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.
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.
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 study of ultracold trapped fermionic gases is a rapidly exploding subject [1-10] which is defining new directions in condensed matter and atomic physics. It has also captured the attention of physicists who study color superconducting quark matter as well as nuclear matter [11-13]. Indeed, it is hard, in recent times, to find a subfield of physics which appeals this broadly to the research community. As we come to understand the gases, experimentalists will move to address fermionic atoms in optical lattices; this will provide important insight to condensed matter physicists as analogue systems for “solving ” intractable many body problems. What makes these gases (and lattices) so important is their remarkable tunability and controllability. Using a Feshbach resonance, one can tune the attractive two-body interaction from weak to strong, and thereby make a smooth crossover from a BCS superfluid to a Bose-Einstein condensation (BEC) [14, 15]. Not only does this allow high transition temperatures Tc (relative to the Fermi energy EF) but it may also provide insights into the high temperature cuprate superconductivity [16-19]. Furthermore, one can tune the population of the two spin states, essentially at will, in this way, allowing exploration [20-22] of exotic polarized phases such as the FFLO [23-25] superfluids, of interest to condensed matter, nuclear and particle physicists. One will be able to tune the lattice parameters such as bandwidth, on-site repulsion, even random disorder, etc and thereby study the famous, and as yet, unsolved Hubbard model Hamiltonian. This paper will concentrate on those issues relating to the effects which are of current interest to experimentalists. In particular, we will study BCS-BEC crossover in atomic Fermi gases, looking at a wide range of different experiments. Our group has been principally interested in the effects of finite temperature [26, 17, 18] and in this review, we will discuss how temperature T enters into the standard crossover theory, and how temperature can be measured, and how temc○
In continuation of our earlier work on the nonextensive form of the Gross–Pitaevskii equation (GPE) [M. Maleki, H. Mohammadzadeh and Z. Ebadi, Int. J. Geom. Methods Mod. Phys. 20 (2023) 2350216], we now delve into its [Formula: see text]-deformed counterpart. GPE is a type of nonlinear partial differential equation that is specifically designed to describe the behavior of a group of particles with Bose–Einstein statistics, such as atoms in a superfluid or a Bose–Einstein condensate (BEC). In some systems, the standard Bose–Einstein or Fermi–Dirac statistics may not apply, and generalized statistics may be needed to describe the behavior of the particles. Therefore in this paper, we investigate the dynamics of a system with particle obeying [Formula: see text]-deformed statistics described by the [Formula: see text]-deformed GPE. First, we use the oscillator algebra and [Formula: see text]-calculus to obtain the well-known Schrödinger equation. By selecting an appropriate Hamiltonian for the condensate phase and minimizing the free energy, we derive the [Formula: see text]-deformed time-independent GPE.
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.
A series of major experimental breakthroughs have been obtained in the researches of ultracold quantum gases in the past years,which include the realization of Bose-Einstein condensation(BEC),the achievements of Fermi degeneracy and the observations of molecular BEC.This renewed the theoretical studies on the characteristics of BEC and degenerate Fermi gases,which have become a topic of great interest in the recent years.Based on the research works of our groups in the last years,the statistical properties of ultracold quantum gases are briefly described.The main contents include:the finite-size effects of quantum gases,roles of external potential,influence of interatomic interaction,low temperature behaviors of q-deformed quantum gas and the nonextensive statistics.The results obtained may be helpful to the further understanding of the low temperature behaviors of quantum gases as well as the underlying nature of various macroscopic quantum effects.
PACS. 03.75.Hh – Static properties of condensates; thermodynamical, statistical and structural properties. PACS. 03.75.Ss – Degenerate Fermi gases. PACS. 05.30.Fk – Fermion systems and electron gas. Abstract. – We present a theory for a superfluid Fermi gas near the BCS-BEC crossover, including pairing fluctuation contributions to the free energy similar to that considered by Nozières and Schmitt-Rink for the normal phase. In the strong coupling limit, our theory is able to recover the Bogoliubov theory of a weakly interacting Bose gas with a molecular scattering length very close to the known exact result. We compare our results with recent Quantum Monte Carlo simulations both for the ground state and at finite temperature. Excellent agreement is found for all interaction strengths where simulation results are available. The recent experimental realization of strongly interacting Fermi gases of 6 Li and 40 K atoms near a Feshbach resonance has opened up the exciting possibility of investigating the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC) [1–5]. In these systems, the inter-atomic interaction strength can be varied by tuning