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

Physical review. B, Condensed matter

Fluctuations in the Bardeen-Cooper-Schrieffer Hamiltonian induced by finite particle number

G. Puddu

Published May 1, 1992• 20 citations
Publisher

Abstract

The fluctuations properties of the BCS Hamiltonian which are induced in finite systems are explored. It is shown that for very large particle numbers, the static-path approximation is an excellent approximation, while for small particle numbers, the random-phase-approximation corrections should be taken into account, especially for particle numbers small enough to cause the normal- to superconducting-phase transition. A correlated mean field containing quantal fluctuations is introduced and compared with the Hartree-Bogoliubov mean field.

DOI
10.1103/physrevb.45.9882

Related papers

New Bardeen-Cooper-Schrieffer-type theory at finite temperature with particle-number conservation

Hitoshi Nakada, K. Tanabe

We formulate a new Bardeen-Cooper-Schrieffer (BCS)-type theory at finite temperature, by deriving a set of variational equations of the free energy after the particle-number projection. With its broad applicability, this theory can be a useful tool for investigating the pairing phase transition in finite systems with the particle-number conservation. This theory provides effects of the symmetry-restoring fluctuation (SRF) for the pairing phenomena in finite fermionic systems, distinctively from those of additional quantum fluctuations. It is shown by numerical calculations that the phase transition is compatible with the conservation in this theory, and that the SRF shifts up the critical temperature (${T}^{\mathrm{cr}}$). This shift of ${T}^{\mathrm{cr}}$ occurs due to reduction of degrees-of-freedom in canonical ensembles, and decreases only slowly as the particle-number increases (or as the level spacing narrows), in contrast to the conventional BCS theory.

arXiv:quant-ph/0603113•Physical Review C•Dec 6, 2006•19 citations

Superfluidity of heated Fermi systems in the static fluctuation approximation

Airat A. Khamzin, А. С. Никитин, Аirat Sitdikov

Superfluidity properties of heated finite Fermi systems are studied in the static fluctuation approximation, which is an original method. This method relies on a single and controlled approximation, which permits taking correctly into account quasiparticle correlations and thereby going beyond the independent-quasiparticle model. A closed self-consistent set of equations for calculating correlation functions at finite temperature is obtained for a finite Fermi system described by the Bardeen–Cooper–Schrieffer Hamiltonian. An equation for the energy gap is found with allowance for fluctuation effects. It is shown that the phase transition to the supefluid state is smeared upon the inclusion of fluctuations.

Physics of Atomic Nuclei•Oct 1, 2015•5 citations

Spectrum of the BCS Reduced Hamiltonian in the Theory of Superconductivity

Yusuke S. Kato

The spectrum and eigenstates of Bardeen-Cooper-Schrieffer's reduced Hamiltonian in the theory of superconductivity are investigated in a mathematically rigorous manner. The Hamiltonian is defined first for a finite number of electrons contained in a finite volume. The interaction Hamiltonian is assumed to take a simple form. The Hamiltonian after the Bogoliubov transformation (transformed Hamiltonian) has a limit in the weak topology of Hilbert space when the volume tends to infinity with the number density of electrons kept constant. In the limit the interaction part vanishes and the free part gives a continuous excitation spectrum of Bogolons (transformed electrons). It should be noticed that the operator norm of the former does not tend to zero and has no strong limit as well as the total Hamilton. An equivalent boson which consists of two Bogolons with opposite momenta and spins is introduced. In the Fock space for the bosons the Hamiltonian has a strong limit for infinite volume. The limit operator has the same excitation spectrum of Bogolons as the free Hamiltonian and the existence of the lower limit of the spectrum is shown for some special class of the interactions. For large volume the limit operator is a good approximation to be transformed Hamiltonian. The ground state of the free part of the transformed Hamiltonian (i.e. the ground state of Bardeen-Cooper-Schrieffer's calculation) is not the exact eigenstate of the transformed total Hamiltonian and the difference does not vanish when the volume tends to infinity.

Progress of Theoretical Physics•Nov 1, 1965•1 citation

Particle-Hole Fluctuations in Superconductors

Mason Protter

In this thesis we investigate how corrections originating from particle-hole interactions may be introduced into the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity using functional integral methods. The attractive Fermionic contact interaction is simultaneously decoupled in the Cooper and exchange channels, with the exchange-channel decoupling field treated as a small fluctuation about its zero mean-field value. Upon integrating out the exchange-channel fluctuation field, the system is described by a new effective action for the Cooper-channel decoupling field, which models the superconducting condensate. The saddle-point condition of this effective action generates the standard mean-field BCS equations with correction terms containing particle-hole polarization bubbles. The corrections obtained in this thesis are similar in spirit, albeit different in form, to those originally found in 1961 by Gor’kov and Melik-Barkhudarov (GMB). An advantage of the technique employed here, as opposed to the GMB approach, is the methodical manner in which fluctuations can be incorporated into superconductors.

•Jan 1, 2019•0 citations

Landau expansion for superconductors with dynamical fluctuations beyond the mean-field theory of Bardeen, Cooper, and Schrieffer (BCS)

Josef Frühauf

We study the extension of the superconducting fluctuation-exchange approximation for the attractive Hubbard model into the ordered phase. We show that including all possible diagrams with anomalous propagators in the Luttinger-Ward functional leads to a mismatch between the critical temperature determined by the instability of the normal phase and that associated with the emergence of anomalous long-range order. We then derive a consistent theory by restricting two-particle propagation in the ladder diagrams in the ordered phase. Our main contributions are a derivation of this consistent approximation and the numerical analysis of its thermodynamic and spectral properties.

•Jan 1, 2026•0 citations

Number-Phase Fluctuations in Isolated Superconductors

Xiaotian Si, Wataru Kohno, Takafumi Kita

We improve the Bardeen-Cooper-Schrieffer wave function with a fixed particle number so as to incorporate many-body correlations beyond the mean-field treatment. It is shown that the correlations lower the ground-state energy far more than Cooper-pair condensation in the weak-coupling region. Moreover, they naturally bring a superposition over the number of condensed particles. Thus, Cooper-pair condensation is special among the various bound-state formations of quantum mechanics in that number fluctuations are necessarily present in the condensate through the dynamical exchange of particles with the non-condensate reservoir. On the basis of this result, we propose $\varDelta N_{\rm con}\cdot \varDelta ϕ\gtrsim 1$ as the uncertainty relation relevant to the number-phase fluctuations in superconductors and superfluids, where the number of condensed particles $N_{\rm con}$ is used instead of the total particle number $N$. The formula implies that a macroscopic phase $ϕ$ can be established even in number-fixed superconductors and superfluids since $\varDelta N_{\rm con}\gg 1$.

arXiv:1809.03708•Journal of the Physical Society of Japan•Sep 10, 2018•2 citations
The fluctuations properties of the BCS Hamiltonian which are induced in finite systems are explored. It is shown that for very large particle numbers, the static-path approximation is an excellent approximation, while for small particle numbers, the random-phase-approximation corrections should be taken into account, especially for particle numbers small enough to cause the normal- to superconducting-phase transition. A correlated mean field containing quantal fluctuations is introduced and compared with the Hartree-Bogoliubov mean field.
We formulate a new Bardeen-Cooper-Schrieffer (BCS)-type theory at finite temperature, by deriving a set of variational equations of the free energy after the particle-number projection. With its broad applicability, this theory can be a useful tool for investigating the pairing phase transition in finite systems with the particle-number conservation. This theory provides effects of the symmetry-restoring fluctuation (SRF) for the pairing phenomena in finite fermionic systems, distinctively from those of additional quantum fluctuations. It is shown by numerical calculations that the phase transition is compatible with the conservation in this theory, and that the SRF shifts up the critical temperature (Tcr{T}^{\mathrm{cr}}Tcr). This shift of Tcr{T}^{\mathrm{cr}}Tcr occurs due to reduction of degrees-of-freedom in canonical ensembles, and decreases only slowly as the particle-number increases (or as the level spacing narrows), in contrast to the conventional BCS theory.
Superfluidity properties of heated finite Fermi systems are studied in the static fluctuation approximation, which is an original method. This method relies on a single and controlled approximation, which permits taking correctly into account quasiparticle correlations and thereby going beyond the independent-quasiparticle model. A closed self-consistent set of equations for calculating correlation functions at finite temperature is obtained for a finite Fermi system described by the Bardeen–Cooper–Schrieffer Hamiltonian. An equation for the energy gap is found with allowance for fluctuation effects. It is shown that the phase transition to the supefluid state is smeared upon the inclusion of fluctuations.
The spectrum and eigenstates of Bardeen-Cooper-Schrieffer's reduced Hamiltonian in the theory of superconductivity are investigated in a mathematically rigorous manner. The Hamiltonian is defined first for a finite number of electrons contained in a finite volume. The interaction Hamiltonian is assumed to take a simple form. The Hamiltonian after the Bogoliubov transformation (transformed Hamiltonian) has a limit in the weak topology of Hilbert space when the volume tends to infinity with the number density of electrons kept constant. In the limit the interaction part vanishes and the free part gives a continuous excitation spectrum of Bogolons (transformed electrons). It should be noticed that the operator norm of the former does not tend to zero and has no strong limit as well as the total Hamilton. An equivalent boson which consists of two Bogolons with opposite momenta and spins is introduced. In the Fock space for the bosons the Hamiltonian has a strong limit for infinite volume. The limit operator has the same excitation spectrum of Bogolons as the free Hamiltonian and the existence of the lower limit of the spectrum is shown for some special class of the interactions. For large volume the limit operator is a good approximation to be transformed Hamiltonian. The ground state of the free part of the transformed Hamiltonian (i.e. the ground state of Bardeen-Cooper-Schrieffer's calculation) is not the exact eigenstate of the transformed total Hamiltonian and the difference does not vanish when the volume tends to infinity.
In this thesis we investigate how corrections originating from particle-hole interactions may be introduced into the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity using functional integral methods. The attractive Fermionic contact interaction is simultaneously decoupled in the Cooper and exchange channels, with the exchange-channel decoupling field treated as a small fluctuation about its zero mean-field value. Upon integrating out the exchange-channel fluctuation field, the system is described by a new effective action for the Cooper-channel decoupling field, which models the superconducting condensate. The saddle-point condition of this effective action generates the standard mean-field BCS equations with correction terms containing particle-hole polarization bubbles. The corrections obtained in this thesis are similar in spirit, albeit different in form, to those originally found in 1961 by Gor’kov and Melik-Barkhudarov (GMB). An advantage of the technique employed here, as opposed to the GMB approach, is the methodical manner in which fluctuations can be incorporated into superconductors.
We study the extension of the superconducting fluctuation-exchange approximation for the attractive Hubbard model into the ordered phase. We show that including all possible diagrams with anomalous propagators in the Luttinger-Ward functional leads to a mismatch between the critical temperature determined by the instability of the normal phase and that associated with the emergence of anomalous long-range order. We then derive a consistent theory by restricting two-particle propagation in the ladder diagrams in the ordered phase. Our main contributions are a derivation of this consistent approximation and the numerical analysis of its thermodynamic and spectral properties.
We improve the Bardeen-Cooper-Schrieffer wave function with a fixed particle number so as to incorporate many-body correlations beyond the mean-field treatment. It is shown that the correlations lower the ground-state energy far more than Cooper-pair condensation in the weak-coupling region. Moreover, they naturally bring a superposition over the number of condensed particles. Thus, Cooper-pair condensation is special among the various bound-state formations of quantum mechanics in that number fluctuations are necessarily present in the condensate through the dynamical exchange of particles with the non-condensate reservoir. On the basis of this result, we propose ΔNcon⋅Δϕ≳1\varDelta N_{\rm con}\cdot \varDelta ϕ\gtrsim 1ΔNcon​⋅Δϕ≳1 as the uncertainty relation relevant to the number-phase fluctuations in superconductors and superfluids, where the number of condensed particles NconN_{\rm con}Ncon​ is used instead of the total particle number NNN. The formula implies that a macroscopic phase ϕϕϕ can be established even in number-fixed superconductors and superfluids since ΔNcon≫1\varDelta N_{\rm con}\gg 1ΔNcon​≫1.