Tensor network states and algorithms in the presence of a global U(1) symmetry

Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. In a recent paper [arXiv:0907.2994v1] we discussed how to incorporate a global internal symmetry, given by a compact, completely reducible group G, into tensor network decompositions and algorithms. Here we specialize to the case of Abelian groups and, for concreteness, to a U(1) symmetry, often associated with particle number conservation. We consider tensor networks made of tensors that are invariant (or covariant) under the symmetry, and explain how to decompose and manipulate such tensors in order to exploit their symmetry. In numerical calculations, the use of U(1) symmetric tensors allows selection of a specific number of particles, ensures the exact preservation of particle number, and significantly reduces computational costs. We illustrate all these points in the context of the multi-scale entanglement renormalization ansatz.Comment: 22 pages, 25 figures, RevTeX

Electrochemical and photo-electrochemical processes of Methylene blue oxidation by Ti/TiO2 electrodes modified with Fe-allophane

Indexación: Scopus.This work reports the degradation of methylene blue (MB) on Ti/TiO2 and Ti/TiO2/Fe-allophane electrodes in a pH 3 using 0.1 M Na2SO4 as support electrolyte. SEM micrographs show a homogeneous distribution of TiO2 over the whole electrode surface forming nanotubes and nanopores. Fe-allophane modified electrode shows the formation of large-grains agglomerate on the electrode surface due to allophane, which provides a greater surface area to the electrode due to meso and micropore structures. Preliminary cyclic voltammetry show that Ti/TiO2 has the typical voltammetric response due to Ti(III)/Ti(IV) pair. Diffusional problems were observed through of the film when the electrode is modified with Fe-allophane modifying the quasi-reversible process Ti(III)/Ti(IV). Different kind of methodologies in the degradation process were used: Electrochemistry (EC), Photochemistry (PC), Photoelectrochemistry (PEC) and Adsorption (Ads). These methods were developing to discard any reaction or interaction that is not of interest. On Ti/TiO2 with PC and Ads methodologies was not observed any activity to MB degradation showing that is not photosensitive and that the interaction between this and surface electrode is low. But with EC and PEC degradation to 55% is reached after 3 hours of electrolysis. With Ti/TiO2-Fe-allophane electrodes are observed a higher activity for all methodologies. The PC and Ads methods show that the MB degradation reaches to ∼20 % of the initial concentration. As mentioned above, the PC and Ads processes no show degradation on Ti/TiO2, therefore the degradation it only due to the adsorption of MB in/on allophane coat behaving as concentrator matrix. A lower improvement is observed with EC process when is incorporated Ti/TiO2-Fe-allophane is due to the barrier of the electrode surface by oxidation products. With PEC is reached the higher degradation value of ∼88 %, showing an improvement of the degradation with the presence of Fe-allophane. The results indicate that the main role of Fe-allophane on the electrode is similar to a concentrator matrix.http://ref.scielo.org/shz7t

Matrix product states for anyonic systems and efficient simulation of dynamics

Matrix product states (MPS) have proven to be a very successful tool to study lattice systems with local degrees of freedom such as spins or bosons. Topologically ordered systems can support anyonic particles which are labeled by conserved topological charges and collectively carry non-local degrees of freedom. In this paper we extend the formalism of MPS to lattice systems of anyons. The anyonic MPS is constructed from tensors that explicitly conserve topological charge. We describe how to adapt the time-evolving block decimation (TEBD) algorithm to the anyonic MPS in order to simulate dynamics under a local and charge-conserving Hamiltonian. To demonstrate the effectiveness of anyonic TEBD algorithm, we used it to simulate (i) the ground state (using imaginary time evolution) of an infinite 1D critical system of (a) Ising anyons and (b) Fibonacci anyons both of which are well studied, and (ii) the real time dynamics of an anyonic Hubbard-like model of a single Ising anyon hopping on a ladder geometry with an anyonic flux threading each island of the ladder. Our results pertaining to (ii) give insight into the transport properties of anyons. The anyonic MPS formalism can be readily adapted to study systems with conserved symmetry charges, as this is equivalent to a specialization of the more general anyonic case.Comment: 18 pages, 15 figue

Simulation of anyons with tensor network algorithms

Interacting systems of anyons pose a unique challenge to condensed matter simulations due to their non-trivial exchange statistics. These systems are of great interest as they have the potential for robust universal quantum computation, but numerical tools for studying them are as yet limited. We show how existing tensor network algorithms may be adapted for use with systems of anyons, and demonstrate this process for the 1-D Multi-scale Entanglement Renormalisation Ansatz (MERA). We apply the MERA to infinite chains of interacting Fibonacci anyons, computing their scaling dimensions and local scaling operators. The scaling dimensions obtained are seen to be in agreement with conformal field theory. The techniques developed are applicable to any tensor network algorithm, and the ability to adapt these ansaetze for use on anyonic systems opens the door for numerical simulation of large systems of free and interacting anyons in one and two dimensions.Comment: Fixed typos, matches published version. 16 pages, 21 figures, 4 tables, RevTeX 4-1. For a related work, see arXiv:1006.247

Optimal distillation of a GHZ state

We present the optimal local protocol to distill a Greenberger-Horne-Zeilinger (GHZ) state from a single copy of any pure state of three qubits.Comment: RevTex, 4 pages, 2 figures. Published version, some references adde

A matriz socio-histórica e o ethos no coração e na força do MPLA na Angola moderna

Proximate to a Weberian perspective, this article argues that the resilience of the Angolan regime is mainly owed to an ethos structured on top of a specific socio-cultural historical matrix (minority at start), evolving since the 16th century. Such matrix was structured on a prevailing Weltanschauung (world and national vision), that has been progressively self-presented, self-assumed, imposed/assimilated as national and modern within a project of identity and power hegemony, even though still and constantly ridden by several internal contradictions and tensions. Dynamics of this process is central to understand the intricacies of the relationship between rulers and ruled, evolving identities as well as the still significant social support to the party in power after more than four decades in the government. The regime’s resilience lays on such ethos in support of hegemonic power and identity project, above and beyond the president and all his political management abilities, beyond the central instrumentality of the national oil company (SONANGOL), beyond the media spotlight on influential names surrounding the presidency, including the president’s men, generals, and beyond authoritarianism.info:eu-repo/semantics/publishedVersio

Variational quantum Monte Carlo simulations with tensor-network states

We show that the formalism of tensor-network states, such as the matrix product states (MPS), can be used as a basis for variational quantum Monte Carlo simulations. Using a stochastic optimization method, we demonstrate the potential of this approach by explicit MPS calculations for the transverse Ising chain with up to N=256 spins at criticality, using periodic boundary conditions and D*D matrices with D up to 48. The computational cost of our scheme formally scales as ND^3, whereas standard MPS approaches and the related density matrix renromalization group method scale as ND^5 and ND^6, respectively, for periodic systems.Comment: 4+ pages, 2 figures. v2: improved data, comparisons with exact results, to appear in Phys Rev Let

Entanglement renormalization, scale invariance, and quantum criticality

The use of entanglement renormalization in the presence of scale invariance is investigated. We explain how to compute an accurate approximation of the critical ground state of a lattice model, and how to evaluate local observables, correlators and critical exponents. Our results unveil a precise connection between the multi-scale entanglement renormalization ansatz (MERA) and conformal field theory (CFT). Given a critical Hamiltonian on the lattice, this connection can be exploited to extract most of the conformal data of the CFT that describes the model in the continuum limit.Comment: 4 pages, 3 figures, RevTeX 4. Revised for greater clarit

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

Optimal simulation of two-qubit Hamiltonians using general local operations

We consider the simulation of the dynamics of one nonlocal Hamiltonian by another, allowing arbitrary local resources but no entanglement nor classical communication. We characterize notions of simulation, and proceed to focus on deterministic simulation involving one copy of the system. More specifically, two otherwise isolated systems $A$ and $B$ interact by a nonlocal Hamiltonian $H \neq H_A+H_B$. We consider the achievable space of Hamiltonians $H'$ such that the evolution $e^{-iH't}$ can be simulated by the interaction $H$ interspersed with local operations. For any dimensions of $A$ and $B$, and any nonlocal Hamiltonians $H$ and $H'$, there exists a scale factor $s$ such that for all times $t$ the evolution $e^{-iH'st}$ can be simulated by $H$ acting for time $t$ interspersed with local operations. For 2-qubit Hamiltonians $H$ and $H'$, we calculate the optimal $s$ and give protocols achieving it. The optimal protocols do not require local ancillas, and can be understood geometrically in terms of a polyhedron defined by a partial order on the set of 2-qubit Hamiltonians.Comment: (1) References to related work, (2) protocol to simulate one two-qudit Hamiltonian with another, and (3) other related results added. Some proofs are simplifie