As can be inferred from present experiments in ultracold gases, the scattering length is a quantity that determines the thermodynamic state of the gas. As such, a conjugate thermodynamic variable to it exists. Here, we show that the recently introduced contact is the conjugate of the inverse of the scattering length. We find that this identification, in addition to well-known approximations, allows for a derivation of three of the most relevant results concerning the contact, namely its relation to the adiabatic change of the energy, the general form of the energy and the generalized virial theorem. We find that the current identification of the contact variable depends on the use of the contact approximation for the intermolecular potential but it is independent of whether the gas is made out of fermions or bosons and of the strength of the interaction.
As can be inferred from present experiments in ultracold gases, the scattering length is a quantity that determines the thermodynamic state of the gas. As such, a conjugate thermodynamic variable to it exists. Here, we show that the recently introduced contact is the conjugate of the inverse of the scattering length. We find that this identification, in addition to well-known approximations, allows for a derivation of three of the most relevant results concerning the contact, namely its relation to the adiabatic change of the energy, the general form of the energy and the generalized virial theorem. We find that the current identification of the contact variable depends on the use of the contact approximation for the intermolecular potential but it is independent of whether the gas is made out of fermions or bosons and of the strength of the interaction.
As can be inferred from present experiments in ultracold gases, the scattering length is a quantity that determines the thermodynamic state of the gas. As such, there exists a conjugate thermodynamic to it. Here, we show that the recently introduced "contact" is the conjugate of the inverse of the scattering length. We find that this identification allows for a derivation of essentially all the known results regarding the contact. Using the mean-field theory for the Bose-Einstein (BEC) to Bardeen-Cooper-Schriefer (BCS) crossover, we also find that the contact is proportional to the square of the gap. We analyze in detail both a homogenous balanced mixture of fermions and its inhomogenous counterpart in a harmonic trap.
Abstract This chapter is devoted to the derivation of various thermodynamic relations, known as Tan relations, involving the so-called contact parameter. This physical quantity, which determines the behaviour of the momentum distribution of an interacting gas at large momenta, plays a crucial role in characterizing various thermodynamic quantities, like the dependence of the energy on the scattering length (sweep theorem), the relation between the energy and the momentum distribution, the behaviour of the static structure factor at large wave vectors, and the virial theorem in the presence of harmonic trapping.
We argue that the energy of a strongly interacting gas in the unitary limit is parametrized by a universal constant β. For fermions, this parameter is known, but not for bosons. Finding β for bosons is the goal of this thesis. We present the basics of scattering theory, including the s-wave scattering length, which is the characteristic length scale for the interactions at low energies and which goes to infinity at unitarity. Shape and Feshbach resonances are explained. The conventional theory for Bose gases, Bogoliubov theory, is presented, which takes into account quantum fluctuations. It is used to derive identities for the groundstate energy, the condensate fraction and the contact. These are given as expansions in the scattering length, which therefore only hold true for weak interactions. To improve on this, we present an effective action from an atom-molecule theory, which also allows bound states. By a mean-field approach we derive universal identities for the associated energy scales, and in the unitary limit an analytical expression is found for the chemical potential, from which the value β = −0.54 is found. However, this mean-field result is qualitatively not reliable, since quantum depletion is neglected. A new approach is introduced, starting with a Jastrow Ansatz, from which the energy is expressed in the Jastrow function f and radial distribution function g. A cluster expansion is made for g, where a diagrammatic analogy is found. The Hyper-Netted Chain approximation is introduced to calculate g systematically by summing infinitely many diagrams. In HNC/0, we minimize the energy by a variational method, which is based on a well-constructed Ansatz for g. Numerical results using the HNC/0 approximation are given not only for the ground-state energy, but also for the condensate density and the contact. For large scattering lengths the calculations became increasingly less accurate, but the results from Bogoliubov theory are correctly included for small a. An extrapolation for the ground-state energy is made to find β ' −0.20, which is expected to be an upper bound. Moreover, quantum depletion is found to be higher when calculated in HNC/0 for large scattering lengths. Also, it is found that the contact at this value for unitarity remains finite and acquires the value of C ' 10.3n4/3, which is remarkably close to the value for fermions, C ' 11n4/3. Cover image courtesy of [18].
The Unitary Fermi Gas (UFG) is one of the most strongly interacting systems known to date, as it saturates the unitarity bound on the quantum mechanical scattering cross section.The UFG corresponds to a two-component Fermi gas in the limit of short interaction range and large scattering length, and is currently realized in ultracold-atom experiments via Feshbach resonances.While easy to define, the UFG poses a challenging quantum many-body problem, as it lacks any characteristic scale other than the density.As a consequence, accurate quantitative predictions of the thermodynamic properties of the UFG require Monte Carlo calculations.However, significant progress has also been made with purely analytical methods.Notably, in 2005 Tan derived a set of exact thermodynamic relations in which a universal quantity known as the "contact" C plays a crucial role.Recently, C has also been found to determine the prefactor of the high-frequency power-law decay of correlators as well as the right-hand-sides of shear-and bulk viscosity sum rules.The contact is therefore a central piece of information on the UFG in equilibrium as well as away from equilibrium.In this talk we describe some of the known aspects of Fermi gases at and around unitarity, show our latest Monte Carlo results for the contact at finite temperature, and summarize the open questions in the field, some of which we are starting to answer using large-scale Monte Carlo calculations by adapting methods from Lattice QCD.
A quantity known as the contact is a fundamental thermodynamic property of quantum many-body systems with short-range interactions. Determination of the temperature dependence of the contact for the unitary Fermi gas of infinite scattering length has been a major challenge, with different calculations yielding qualitatively different results. Here we use finite-temperature auxiliary-field quantum Monte Carlo (AFMC) methods on the lattice within the canonical ensemble to calculate the temperature dependence of the contact for the homogeneous spin-balanced unitary Fermi gas. We extrapolate to the continuum limit for 40, 66, and 114 particles, eliminating systematic errors due to finite-range effects. We observe a dramatic decrease in the contact as the superfluid critical temperature is approached from below, followed by a gradual weak decrease as the temperature increases in the normal phase. Our theoretical results are in excellent agreement with the most recent precision ultracold atomic gas experiments. We also present results for the energy as a function of temperature in the continuum limit.
The Unitary Fermi Gas (UFG) is one of the most strongly interacting systems known to date, as it saturates the unitarity bound on the quantum mechanical scattering cross section. The UFG corresponds to a two-component Fermi gas in the limit of short interaction range and large scattering length, and is currently realized in ultracold-atom experiments via Feshbach resonances. While easy to define, the UFG poses a challenging quantum many-body problem, as it lacks any characteristic scale other than the density. As a consequence, accurate quantitative predictions of the thermodynamic properties of the UFG require Monte Carlo calculations. However, significant progress has also been made with purely analytical methods. Notably, in 2005 Tan derived a set of exact thermodynamic relations in which a universal quantity known as the "contact" C plays a crucial role. Recently, C has also been found to determine the prefactor of the high- frequency power-law decay of correlators as well as the right-hand-sides of shear- and bulk viscosity sum rules. The contact is therefore a central piece of information on the UFG in equilibrium as well as away from equilibrium. In this talk we describe some of the known aspects of Fermi gases at and around unitarity, show our latest Monte Carlo results for the contact at finite temperature, and summarize the open questions in the field, some of which we are starting to answer using large-scale Monte Carlo calculations by adapting methods from Lattice QCD.