Three-Spin Interactions in Optical Lattices and Criticality in Cluster Hamiltonians

Abstract

We demonstrate that in a triangular configuration of an optical lattice of two atomic species a variety of novel spin-1/2 Hamiltonians can be generated. They include effective three-spin interactions resulting from the possibility of atoms tunneling along two different paths. This motivates the study of ground state properties of various three-spin Hamiltonians in terms of their two-point and n-point correlations as well as the localizable entanglement. We present a Hamiltonian with a finite energy gap above its unique ground state for which the localizable entanglement length diverges for a wide interval of applied external fields, while at the same time the classical correlation length remains finite.PACS numbers: 03.67.Mn, The combination of cold atom technology with optical lattices [1,2] gives rise to a variety of possibilities for constructing spin Hamiltonians [3,4]. This is particularly appealing as the high degree of isolation from the environment that can be achieved in these systems allows for the study of these Hamiltonians under idealised laboratory conditions. In parallel, techniques have been developed for minimising imperfections and impurities [5,6] in the implementation of the desired structures and for their subsequent probing and measurement [7]. These achievements permit the experimental investigation of Hamiltonians that are of interest in areas such as quantum information or condensed matter physics with the added advantage of a remarkable freedom in the choice of external parameters. Presently, attention both in condensed matter physics and in cold atom research is focusing on two-spin interactions as these are most readily accessible experimentally. However, the unique experimental capability provided by cold atom technology allows us to relax this restriction. Here we demonstrate that cold atom technology provides a laboratory to generate and study higher order effects such as three-spin interactions that give rise to unique entanglement properties.The present work serves two purposes. Firstly, it demonstrates that in a two species Bose-Hubbard model in a triangular configuration a wide range of Hamilton operators can be generated that include effective three-spin interactions. They result from the possibility of atomic tunneling through different paths from one vertex to the other. This can be extended to a one dimensional spin chain with three-spin interactions. Secondly, we take this novel experimental capability as a motivation to study unique ground state properties of Hamiltonians that include three-spin interactions. In this context one can study possible quantum phase transitions by considering both the classical correlation properties as well as the entanglement properties of these systems. Specifically we consider the so-called cluster Hamiltonian and its ground state, the cluster state which has previously been shown to play an important role as a resource in the context of quantum computation [20]. Subject to an additional Zeeman term the combined Hamiltonian possesses a fi...

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

“…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;…”

Geometric Phases and Criticality in Spin-Chain Systems

“…The insights provided here shed light into the understanding of more general systems, where analytic solutions might not be available. A physical implementation is proposed with ultra-cold atoms superposed by optical lattices [11,12]. It utilizes Raman activated tunneling transitions as well as coherent drive of the system via Bragg scattering.…”

“…Topological phases are inherently resilient against control errors, a property that can be proved to be of a great advantage when considering many-body systems. Such a study can be theoretically performed on any system which can be analytically elaborated such as the case of the cluster Hamiltonian [11], or exploited numerically when analytic solutions are not known. The generalization of these results to a wide variety of critical phenomena and their relation to the critical exponents is a promising and challenging question which deserves extensive future investigation.…”

Remarks on duality transformations and generalized stabilizer states

“…Such a property requires proof (in fact and perhaps not too surprising this is not correct in the extreme cases J = 0 and J = ∞) even though one can expect it to hold in most physically reasonable cases. Now we move beyond self-duality and link the cluster Hamiltonian [12,13] to the an-isotropic XY model. The cluster Hamiltonian, exhibiting some interesting critical behaviour [12] and strong finite size effects [13], is defined as…”

“…Now we move beyond self-duality and link the cluster Hamiltonian [12,13] to the an-isotropic XY model. The cluster Hamiltonian, exhibiting some interesting critical behaviour [12] and strong finite size effects [13], is defined as…”