Phase diagram of the SO(n) bilinear-biquadratic chain from many-body entanglement

Here we investigate the phase diagram of the SO(n) bilinear-biquadratic quantum spin chain by studying the global quantum correlations of the ground state. We consider the cases of n=3,4 and 5 and focus on the geometric entanglement in the thermodynamic limit. Apart from capturing all the known phase transitions, our analysis shows a number of novel distinctive behaviors in the phase diagrams which we conjecture to be general and valid for arbitrary n. In particular, we provide an intuitive argument in favor of an infinite entanglement length in the system at a purely-biquadratic point. Our results are also compared to other methods, such as fidelity diagrams.Comment: 7 pages, 4 figures. Revised version. To appear in PR

Simulation of strongly correlated fermions in two spatial dimensions with fermionic Projected Entangled-Pair States

We explain how to implement, in the context of projected entangled-pair states (PEPS), the general procedure of fermionization of a tensor network introduced in [P. Corboz, G. Vidal, Phys. Rev. B 80, 165129 (2009)]. The resulting fermionic PEPS, similar to previous proposals, can be used to study the ground state of interacting fermions on a two-dimensional lattice. As in the bosonic case, the cost of simulations depends on the amount of entanglement in the ground state and not directly on the strength of interactions. The present formulation of fermionic PEPS leads to a straightforward numerical implementation that allowed us to recycle much of the code for bosonic PEPS. We demonstrate that fermionic PEPS are a useful variational ansatz for interacting fermion systems by computing approximations to the ground state of several models on an infinite lattice. For a model of interacting spinless fermions, ground state energies lower than Hartree-Fock results are obtained, shifting the boundary between the metal and charge-density wave phases. For the t-J model, energies comparable with those of a specialized Gutzwiller-projected ansatz are also obtained.Comment: 25 pages, 35 figures (revised version

Classical simulation of infinite-size quantum lattice systems in two spatial dimensions

We present an algorithm to simulate two-dimensional quantum lattice systems in the thermodynamic limit. Our approach builds on the {\em projected entangled-pair state} algorithm for finite lattice systems [F. Verstraete and J.I. Cirac, cond-mat/0407066] and the infinite {\em time-evolving block decimation} algorithm for infinite one-dimensional lattice systems [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)]. The present algorithm allows for the computation of the ground state and the simulation of time evolution in infinite two-dimensional systems that are invariant under translations. We demonstrate its performance by obtaining the ground state of the quantum Ising model and analysing its second order quantum phase transition.Comment: 4 pages, 6 figures, 1 table. Revised version, with new diagrams and plots. The results on classical systems can now be found at arXiv:0711.396

Universality of Entanglement and Quantum Computation Complexity

We study the universality of scaling of entanglement in Shor's factoring algorithm and in adiabatic quantum algorithms across a quantum phase transition for both the NP-complete Exact Cover problem as well as the Grover's problem. The analytic result for Shor's algorithm shows a linear scaling of the entropy in terms of the number of qubits, therefore difficulting the possibility of an efficient classical simulation protocol. A similar result is obtained numerically for the quantum adiabatic evolution Exact Cover algorithm, which also shows universality of the quantum phase transition the system evolves nearby. On the other hand, entanglement in Grover's adiabatic algorithm remains a bounded quantity even at the critical point. A classification of scaling of entanglement appears as a natural grading of the computational complexity of simulating quantum phase transitions.Comment: 30 pages, 17 figures, accepted for publication in PR

Mobile word of mouth (m-WOM): analysing its negative impact on webrooming in omnichannel retailing

Purpose: The purpose of this research is to analyse the influence of mobile word of mouth (m-WOM), received at the physical store, which “challenges” the consumer's preferences in a webrooming experience. The impacts of the social relationship between the sender and the receiver of the m-WOM and product category (electronics versus fashion accessories) are examined. Design/methodology/approach: An online experiment was carried out which manipulated the presence and type of challenging m-WOM, and product category, in a 3 × 2 between-subjects factorial design. The participants were 204 consumers recruited through a market research agency. Their perceptions about the helpfulness of the m-WOM, and their product preferences and choices, were analysed. Findings: Receiving in-store m-WOM was perceived as helpful by webroomers and affected their preferences and choices. For electronics online reviews posted by anonymous customers were more influential than friends' opinions, whereas the opposite was the case with fashion accessories. The trustworthiness and expertise of the m-WOM source may explain the effects of m-WOM. Practical implications: m-WOM entails challenges and opportunities for retailers in the omnichannel era. The findings suggest that allowing customers to access m-WOM may be beneficial; however, retailers must consider the type of m-WOM that may be most suitable for their businesses. Recommendations for referral and review sites are also offered. Originality/value: This study examines the impact of challenging m-WOM on shopping experiences, combining online, mobile and physical channels. The results revealed the importance of the information source and product category in the determination of consumers' perceptions of helpfulness, preferences and choice

Entropy and Exact Matrix Product Representation of the Laughlin Wave Function

An analytical expression for the von Neumann entropy of the Laughlin wave function is obtained for any possible bipartition between the particles described by this wave function, for filling fraction nu=1. Also, for filling fraction nu=1/m, where m is an odd integer, an upper bound on this entropy is exhibited. These results yield a bound on the smallest possible size of the matrices for an exact representation of the Laughlin ansatz in terms of a matrix product state. An analytical matrix product state representation of this state is proposed in terms of representations of the Clifford algebra. For nu=1, this representation is shown to be asymptotically optimal in the limit of a large number of particles

Adiabatic quantum computation and quantum phase transitions

We analyze the ground state entanglement in a quantum adiabatic evolution algorithm designed to solve the NP-complete Exact Cover problem. The entropy of entanglement seems to obey linear and universal scaling at the point where the mass gap becomes small, suggesting that the system passes near a quantum phase transition. Such a large scaling of entanglement suggests that the effective connectivity of the system diverges as the number of qubits goes to infinity and that this algorithm cannot be efficiently simulated by classical means. On the other hand, entanglement in Grover's algorithm is bounded by a constant.Comment: 5 pages, 4 figures, accepted for publication in PR

Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems

In this paper we explore the practical use of the corner transfer matrix and its higher-dimensional generalization, the corner tensor, to develop tensor network algorithms for the classical simulation of quantum lattice systems of infinite size. This exploration is done mainly in one and two spatial dimensions (1d and 2d). We describe a number of numerical algorithms based on corner matri- ces and tensors to approximate different ground state properties of these systems. The proposed methods make also use of matrix product operators and projected entangled pair operators, and naturally preserve spatial symmetries of the system such as translation invariance. In order to assess the validity of our algorithms, we provide preliminary benchmarking calculations for the spin-1/2 quantum Ising model in a transverse field in both 1d and 2d. Our methods are a plausible alternative to other well-established tensor network approaches such as iDMRG and iTEBD in 1d, and iPEPS and TERG in 2d. The computational complexity of the proposed algorithms is also considered and, in 2d, important differences are found depending on the chosen simulation scheme. We also discuss further possibilities, such as 3d quantum lattice systems, periodic boundary conditions, and real time evolution. This discussion leads us to reinterpret the standard iTEBD and iPEPS algorithms in terms of corner transfer matrices and corner tensors. Our paper also offers a perspective on many properties of the corner transfer matrix and its higher-dimensional generalizations in the light of novel tensor network methods.Comment: 25 pages, 32 figures, 2 tables. Revised version. Technical details on some of the algorithms have been moved to appendices. To appear in PR

Universal geometric entanglement close to quantum phase transitions

Under successive Renormalization Group transformations applied to a quantum state $\ket{\Psi}$ of finite correlation length $\xi$, there is typically a loss of entanglement after each iteration. How good it is then to replace $\ket{\Psi}$ by a product state at every step of the process? In this paper we give a quantitative answer to this question by providing first analytical and general proofs that, for translationally invariant quantum systems in one spatial dimension, the global geometric entanglement per region of size $L \gg \xi$ diverges with the correlation length as $(c/12) \log{(\xi/\epsilon)}$ close to a quantum critical point with central charge $c$, where $\epsilon$ is a cut-off at short distances. Moreover, the situation at criticality is also discussed and an upper bound on the critical global geometric entanglement is provided in terms of a logarithmic function of $L$.Comment: 4 pages, 3 figure