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Journal of Mathematical Physics

Diagonalization methods for the general bilinear Hamiltonian of an assembly of bosons

Constantino Tsallis

Published Jan 1, 1978• 53 citations
Publisher

Abstract

The problem of the exact diagonalization of the Hamiltonian of an assembly of N bilinearly interacting bosons is discussed in what concerns the eigenvalues as well as for the expression of the new boson operators in terms of the old ones. The method is equivalent to the standard equation-of-motion approach, nevertheless sensibly more concise. Three sets of operational rules are indicated, and their use is exhibited in some examples. In some particular cases of practical importance (for example, when all the coefficients of the Hamiltonian are real), the research of the eigenvalues has been compacted as much as possible.

DOI
10.1063/1.523549

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The problem of the exact diagonalization of the Hamiltonian of an assembly of N bilinearly interacting bosons is discussed in what concerns the eigenvalues as well as for the expression of the new boson operators in terms of the old ones. The method is equivalent to the standard equation-of-motion approach, nevertheless sensibly more concise. Three sets of operational rules are indicated, and their use is exhibited in some examples. In some particular cases of practical importance (for example, when all the coefficients of the Hamiltonian are real), the research of the eigenvalues has been compacted as much as possible.
We take the Bose-Hubbard model to illustrate exact diagonalization techniques in a pedagogical way. We follow the road of first generating all the basis vectors, then setting up the Hamiltonian matrix with respect to this basis, and finally using the Lanczos algorithm to solve low lying eigenstates and eigenvalues. Emphasis is placed on how to enumerate all the basis vectors and how to use the hashing trick to set up the Hamiltonian matrix or matrices corresponding to other quantities. Although our route is not necessarily the most efficient one in practice, the techniques and ideas introduced are quite general and may find use in many other problems.
We calculate formal time derivatives of the mean values of the standard position, velocity, and mechanical momentum operators, i. e., the Ehrenfest theorem for a one-dimensional Dirac particle in the coordinate representation.We show that these derivatives contain boundary terms that essentially depend on the values taken there by characteristic bilinear densities.We do not automatically take the boundary terms to vanish (as is usually done); nevertheless, we relate the boundary terms to similar terms that must be zero if one requires the hermiticity of certain specific unbounded operators.Throughout the article, we thoroughly discuss and illustrate all these aspects, which include the relations to certain boundary conditions.To clarify, we call our approach formal because all operations involving operators (for example, some operators products) are performed without respecting the restrictions imposed by the sets of functions on which the self-adjoint operators can act.Moreover, the Dirac Hamiltonian that we consider in our calculations contains a potential that is the time component of a Lorentz two-vector; nevertheless, we also obtain and concisely discuss the Ehrenfest theorem for a Hamiltonian with the most general Lorentz potential in (1+1) dimensions.
Despite the advances in the development of numerical methods analytical approaches still play the key role on the way towards a deeper understanding of many-particle systems. In this regard, diagonalization schemes for Hamiltonians represent an important direction in the field. Among these techniques the method, presented here, might be that approach with the widest range of possible applications: We demonstrate that both stepwise and continuous unitary transformations to diagonalize the many-particle Hamiltonian as well as perturbation theory and also nonperturbative treatments can be understood within the same theoretical framework. The new method is based on the introduction of generalized projection operators and allows us to develop a renormalization scheme which is used to evaluate directly the physical quantities of a many-particle system. The applicability of this approach is shown for two important elementary many-particle problems.