Renormalization group constructions of topological quantum liquids and beyond

Abstract

We give a detailed physical argument for the area law for entanglement entropy in gapped phases of matter arising from local Hamiltonians. Our approach is based on renormalization group (RG) ideas and takes a resource oriented perspective. We report four main results. First, we argue for the "weak area law": any gapped phase with a unique ground state on every closed manifold obeys the area law. Second, we introduce an RG based classification scheme and give a detailed argument that all phases within the classification scheme obey the area law. Third, we define a special sub-class of gapped phases, \textit{topological quantum liquids}, which captures all examples of current physical relevance, and we rigorously show that TQLs obey an area law. Fourth, we show that all topological quantum liquids have MERA representations which achieve unit overlap with the ground state in the thermodynamic limit and which have a bond dimension scaling with system size $L$ as $e^{c{\log }^{d(1+\delta )}(L)}$ for all $\delta >0$. For example, we show that chiral phases in $d=2$ dimensions have an approximate MERA with bond dimension $e^{c{\log }^{2(1+\delta )}(L)}$. We discuss extensively a number of subsidiary ideas and results necessary to make the main arguments, including field theory constructions. While our argument for the general area law rests on physically-motived assumptions (which we make explicit) and is therefore not rigorous, we may conclude that "conventional" gapped phases obey the area law and that any gapped phase which violates the area law must be a dragon.

91 pages, 8 figures

References in corpus (48)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Insulators with Inversion Symmetry
• A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator
• Topological Field Theory of Time-Reversal Invariant Insulators
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• A class of quantum many-body states that can be efficiently simulated
• Local stabilizer codes in three dimensions without string logical operators
• Area laws in quantum systems: mutual information and correlations
• Efficient quantum state tomography
• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• Criticality, the area law, and the computational power of PEPS
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Gravitation from Entanglement in Holographic CFTs
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Bipartite entanglement and entropic boundary law in lattice spin systems
• A Sharp Fannes-type Inequality for the von Neumann Entropy
• General entanglement scaling laws from time evolution
• Entropy and Entanglement in Quantum Ground States
• Entanglement renormalization and topological order
• The Dynamics of 1D Quantum Spin Systems Can Be Approximated Efficiently
• Approximating Gibbs states of local Hamiltonians efficiently with PEPS
• Universality of Gravity from Entanglement
• Generalizations and limitations of string-net models
• Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity
• Exact entanglement renormalization for string-net models
• Stochastic Error Cancellation in Analog Quantum Simulation
• Topology driven quantum phase transitions in time-reversal invariant anyonic quantum liquids
• Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes
• Short-range entanglement and invertible field theories
• Locality in Quantum Systems
• Constructing holographic spacetimes using entanglement renormalization
• Area law and vacuum reordering in harmonic networks
• Symmetries and boundary theories for chiral Projected Entangled Pair States
• Reconstructing quantum states from local data
• Real-space renormalization yields finite correlations
• Grassmann tensor network states and its renormalization for strongly correlated fermionic and bosonic states
• Sufficient Condition for Entanglement Area Laws in Thermodynamically Gapped Spin Systems
• Local tensor network for strongly correlated projective states
• Topology and Phases in Fermionic Systems
• Stability of the Area Law for the Entropy of Entanglement
• Approximating the ground state of gapped quantum spin systems
• Quantum Markov Networks and Commuting Hamiltonians
• Frustration, Area Law, and Interference in Quantum Spin Models
• Determining the structure of real-space entanglement spectrum from approximate conditional independence
• Tensor-product representations for string-net condensed states
• Notes on Some Questions in Mathematical Physics and Quantum Information

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