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arXiv:1001.1121 · Journal of Statistical Mechanics Theory and Experiment

S-matrix approach to quantum gases in the unitary limit: I. The two-dimensional case

Pye-Ton How, André LeClair

Published Mar 26, 2010• 8 citations
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Abstract

In three spatial dimensions, in the unitary limit of a non-relativistic quantum Bose or Fermi gas, the scattering length diverges. This occurs at a renormalization group fixed point. Thus these systems present interesting examples of interacting scale-invariant models with dynamical exponent z = 2. We study this problem in two and three spatial dimensions using the S -matrix-based approach to the thermodynamics we recently developed. It is well suited to the unitary limit where the S matrix S = − 1, since it allows an expansion in the inverse coupling. We define a meaningful scale-invariant, unitary limit in two spatial dimensions, where again the scattering length diverges. In the two-dimensional case, the integral equation for the pseudo-energy becomes transcendentally algebraic and we can easily compute the various universal scaling functions as a function of μ/ T , such as the energy per particle. The ratio of the shear viscosity to the entropy density η/ s is above the conjectured lower bound of for all cases except attractive bosons. For attractive two-component fermions, , whereas for attractive bosons .

DOI
10.1088/1742-5468/2010/03/p03025

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In three spatial dimensions, in the unitary limit of a non-relativistic quantum Bose or Fermi gas, the scattering length diverges. This occurs at a renormalization group fixed point. Thus these systems present interesting examples of interacting scale-invariant models with dynamical exponent z = 2. We study this problem in two and three spatial dimensions using the S -matrix-based approach to the thermodynamics we recently developed. It is well suited to the unitary limit where the S matrix S = − 1, since it allows an expansion in the inverse coupling. We define a meaningful scale-invariant, unitary limit in two spatial dimensions, where again the scattering length diverges. In the two-dimensional case, the integral equation for the pseudo-energy becomes transcendentally algebraic and we can easily compute the various universal scaling functions as a function of μ/ T , such as the energy per particle. The ratio of the shear viscosity to the entropy density η/ s is above the conjectured lower bound of for all cases except attractive bosons. For attractive two-component fermions, , whereas for attractive bosons .
A new analytic treatment of three-dimensional homogeneous Bose and Fermi gases in the unitary limit of negative infinite scattering length is presented, based on the S -matrix approach to statistical mechanics we recently developed. The unitary limit occurs at a fixed point of the renormalization group with dynamical exponent z = 2 where the S -matrix equals − 1. For fermions we find T c / T F ≈0.1. For bosons we present evidence that the gas does not collapse, but rather has a critical point that is a strongly interacting form of Bose–Einstein condensation. This bosonic critical point occurs at n λ T 3 ≈1.3, where n is the density and λ T the thermal wavelength, which is lower than the ideal gas value of ζ(3/2) = 2.61.
A novel formalism of quantum statistical mechanics, based on the zero-temperature S-matrix of the quantum system, is presented in this thesis. In our new formalism, the lowest order approximation ("two-body approximation") corresponds to the exact resummation of all binary collision terms, and can be expressed as an integral equation reminiscent of the thermodynamic Bethe Ansatz (TBA). Two applications of this formalism are explored: the critical point of a weakly-interacting Bose gas in two dimensions, and the scaling behavior of quantum gases at the unitary limit in two and three spatial dimensions. We found that a weakly-interacting 2D Bose gas undergoes a superfluid transition at Tc [ALMOST EQUAL TO] 2[pi]n/[m log(2[pi]/mg )], where n is the number density, m the mass of a particle, and g the coupling. In the unitary limit where the coupling g diverges, the two-body kernel of our integral equation has simple forms in both two and three spatial dimensions, and we were able to solve the integral equation numerically. Various scaling functions in the unitary limit are defined (as functions of [MICRO SIGN]/T ) and computed from the numerical solutions. For bosons in three spatial dimensions, we present evidence that the gas undergoes 3 a strongly interacting version of Bose-Einstein condensation at n[lamda]T [ALMOST EQUAL TO] 1.3, where n is the number density and [lamda]T is the thermal wavelength. Finally, we look at the ratio of shear viscosity to entropy density [eta]/s of the unitary quantum gas, which has a conjectured lower bound of [eta]/s [GREATER-THAN OR EQUAL TO] /4[pi]kb based on the AdS/CFT correspondence of a strongly coupled Yang-Mills theory.
We calculate the ratio of the viscosity to the entropy density for both Bose and Fermi gases in the unitary limit using a new approach to the quantum statistical mechanics of gases based on the S -matrix. In the unitary limit the scattering length diverges and the S -matrix equals − 1. For the fermion case, we obtain η/ s >4.7 times the proposed lower bound of ℏ/4π k B , which came from the anti-Desitter space/conformal field theory correspondence (AdS/CFT) for gauge theories, consistent with the most recent experiments. For the bosonic case, we present evidence that the gas undergoes a phase transition to a strongly interacting Bose–Einstein condensate and is a more perfect fluid, with η/ s <1.3 times the bound.
We elucidate universal many-body properties of a one-dimensional, two-component ultracold Fermi gas near the ppp-wave Feshbach resonance. The low-energy scattering in this system can be characterized by two parameters, that is, ppp-wave scattering length and effective range. At the unitarity limit where the ppp-wave scattering length diverges and the effective range is reduced to zero without conflicting with the causality bound, the system obeys universal thermodynamics as observed in a unitary Fermi gas with contact sss-wave interaction in three dimensions. It is in contrast to a Fermi gas with the ppp-wave resonance in three dimensions in which the effective range is inevitably finite. We present the universal equation of state in this unitary ppp-wave Fermi gas within the many-body TTT-matrix approach as well as the virial expansion method. Moreover, we examine the single-particle spectral function in the high-density regime where the virial expansion is no longer valid. On the basis of the Hartree-like self-energy shift at the divergent scattering length, we conjecture that the equivalence of the Bertsch parameter across spatial dimensions holds even for a one-dimensional unitary ppp-wave Fermi gas.
Abstract We revisit the properties of the two-component Fermi gas with short-range interactions in three dimensions, in the limit where the s -wave scattering length diverges. Such a unitary Fermi gas possesses universal thermodynamic and dynamical observables that are independent of any interaction length scale. Focusing on trapped systems of N fermions, where N ≤ 10 , we investigate how well we can determine the zero-temperature behavior of the many-body system from published few-body data on the ground-state energy and the contact. For the unpolarized case, we find that the Bertsch parameters extracted from trapped few-body systems all lie within 15% of the established value. Furthermore, the few-body values for the contact are well within the range of values determined in the literature for the many-body system. In the limit of large spin polarization, we obtain a similar accuracy for the polaron energy, and we estimate the polaron’s effective mass from the dependence of its energy on N . We also compute an upper bound for the squared wave-function overlap between the unitary Fermi system and the non-interacting ground state, both for the trapped and uniform cases. This allows us to prove that the trapped unpolarized ground state at unitarity has zero overlap with its non-interacting counterpart in the many-body limit N → ∞ .