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arXiv:1611.01140 · Physical review. B./Physical review. B

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

Zhu-Xi Luo, Ethan Lake, Yong-Shi Wu

Published Jul 5, 2017• 11 citations
arXivHTMLPDF

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.

DOI
10.1103/physrevb.96.035101

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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.
We show that the path integral of conformal field theories in D dimensions ( CFT D ) can be constructed by solving for eigenstates of a renormalization group (RG) operator following from the Turaev-Viro formulation of a topological field theory (topological quantum field theory) (TQFT) in D + 1 dimensions ( TQFT D + 1 ), explicitly realizing the holographic sandwich relation between a symmetric theory and a TQFT. Generically, exact eigenstates corresponding to symmetric TQFT D follow from Frobenius algebra in TQFT D + 1 . For D = 2 , we construct eigenstates that produce 2D rational CFT path integrals exactly, which curiously connect a continuous-field theoretic path integral with the Turaev-Viro state sum. We also devise and illustrate numerical methods for D = 2 , 3 to search for CFT D as phase transition points between symmetric TQFT D . Finally, since the RG operator is in fact an exact analytic holographic tensor network, we compute “bulk-boundary” correlators and compare them with the AdS/CFT dictionary at D = 2 . Promisingly, they are numerically compatible given our accuracy, although further works will be needed to explore the precise connection to the AdS/CFT correspondence. Published by the American Physical Society 2024
In the holographic correspondence of quantum gravity, a global on-site symmetry at the boundary generally translates to a local gauge symmetry in the bulk. We describe one way how the global boundary on-site symmetries can be gauged within the formalism of the multiscale renormalization ansatz (MERA), in light of the ongoing discussion between tensor networks and holography. We describe how to ``lift'' the MERA representation of the ground state of a generic one dimensional (1D) local Hamiltonian, which has a global on-site symmetry, to a dual quantum state of a 2D ``bulk'' lattice on which the symmetry appears gauged. The 2D bulk state decomposes in terms of spin network states, which label a basis in the gauge-invariant sector of the bulk lattice. This decomposition is instrumental to obtain expectation values of gauge-invariant observables in the bulk, and also reveals that the bulk state is generally entangled between the gauge and the remaining (``gravitational'') bulk degrees of freedom that are not fixed by the symmetry. We present numerical results for ground states of several 1D critical spin chains to illustrate that the bulk entanglement potentially depends on the central charge of the underlying conformal field theory. We also discuss the possibility of emergent topological order in the bulk using a simple example, and also of emergent symmetries in the nongauge (gravitational) sector in the bulk. More broadly, our holographic model translates the MERA, a tensor network state, to a superposition of spin network states, as they appear in lattice gauge theories in one higher dimension.
It is believed that most (perhaps all) gapped phases of matter can be described at long distances by topological quantum field theory (TQFT). On the other hand, it has been rigorously established that in 1+1d ground states of gapped Hamiltonians can be approximated by matrix product states (MPS). We show that the state-sum construction of 2d TQFT naturally leads to MPS in their standard form. In the case of systems with a global symmetry GGG, this leads to a classification of gapped phases in 1+1d in terms of Morita-equivalence classes of GGG-equivariant algebras. Nonuniqueness of the MPS representation is traced to the freedom of choosing an algebra in a particular Morita class. In the case of short-range entangled phases, we recover the group cohomology classification of SPT phases.
One of the apparent advantages of quantum computers over their classical counterparts is their ability to efficiently contract tensor networks. In this article, we study some implications of this fact in the case of topological tensor networks. The graph underlying these networks is given by the triangulation of a manifold, and the structure of the tensors ensures that the overall tensor is independent of the choice of internal triangulation. This leads to quantum algorithms for additively approximating certain invariants of triangulated manifolds. We discuss the details of this construction in two specific cases. In the first case, we consider triangulated surfaces, where the triangle tensor is defined by the multiplication operator of a finite group; the resulting invariant has a simple closed-form expression involving the dimensions of the irreducible representations of the group and the Euler characteristic of the surface. In the second case, we consider triangulated 3-manifolds, where the tetrahedral tensor is defined by the so-called Fibonacci anyon model; the resulting invariant is the well-known Turaev-Viro invariant of 3-manifolds.
One of the apparent advantages of quantum computers over their classical counterparts is their ability to efficiently contract tensor networks. In this article, we study some implications of this fact in the case of topological tensor networks. The graph underlying these networks is given by the triangulation of a manifold, and the structure of the tensors ensures that the overall tensor is independent of the choice of internal triangulation. This leads to quantum algorithms for additively approximating certain invariants of triangulated manifolds. We discuss the details of this construction in two specific cases. In the first case, we consider triangulated surfaces, where the triangle tensor is defined by the multiplication operator of a finite group; the resulting invariant has a simple closed-form expression involving the dimensions of the irreducible representations of the group and the Euler characteristic of the surface. In the second case, we consider triangulated 3-manifolds, where the tetrahedral tensor is defined by the so-called Fibonacci anyon model; the resulting invariant is the well-known Turaev-Viro invariant of 3-manifolds.
We consider Abelian topological quantum field theories (TQFTs) in 3D and show that gaugings of invertible global symmetries naturally give rise to additive codes. These codes emerge as nonanomalous subgroups of the 1-form symmetry group, parametrizing the fusion rules of condensable TQFT anyons. The boundary theories dual to TQFTs with a maximal symmetry subgroup gauged, i.e., with the corresponding anyons condensed, are "code" conformal field theories (CFTs). This observation bridges together, in the holographic sense, results on 1-form symmetries of 3D TQFTs with developments connecting codes to 2D CFTs. Building on this relationship, we proceed to consider the ensemble of maximal gaugings (topological boundary conditions) in a general, not necessarily Abelian 3D TQFT, and propose that the resulting ensemble of boundary CFTs has a holographic description as a gravitational theory: the bulk TQFT summed over topologies.