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arXiv:2410.08467 · Progress of Theoretical and Experimental Physics

Lattice Fermions with Solvable Wide Range Interactions

Ryu Sasaki B

Published Jan 1, 2025• 2 citations
arXivHTMLPDF

Abstract

Abstract Exactly solvable (spinless) lattice fermions with wide range interactions are constructed explicitly based on exactly solvable stationary and reversible Markov chains $\mathcal {K}^R$ reported a few years earlier by Odake and myself. The reversibility of $\mathcal {K}^R$ with the stationary distribution $\pi$ leads to a positive classical Hamiltonian $\mathcal {H}^R$. The exact solvability of $\mathcal {H}^R$ warrants that of a spinless lattice fermion $c_x$, $c_x^\dagger$, $\mathcal {H}^R_f=\sum _{x,y\in \mathcal {X}}c_x^\dagger \mathcal {H}^R(x,y) c_y$ based on the principle advocated recently by myself. The reversible Markov chains $\mathcal {K}^R$ are constructed by convolutions of the orthogonality measures of the discrete orthogonal polynomials of the Askey scheme. Several explicit examples of the fermion systems with wide range interactions are presented.

DOI
10.1093/ptep/ptae190

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Abstract By generalizing the one-to-one correspondence between exactly solvable Hermitian matrices H=H†\mathcal {H}=\mathcal {H}^\daggerH=H† and exactly solvable spinless fermion systems Hf=∑x,ycx†H(x,y)cy\mathcal {H}_f=\sum _{x,y}c_x^\dagger \mathcal {H}(x,y)c_yHf​=∑x,y​cx†​H(x,y)cy​, 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 H\mathcal {H}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 K\mathcal {K}K, from which real symmetric matrices H\mathcal {H}H are uniquely defined by the similarity transformation in terms of the square root of the stationary distribution. The fermions have wide-range interactions.
Abstract 15 exactly solvable inhomogeneous (spinless) fermion systems on 1D lattices are constructed explicitly based on the discrete orthogonal polynomials of the Askey scheme, e.g. Krawtchouk, Hahn, Racah, Meixner, and q-Racah polynomials. The Schrödinger 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 multipoint correlation functions are obtained by Wick’s theorem. 15 corresponding exactly solvable XX spin systems are also displayed. They all have nearest-neighbor interactions. The exact solvability of the Schrödinger equation means that the corresponding Fokker–Planck equation is also exactly solvable. This leads to 15 exactly solvable birth and death fermions and 15 birth and death spin models. These provide plenty of material for calculating interesting quantities, e.g. entanglement entropy, etc.
A coinless quantisation procedure of general reversible Markov chains on graphs is presented. A quantum Hamiltonian H is obtained by a similarity transformation of the fundamental transition probability matrix K in terms of the square root of the reversible distribution. The evolution of the classical and quantum Markov chains is described by the solutions of the eigenvalue problem of the quantum Hamiltonian H. About twenty plus exactly solvable Markov chains based on the hypergeometric orthogonal polynomials of Askey scheme, derived by Odake-Sasaki, would provide a good window for scrutinising the quantum/classical contrast of Markov chains. Among them five explicit examples, related to the Krawtchouk, Hahn, q-Hahn, Charlier and Meixner, are demonstrated to illustrate the actual calculations.
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