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arXiv:1412.5112 · Canadian Journal of Physics

Fisher zeros of a unitary Bose gas

W. van Dijk, Calvin Lobo, Allison MacDonald, Rajat K. Bhaduri

Published Dec 19, 2014• 21 citations
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Abstract

For real inverse temperature β, the canonical partition function is always positive, being a sum of positive terms. There are zeros, however, on the complex β plane that are called Fisher zeros. In the thermodynamic limit, the Fisher zeros coalesce into continuous curves. In case there is a phase transition, the zeros tend to pinch the real-β axis. For an ideal trapped Bose gas in an isotropic three-dimensional harmonic oscillator, this tendency is clearly seen, signalling Bose–Einstein condensation (BEC). The calculation can be formulated exactly in terms of the virial expansion with temperature-dependent virial coefficients. When the second virial coefficient of a strongly interacting attractive unitary gas is included in the calculation, BEC seems to survive, with the condensation temperature shifted to a lower value for the unitary gas. This shift is consistent with a direct calculation of the heat capacity from the canonical partition function of the ideal and the unitary gas.

DOI
10.1139/cjp-2014-0585

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For real inverse temperature β, the canonical partition function is always positive, being a sum of positive terms. There are zeros, however, on the complex β plane that are called Fisher zeros. In the thermodynamic limit, the Fisher zeros coalesce into continuous curves. In case there is a phase transition, the zeros tend to pinch the real-β axis. For an ideal trapped Bose gas in an isotropic three-dimensional harmonic oscillator, this tendency is clearly seen, signalling Bose–Einstein condensation (BEC). The calculation can be formulated exactly in terms of the virial expansion with temperature-dependent virial coefficients. When the second virial coefficient of a strongly interacting attractive unitary gas is included in the calculation, BEC seems to survive, with the condensation temperature shifted to a lower value for the unitary gas. This shift is consistent with a direct calculation of the heat capacity from the canonical partition function of the ideal and the unitary gas.
We present a detailed description of a classification scheme for phase transitions in finite systems based on the distribution of Fisher zeros of the canonical partition function in the complex temperature plane. We apply this scheme to finite Bose systems in power-law traps within a semi-analytic approach with a continuous one-particle density of states $\ensuremath{\Omega}(E)\ensuremath{\sim}{E}^{d\ensuremath{-}1}$ for different values of d and to a three-dimensional harmonically confined ideal Bose gas with discrete energy levels. Our results indicate that the order of the Bose-Einstein condensation phase transition sensitively depends on the confining potential.
Noninteracting fermions, placed in a system with a continuous density of states, may have zeros in the N -fermion canonical partition function on the positive real β axis (or very close to it), even for a small number of particles. This results in a singular free energy, and instability in other thermal properties of the system. In the context of trapped fermions in a harmonic oscillator, these zeros are shown to be unphysical. Our results are also applicable to mean-field canonical calculations for fermions. By contrast, similar bosonic calculations with continuous density of states yield sensible results.
We study the thermodynamic behavior of an ideal gas of bosons trapped in a three-dimensional anisotropic harmonic-oscillator potential. The condensate fraction as well as the specific heat is calculated using the Euler-Maclaurin approximation. For a finite number of particles there is no phase transition, but there is a well-defined temperature at which the condensation starts. We also consider condensation in lower dimensions, and, for one-dimensional systems, we discuss the dependence of the condensate fraction and heat capacity on the ensemble used.
We investigate the thermodynamic geometry of an ideal quantum gas obeying inclusion statistics, characterized by a negative statistical parameter g<0g < 0g<0. In this framework the grand-canonical partition function admits a finite maximum fugacity, and the thermodynamic scalar curvature RRR is strictly positive for all g<0g < 0g<0, reflecting effective attractive statistical interactions analogous to those of a bosonic system. As the fugacity approaches its maximum value, RRR diverges, signaling a phase transition of the Bose-Einstein condensation type. A key distinction from the ordinary ideal Bose gas is that the condensation temperature is elevated relative to the bosonic case, and finite-temperature condensation occurs even in the dimensional regime 1/2<D/σ≤11/2 < D/σ\leq 11/2<D/σ≤1 where standard bosons do not condense, while for D/σ≤1/2D/σ\leq 1/2D/σ≤1/2 the transition temperature vanishes. Three independent criteria; divergence of RRR, the maximum fugacity singularity, and the non-analytic cusp in the specific heat, coincide at the same condensation point, confirming the thermodynamic consistency of the transition.
By setting the inverse temperature β loose to occupy the complex plane, Fisher showed that the zeros of the complex partition function Z , if approaching the real β axis, reveal a thermodynamic phase transition. More recently, Fisher zeros were used to mark the dynamical phase transition in quench dynamics. It remains unclear, however, how Fisher zeros can be employed to better understand quantum phase transitions or the nonunitary dynamics of open quantum systems. Here we answer this question by a comprehensive analysis of the analytically continued one-dimensional transverse field Ising model. We exhaust all the Fisher zeros to show that in the thermodynamic limit they congregate into a remarkably simple pattern in the form of continuous open or closed lines. These Fisher lines evolve smoothly as the coupling constant is tuned, and a qualitative change identifies the quantum critical point. By exploiting the connection between Z and the thermofield double states, we obtain analytical expressions for the short- and long-time dynamics of the survival amplitude, including its scaling behavior at the quantum critical point. We point out Z can be realized and probed in monitored quantum circuits. The exact analytical results are corroborated by the numerical tensor renormalization group. We further show that similar patterns of Fisher zeros also emerge in other spin models. Therefore, the approach outlined may serve as a powerful tool for interacting quantum systems. Published by the American Physical Society 2024
Abstract We have proposed an analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, two-dimensional, and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a series of q -digamma functions in mathematics. We employ the analytical solution to calculate the internal energy E , the entropy S , the Helmholtz free energy F , and the heat capacity at constant number C N of ideal Bose atoms in a two-dimensional isotropic harmonic trap. The first main finding in this paper is that the variation with temperature of the internal energy of a finite number of ideal Bose atoms has an inflection point, which occurs at the transition temperature T c . The second main finding is that the internal energy of a finite number of ideal Bose atoms has a classic limit E = 2 N k B T as T ≫ T c . The third main finding is that the variation with temperature of the heat capacity of a finite number of ideal Bose atoms has a maximal value, which occurs at T c . The fourth main finding is that the heat capacity of a finite number of ideal Bose atoms has a classic limit C N / k B = 2 N as T ≫ T c . The fifth main finding is that in the thermodynamic limit, at critical temperature T c the heat capacity at constant number shows a cusp singularity, which is analogous to the λ -transition of liquid helium. Consequently, the transition between normal and condensed states is a second-order phase transition.