Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:nlin/0512060 · Journal of Physics A Mathematical and General

Relaxation of nonlinear oscillations in BCS superconductivity

Razvan Teodorescu

Published Aug 2, 2006• 2 citations
arXivHTMLPDF

Abstract

The diagonal case of the $sl(2)$ Richardson-Gaudin quantum pairing model \cite{Richardson1,Richardson2,Richardson3,Richardson4,Richardson5,Richardson6,G audin76} is known to be solvable as an Abel-Jacobi inversion problem \cite{SOV,Kuznetzov,Kuz1,Kuz2,Kuz3,Kuz4,Kuz5,YAKE04}. This is an isospectral (stationary) solution to a more general integrable hierarchy, in which the full time evolution can be written as isomonodromic deformations. Physically, the more general solution is appropriate when the single-particle electronic spectrum is subject to external perturbations. The asymptotic behavior of the nonlinear oscillations in the case of elliptic solutions is derived.

DOI
10.1088/0305-4470/39/33/007

Related papers

EXACTLY SOLVABLE PAIRING HAMILTONIANS

J. Dukelsky, B. Errea, Sergio Lerma-Hernández +1 more

We review the development of the microscopic BCS theory of superconductivity and the exact solution of the pairing Hamiltonian given by Richardson in 1963. We then introduce the generalized Richardson-Gaudin exactly-solvable pairing models for SU(2) (pairing between like particles), SO(5) ( T =1 pairing) and SO(8) ( T =0,1 pairing) algebras.

International Journal of Modern Physics E•Feb 1, 2007•2 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:1111.1519•Physical Review B•Dec 14, 2011•19 citations

Exact Solutions for Pairing Interactions

J. Dukelsky, Stuart Pittel

The exact solution of the BCS pairing Hamiltonian was found by Richardson in 1963. While little attention was paid to this exactly solvable model in the remainder of the 20th century, there was a burst of work at the beginning of this century focusing on its applications in different areas of quantum physics. We review the history of this exact solution and discuss recent developments related to the Richardson-Gaudin class of integrable models, focussing on the role of these various models in nuclear physics.

arXiv:1204.2950•WORLD SCIENTIFIC eBooks•Mar 1, 2013•1 citation

Nonperturbative quantum dynamics of the order parameter in the BCS pairing model

Victor M. Galitski

We consider quantum dynamics of the order parameter in the discrete pairing model (Richardson model) in thermodynamic equilibrium. The integrable Richardson Hamiltonian is represented as a direct sum of Hamiltonians acting in different Hilbert spaces of single-particle and paired/empty states. This allows us to factorize the full thermodynamic partition function into a combination of simple terms associated with real spins on singly occupied states and the partition function of the quantum $XY$ model for Anderson pseudospins associated with the paired/empty states. Using coherent-state path integral, we calculate the effects of superconducting phase fluctuations exactly. The contribution of superconducting amplitude fluctuations to the partition function in the broken-symmetry phase is shown to follow from the Bogoliubov-de Gennes equations in imaginary time. These equations in turn allow several interesting mappings, e.g., they are shown to be in a one-to-one correspondence with the one-dimensional Schr\"odinger equation in supersymmetric quantum mechanics. However, the most practically useful approach to calculate functional determinants is found to be via an analytical continuation of the quantum order parameter to real time, $\ensuremath{\Delta}(\ensuremath{\tau}\ensuremath{\rightarrow}it)$, such that the problem maps onto that of a driven two-level system. The contribution of a particular dynamic order parameter, $\ensuremath{\Delta}(\ensuremath{\tau})$, to the partition function is shown to correspond to the sum of the Berry phase and dynamic phase accumulated by the pseudospin. We also examine a family of exact solutions for two-level-system dynamics on a class of elliptic functions and suggest a compact expression to estimate the functional determinants on such trajectories. The possibility of having quantum soliton solutions coexisting with classical BCS mean field is discussed.

arXiv:1003.2237•Physical Review B•Aug 10, 2010•14 citations

Exactly solvable Richardson–Gaudin models and their applications

Stuart Pittel, J. Dukelsky

We first show that the quantum pairing problem can be mapped exactly on to a classical electrostatic problem in two dimensions and then use this analogy to obtain a pictorial representation of how superconductivity arises in a finite fermionic system. Specific application to the nuclei 114–116 Sn suggests some new insight into the evolution of pairing correlations in a quantum system with few active particles. We also summarize other recent work on exactly solvable pairing models, including their applications in a wide variety of strongly correlated quantum systems.

Physica Scripta•Jun 28, 2006•4 citations

Classical r-matrices, “elliptic” BCS and Gaudin-type hamiltonians and spectral problem

T. Skrypnyk

In this article we make a brief review of the generalized Gaudin spin chains with [21] and without [18], [19] external magnetic field in the simplest case of g=so(3) and provide the r-matrix interpretation and generalization of the recent results of [38], [39], [40], on their exact solution. We present the classification theorem for the r-matrices that are diagonal in the natural basis and for the “shift elements” playing the role of the interaction strengths and external magnetic fields in the corresponding Gaudin-type models. We apply these results to the description, classification and solution of BCS-Richardson-type models. In particular, we introduce “elliptic” BCS-Richardson hamiltonian corresponding to non-skew-symmetric elliptic r-matrix and calculate its spectrum. We show that the “closed” [23], [30] and the “open” [36] BCS-Richardson hamiltonian of p+ip-type coincide with its trigonometric degenerations.

Nuclear Physics B•Feb 19, 2019•27 citations
The diagonal case of the sl(2)sl(2)sl(2) Richardson-Gaudin quantum pairing model \cite{Richardson1,Richardson2,Richardson3,Richardson4,Richardson5,Richardson6,G audin76} is known to be solvable as an Abel-Jacobi inversion problem \cite{SOV,Kuznetzov,Kuz1,Kuz2,Kuz3,Kuz4,Kuz5,YAKE04}. This is an isospectral (stationary) solution to a more general integrable hierarchy, in which the full time evolution can be written as isomonodromic deformations. Physically, the more general solution is appropriate when the single-particle electronic spectrum is subject to external perturbations. The asymptotic behavior of the nonlinear oscillations in the case of elliptic solutions is derived.
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).
The exact solution of the BCS pairing Hamiltonian was found by Richardson in 1963. While little attention was paid to this exactly solvable model in the remainder of the 20th century, there was a burst of work at the beginning of this century focusing on its applications in different areas of quantum physics. We review the history of this exact solution and discuss recent developments related to the Richardson-Gaudin class of integrable models, focussing on the role of these various models in nuclear physics.
We consider quantum dynamics of the order parameter in the discrete pairing model (Richardson model) in thermodynamic equilibrium. The integrable Richardson Hamiltonian is represented as a direct sum of Hamiltonians acting in different Hilbert spaces of single-particle and paired/empty states. This allows us to factorize the full thermodynamic partition function into a combination of simple terms associated with real spins on singly occupied states and the partition function of the quantum XYXYXY model for Anderson pseudospins associated with the paired/empty states. Using coherent-state path integral, we calculate the effects of superconducting phase fluctuations exactly. The contribution of superconducting amplitude fluctuations to the partition function in the broken-symmetry phase is shown to follow from the Bogoliubov-de Gennes equations in imaginary time. These equations in turn allow several interesting mappings, e.g., they are shown to be in a one-to-one correspondence with the one-dimensional Schr\"odinger equation in supersymmetric quantum mechanics. However, the most practically useful approach to calculate functional determinants is found to be via an analytical continuation of the quantum order parameter to real time, $\ensuremath{\Delta}(\ensuremath{\tau}\ensuremath{\rightarrow}it)$, such that the problem maps onto that of a driven two-level system. The contribution of a particular dynamic order parameter, $\ensuremath{\Delta}(\ensuremath{\tau})$, to the partition function is shown to correspond to the sum of the Berry phase and dynamic phase accumulated by the pseudospin. We also examine a family of exact solutions for two-level-system dynamics on a class of elliptic functions and suggest a compact expression to estimate the functional determinants on such trajectories. The possibility of having quantum soliton solutions coexisting with classical BCS mean field is discussed.
We first show that the quantum pairing problem can be mapped exactly on to a classical electrostatic problem in two dimensions and then use this analogy to obtain a pictorial representation of how superconductivity arises in a finite fermionic system. Specific application to the nuclei 114–116 Sn suggests some new insight into the evolution of pairing correlations in a quantum system with few active particles. We also summarize other recent work on exactly solvable pairing models, including their applications in a wide variety of strongly correlated quantum systems.
In this article we make a brief review of the generalized Gaudin spin chains with [21] and without [18], [19] external magnetic field in the simplest case of g=so(3) and provide the r-matrix interpretation and generalization of the recent results of [38], [39], [40], on their exact solution. We present the classification theorem for the r-matrices that are diagonal in the natural basis and for the “shift elements” playing the role of the interaction strengths and external magnetic fields in the corresponding Gaudin-type models. We apply these results to the description, classification and solution of BCS-Richardson-type models. In particular, we introduce “elliptic” BCS-Richardson hamiltonian corresponding to non-skew-symmetric elliptic r-matrix and calculate its spectrum. We show that the “closed” [23], [30] and the “open” [36] BCS-Richardson hamiltonian of p+ip-type coincide with its trigonometric degenerations.