Matrix Product Density Operators: Renormalization Fixed Points and Boundary Theories

Abstract

We consider the tensors generating matrix product states and density operators in a spin chain. For pure states, we revise the renormalization procedure introduced by F. Verstraete et al. in 2005 and characterize the tensors corresponding to the fixed points. We relate them to the states possessing zero correlation length, saturation of the area law, as well as to those which generate ground states of local and commuting Hamiltonians. For mixed states, we introduce the concept of renormalization fixed points and characterize the corresponding tensors. We also relate them to concepts like finite correlation length, saturation of the area law, as well as to those which generate Gibbs states of local and commuting Hamiltonians. One of the main result of this work is that the resulting fixed points can be associated to the boundary theories of two-dimensional topological states, through the bulk-boundary correspondence introduced by Cirac et al. in 2011.

63 pages, Annals of Physics (2016). This version corrects an error in the statement and proof of Theorem 4.9 by slightly generalizing Definition 4.8

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• Symmetry-enforced minimal entanglement and correlation in quantum spin chains
• Entanglement in finite quantum systems under twisted boundary conditions
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