Supersymmetry on the lattice: Geometry, Topology, and Flat Bands

Abstract

In quantum mechanics, supersymmetry (SUSY) posits an equivalence between two elementary degrees of freedom, bosons, and fermions defined by local rules. Here we apply it to find connections between bosonic and fermionic lattice models in the realm of condensed matter physics and uncover a novel 5-fold way topology it demands in these systems. At the single-particle level, our connections pair a bosonic and fermionic lattice model, either describing the hopping of number-conserving particles or local couplings between fermion parity-conserving particles. The pair are isospectral except for zero modes, such as flat bands, quadratic band touchings, and nexus points, whose existence is undergirded by the Witten index of the SUSY theory. We develop a unifying framework to formulate these SUSY connections in terms of general lattice graph correspondences. Notably, in this framework, the supercharge operator that generates SUSY is Hermitian and can itself be interpreted as a hopping Hamiltonian on a bipartite lattice, a feature that enables the discovery of materials or model lattices hosting the SUSY partners. To illustrate the power of SUSY, we present 16 use cases of SUSY, that span topics including frustrated magnets, Kitaev spin liquids, and topological superconductors, the majority of which turn out to provide insights into the discovery and design of flat bands and topological materials.

arXiv comment39 pages, 30 figures

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• Kagome metals
• Instanton-Induced Supersymmetry Breaking in Topological Semimetals
• Exceptionally deficient topological square-root insulators
• SUSY design of smooth quantum rings in graphene
• Decorated cluster lattices: a natural framework for classical spin liquids and flat bands
• Emergent spacetime supersymmetry at 2D fractionalized quantum criticality
• Exceptional horns in $n$-root graphene and Lieb photonic ring lattices