Entanglement in a Valence-Bond Solid State

Abstract

We study entanglement in Valence-Bond-Solid state, which describes the ground state of an AKLT quantum spin chain, consisting of bulk spin-1's and two spin-1/2's at the ends. We characterize entanglement between various subsystems of the ground state by mostly calculating the entropy of one of the subsystems; when appropriate, we evaluate concurrences as well. We show that the reduced density matrix of a continuous block of bulk spins is independent of the size of the chain and the location of the block relative to the ends. Moreover, we show that the entanglement of the block with the rest of the sites approaches a constant value exponentially fast, as the size of the block increases. We also calculate the entanglement of (i) any two bulk spins with the rest, and (ii) the end spin-1/2's (together and separately) with the rest of the ground state. For example, we show that (i) any two bulk spins become maximally entangled with the rest of the ground state exponentially fast in their separation distance, (ii) the two end spin-1/2's share no entanglement, and (iii) each end spin-1/2 is maximally entangled with the rest. There is considerable current interest in quantifying entanglement in various quantum systems. Entanglement in spin chains, correlated electrons, interacting bosons and other models was studied [3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. Entanglement is a fundamental measure of how much quantum effects we can observe and use, and it is the primary resource in quantum computation and quantum information processing [1,2]. Also entanglement plays a role in the quantum phase transitions [3,4], and it has been experimentally demonstrated that the entanglement may affect macroscopic properties of solids [5,6].In this Letter, we will study a spin chain introduced by Affleck, Kennedy, Lieb, and Tasaki (AKLT model) [28,29]. The ground state of the model is a unique pure state. It is known as Valence-Bond Solid (VBS), and plays a central role in condensed matter physics. Haldane [31] conjectured that an anti-ferromagnetic Hamiltonian describing half-odd-integer spins is gap-less, but for integer spins it has a gap. AKLT model describing interaction of spin-1's in the bulk agrees with the conjecture. An implementation of AKLT in optical lattices was proposed recently [32], and the use of AKLT model for universal quantum computation was discussed in [30]. VBS is also closely related to Laughlin ansatz [33] and to fractional quantum Hall effect [34].We investigate the seminal AKLT model from the new perspective of quantum information, and evaluate the entanglement (in terms of entropy) of various subsystems of the VBS. The results and the methodologies adopted herein have several implications from the perspective of both quantum information and condensed matter. For example, while the entanglement in spin chains with periodic boundary conditions has been studied extensively, our results provide entanglement calculations for spin chains with open boundary conditions. For critical gap-les...

Entanglement and Density Matrix of a Block of Spins in AKLT Model

“…After integrating over these variables, the density matrix ρ L turns out to be independent of both the starting site k and the total length L of the block. This property has been proved in Fan, et al (2004) for spin S = 1 (using a different representation, namely the maximally entangled states, see §4) and generalized in Katsura, et al (2007a) for generic spin-S. Therefore, we can choose k = 1 and the density matrix takes the form…”

“…It contains information of all correlation functions (Jin & Korepin 2004, Katsura, et al 2007a, Arovas, et al 1988. Moreover, it has been shown in Fan, et al (2004), Katsura, et al (2007a) that the density matrix is independent of the size of the chain and the location of the block relative to the ends. Therefore we can take the length of the block equal to the length of the whole chain.…”

“…The density matrix of a contiguous block of bulk spins (we call it the density matrix later for short) has been studied extensively in Kirillov & Korepin (1990), Verstraete, et al (2004a), Fan, et al (2004), Katsura, et al (2007a), Freitag & Muller-Hartmann (1991). It contains information of all correlation functions (Jin & Korepin 2004, Katsura, et al 2007a, Arovas, et al 1988.…”

“…By equivalent we mean that all non-zero eigenvalues are the same. Using this method, eigenvalues of the density matrix as well as entanglement entropies were obtained (Fan, et al 2004;Katsura, et al 2007a;Freitag & Muller-Hartmann 1991) without knowing the eigenvectors explicitly.…”

“…Extensive research has been undertaken to understand quantum entanglement for spin chains, correlated electrons, interacting bosons as well as other models, see Amico, et al (2007), Audenaert, et al (2002), Fan & Korepin (2008), Katsura, et al (2007b), Fan, et al (2007), Arnesen, et al (2001), Korepin (2004), Verstraete, et al (2004a, b), Campos Venuti, et al (2006), Jin & Korepin (2004), Vedral (2004), Latorre, et al (2004aLatorre, et al ( , b, 2005, Orus (2005), Orus & Latorre (2004), Pachos & Plenio (2004), Plenio et al (2004), Fan & Lloyd (2005), Chen, et al (2004), Zanardi & Rasetti (1999), Popkov & Salerno (2004), Keating & Mezzadri (2004), Gu, et al (2003Gu, et al ( , 2004, , , Holzhey, et al (1994), Calabrese & Cardy (2004), Levin & Wen (2006), Kitaev & Preskill (2006), Ryu & Hatsugai (2006), Hirano & Hatsugai (2007) for reviews and references. Characteristic functions of quantum entanglement, such as von Neumann entropy and Renyi entropy, are obtained and discussed through studying reduced density matrices of subsystems (Fan, et al 2004;Katsura, et al 2007a;Its, et al 2005;Franchini, et al 2007Franchini, et al , 2008Vidal, et al 2003). An area law for the von Neumann entropy in harmonic lattice systems has been extensively studied …”