Matrix Product Density Operators: Renormalization Fixed Points and Boundary Theories

TL;DR

This paper studies tensor network representations of quantum states in spin chains, characterizing fixed points under renormalization and linking them to boundary theories of topological states.

Contribution

It revises the renormalization procedure for matrix product states and density operators, characterizing fixed points and connecting them to boundary theories of topological phases.

Findings

Fixed points correspond to zero correlation length states.

Fixed points relate to ground states of local, commuting Hamiltonians.

Boundary theories of 2D topological states are associated with these fixed points.

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.