Entanglement, fractional magnetization, and long-range interactions

Cadarso, Andrea; Sanz, Mikel; Wolf, Michael M.; Cirac, J. I.; Pérez-Garcı́a, David

doi:10.1103/physrevb.87.035114

Cited by 18 publications

(19 citation statements)

References 21 publications

(49 reference statements)

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“…This result is in contradiction with the theorem proved in [17], the reason is that the ground state of our longrange hamiltonian does not necessarily related to the ground state of any short-range hamiltonian! This is simply because if it was possible to approximate the ground state of our hamiltonian with ground state of a short-range hamiltonian we would expect conformal symmetry which is not the case in our model.…”

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confidence: 54%

“…In particular they found that the area law should be valid in higher dimensions even for long-range interacting harmonic oscillators, and most recently the entanglement entropy for the long range anti-ferromagnetic Ising chain is calculated by numerical means [16]. Finally we should mention that in [17] based on the matrix product states it was argued that for those long-range systems that do not have any short-range counterparts, in other words one can not approximate the ground state of the long-range model with the ground state of another shortrange model, the presence of the long-range interaction implies larger entanglement entropy or the volume scaling of the entropy.…”

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confidence: 99%

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Entanglement entropy in long-range harmonic oscillators

Nezhadhaghighi¹,

Rajabpour²

2012

EPL

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-We study the Von Neumann and Rényi entanglement entropy of long-range harmonic oscillators (LRHO) by both theoretical and numerical means. We show that the entanglement entropy in massless harmonic oscillators increases logarithmically with the sub-system size as S = c ef f 3 log l. Although the entanglement entropy of LRHO's shares some similarities with the entanglement entropy at conformal critical points we show that the Rényi entanglement entropy presents some deviations from the expected conformal behaviour. In the massive case we demonstrate that the behaviour of the entanglement entropy with respect to the correlation length is also logarithmic as the short range case.Introduction. -While the entanglement entropy of short-range interacting quantum systems in one dimension is well studied with many different techniques [1], such as hamiltonian techniques [2][3][4][5], euclidean methods [6,7] and conformal field theory [8,9] there are few results concerning the long-range interacting systems. The main difficulty is the lack of exact solution for these systems which makes the problem much harder than the short-range counterparts. Because of the presence of non-trivial dynamical exponents the common euclidean techniques are usually useless in calculating the entanglement entropy in these systems. It is worth mentioning that although having a nontrivial dynamical exponents is not restricted to just longrange interacting systems, see for example [10,11], they usually provide a very natural knob to change it arbitrarily. Because of the huge finite size effects in these systems even the numerical calculations are not easy as their shortrange counterparts. Nevertheless, recently much progress has been made in different directions: In [12] the entanglement entropy in Lipkin-Meshkov-Glick (LMG) model is studied. This model has hamiltonian similar to the XY model but while in the latter model the interaction only takes place between nearest neighbors, in the LMG model, all spins interact among themselves. In [13], the interactions are restricted to the Ising-type without external magnetic field which allows to study both the static and the dynamical entanglement properties of the system. Eisert et. all [14] found a logarithmically divergent geometric en-

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confidence: 54%

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confidence: 99%

Entanglement entropy in long-range harmonic oscillators

Nezhadhaghighi¹,

Rajabpour²

2012

EPL

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“…We proceed with the Eq. (22). To evaluate this integral we must consider B ∈ (−∞ < z < 0) ∪ (l < z < ∞) as the complement of the sub-region l. The matrix Λ becomes:…”

Section: Von Neumann and Rényi Entanglement Entropymentioning

confidence: 99%

Quantum entanglement entropy and classical mutual information in long-range harmonic oscillators

Nezhadhaghighi

Rajabpour

2013

Phys. Rev. B

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We study different aspects of quantum von Neumann and Rényi entanglement entropy of one dimensional long-range harmonic oscillators that can be described by well-defined non-local field theories. We show that the entanglement entropy of one interval with respect to the rest changes logarithmically with the number of oscillators inside the subsystem. This is true also in the presence of different boundary conditions. We show that the coefficients of the logarithms coming from different boundary conditions can be reduced to just two different universal coefficients. We also study the effect of the mass and temperature on the entanglement entropy of the system in different situations. The universality of our results is also confirmed by changing different parameters in the coupled harmonic oscillators. We also show that more general interactions coming from general singular Toeplitz matrices can be decomposed to our long-range harmonic oscillators. Despite the long-range nature of the couplings we show that the area law is valid in two dimensions and the universal logarithmic terms appear if we consider subregions with sharp corners. Finally we study analytically different aspects of the mutual information such as its logarithmic dependence to the subsystem, effect of mass and influence of the boundary. We also generalize our results in this case to general singular Toeplitz matrices and higher dimensions. Contents

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“…While dipolar interactions did not destroy the SPT phases in our example, quantum magnets with long-range interactions have recently been shown to harbor unusual and often dimension-specific physics. [90][91][92][93][94][95][96] The polar-molecule experiment we propose could therefore help guide the theoretical understanding of these effects in two-dimensional and threedimensional systems, including SPT phases, where efficient numerical methods are lacking. In fact, the classification of SPT phases is yet to be extended to models with long-range interactions.…”

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Topological phases in ultracold polar-molecule quantum magnets

et al. 2013

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We show how to use polar molecules in an optical lattice to engineer quantum spin models with arbitrary spin S 1/2 and with interactions featuring a direction-dependent spin anisotropy. This is achieved by encoding the effective spin degrees of freedom in microwave-dressed rotational states of the molecules and by coupling the spins through dipolar interactions. We demonstrate how one of the experimentally most accessible anisotropies stabilizes symmetry protected topological phases in spin ladders. Using the numerically exact density matrix renormalization group method, we find that these interacting phases-previously studied only in the nearestneighbor case-survive in the presence of long-range dipolar interactions. We also show how to use our approach to realize the bilinear-biquadratic spin-1 and the Kitaev honeycomb models. Experimental detection schemes and imperfections are discussed. have spurred tremendous interest in exotic strongly correlated many-body phenomena arising from anisotropic, long-ranged dipole-dipole interactions. The types of anisotropies realizable with these interactions are typically limited to simple changes of the interaction sign and magnitude according to the spherical harmonic Y 2,0 ∝ 1 − 3 cos 2 θ , where (θ,φ) are the spherical coordinates of the vector connecting the two interacting dipoles. 11,13,32,33 In this Rapid Communication we show, in the context of polar molecules, that microwave dressing provides a tremendous degree of simultaneous control over five independent dipoledipole interaction terms whose angular dependences are given by the rank-2 spherical harmonics. This opens the door to simulating well-known models including the spin-1/2 XXZ model with a direction-dependent spin anisotropy, the spin-1 bilinear-biquadratic model, 47 and the Kitaev honeycomb model. 48 Thanks to the use of direct dipole-dipole coupling, the resulting interactions are stronger and hence easier to observe experimentally than other-potentially direction-dependentspin-spin interactions such as superexchange in ultracold atoms 49 or perturbative dipole-dipole-mediated couplings between polar molecules. 16,17 As a specific example demonstrating the reach of our method, we show how to design a spin-1/2 XXZ model with direction-dependent spin anisotropy using a minimal and experimentally reasonable microwave configuration. In a two-legged ladder geometry with nearest-neighbor interactions, this model has been shown to exhibit symmetry protected topological (SPT) phases. 50 These phases are exotic gapped states of matter distinct from trivial gapped phases when specific symmetries are present. A lattice of polar molecules in the XY plane is subjected to a dc electric field alongẑ. We define the xyz coordinate system as the rotation of the XYZ coordinate system around Z by 0 and then aroundŷ by 0 . A vector R with polar coordinates (R, ) in the XY plane has spherical coordinates (R,θ,φ) in the xyz coordinate system. method (DMRG), 64 we compute the phase diagram of the two-legged-ladder model obtai...

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Entanglement, fractional magnetization, and long-range interactions

Abstract: Based on the theory of matrix product states, we give precise statements and complete analytical proofs of the following claim: A large fractionalization in the magnetization or the need of long-range interactions imply large entanglement in the state of a quantum spin chain.

Cited by 18 publications

References 21 publications

Entanglement entropy in long-range harmonic oscillators

Entanglement entropy in long-range harmonic oscillators

Quantum entanglement entropy and classical mutual information in long-range harmonic oscillators

Topological phases in ultracold polar-molecule quantum magnets

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