arXiv:1208.1088 · Progress of Theoretical Physics

Excited States in Richardson Pairing Model: `Probabilistic' Approach

W. V. Pogosov

Published Jul 1, 2012• 2 citations

Abstract

Richardson equations can be mapped on the classical electrostatic problem in two dimensions. We have recently suggested a new analytical approach to these equations in the thermodynamical limit, which is based on the `probability' of the system of charges to be in a given configuration at the effective temperature equal to the interaction constant. In this paper, we apply this approach to excited states of the Richardson pairing model. We focus on the equally-spaced situation and address arbitrary fillings of the energy layer, where interaction acts. The `partition function' for the classical problem on the plane, which is given by the Selberg-type integral, is evaluated exactly. Three regimes for the energy gap are identified, which can be treated as the dilute regime of pairs, BCS regime, and dilute regime of holes.

DOI: 10.1143/ptp.128.1

Related papers

‘Probabilistic’ approach to Richardson equations

W. V. Pogosov

It is known that solutions of Richardson equations can be represented as stationary points of the 'energy' of classical free charges on the plane. We suggest considering the 'probabilities' of the system of charges occupying certain states in the configurational space at the effective temperature given by the interaction constant, which goes to zero in the thermodynamical limit. It is quite remarkable that the expression of 'probability' has similarities with the square of the Laughlin wavefunction. Next, we introduce the 'partition function', from which the ground state energy of the initial quantum-mechanical system can be determined. The 'partition function' is given by a multidimensional integral, which is similar to the Selberg integrals appearing in conformal field theory and random-matrix models. As a first application of this approach, we consider a system with the constant density of energy states at arbitrary filling of the energy interval where potential acts. In this case, the 'partition function' is rather easily evaluated using properties of the Vandermonde matrix. Our approach thus yields a quite simple and short way to find the ground state energy, which is shown to be described by a single expression all over from the dilute to the dense regime of pairs. It also provides additional insight into the physics of Cooper-paired states.

arXiv:1111.2907Journal of Physics Condensed MatterFeb 2, 201219 citations

A Microcanonical Comparison of Richardsons Equations With The BCS Mean Field Approximation

David From

The exact solution of the reduced pairing Hamiltonian is compared to the BCS mean field approximation for low particle numbers in finite systems. A combination of two different methods for solving the Richardson equations is presented. One of the methods originally intended to be solved analytically is instead adapted to a numerical scheme. The comparison includes the pairing gap parameter, as approximated by the three-point formula in the exact solution, and a microcanonical investigation of systems with levels of varying degeneracies. The gaps as calculated within the two approaches is found to deviate more and more from each other with increasing pairing strength. However testing the three-point-formula with the BCS approximation suggests that this might be due to the formula itself. Furthermore the BCS approximation, when re-evaluating the gap for each excitation, does a very good job at reproducing the energy spectra as obtained for the Richardson solution of the highly excited states where the pairing effects are small. However for the low lying excitations the assessment is not so good. When keeping the gap constant for all excitations the excitation spectra of the BCS approximation broaden a lot more than that of the exact method. This procedure gives unreasonable results for the systems and low particle numbers considered.

Jan 1, 20160 citations

Electrostatic analogy for integrable pairing force Hamiltonians

Luigi Amico A, Antonio Di Lorenzo, Andrea Mastellone +2 more

For the exactly solved reduced BCS model an electrostatic analogy exists; in particular it served to obtain the exact thermodynamic limit of the model from the Richardson Bethe ansatz equations. We present an electrostatic analogy for a wider class of integrable Hamiltonians with pairing force interactions. We apply it to obtain the exact thermodynamic limit of this class of models. To verify the analytical results, we compare them with numerical solutions of the Bethe ansatz equations for finite systems at half-filling for the ground state.

Jan 1, 200219 citations

Exact expectation values within Richardson's approach for the pairing Hamiltonian in a macroscopic system

G. Gorohovsky, Eldad Bettelheim

BCS superconductivity is explained by a simple Hamiltonian describing an attractive pairing interaction between pairs of electrons. The Hamiltonian may be treated using a mean-field method, which is adequate to study equilibrium properties and a variety of nonequilibrium effects. Nevertheless, in certain nonequilibrium situations, even in a macroscopic rather than a microscopic superconductor, the application of mean-field theory may not be valid. In such cases, one may resort to the full solution of the Hamiltonian, as given by Richardson in the 1960s. The relevance of Richardson's solution to macroscopic nonequilibrium superconductors was pointed out recently based on the existence of quantum instabilities out of equilibrium. It is then of interest to obtain analytical expressions for expectation values between exact eigenvalues of the pairing Hamiltonian within the Richardson approach for macroscopic systems. We undertake this task in the current paper. It should be noted that Richardson's approach yields the full set of eigenvalues of the Hamiltonian, while BCS theory yields only a subset. The results obtained here, then, generalize the familiar BCS expressions (e.g., for the electron occupation or pairing correlations) to cases where the spectrum of excitations diverges from BCS theory (e.g., in cases where the spectrum exhibits multiple gaps).

arXiv:1111.1519Physical Review BDec 14, 201119 citations

Improvement to the Projected BCS Approximation

Stuart Pittel, J. Dukelsky, Carlos Esebbag

We consider the structure of the number projected BCS wave function in the particle-hole basis, and use it to study several approximate treatments of pairing. The analysis is carried out for the exactly solvable Richardson model involving a pure pairing hamiltonian acting in a space of equally spaced doubly degenerate levels at half filling.

Journal of Physics Conference SeriesSep 14, 20150 citations

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

W. V. Pogosov

Authorship and verification statement. The scientific problem, the original methodological direction, and the research objectives were formulated by W. V. Pogosov. The analytical derivations and numerical computations in this project were carried out by an advanced large language model (LLM), through an iterative interaction with the author. The author has not exhaustively checked all calculations and proofs. This is a working manuscript with explicit unresolved assumptions, supplied for inspection, reproduction, and further development. The manuscript develops a Selberg–residue approach to the equally spaced Richardson–Gaudin pairing model. Meromorphic contour periods are reduced to finite determinant expressions. Complementary kernels and sign-corrected minor expansions organize ground-state, marked, blocked, and nested-contour sectors. The text studies their thermodynamic asymptotics and candidate finite-size energy corrections while distinguishing exact period identities from physical eigenenergy interpretations. 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. Within this line of work, it also first formulated explicitly the Hamiltonian origin of electron–hole symmetry in the exact Richardson analysis; its Appendix A acknowledges earlier observations of spectral duality. 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. Version 1.0 incorporates a repair of the confluent lattice-minor argument, using a saddle-localized continuation and an exponentially small error controlled on integer indices. Important open inputs remain: contour selection, uniform Gaussian energy control, the proposed Hessian-to-spectral-shift identification, and finite-defect comparisons. The correspondence between the period endpoint and the physical real-root density is also conjectural. The deposit contains the full manuscript, LaTeX sources, bibliography, fifteen supporting scientific programs, fourteen labelled historical reports, and a current separately implemented lattice-check report. The latter records 96 small exact-arithmetic comparisons and 24 larger minor and derivative checks. Independence here concerns implementations within the LLM-assisted workflow, not independent human verification. The earlier program collection has not been rerun in full for this release, and finite tests do not prove the unresolved analytical assumptions.

Sep 9, 20260 citations