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arXiv:1804.02376 · Journal of Statistical Mechanics Theory and Experiment

q -deformed fermion in many-particle systems and its application to BCS theory

Xu-Yang Hou, Xun Huang, Yan He, Hao Guo

Published Dec 1, 2018• 10 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1742-5468/aaeb43

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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.
During the last three decades, nonstandard statistics for indistinguishable quantum particles has attracted wide attention and research interests from many institutions. Among these new types of statistics, the [Formula: see text]-deformed Bose and Fermi statistics, originated from the study of quantum algebra, are being applied in more and more physical systems. In this paper, we construct a [Formula: see text]-deformed generalization of the BCS-Leggett theory for ultracold Fermi gases based on our previously constructed [Formula: see text]-deformed BCS theory. Some interesting features of this [Formula: see text]-deformed interacting quantum gas are obtained by numerical analysis. For example, in the ordinary Bose–Einstein Condensation regime, the gas presents a fermionic feature instead of bosonic feature if the deformation parameter is tuned suitably, which might be referred to as the [Formula: see text]-induced “Bose–Fermi” crossover. Conversely, a weak sign of the “Fermi–Bose” crossover is also found in the ordinary weak fermionic regime.
A q-deformed field theory at finite temperature is presented and its related thermo field dynamics is constructed. This enables us to introduce the quon field, whose Lagrangian leads to a q-deformed partition function. Accordingly, the general thermostatistical properties of a gas model of quon fields, such as the q-deformed statistical distribution function describing an intermediate-statistics behavior and the equation of state in two and three dimensions, are investigated. For low temperatures, the conditions under which Bose–Einstein-like condensation would occur in the quon gas model are discussed. It is shown that the critical temperature of such a gas is higher than that of the usual Bose gas for values of the model parameter q in the range [Formula: see text]. For high temperatures, possible anyonic behavior of the present quon gas model in two spatial dimensions is studied through an analysis of the effect of deformation on the second and third virial coefficients in the equation of state. The results obtained in this work reveal the ability of the model for analyzing a parastatistical behavior of systems with quasiparticles and put forward it as a possible candidate for effectively modeling the properties of some exotic quantum states, such as in the case of dark matter constituents.
An outstanding idea originally introduced by Greenberg is to investigate whether there is equivalence between intermediate statistics, which may be different from anyonic statistics, and q-deformed particle algebra. Also, a model to be studied for addressing such an idea could possibly provide us some new consequences about the interactions of particles as well as their internal structures. Motivated mainly by this idea, in this work, we consider a q-deformed Fermi gas model whose statistical properties enable us to effectively study interpolating statistics. Starting with a generalized Fermi-Dirac distribution function, we derive several thermostatistical functions of a gas of these deformed fermions in the thermodynamical limit. We study the high-temperature behavior of the system by analyzing the effects of q deformation on the most important thermostatistical characteristics of the system such as the entropy, specific heat, and equation of state. It is shown that such a deformed fermion model in two and three spatial dimensions exhibits the interpolating statistics in a specific interval of the model deformation parameter 0 < q < 1. In particular, for two and three spatial dimensions, it is found from the behavior of the third virial coefficient of the model that the deformation parameter q interpolates completely between attractive and repulsive systems, including the free boson and fermion cases. From the results obtained in this work, we conclude that such a model could provide much physical insight into some interacting theories of fermions, and could be useful to further study the particle systems with intermediate statistics.
The statistics of q-oscillators, quons, and, to some extent, of anyons are studied and the basic differences among these objects are pointed out In particular, the statistical distributions for different bosonic and fermionic q-oscillators are found for their corresponding Fock space representations in the case when the Hamiltonian is identified with the number operator. In this case and for non-relativistic particles, the single-particle temperature Green function is defined with q-deformed periodicity conditions. The equations of state for nonrelativistic and ultrarelativistic bosonic q-gases in an arbitrary space dimension are found near Bose statistics, as well as that for an anyonic gas near Bose and Fermi statistics. The first corrections to the second virial coefficients are also evaluated.
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 ($k=\ensuremath{-}1$) 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 $q\ensuremath{\rightarrow}0$, 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.
Thermodynamical properties of a model quon gas are reported. The quons considered are the objects obeying the deformed commutator aa†−qa†a=1 with −1⩽q⩽1. Although q=1 and q=−1 correspond respectively to boson and fermion statistics, it is not trivial to generate a quon gas which interpolates between the boson gas and the fermion gas without Pauli’s exclusion principle. The proposed model, interpolating between the two limits, can realize the Maxwell-Boltzmann limit at q=0 and simulate a free anyon gas. Under the restriction −1⩽q⩽1, the gas is deformed but bosonic in nature. It shows the Bose-Einstein condensation even in two dimensions.