Ground-state properties of a supersymmetric fermion chain

TL;DR

This paper investigates the ground state of a supersymmetric fermion chain, revealing exact results, dualities, and polynomial structures related to the Painlevé VI equation, advancing understanding of strongly interacting quantum systems.

Contribution

It introduces exact expressions for order parameters, uncovers a duality between weak and strong couplings, and links ground state polynomials to Painlevé VI, providing new analytical tools for supersymmetric fermion chains.

Findings

Exact formulas for order parameters and critical exponents.

Identification of polynomial structures related to Painlevé VI.

Explicit verification of polynomial relations up to 24 sites.

Abstract

We analyze the ground state of a strongly interacting fermion chain with a supersymmetry. We conjecture a number of exact results, such as a hidden duality between weak and strong couplings. By exploiting a scale free property of the perturbative expansions, we find exact expressions for the order parameters, yielding the critical exponents. We show that the ground state of this fermion chain and another model in the same universality class, the XYZ chain along a line of couplings, are both written in terms of the same polynomials. We demonstrate this explicitly for up to N = 24 sites, and provide consistency checks for large N. These polynomials satisfy a recursion relation related to the Painlev\'e VI differential equation, and using a scale-free property of these polynomials, we derive a simple and exact formula for their limit as N goes to infinity.