The structure of fixed-point tensor network states characterizes the patterns of long-range entanglement

2016/11/30 by Zhu-Xi Luo, Ethan Lake, Yong-Shi Wu
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Boundary (topology) #Combinatorics #Cosmology and Gravitation Theories #Exact solutions in general relativity #Field (mathematics) #Fixed point #Mathematical analysis #Mathematics #Noncommutative and Quantum Gravity Theories #Physics #Pure mathematics #Quantum #Quantum entanglement #Quantum mechanics #Symmetric tensor #Tensor (intrinsic definition) #Topological quantum field theory #Topology (electrical circuits) #Unitary state #cond-mat.str-el

paper · pdf · doi:10.1103/physrevb.96.035101

published as Phys. Rev. B 96, 035101 (2017) · version 3, figures added

openalex created_date 2016/11/11 · arxiv created 2017/03/16 · openalex publication_date 2017/07/05 · arxiv updated 2017/07/12 · openalex updated_date 2026/08/05

Abstract

The algebraic structure of representation theory naturally arises from 2D fixed-point tensor network states, and conceptually formulates the pattern of long-range entanglement realized in such states. In 3D, the same underlying structure is also shared by Turaev-Viro state-sum topological quantum field theory (TQFT). We show that a 2D fixed-point tensor network state arises naturally on the boundary of the 3D manifold on which the TQFT is defined, and the fact that exactly the same information is needed to construct either the tensor network or the TQFT is made explicit in a form of holography. Furthermore, the entanglement of the fixed-point states leads to an emergence of pregeometry in the 3D TQFT bulk. We further extend these ideas to the case where an additional global on-site unitary symmetry is imposed on the tensor network states.

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