Odd-even statistical staggering in a Lipkin-like few fermions model has been recently encountered. Of course, staggering in nuclear binding energies is a well established fact. Similar effects are detected in other finite fermion systems as well, as for example, ultra small metallic grains and metal clusters. We work in this effort with the above-mentioned Lipkin-like, two-level fermion model and show that statistical staggering effects can be detailedly explained by recourse to a straightforward analysis of the associated energy-spectra.Instituto de Física La Plat

Ferri, Gustavo L.

Plastino, Ángel Luis

Plastino, Ángel Ricardo

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quantum reportsArticleSpectral Explanation for Statistical Odd-Even Staggering in FewFermions SystemsAngelo Plastino 1,* , Gustavo Luis Ferri 2 and Angel Ricardo Plastino 3Citation: Plastino, A., Ferri, G.L.;Plastino, A.R. Spectral Explanationfor Statistical Odd-Even Staggering inFew Fermions Systems. Quantum Rep.2021, 3, 166–172. https://doi.org/10.3390/quantum3010010Academic Editor: Michel PlanatReceived: 20 January 2021Accepted: 10 February 2021Published: 16 February 2021Publisher’s Note: MDPI stays neutralwith regard to jurisdictional claims inpublished maps and institutional affil-iations.Copyright: © 2021 by the authors.Licensee MDPI, Basel, Switzerland.This article is an open access articledistributed under the terms andconditions of the Creative CommonsAttribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).1 Instituto de Física La Plata–CCT-CONICET, Universidad Nacional de La Plata, C.C. 727,La Plata 1900, Argentina2 Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de La Pampa,Santa Rosa, La Pampa 6300, Argentina; gustavoluisferri@gmail.com3 CeBio y Departamento de Ciencias Basicas, Universidad Nacional del Noroeste de la Prov. de Buenos Aires,UNNOBA, CONICET, Roque Saenz Pena 456, Junin 6000, Argentina; arplastino@unnoba.edu.ar* Correspondence: plastino@fisica.unlp.edu.arAbstract: Odd-even statistical staggering in a Lipkin-like few fermions model has been recentlyencountered. Of course, staggering in nuclear binding energies is a well established fact. Similareffects are detected in other finite fermion systems as well, as for example, ultra small metallicgrains and metal clusters. We work in this effort with the above-mentioned Lipkin-like, two-levelfermion model and show that statistical staggering effects can be detailedly explained by recourse toa straightforward analysis of the associated energy-spectra.Keywords: statistical complexity; exactly solvable models; even-odd effects1. IntroductionA well known fermion phenomenon is odd-even statistical staggering. It is foundin few (not necessarily nuclear) fermion systems. The effect is empirically well known innuclear physics and metal clusters ([1–3], and references therein). The associated odd-evendifferences in nuclear masses are also influenced by mean-field and odd-nucleon blockingeffects [4,5].We will deal below with an exactly solvable (interacting) fermions-model of the Lipkinkind that does not appeal to pairing interactions. This is relevant as paring forces arebelieved to be responsible for the effect in nuclei [3]. It was shown in [6] that odd-even dif-ferences are an intrinsic consequence of the dynamics of interacting fermions. The ensuingodd-even differences were incorporated [6] into an order–disorder environment that has asa protagonist the so-called statistical complexity concept [7], where “order” is produced bythe fermion–fermion interaction while disorder is generated by the temperature T.The above-mentioned order–disorder interplay was described in [6] via Gibbs’ canonicalensemble considerations. In them, the concomitant probability distribution is proportional toexp (−βĤ), with Ĥ standing for the Hamiltonian and β for the inverse temperature.We will see below that the staggering effect is translated from the energy (in the nuclearinstance) to thermal quantifiers like the statistical complexity or the entropy. Interestinglyenough, it manifests itself also in the variation of the mean energy with the temperatures,i.e., the specific heat.More importantly, we will explicitly demonstrate that odd-even differences are entirelyattributable to the characteristics of the associated energy-spectra of either odd or evensystems.2. The ModelThe Lipkin Model (LM) [8] was very useful in research that revolved around thevalidity and/or usefulness of several theoretical techniques devised for investigating theQuantum Rep. 2021, 3, 166–172. https://doi.org/10.3390/quantum3010010 https://www.mdpi.com/journal/quantumrepQuantum Rep. 2021, 3 167multiple facets of the fermion many-body problem. The LM is based on an SU(2) algebraand produces easily accessible exact solutions.We will occupy ourselves here with a simplified LM-version proposed in Reference [9].The models of [8,9] deal with of N fermions distributed between (2N)-fold degeneratesingle-particle (sp) levels separated by an sp energy gap ε. Two quantum numbers (µ andp) are assigned to a general single sp. The first takes the values µ = −1 (lower level) andµ = +1 (upper level). The p−, often denominated quasi-spin or pseudo spin, picks outa specific belonging to the N-fold degeneracy. The couple p, µ may be viewed as a “site”that is occupied or empty. Double site-occupancy is forbidden. We haveN = 2J, (1)where J stands for an “angular momentum”. In the wake of Lipkin et al. [8] we define thequasi-spin operatorsĴ+ = ∑pC†p,+Cp,−, (2)Ĵ− = ∑pC†p,−Cp,+, (3)Ĵz = ∑p,µµ C†p,µCp,µ, (4)Ĵ2 = Ĵ2z +12( Ĵ+ Ĵ− + Ĵ− Ĵ+), (5)where the eigenvalues of Ĵ2 are of the form J(J + 1).The model’s Hamiltonian of Reference [9] isĤ = ε Ĵz −Vs(12( Ĵ+ Ĵ− + Ĵ− Ĵ+)− Ĵ). (6)Simplifyng with either V = Vs/ε or ε = 1, we writeĤ = Ĵz −V(12( Ĵ+ Ĵ− + Ĵ− Ĵ+)− Ĵ), (7)and the unperturbed ground state (gs) for V = 0 is, on account of Equation (1),|J, Jz〉 = |J,−N/2〉, (8)endowed with energyEo = −N/2. (9)The interaction between two fermions here is of a very simple nature. We have justforward scattering [9]. In interacting, the states of the two incoming fermions are identicalto those of the two outgoing ones [9].An important fact, we reiterate, is that doubly occupied p−sites are not allowed for.The Hamiltonian commutes with the pair of operators Ĵ2 and Ĵz.Accordingly, the exact solution must be encountered within the J-multiplet of theunperturbed ground state. The states of such multiplet will be denoted as |J, M〉. One ofthem has then to minimize the energy. The ensuing M value should depend on the strengthV of the interaction.A remarkable trait of our model [9] is that, as V increases from zero, Eo is not at oncemodified. It maintains its value till a critical V−specific figure is reached, of 1/(N − 1). Atthis juncture, the interacting ground state abruptly becomes |J,−N/2 + 1〉. If V continuesaugmenting, additional “crossing” transitions (ct) take place (the Jz-value characterizing theground state changes). The ct between Jz = −k and Jz = −k+ 1 happens at V = 1/(2k− 1).Quantum Rep. 2021, 3 168This ct-series ends as the interacting ground state becomes either Jz = 0 (Vcrit = 1 forinteger J), or Jz = −1/2 (Vcrit = 1/2 for odd J). In such circumstances one has, regardlessof the value J [9]Vcrit = 1/2, (10)for half-integer J andVcrit = 1, (11)for integer J.Our Model at Finite TemperaturesOne needs to compare model-results for distinct J values, which is facilitated becausedouble occupancy of a p-site is strictly forbidden. Accordingly, the Hamiltonian matrixshould be the (2J + 1)× (2J + 1) one of the Jz = −N/2 multiplet, with N = 2J [8].One has for the free energy F(J), in terms of the partition function Z(J),F = −T ln Z = −T ln Tr(exp (−βĤ)), (12)where, hereafter, we set the Boltzmann constant equal to unity. For each different J thetrace operation is a sum over the Jz quantum number m andZ(J) =m=J∑m=−Jexp (−βEJm), (13)with an energy EJm [9]EJm = m−V[J(J + 1)−m2 − J]. (14)The pertinent Gibbs’ probabilities PJm then become [9]PJm =exp (−βEJm)Z(J), (15)for all m = −J,−J + 1, . . . , J − 1, J. Thus, the Boltzmann–Gibbs S entropy is ([10] pp. 202–211)S(J) = −m=J∑m=−JPJm ln PJm, (16)As for the number of micro-states m one has, of course,O(J) = 2J + 1, (17)so that the pertinent uniform probabilities (the same for any m) should readP(uJ) = 1/O(J). (18)3. LMC Statistical ComplexityThis notion is of the essence here [7,11–24]. That of complexity is a pervasive notionin modern science. Any complex system is customarily linked with a mixture of order anddisorder as well as to emergent phenomena. However, a universal and precise definitionof complexity is still missing. There are different ways to calculate it, like the so-calledalgorithmic complexity, introduced by Kolmogorov so as to avoid the probabilistic inter-pretation of Shannon. Nevertheless, a successful and much utilized expression for it isthat advanced by L. Ruiz, Mancini, and Calbet (LMC) [7] that we will employ in this work.Mathematically, it is the product of the ordinary entropy S times a distance in probabilityspace between the current probability distribution and the uniform one. This distance isQuantum Rep. 2021, 3 169called the disequilibrium D. D is a measure of “order” as the larger it becomes, the largerthe quantity of “privileged” states in our sample state’s space (our J multiplet here). It isdefined as [7]D(J) =m=J∑m=−J[PJm − P(uJ)]2. (19)D tells us about the degree of “order” in our system. This order grows with D. Tolearn about more details, features, and applications of the disequilibrium the reader isdirected to References [23,24]. We reiterate that LMC defines the statistical complexity C as[7],C = S D. (20)In Reference [6], it was discovered that the three quantifiers (S, D, and C) display theodd-even effect. An example is displayed below in Figure 1, where it is clearly appreciated.N3 4 5 6 7 8 900.20.40.60.8T0=1;  V=10SDCFigure 1. Statistical quantifiers (entropy S, disequilibrium D, and complexity C versus fermion-number N) at temperature = 1 Kelvin and coupling constant V = 10. The even-odd staggering effectis plainly visible.4. Results4.1. A Staggering ExampleAs just promised, Figure 1 exhibits our odd-even effect. Statistical quantifiers (entropyS, disequilibrium D, and complexity C) are plotted versus the fermion-number N at atemperature = 1 Kelvin, with the coupling constant V = 10. The even-odd staggeringeffect is plainly visible.4.2. Model’s Level-SpectraWe pass now to illustrate the behavior of our model’s low-lying energy levels. Fordidactic purposes, we shift the energy scale so that the ground-state (GS) energy alwaysequals zero. The temperature will always equal 1 Kelvin in our graphs.Consider now a case in which the number N of fermions is even, for example N = 6in Figure 2. We plainly see that the energy of the GS and that of the first excited state (1es)always lie close together as the coupling constant (cc) V augments.Quantum Rep. 2021, 3 170Contrary wise, look at a case with an odd fermion number N = 7 in Figure 3. Thingslook quite different now, since only the GS separates itself from the remaining low ly-ing states.The same even-odd GS-1es pattern reported above is repeated for the pairs N = k, k+ 1in the next four graphs, for k = 10, 16.V0 0.5 1 1.5 2 2.5 3Energy Levels05101520N=6GSM=-5/2M=-3/2M=-1/2M=1/2M=3/2M=5/2Figure 2. Level energies for the ground state (GS) and the first five excited states versus the couplingconstant V of our fermion model. The number of fermions is N = 6. Energies are re-normalizedso that the one for the ground state always equals zero. We note that the energy of the first excited(1ex) state (red line) is, for almost every V−value, very close to that of the GS. This fact results in twoBoltzmann exponential factors of quite similar nature, i.e., similar probabilities. The remaining fourexcites states’ energies (second to fifth) grow very rapidly with V and this distances them more andmore from the almost constant energy pair [GS, 1ex]. This makes the respective probabilities of thesefour excited states grow progressively smaller. Probabilistically then, our scenario is dominated bythe pair [GS, 1ex].V0 0.5 1 1.5 2 2.5 3Energy Levels-25-20-15-10-505M=-3M=-2M=-1M=0M=1M=2M=3N=7Figure 3. Model-energy levels versus coupling constant V for N = 7 fermions. Energies are re-normalized so that the one for the ground state equals zero always. Characterization of levels is likein Figure 2. It is clearly seen that, as V grows, the energetic distance between the ground state and allthe excited states steadily augments.Quantum Rep. 2021, 3 1714.3. ExplanationIt is clear that the level-probabilities behave in the following manner• For odd N, the largest probability in the partition function is by far that associatedto the GS, E0. Let us call it P0, the energy of which is much lower than that of all theremaining states of energy Ei >> E0. Thus, for all excited i, Pi << P0.• For even N, two different probabilities (those for the GS and for the 1st excited state)become appreciably larger than the ones for the rest of the energy levels. Thus, theentropy augments, as compared to that of the N-odd instance, and D consequentlydiminishes in the same fashion.• Statistical odd-even staggering arises then because the entropy, for a given even N, islarger than that for its two neighbors N − 1 and N + 1.• Odd-even staggering emerges thus as a spectral feature. In other words, it is aconsequence of the Hamiltonian’s nature. This nature is that of forward-scatteringbetween the model’s fermions [9]. Kramer’s degeneracy is probably at play [25]. Thisis our main result.5. ConclusionsWe have shown that statistical odd-even staggering can be understood on the basis ofthe behavior of the low lying energy levels of our finite fermions model.In other words, we showed that odd-even staggering (in our fermion model) is justthe necessary consequence of spectral differences between odd or even systems.These differences are then attributable to the pertinent Hamiltonian that describesforward scattering. Given the simplicity of the fermion–fermion interaction, our findingssuggest that the observed odd-even differences are intrinsic, essential features of thefermionic nature.Author Contributions: Conceptualization, the three authors; methodology, the three authors; thethree authors; validation, the three authors; formal analysis, the three authors; investigation, thethree authors; resources, the three authors; data curation, the three authors; writing—original draftpreparation, the three authors; writing—review and editing, the three authors; visualization, the threeauthors; supervision, the three authors; project administration, the three authors; funding acquisition,the three authors. All authors have read and agreed to the published version of the manuscript.Funding: Grant from Conicet, Argentina.Data Availability Statement: Available on request.Conflicts of Interest: The authors declare no conflict of interest.References1. Satula‚ W.; Dobaczewski, J.; Nazarewicz, W. 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Spectral Explanation for Statistical Odd-Even Staggering in Few Fermions Systems

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Spectral Explanation for Statistical Odd-Even Staggering in Few Fermions Systems

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