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.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 eBooksDec 1, 20150 citations

General pairing mechanisms in the BCS-theory of superconductivity

Christian Hainzl, Michael Loss

arXiv:1603.07659The European Physical Journal BApr 28, 20177 citations

Richardson-Gaudin integrability in the contraction limit of the quasispin

Stijn De Baerdemacker

Background: The reduced, level-independent, Bardeen-Cooper-Schrieffer Hamiltonian is exactly diagonalizable by means of a Bethe ansatz wave function, provided the free variables in the ansatz are the solutions of the set of Richardson-Gaudin equations. On the one side, the Bethe ansatz is a simple product state of generalized pair operators. On the other hand, the Richardson-Gaudin equations are strongly coupled in a nonlinear way, making them prone to singularities. Unfortunately, it is nontrivial to give a clear physical interpretation to the Richardson-Gaudin variables because no physical operator is directly related to the individual variables.Purpose: The purpose of this paper is to shed more light on the singular behavior of the Richardson-Gaudin equations, and how this is related to the product wave structure of the Bethe ansatz.Method: A pseudodeformation of the quasispin algebra is introduced, leading towards a Heisenberg-Weyl algebra in the contraction limit of the deformation parameter. This enables an adiabatic connection of the exact Bethe ansatz eigenstates with pure bosonic multiphonon states. The physical interpretation of this approach is an adiabatic suppression of the Pauli exclusion principle.Results: The method is applied to a so-called ``picket-fence'' model for the BCS Hamiltonian, displaying a typical critical behavior in the Richardson-Gaudin variables. It was observed that the associated bosonic multiphonon states change collective nature at the critical interaction strengths of the Richardson-Gaudin equations.Conclusions: The Pauli exclusion principle is the main responsible for the critical behavior of the Richardson-Gaudin equations, which can be suppressed by means of a pseudodeformation of the quasispin algebra.Background: The reduced, level-independent, Bardeen-Cooper-Schrieffer Hamiltonian is exactly diagonalizable by means of a Bethe ansatz wave function, provided the free variables in the ansatz are the solutions of the set of Richardson-Gaudin equations. On the one side, the Bethe ansatz is a simple product state of generalized pair operators. On the other hand, the Richardson-Gaudin equations are strongly coupled in a nonlinear way, making them prone to singularities. Unfortunately, it is nontrivial to give a clear physical interpretation to the Richardson-Gaudin variables because no physical operator is directly related to the individual variables.Purpose: The purpose of this paper is to shed more light on the singular behavior of the Richardson-Gaudin equations, and how this is related to the product wave structure of the Bethe ansatz.Method: A pseudodeformation of the quasispin algebra is introduced, leading towards a Heisenberg-Weyl algebra in the contraction limit of the deformation parameter. This enables an adiabatic connection of the exact Bethe ansatz eigenstates with pure bosonic multiphonon states. The physical interpretation of this approach is an adiabatic suppression of the Pauli exclusion principle.Results: The method is applied to a so-called ``picket-fence'' model for the BCS Hamiltonian, displaying a typical critical behavior in the Richardson-Gaudin variables. It was observed that the associated bosonic multiphonon states change collective nature at the critical interaction strengths of the Richardson-Gaudin equations.Conclusions: The Pauli exclusion principle is the main responsible for the critical behavior of the Richardson-Gaudin equations, which can be suppressed by means of a pseudodeformation of the quasispin algebra.

arXiv:1207.3417Physical Review COct 24, 201239 citations

a Generalized BCS Approach to the Theory of Superconductivity

Yibin Zheng

Jan 1, 19920 citations

Quantum Correlations of Ideal Bose and Fermi Gases in the Canonical Ensemble

Kazumasa Tsutsui, Takafumi Kita

We derive an expression for the reduced density matrices of ideal Bose and Fermi gases in the canonical ensemble, which corresponds to the Bloch--De Dominicis (or Wick's) theorem in the grand canonical ensemble for normal-ordered products of operators. Using this expression, we study one- and two-body correlations of homogeneous ideal gases with $N$ particles. The pair distribution function $g^{(2)}(r)$ of fermions clearly exhibits antibunching with $g^{(2)}(0)=0$ due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with $g^{(2)}(0)\approx 2$ , corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature $T_0$ , an off-diagonal long-range order develops in the one-particle density matrix to reach $g^{(1)}(r)=1$ at $T=0$ , and the pair correlation starts to decrease towards $g^{(2)}(r)\approx 1$ at $T=0$ . The results for $N\rightarrow \infty$ are seen to converge to those of the grand canonical ensemble obtained by assuming the average $\langle\hatψ({\bf r})\rangle$ of the field operator $\hatψ({\bf r})$ below $T_0$ . This fact justifies the introduction of the "anomalous" average $\langle\hatψ({\bf r})\rangle\neq 0$ below $T_0$ in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.We derive an expression for the reduced density matrices of ideal Bose and Fermi gases in the canonical ensemble, which corresponds to the Bloch--De Dominicis (or Wick's) theorem in the grand canonical ensemble for normal-ordered products of operators. Using this expression, we study one- and two-body correlations of homogeneous ideal gases with $N$ particles. The pair distribution function $g^{(2)}(r)$ of fermions clearly exhibits antibunching with $g^{(2)}(0)=0$ due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with $g^{(2)}(0)\approx 2$, corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature $T_0$, an off-diagonal long-range order develops in the one-particle density matrix to reach $g^{(1)}(r)=1$ at $T=0$, and the pair correlation starts to decrease towards $g^{(2)}(r)\approx 1$ at $T=0$. The results for $N\rightarrow \infty$ are seen to converge to those of the grand canonical ensemble obtained by assuming the average $\langle\hatψ({\bf r})\rangle$ of the field operator $\hatψ({\bf r})$ below $T_0$. This fact justifies the introduction of the "anomalous" average $\langle\hatψ({\bf r})\rangle\neq 0$ below $T_0$ in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.

arXiv:1610.03617Journal of the Physical Society of JapanOct 12, 20164 citations

BCS ansatz for superconductivity in the light of the Bogoliubov approach and the Richardson–Gaudin exact wave function

Monique Combescot, W. V. Pogosov, Odile Betbeder-Matibet

arXiv:1111.4781Physica C SuperconductivityNov 7, 201212 citations

N

particles. The pair distribution function

g^{(2)}(r)

of fermions clearly exhibits antibunching with

g^{(2)}(0)=0

due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with

g^{(2)}(0)\approx 2

, corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature

T_0

, an off-diagonal long-range order develops in the one-particle density matrix to reach

g^{(1)}(r)=1

T=0

, and the pair correlation starts to decrease towards

g^{(2)}(r)\approx 1

T=0

. The results for

N\rightarrow \infty

are seen to converge to those of the grand canonical ensemble obtained by assuming the average

\langle\hatψ({\bf r})\rangle

of the field operator

\hatψ({\bf r})

below

T_0

. This fact justifies the introduction of the "anomalous" average

\langle\hatψ({\bf r})\rangle\neq 0

below

T_0

in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.