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arXiv:2506.03228 · Quantum Information Processing

A simple mechanism for unstable degeneracies in local Hamiltonians

José Garre-Rubio

Published Nov 10, 2025• 0 citations
arXivHTMLPDF

Abstract

Abstract If a local Hamiltonian eigenstate is mapped to another state by local operators commuting with the Hamiltonian terms, the latter is also an eigenstate. This basic observation implies a no-go result for both being a unique ground state and having a degeneracy protected against local perturbations.
DOI
10.1007/s11128-025-04987-0

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Strongly correlated quantum matter is fundamentally defined by the tension between non-commuting quantum operators. Hamiltonians exhibiting macroscopic degeneracies are of general interest in this field because they imply an infinite susceptibility to any non-commuting perturbation. In moiré heterostructures, engineering such extensive degeneracies in the kinetic Hamiltonian creates a fertile garden for exotic strongly correlated phases of matter to emerge from the resulting flat bands. Here, we introduce a prescription to construct an infinite family of exact flat band Hamiltonians supported on the faces of arbitrary graphs. We demonstrate this algorithm on the faces of four Bravais lattices. Using a discrete graph generalization of the Atiyah-Singer index theorem, we prove that the extensive degeneracies of such face-graph Hamiltonians are protected by the local topology of the face-graph connectivity. The resulting macroscopic null spaces yield compact localised states that remain localised over time due to frustration in hopping pathways. We discuss the broad implications of such non-dispersing quantum modes in diverse settings, from arrested dynamics in quantum networks and quantum machine learning algorithms to Majorana-free topological quantum computation.
Abstract The usual way to proceed in quantum mechanics is to solve Schrödinger’s equation and obtain all observables of interest from the resulting eigenstates. Schrödinger’s equation depends on a given potential, which establishes a map from potentials to observables, one of them being the probability density. This chapter introduces the main idea of DFT, namely, that this map can be inverted. An example, for a one-electron system, shows how it is possible to construct, from a given density, that potential which produces that density as its ground-state density. This then establishes a formal map from ground-state densities to Hamiltonians and, hence, to observables. It is then discussed what these findings imply, and how they need to be generalized to establish the DFT formal framework.