Exactly solvable multicomponent spinless fermions

TL;DR

The work constructs four exactly solvable, multicomponent spinless fermion systems from multivariate hypergeometric orthogonal polynomials: multivariate Krawtchouk (2), multivariate Meixner (3), and two Rahman-like families (type 1 and type 2). Each polynomial family serves as the eigenbasis for a corresponding real symmetric or Markov-chain operator, whose spectrum is linear or multiplicative and determined by roots of associated matrices (F(p), F(c), F^(1), F^(2)). By forming Hermitian Hamiltonians via similarity transformations tied to stationary distributions, the authors obtain exactly solvable fermionic lattices with either nearest-neighbor or wide-range interactions, and provide explicit diagonalizations through momentum-space representations. These constructions offer a rich, solvable framework for exploring many-body quantum phenomena and the roles of multidimensional orthogonal polynomials in quantum Hamiltonians.

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.