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

Science China Mathematics

U-MATRIX THEORY, BOGOLIUBOV TRANSFORMATION AND BCS TRIAL WAVE FUNCTION

Lam A

Published Jan 1, 1989• 2 citations
OpenAlex

Abstract

The U-matrlx theory in quantum mechanics recently developed has been appliedto study a superconductor system described by the BCS pairing Hamiltonian . It is shown that in general the -operator in our theory specifies a canonical transformation. Using the to describe the system, it is proved that if the Hartrec type approximation is adopted, or the c-number approach is used, the Bogoliubov transformation can be derived via the U-matrix theory if the perturbing Hamiltonian is turned on 'suddenly'. It is also shown that based on the U-matrix, the BCS ground state wave function of a superconducting system can be deduced under the sudden switching condition.

Related papers

The Bogoliubov Approach

Monique Combescot, Shiue-Yuan Shiau

Chapter 8 deals with the Bogoliubov approach to the BCS problem. This approach is of fundamental interest because it gives a mathematical support to the BCS ansatz, which was simply proposed by Bardeen, Cooper, and Schrieffer as a trial wave function. This chapter shows that the BCS ansatz corresponds to the ground-state wave function of the one-body Bogoliubov Hamiltonian constructed from the BCS Hamiltonian. This mathematical support is highly welcome because the BCS ansatz contains the questionable idea that composite bosons condense, like elementary bosons, all into the same state, even if the product of N identical electron pair operators substantially differs from that of N single pair operators when N gets large, owing to the moth-eaten effect induced by the Pauli exclusion principle.

Oxford University Press eBooks•Dec 1, 2015•0 citations

Extended BCS Theories

Roger Balian, H. Flocard, M. Vénéroni

Extensions of the Hartree-Fock-Bogoliubov theory are worked out which are tailored for, (i) the consistent evaluation of fluctuations and correlations and (ii) the restoration through projection of broken symmetries. For both purposes we rely on a single variational principle which optimizes the characteristic function. The Bloch equation is used as a constraint to define the equilibrium state, and the trial quantities are a density operator and a Lagrangian multiplier matrix which is akin to an observable. The conditions of stationarity are respectively a Schrödinger-like equation and a Heisenberg-- like equation with an imaginary time running backwards. General conditions for the trial spaces are stated that warrant the preservation of thermodynamic relations. When the trial spaces are chosen to be of the independent-quasi-particle type, the ensuing coupled equations provide an extension of the Hartree-Fock-Bogoliubov approximation, which optimizes the characteristic function. Variational expressions for thermodynamic quantities or characteristic functions are also obtained with projected trial states, whether an invariance symmetry is broken or not. In particular, the projection on even or odd particle number is worked out for a pairing Hamiltonian, which leads to new equations replacing the BCS ones. Qualitative differences between even and odd systems, depending on the temperature $T$, the level density and the strength of the pairing force, are investigated analytically and numerically.

arXiv:nucl-th/9706041•Jun 17, 1997•0 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

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

Numerical Simulation Solution of the BCS Pairing Problem

Feng Xu, An-Min Wang, Xiao-Dong Yang +1 more

We propose a new simulation computational method to solve the reduced BCS Hamiltonian based on spin analogy and submatrix diagonalization. Then we further apply this method to solve superconducting energy gap and the results are well consistent with those obtained by Bogoliubov transformation method. The exponential problem of 2 N -dimensional matrix is reduced to the polynomial problem of N -dimensional matrix. It is essential to validate this method on a real quantum computer and is helpful to understanding the many-body quantum theory.

arXiv:quant-ph/0407100•Communications in Theoretical Physics•Jul 1, 2005•4 citations

Numerical Simulation Solution of the BCS Pairing Problem

Xufeng, Feng, Wang +5 more

We propose a new simulation computational method to solve the reduced BCS Hamiltonian based on spin analogy and submatrix diagonalization. Then we further apply this method to solve superconducting energy gap and the results are well consistent with those obtained by Bogoliubov transformation method. The exponential problem of 2N-dimensional matrix is reduced to the polynomial problem of N-dimensional matrix. It is essential to validate this method on a real quantum computer and is helpful to understanding the many-body quantum theory.

理论物理通讯：英文版•Jan 1, 2005•0 citations
The U-matrlx theory in quantum mechanics recently developed has been appliedto study a superconductor system described by the BCS pairing Hamiltonian . It is shown that in general the -operator in our theory specifies a canonical transformation. Using the to describe the system, it is proved that if the Hartrec type approximation is adopted, or the c-number approach is used, the Bogoliubov transformation can be derived via the U-matrix theory if the perturbing Hamiltonian is turned on 'suddenly'. It is also shown that based on the U-matrix, the BCS ground state wave function of a superconducting system can be deduced under the sudden switching condition.
Chapter 8 deals with the Bogoliubov approach to the BCS problem. This approach is of fundamental interest because it gives a mathematical support to the BCS ansatz, which was simply proposed by Bardeen, Cooper, and Schrieffer as a trial wave function. This chapter shows that the BCS ansatz corresponds to the ground-state wave function of the one-body Bogoliubov Hamiltonian constructed from the BCS Hamiltonian. This mathematical support is highly welcome because the BCS ansatz contains the questionable idea that composite bosons condense, like elementary bosons, all into the same state, even if the product of N identical electron pair operators substantially differs from that of N single pair operators when N gets large, owing to the moth-eaten effect induced by the Pauli exclusion principle.
Extensions of the Hartree-Fock-Bogoliubov theory are worked out which are tailored for, (i) the consistent evaluation of fluctuations and correlations and (ii) the restoration through projection of broken symmetries. For both purposes we rely on a single variational principle which optimizes the characteristic function. The Bloch equation is used as a constraint to define the equilibrium state, and the trial quantities are a density operator and a Lagrangian multiplier matrix which is akin to an observable. The conditions of stationarity are respectively a Schrödinger-like equation and a Heisenberg-- like equation with an imaginary time running backwards. General conditions for the trial spaces are stated that warrant the preservation of thermodynamic relations. When the trial spaces are chosen to be of the independent-quasi-particle type, the ensuing coupled equations provide an extension of the Hartree-Fock-Bogoliubov approximation, which optimizes the characteristic function. Variational expressions for thermodynamic quantities or characteristic functions are also obtained with projected trial states, whether an invariance symmetry is broken or not. In particular, the projection on even or odd particle number is worked out for a pairing Hamiltonian, which leads to new equations replacing the BCS ones. Qualitative differences between even and odd systems, depending on the temperature TTT, the level density and the strength of the pairing force, are investigated analytically and numerically.
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.
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.
We propose a new simulation computational method to solve the reduced BCS Hamiltonian based on spin analogy and submatrix diagonalization. Then we further apply this method to solve superconducting energy gap and the results are well consistent with those obtained by Bogoliubov transformation method. The exponential problem of 2 N -dimensional matrix is reduced to the polynomial problem of N -dimensional matrix. It is essential to validate this method on a real quantum computer and is helpful to understanding the many-body quantum theory.
We propose a new simulation computational method to solve the reduced BCS Hamiltonian based on spin analogy and submatrix diagonalization. Then we further apply this method to solve superconducting energy gap and the results are well consistent with those obtained by Bogoliubov transformation method. The exponential problem of 2N-dimensional matrix is reduced to the polynomial problem of N-dimensional matrix. It is essential to validate this method on a real quantum computer and is helpful to understanding the many-body quantum theory.