Exactly Solvable Multicomponent Spinless Fermions

Abstract By generalizing 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 Krawtchouk, 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-neighbor 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.