Exactly solvable inhomogeneous fermion systems

TL;DR

This paper constructs and solves 15 inhomogeneous fermion systems on 1D lattices using discrete orthogonal polynomials, providing explicit eigenvalues, eigenstates, and correlation functions, with applications to related spin and stochastic models.

Contribution

It introduces a new class of exactly solvable inhomogeneous fermion and spin systems based on Askey scheme polynomials, with explicit solutions and correlations.

Findings

Explicit eigenvalues and eigenstates for 15 models

Derived ground state two-point correlation functions

Established solvability of related stochastic models

Abstract

15 exactly solvable inhomogeneous (spinless) fermion systems on one-dimensional lattices are constructed explicitly based on the discrete orthogonal polynomials of Askey scheme, e.g. the Krawtchouk, Hahn, Racah, Meixner, $q$-Racah polynomials. The Schr\"odinger and Heisenberg equations are solved explicitly, as the entire set of the eigenvalues and eigenstates are known explicitly. The ground state two point correlation functions are derived explicitly. The multi point correlation functions are obtained by Wick's Theorem. Corresponding 15 exactly solvable XX spin systems are also displayed. They all have nearest neighbour interactions. The exact solvability of Schr\"odinger equation means that of the corresponding Fokker-Planck equation. This leads to 15 exactly solvable Birth and Death fermions and 15 Birth and Death spin models. These provide plenty of materials for calculating interesting quantities, e.g. entanglement entropy, etc.