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Communications in Theoretical Physics

An Extension of the Pseudo-Fermion Method

Ming Shaw, Fu-sui Liu

Published Sep 1, 1993• 8 citations
Publisher

Abstract

Using Abrikosov's pseudo-fermion representation of a single spin and Larsen's note with pseudo-fermion energy , we describe a procedure which allows calculation of spin-spin correlation as well as spin-electron (hole) correlation using standard methods including the linked cluster theorem of diagrammatic perturbation theory.
DOI
10.1088/0253-6102/20/2/147

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The possibility of a perturbation treatment for quantum crystals is investigated. It is found that the usual form of the unperturbed Hamiltonian, being a single-particle operator, is not suitable for this situation, since it does not produce correlations necessary in a theory of quantum crystals. These correlations prevent the unperturbed ground state or low excited states from containing multiply occupied lattice sites. An unperturbed Hamiltonian is proposed which generates these correlations but is no longer a single-particle operator. The rules for a diagrammatic representation of a perturbation expansion starting from this unperturbed Hamiltonian are given. They differ from the usual rules for Goldstone diagrams. In particular, the linked-cluster theorem is not valid. Adding and subtracting pairs of mutually cancelling diagrams, in a manner similar to that in which Pauli-principle-violating diagrams are treated in the standard perturbation theory for fermions, we find new rules which, at least at low temperatures, are identical to those for spinless fermions. This result is essentially independent of the statistics of the actual lattice particles. In the case where the lattice particles are actually fermions with spin \textonehalf{}, e.g., 3He^{3}\mathrm{He}3He, we find an expansion formally identical to the problem of spinless fermions interacting among themselves and with a set of spins \textonehalf{} localized at lattice sites.
Electron-hole or quasiparticle representation plays a central role in describing electronic excitations in many-electron systems. For charge-neutral excitation, the electron-hole interaction kernel is the quantity of interest for calculating important excitation properties such as optical gap, optical spectra, electron-hole recombination, and electron-hole binding energies. The electron-hole interaction kernel can be formally derived from the density-density correlation function using both Green's function and time-dependent density functional theory (TDDFT) formalism. The accurate determination of the electron-hole interaction kernel remains a significant challenge for precise calculations of optical properties in the GW+BSE formalism. From the TDDFT perspective, the electron-hole interaction kernel has been viewed as a path to systematic development of frequency-dependent exchange-correlation functionals. Traditional approaches, such as many-body perturbation theory formalism, use unoccupied states (which are defined with respect to Fermi vacuum) to construct the electron-hole interaction kernel. However, the inclusion of unoccupied states has long been recognized as the leading computational bottleneck that limits the application of this approach for larger finite systems. In this work, an alternative derivation that avoids using unoccupied states to construct the electron-hole interaction kernel is presented. The central idea of this approach is to use explicitly correlated geminal functions for treating electron-electron correlation for both ground and excited state wave functions. Using this ansatz, it is derived using both diagrammatic and algebraic techniques that the electron-hole interaction kernel can be expressed only in terms of linked closed-loop diagrams. It is proved that the cancellation of unlinked diagrams is a consequence of linked-cluster theorem in real-space representation. The electron-hole interaction kernel derived in this work was used to calculate excitation energies in many-electron systems, and results were found to be in good agreement with the EOM-CCSD and GW+BSE methods. The numerical results highlight the effectiveness of the developed method for overcoming the computational barrier of accurately determining the electron-hole interaction kernel to applications of large finite systems such as quantum dots and nanorods.
Using a dual representation of lattice fermion models that is based on spin-charge transformation and fermionization of the original description, I derive an algorithm for diagrammatic Monte Carlo simulation of strongly correlated systems. This scheme allows eliminating large expansion parameters, as well as large corrections to the density matrix that generally prevent diagrammatic methods from being efficient in this regime. As an example, I compute the filling factor for the Hubbard model at infinite on-site repulsion and compare the results to controllable data obtained from numerical linked-cluster expansion. I find excellent agreement between the two methods, as well as rapid convergence of the diagrammatic series. I also report results for the momentum distribution and kinetic energy of the electrons.