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arXiv:1307.6905 · Physics Letters B

Matrix elements of one-body and two-body operators between arbitrary HFB multi-quasiparticle states

Qing-Li Hu, Zao-Chun Gao, Y.S. Chen

Published May 22, 2014• 35 citations
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Abstract

We present new formulae for the matrix elements of one-body and two-body physical operators, which are applicable to arbitrary Hartree–Fock–Bogoliubov wave functions, including those for multi-quasiparticle excitations. The testing calculations show that our formulae may substantially reduce the computational time by several orders of magnitude when applied to many-body quantum system in a large Fock space.

DOI
10.1016/j.physletb.2014.05.045

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We present new formulae for the matrix elements of one-body and two-body physical operators, which are applicable to arbitrary Hartree–Fock–Bogoliubov wave functions, including those for multi-quasiparticle excitations. The testing calculations show that our formulae may substantially reduce the computational time by several orders of magnitude when applied to many-body quantum system in a large Fock space.
We present a generalization of the Hartree-Fock Bogoliubov (HFB) theory in which the coupling between one and two quasi-particles is taken into account.This is done by writing the excitation operators as linear combinations of one and two HFB quasi-particles. The excitation energies and the quasi-particle amplitudes are given by generalized Bogoliubov equations. The excitation spectrum has two branches. The first one is a discrete branch which is gapless and has a phonon character at large wave-length and, contrarily to HFB, is always stable. This branch is detached from a continuum branch whose threshold, at fixed total momentum, coincides with the two quasi-particle threshold of the HFB theory. The gap between these two branches at P=0 is equal to two times the HFB gap, which then provides for the relevant energy scale. We also give numerical results for a specific case.
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We discuss exact functional integral expressions for many-body matrix elements of the type ⟨ψ, A |e −βĤ |ψ, A ⟩, between particle number projected Hartree–Fock–Bogoliubov (HFB) wavefunctions with time-reversal symmetry. A proof of positivity is given for a class of Hamiltonians and when the HFB wavefunctions with time-reversal symmetry are particle number projected to an even number of particles. We show explicitly how to reduce the propagator in the functional integral to a Hermitian positive definite propagator for particle pairs. This result generalizes that previously obtained using Bardeen–Cooper–Schrieffer wavefunctions.
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Numerical difficulties associated with computing matrix elements of operators between Hartree-Fock-Bogoliubov (HFB) wavefunctions have plagued the development of HFB-based many-body theories for decades. The problem arises from divisions by zero in the standard formulation of the nonorthogonal Wick's theorem in the limit of vanishing HFB overlap. In this Communication, we present a robust formulation of Wick's theorem that stays well-behaved regardless of whether the HFB states are orthogonal or not. This new formulation ensures cancellation between the zeros of the overlap and the poles of the Pfaffian, which appears naturally in fermionic systems. Our formula explicitly eliminates self-interaction, which otherwise causes additional numerical challenges. A computationally efficient version of our formalism enables robust symmetry-projected HFB calculations with the same computational cost as mean-field theories. Moreover, we avoid potentially diverging normalization factors by introducing a robust normalization procedure. The resulting formalism treats even and odd number of particles on equal footing and reduces to Hartree-Fock as a natural limit. As proof of concept, we present a numerically stable and accurate solution to a Jordan-Wigner-transformed Hamiltonian, whose singularities motivated the present work. Our robust formulation of Wick's theorem is a most promising development for methods using quasiparticle vacuum states.