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arXiv:cond-mat/0411094 · Europhysics Letters (EPL)

Sufficient conditions for topological order in insulators

M. B Hastings

Published May 12, 2005• 82 citations
arXivHTMLPDF

Abstract

We prove the existence of low-energy excitations in insulating systems at general filling factor under certain conditions, and discuss in which cases these may be identified as topological excitations. In the specific case of half-filling this proof provides a significantly shortened proof of the recent higher-dimensional Lieb-Schultz-Mattis theorem.

DOI
10.1209/epl/i2005-10046-x

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We prove the existence of low-energy excitations in insulating systems at general filling factor under certain conditions, and discuss in which cases these may be identified as topological excitations. In the specific case of half-filling this proof provides a significantly shortened proof of the recent higher-dimensional Lieb-Schultz-Mattis theorem.
We consider two-dimensional electron systems in zero magnetic field at fractional filling. For such systems a Lieb-Schultz-Mattis theorem applies, forbidding the existence of a trivial insulator. However, the theorem does not distinguish between bosonic and fermionic systems. In this Rapid Communication we argue that in the case of fermionic systems, the topological orders that are compatible with the microscopic constraints are in general different from the bosonic case. We find different results in the case of even and odd denominator fillings, with even denominator fillings deviating more strongly from the bosonic case. Parts of our results also hold in three dimensions.
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.
Following recent developments in the classification of bosonic short-range entangled phases, we examine many-body quantum systems whose ground state fractionalization obeys the Lieb-Schultz-Mattis (LSM) theorem. We generalize the topological classification of such phases by LSM anomalies (arXiv:1907.08204) to take magnetic and non-symmorphic lattice effects into account, and provide direct computations of the LSM anomaly in specific examples. We show that the anomaly-free condition coincides with established filling constraints (arXiv:1705.09298, arXiv:1505.04193), and we also derived new ones on novel crystalline quantum systems.
Abstract. For a large class of finite-range quantum spin models with half-integer spins, we prove that uniqueness of the ground state implies the existence of a low-lying excited state. For systems of linear size L, of arbitrary finite dimension, we obtain an upper bound on the excitation energy (i.e., the gap above the ground state) of the form (C log L)/L. This result can be regarded as a multi-dimensional Lieb-Schultz-Mattis theorem [7] and provides a rigorous proof of the main result in [4]. 1. Introduction and
The Lieb-Schultz-Mattis theorem for spin chains is generalized to a wide range of models of interacting electrons and localized spins on a one dimensional lattice. The existence of a low-energy state is generally proved except for special commensurate fillings where a gap may occur. Moreover, the crystal momentum of the constructed low-energy state is 2kF{2k}_{F}2kF​, where kF{k}_{F}kF​ is the Fermi momentum of the noninteracting model, corresponding to Luttinger's theorem. For the Kondo lattice model, our result implies that kF{k}_{F}kF​ must be calculated by regarding the localized spins as additional electrons.
We consider a general one-dimensional tight-binding electron model which has a period PPP. For any filling factor ννν such that PνPνPν is non-integral, we prove that the model in the infinite volume limit has either a symmetry breaking or a unique ground state with gapless excitations. The proof is based on the idea of Yamanaka, Oshikawa and Affleck (cond-mat/9701141), who extended the Lieb-Schultz-Mattis argument to electron systems.