Exact Results for the Tavis-Cummings and H$\boldsymbol{\ddot{u}}$ckel Hamiltonians with Diagonal Disorder

TL;DR

This paper introduces an exact method to compute electronic states in disordered Hamiltonians, revealing how disorder affects polaritonic states and providing solutions for Hückel models with Cauchy disorder.

Contribution

It develops a deterministic complex Hamiltonian approach to exactly solve disordered one-electron Hamiltonians, including Tavis-Cummings and Hückel models, with implications for molecular energy level analysis.

Findings

Exact solutions for molecular states in microcavities for any N

Disorder influences the width of polaritonic states based on energy distribution

Method applicable to Hückel Hamiltonians with on-site Cauchy disorder

Abstract

We present an exact method to calculate the electronic states of one electron Hamiltonians with diagonal disorder. We show that the disorder averaged one particle Green's function can be calculated directly, using a deterministic complex (non-Hermitian) Hamiltonian. For this, we assume that the molecular states have a Cauchy (Lorentz) distribution and use the supersymmetric method which has already been used in problems of solid state physics. Using the method we find exact solutions to the states of $N$ molecules, confined to a microcavity, for any value of $N$. Our analysis shows that the width of the polaritonic states as a function of $N$ depend on the nature of disorder, and hence can be used to probe the way molecular energy levels are distributed. We also show how one can find exact results for H$\ddot{u}$ckel type Hamiltonians with on-site, Cauchy disorder and demonstrate its use.