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Lieb-Schultz-Mattis theorem in long-range interacting systems and generalizations

by Ruizhi Liu, Jinmin Yi, Shiyu Zhou, Liujun Zou

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Authors (as registered SciPost users):

Liujun Zou

Abstract

In a unified fashion, we establish Lieb-Schultz-Mattis theorem in long-range interacting systems and its generalizations. We show that, for a quantum spin chain, if the multi-spin interactions decay fast enough as their ranges increase and the Hamiltonian has an anomalous symmetry, the Hamiltonian cannot have a unique gapped symmetric ground state. If the Hamiltonian contains only 2-spin interactions, this theorem holds when the interactions decay faster than $1/r^2$, with $r$ the distance between the two interacting spins. Moreover, any pure state with an anomalous symmetry, which may not be a ground state of any natural Hamiltonian, must be long-range entangled. The symmetries we consider include on-site internal symmetries combined with lattice translation symmetries, and they can also extend to purely internal but non-on-site symmetries. Moreover, these internal symmetries can be discrete or continuous. We explore the applications of the theorems through various examples.

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