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

We study a model of interacting spinless fermions in a one dimensional lattice with supersymmetry (SUSY). The Hamiltonian is given by the anticommutator of two supercharges $Q$ and $Q^{†}$, 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 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.

Figure 1

Schematic representation of (a) the hopping term ($H_{\mathrm{hop}}$), (b) the nearest-neighbor attractive and the next-nearest neighbor repulsive interactions ($H_{\text{charge}}$), and (c) the pair hopping term ($H_{\mathrm{pair}}$). Green (gray) balls represent spinless fermions.

Figure 2

Finite-size scaling of the lowest excitation energy $\mathrm{\Delta }E$ vs $1/N$ for $g=2$, 4, 6, 8. The lines are linear fits to the data.

Figure 3

The TL parameter $K$ as a function of $g$ for $N=16$, 18, 20, 22. The curves are fits to the five data points. As $N$ increases, $K$ converges to a value labeled as extrapolation.

Figure 4

TL parameter $K$ vs system length $N$ for $g=2$, 3, 4, 6, 8. The lines are fits to the data points. Their intercepts are the TL parameters $K$ in the infinite-size limit.