Lieb-Schultz-Mattis in higher dimensions

By M. B. Hastings. 2003

577 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics

View Paper: https://www.semanticscholar.org/paper/d45f7223ad0975edc42897616dc22951c75bf2b1

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In Atomic and Molecular Physics, and Optics

In Condensed Matter Physics

Citations in the corpus

The Hubbard Model (2021, methodology): “While the LSM argument does not apply in dimensions d > 1, it was extended by Oshikawa (18) and subsequently rigorized by Hastings (19) to d-dimensional systems with periodic boundary conditions (i.e. on a d-torus) and a finite excitation gap.”
Spectral Gap and Exponential Decay of Correlations (2005, methodology): “For this purpose, we use the following lemma [3]: Lemma 3.”
Translational symmetry and microscopic constraints on symmetry-enriched topological phases: a view from the surface (2015, background): “The celebrated Lieb-Schultz-Mattis (LSM) theorem [1, 2], including the higher-dimensional generalizations by Oshikawa [3] and Hastings [4], shows that translationally invariant spin systems with an odd number of S = 1/2 moments per unit cell cannot have a symmetric, gapped, and nondegenerate ground ”
Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect (2012, methodology): “This feature is reminiscent of the Lieb-SchultzMattis (LSM) theorem [6] and its generalization [28,29] to states of bosonic systems at a fractional filling on clean 2D lattices.”
Towards the fast scrambling conjecture (2011, methodology): “Hastings improved the original Lieb-Robinson technique so as to produce dimension-independent bounds [54] and Nachtergaele-Sims showed how to adapt it to general graphs [27].”
Quantum frustration in organic Mott insulators: from spin liquids to unconventional superconductors (2010, background): “The repeat unit in this lattice consists of a tetrahedron of four spins (giving an integer total spin) and so the Lieb-Schultz-Mattis-Hastings theorem (Hastings, 2004) which can preclude gapped spin liquid ground states does not apply.”
Exponential Decay of Correlations Implies Area Law (2012, background): “…local Hamiltonian exhibits exponential decay of correlations, with l 0 of order unit and the correlation length ξ of order of the spectral gap of the model [4] (see also [3, 5–7] for subsequent developments and [1, 2] for a previous analogous result in the context of relativistic quantum systems).”