find a remarkably simple parametrization of the density profiles of the attractively interacting case, which we generalize to the one-body density matrix. While the total momentum is not a conserved quantity in the presence of hard walls, the magnitude of the momentum does provide a good quantum number. We are therefore able to provide a detailed characterization of the (quasi-)momentum distribution, which displays rather robust discontinuity at the Fermi surface. In addition, we determine the spatially varying on-site density-density correlation, which in turn yields Tan's contact density and, upon integration, Tan's contact. As is well known, the latter fully determines the short-range correlations and plays a crucial role in a multitude of equilibrium and non-equilibrium sum rules.

cond-mat.quant-gas↗