On the similarity transformations in second quantized fermion-boson interacting hamiltonian and the BCS theory

C.A Dartora,Fillipi Klos Rodrigues De Campos

doi:10.1590/1806-9126-rbef-2015-0032

C.A Dartora, Fillipi Klos Rodrigues De Campos

Open Access

https://doi.org/10.1590/1806-9126-rbef-2015-0032

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Abstract

A similarity transformation is an equivalence relation between square matrices which preserves determinant, trace and eigenvalues, playing a key role in quantum mechanics in simplifying complex hamiltonian systems and improving analytical results attainable from the use of perturbation theory. As a prototypical example, the conventional BCS theory of superconductivity is usually derived from a similarity transformation of the original electron-phonon hamiltonian, written in second quantized version. Here we discuss the general method for writing the similarity transformation operator in second quantized form, allowing one to recast a hamiltonian describing an interacting fermion-boson system into an effective theory in which only the desired degrees of freedom are kept after the transformation.

Highlights

There is little room for controversy in stating that fermion-boson interacting systems are among the most general problems in physics, since they can be used to describe almost everything in the real world, at least at a quantum-mechanical level
Despite the apparent simplicity of fermion-boson models when written in second quantization formalism, the vast majority of these problems are not solvable in an exact manner and, demand clever methods and mathematical tricks in order to obtain an analytically tractable perturbation scheme, allowing one to predict meaningful physical results
The BCS theory successfully explains the conventional superconductivity of materials, by considering attractive electron-electron interactions mediated by the exchange of virtual phonons(the quantized excitations of lattice vibrations), which leads to the formation of the so-called Cooper pairs, a bound state of electrons in momentum space

Summary

Introduction

There is little room for controversy in stating that fermion-boson interacting systems are among the most general problems in physics, since they can be used to describe almost everything in the real world, at least at a quantum-mechanical level. Despite the apparent simplicity of fermion-boson models when written in second quantization formalism, the vast majority of these problems are not solvable in an exact manner and, demand clever methods and mathematical tricks in order to obtain an analytically tractable perturbation scheme, allowing one to predict meaningful physical results. On the similarity transformations in second quantized fermion-boson interacting hamiltonian. In the canonical formalism, the conventional BCS theory of superconductivity is usually derived from a similarity transformation of the original electron-phonon hamiltonian [12]. We discuss the general method for writing similarity transformation operators in second quantized form, allowing one to recast a hamiltonian describing an interacting fermionboson system into an effective theory, in which only the desired degrees of freedom are kept after the transformation.

The general form of fermion-boson interacting hamiltonians

The general method of similarity transformation

The conventional BCS superconductivity theory

Conclusion