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arXiv:1806.04049 · International Journal of Modern Physics B

Rényi entropy of highly entangled spin chains

Fumihiko Sugino, V. E. Korepin

Published Oct 30, 2018• 19 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s021797921850306x

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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.
Abstract We introduce a new spin chain which is a deformation of the Fredkin spin chain and has a phase transition between bounded and extensive entanglement entropy scaling. In this chain, spins have a local interaction of three nearest neighbors. The Hamiltonian is frustration-free and its ground state can be described analytically as a weighted superposition of Dyck paths that depends on a deformation parameter t . In the purely spin 1/2 case, whenever t ≠ 1 , the entanglement entropy obeys an area law: it is bounded from above by a constant, when the size of the block n increases. When a local color degree of freedom is introduced and t > 1 the entanglement entropy increases linearly, while for t < 1 an area law is obeyed. The half-chain entanglement entropy is tightly bounded by n log ⁡ s where 2 n is the length of the chain, and s is the number of colors. Our chain fosters a new example for a significant boost to entropy and for the existence of the associated critical rainbow phase where the entanglement entropy scales with volume that discovered by Zhang, Ahmadain and Klich ( Proc. Natl Acad. Sci 2017).
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.
The Rényi entanglement entropy is widely used in studying quantum entanglement properties in strongly correlated systems, whose analytic continuation as the Rényi index n→1n \to 1n→1 is often believed to yield the von Neumann entanglement entropy. However, earlier findings indicate that this process exhibits a singularity for the colored Motzkin spin chain problem, leading to different scaling behaviors of ∼l\sim \sqrt{l}∼l​ and ∼log⁡l\sim \log{l}∼logl for the von Neumann and Rényi entropies, respectively. Our analytical and numerical calculations confirm this transition, which can be explained by the exponentially increasing density of states in the entanglement spectrum that we extract numerically. Disorder operators are further employed under various symmetries to study such a system. Both analytical and numerical results demonstrate that the scaling of the disorder operators also follows log⁡l\log{l}logl as the leading behavior, matching that of the Rényi entropy. We propose that the coefficient of the term log⁡l\log{l}logl is a universal constant shared by both the Rényi entropies and disorder operators. This universal constant could potentially help capture the underlying constraint physics of Motzkin walks.
Thanks to Pfaffian techniques, we study the Rényi entanglement entropies and the entanglement spectrum of large subsystems for two-dimensional Rokhsar–Kivelson wavefunctions constructed from a dimer model on the triangular lattice. By including a fugacity t on some suitable bonds, one interpolates between the triangular lattice ( t = 1) and the square lattice ( t = 0). The wavefunction is known to be a massive topological liquid for t > 0 whereas it is a gapless critical state at t = 0. We mainly consider two geometries for the subsystem: that of a semi-infinite cylinder and the disc-like set-up proposed by Kitaev and Preskill (2006 Phys. Rev. Lett. 96 110404). In the cylinder case, the entropies contain an extensive term—proportional to the length of the boundary—and a universal subleading constant s n ( t ). Fitting these cylinder data (up to a perimeter of L = 32 sites) provides s n with a very high numerical accuracy (10 − 9 at t = 1 and 10 − 6 at t = 0.5). In the topological liquid phase we find s n ( t > 0) = − ln2, independent of the fugacity t and the Rényi parameter n . At t = 0 we recover a previously known result, for n < 1 and s n ( t = 0) = − ln(2)/( n − 1) for n > 1. In the disc-like geometry—designed to get rid of the boundary contributions—we find an entropy s n KP ( t 0) = − ln2 in the whole massive phase whatever n > 0, in agreement with the result of Flammia et al (2009 Phys. Rev. Lett. 103 261601). Some results for the gapless limit are discussed.
The entanglement entropy measures quantum correlations and it can be seen as the uncertainty on a quantum state. In one spatial dimension, the entanglement entropy scales as the boundary that divides two subsystems, so an area law has been proposed. However, the entanglement entropy diverges logarithmically at conformally invariant critical points, so the area law does not hold. The purpose of the work is to find a way to get more information about a critical state. The ground state of the Heisenberg XXZ model at criticality is analyzed by means of critical Ising eigenstates. Two ways of analysis are followed: a basis made of Ising eigenstates is built up and used to represent the XXZ ground state, then the Shannon entropy in the new basis is computed; the adiabatic evolution from the Ising ground state to the XXZ ground state. The result is that the Shannon entropy in the Ising basis scales linearly with the length of the system, while a phase transition is encountered during the adiabatic evolution. The conclusion is that there is no net gain in information after the procedure and possibly it is related to the fact the two systems stand in different phases.
Abstract Area law violations for entanglement entropy in the form of a square root have recently been studied for one-dimensional frustration-free quantum systems based on the Motzkin walks and their variations. Here we consider a Motzkin walk with a different Hilbert space on each step of the walk spanned by the elements of a symmetric inverse semigroup with the direction of each step governed by its algebraic structure. This change alters the number of paths allowed in the Motzkin walk and introduces a ground state degeneracy that is sensitive to boundary perturbations. We study the frustration-free spin chains based on three symmetric inverse semigroups, S 1 3 , S 2 3 and S 1 2 . The system based on S 1 3 and S 2 3 provides examples of quantum phase transitions in one dimension, with the former exhibiting a transition between the area law and a logarithmic violation of the area law and the latter providing an example of transition from logarithmic scaling to a square root scaling in the system size, mimicking a colored S 1 3 system. The system with S 1 2 is much simpler and produces states that continue to obey the area law.