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Ukrains’kyi Matematychnyi Zhurnal

Спектр та стани БКШ гамільтоніана з джерелами

D. Ya. Petrina

Published Sep 25, 2008• 0 citations
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Abstract

We consider the BCS Hamiltonian with sources, as proposed by Bogolyubov and Bogolyubov, Jr. We prove that the eigenvectors and eigenvalues of the BCS Hamiltonian with sources can be exactly determined in the thermodynamic limit. Earlier, Bogolyubov proved that the energies per volume of the BCS Hamiltonian with sources and the approximating Hamiltonian coincide in the thermodynamic limit.

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The BCS Hamiltonian in a finite cube with periodic boundary condition is considered. The special subspace of pairs of particles with opposite momenta and spin is introduced. It is proved that, in this subspace, the spectrum of the BCS Hamiltonian is defined exactly for one pair, and for n pairs the spectrum is defined through the eigenvalues of one pair and a term that tends to zero as the volume of the cube tends to infinity. On the subspace of pairs, the BCS Hamiltonian can be represented as a sum of two operators. One of them describes the spectra of noninteracting pairs and the other one describes the interaction between pairs that tends to zero as the volume of the cube tends to infinity. It is proved that, on the subspace of pairs, as the volume of the cube tends to infinity, the BCS Hamiltonian tends to the approximating Hamiltonian, which is a quadratic form with respect to the operators of creation and annihilation.

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Chapter 6

Symmetry Breaking

The Hamiltonian for the many-electron system in finite volume [0, L]d is given by H = H0 + Hint where H0 = 1Ld ∑ eka + kσakσ (6.1) ek = k 2 2m − µ, and Hint = 1L3d ∑ ∑ V (k− p)a+kσa+q−k,τaq−p,τapσ (6.2) As usual, the interacting part may be represented by the following diagram: Since there is conservation of momentum, there are three independent momenta. One may consider three natural limiting cases with two independent momenta: forward exchange BCS In chapter 8 where we prove rigorous bounds on Feynman diagrams we show that the most singular contributions to the perturbation series come from the the third diagram in the above figure. To retain only this term, the q = 0 term of Hint given by (6.2), is the basic approximation of BCS theory. Furthermore V (k−p) is substituted by its value on the Fermi surface, V (k − p) ≈ V (kˆ − pˆ) where kˆ = kF k‖k‖ , kF = √ 2mµ, and the momenta k,p are restricted to values close to the Fermi surface, |ek|, |ep| ≤ ωD, where the cutoff ωD is referred to as the Debye frequency. Thus the BCS approximation reads Hint ≈ HBCS (6.3) where HBCS = 1L3d ∑ ∑ V (kˆ− pˆ) a+kσa+−k,τa−p,τapσ (6.4) We first make a comment on the volume factors. In (6.4) we retained only the q = 0 term of (6.2), but we did not cancel a volume factor. It is not obvious that (6.4) is still proportional to the volume. However, in the case of the forward approximation which is defined by putting k = p in (6.2) without canceling a volume factor, Hint ≈ Hforw := 1L3d ∑ ∑ V (0)a+kσa + p,τap,τakσ (6.5) there is an easy argument which shows that (6.5) is still proportional to the volume (for constant density). Namely, on a fixed N -particle space FN the interacting part Hint is a multiplication operator given by N∑ V (xi − xj) (6.6) Let δy(x) = δ(x − y). Then ϕ(x1, · · · ,xN ) := δy1 ∧ · · · ∧ δyN (x1, · · · ,xN ) is an eigenfunction of Hint with eigenvalue E = 12 N∑ V (yi − yj) = 12Ld N∑ ∑ e−i(yi−yj)qV (q) (6.7) which is, for V (x) ∈ L1, proportional to N or to the volume for constant density. One finds that ϕ is also an eigenvector of the forward term, Hforwϕ = Eforwϕ where Eforw is obtained from (6.7) by putting q = 0 without canceling a volume factor, V (q = 0) (6.8) Although the model defined by the Hamiltonian HBCS in (6.4) is still quartic in the annihilation and creation operators, it can be solved explicitly, without making a quadratic mean field approximation as it is usually done. The important point is that the approximation ‘putting q = 0 without canceling a volume factor’ has the effect that in the bosonic functional integral representation the volume factors enter the formulae in such a way that in the thermodynamic limit the integration variables are forced to take values at the global minimum of the effective potential. That is, fluctuations around the minimum configuration are suppressed for Ld →∞.

Chapman & Hall/CRC eBooks•Aug 30, 2004•0 citations
We consider the BCS Hamiltonian with sources, as proposed by Bogolyubov and Bogolyubov, Jr. We prove that the eigenvectors and eigenvalues of the BCS Hamiltonian with sources can be exactly determined in the thermodynamic limit. Earlier, Bogolyubov proved that the energies per volume of the BCS Hamiltonian with sources and the approximating Hamiltonian coincide in the thermodynamic limit.
We investigate the spectrum of a model Hamiltonian with BCS and mean-field interaction in a finite domain under periodic boundary conditions. The model Hamiltonian is considered on the states of pairs and waves of density charges and their excitations. It is represented as the sum of three operators that describe noninteracting pairs, the interaction between pairs, and the interaction between pairs and waves of density charges. The last two operators tend to zero in the thermodynamic limit, and the spectrum of the model Hamiltonian coincides with the spectrum of noninteracting pairs with chemical potential shifted by mean-field interaction. It is shown that the model and approximating Hamiltonians coincide in the thermodynamic limit on their ground and excited states and both have two branches of eigenvalues and eigenvectors.
We consider the model and approximating Hamiltonians directly in the case of infinite volume. We show that each of these Hamiltonians has two branches of the spectrum and two systems of eigenvectors, which represent excitations of the ground states of the model and approximating Hamiltonians as well as the ground states themselves. On both systems of eigenvectors, the model and approximating Hamiltonians coincide with one another. In both branches of the spectrum, there is a gap between the eigenvalues of the ground and excited states.
The BCS Hamiltonian in a finite cube with periodic boundary condition is considered. The special subspace of pairs of particles with opposite momenta and spin is introduced. It is proved that, in this subspace, the spectrum of the BCS Hamiltonian is defined exactly for one pair, and for n pairs the spectrum is defined through the eigenvalues of one pair and a term that tends to zero as the volume of the cube tends to infinity. On the subspace of pairs, the BCS Hamiltonian can be represented as a sum of two operators. One of them describes the spectra of noninteracting pairs and the other one describes the interaction between pairs that tends to zero as the volume of the cube tends to infinity. It is proved that, on the subspace of pairs, as the volume of the cube tends to infinity, the BCS Hamiltonian tends to the approximating Hamiltonian, which is a quadratic form with respect to the operators of creation and annihilation.
The Hamiltonian for the many-electron system in finite volume [0, L]d is given by H = H0 + Hint where H0 = 1Ld ∑ eka + kσakσ (6.1) ek = k 2 2m − µ, and Hint = 1L3d ∑ ∑ V (k− p)a+kσa+q−k,τaq−p,τapσ (6.2) As usual, the interacting part may be represented by the following diagram: Since there is conservation of momentum, there are three independent momenta. One may consider three natural limiting cases with two independent momenta: forward exchange BCS In chapter 8 where we prove rigorous bounds on Feynman diagrams we show that the most singular contributions to the perturbation series come from the the third diagram in the above figure. To retain only this term, the q = 0 term of Hint given by (6.2), is the basic approximation of BCS theory. Furthermore V (k−p) is substituted by its value on the Fermi surface, V (k − p) ≈ V (kˆ − pˆ) where kˆ = kF k‖k‖ , kF = √ 2mµ, and the momenta k,p are restricted to values close to the Fermi surface, |ek|, |ep| ≤ ωD, where the cutoff ωD is referred to as the Debye frequency. Thus the BCS approximation reads Hint ≈ HBCS (6.3) where HBCS = 1L3d ∑ ∑ V (kˆ− pˆ) a+kσa+−k,τa−p,τapσ (6.4) We first make a comment on the volume factors. In (6.4) we retained only the q = 0 term of (6.2), but we did not cancel a volume factor. It is not obvious that (6.4) is still proportional to the volume. However, in the case of the forward approximation which is defined by putting k = p in (6.2) without canceling a volume factor, Hint ≈ Hforw := 1L3d ∑ ∑ V (0)a+kσa + p,τap,τakσ (6.5) there is an easy argument which shows that (6.5) is still proportional to the volume (for constant density). Namely, on a fixed N -particle space FN the interacting part Hint is a multiplication operator given by N∑ V (xi − xj) (6.6) Let δy(x) = δ(x − y). Then ϕ(x1, · · · ,xN ) := δy1 ∧ · · · ∧ δyN (x1, · · · ,xN ) is an eigenfunction of Hint with eigenvalue E = 12 N∑ V (yi − yj) = 12Ld N∑ ∑ e−i(yi−yj)qV (q) (6.7) which is, for V (x) ∈ L1, proportional to N or to the volume for constant density. One finds that ϕ is also an eigenvector of the forward term, Hforwϕ = Eforwϕ where Eforw is obtained from (6.7) by putting q = 0 without canceling a volume factor, V (q = 0) (6.8) Although the model defined by the Hamiltonian HBCS in (6.4) is still quartic in the annihilation and creation operators, it can be solved explicitly, without making a quadratic mean field approximation as it is usually done. The important point is that the approximation ‘putting q = 0 without canceling a volume factor’ has the effect that in the bosonic functional integral representation the volume factors enter the formulae in such a way that in the thermodynamic limit the integration variables are forced to take values at the global minimum of the effective potential. That is, fluctuations around the minimum configuration are suppressed for Ld →∞.