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International Journal of Modern Physics A

FLOW EQUATIONS FOR HAMILTONIANS

Franz Wegner

Published Apr 30, 2001• 1 citation
Publisher

Abstract

A method to diagonalize or block-diagonalize Hamiltonians by means of an appropriate continuous unitary transformation is reviewed. We will outline (i) the procedure for the elimination of the electron-phonon interaction and the construction of the effective attractive electron-electron interaction, and (ii) the application to some systems with electron-electron interaction (n-orbital model and Hubbard model).

DOI
10.1142/s0217751x0100458x

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arXiv:nucl-th/0305052•May 19, 2003•3 citations
A method to diagonalize or block-diagonalize Hamiltonians by means of an appropriate continuous unitary transformation is reviewed. We will outline (i) the procedure for the elimination of the electron-phonon interaction and the construction of the effective attractive electron-electron interaction, and (ii) the application to some systems with electron-electron interaction (n-orbital model and Hubbard model).
A recently [9] proposed method of continuous unitary transformations is used to eliminate the interaction between electrons and phonons. The differential equations for the couplings represent an infinitesimal formulation of a sequence of Fröhlichtransformations [5]. The two approaches are compared. Our result will turn out to be less singular than Fröhlich’s. Furthermore the interaction between electrons belonging to a Cooper-pair will always be attractive in our approach. Even in the case where Fröhlich’s transformation is not defined (Fröhlich actually excluded these regions from the transformation), we obtain an elimination of the electron-phonon interaction. This is due to a sufficiently slow change of the phonon energies as a function of the flow parameter. 1 New address: Max-Planck-Institut für Kolloid- und Grenzflächenforschung, Kantstraße 55,
In the present time we observe a growing number of publications where the, so-called, flow equations are successfully used to diagonalize Hamiltonians by means of an appropriate unitary transformation. Here we discuss and compare the flow equations (FE) method (proposed in 1994) with the method of one step continuous unitary transformations (OS CUT) (proposed in 1982). It is shown that the FE method can be considered as a generalization of the OS CUT approach to the case of parameter dependent generator. The OS CUT method gives linear differential equations for the diagonalization procedure. In the FE method the system of differential equations is nonlinear. Finally we discuss the generalization of idea of continuous unitary transformations for the case of quantum equations of motion (Heisenberg picture and density matrix).
This thesis presents an overview of the flow equations recently introduced by Wegner. The little known mathematical framework of the flow in the manifold of unitarily equivalent matrices, as discovered in the mathematical literature before Wegner's paper, is established in the initial chapter and used as a background for the entire presentation. The application of flow equations to the Foldy-Wouthuysen transformation and to the elimination of the electron-phonon coupling in a solid is reviewed. Recent flow equations approaches to the Lipkin model are examined thoroughly, paying special attention to their utility near the phase change boundary. We present more robust schemes by requiring that expectation values be flow dependent; either through a variational or self-consistent calculation. The similarity renormalization group equations recently developed by Glazek and Wilson are also reviewed. Their relationship to Wegner's flow equations is investigated through the aid of an instructive model.