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arXiv:2605.05431

A transition in the hole probability at finite temperature for free fermions in $d$ dimensions

Giuseppe Del Vecchio Del Vecchio, Pierre Le Doussal, Grégory Schehr

Published May 6, 2026• 0 citations
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Abstract

In a free Fermi gas at temperature $T$ much higher than the Fermi temperature one expects that the fluctuations of the number of particles in a given region has Poissonian/classical statistics. On the other hand at low temperature the Pauli exclusion principle leads to non trivial counting statistics. It is of great interest from a theoretical and experimental point of view to characterize the crossover between these two limits. Here we focus on the hole probability $P(R,T)$, i.e. the probability that a region of size $R$ is devoid of particles, in dimension $d$, and on the case of a spherical region of large radius $R$. We show that at low temperature it takes the scaling form $P(R,T)\sim \exp\big[-(k_F R)^{d+1}Φ_d(u=2R\,T/k_F)\big],$ where $k_F$ is the Fermi momentum. By mapping the problem to an effective Coulomb gas, we compute exactly the scaling function $Φ_d(u)$ in any dimension. Remarkably, it exhibits a transition of order $\tfrac{3}{2}(d+1)$ at the universal critical value $u_c=2/π$, signaling a sharp change in the mechanism of rare fluctuations, associated with the emergence of a macroscopic gap in the optimal density of the associated Coulomb gas. Our analytical predictions are supported by precise numerical evaluations of the corresponding Fredholm determinants.

DOI
10.48550/arxiv.2605.05431

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A transition in the hole probability at finite temperature for free fermions in ddd dimensions
In a free Fermi gas at temperature TTT much higher than the Fermi temperature one expects that the fluctuations of the number of particles in a given region has Poissonian/classical statistics. On the other hand at low temperature the Pauli exclusion principle leads to non trivial counting statistics. It is of great interest from a theoretical and experimental point of view to characterize the crossover between these two limits. Here we focus on the hole probability P(R,T)P(R,T)P(R,T), i.e. the probability that a region of size RRR is devoid of particles, in dimension ddd, and on the case of a spherical region of large radius RRR. We show that at low temperature it takes the scaling form P(R,T)∼exp⁡[−(kFR)d+1Φd(u=2R T/kF)],P(R,T)\sim \exp\big[-(k_F R)^{d+1}Φ_d(u=2R\,T/k_F)\big],P(R,T)∼exp[−(kF​R)d+1Φd​(u=2RT/kF​)], where kFk_FkF​ is the Fermi momentum. By mapping the problem to an effective Coulomb gas, we compute exactly the scaling function Φd(u)Φ_d(u)Φd​(u) in any dimension. Remarkably, it exhibits a transition of order 32(d+1)\tfrac{3}{2}(d+1)23​(d+1) at the universal critical value uc=2/πu_c=2/πuc​=2/π, signaling a sharp change in the mechanism of rare fluctuations, associated with the emergence of a macroscopic gap in the optimal density of the associated Coulomb gas. Our analytical predictions are supported by precise numerical evaluations of the corresponding Fredholm determinants.
Abstract The hole probability, i.e. , the probability that a region is void of particles, is a benchmark of correlations in many-body systems. We compute analytically this probability P ( R ) for a sphere of radius R in the case of N noninteracting fermions in their ground state in a d -dimensional trapping potential. Using a connection to the Laguerre-Wishart ensembles of random matrices, we show that, for large N and in the bulk of the Fermi gas, P ( R ) is described by a universal scaling function of k F R , for which we obtain an exact formula ( k F being the local Fermi wave vector). It exhibits a super-exponential tail where is a universal amplitude, in good agreement with existing numerical simulations. When R is of the order of the radius of the Fermi gas, the hole probability is described by a large deviation form which is not universal and which we compute exactly for the harmonic potential. Similar results also hold in momentum space.
We develop a first-principles approach to compute the counting statistics in the ground state of N noninteracting spinless fermions in a general potential in arbitrary dimensions d (central for d>1). In a confining potential, the Fermi gas is supported over a bounded domain. In d=1, for specific potentials, this system is related to standard random matrix ensembles. We study the quantum fluctuations of the number of fermions N_{D} in a domain D of macroscopic size in the bulk of the support. We show that the variance of N_{D} grows as N^{(d-1)/d}(A_{d}logN+B_{d}) for large N, and obtain the explicit dependence of A_{d},B_{d} on the potential and on the size of D (for a spherical domain in d>1). This generalizes the free-fermion results for microscopic domains, given in d=1 by the Dyson-Mehta asymptotics from random matrix theory. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem D, in any dimension, supported by exact results for d=1.
The Wigner function WN(x,p){W}_{N}(\mathbf{x},\mathbf{p})WN​(x,p) is a useful quantity to characterize the quantum fluctuations of an NNN-body system in its phase space. Here we study WN(x,p){W}_{N}(\mathbf{x},\mathbf{p})WN​(x,p) for NNN noninteracting spinless fermions in a ddd-dimensional spherical hard box of radius RRR at temperature T=0T=0T=0. In the large-NNN limit, the local-density approximation predicts that ${W}_{N}(\mathbf{x},\mathbf{p})\ensuremath{\approx}1/{(2\ensuremath{\pi}\ensuremath{\hbar})}^{d}$ inside a finite region of the (x,p)(\mathbf{x},\mathbf{p})(x,p) plane, namely, for ∣x∣<R|\mathbf{x}|<R∣x∣<R and ∣p∣<kF|\mathbf{p}|<{k}_{F}∣p∣<kF​, where kF{k}_{F}kF​ is the Fermi momentum, while WN(x,p){W}_{N}(\mathbf{x},\mathbf{p})WN​(x,p) vanishes outside this region, or droplet, on a scale determined by quantum fluctuations. In this paper we investigate systematically, in this quantum region, the structure of the Wigner function along the edge of this droplet, called the Fermi surf. In one dimension, we find that there are three distinct edge regions along the Fermi surf and we compute exactly the associated nontrivial scaling functions in each regime. We also study the momentum distribution ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{N}(p)$ and find a striking algebraic tail for very large momenta ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{N}(p)\ensuremath{\propto}1/{p}^{4}$, well beyond kF{k}_{F}kF​, reminiscent of a similar tail found in interacting quantum systems (discussed in the context of Tan's relation). We then generalize these results to higher ddd and find, remarkably, that the scaling function close to the edge of the box is universal, i.e., independent of the dimension ddd.
We study the distribution of particle number in extended subsystems of a one-dimensional noninteracting Fermi gas confined in a potential well at zero temperature. Universal features are identified in the scaled bulk and edge regions of the trapped gas where the full counting statistics are given by the corresponding limits of the eigenvalue statistics in Gaussian unitary random matrix ensembles. The universal limiting behavior is confirmed by the bulk and edge scaling of the particle number fluctuations and the entanglement entropy.
We investigate the ground-state properties of trapped fermion systems described by the Hubbard model with an external confining potential. We discuss the universal behaviors of systems in different regimes: from few particles, i.e., in dilute regimes, to the trap thermodynamic limit. The asymptotic trap-size (TS) dependence in the dilute regime (increasing the trap size $\ensuremath{\ell}$ keeping the particle number NNN fixed) is described by a universal TS scaling controlled by the dilute fixed point associated with the metal-to-vacuum quantum transition. This scaling behavior is numerically checked by DMRG simulations of the one-dimensional (1D) Hubbard model. In particular, the particle density and its correlations show crossovers among different regimes: for strongly repulsive interactions they approach those of a spinless Fermi gas, for weak interactions those of a free Fermi gas, and for strongly attractive interactions they match those of a gas of hard-core bosonic molecules. The large-NNN limit keeping the ratio $N/\ensuremath{\ell}$ fixed corresponds to a 1D trap thermodynamic limit. We address issues related to the accuracy of the local density approximation (LDA). We show that the particle density approaches its LDA in the large-$\ensuremath{\ell}$ limit. When the trapped system is in the metallic phase, corrections at finite $\ensuremath{\ell}$ are $O({\ensuremath{\ell}}^{\ensuremath{-}1})$ and oscillating around the center of the trap. They become significantly larger at the boundary of the fermion cloud, where they get suppressed as $O({\ensuremath{\ell}}^{\ensuremath{-}1/3})$ only. This anomalous behavior arises from the nontrivial scaling at the metal-to-vacuum transition occurring at the boundaries of the fermion cloud.
Abstract Studying the electronic properties of small metallic particles Kubo and subsequently Denton, Mühlschlegel and Scalapino recognized the surprising effect that the mean occupation numbers of noninteracting fermions (electrons) are not given within the canonical ensemble by the usual expression of Fermi‐Dirac statistics. The fermions behave as if they are colder and observed deviations are most significant in the vicinity of the Fermi level. The results of our Monte Carlo simulations, as well as our complementary numerical and analytical results are in very good agreement with the predictions of Denton et al. We found that the effect vanishes for very small and very large spacing between the energy levels in comparison to the thermal energy kBT. Associated with this effect are the particle‐hole correlations. Extending the method used by Denton et al. we derive an analytical formula for the particle‐hole correlation functions whose predictions are again in very good agreement with our numerical results. The low‐temperature aspects of the effect are considered by applying a two‐state model. The role of the finite‐size effect is also discussed.