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arXiv:1208.0735

Scaling of the Rényi entropies in gapped quantum spin systems: Entanglement-driven order beyond symmetry breaking

S. M. Giampaolo, Simone Montangero, Fabio Dell’Anno, Silvio De Siena, Fabrizio Illuminati

Published Aug 3, 2012• 1 citation
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Abstract

We investigate the scaling of the Rényi $α$-entropies in one-dimensional gapped quantum spin models. We show that the block entropies with $α> 2$ violate the area law monotonicity and exhibit damped oscillations. Depending on the existence of a factorized ground state, the oscillatory behavior occurs either below factorization or it extends indefinitely. The anomalous scaling corresponds to an entanglement-driven order that is independent of ground-state degeneracy and is revealed by a nonlocal order parameter defined as the sum of the single-copy entanglement over all blocks.

DOI
10.48550/arxiv.1208.0735

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We investigate the scaling of the Rényi ααα-entropies in one-dimensional gapped quantum spin models. We show that the block entropies with α>2α> 2α>2 violate the area law monotonicity and exhibit damped oscillations. Depending on the existence of a factorized ground state, the oscillatory behavior occurs either below factorization or it extends indefinitely. The anomalous scaling corresponds to an entanglement-driven order that is independent of ground-state degeneracy and is revealed by a nonlocal order parameter defined as the sum of the single-copy entanglement over all blocks.
We investigate the scaling of the entanglement spectrum and of the R\'enyi block entropies and determine its universal aspects in the ground state of critical and noncritical one-dimensional quantum spin models. In all cases, the scaling exhibits an oscillatory behavior that terminates at the factorization point and whose frequency is universal. Parity effects in the scaling of the R\'enyi entropies for gapless models at zero field are thus shown to be a particular case of such universal behavior. Likewise, the absence of oscillations for the Ising chain in transverse field is due to the vanishing value of the factorizing field for this particular model. In general, the transition occurring at the factorizing field between two different scaling regimes of the entanglement spectrum corresponds to a quantum transition to the formation of finite-range, ordered structures of quasidimers, quasitrimers, and quasipolymers. This entanglement-driven transition is superimposed to and independent of the long-range magnetic order in the broken symmetry phase. Therefore, it conforms to recent generalizations that identify and classify the quantum phases of matter according to the structure of ground-state entanglement patterns. We characterize this form of quantum order by a global order parameter of entanglement defined as the integral, over blocks of all lengths, of the R\'enyi entropy of infinite order. Equivalently, it can be defined as the integral of the bipartite single-copy or geometric entanglement. The global entanglement order parameter remains always finite at fields below the factorization point and vanishes identically above it.
Entanglement is one of the most intriguing features of quantum theory and a main resource in quantum information science. Ground states of quantum many-body systems with local interactions typically obey an “area law” which means that the entanglement entropy is proportional to the boundary length. It is exceptional when the system is gapless, and the area law had been believed to be violated by at most a logarithm over two decades. Recent discovery of Motzkin and Fredkin spin chain models is striking, since these models provide significant violation of the entanglement beyond the belief, growing as a square root of the volume in spite of local interactions. In this paper, we first analytically compute the Rényi entropy of the Motzkin and Fredkin models by careful treatment of asymptotic analysis. The Rényi entropy is an important quantity, since the whole spectrum of an entangled subsystem is reconstructed once the Rényi entropy is known as a function of its parameter. We find nonanalytic behavior of the Rényi entropy with respect to the parameter, which is a novel phase transition never seen in any other spin chain studied so far. Interestingly, similar behavior is seen in the Rényi entropy of Rokhsar–Kivelson states in two dimensions.
The Rényi area law conjecture is false below α=1α=1α=1 in high dimensions
Area laws express the expectation that entanglement in gapped quantum ground states scales with a boundary rather than a volume. In one dimension this holds for every Rényi entropy, and the Rényi area-law conjecture asserts the same in higher dimensions for each fixed 0<α<10<α<10<α<1. Here we disprove this conjecture in every spatial dimension two and higher. We construct finite-range Hamiltonians with fixed local dimension, bounded interaction strength, unique ground states and a uniform spectral gap, whose ground states exhibit volume-law Rényi entanglement while obeying a von Neumann area law and remaining connected to product states through uniformly gapped local paths. The mechanism is error-space complexity: locally identifiable components of arbitrarily small weight can encode states with high entanglement complexity. We also establish the worst-case rank-error trade-off for approximate ground-state projectors, a key tool in one-dimensional area-law proofs. At fixed gap, the optimal Schmidt rank at projection error εεε across a planar cut of cross-section AAA scales as exp⁡[Θ~((Alog⁡ε−1)2/3)]\exp[\tildeΘ((A\logε^{-1})^{2/3})]exp[Θ~((Alogε−1)2/3)] when log⁡ε−1≳A\logε^{-1}\gtrsim Alogε−1≳A. Thus α=1α=1α=1 is the sharp threshold for a universal Rényi area law within this adiabatically connected class.
An area law is proved for the Renyi entanglement entropy of possibly degenerate ground states in one-dimensional gapped quantum systems. Suppose in a chain of nnn spins the ground states of a local Hamiltonian with energy gap εεε are constant-fold degenerate. Then, the Renyi entanglement entropy $R_α(0
An area law is proved for the Renyi entanglement entropy of possibly degenerate ground states in one-dimensional gapped quantum systems. Suppose in a chain of nnn spins the ground states of a local Hamiltonian with energy gap ϵ\epsilonϵ are constant-fold degenerate. Then, the Renyi entanglement entropy Rα(0<α<1)R_\alpha(0<\alpha<1)Rα​(0<α<1) of any ground state across any cut is upper bounded by O~(α−3/ϵ)\tilde O(\alpha^{-3}/\epsilon)O~(α−3/ϵ), and any ground state can be well approximated by a matrix product state of subpolynomial bond dimension 2O~(ϵ−1/4log⁡3/4n)2^{\tilde O(\epsilon^{-1/4}\log^{3/4}n)}2O~(ϵ−1/4log3/4n).
Generic quantum states in the Hilbert space of a many-body system are nearly maximally entangled whereas low-energy physical states are not; the so-called area laws for quantum entanglement are widespread. In this paper we introduce the novel concept of entanglement susceptibility by expanding the 2-Rényi entropy in the boundary couplings. We show how this concept leads to the emergence of area laws for bi-partite quantum entanglement in systems ruled by local gapped Hamiltonians. Entanglement susceptibility also captures quantitatively which violations one should expect when the system becomes gapless. We also discuss an exact series expansion of the 2-Rényi entanglement entropy in terms of connected correlation functions of a boundary term. This is obtained by identifying Rényi entropy with ground state fidelity in a doubled and twisted theory.