Analysis of the Hubbard ring using counting operators

Abstract

We prove three theorems about the use of a counting operator to study the spectrum of model Hamiltonians. We analytically calculate the eigenvalues of the Hubbard ring with four lattice positions and apply our theorems to describe the observed level crossings.

A multiplet analysis of spectra in the presence of broken symmetries

“…The simplest one is a uniform field in the x-direction follows immediately that M = L z is a generalized symmetry of H, with γ = − . The method of classifying eigenvalues into multiplets, as worked out in [7], works well if a basis of common eigenvectors of H 0 and M is explicitly known. This is only partially the case for the hamiltonian of the hydrogen atom with M = L z since the spectrum of these operators is partly discrete, partly continuous.…”

“…This special case has been investigated in [7,8]. In a slightly different context ( 19) is called the Dolan-Grady condition [9].…”

“…From these studies it is clear that supersymmetry is a rather exceptional phenomenon not present in the more common models of solid state physics. On the other hand it was pointed out in [7,8] that many hamiltonians H satisfy a higher order commutator relation with respect to certain hermitian operators M . In the present paper such an operator M is called a generalised symmetry of the hamiltonian H. This is motivated by showing that when supersymmetry is broken by adding a perturbation term linear in the supercharges then some symmetry operators become a generalised symmetry of the new hamiltonian.…”

“…In [7,8] the ladder operator R and its conjugate R † are defined as operators satisfying [R, M ] = γR. The parameter γ is assumed to be real.…”

“…A point of discussion is whether one should add in the above definition the requirement that the operators R † R and RR † commute with H 0 . This is the case in many examples and has been used in [7] to simplify the analysis of the spectrum of model hamiltonians.…”