Entanglement renormalization

In the context of real-space renormalization group methods, we propose a novel scheme for quantum systems defined on a D-dimensional lattice. It is based on a coarse-graining transformation that attempts to reduce the amount of entanglement of a block of lattice sites before truncating its Hilbert space. Numerical simulations involving the ground state of a 1D system at criticality show that the resulting coarse-grained site requires a Hilbert space dimension that does not grow with successive rescaling transformations. As a result we can address, in a quasi-exact way, tens of thousands of quantum spins with a computational effort that scales logarithmically in the system's size. The calculations unveil that ground state entanglement in extended quantum systems is organized in layers corresponding to different length scales. At a quantum critical point, each rellevant length scale makes an equivalent contribution to the entanglement of a block with the rest of the system.Comment: 4 pages, 4 figures, updated versio

Characterizing topological order by studying the ground states of an infinite cylinder

Given a microscopic lattice Hamiltonian for a topologically ordered phase, we describe a tensor network approach to characterize its emergent anyon model and, in a chiral phase, also its gapless edge theory. First, a tensor network representation of a complete, orthonormal set of ground states on a cylinder of infinite length and finite width is obtained through numerical optimization. Each of these ground states is argued to have a different anyonic flux threading through the cylinder. In a chiral phase, the entanglement spectrum of each ground state is seen to reveal a different sector of the corresponding gapless edge theory. A quasi-orthogonal basis on the torus is then produced by chopping off and reconnecting the tensor network representation on the cylinder. Elaborating on the recent proposal of [Y. Zhang et al. Phys. Rev. B 85, 235151 (2012)], a rotation on the torus yields an alternative basis of ground states and, through the computation of overlaps between bases, the modular matrices S and U (containing the mutual and self statistics of the different anyon species) are extracted. As an application, we study the hard-core boson Haldane model by using the two-dimensional density matrix renormalization group. A thorough characterization of the universal properties of this lattice model, both in the bulk and at the edge, unambiguously shows that its ground space realizes the \nu=1/2 bosonic Laughlin state.Comment: 10 pages, 11 figure

Hydrogen column density evaluations toward Capella: consequences on the interstellar deuterium abundance

The deuterium abundance evaluation in the direction of Capella has for a long time been used as a reference for the local interstellar medium (ISM) within our Galaxy. We show here that broad and weak HI components could be present on the Capella line of sight, leading to a large new additional systematic uncertainty on the N(HI) evaluation. The D/H ratio toward Capella is found to be equal to 1.67 (+/-0.3)x10^-5 with almost identical chi^2 for all the fits (this range includes only the systematic error; the 2 sigma statistical one is almost negligible in comparison). It is concluded that D/H evaluations over HI column densities below 10^19 cm^-2 (even perhaps below 10^20 cm^-2 if demonstrated by additional observations) may present larger uncertainties than previously anticipated. It is mentionned that the D/O ratio might be a better tracer for DI variations in the ISM as recently measured by the Far Ultraviolet Spectroscopic Explorer (FUSE).Comment: Accepted for publication in the Astrophysical Journal Letter

Entanglement renormalization in fermionic systems

We demonstrate, in the context of quadratic fermion lattice models in one and two spatial dimensions, the potential of entanglement renormalization (ER) to define a proper real-space renormalization group transformation. Our results show, for the first time, the validity of the multi-scale entanglement renormalization ansatz (MERA) to describe ground states in two dimensions, even at a quantum critical point. They also unveil a connection between the performance of ER and the logarithmic violations of the boundary law for entanglement in systems with a one-dimensional Fermi surface. ER is recast in the language of creation/annihilation operators and correlation matrices.Comment: 5 pages, 4 figures Second appendix adde

Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law

This work explores the use of a tree tensor network ansatz to simulate the ground state of a local Hamiltonian on a two-dimensional lattice. By exploiting the entropic area law, the tree tensor network ansatz seems to produce quasi-exact results in systems with sizes well beyond the reach of exact diagonalisation techniques. We describe an algorithm to approximate the ground state of a local Hamiltonian on a L times L lattice with the topology of a torus. Accurate results are obtained for L={4,6,8}, whereas approximate results are obtained for larger lattices. As an application of the approach, we analyse the scaling of the ground state entanglement entropy at the quantum critical point of the model. We confirm the presence of a positive additive constant to the area law for half a torus. We also find a logarithmic additive correction to the entropic area law for a square block. The single copy entanglement for half a torus reveals similar corrections to the area law with a further term proportional to 1/L.Comment: Major rewrite, new version published in Phys. Rev. B with highly improved numerical results for the scaling of the entropies and several new sections. The manuscript has now 19 pages and 30 Figure

Quantum mechanical analysis of the elastic propagation of electrons in the Au/Si system: application to Ballistic Electron Emission Microscopy

We present a Green's function approach based on a LCAO scheme to compute the elastic propagation of electrons injected from a STM tip into a metallic film. The obtained 2D current distribution in real and reciprocal space furnish a good representation of the elastic component of Ballistic Electron Emission Microscopy (BEEM) currents. Since this component accurately approximates the total current in the near threshold region, this procedure allows --in contrast to prior analyses-- to take into account effects of the metal band structure in the modeling of these experiments. The Au band structure, and in particular its gaps appearing in the [111] and [100] directions provides a good explanation for the previously irreconcilable results of nanometric resolution and similarity of BEEM spectra on both Au/Si(111) and Au/Si(100).Comment: 12 pages, 9 postscript figures, revte

Chiral spin liquid and emergent anyons in a Kagome lattice Mott insulator

Topological phases in frustrated quantum spin systems have fascinated researchers for decades. One of the earliest proposals for such a phase was the chiral spin liquid put forward by Kalmeyer and Laughlin in 1987 as the bosonic analogue of the fractional quantum Hall effect. Elusive for many years, recent times have finally seen a number of models that realize this phase. However, these models are somewhat artificial and unlikely to be found in realistic materials. Here, we take an important step towards the goal of finding a chiral spin liquid in nature by examining a physically motivated model for a Mott insulator on the Kagome lattice with broken time-reversal symmetry. We first provide a theoretical justification for the emergent chiral spin liquid phase in terms of a network model perspective. We then present an unambiguous numerical identification and characterization of the universal topological properties of the phase, including ground state degeneracy, edge physics, and anyonic bulk excitations, by using a variety of powerful numerical probes, including the entanglement spectrum and modular transformations.Comment: 9 pages, 9 figures; partially supersedes arXiv:1303.696

Boundary quantum critical phenomena with entanglement renormalization

We extend the formalism of entanglement renormalization to the study of boundary critical phenomena. The multi-scale entanglement renormalization ansatz (MERA), in its scale invariant version, offers a very compact approximation to quantum critical ground states. Here we show that, by adding a boundary to the scale invariant MERA, an accurate approximation to the critical ground state of an infinite chain with a boundary is obtained, from which one can extract boundary scaling operators and their scaling dimensions. Our construction, valid for arbitrary critical systems, produces an effective chain with explicit separation of energy scales that relates to Wilson's RG formulation of the Kondo problem. We test the approach by studying the quantum critical Ising model with free and fixed boundary conditions.Comment: 8 pages, 12 figures, for a related work see arXiv:0912.289

Hot electron transport in Ballistic Electron Emission Spectroscopy: band structure effects and k-space currents

Using a Green's function approach, we investigate band structure effects in the BEEM current distribution in reciprocal space. In the elastic limit, this formalism provides a 'parameter free' solution to the BEEM problem. At low temperatures, and for thin metallic layers, the elastic approximation is enough to explain the experimental I(V) curves at low voltages. At higher voltages inelastic effects are approximately taken into account by introducing an effective RPA-electron lifetime, much in similarity with LEED theory. For thick films, however, additional damping mechanisms are required to obtain agreement with experiment.Comment: 4 pages, 3 postscript figures, revte

Observation of enhanced transmission for s-polarized light through a subwavelength slit

Enhanced optical transmission (EOT) through subwavelength apertures is usually obtained for p-polarized light. The present study experimentally investigates EOT for s-polarized light. A subwavelength slit surrounded on each side by periodic grooves has been fabricated in a gold film and covered by a thin dielectric layer. The excitation of s-polarized dielectric waveguide modes inside the dielectric film strongly increases the s-polarized transmission. Transmission measurements are compared with a coupled mode model and show good qualitative agreement. Adding a waveguide can improve light transmission through subwavelength apertures, as both s and p-polarization can be efficiently transmitted.Comment: 11 pages, 3 figures, submitted to Applied Physics Letter