sl(M+1) Construction of Quasi-solvable Quantum M-body Systems

TL;DR

This paper introduces a systematic method for constructing permutation-symmetric quasi-solvable quantum many-body systems using sl(M+1) algebra, classifies models, and explores M-body interactions without losing solvability.

Contribution

It presents a novel algebraic framework for building and classifying quasi-solvable quantum many-body systems with permutation symmetry, including generalizations to M-body interactions.

Findings

Derived a systematic construction method for quasi-solvable models.

Classified five inequivalent Inozemtsev-type models.

Explored inclusion of M-body interactions preserving quasi-solvability.

Abstract

We propose a systematic method to construct quasi-solvable quantum many-body systems having permutation symmetry. By the introduction of elementary symmetric polynomials and suitable choice of a solvable sector, the algebraic structure of sl(M+1) naturally emerges. The procedure to solve the canonical-form condition for the two-body problem is presented in detail. It is shown that the resulting two-body quasi-solvable model can be uniquely generalized to the M-body system for arbitrary M under the consideration of the GL(2,K) symmetry. An intimate relation between quantum solvability and supersymmetry is found. With the aid of the GL(2,K) symmetry, we classify the obtained quasi-solvable quantum many-body systems. It turns out that there are essentially five inequivalent models of Inozemtsev type. Furthermore, we discuss the possibility of including M-body (M>=3) interaction terms without destroying the quasi-solvability.