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

Progress of Theoretical Physics

The Theory of Superconductivity in the Tamm-Dancoff Approximation

B. P. Nigam

Published Mar 1, 1961• 0 citations
Publisher

Abstract

The Tamm-Dancoff approximation is applied to the interaction Hamiltonian used by Bogoliubov in his theory of superconductivity. It is shown that (i) the requirement that there is no virtual production of a single pair of electron and hole without phonons, or (ii) the electron-phonon coupling parameter tends to zero, leads to essentially the same compensating equation as obtained by Bogoliubov.

DOI
10.1143/ptp.25.436

Related papers

On the “new Tamm-Dancoff” approximation treatment of the superconductivity Hamiltonian

Kazuo Yamazaki

Nuclear Physics•Feb 1, 1961•6 citations

The Tamm-Dancoff Approximation as the boson limit of the Richardson-Gaudin equations for pairing

Stijn De Baerdemacker

A connection is made between the exact eigen states of the BCS Hamiltonian and the predictions made by the Tamm-Dancoff Approximation. This connection is made by means of a parametrised algebra, which gives the exact quasi-spin algebra in one limit of the parameter and the Heisenberg-Weyl algebra in the other. Using this algebra to construct the Bethe Ansatz solution of the BCS Hamiltonian, we obtain parametrised Richardson-Gaudin equations, leading to the secular equation of the Tamm-Dancoff Approximation in the bosonic limit. An example is discussed in depth.

arXiv:1011.3691•Journal of Physics Conference Series•Mar 1, 2011•4 citations

Applications of the Pairing Theory

J. R. Schrieffer

This chapter reviews the calculation of a number of system properties within the pairing approximation. Since the BCS theory was originally proposed, attempts to justify the pairing correlations basic to the theory have proceeded along two lines. The first approach has been to apply the BCS theory to a wide variety of phenomena in superconductors and check the theoretical predictions of the pairing approximation against experiment. The second approach has been to treat by various approximate methods the residual interactions neglected within the pairing scheme, hoping to show that these residual interactions introduce no major change in the predicted properties of the system. The interaction Hamiltonian, being a one-body operator, can at most change the state of one electron. While the majority of the electron–phonon interaction has been accounted for in forming the wave functions for the normal and superconducting states, there remains the part corresponding to resonant phonon absorption and emission processes.

CRC Press eBooks•Mar 5, 2018•0 citations

THE TAMM-DANCOFF APPROXIMATION AS THE CONTRACTION LIMIT OF THE RICHARDSON-GAUDIN EQUATIONS

Stijn De Baerdemacker

A connection is made between the exact eigen states of the level-independent Bardeen-Cooper-Schrieffer (BCS) Hamiltonian and its Tamm-Dancoff Approximation (TDA). This is done by means of a deformation of the quasi-spin algebra, which connects the Bethe Ansatz states with a unique multi-phonon mode of the TDA. The procedure is illustrated with a model describing neutron superfluidity in 56 Fe.

International Symposium on Capture Gamma-Ray Spectroscopy and Related Topics (CGS)•Apr 28, 2013•0 citations

Coupled Bogoliubov equations for electrons and phonons

C. Y. Wang, Sangeeta Sharma, T. Müller +2 more

The interaction between fermions and bosons is central to modern physics. To describe systems with such interactions, the authors apply Wick's theorem to the square of the Fr\"ohlich Hamiltonian and keep all terms up to quadratic order. What emerges are two coupled Bogoliubov equations: one for electrons and one for phonons. Remarkably, the self-consistent solution to these equations accurately describes both the renormalized gap in insulators and the superconducting gap in metals.

Physical review. B./Physical review. B•May 16, 2022•3 citations

The Constrained Hartree-Fock-Bogoliubov Approximation at Finite Temperature for Superconductors -Proposal of the Formalism and Its Numerical Method-

K. Tanabe, Mitsuru Ogoshi

To describe comprehensively the superconducting phase together with the other phases realized in the copper oxcides, the normal phase, the antiferro-magnetic and the spin-glass phases, an application of the constrained thermal Hartree-Fock-Bogoliubov (CTHFB) approximation is proposed. A set of the electron number- and the magnetization-constrained formalism is exemplified by employing the one-band Hubbard model. The CTHFB equation is applicable to the microscopic interaction whether it includes the phonon-mediated pairing interaction, or it provides a self-sustaining mechanism of the Cooper pair formation. A new type of the order parameter is introduced by extending the generalized Bogoliubov transformation to mix electron spin projections onto the quantization axis (i.e. \(\hat{S}_{z}\)). This yields the lowered minimum of the grand potential, which corresponds to the physical solution violating time-reversal invariance. The gradient method is proposed to solve the CTHFB equation, the non-linear eigenvalue broblem with constraints.

Journal of the Physical Society of Japan•Sep 15, 1990•3 citations
The Tamm-Dancoff approximation is applied to the interaction Hamiltonian used by Bogoliubov in his theory of superconductivity. It is shown that (i) the requirement that there is no virtual production of a single pair of electron and hole without phonons, or (ii) the electron-phonon coupling parameter tends to zero, leads to essentially the same compensating equation as obtained by Bogoliubov.
A connection is made between the exact eigen states of the BCS Hamiltonian and the predictions made by the Tamm-Dancoff Approximation. This connection is made by means of a parametrised algebra, which gives the exact quasi-spin algebra in one limit of the parameter and the Heisenberg-Weyl algebra in the other. Using this algebra to construct the Bethe Ansatz solution of the BCS Hamiltonian, we obtain parametrised Richardson-Gaudin equations, leading to the secular equation of the Tamm-Dancoff Approximation in the bosonic limit. An example is discussed in depth.
This chapter reviews the calculation of a number of system properties within the pairing approximation. Since the BCS theory was originally proposed, attempts to justify the pairing correlations basic to the theory have proceeded along two lines. The first approach has been to apply the BCS theory to a wide variety of phenomena in superconductors and check the theoretical predictions of the pairing approximation against experiment. The second approach has been to treat by various approximate methods the residual interactions neglected within the pairing scheme, hoping to show that these residual interactions introduce no major change in the predicted properties of the system. The interaction Hamiltonian, being a one-body operator, can at most change the state of one electron. While the majority of the electron–phonon interaction has been accounted for in forming the wave functions for the normal and superconducting states, there remains the part corresponding to resonant phonon absorption and emission processes.
To describe comprehensively the superconducting phase together with the other phases realized in the copper oxcides, the normal phase, the antiferro-magnetic and the spin-glass phases, an application of the constrained thermal Hartree-Fock-Bogoliubov (CTHFB) approximation is proposed. A set of the electron number- and the magnetization-constrained formalism is exemplified by employing the one-band Hubbard model. The CTHFB equation is applicable to the microscopic interaction whether it includes the phonon-mediated pairing interaction, or it provides a self-sustaining mechanism of the Cooper pair formation. A new type of the order parameter is introduced by extending the generalized Bogoliubov transformation to mix electron spin projections onto the quantization axis (i.e. S^z\hat{S}_{z}S^z​). This yields the lowered minimum of the grand potential, which corresponds to the physical solution violating time-reversal invariance. The gradient method is proposed to solve the CTHFB equation, the non-linear eigenvalue broblem with constraints.