Exactly solvable multicomponent spinless fermions

TL;DR

This paper constructs four exactly solvable multicomponent spinless fermion systems by generalizing the correspondence with certain matrices, relating them to special polynomials and Markov chains, and analyzing their interaction types.

Contribution

It explicitly constructs four types of exactly solvable multicomponent fermion systems linked to special polynomials and Markov chain matrices, expanding the class of solvable models.

Findings

Fermion systems related to multivariate Krawtchouk, Meixner, and Rahman-like polynomials.

Eigenvectors of real symmetric matrices and Markov chain matrices used to define fermion interactions.

Fermions exhibit nearest neighbour or wide-range interactions depending on the polynomial type.

Abstract

By generalising the one to one correspondence between exactly solvable hermitian matrices $\mathcal{H}=\mathcal{H}^\dagger$ and exactly solvable spinless fermion systems $\mathcal{H}_f=\sum_{x,y}c_x^\dagger\mathcal{H}(x,y)c_y$, four types of exactly solvable multicomponent fermion systems are constructed explicitly. They are related to the multivariate Krawtcouk, Meixner and two types of Rahman like polynomials, constructed recently by myself. The Krawtchouk and Meixner polynomials are the eigenvectors of certain real symmetric matrices $\mathcal{H}$ which are related to the difference equations governing them. The corresponding fermions have nearest neighbour interactions. The Rahman like polynomials are eigenvectors of certain reversible Markov chain matrices $\mathcal{K}$, from which real symmetric matrices $\mathcal{H}$ are uniquely defined by the similarity transformation in terms of the square root of the stationary distribution. The fermions have wide range interactions.