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arXiv2014

Low-dimensional dynamics of populations of pulse-coupled oscillators

Large communities of biological oscillators show a prevalent tendency to self-organize in time. This cooperative phenomenon inspired Winfree to formulate a mathematical model that originated the theory of macroscopic synchronization. Despite its fundamental importance, a complete mathematical analysis of the model proposed by Winfree ---consisting of a large population of all-to-all pulse-coupled oscillators--- is still missing. Here we show that the dynamics of the Winfree model evolves into the so-called Ott-Antonsen manifold. This important property allows for an exact description of this high-dimensional system in terms of a few macroscopic variables, and the full investigation of its dynamics. We find that brief pulses are capable of synchronizing heterogeneous ensembles which fail to synchronize with broad pulses, specially for certain phase response curves. Finally, to further illustrate the potential of our results, we investigate the possibility of `chimera' states in populations of identical pulse-coupled oscillators. Chimeras are self-organized states in which the symmetry of a population is broken into a synchronous and an asynchronous part. Here we derive three ordinary differential equations describing two coupled populations, and uncover a variety of chimera states, including a new class with chaotic dynamics.

nlin.AO↗

arXiv2014

Hierarchical Block Structures and High-resolution Model Selection in Large Networks

Discovering and characterizing the large-scale topological features in empirical networks are crucial steps in understanding how complex systems function. However, most existing methods used to obtain the modular structure of networks suffer from serious problems, such as being oblivious to the statistical evidence supporting the discovered patterns, which results in the inability to separate actual structure from noise. In addition to this, one also observes a resolution limit on the size of communities, where smaller but well-defined clusters are not detectable when the network becomes large. This phenomenon occurs not only for the very popular approach of modularity optimization, which lacks built-in statistical validation, but also for more principled methods based on statistical inference and model selection, which do incorporate statistical validation in a formally correct way. Here we construct a nested generative model that, through a complete description of the entire network hierarchy at multiple scales, is capable of avoiding this limitation, and enables the detection of modular structure at levels far beyond those possible with current approaches. Even with this increased resolution, the method is based on the principle of parsimony, and is capable of separating signal from noise, and thus will not lead to the identification of spurious modules even on sparse networks. Furthermore, it fully generalizes other approaches in that it is not restricted to purely assortative mixing patterns, directed or undirected graphs, and ad hoc hierarchical structures such as binary trees. Despite its general character, the approach is tractable, and can be combined with advanced techniques of community detection to yield an efficient algorithm that scales well for very large networks.

physics.data-an↗

arXiv2013

The Hilbert-glass transition: new universality of temperature-tuned many-body dynamical quantum criticality

We consider a new class of unconventional critical phenomena that is characterized by singularities only in dynamical quantities and has no thermodynamic signatures. A possible example is the recently proposed many-body localization transition, in which transport coefficients vanish at a critical temperature. Describing this unconventional quantum criticality has been technically challenging as understanding the finite-temperature dynamics requires the knowledge of a large number of many-body eigenstates. Here we develop a real-space renormalization group method for excited state (RSRG-X), that allow us to overcome this challenge, and establish the existence and universal properties of such temperature-tuned dynamical phase transitions. We characterize a specific example: the 1D disordered transverse field Ising model with interactions. Using RSRG-X, we find a finite-temperature transition, between two localized phases, characterized by non-analyticities of the dynamic spin correlation function and the low frequency heat conductivity.

cond-mat.str-el↗

arXiv2014

Dynamical Conductivity Across The Disorder-Tuned Superconductor-Insulator Transition

A quantum phase transition is a dramatic event marked by large spatial and temporal fluctuations, where one phase of matter with its ground state and tower of excitations reorganizes into a completely different phase. We provide new insight into the disorder-driven superconductor-insulator transition (SIT) in two dimensions, a problem of great theoretical and experimental interest, with the dynamical conductivity σ(ω) and the bosonic (pair) spectral function P(ω) calculated from quantum Monte Carlo simulations. We identify characteristic energy scales in the superconducting and insulating phases that vanish at the transition due to enhanced quantum fluctuations, despite the persistence of a robust fermionic gap across the SIT. Disorder leads to enhanced absorption in σ(ω) at low frequencies compared to the SIT in a clean system. Disorder also expands the quantum critical region, due to a change in the universality class, with an underlying T=0 critical point with a finite low-frequency conductivity.

cond-mat.supr-con↗

arXiv2014

Glassy Chimeras could be blind to quantum speedup: Designing better benchmarks for quantum annealing machines

Recently, a programmable quantum annealing machine has been built that minimizes the cost function of hard optimization problems by adiabatically quenching quantum fluctuations. Tests performed by different research teams have shown that, indeed, the machine seems to exploit quantum effects. However experiments on a class of random-bond instances have not yet demonstrated an advantage over classical optimization algorithms on traditional computer hardware. Here we present evidence as to why this might be the case. These engineered quantum annealing machines effectively operate coupled to a decohering thermal bath. Therefore, we study the finite-temperature critical behavior of the standard benchmark problem used to assess the computational capabilities of these complex machines. We simulate both random-bond Ising models and spin glasses with bimodal and Gaussian disorder on the D-Wave Chimera topology. Our results show that while the worst-case complexity of finding a ground state of an Ising spin glass on the Chimera graph is not polynomial, the finite-temperature phase space is likely rather simple: Spin glasses on Chimera have only a zero-temperature transition. This means that benchmarking optimization methods using spin glasses on the Chimera graph might not be the best benchmark problems to test quantum speedup. We propose alternative benchmarks by embedding potentially harder problems on the Chimera topology. Finally, we also study the (reentrant) disorder-temperature phase diagram of the random-bond Ising model on the Chimera graph and show that a finite-temperature ferromagnetic phase is stable up to 19.85(15)% antiferromagnetic bonds. Beyond this threshold the system only displays a zero-temperature spin-glass phase. Our results therefore show that a careful design of the hardware architecture and benchmark problems is key when building quantum annealing machines.

cond-mat.dis-nn↗

arXiv2018

Classification and Geometry of General Perceptual Manifolds

Perceptual manifolds arise when a neural population responds to an ensemble of sensory signals associated with different physical features (e.g., orientation, pose, scale, location, and intensity) of the same perceptual object. Object recognition and discrimination requires classifying the manifolds in a manner that is insensitive to variability within a manifold. How neuronal systems give rise to invariant object classification and recognition is a fundamental problem in brain theory as well as in machine learning. Here we study the ability of a readout network to classify objects from their perceptual manifold representations. We develop a statistical mechanical theory for the linear classification of manifolds with arbitrary geometry revealing a remarkable relation to the mathematics of conic decomposition. Novel geometrical measures of manifold radius and manifold dimension are introduced which can explain the classification capacity for manifolds of various geometries. The general theory is demonstrated on a number of representative manifolds, including L2 ellipsoids prototypical of strictly convex manifolds, L1 balls representing polytopes consisting of finite sample points, and orientation manifolds which arise from neurons tuned to respond to a continuous angle variable, such as object orientation. The effects of label sparsity on the classification capacity of manifolds are elucidated, revealing a scaling relation between label sparsity and manifold radius. Theoretical predictions are corroborated by numerical simulations using recently developed algorithms to compute maximum margin solutions for manifold dichotomies. Our theory and its extensions provide a powerful and rich framework for applying statistical mechanics of linear classification to data arising from neuronal responses to object stimuli, as well as to artificial deep networks trained for object recognition tasks.

cond-mat.dis-nn↗

arXiv2013

Correcting beliefs in the mean-field and Bethe approximations using linear response

Approximating marginals of a graphical model is one of the fundamental problems in the theory of networks. In a recent paper a method was shown to construct a variational free energy such that the linear response estimates, and maximum entropy estimates (for beliefs) are in agreement, with implications for direct and inverse Ising problems[1]. In this paper we demonstrate an extension of that method, incorporating new information from the response matrix, and we recover the adaptive-TAP equations as the first order approximation[2]. The method is flexible with respect to applications of the cluster variational method, special cases of this method include Naive Mean Field (NMF) and Bethe. We demonstrate that the new framework improves estimation of marginals by orders of magnitude over standard implementations in the weak coupling limit. Beyond the weakly coupled regime we show there is an improvement in a model where the NMF and Bethe approximations are known to be poor for reasons of frustration and short loops.

cond-mat.dis-nn↗

arXiv2013

Boolean and Non-Boolean Computation With Spin Devices

Recently several device and circuit design techniques have been explored for applying nano-magnets and spin torque devices like spin valves and domain wall magnets in computational hardware. However, most of them have been focused on digital logic, and, their benefits over robust and high performance CMOS remains debatable. Ultra-low voltage, current-switching operation of magneto-metallic spin torque devices can potentially be more suitable for non-Boolean computation schemes that can exploit current-mode analog processing. Device circuit co-design for different classes of non-Boolean-architectures using spin-torque based neuron models in spin-CMOS hybrid circuits show that the spin-based non-Boolean designs can achieve 15X-100X lower computation energy for applications like, image-processing, data-conversion, cognitive-computing, pattern matching and programmable-logic, as compared to state of art CMOS designs.

cond-mat.dis-nn↗

arXiv2011

Zero forcing, linear and quantum controllability for systems evolving on networks

We study the dynamics of systems on networks from a linear algebraic perspective. The control theoretic concept of controllability describes the set of states that can be reached for these systems. Under appropriate conditions, there is a connection between the quantum (Lie theoretic) property of controllability and the linear systems (Kalman) controllability condition. We investigate how the graph theoretic concept of a zero forcing set impacts the controllability property. In particular, we prove that if a set of vertices is a zero forcing set, the associated dynamical system is controllable. The results open up the possibility of further exploiting the analogy between networks, linear control systems theory, and quantum systems Lie algebraic theory. This study is motivated by several quantum systems currently under study, including continuous quantum walks modeling transport phenomena. Additionally, it proposes zero forcing as a new notion in the analysis of complex networks.

quant-ph↗

arXiv2013

Subgap states in disordered superconductors

We revise the problem of the density of states in disordered superconductors. Randomness of local sample characteristics translates to the quenched spatial inhomogeneity of the spectral gap, smearing the BCS coherence peak. We show that various microscopic models of potential and magnetic disorder can be reduced to a universal phenomenological random order parameter model, whereas the details of the microscopic description are encoded in the correlation function of the order parameter fluctuations. The resulting form of the density of states is generally described by two parameters: the width Gamma measuring the broadening of the BCS peak, and the energy scale Gamma_{tail} which controls the exponential decay of the density of the subgap states. We refine the existing instanton approaches for determination of Gamma_{tail} and show that they appear as limiting cases of a unified theory of optimal fluctuations in a nonlinear system. Application to various types of disorder is discussed.

cond-mat.supr-con↗

arXiv2013

Effect of bulk charged impurities on the bulk and surface transport in three-dimensional topological insulators

In the three-dimensional topological insulator (TI), the physics of doped semiconductors exists literally side-by-side with the physics of ultra-relativistic Dirac fermions. This unusual pairing creates a novel playground for studying the interplay between disorder and electronic transport. In this mini-review we focus on the disorder caused by the three-dimensionally distributed charged impurities that are ubiquitous in TIs, and we outline the effects it has on both the bulk and surface transport in TIs. We present self-consistent theories for Coulomb screening both in the bulk and at the surface, discuss the magnitude of the disorder potential in each case, and present results for the conductivity. In the bulk, where the band gap leads to thermally activated transport, we show how disorder leads to a smaller-than-expected activation energy that gives way to VRH at low temperatures. We confirm this enhanced conductivity with numerical simulations that also allow us to explore different degrees of impurity compensation. For the surface, where the TI has gapless Dirac modes, we present a theory of disorder and screening of deep impurities, and we calculate the corresponding zero-temperature conductivity. We also comment on the growth of the disorder potential as one moves from the surface of the TI into the bulk. Finally, we discuss how the presence of a gap at the Dirac point, introduced by some source of time-reversal symmetry breaking, affects the disorder potential at the surface and the mid-gap density of states.

cond-mat.mes-hall↗

arXiv2013

Metastability in a stochastic neural network modeled as a velocity jump Markov process

One of the major challenges in neuroscience is to determine how noise that is present at the molecular and cellular levels affects dynamics and information processing at the macroscopic level of synaptically coupled neuronal populations. Often noise is incorprated into deterministic network models using extrinsic noise sources. An alternative approach is to assume that noise arises intrinsically as a collective population effect, which has led to a master equation formulation of stochastic neural networks. In this paper we extend the master equation formulation by introducing a stochastic model of neural population dynamics in the form of a velocity jump Markov process. The latter has the advantage of keeping track of synaptic processing as well as spiking activity, and reduces to the neural master equation in a particular limit. The population synaptic variables evolve according to piecewise deterministic dynamics, which depends on population spiking activity. The latter is characterised by a set of discrete stochastic variables evolving according to a jump Markov process, with transition rates that depend on the synaptic variables. We consider the particular problem of rare transitions between metastable states of a network operating in a bistable regime in the deterministic limit. Assuming that the synaptic dynamics is much slower than the transitions between discrete spiking states, we use a WKB approximation and singular perturbation theory to determine the mean first passage time to cross the separatrix between the two metastable states. Such an analysis can also be applied to other velocity jump Markov processes, including stochastic voltage-gated ion channels and stochastic gene networks.

cond-mat.dis-nn↗

arXiv2013

Dynamical symmetry breaking in a 2D electron gas with a spectral node

We study a disordered 2D electron gas with a spectral node in a vicinity of the node. After identifying the fundamental dynamical symmetries of this system, the spontaneous breaking of the latter by a Grassmann field is studied within a nonlinear sigma model approach. This allows us to reduce the average two-particle Green's function to a diffusion propagator with a random diffusion coefficient. The latter has non-degenerate saddle points and is treated by the conventional self-consistent Born approximation. This leads to a renormalized chemical potential and a renormalized diffusion coefficient, where the DC conductivity increases linearly with the density of quasiparticles. Applied to the special case of Dirac fermions, our approach provides a comprehensive description of the minimal conductivity at the Dirac node as well as for the V-shape conductivity inside the bands.

cond-mat.dis-nn↗

arXiv2013

Detecting hierarchical and overlapping network communities using locally optimal modularity changes

Agglomerative clustering is a well established strategy for identifying communities in networks. Communities are successively merged into larger communities, coarsening a network of actors into a more manageable network of communities. The order in which merges should occur is not in general clear, necessitating heuristics for selecting pairs of communities to merge. We describe a hierarchical clustering algorithm based on a local optimality property. For each edge in the network, we associate the modularity change for merging the communities it links. For each community vertex, we call the preferred edge that edge for which the modularity change is maximal. When an edge is preferred by both vertices that it links, it appears to be the optimal choice from the local viewpoint. We use the locally optimal edges to define the algorithm: simultaneously merge all pairs of communities that are connected by locally optimal edges that would increase the modularity, redetermining the locally optimal edges after each step and continuing so long as the modularity can be further increased. We apply the algorithm to model and empirical networks, demonstrating that it can efficiently produce high-quality community solutions. We relate the performance and implementation details to the structure of the resulting community hierarchies. We additionally consider a complementary local clustering algorithm, describing how to identify overlapping communities based on the local optimality condition.

physics.soc-ph↗

arXiv2012

Pinning theory of domain walls in helical magnets

The theory of elasticity and pinning of domain walls in helical magnets is presented. Domain walls perpendicular to the helical axis show non-local elasticity and are marginally pinned by local disorder. Weak anisotropy combined with magnetic dilution leads however to a non-local bulk pinning effect. Domain walls with other orientations include generically vortex arrays, similar to type-II superconductors. Their pinning force is calculated as a function of wall orientation, pitch angle and impurity concentration. It is shown that metastable domains can vary between needle and pancake like shape.

cond-mat.mes-hall↗

arXiv2013

Anderson Localization for a Multi-Particle Quantum Graph

We study a multi-particle quantum graph with random potential. Taking the approach of multiscale analysis we prove exponential and strong dynamical localization of any order in the Hilbert-Schmidt norm near the spectral edge. Apart from the results on multi-particle systems, we also prove Lifshitz-type asymptotics for single-particle systems. This shows in particular that localization for single-particle quantum graphs holds under a weaker assumption on the random potential than previously known.

math-ph↗

arXiv2012

A remark on the trace-map for the Silver mean sequence

In this work we study the Silver mean sequence based on substitution rules by means of a transfer-matrix approach. Using transfer-matrix method we find a recurrence relation for the traces of general transfer-matrices which characterizes electronic properties of the quasicrystal in question. We also find an invariant of the trace-map.

cond-mat.dis-nn↗

arXiv2014

Benchtop Fabrication of Memristive Atomic Switch Networks

Recent advances in nanoscale science and technology provide possibilities to directly self-assemble and integrate functional circuit elements within the wiring scheme of devices with potentially unique architectures. Electroionic resistive switching circuits comprising highly interconnected fractal electrodes and metal-insulator-metal interfaces, known as atomic switch networks, have been fabricated using simple benchtop techniques including solution-phase electroless deposition. These devices are shown to activate through a bias-induced forming step that produces the frequency dependent, nonlinear hysteretic switching expected for gapless-type atomic switches and memristors. By eliminating the need for complex lithographic methods, such an approach toward device fabrication provides a more accessible platform for the study of ionic resistive switches and memristive systems.

cond-mat.mtrl-sci↗