Supersymmetry breaking and Nambu-Goldstone fermions in an extended Nicolai model

TL;DR

This paper investigates a supersymmetric model of interacting spinless fermions in one dimension, revealing spontaneous SUSY breaking and identifying massless Dirac fermions as Nambu-Goldstone fermions in the low-energy spectrum.

Contribution

It provides a rigorous analysis of SUSY breaking and low-energy excitations in an extended Nicolai model with a tunable parameter, connecting ground state properties to Nambu-Goldstone fermions.

Findings

SUSY is spontaneously broken for any positive coupling g.

Ground-state energy density remains non-zero at large g in the infinite limit.

Low-energy excitations are massless Dirac fermions, identified as Nambu-Goldstone fermions.

Abstract

We study a model of interacting spinless fermions in a one dimensional lattice with supersymmetry (SUSY). The Hamiltonian is given by the anti-commutator of two supercharges $Q$ and $Q^\dagger$, each of which is comprised solely of fermion operators and possesses one adjustable parameter $g$. When the parameter $g$ vanishes, the model is identical to the one studied by Nicolai [H. Nicolai, J. Phys. A: Math. Gen. \textbf{9}, 1497 (1976)], where the zero-energy ground state is exponentially degenerate. On the other hand, in the large-$g$ limit the model reduces to the free-fermion chain with a four-fold degenerate ground state. We show that for finite chains SUSY is spontaneously broken when $g > 0$. We also rigorously prove that for sufficiently large $g$ the ground-state energy density is nonvanishing in the infinite-volume limit. We further analyze the nature of the low-energy excitations by employing various techniques such as rigorous inequalities, exact numerical diagonalization, and renormalization group method with bosonization. The analysis reveals that the low-energy excitations are described by massless Dirac fermions (or Thirring fermions more generally), which can be thought of as Nambu-Goldstone fermions from the spontaneous SUSY breaking.