Local symmetries and extensive ground-state degeneracy of a 1D supersymmetric fermionic chain

We study a $1$D supersymmetric (SUSY) hard-core fermion model first proposed by Fendley, Schoutens, and de Boer [Phys. Rev. Lett. 90, 120402 (2003)]. We focus on the full Hilbert space instead of a restricted subspace. Exact diagonalization shows the degeneracy of zero-energy states scales exponentially with size of the system, with a recurrence relation between different system sizes. We solve the degeneracy problem by showing the ground states can be systematically constructed by inserting immobile walls of fermions into the chain. Mapping the counting problem to a combinatorial one and obtaining the exact generating function, we prove the recurrence relation on both open and periodic chains. We also provide an explicit mapping between ground states, giving a combinatorial explanation of the recurrence relation.

💡 Research Summary

The paper investigates the one‑dimensional supersymmetric (𝒩 = 2) hard‑core fermion chain originally introduced by Fendley, Schoutens, and de Boer (the FSD model). While the original analysis was confined to a restricted Hilbert space where neighboring sites could never both be occupied, the authors lift this constraint and study the full Hilbert space. The supersymmetry generator is
(Q^{\dagger}= \sum_{i} P_{i-1},c^{\dagger}{i},P{i+1})
with (P_{i}=1-n_{i}) the projector onto an empty site and (c^{\dagger}_{i}) the fermionic creation operator. The Hamiltonian is the usual supersymmetric form (H={Q,Q^{\dagger}}). Because (Q^{\dagger}) creates a fermion only when both neighbours are empty, the dynamics in the unrestricted space allow many more configurations than in the constrained case.

Exact diagonalization for chains up to moderate length reveals that the number of zero‑energy (ground) states, denoted (a(L)), grows exponentially with the system size (L). For open boundary conditions the sequence begins (starting at (L=5)) as 4, 4, 8, 16, 24, 40, 72, 120,… and satisfies the simple linear recurrence
(a_{n}=a_{n-1}+2a_{n-3}) for (n\ge 4).
For periodic chains the same recurrence holds when (n\equiv 3m-1) (mod 3); otherwise additional correction terms appear, but the overall exponential growth persists.

The central conceptual advance is the introduction of “immobile walls”: a pair of adjacent occupied sites (the pattern “11”) is invariant under both (Q) and (Q^{\dagger}). Consequently any ground‑state configuration can be viewed as a sequence of such walls separated by “empty intervals”. Within each interval the dynamics reduce to the original constrained FSD model, whose ground‑state structure is already known. Therefore the full ground‑state manifold is generated by (i) choosing a placement of walls (equivalently a partition of the chain length) and (ii) selecting a ground‑state configuration for each interval. This mapping reduces the counting problem to a combinatorial partition problem.

By translating the wall‑placement problem into integer partitions with parts of size 1 and 2 (reflecting the possible lengths of empty intervals modulo 3), the authors derive an exact generating function for the degeneracy: \