We propose and study a generalized quantum statistical framework, referred to as \emph{alpha statistics}, that continuously interpolates between Bose--Einstein and Fermi--Dirac statistics and naturally extends into the hyperbosonic regime for $α< 0$. Inspired by Haldane's exclusion statistics, this formulation introduces a modified occupation weight function that encodes effective statistical interactions via the parameter $α$. Using thermodynamic geometry, we analyze the sign and singular behavior of the thermodynamic curvature as a diagnostic of underlying interactions and phase structures. A crossover temperature $T^{*}$, at which the curvature changes sign, marks the transition between effectively attractive (Bose-like) and repulsive (Fermi-like) statistical regimes. When expressed relative to the Bose--Einstein condensation temperature $T_{c}$, the ratio $T^{*}/T_{c}$ depends universally on $α$. For negative $α$, corresponding to hyperbosonic statistics, we find curvature singularities at specific fugacities, indicating modified condensation phenomena distinct from conventional Bose condensation. These results highlight the geometric and thermodynamic consequences of alpha statistics and establish a link between fractional exclusion principles and curvature-induced interaction signatures in statistical thermodynamics.
We propose and study a generalized quantum statistical framework, referred to as \emph{alpha statistics}, that continuously interpolates between Bose--Einstein and Fermi--Dirac statistics and naturally extends into the hyperbosonic regime for α<0. Inspired by Haldane's exclusion statistics, this formulation introduces a modified occupation weight function that encodes effective statistical interactions via the parameter α. Using thermodynamic geometry, we analyze the sign and singular behavior of the thermodynamic curvature as a diagnostic of underlying interactions and phase structures. A crossover temperature T∗, at which the curvature changes sign, marks the transition between effectively attractive (Bose-like) and repulsive (Fermi-like) statistical regimes. When expressed relative to the Bose--Einstein condensation temperature Tc, the ratio T∗/Tc depends universally on α. For negative α, corresponding to hyperbosonic statistics, we find curvature singularities at specific fugacities, indicating modified condensation phenomena distinct from conventional Bose condensation. These results highlight the geometric and thermodynamic consequences of alpha statistics and establish a link between fractional exclusion principles and curvature-induced interaction signatures in statistical thermodynamics.
We investigate the thermodynamic geometry of an ideal quantum gas obeying inclusion statistics, characterized by a negative statistical parameter g<0. In this framework the grand-canonical partition function admits a finite maximum fugacity, and the thermodynamic scalar curvature R is strictly positive for all g<0, reflecting effective attractive statistical interactions analogous to those of a bosonic system. As the fugacity approaches its maximum value, R 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≤1 where standard bosons do not condense, while for D/σ≤1/2 the transition temperature vanishes. Three independent criteria; divergence of R, 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.
Abstract We propose a generalized quantum statistical framework, termed Max–Min statistics, which imposes simultaneous lower ( N min ) and upper ( N max ) bounds on the occupation numbers of single-particle quantum levels. The formalism is developed using two complementary but equivalent approaches: a combinatorial microstate-counting scheme based on modified statistical weights, and an operator-algebraic construction employing a generalized number operator with constrained ladder operations. The resulting distribution function provides a unified description that continuously interpolates between Bose–Einstein, Fermi–Dirac, Gentile, and Ewkon statistics as limiting cases. Thermodynamic consequences of the occupation constraints are examined within the grand canonical ensemble, and the geometric properties of the corresponding thermodynamic state space are analyzed using Ruppeiner geometry. The scalar curvature is shown to vary systematically with the allowable occupation range Δ N = N max − N min , reflecting changes in statistical correlations induced by the imposed bounds. In particular, non-zero lower occupancy constraints lead to distinct thermodynamic behaviors that require regularization via an energy cutoff. These results establish the Max–Min framework as a flexible and mathematically consistent tool for exploring bounded and intermediate quantum statistics from a formal and phenomenological perspective.
Anyons are quasiparticles with quantum statistics interpolating between those of bosons and fermions. Two distinct manifestations of anyonic behaviour have been theoretically established: fractional exchange statistics where particle exchange can produce any phase, and generalized exclusion statistics which extends the Pauli exclusion principle. While anyons exhibiting fractional exchange statistics have been observed in diverse platforms, experimental realizations of generalized exclusion statistics and direct measurements of its thermodynamic signatures have remained elusive. Here, we realize an anyonic thermodynamic ensemble obeying generalized exclusion statistics and detect its anyonic thermodynamics in a one-dimensional strongly interacting quantum gas. To achieve this, we exploit the bijective mapping between dynamical and statistical interactions in one dimension. By tuning interaction strength and temperature over a wide range, we measure the equation of state and identify clear departures from Bose-Einstein and Fermi-Dirac statistics. These deviations are quantitatively captured by generalized exclusion statistics, providing direct evidence for the generalized Pauli principle. Independent probes of other thermodynamic quantities including pressure and the Tan contact further validate this framework. Our results establish a versatile platform for engineering anyonic exclusion statistics and open the door to thermodynamic applications of anyons in quantum technologies.
A formal thermodynamic mapping is established between the attractive Fermi–Hubbard model and the repulsive Bose–Hubbard model at finite temperature and at imaginary chemical potential μ=iθ. By utilizing a large N-expansion, it is shown that the partition functions of the two models are related by a plain shift θ→θ+π. This condition maps the BCS–BEC crossover of attractive fermions to a Bose–Fermi crossover (fermion-like occupation) of repulsive bosons. A central feature of this correspondence is the thermal kernel g(βE,ϕ) (with β the inverse absolute temperature, E the energy scale, and ϕ the phase angle), whose analytic continuation gB(βE,ϕ)=gF(βE,ϕ+π) governs the bosonic (B) and fermionic (F) sectors. Interestingly, the particular angles ϕ=2π/3 and 4π/3 for fermions correspond to ϕ=π/3 and 5π/3 for bosons, marking the boundaries of an universal thermal window. It is further argued that the present mechanism shows how an emergent, fermionization-like phenomenon can occur at finite interaction strength through a thermodynamic effect induced by the imaginary chemical potential. It is emphasized that this does not imply a transmutation of quantum statistics at the operator level, but rather a thermodynamic exclusion-like behavior driven by the imaginary chemical potential, unlike the Tonks–Girardeau limit, where fermionization arises from an infinite repulsive interaction and anyonic or Floquet-engineered systems where transmutation emerges from modified statistics or dynamics. Effectively, the phase ϕ is a statistical parameter; by twisting the thermal phase, it generates fermion-like behavior without hard-core constraints or infinite repulsion through purely thermodynamic mechanisms. The gap equation and number equation for the bosonic model are derived, highlighting the role of the imaginary chemical potential as a statistical regulator. The results obtained here provide a unified framework for understanding crossovers in interacting lattice systems.
We find a mapping between the attractive Fermi-Hubbard model and the repulsive Bose-Hubbard model at finite temperature and at imaginary chemical potential μ=iθ. We show, by using a large N-expansion, that the partition functions of the two models are related by a simple shift θ→θ+π. This condition maps the BCS--BEC crossover of attractive fermions to a Bose--Fermi crossover (fermion-like occupation) of repulsive bosons. Central feature of this correspondence plays the thermal kernel g(βE,ϕ), whose analytic continuation gB(βE,ϕ)=gF(βE,ϕ+π) governs the bosonic and fermionic sectors. Interestingly, we are able to find that the special angles ϕ=2π/3,4π/3 for fermions correspond to ϕ=π/3,5π/3 for bosons, marking the boundaries of a universal thermal window. We further argue that the present mechanism shows that fermionization can occur at finite interaction strength through a thermodynamic effect induced by the imaginary chemical potential. This suggests that it is a new way of fermionization (not a change in statistics but a fermion-like behaviour) unlike the Tonks--Girardeau limit, where fermionization arises from an infinite repulsive interaction and anyonic or Floquet-engineered systems where transmutation emerges from modified statistics or dynamics. Essentially, the phase ϕ is a statistical parameter; by twisting the thermal phase, it generates fermion-like behaviour without hard-core constraints or infinite repulsion but only by using thermodynamics. We derive the gap equation and number equation for the bosonic model, highlighting the role of the imaginary chemical potential as a statistical regulator. Our results provide a unified framework for understanding crossovers in interacting lattice systems.
This paper establishes and systematically develops the “Theory of Characteristic Equations” for Bose-Einstein condensates (BECs), revealing a profound correspondence between diverse types of equations – polynomial (partition functions), differential (Gross-Pitaevskii and Bogoliubov-de Gennes equations), integral (path integrals and dimensional regularization), difference (lattice BECs), sum (antidifferences), variational (energy functionals), inverse variational (effective actions), and transcendental (BKT transition and Efimov effect) – and geometric/topological structures of quantum degenerate Bose gases. This provides a complete geometric characteristic representation for BECs, thereby extending and upgrading classical mean-field theory, Bogoliubov theory, and the theory of quantum phase transitions to a unified framework. Starting from the observation that the partition function of an ideal Bose gas is a characteristic polynomial that reduces to the binomial theorem via the Euler- Maclaurin formula, we propose the Dimensional Extension Principle: integer parameters such as spatial dimension, particle number, and interaction range can be analytically continued via the Gamma function and generalized binomial coefficients, thereby generalizing integer-dimensional BEC theory to real and even complex dimensions. We rigorously construct negative-dimensional BEC models, compute their homology via Koszul complexes and BGG resolutions, and establish duality relations. We prove equivalence between dimensional regularization in quantum field theory and characteristic integral equations for BECs. We interpret the Bogoliubov excitation spectrum as a characteristic differential equation, extending it to fractional-dimensional lattices Zα whose heat kernel coefficients are polynomials in α with binomial structure. We incorporate variational principles (Gross-Pitaevskii functionals) and prove analytic dependence on dimension for condensate ground states. We extend difference equations to fractional-dimensional optical lattices via interpolation, with spectra depending continuously on the dimension. We develop sum equations (antidifferences) capturing the discrete partition function and its analytic continuation, leading to α-dimensional BEC models whose characteristic equation reduces to the Euler-binomial identity in the Abel sense for Re(α) > −1. We introduce modular and elliptic BEC networks (e.g., BECs in frustrated lattices and the BKT transition), whose characteristic equations reduce to modular equations. All major BEC invariants – partition functions, heat-kernel coefficients, and their fractional extensions – are unified under this framework. We prove a Strong Unification Theorem showing that all these invariants are manifestations of a single underlying structure: the binomial theorem and its analytic continuations. We establish a fractional-dimensional index theorem for Dirac operators on bipartite BEC networks and extend the Atiyah-Singer theorem to non-integer dimensions. The work demonstrates that characteristic equations provide a complete representation of BEC features, upgrading spectral, statistical, and hydrodynamic theories to a generalized geometry capable of handling non-integer and negative parameters, as well as various types of equations, offering a unified language for many branches of condensed matter physics, quantum field theory, and mathematical physics.