Selberg Integrals and Particle–Hole Duality in the Richardson–Gaudin Pairing Model

Abstract

Authorship, provenance, and verification. The scientific problem, original method, and research direction belong to W. V. Pogosov. This project builds on his published and unpublished work developed before the involvement of an advanced large language model (LLM). During the subsequent LLM-assisted phase, the LLM produced detailed derivations, calculations, and verification programs through an extended collaboration with the author. The author has not exhaustively verified every calculation and proof. The research developed through many iterations and branching lines of enquiry over a substantial period of time. The author posed and refined questions, assessed proposed directions, and guided the revision, rejection, and combination of intermediate approaches; the LLM supplied calculations and arguments that prompted further questions and revisions. The resulting manuscript is a record of this reciprocal, sustained research process. Selberg Integrals and Particle–Hole Duality in the Richardson–Gaudin Pairing Model. Subtitle: Exact Periods, Self-Dual States, and Finite-Size Crossover. Version 1.3, the fourth prepared edition, combines two lines of analysis of the reduced pairing Hamiltonian: exact Selberg-type contour periods and a direct particle–hole/self-dual polynomial construction. The title has been updated to reflect this broader scope; the preceding twenty-two chapters are retained. The method originated in W. V. Pogosov, “'Probabilistic' approach to Richardson equations,” Journal of Physics: Condensed Matter 24, 075701 (2012), doi:10.1088/0953-8984/24/7/075701 (https://doi.org/10.1088/0953-8984/24/7/075701), arXiv:1111.2907 (https://arxiv.org/abs/1111.2907). That work treated the ground state and obtained the required coupling dependence without a full direct evaluation of the determinant normalization. Its Appendix A explicitly derives the electron–hole rewriting of the Hamiltonian and acknowledges earlier observations of spectral duality. This is a starting point in the genealogy of the present project, not a claim that particle–hole symmetry itself was first recognized in 2012. The subsequent paper, “Excited states in Richardson pairing model: 'probabilistic' approach,” Progress of Theoretical Physics 128, 1–14 (2012), doi:10.1143/PTP.128.1 (https://doi.org/10.1143/PTP.128.1), arXiv:1208.1088 (https://arxiv.org/abs/1208.1088), extended the approach to excited states, again leaving a common determinant normalization unevaluated directly. The author's 2014 doctoral dissertation, Superconductors and dilute superfluid Bose systems: From the microscopic to the macroscopic scale (in Russian), developed direct determinant evaluations by successive row differences and triangular reduction. See the Lebedev Physical Institute dissertation-council record (https://www.lebedev.ru/ru/fian-dissertation-councils/posts.html?id=188) and the complete thesis on ResearchGate (https://www.researchgate.net/publication/271840408_Sverhprovodniki_i_razrezennye_sverhtekucie_boze-sistemy_ot_mikro-_k_makrourovnu) (doi:10.13140/2.1.1066.5766 (https://doi.org/10.13140/2.1.1066.5766)). From the perspective of the broader program pursued here, these earlier results constituted a partial solution. The present project attempts to carry that program through; completion is not claimed. The author's two 2012 papers, 2014 dissertation, the joint paper on particle–hole duality published in 2015, and his 2015 analysis of BCS applicability all belong to the pre-LLM research programme. Further author-developed results and working materials remained unpublished. Only after this published and unpublished corpus had accumulated was an advanced LLM introduced to the project; none of that earlier research is retrospectively attributed to the LLM. The duality line develops the direct rapidity and polynomial route proposed by L. V. Bork and W. V. Pogosov, “Particle–hole duality, integrability, and Russian doll BCS model,” Nuclear Physics B 897, 405–420 (2015), doi:10.1016/j.nuclphysb.2015.05.031 (https://doi.org/10.1016/j.nuclphysb.2015.05.031), arXiv:1412.1016 (https://arxiv.org/abs/1412.1016). Their self-dual/mutually dual classification is distinguished from the eigenvalue-based line already developed before 2015: A. Faribault and collaborators, Physical Review B 83, 235124 (2011), doi:10.1103/PhysRevB.83.235124 (https://doi.org/10.1103/PhysRevB.83.235124), and A. Faribault and D. Schuricht, Journal of Physics A 45, 485202 (2012), doi:10.1088/1751-8113/45/48/485202 (https://doi.org/10.1088/1751-8113/45/48/485202). No literature-wide claim that the alternative equations had never been studied is made. Root reconstruction, dual Bethe representations, and determinant formulas for overlaps are related tools with different purposes. The distinction between BCS validity criteria for condensation energy and excitation gaps also belongs to the earlier physical programme. W. V. Pogosov, “Applicability of Bardeen–Cooper–Schrieffer theory to small-sized superconductors: Role of Cooper-pair binding energy,” Solid State Communications 207, 1–4 (2015), doi:10.1016/j.ssc.2014.12.025 (https://doi.org/10.1016/j.ssc.2014.12.025), arXiv:1412.1015 (https://arxiv.org/abs/1412.1015), emphasized the role of the Cooper-pair binding scale. Antecedents include M. Schechter, Y. Imry, Y. Levinson and J. von Delft, “Thermodynamic properties of a small superconducting grain,” Physical Review B 63, 214518 (2001), doi:10.1103/PhysRevB.63.214518 (https://doi.org/10.1103/PhysRevB.63.214518), and E. A. Yuzbashyan, A. A. Baytin and B. L. Altshuler, “Finite-size corrections for the pairing Hamiltonian,” Physical Review B 71, 094505 (2005), doi:10.1103/PhysRevB.71.094505 (https://doi.org/10.1103/PhysRevB.71.094505). The present self-dual route supplies finite identities, bounds and selected crossover calculations for this established physical question; it does not present the distinction of the two criteria as a newly discovered effect. New Chapter 23 develops finite self-dual polynomial identities, their real nonlocal boundary representation, energy and correlation-energy formulas, and an effective spin representation of the self-dual sector at symmetric half filling. The resulting ground-state route and BCS limits do not assume a continuum arc of Richardson roots. The chapter includes uniqueness of the analytic ground polynomial, convergence of a monotone ground-state Newton scheme for fixed finite parameters, and explicit two-sided energy bounds yielding the leading BCS energy at fixed coupling. These statements do not claim a uniform crossover solution or the exact next asymptotic coefficient. The effective Hamiltonian is related explicitly to the integrable spin/boson model of J. Links, Nuclear Physics B 916 (2017), doi:10.1016/j.nuclphysb.2017.01.005 (https://doi.org/10.1016/j.nuclphysb.2017.01.005); the Hamiltonian itself is not claimed as a new model. New Chapter 24 studies blocked defects, coalescence of the central levels, the first gap within the self-dual sector, and the different BCS validity scales for condensation energy and excitation gaps. The reference Fermi-sea energy includes the diagonal pairing interaction. Controlled finite-system bounds and joint weak-coupling/large-size regimes are distinguished from observations and from unresolved questions. In particular, equality of the first self-dual gap with the minimum of the complete pair-preserving spectrum is not asserted as a theorem for all sizes and couplings. New Chapter 25 separates the single positive rapidity of the selected first self-dual excitation from the rest of the system. At finite size, the selected physical branch obeys a real nonlinear response equation, one scalar root condition, and an exact energy formula involving the response integral. Uniqueness of the response at a fixed root does not prove global uniqueness of the scalar root. The accuracy of a linear response followed by one Newton correction is a numerical observation in the tested regime. A complete universal excited-state crossover limit and uniform one-correction accuracy remain open. Cancellation of the explicit inverse-size term in certain digamma sums is not a convergence-rate theorem for the full excitation gap. The retained period line includes the Case A Pfaffian construction, the completely normalized confluent Jack hypergeometric representation at arbitrary even auxiliary beta, a finite-system ground-state limit as the auxiliary multiplier p tends to infinity, and the formal all-order cancellation of higher extensive terms in the beta = 2 one-pair saddle expansion. These results keep their original distinctions between exact periods, formal series, and physical eigenenergies. The new self-dual route does not automatically settle the remaining contour-selection, Gaussian-remainder, or finite-defect bridges in that line. General-beta Case B and real nonzero-temperature thermodynamics are outside this update. The deposit contains twenty-five chapters and fourteen appendices, LaTeX sources, bibliography, scientific programs, retained historical reports, the unmodified crossover checkpoint v0.1, and the recovered selfdual_iteration5 dependency used by the earlier branch-ordering arguments. A separate version 1.3 audit verifies the checkpoint manifests and repeats two finite symbolic tests, seven polynomial-root comparisons, thirty digamma checks, fifty-five small diagonalization cases, twelve independently assembled finite-spectrum comparisons, five crossover points, and one scalar-response regime at two Fourier grid sizes. The exact test scope, environment, residuals, and commands are recorded in revision/DUALITY_NUMERICS_v1_3.md. Full historical scans are not presented as freshly rerun. Independence describes separate implementations within the LLM-assisted workflow, not independent human verification. Finite tests do not prove the remaining analytical assumptions.