PEPS as ground states: Degeneracy and topology

By Norbert Schuch, I. Cirac, David Pérez-García. 2010

274 citations recorded by the source corpus.

We introduce a framework for characterizing Matrix Product States (MPS) and Projected Entangled Pair States (PEPS) in terms of symmetries. This allows us to understand how PEPS appear as ground states of local Hamiltonians with finitely degenerate ground states and to characterize the ground state subspace. Subsequently, we apply our framework to show how the topological properties of these ground states can be explained solely from the symmetry: We prove that ground states are locally indistinguishable and can be transformed into each other by acting on a restricted region, we explain the origin of the topological entropy, and we discuss how to renormalize these states based on their symmetries. Finally, we show how the anyonic character of excitations can be understood as a consequence of the underlying symmetries.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics

View Paper: https://www.semanticscholar.org/paper/8652d080f5dc8be961b2b2fad9a9b96249e93fce

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In Atomic and Molecular Physics, and Optics

In Condensed Matter Physics

Citations in the corpus

Stability of Frustration-Free Hamiltonians (2011, background): “In fact, it is possible in many cases (e.g. Hamiltonians with a unique Matrix Product State as their groundstate, or injective PEPS [36], [40], [41]) to modify a frustration-free Hamiltonian H0 to a different Hamiltonian H1, satisfying 4 Local-TQO without changing the global ground state subspace, o”
Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems (2014, methodology): “These two equations imply that the resulting PEPS cannot be injective, but below we prove it to be G injective [44] instead.”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions (2016, unknown): “In the PEPS and PEPO, the ranks are kept considerably smaller at a cost of employing 5th- or even 6th-order core tensors and the associated higher computational complexity with respect to the order (Verstraete et al., 2008; Evenbly and Vidal, 2009; Schuch et al., 2010).”
Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence (2015, background): “(10)Here we limit ourselves to provide the simplest example which conveys the general spirit of deriving global symmetries from local tensor symmetries [62].”
Hand-waving and interpretive dance: an introductory course on tensor networks (2016, background): “Despite this, physically relevant states often possess additional structure not found in arbitrary states, and as such do not exhibit this pathological complexity, allowing them to be efficiently described and studied.”
Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, methodology): “All of these features can succinctly be understood from the fact that virtual MPO strings are attached to them, which are immaterial as they can be moved at will through the lattice (Bultinck et al., 2017; Schuch et al., 2010).”