Statistical Physics and Condensed Matter

# Flow Equations for Hamiltonians

## Introduction

The method of flow equations is a non-perturbative method that can be used to diagonalize, block-diagonalize or renormalize a given Hamiltonian. A continuous unitary transformation applied to given Hamiltonian yields flow equations. The method is especially useful for systems with different energy scales. It has been applied to various problems in Condensed Matter Physics including
• Electron-phonon coupling
• Dissipative quantum systems
• Interacting fermions, Hubbard model
• Spin and pseudo-spin models
• Impurity problems
In many cases the method yields very good results for the spectrum of the Hamiltonian and for dynamic correlation functions.

The flow equation desribes the flow of a Hamiltonian H as a function of the flow-parameter l and has the form dH(l)/dl = [η(l),H(l)], where the generator η(l) of the unitary transformation has typically the form η(l)=[H(l),X]. Here X may or may not depend on H(l). H(0) is the initial Hamiltonian and H(infinity) has normally the desired diagonal or block-diagonal form.

A list of papers from our institute using flow equations for Hamiltonians is provided.
This list is also available with abstracts.
Book on Flow Equations Stefan Kehrein, The Flow Equation Approach to Many-Particle Systems
Springer Tracts in Modern Physics 217

A similar method using continuous unitary transformations called similarity renormalization has been developed by Stanislaw Glazek and Kenneth G. Wilson and has been applied to problems in quantum chromodynamics and quantum electrodynamics using light front Hamiltonians. We list some references on this subject as well as other contributions from other groups.

Meanwhile (2002) we have learned from Volker Bach, that such a method called double bracket flow and isospectral flow, resp., has also been developed by the mathematicians Roger W. Brockett, Moody T. Chu, and Kenneth R. Driessel.

Some papers with similar ideas, which however do not yet implement, that the generator of the transformation is continuous and depends on the running Hamiltonian itself are listed here.

## International Workshop, March 10-21, 2003 on

### Functional Renormalization of Interacting Quantum Many-Body Systems at the Max-Planck-Institute for the Physics of Complex Systems in Dresden Bibliography to the Workshop compiled by Manfred Salmhofer Scientific Coordinators of the Workshop: Stefan Kehrein, Andreas Mielke, Götz S. Uhrig, Franz Wegner.

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December 2003