Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems

Topological phases protected by symmetry can occur in gapped and---surprisingly---in critical systems. We consider non-interacting fermions in one dimension with spinless time-reversal symmetry. It is known that the phases are classified by a topological invariant $\omega$ and a central charge $c$. We investigate the correlations of string operators, giving insight into the interplay between topology and criticality. In the gapped phases, these non-local string order parameters allow us to extract $\omega$. Remarkably, ratios of correlation lengths are universal. In the critical phases, the scaling dimensions of these operators serve as an order parameter, encoding $\omega$ and $c$. We derive exact asymptotics of these correlation functions using Toeplitz determinant theory. We include physical discussion, e.g., relating lattice operators to the conformal field theory. Moreover, we discuss the dual spin chains. Using the aforementioned universality, the topological invariant of the spin chain can be obtained from correlations of local observables.Comment: 35 pages, 5 page appendi

A Weighted Correlation Index for Rankings with Ties

Understanding the correlation between two different scores for the same set of items is a common problem in information retrieval, and the most commonly used statistics that quantifies this correlation is Kendall's $\tau$. However, the standard definition fails to capture that discordances between items with high rank are more important than those between items with low rank. Recently, a new measure of correlation based on average precision has been proposed to solve this problem, but like many alternative proposals in the literature it assumes that there are no ties in the scores. This is a major deficiency in a number of contexts, and in particular while comparing centrality scores on large graphs, as the obvious baseline, indegree, has a very large number of ties in web and social graphs. We propose to extend Kendall's definition in a natural way to take into account weights in the presence of ties. We prove a number of interesting mathematical properties of our generalization and describe an $O(n\log n)$ algorithm for its computation. We also validate the usefulness of our weighted measure of correlation using experimental data

A primer on quantum fluids

This book introduces the theoretical description and properties of quantum fluids. The focus is on gaseous atomic Bose-Einstein condensates and, to a minor extent, superfluid helium, but the underlying concepts are relevant to other forms of quantum fluids such as polariton and photonic condensates. The book is pitched at the level of advanced undergraduates and early postgraduate students, aiming to provide the reader with the knowledge and skills to develop their own research project on quantum fluids. Indeed, the content for this book grew from introductory notes provided to our own research students. It is assumed that the reader has prior knowledge of undergraduate mathematics and/or physics; otherwise, the concepts are introduced from scratch, often with references for directed further reading.Comment: 132 pages. Published as SpringerBriefs in Physics book. Typos corrected in this versio

Sequence effects in categorization of simple perceptual stimuli

Categorization research typically assumes that the cognitive system has access to a (more or less noisy) representation of the absolute magnitudes of the properties of stimuli and that this information is used in reaching a categorization decision. However, research on identification of simple perceptual stimuli suggests that people have very poor representations of absolute magnitude information and that judgments about absolute magnitude are strongly influenced by preceding material. The experiments presented here investigate such sequence effects in categorization tasks. Strong sequence effects were found. Classification of a borderline stimulus was more accurate when preceded by a distant member of the opposite category than by a distant member of the same category. It is argued that this category contrast effect cannot be accounted for by extant exemplar or decision-bound models of categorization. The effect suggests the use of relative magnitude information in categorization. A memory and contrast model illustrates how relative magnitude information may be used in categorization

Consensus Acceleration in Multiagent Systems with the Chebyshev Semi-Iterative Method

We consider the fundamental problem of reaching consensus in multiagent systems; an operation required in many applications such as, among others, vehicle formation and coordination, shape formation in modular robotics, distributed target tracking, and environmental modeling. To date, the consensus problem (the problem where agents have to agree on their reported values) has been typically solved with iterative decentralized algorithms based on graph Laplacians. However, the convergence of these existing consensus algorithms is often too slow for many important multiagent applications, and thus they are increasingly being combined with acceleration methods. Unfortunately, state-of-the-art acceleration techniques require parameters that can be optimally selected only if complete information about the network topology is available, which is rarely the case in practice. We address this limitation by deriving two novel acceleration methods that can deliver good performance even if little information about the network is available. The first proposed algorithm is based on the Chebyshev semi-iterative method and is optimal in a well defined sense; it maximizes the worst-case convergence speed (in the mean sense) given that only rough bounds on the extremal eigenvalues of the network matrix are available. It can be applied to systems where agents use unreliable communication links, and its computational complexity is similar to those of simple Laplacian-based methods. This algorithm requires synchronization among agents, so we also propose an asynchronous version that approximates the output of the synchronous algorithm. Mathematical analysis and numerical simulations show that the convergence speed of the proposed acceleration methods decrease gracefully in scenarios where the sole use of Laplacian-based methods is known to be impractical