Power law violation of the area law in quantum spin chains

Abstract

The sub-volume scaling of the entanglement entropy with the system's size, $n$, has been a subject of vigorous study in the last decade [1]. The area law provably holds for gapped one dimensional systems [2] and it was believed to be violated by at most a factor of $\log \left(n\right)$ in physically reasonable models such as critical systems. In this paper, we generalize the spin$-1$ model of Bravyi et al [3] to all integer spin-$s$ chains, whereby we introduce a class of exactly solvable models that are physical and exhibit signatures of criticality, yet violate the area law by a power law. The proposed Hamiltonian is local and translationally invariant in the bulk. We prove that it is frustration free and has a unique ground state. Moreover, we prove that the energy gap scales as $n^{-c}$, where using the theory of Brownian excursions, we prove $c\ge 2$. This rules out the possibility of these models being described by a conformal field theory. We analytically show that the Schmidt rank grows exponentially with $n$ and that the half-chain entanglement entropy to the leading order scales as $\sqrt{n}$ (Eq. 16). Geometrically, the ground state is seen as a uniform superposition of all $s-$colored Motzkin walks. Lastly, we introduce an external field which allows us to remove the boundary terms yet retain the desired properties of the model. Our techniques for obtaining the asymptotic form of the entanglement entropy, the gap upper bound and the self-contained expositions of the combinatorial techniques, more akin to lattice paths, may be of independent interest.

v3: 10+33 pages. In the PNAS publication, the abstract was rewritten and title changed to "Supercritical entanglement in local systems: Counterexample to the area law for quantum matter". The content is same otherwise. v2: a section was added with an external field to include a model with no boundary terms (open and closed chain). Asymptotic technique is improved. v1:37 pages, 10 figures. Proc. Natl. Acad. Sci. USA, (Nov. 2016)

References in corpus (11)

• The Quantum Internet
• Quantum criticality
• Correcting Quantum Errors with Entanglement
• Aspects of generic entanglement
• Measurement of the production of a W boson in association with a charm quark in pp collisions at sqrt(s)=7 TeV with the ATLAS detector
• Entanglement over the rainbow
• Volume Law for the Entanglement Entropy in Non-local QFTs
• Explicit Hamiltonians Inducing Volume Law for Entanglement Entropy in Fermionic Lattices
• Entanglement and correlation functions of the quantum Motzkin spin-chain
• Generic Local Hamiltonians are Gapless
• Eigenvalues and Low Energy Eigenvectors of Quantum Many-Body Systems

Cited by in corpus (92)

• Adiabatic Quantum Computing
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Quantum Many-Body Scar States with Emergent Kinetic Constraints and Finite-Entanglement Revivals
• Lecture Notes of Tensor Network Contractions
• Fractal, logarithmic and volume-law entangled non-thermal steady states via spacetime duality
• Multipartite entanglement in topological quantum phases
• A Holographic Derivation of the Weak Gravity Conjecture
• Long-range topological insulators and weakened bulk-boundary correspondence
• Hilbert Space Fragmentation and Exact Scars of Generalized Fredkin Spin Chains
• Anomalous hydrodynamics in a class of scarred frustration-free Hamiltonians
• Subdiffusion and many-body quantum chaos with kinetic constraints
• Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains
• Quantum phase transition from bounded to extensive entanglement entropy in a frustration-free spin chain
• Deformed Fredkin Spin Chain with Extensive Entanglement
• Gapless quantum spin chains: multiple dynamics and conformal wavefunctions
• Spectral gaps of frustration-free spin systems with boundary
• Entanglement as geometry and flow
• Quantum spin chains with multiple dynamics
• Temperature driven quenches in the Ising model: appearance of negative Rényi mutual information
• Undecidability of the Spectral Gap in One Dimension
• Unusual area-law violation in random inhomogeneous systems
• Imprints of Barrow-Tsallis Cosmology in Primordial Gravitational Waves
• The Feynman-Kitaev computer's clock: bias, gaps, idling and pulse tuning
• Correlation Length versus Gap in Frustration-Free Systems
• Holographic rainbow networks for colorful Motzkin and Fredkin spin chains
• Entanglement and correlation functions of the quantum Motzkin spin-chain
• Entropy, gap and a multi-parameter deformation of the Fredkin spin chain
• Finite-size Gap, Magnetization, and Entanglement of Deformed Fredkin Spin Chain
• The Complexity of Translationally-Invariant Spin Chains with Low Local Dimension
• How much delocalisation is needed for an enhanced area law of the entanglement entropy?
• Higher-curvature corrections to holographic entanglement entropy in geometries with hyperscaling violation
• Haldane Topological Orders in Motzkin Spin Chains
• Exact holographic tensor networks for the Motzkin spin chain
• Renyi entropy of highly entangled spin chains
• The gap of the area-weighted Motzkin spin chain is exponentially small
• Entanglement Transitions in Unitary Circuit Games
• Spin Multiplicities
• Classification of Non-local Actions: Area versus Volume Entanglement Entropy
• Energy level splitting for weakly interacting bosons in a harmonic trap
• Glassy word problems: ultraslow relaxation, Hilbert space jamming, and computational complexity
• Hidden Bethe states in a partially integrable model
• Entanglement entropy of the long-range Dyson hierarchical model
• The frustration-free fully packed loop model
• Long-distance entanglement in Motzkin and Fredkin spin chains
• Analytic model of thermalization: Quantum emulation of classical cellular automata
• Nodal Line Entanglement Entropy: Generalized Widom Formula from Entanglement Hamiltonians
• The Fredkin staircase: An integrable system with a finite-frequency Drude peak
• The pair-flip model: a very entangled translationally invariant spin chain
• Quantum ground state isoperimetric inequalities for the energy spectrum of local Hamiltonians
• Sublogarithmic behaviour of the entanglement entropy in fermionic chains
• Area Law Violations and Quantum Phase Transitions in Modified Motzkin Walk Spin Chains
• Stability of the enhanced area law of the entanglement entropy
• Boundary Description of Microstates of the Two-Dimensional Black Hole
• Production of genuine multimode entanglement in circular waveguides with long-range interactions
• Random translation-invariant Hamiltonians and their spectral gaps
• Coupled Fredkin and Motzkin chains from quantum six- and nineteen-vertex models
• On the Entanglement Entropy of Maxwell Theory: A Condensed Matter Perspective
• Deforming the Fredkin spin chain away from its frustration-free point
• Shor-Movassagh chain leads to unusual integrable model
• Can a spin chain relate combinatorics to number theory?
• Entanglement entropy of the $Q\ge 4$ quantum Potts chain
• Deformed Fredkin model for the $\nu =5/2$ Moore-Read state on thin cylinders
• Subdiffusive transport in the Fredkin dynamical universality class
• Classifying Data with Local Hamiltonians
• Entanglement between random and clean quantum spin chains
• Power-law entanglement and Hilbert space fragmentation in non-reciprocal quantum circuits
• Entanglement blossom in a simplex matryoshka
• Entanglement subvolume law for 2D frustration-free spin systems
• Interplay between local and non-local frustration in the 1D ANNNI chain I -- The even case
• Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians
• Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction
• Entanglement Fractalization
• Bicolor loop models and their long range entanglement
• Dynamics and correlations in Motzkin and Fredkin spin chains
• Periodic Motzkin chain: Ground states and symmetries
• Tunable quantum spin chain with three-body interactions
• Quantum lozenge tiling and entanglement phase transition
• The c-d conjecture
• Sample complexity of matrix product states at finite temperature
• Non-Interacting Motzkin Chain - Periodic Boundary Conditions
• Symmetries of the periodic Fredkin chain
• Sequential Generation of Two-dimensional Super-area-law States with Local Parent Hamiltonian
• Anomalous Enhancement of Entanglement Entropy in Nonequilibrium Steady States Driven by Zero-Temperature Reservoirs
• Critical Fermions are Universal Embezzlers
• A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems
• Gapped and gapless phases of frustration-free spin-1/2 chains
• Multipartite Entanglement in Crossing the Quantum Critical Point
• Optimal bound on long-range distillable entanglement
• Exact quantification of bipartite entanglement in unresolvable spin ensembles
• Absence of local conserved charges of the Fredkin spin chain and its truncated versions
• Highly entangled spin chains and 2D quantum gravity
• Knabe's spectral gap method for open boundary conditions