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Physical Review A

Quantum many-body theory for q -deformed fermions and bosons

Habib Esmaili, Hosein Mohammadzadeh, Mehdi Biderang, Morteza NattaghNajafi

Published Feb 20, 2025• 6 citations
Publisher

Abstract

We present a comprehensive quantum many-body theory for $q$-deformed fermions and bosons, offering a novel framework that relates particle statistics directly to effective interaction strength. Deformed by the parameters $k$ and $q$, these particles exhibit statistical behaviors that interpolate between conventional bosonic and fermionic systems, enabling us to model complex interactions via statistical modifications. We develop a generalized Wick theorem and extended Feynman diagrammatics tailored to $kq$ particles, allowing us to calculate two types of Green's functions. Explicit expressions for these Green's functions are derived in both direct and momentum spaces, providing key insights into the collective properties of $q$-fermionic and bosonic systems. Using a random-phase approximation, we estimate the dielectric function for $q$-fermion gas ($k=\ensuremath{-}1$) and analyze the Friedel oscillations, the plasmon excitations, and the energy-loss function. Our results demonstrate that the effective interaction is tuned by the value of $q$ so that a noninteracting limit is obtained as $q\ensuremath{\rightarrow}0$, where the Friedel and the plasma oscillations disappear. An optimal value of $q$, namely, ${q}^{*}$, the plasma frequency, and the energy-loss function show an absolute maximum and the effective interaction changes behavior.

DOI
10.1103/physreva.111.022223

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We present a comprehensive quantum many-body theory for qqq-deformed fermions and bosons, offering a novel framework that relates particle statistics directly to effective interaction strength. Deformed by the parameters kkk and qqq, these particles exhibit statistical behaviors that interpolate between conventional bosonic and fermionic systems, enabling us to model complex interactions via statistical modifications. We develop a generalized Wick theorem and extended Feynman diagrammatics tailored to kqkqkq particles, allowing us to calculate two types of Green's functions. Explicit expressions for these Green's functions are derived in both direct and momentum spaces, providing key insights into the collective properties of qqq-fermionic and bosonic systems. Using a random-phase approximation, we estimate the dielectric function for qqq-fermion gas (k=\ensuremath−1k=\ensuremath{-}1k=\ensuremath−1) and analyze the Friedel oscillations, the plasmon excitations, and the energy-loss function. Our results demonstrate that the effective interaction is tuned by the value of qqq so that a noninteracting limit is obtained as q\ensuremath→0q\ensuremath{\rightarrow}0q\ensuremath→0, where the Friedel and the plasma oscillations disappear. An optimal value of qqq, namely, q∗{q}^{*}q∗, the plasma frequency, and the energy-loss function show an absolute maximum and the effective interaction changes behavior.
We present a comprehensive quantum many body theory for kq deformed particles, offering a novel framework that relates particle statistics directly to effective interaction strength. Deformed by the parameters k and q, these particles exhibit statistical behaviors that interpolate between conventional bosonic and fermionic systems, enabling us to model complex interactions via statistical modifications. We develop a generalized Wick's theorem and extended Feynman diagrammatic tailored to kq-particles, allowing us to calculate two types of Green functions. Explicit expressions for these Green functions are derived in both direct and momentum spaces, providing key insights into the collective properties of kq-deformed systems. Using a random phase approximation (RPA), we estimate the dielectric function for q-fermion gas, and analyze the Friedel oscillations, the plasmon excitations, and the energy loss function. Our results demonstrate that the effective interaction is tuned by the value of q, so that a non interacting limit is obtained as q goes to zero, where the Friedel as well as the plasma oscillations disappear. There is an optimal value of q, the plasma frequency, as well as the energy loss function show an absolute maximum, and the effective interaction changes behavior.
We develop a comprehensive many-body theory for systems of particles obeying quantum unified statistics, or quons. After exploring the properties of the Fock space of this system, we formulate a systematic S-matrix expansion and a generalized Wick's theorem. A consistent Green-function formalism is constructed at both zero and finite temperatures, accompanied by a generalized Wick's theorem appropriate for infinite-statistics operator algebras. Within this framework, we establish diagrammatic rules for interacting quon systems. Employing the random phase approximation, we derive the dielectric function and reveal the emergence of anomalous plasmon modes that have no direct counterpart in conventional Bose or Fermi systems. We further analyze the ground-state energy, energy-loss function, generalized Thomas-Fermi screening wave vectors, and Friedel oscillations, elucidating how infinite statistics qualitatively modifies collective behavior and screening properties.
Abstract Green-function techniques are among the most versatile tools for investigating quantum many-body systems at finite temperature. Yet, standard textbook treatments often fragment the formalism—separating bosons and fermions, systems with massless and massive particles, or restricting proofs to special Hamiltonians—thereby obscuring the structural unity of the theory. This work develops a general, Hamiltonian-independent framework for both real- and imaginary-time Green-functions, valid for any combination of bosonic and fermionic degrees of freedom, including mixed systems such as interacting electron–phonon models. The key conceptual advance is the nature index η , which complements the conventional statistical sign ζ . While ζ encodes particle statistics, η distinguishes canonical quantization of systems with massless particles ( η = 0) from second quantization of systems with massive particles ( η = 1). This classification removes ambiguities in chemical-potential terms and unifies the treatment of photons, phonons, electrons, and their mixtures under a single notation. From this basis, we formulate generalized ensemble constructions, generalized Lehmann-representations, and a fluctuation-dissipation theorem valid for arbitrary Hamiltonians. Several results—such as the monotonicity of particle number in the chemical potential, an explicit formula for the quadratic contribution to the internal energy from the retarded Green-function, and the Hartree-contribution to the self-energy in electron–phonon systems—are either absent from, or only sparsely mentioned in, existing literature. We further introduce Green’s Chart, a visual and algebraic map that systematically connects all Green-function types through analytic continuation, Fourier-transforms, and Wick-rotations, enabling all to be generated from the Matsubara Green-function. Applications to both non-interacting and interacting models demonstrate the framework’s ability to compute densities of states, thermodynamic potentials, and spectral functions with a consistent formalism. The results provide a complete and reproducible toolkit for equilibrium many-body theory, consolidating scattered results into a unified structure suitable for both pedagogical and computational purposes.
We study the high-temperature behavior of an intermediate-statistics fermionic gas model whose quantum statistical properties enable us to effectively deduce the details about both the interaction among deformed (quasi)particles and their anyonic behavior. Starting with a deformed fermionic grand partition function, we calculate, in the thermodynamical limit, several thermostatistical functions of the model such as the internal energy and the entropy by means of a formalism of the fermionic q calculus. For high temperatures, a virial expansion of the equation of state for the system is obtained in two and three dimensions and the first five virial coefficients are derived in terms of the model deformation parameter q. From the results obtained by the effect of fermionic deformation, it is found that the model parameter q interpolates completely between bosonlike and fermionic systems via the behaviors of the third and fifth virial coefficients in both two and three spatial dimensions and in addition it characterizes effectively the interaction among quasifermions. Our results reveal that the present deformed (quasi)fermion model could be very efficient and effective in accounting for the nonlinear behaviors in interacting composite particle systems.
Abstract In recent decades, there have been increasing interests in quantum statistics beyond the standard Fermi–Dirac and Bose–Einstein statistics, such as the fractional statistics, quon statistics, anyon statistics and quantum groups, since they can provide new insights into the cosmology, nuclear physics and condensed matter. In this paper, we study the many-particle system formed by the q -deformed fermions ( q -fermion), which is realized by deforming the quantum algebra of the anticommutation relations. From the standard perspective of the finite temperature field theory, we first construct the finite temperature Green’s function formalism for the free many- q -fermion system, then generalize it to the well known interacting fermionic system, the superconductor, and finally obtain a consistent q -deformed BCS ( q BCS) theory. Interesting features can be implied from this model. For example, it predicts a Sarma-like ordered phase, which we call the q -deformed Sarma phase, at low temperature. It also presents a symmetric phase diagram in the parameter space and modified thermodynamic relations.
The idea that a system obeying interpolating statistics can be described by a deformed oscillator algebra, or quantum groups, has been an outstanding issue. We are able to demonstrate that a q-deformed oscillator algebra can be used to describe the statistics of particles which provide a continuous interpolation between Bose and Fermi statistics. We show that the generalized intermediate statistics splits into Boson-like and Fermion-like regimes, each described by a unique oscillator algebra. The thermostatistics of Boson-like particles is described by employing q-calculus based on the Jackson derivative while the Fermion-like particles are described by ordinary derivatives of thermodynamics. Thermodynamic functions for systems of both types are determined and examined.