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arXiv:cond-mat/0409400

A way to solve BCS-type pairing model

An Min Wang, Feng Xu

Published Sep 15, 2004• 0 citations
arXivHTMLPDF

Abstract

We propose a way to solve BCS-type pairing model by to exactly solve its spin-analogy in the subspace. The advantages of our method are to avoid to directly deal with the approximate procedure and to transfer an exponentially complicated problem to a polynomial problem. Moreover, it builds the theoretical foundation of quantum simulation of pairing model and can be used to check the precision of quantum simulation. It is also helpful to understand the many-body quantum theory.

DOI
10.48550/arxiv.cond-mat/0409400

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We propose a way to solve BCS-type pairing model by to exactly solve its spin-analogy in the subspace. The advantages of our method are to avoid to directly deal with the approximate procedure and to transfer an exponentially complicated problem to a polynomial problem. Moreover, it builds the theoretical foundation of quantum simulation of pairing model and can be used to check the precision of quantum simulation. It is also helpful to understand 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.
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.
The solvable two-level pairing model is used to test two new number-conserving approximations. These approximations are compared with the exact results, with perturbation theory, with the BCS method, and with the projected BCS method of Kerman, Lawson, and Macfarlane. It is found that second-order perturbation theory and one of the tested approximations work rather well for not too high values of the pairing-force strength, while for higher values the other approximation turns out to be quite good.
Chapter 10 deals with the exact solution to the BCS problem, as obtained by analytically solving Richardson-Gaudin equations. The chapter first provides a simple derivation of these equations and shows how to solve them for arbitrary N. The explicit N dependence of the N–Cooper pair energy obtained from this resolution provides invaluable insights into the role of the Pauli exclusion principle in paired fermion systems because the reduced BCS potential does not directly act between Cooper pairs. Revelation of this subtle point makes it possible to understand (i) why Cooper pairs can strongly overlap without breaking, whereas, at large density, an exciton system dissociates into an electron-hole plasma, and (ii) why the gap energy is far larger than the average pair energy. This understanding is further supported by the Cooper pair many-body parameter, which has the same form as the ones for Wannier or Frenkel excitons do.