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(2020) Theory of Noninteracting Fermions and Bosons in the Canonical Ensemble

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(2020) Theory of Noninteracting Fermions and Bosons in the Canonical Ensemble PHYSICAL REVIEW RESEARCH 2, 043206 (2020) Theory of noninteracting fermions and bosons in the canonical ensemble Hatem Barghathi ,1,2 Jiangyong Yu,1 and Adrian Del Maestro 3,4,1 1Department of Physics, University of Vermont, Burlington, Vermont 05405, USA 2Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409, USA 3Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA 4Min H. Kao Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA (Received 10 August 2020; accepted 18 October 2020; published 9 November 2020) We present a self-contained theory for the exact calculation of particle number counting statistics of nonin- teracting indistinguishable particles in the canonical ensemble. This general framework introduces the concept of auxiliary partition functions, and represents a unification of previous distinct approaches with many known results appearing as direct consequences of the developed mathematical structure. In addition, we introduce a general decomposition of the correlations between occupation numbers in terms of the occupation numbers of individual energy levels, that is valid for both nondegenerate and degenerate spectra. To demonstrate the applicability of the theory in the presence of degeneracy, we compute energy-level correlations up to fourth order in a bosonic ring in the presence of a magnetic field. DOI: 10.1103/PhysRevResearch.2.043206 I. INTRODUCTION the symmetry resolved entanglement which has recently been studied in a variety of physical systems [26–37]. In more In quantum statistical physics, the analysis of a fixed general settings, the presence of conservation laws could de- number N of indistinguishable particles is difficult, even mand a canonical treatment, as in the case of nuclear statistical in the noninteracting limit. In such a canonical ensemble, models [38–56]. the constraint on the total number of particles gives rise to Given its pedagogical and now practical importance, as correlations between the occupation probability and the corre- well as the long history of the problem of noninteracting sponding occupation numbers of the available energy levels. quantum particles, the last 50 years has provided a host of As a result, the canonical treatment of noninteracting fermions results, varying from general recursive relations that govern and bosons is in general avoided, and it is often completely the canonical partition functions and the corresponding oc- absent in introductory texts addressing quantum statistical cupation numbers [5,57–67], to approximate [4,68–72] and physics [1–3]. The standard approach is to relax the fixed exact results for some special cases [73–81]. More recently, N constraint and instead describe the system using the grand for the case of nondegenerate energy spectra, the exact de- canonical ensemble. While in most cases this approximation composition of higher-order occupation number correlations can be trusted in the large-N limit, it can explicitly fail in in terms of the occupation numbers of individual bosonic and the low-temperature regime [4,5]. The situation is even worse fermionic energy levels has been reported [63,64]. if the low-temperature system under study is mesoscopic, In this paper, we present a unified framework for the calcu- containing a relatively small number of particles. lation of physical observables in the canonical ensemble for N Recent advances in ultracold atom experiments [6–15]are fermions or bosons that is applicable to general noninteracting making such conditions the rule rather than the exception, Hamiltonians that may contain degenerate energy levels. We where the interactions between particles are often ignored, present an analysis of the mathematical structure of bosonic especially in the fermionic case [16]. Thus an accurate sta- and fermionic canonical partition functions in the noninter- tistical representation of these systems should be canonical, acting limit that leads to a set of recursion relations for enforcing the existence of fixed N. In a different context, exactly calculating the energy-level occupation probabilities particle number conservation can reduce the amount of quan- and average occupation numbers. Using argument from linear tum information (entanglement) that can be extracted from a algebra, we show how higher-order correlations between the quantum state [17–25] and thus the fixed N constraint requires occupation numbers can be factorized, allowing them to be a canonical treatment. This is reflected in the calculation of obtained from the knowledge of the occupation numbers of the corresponding energy levels, a canonical generalization of Wick’s theorem [82]. The key observation yielding sim- plification of calculations in the canonical ensemble is that Published by the American Physical Society under the terms of the occupation probabilities and occupation numbers can be ex- Creative Commons Attribution 4.0 International license. Further pressed via auxiliary partition functions (APFs)—canonical distribution of this work must maintain attribution to the author(s) fermionic or bosonic partition functions that correspond to a and the published article’s title, journal citation, and DOI. set of energy levels that are obtained from the full spectrum 2643-1564/2020/2(4)/043206(16) 043206-1 Published by the American Physical Society BARGHATHI, YU, AND DEL MAESTRO PHYSICAL REVIEW RESEARCH 2, 043206 (2020) of the targeted system by making a subset of the energy levels tinguishable particles is defined by degenerate (increasing its degeneracy if it is already degener- ate) or alternatively by excluding it from the spectrum. Results ZN = X (n|N ), (1) obtained via auxiliary partition functions are validated by n|N demonstrating that a number of previously reported formulas can be naturally recovered in a straightforward fashion within where this framework. −β | = ini In a noninteracting system, observables such as the aver- X (n N ) e (2) age energy and magnetization can be obtained solely from i∈S the knowledge of average occupation numbers; however, the are the Boltzmann factors at inverse temperature β = 1/k T calculation of the corresponding statistical fluctuations in such B and the components {n } of the vector n| = (n ,...,n )| quantities, i.e., specific heat and magnetic susceptibility, re- i N 1 M N are the occupation numbers for the corresponding energy lev- quires knowledge of the fluctuations in occupation numbers els satisfying n = N. The summation in Eq. (1) runs and (in the canonical ensemble) correlations between them. i∈S i over all the possible occupation vectors n| which, in addition Therefore, the factorization of correlations between occupa- N to conserving the total number of particles N, obeys occupa- tion numbers in terms of the average occupation numbers tion limits for each of the energy levels : n nmax, thus of individual energy levels provides a simplified approach to i i i N N = nmax. calculate quantities such as specific heat and magnetic sus- max i i Consider the spectrum defined by S as the union of disjoint ceptibility of a given system. Here, we highlight a method to subsets (subspectra) S (1) and S (2) = S \ S(1), i.e., S = S (1) ∪ compute such correlations by considering a degenerate system S (2). Under this decomposition the Boltzmann factors in S can of a finite periodic chain of noninteracting bosons that is (1) (2) (1) be factorized as X (n|N ) = X (n |k )X (n |N−k ), where n |k influenced by an external uniform magnetic field. (2) (1) and n | − represent occupation vectors of k = (1) n The auxiliary partition function approach presented here N k i∈S i and N − k particles in S (1) and S (2), respectively. Summing provides a set of tools that can be used to analyze experi- (1) (2) X (n| ) over all possible n | and n | − gives mental data in low-density atomic gases where the number N k N k of particles is fixed. The application of the physically rele- (1) (2) X (n| ) = Z S Z − S , (3) vant canonical ensemble can eliminate errors introduced via N k N k (1)| (2)| the grand canonical approximation (especially in the inferred n k n N−k temperature) and lead to a more accurate interpretation of where we have introduced the auxiliary partition functions experimental results, including an improved diagnosis of the (APFs) role of weak interaction effects. It is hoped that the relative simplicity of the mathematical approach presented in this pa- Z S (1) = X n(1)| , (4) per may encourage the inclusion of the interesting topic of the k k (1)| canonical treatment of Fermi and Bose gases in college-level nk (2) (2) textbooks. ZN−k S = X n |N−k . (5) (2) In Sec. II we present the general formalism of the theory, n |N−k where we introduce APFs and write occupation probability distributions, occupation numbers, and their correlations in For Eqs. (4) and (5) to satisfy the restrictions imposed by the { max} terms of the APFs. In the same section, we also show how per-energy-level maximum occupancies ni and the fixed , − max , max some of the previously known results can be recovered in a N, k must satisfy max(0 N N2 ) k min(N N1 ), max max straightforward fashion. In Sec. III, considering fermionic and where N1 and N2 are the maximum numbers of parti- S (1) S (2) bosonic systems, we derive the decomposition of higher-order cles allowed in and , respectively. Additionally, any | occupation number correlations into individual energy-level vector n N under these constraints can be decomposed into (1)| (2)| occupation numbers for nondegenerate and degenerate energy two allowed vectors n k and n N−k and vice versa. Thus the sum in Eq. (1) can be similarly decomposed as | ≡ spectra, alike. To illustrate the applicability of the theory, in n N kmax (2) Sec. IV, we consider a bosonic ring in the presence of a (1) (2) where k = max(0, N − N ) and k=kmin n |k n |N−k min max magnetic field.
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