Supersymmetry breaking and Nambu-Goldstone fermions with cubic dispersion

TL;DR

This work presents a one-dimensional lattice model of spinless fermions with supersymmetry such that the Hamiltonian is the anticommutator of two nilpotent supercharges $Q$ and $Q^\dagger$, and the particle-number symmetry is absent for generic $g$. At $g=0$ the model has an extensive zero-energy ground-state degeneracy, while for nonzero $g$ SUSY is spontaneously broken in finite systems and, above a finite threshold $g>4/\pi$, in the infinite-volume limit; the authors provide rigorous and numerical evidence for spontaneous breaking and analyze low-energy excitations. They construct variational states to bound the NG fermion energy and show the existence of gapless modes, which, in the large-$g$ limit, display cubic dispersion $\omega(p)\propto |p|^3$, confirmed by exact diagonalization. This cubic NG fermion dispersion constitutes a novel non-relativistic realization of SUSY breaking, stable against SUSY-preserving perturbations, and extendable to two dimensions where similar cubic NG modes appear on a triangular lattice, linking to Majorana-like excitations in quantum spin liquids.

Abstract

We introduce a lattice fermion model in one spatial dimension with supersymmetry (SUSY) but without particle number conservation. The Hamiltonian is defined as the anticommutator of two nilpotent supercharges $Q$ and $Q^\dagger$. Each supercharge is built solely from spinless fermion operators and depends on a parameter $g$. The system is strongly interacting for small $g$, and in the extreme limit $g=0$, the number of zero-energy ground states grows exponentially with the system size. By contrast, in the large-$g$ limit, the system is non-interacting and SUSY is broken spontaneously. We study the model for modest values of $g$ and show that under certain conditions spontaneous SUSY breaking occurs in both finite and infinite chains. We analyze the low-energy excitations both analytically and numerically. Our analysis suggests that the Nambu-Goldstone fermions accompanying the spontaneous SUSY breaking have cubic dispersion at low energies.

Figures (5)

• Figure 1: Schematics of individual terms in the Hamiltonian $H$. (a) the paring term ($H_{\rm free}$), (b) the nearest-neighbor and the next-nearest-neighbor repulsive interactions ($H_1$), and (c) the first line of the third term ($H_2$). Green (gray) balls represent spinless fermions.
• Figure 2: Dispersion relation of the $\mathbb{Z}_2$ Nicolai model for $g=4$. We plot energy spectrum $\varepsilon(p)$ for $N=11,13,15$. Here, $p$ is the wave number. The gray solid curve indicates the one-particle dispersion relation of $H_{\rm free}$ and is described by $2g|f(p)|$. Gray dotted curve is described by $4g|f(p/2)|$ and indicates the dispersion of two-particle bound states of $H_{\rm free}$.
• Figure 3: Dispersion relation of the $\mathbb{Z}_2$ Nicolai model for $g=4$. We plot energy spectrum $\varepsilon(p)$ for $N=10,12,14$. Here, $p$ is the wave number. The gray solid curve indicates the one-particle dispersion relation of $H_{\rm free}$ and is described by $2g|f(p)|$. The gray dotted and dashed curves are described by $4g|f(p/2)|$ and $2g|f(\pi-p)|$, respectively. They indicate the dispersion of two-particle bound states of $H_{\rm free}$.
• Figure 4: Energy difference between the first excited and the ground states as a function of $1/N^3$ for $g=2,4,6,8$. Lines are fits to the data of $N=15,\dots,20$.
• Figure 5: The lowest excitation energy $\Delta E$ of $H$ [Eq. (\ref{['eq:quartHam']})] as a function of $1/N^3$ for $g=2,4,6,8$, $g_3=1/3$ and $g_5=1/5$. Lines are fits to the data of $N=17,\dots,20$.