Locality in Quantum and Markov Dynamics on Lattices and Networks

Abstract

We consider gapped systems governed by either quantum or Markov dynamics, with the low-lying states below the gap being approximately degenerate. For a broad class of dynamics, we prove that ground or stationary state correlation functions can be written as a piece decaying exponentially in space plus a term set by matrix elements between the low-lying states. The key to the proof is a local approximation to the negative energy, or annihilation, part of an operator in a gapped system. Applications to numerical simulation of quantum systems and to networks are discussed.

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References in corpus (10)

• Statistical mechanics of complex networks
• Finding and evaluating community structure in networks
• Lieb-Schultz-Mattis in Higher Dimensions
• Diverging Entanglement Length in Gapped Quantum Spin Systems
• Order versus Disorder in the Quantum Heisenberg Antiferromagnet on the Kagom{é} lattice: an approach through exact spectra analysis
• Three-spin interactions in optical lattices and criticality in cluster Hamiltonians
• Expansion algorithm for the density matrix
• Quantum disorder in the two-dimensional pyrochlore Heisenberg antiferromagnet
• Mean-Field and Anomalous Behavior on a Small-World Network
• Random Vibrational Networks and Renormalization Group

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