Entanglement Entropy of One-dimensional Gapped Spin Chains

We investigate in detail the behavior of the bipartite fluctuations of particle numberN and spin S z in many-body quantum systems, focusing on systems where such U(1) charges are both conserved and fluctuate within subsystems due to exchange of charges between subsystems. We propose that the bipartite fluctuations are an effective tool for studying many-body physics, particularly its entanglement properties, in the same way that noise and Full Counting Statistics have been used in mesoscopic transport and cold atomic gases. For systems that can be mapped to a problem of non-interacting fermions we show that the fluctuations and higher-order cumulants fully encode the information needed to determine the entanglement entropy as well as the full entanglement spectrum through the Rényi entropies. In this connection we derive a simple formula that explicitly relates the eigenvalues of the reduced density matrix to the Rényi entropies of integer order for any finite density matrix. In other systems, particularly in one dimension, the fluctuations are in many ways similar but not equivalent to the entanglement entropy. Fluctuations are tractable analytically, computable numerically in both density matrix renormalization group and quantum Monte Carlo calculations, and in principle accessible in condensed matter and cold atom experiments. In the context of quantum point contacts, measurement of the second charge cumulant showing a logarithmic dependence on time would constitute a strong indication of many-body entanglement.

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“…Exact expressions for the von Neumann entropy have already been derived [127][128][129][130]. For OBCs the entangle- ment entropy is given by…”

Section: Gapped Systemsmentioning

confidence: 99%

“…14, perfectly matches the DMRG data (green dots). For PBCs, the von Neumann entropy is given by [128,129] 48) where…”

Section: Gapped Systemsmentioning

confidence: 99%

Bipartite fluctuations as a probe of many-body entanglement

et al. 2012

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“…Entanglement properties also play an important role in condensed matter physics, such as phase transitions (Osterloh, et al 2002;Osborne & Nielsen 2002) and macroscopic properties of solids (Ghosh, et al 2003;Vedral 2004). 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;…”

Section: Introductionmentioning

confidence: 99%

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

Katsura

Hirano

et al. 2008

J Stat Phys

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We study a 1-dimensional AKLT spin chain, consisting of spins S in the bulk and S/2 at both ends. The unique ground state of this AKLT model is described by the Valence-Bond-Solid (VBS) state. We investigate the density matrix of a contiguous block of bulk spins in this ground state. It is shown that the density matrix is a projector onto a subspace of dimension (S + 1)2 . This subspace is described by nonzero eigenvalues and corresponding eigenvectors of the density matrix. We prove that for large block the von Neumann entropy coincides with Renyi entropy and is equal to ln (S + 1) 2 .

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“…Examples of such systems are quantum Hall liquids 5,6,7 , Haldane spin systems 8,9 , polyacetylene 10 , and spin-Peierls system 11 . Recently it has become clear that the bulk-edge correspondence 6,7 has an intimate relation to the entanglement entropy 12,13,14 , which has been discussed to detect non trivial structures of topologically ordered states 13,14,15,16 .…”

Section: Introductionmentioning

confidence: 99%

Topological classification of gapped spin chains: Quantized Berry phase as a local order parameter

2008

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We characterize several phases of gapped spin systems by local order parameters defined by quantized Berry phases 1 . This characterization is topologically stable against any small perturbation as long as the energy gap remains finite. The models we pick up are S = 1, 2 dimerized Heisenberg chains and S = 2 Heisenberg chains with uniaxial single-ion-type anisotropy. Analytically we also evaluate the topological local order parameters for the generalized Affleck-Kennedy-Lieb-Tasaki (AKLT) model. The relation between the present Berry phases and the fractionalization in the integer spin chains are discussed as well.

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Entanglement Entropy of One-dimensional Gapped Spin Chains

Cited by 22 publications

References 30 publications

Bipartite fluctuations as a probe of many-body entanglement

Bipartite fluctuations as a probe of many-body entanglement

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

Topological classification of gapped spin chains: Quantized Berry phase as a local order parameter

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