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Lieb-Schultz-Mattis in Higher Dimensions

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arxiv cond-mat/0305505 v6 pith:WY4HO5HC submitted 2003-05-21 cond-mat.str-el hep-thmath-phmath.MP

M. B. Hastings This is my paper · ORCID
classification cond-mat.str-elhep-thmath-phmath.MP
keywords caseoperatorsspinsystemsboundsexpectationhigherlieb-schultz-mattis
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A generalization of the Lieb-Schultz-Mattis theorem to higher dimensional spin systems is shown. The physical motivation for the result is that such spin systems typically either have long-range order, in which case there are gapless modes, or have only short-range correlations, in which case there are topological excitations. The result uses a set of loop operators, analogous to those used in gauge theories, defined in terms of the spin operators of the theory. We also obtain various cluster bounds on expectation values for gapped systems. These bounds are used, under the assumption of a gap, to rule out the first case of long-range order, after which we show the existence of a topological excitation. Compared to the ground state, the topologically excited state has, up to a small error, the same expectation values for all operators acting within any local region, but it has a different momentum.

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