Spectral Gap and Exponential Decay of Correlations

By M. B. Hastings, T. Koma. 2005

677 citations recorded by the source corpus.

We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain self-similarity in (fractal) dimensions D < 2. In particular, if the system is translationally invariant in one of the spatial directions, then this self-similarity condition is automatically satisfied. We also treat systems with long-range, power-law decaying interactions.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8d9d43734658f297640ed29c859ba180485c85b8

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In Artificial Intelligence

In Atomic and Molecular Physics, and Optics

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Citations in the corpus

The Physics of Communicability in Complex Networks (2011, methodology): “…theory, where it has been used, for example, to establish area laws for the entanglement entropy; see, e.g., (Cramer and Eisert, 2006; Cramer et al., 2006; Hastings, 2004; Hastings and Koma, 2006; Schuch, 2007; Schuch et al., 2006) and especially (Eisert et al., 2010) for a comprehensive survey.”
A Short Proof of Stability of Topological Order under Local Perturbations (2010, methodology): “It will be applicable only to evolution under Hamiltonians H having a finite Lieb-Robinson velocity [4, 5, 6, 24], for example, Hamiltonians with exponentially decaying interactions.”
Locality of temperature (2013, methodology): “This is rigorously proven using information theory inspired methods such as Lieb-Robinson bounds and quasiadiabatic continuation [27–29].”
Towards the fast scrambling conjecture (2011, methodology): “Specifically, the version of the bound proved in [28] ensures that ∥∥[OA(t), OB]∥∥ ≤ const.”
Chaos in quantum channels (2015, background): “Such behavior is reminiscent of the Lieb-Robinson bound on the commutator of local operators separated in time for systems with local interactions [29, 30, 31] and suggests identifying vB with the LiebRobinson velocity.”