Non-local space-time supersymmetry on the lattice

TL;DR

This paper reveals hidden nonlocal supersymmetry in certain one-dimensional quantum systems, enabling bounds and exact solutions for ground state energies, and discusses potential generalizations to other algebraic structures.

Contribution

It demonstrates the presence of nonlocal supersymmetry in quantum chains and models, providing new insights into their solvability and spectral properties.

Findings

Supersymmetry provides bounds on ground state energies.

Exact ground state energies are derived for specific boundary conditions.

Space-time supersymmetry coexists with known algebraic symmetries in models.

Abstract

We show that several well-known one-dimensional quantum systems possess a hidden nonlocal supersymmetry. The simplest example is the open XXZ spin chain with \Delta=-1/2. We use the supersymmetry to place lower bounds on the ground state energy with various boundary conditions. For an odd number of sites in the periodic chain, and with a particular boundary magnetic field in the open chain, we can derive the ground state energy exactly. The supersymmetry thus explains why it is possible to solve the Bethe equations for the ground state in these cases. We also show that a similar space-time supersymmetry holds for the t-J model at its integrable ferromagnetic point, where the space-time supersymmetry and the Hamiltonian it yields coexist with a global u(1|2) graded Lie algebra symmetry. Possible generalizations to other algebras are discussed.