Soliton pairs and hidden supersymmetry in a chain of interacting Majorana particles
This paper studies a one-dimensional chain of interacting Majorana fermions that realizes a supersymmetric phase known as the tricritical Ising (TCI) point and the nearby gapped, ordered phase. The authors ask how supersymmetry (SUSY) shows up once the system is pushed away from the critical point and what the lowest-energy excitations look like in the gapped phase. Using a lattice model introduced by O’Brien and Fendley, they combine analytical reasoning and numerical calculations to probe these questions. The model they study has two basic ingredients. Neighboring Majorana operators hybridize with amplitude t, and four-Majorana interaction terms have strength g. Majorana fermions are particles that are their own antiparticles and are often discussed in condensed-matter settings. The authors focus on periodic boundary conditions (the Ramond sector) and work near the point in parameter space where the system is known to realize the supersymmetric TCI field theory. Numerically, they use the density matrix renormalization group (DMRG) and finite-size scaling to follow how signatures of SUSY change as g/t is varied. To detect SUSY away from the critical point the authors analyze a ratio R built from matrix elements of a lattice version of the supercharge Q. The supercharge is an operator that converts fermionic excitations to bosonic ones and vice versa. At the TCI point the field theory predicts R=1/7. The paper reports that R stays finite near the TCI point on the gapped side but shows a sharp upturn that moves toward the TCI point with increasing system size on the Ising side. In the thermodynamic limit R appears to diverge on the Ising side, signaling spontaneous breaking of SUSY there. Deeper into the gapped phase R decays continuously toward zero, which the authors interpret as the gradual disappearance of the SUSY signature while SUSY still survives close to the transition. Turning to the low-energy excitations, the authors present evidence that the first excited states in the supersymmetric gapped regime are soliton–antisoliton pairs. A soliton here is a localized domain wall that separates regions with the two different ordered patterns of the ground states. Each soliton or antisoliton binds an emergent, localized Majorana mode. Together the two localized Majorana modes form a single nonlocal Dirac fermion. Whether that Dirac fermion is occupied or not distinguishes eigenstates with even or odd total fermion parity. With periodic boundary conditions each eigenstate therefore has a partner of opposite fermion parity, giving at least a twofold degeneracy. There are important caveats. The lattice supercharges used do not strictly commute like their continuum counterparts, so the mapping to the field theory works under renormalization rather than exactly on the lattice. The precise position of the TCI point has finite-size uncertainties, and the R diagnostic shows finite-size features that must be extrapolated carefully. The evidence presented is numerical and interpretive: it builds a consistent picture that soliton–antisoliton pairs and emergent Majorana modes explain the low-energy spectrum near the supersymmetric gapped phase, but it does not constitute an exact, model-independent proof.