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arXiv:math-ph/0611043 · International Journal of Modern Physics A

INTERACTING BOSE AND FERMI GASES IN LOW DIMENSIONS AND THE RIEMANN HYPOTHESIS

André LeClair

Published Apr 10, 2008• 14 citations
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Abstract

We apply the S-matrix based finite temperature formalism to nonrelativistic Bose and Fermi gases in 1+1 and 2+1 dimensions. For the (2+1)-dimensional case in the constant scattering length approximation, the free energy is given in terms of Roger's dilogarithm in a way analagous to the thermodynamic Bethe ansatz for the relativistic (1+1)-dimensional case. The 1d fermionic case with a quasiperiodic two-body potential is closely connected with the Riemann hypothesis.

DOI
10.1142/s0217751x08039451

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A new treatment of the critical point of the two-dimensional interacting Bose gas is presented. In the lowest order approximation we obtain the critical temperature Tc≈2πn/[mlog(2π/mg)], where n is the density, m the mass, and g the coupling. This result is based on a new formulation of interacting gases at finite density and temperature which is reminiscent of the thermodynamic Bethe ansatz in one dimension. In this formalism, the basic thermodynamic quantities are expressed in terms of a pseudo-energy. Consistent resummation of 2-body scattering leads to an integral equation for the pseudo-energy with a kernel based on the logarithm of the exact 2-body S-matrix.
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We study both Bose and Fermi gases at finite temperature and density in an approximation that sums an infinite number of many-body processes that are reducible to two-body scatterings. This is done for arbitrary negative scattering length, which interpolates between the ideal and unitary gas limits. In the unitary limit, we compute the first four virial coefficients within our approximation. The second virial coefficient is exact, and we extend the previously known result for fermions to bosons and also for both bosons and fermions for the upper branch on the other side of unitarity (infinitely large positive scattering length). Assuming bosons can exist in a metastable state before undergoing mechanical collapse, we map out the critical temperatures for strongly coupled Bose-Einstein condensation as a function of scattering length.
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Abstract Using a formalism based on the two-body S-matrix we study two-dimensional Bose and Fermi gases with both attractive and repulsive interactions. Approximate analytic expressions, valid at weak coupling and beyond, are developed and applied to the Berezinskii–Kosterlitz–Thouless (BKT) transition. We successfully recover the correct logarithmic functional form of the critical chemical potential and density for the Bose gas. For fermions, the BKT critical temperature is calculated in BCS and BEC regimes through consideration of Tan’s contact.