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arXiv:1410.7370 · Physical Review A

Universality in one-dimensional fermions at finite temperature: Density, pressure, compressibility, and contact

M. D. Hoffman, P. D. Javernick, Andrew C. Loheac, William J. Porter, E. R. Anderson, Joaquín E. Drut

Published Mar 16, 2015• 34 citations
arXivHTMLPDF

Abstract

We present finite-temperature, lattice Monte Carlo calculations of the particle number density, compressibility, pressure, and Tan's contact of an unpolarized system of short-range, attractively interacting spin-1/2 fermions in one spatial dimension, i.e., the Gaudin-Yang model. In addition, we compute the second-order virial coefficients for the pressure and the contact, both of which are in excellent agreement with the lattice results in the low-fugacity regime. Our calculations yield universal predictions for ultracold atomic systems with broad resonances in highly constrained traps. We cover a wide range of couplings and temperatures and find results that support the existence of a strong-coupling regime in which the thermodynamics of the system is markedly different from the noninteracting case. We compare and contrast our results with identical systems in higher dimensions.

DOI
10.1103/physreva.91.033618

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We present finite-temperature, lattice Monte Carlo calculations of the particle number density, compressibility, pressure, and Tan's contact of an unpolarized system of short-range, attractively interacting spin-1/2 fermions in one spatial dimension, i.e., the Gaudin-Yang model. In addition, we compute the second-order virial coefficients for the pressure and the contact, both of which are in excellent agreement with the lattice results in the low-fugacity regime. Our calculations yield universal predictions for ultracold atomic systems with broad resonances in highly constrained traps. We cover a wide range of couplings and temperatures and find results that support the existence of a strong-coupling regime in which the thermodynamics of the system is markedly different from the noninteracting case. We compare and contrast our results with identical systems in higher dimensions.
Using ab initio lattice methods, we calculate the finite temperature thermodynamics of homogeneous two-dimensional spin-1/2 fermions with attractive short-range interactions. We present results for the density, pressure, compressibility, and quantum anomaly (i.e., Tan's contact) for a wide range of temperatures (mostly above the superfluid phase, including the pseudogap regime) and coupling strengths, focusing on the unpolarized case. Within our statistical and systematic uncertainties, our prediction for the density equation of state differs quantitatively from the prediction by Luttinger-Ward theory in the strongly coupled region of parameter space, but otherwise agrees well with it. We also compare our calculations with the second- and third-order virial expansion, with which they are in excellent agreement in the low-fugacity regime.
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(2)-, 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(NfN_fNf​) 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. These correlations 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 The results presented here are a prediction for future experiments in one dimension with atoms of high nuclear spin.
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(Nf{N}_{f}Nf​) 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 $\ensuremath{\beta}\ensuremath{\mu}$, where $\ensuremath{\beta}$ is the inverse temperature and $\ensuremath{\mu}$ 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.
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.
We present a nonperturbative computation of the equation of state of polarized, attractively interacting, nonrelativistic fermions in one spatial dimension at finite temperature. We show results for the density, spin magnetization, magnetic susceptibility, and Tan's contact. We compare with the second-order virial expansion, a next-to-leading-order lattice perturbation theory calculation, and interpret our results in terms of pairing correlations. Our lattice Monte Carlo calculations implement an imaginary chemical potential difference to avoid the sign problem. The thermodynamic results on the imaginary side are analytically continued to obtain results on the real axis. We focus on an intermediate- to strong-coupling regime, and cover a wide range of temperatures and spin imbalances.
We determine the ground-state energy and Tan's contact of attractively interacting few-fermion systems in a one-dimensional harmonic trap, for a range of couplings and particle numbers. Complementing those results, we show the corresponding density profiles. The calculations were performed with a lattice Monte Carlo approach based on a nonuniform discretization of space, defined via Gauss-Hermite quadrature points and weights. This particular coordinate basis is natural for systems in harmonic traps, and can be generalized to traps of other shapes. In all cases, it yields a position-dependent coupling and a corresponding nonuniform Hubbard-Stratonovich transformation. The resulting path integral is performed with hybrid Monte Carlo as a proof of principle for calculations at finite temperature and in higher dimensions. We present results for N=4,...,20N=4,...,20N=4,...,20 particles (although the method can be extended beyond that) to cover the range from few- to many-particle systems. This method is exact up to statistical and systematic uncertainties, which we account for---and thus also represents an ab initio calculation of this system, providing a benchmark for other methods and a prediction for ultracold-atom experiments.