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.
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.
One-particle properties of noninteracting fermions in a one-dimensional harmonic trap and at zero temperature are studied. Exact expressions and asymptotic results for a large fermion number N are given for the particle density distribution
n0(z,N). For large N and near the classical boundary at the Fermi energy, the density displays increasing fluctuations. A simple scaling of these tails of the density distribution with respect to N is established. The Fourier transform of the density distribution is calculated exactly. It displays a small but characteristic hump near
2kF, with
kF being a properly defined Fermi wave number. This is due to Friedel oscillations, which are identified and discussed. These quantum effects are missing in the semiclassical approximation. Momentum distributions are also evaluated and discussed. As an example of a time-dependent one-particle problem, we calculate exactly the evolution of the particle density when the trap is suddenly switched off, and find a simple scaling behavior, in agreement with recent general results.
Beyond the analysis of the Lee-Yang (LY) zero of
ξ at
0 K presented by our previous work [He et. al. Phys. Rev. E 113, 24115 (2026)], it is important but intricate to understand how these zeros evolve with temperature (
T). Here, we use an analytically solvable noninteracting one-dimensional particle-on-a-ring model to address this. We determine the trajectories of these zeros and analyze how their evolution with
T reshapes the analytic structure of the partition function. In particular, the zero originating from
ξ=−1 at
T=0 remains close to
−1 at low
T, where it governs the sign factor and strongly constrains continuation along the real
ξ axis. This explains why both direct extrapolation and implicit schemes such as contour-based fitting can fail in the low-
T regime, even at high fitting order, while becoming reasonable again once the relevant zeros move away at higher
Ts. Furthermore, based on the polynomial structure of the partition function, we propose a new fitting strategy for low-
T fermionic properties. The key is to first obtain reliable high-
T fermionic properties by continuing sign-problem-free data in
ξ∈[0,1] to
ξ=−1, and then extend this information toward lower
T through
T-fitting of the
ξ-independent remainder
ϕ(β)=ZF. These results provide a solvable benchmark for diagnosing the validity of analytic continuation and suggest a possible route toward treating more realistic interacting fermionic systems.
We study a system of
N noninteracting spinless fermions in a confining double-well potential in one dimension. We show that when the Fermi energy is close to the value of the potential at its local maximum, physical properties, such as the average density and the fermion position correlation functions, display a universal behavior that depends only on the local properties of the potential near its maximum. This behavior describes the merging of two Fermi gases, which are disjoint at sufficiently low Fermi energies. We describe this behavior in terms of a correlation kernel that we compute analytically and we call it the inverted parabola kernel. As an application, we calculate the mean and variance of the number of particles in an interval of size
2L centered around the position of the local maximum, for sufficiently small
L. We discuss the possibility of observing our results in experiments, as well as extensions to nonzero temperature and to higher space dimensions.
While the thermodynamics of a system of non-interacting fermions can be straightforwardly determined in the grand canonical ensemble, results for a specific number of N particles are more difficult to obtain. In this talk we will present a general recursion scheme for the canonical partition function of free fermions with a quadratic energy level spacing as might be present for ultracold fermionic atoms confined inside a box trap. Exact results for the entropy, specific heat and one and two particle occupation probabilities are numerically obtained to arbitrary precision and compared with their corresponding grand canonical values. The numerical stability of the recursion relation allows us to quantify deviations from Wicks theorem in the canonical ensemble for a variety of temperatures and densities.
We present closed analytical expressions for the particle and kinetic-energy spatial densities at finite temperatures for a system of noninteracting fermions (bosons) trapped in a d-dimensional harmonic-oscillator potential. For
d=2 and
3, exact expressions for the N-particle densities are used to calculate perturbatively the temperature dependence of the splittings of the energy levels in a given shell due to a very weak interparticle interaction in a dilute Fermi gas. In two dimensions, we obtain analytically the surprising result that the l degeneracy in a harmonic-oscillator shell is not lifted in the lowest order even when the exact, rather than the Thomas-Fermi expression for the particle density is used. We also demonstrate rigorously (in two dimensions) the reduction of the exact zero-temperature fermionic expressions to the Thomas-Fermi form in the large-N limit.