Hubbard Model
The Hubbard model is the simplest model for interacting particles in a lattice. It accounts for short range Coulomb repulsion between nearby clectrons with with an effective cnergy value U and effective kinetic energy of an electron capable of tunneling, often called "hopping", as t.
We will be concerned with two electrons in a system with two orbitals. This corresponds to 4 possible states:
-(⬆,⬇I =(1 0 0 0)spin-up in orbital 1 and spin-down in orbital 2
-(⬇,⬆I =(0 1 0 0) spin-up and spin-down electrons in orbital 2
-(⬆⬇,.I=(0 0 1 0) spin up and spin down electrons in orbital 1
-(.,⬆⬇I=(0 0 0 1), spin-up and spin-down electrons in orbital 2
The states with two electrons in the same orbital are called "ionic" and the states with electrons in different orbitals are "covalent."
The Hamiltonian representing the system in the above basis is given as
hat
(H)=ℏω[0,0,-t,-t]
[0,0,t,t]
[-t,t,U,0]
[-t,t,0,U]
Calculate the eigenvalues and (normalized) eigenvectors of hat(H).