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Modern Physics Letters B

Free energy behavior in exactly solvable many-fermion interacting systems

Angelo Plastino, Diana Monteoliva, Angel Ricardo Plastino

Published Jul 20, 2023• 0 citations
Publisher

Abstract

The objective of this work is to show that simple modifications in the form of the fermion–fermion interacting potential generate widely different thermodynamic behaviors, with macroscopic manifestations reflected by the thermal free energy. This is done by appealing to quantum statistical mechanics, in its Gibbs’ ensemble formalism, employing two different SU2 exactly solvable models. Some intriguing aspects of the fermion dynamics are uncovered. We conclude that, in investigating fermion dynamics, minute changes in the Hamiltonian may lead to dramatic consequences in the behavior of the Helmholtz’ free energy.

DOI
10.1142/s0217984923501841

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The objective of this work is to show that simple modifications in the form of the fermion–fermion interacting potential generate widely different thermodynamic behaviors, with macroscopic manifestations reflected by the thermal free energy. This is done by appealing to quantum statistical mechanics, in its Gibbs’ ensemble formalism, employing two different SU2 exactly solvable models. Some intriguing aspects of the fermion dynamics are uncovered. We conclude that, in investigating fermion dynamics, minute changes in the Hamiltonian may lead to dramatic consequences in the behavior of the Helmholtz’ free energy.
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Abstract Chapter 6 introduces quantum-mechanical ensemble theory by proving the asymptotic equivalence of the quantum-mechanical, microcanonical ensemble average with the quantum grand canonical ensemble average for many-particle systems, based on the method of Darwin and Fowler. The procedures involved identify the grand partition function, entropy and other statistical thermodynamic variables, including the grand potential, Helmholtz free energy, thermodynamic potential, Gibbs free energy, Enthalpy and their relations in accordance with the fundamental laws of thermodynamics. Accompanying saddle-point integrations define temperature (inverse thermal energy) and chemical potential (Fermi energy). The concomitant emergence of quantum statistical mechanics and Bose–Einstein and Fermi–Dirac distribution functions are discussed in detail (including Bose condensation). The magnetic moment is derived from the Helmholtz free energy and is expressed in terms of a one-particle retarded Green’s function with an imaginary time argument related to inverse thermal energy. This is employed in a discussion of diamagnetism and the de Haas-van Alphen effect.
We deal here with the notion of statistical order and apply it to a system of interacting fermions endowed with an SU2 × SU2 symmetry. The discussion takes place in a thermal quantum statistical scenario. Two distinct fermion–fermion interactions are at play. One of them is a well-known spin–flip interaction. The other is the pairing interaction responsible for nuclear superconductivity. We used novel statistical quantifiers that yield insights regarding changes in the statistical order produced when the values of the pertinent coupling constants vary. In particular, we show that judicious manipulation of the energy cost associated with statistical order variations with the fermion number is the key to understanding important details of the associated dynamics.
This work scrutinizes, using statistical mechanics indicators, important traits displayed by quantum many-body systems. Our statistical mechanics quantifiers are employed, in the context of Gibbs’ canonical ensemble at temperature T. A new quantifier of this sort is also presented here. The present discussion focuses attention on the role played by the fermion number N in many-fermion dynamics, that is, N is our protagonist. We have discovered discovers particular values of N for which the thermal indicators exhibit unexpected abrupt variations. Such a fact reflects an unanticipated characteristic of fermionic dynamics.
We employ the quench action method for a recently considered geometrical quantum quench: two free fermionic chains initially at different temperatures are joined together in the middle and let evolve unitarily with a translation invariant Hamiltonian. We show that two different stationary regimes are reached at long times, depending on the interplay between the observation time scale TTT and the total length LLL of the system. We show the emergence of a nonequilibrium steady state supporting an energy current for observation time TTT much smaller than the system size LLL. We then identify a longer time scale for which thermalization occurs in a generalized Gibbs ensemble.
Investigation on foundational aspects of quantum statistical mechanics recently entered a renaissance period due to novel intuitions from quantum information theory and to increasing attention on the dynamical aspects of single quantum systems. In the present contribution a simple but effective theoretical framework is introduced to clarify the connections between a purely mechanical description and the thermodynamic characterization of the equilibrium state of an isolated quantum system. A salient feature of our approach is the very transparent distinction between the statistical aspects and the dynamical aspects in the description of isolated quantum systems. Like in the classical statistical mechanics, the equilibrium distribution of any property is identified on the basis of the time evolution of the considered system. As a consequence equilibrium properties of quantum system appear to depend on the details of the initial state due to the abundance of constants of the motion in the Schrodinger dynamics. On the other hand the study of the probability distributions of some functions, such as the entropy or the equilibrium state of a subsystem, in statistical ensembles of pure states reveals the crucial role of typicality as the bridge between macroscopic thermodynamics and microscopic quantum dynamics. We shall consider two particular ensembles: the random pure state ensemble and the fixed expectation energy ensemble. The relation between the introduced ensembles, the properties of a given isolated system, and the standard quantum statistical description are discussed throughout the presentation. Finally we point out the conditions which should be satisfied by an ensemble in order to get meaningful thermodynamical characterization of an isolated quantum system.