Showing papers in "Journal of Statistical Mechanics: Theory and Experiment in 2012"

ABJM theory as a Fermi gas

Abstract: The partition function on the three-sphere of many supersymmetric Chern-Simons-matter theories reduces, by localization, to a matrix model. We develop a new method to study these models in the M-theory limit, but at all orders in the 1/N expansion. The method is based on reformulating the matrix model as the partition function of an ideal Fermi gas with a non-trivial, one-particle quantum Hamiltonian. This new approach leads to a completely elementary derivation of the N^{3/2} behavior for ABJM theory and N=3 quiver Chern-Simons-matter theories. In addition, the full series of 1/N corrections to the original matrix integral can be simply determined by a next-to-leading calculation in the WKB or semiclassical expansion of the quantum gas, and we show that, for several quiver Chern-Simons-matter theories, it is given by an Airy function. This generalizes a recent result of Fuji, Hirano and Moriyama for ABJM theory. It turns out that the semiclassical expansion of the Fermi gas corresponds to a strong coupling expansion in type IIA theory, and it is dual to the genus expansion. This allows us to calculate explicitly non-perturbative effects due to D2-brane instantons in the AdS background.

Ordering in bidirectional pedestrian flows and its influence on the fundamental diagram

Abstract: Experiments under laboratory conditions were carried out to study the ordering in bidirectional pedestrian streams and its influence on the fundamental diagram (density–speed–flow relation). The Voronoi method is used to resolve the fine structure of the resulting velocity–density relations and spatial dependence of the measurements. The data show that the specific flow concept is applicable also for bidirectional streams. For various forms of ordering in bidirectional streams, no large differences among density–flow relationships are found in the observed density range. The fundamental diagrams of bidirectional streams with different forms of ordering are compared with those of unidirectional streams. The result shows differences in the shape of the relation for ρ > 1.0 m − 2. The maximum of the specific flow in unidirectional streams is significantly larger than that in all bidirectional streams examined.

Probabilistic reconstruction in compressed sensing: algorithms, phase diagrams, and threshold achieving matrices

Abstract: Compressed sensing is a signal processing method that acquires data directly in a compressed form. This allows one to make fewer measurements than were considered necessary to record a signal, enabling faster or more precise measurement protocols in a wide range of applications. Using an interdisciplinary approach, we have recently proposed in Krzakala et?al (2012 Phys. Rev. X 2 021005) a strategy that allows compressed sensing to be performed at acquisition rates approaching the theoretical optimal limits. In this paper, we give a more thorough presentation of our approach, and introduce many new results. We present the probabilistic approach to reconstruction and discuss its optimality and robustness. We detail the derivation of the message passing algorithm for reconstruction and expectation maximization learning of signal-model parameters. We further develop the asymptotic analysis of the corresponding phase diagrams with and without measurement noise, for different distributions of signals, and discuss the best possible reconstruction performances regardless of the algorithm. We also present new efficient seeding matrices, test them on synthetic data and analyze their performance asymptotically.

Parafermionic edge zero modes inZn-invariant spin chains

Abstract: A sign of topological order in a gapped one-dimensional quantum chain is the existence of edge zero modes. These occur in the Z2-invariant Ising/Majorana chain, where they can be understood using free-fermion techniques. Here I discuss their presence in spin chains with Zn symmetry, and prove that for appropriate couplings they are exact, even in this strongly interacting system. These modes are naturally expressed in terms of parafermions, generalizations of fermions to the Zn case. I show that parafermionic edge zero modes do not occur in the usual ferromagnetic and antiferromagnetic cases, but rather only when the interactions are chiral, so that spatial-parity and time-reversal symmetries are broken.

Quantum quench in the transverse field Ising chain: I. Time evolution of order parameter correlators

Abstract: We consider the time evolution of order parameter correlation functions after a sudden quantum quench of the magnetic field in the transverse field Ising chain. Using two novel methods based on determinants and form factor sums respectively, we derive analytic expressions for the asymptotic behaviour of one-?and two-point correlators. We discuss quenches within the ordered and disordered phases as well as quenches between the phases and to the quantum critical point. We give detailed accounts of both methods.

Quantum quenches in the transverse field Ising chain: II. Stationary state properties

Abstract: We consider the stationary state properties of the reduced density matrix as well as spin–spin correlation functions after a sudden quantum quench of the magnetic field in the transverse field Ising chain. We demonstrate that stationary state properties are described by a generalized Gibbs ensemble. We discuss the approach to the stationary state at late times.

Comparative evaluation of community detection algorithms: a topological approach

Abstract: Community detection is one of the most active fields in complex network analysis, due to its potential value in practical applications. Many works inspired by different paradigms are devoted to the development of algorithmic solutions allowing the network structure in such cohesive subgroups to be revealed. Comparative studies reported in the literature usually rely on a performance measure considering the community structure as a partition (Rand index, normalized mutual information, etc). However, this type of comparison neglects the topological properties of the communities. In this paper, we present a comprehensive comparative study of a representative set of community detection methods, in which we adopt both types of evaluation. Community-oriented topological measures are used to qualify the communities and evaluate their deviation from the reference structure. In order to mimic real-world systems, we use artificially generated realistic networks. It turns out there is no equivalence between the two approaches: a high performance does not necessarily correspond to correct topological properties, and vice versa. They can therefore be considered as complementary, and we recommend applying both of them in order to perform a complete and accurate assessment.

Form factor approach to dynamical correlation functions in critical models

Abstract: We develop a form factor approach to the study of dynamical correlation functions of quantum integrable models in the critical regime. As an example, we consider the quantum non-linear Schr?dinger model. We derive the long-distance/long-time asymptotic behavior of various two-point functions of this model. We also compute edge exponents and amplitudes characterizing the power-law behavior of dynamical response functions on the particle?hole excitation thresholds. These last results confirm predictions based on the non-linear Luttinger liquid method. Our results rely on a first principles derivation, based on a microscopic analysis of the model, without invoking, at any stage, any correspondence with a continuous field theory. Furthermore, our approach only makes use of certain general properties of the model, so that it should be applicable, possibly with minor modifications, to a wide class of (not necessarily integrable) gapless one-dimensional Hamiltonians.

The KPZ equation with flat initial condition and the directed polymer with one free end

Abstract: We study the directed polymer (DP) of length t in a random potential in dimension 1 + 1 in the continuum limit, with one end fixed and one end free. This maps onto the Kardar?Parisi?Zhang growth equation in time t, with flat initial conditions. We use the Bethe ansatz solution for the replicated problem, which is an attractive bosonic model. The problem is more difficult than the previous solution of the fixed endpoint problem as it requires regularization of the spatial integrals over the Bethe eigenfunctions. We use either a large fixed system length or small finite slope KPZ initial conditions (wedge). The latter allows one to take properly into account non-trivial contributions, which appear as deformed strings in the former. By considering a half-space model in a proper limit we obtain an expression for the generating function of all positive integer moments of the directed polymer partition function. We obtain the generating function of the moments of the DP partition sum as a Fredholm Pfaffian. At large time, this Fredholm Pfaffian, valid for all time t, exhibits convergence of the free energy (i.e.?KPZ height) distribution to the GOE Tracy?Widom distribution.

Dynamics in the Ising field theory after a quantum quench

Abstract: We study the real-time dynamics of the order parameter σ(t) in the Ising field theory after a quench in the fermion mass, which corresponds to a quench in the transverse field of the corresponding transverse field Ising chain. We focus on quenches within the ordered phase. The long-time behaviour is obtained analytically by a resummation of the leading divergent terms in a form-factor expansion for σ(t). Our main result is the development of a method for treating divergences associated with working directly in the field theory limit. We recover the scaling limit of the corresponding result by Calabrese et al (2011 Phys. Rev. Lett. 106 227203), which was obtained for the lattice model. Our formalism generalizes to integrable quantum quenches in other integrable models.

Entanglement of excited states in critical spin chains

Abstract: R?nyi and von Neumann entropies quantifying the amount of entanglement in ground states of critical spin chains are known to satisfy a universal law which is given by the conformal field theory (CFT) describing their scaling regime. This law can be generalized to excitations described by primary fields in CFT, as was done by?Alcaraz et al in 2011 (see reference?[1], of which this work is a completion). An alternative derivation is presented, together with numerical verifications of our results in different models belonging to the c = 1, 1/2 universality classes. Oscillations of the R?nyi entropy in excited states are also discussed.

Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: complete spectrum and matrix elements of some quasi-local operators

Abstract: The integrable quantum models, associated with the transfer matrices of the 6-vertex reflection algebra for spin-1/2 representations, are studied in this paper. In the framework of Sklyanin’s quantum separation of variables (SOV), we provide the complete characterization of the eigenvalues and eigenstates of the transfer matrix and the proof of the simplicity of the transfer matrix spectrum. Moreover, we use these integrable quantum models as further key examples for which to develop a method in the SOV framework to compute matrix elements of local operators. This method is based on the resolution of the quantum inverse problem (i.e. the reconstruction of local operators in terms of the quantum separate variables) plus the computation of the action of separate covectors on separate vectors. In particular, for these integrable quantum models, which in the homogeneous limit reproduce the open spin-1/2 XXZ quantum chains with non-diagonal boundary conditions, we have obtained the SOV-reconstruction for a class of quasi-local operators and determinant formulae for the covector–vector actions. As a consequence of these findings we provide one determinant formula for the matrix elements of this class of reconstructed quasi-local operator on transfer matrix eigenstates.

Zamolodchikov–Faddeev algebra and quantum quenches in integrable field theories

Abstract: We analyze quantum quenches in integrable models and in particular we determine the initial state in the basis of eigenstates of the post-quench Hamiltonian. This leads us to consider the set of transformations of creation and annihilation operators that respect the Zamolodchikov–Faddeev algebra satisfied by integrable models. We establish that the Bogoliubov transformations hold only in the case of quantum quenches in free theories. For the most general case of interacting theories, we identify two classes of transformations. The first class induces a change in the S-matrix of the theory but not in its ground state, whereas the second class results in a 'dressing' of the operators. We consider as examples of our approach the transformations associated with a change of the interaction in the sinh–Gordon model and the Lieb–Liniger model.

The Bethe approximation for solving the inverse Ising problem: a comparison with other inference methods

Abstract: The inverse Ising problem consists in inferring the coupling constants of an Ising model given the correlation matrix. The fastest methods for solving this problem are based on mean-field approximations, but which one performs better in the general case is still not completely clear. In the first part of this work, I summarize the formulas for several mean-field approximations and I derive new analytical expressions for the Bethe approximation, which allow one to solve the inverse Ising problem without running the susceptibility propagation algorithm (thus avoiding the lack of convergence). In the second part, I compare the accuracy of different mean-field approximations on several models (diluted ferromagnets and spin glasses) defined on random graphs and regular lattices, showing which one is in general more effective. A simple improvement over these approximations is proposed. Also a fundamental limitation is found in using methods based on TAP and Bethe approximations in the presence of an external field.

Entanglement Entropy of Two Disjoint Intervals from Fusion Algebra of Twist Fields

Abstract: We study the entanglement and Renyi entropies of two disjoint intervals in minimal models of conformal field theory. We use the conformal block expansion and fusion rules of twist fields to define a systematic expansion in the elliptic parameter of the trace of the nth power of the reduced density matrix. Keeping only the first few terms we obtain an approximate expression that is easily analytically continued to , leading to an approximate formula for the entanglement entropy. These predictions are checked against some known exact results as well as against existing numerical data.

Multiscale analysis of spreading in a large communication network

Abstract: In temporal networks, both the topology of the underlying network and the timings of interaction events can be crucial in determining how a dynamic process mediated by the network unfolds. We have explored the limiting case of the speed of spreading in the SI model, set up such that an event between an infectious and a susceptible individual always transmits the infection. The speed of this process sets an upper bound for the speed of any dynamic process that is mediated through the interaction events of the network. With the help of temporal networks derived from large-scale time-stamped data on mobile phone calls, we extend earlier results that indicate the slowing-down effects of burstiness and temporal inhomogeneities. In such networks, links are not permanently active, but dynamic processes are mediated by recurrent events taking place on the links at specific points in time. We perform a multiscale analysis and pinpoint the importance of the timings of event sequences on individual links, their correlations with neighboring sequences, and the temporal pathways taken by the network-scale spreading process. This is achieved by studying empirically and analytically different characteristic relay times of links, relevant to the respective scales, and a set of temporal reference models that allow for removing selected time-domain correlations one by one. Our analysis shows that for the spreading velocity, time-domain inhomogeneities are as important as the network topology, which indicates the need to take time-domain information into account when studying spreading dynamics. In particular, results for the different characteristic relay times underline the importance of the burstiness of individual links.

Exact theory of dense amorphous hard spheres in high dimension I. The free energy

Abstract: We consider the theory of the glass transition and jamming of hard spheres in the large space dimension limit. Previous investigations were based on the assumption that the probability distribution within a 'cage' is Gaussian, which is not fully consistent with numerical results. Here we perform a replica calculation without making any assumption on the cage shape. We show that thermodynamic functions turn out to be exact within the Gaussian ansatz—provided one allows for arbitrary replica symmetry breaking—and indeed agree well with numerical results. The actual structure function (the so-called non-ergodic parameter) is not Gaussian, an apparent paradox which we discuss. In this paper we focus on the free energy, future papers will present the results for the structure functions and a detailed comparison with numerical results.

The algebraic Bethe ansatz for scalar products in SU(3)-invariant integrable models

Abstract: We study SU(3)-invariant integrable models solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for particular case of scalar products of Bethe vectors. This representation can be used for the calculation of form factors and correlation functions of XXX SU(3)-invariant Heisenberg chain.

Probabilistic interconnection between interdependent networks promotes cooperation in the public goods game

Abstract: Most previous works study the evolution of cooperation in a structured population by commonly employing an isolated single network. However, realistic systems are composed of many interdependent networks coupled with each other, rather than an isolated single one. In this paper, we consider a system including two interacting networks with the same size, entangled with each other by the introduction of probabilistic interconnections. We introduce the public goods game into such a system, and study how the probabilistic interconnection influences the evolution of cooperation of the whole system and the coupling effect between two layers of interdependent networks. Simulation results show that there exists an intermediate region of interconnection probability leading to the maximum cooperation level in the whole system. Interestingly, we find that at the optimal interconnection probability the fraction of internal links between cooperators in two layers is maximal. Also, even if initially there are no cooperators in one layer of interdependent networks, cooperation can still be promoted by probabilistic interconnection, and the cooperation levels in both layers can more easily reach an agreement at the intermediate interconnection probability. Our results may be helpful in understanding cooperative behavior in some realistic interdependent networks and thus highlight the importance of probabilistic interconnection on the evolution of cooperation.

Finite-temperature mutual information in a simple phase transition

Abstract: We study the finite-temperature behavior of the Lipkin–Meshkov–Glick model with a focus on correlation properties as measured by the mutual information. The latter, which quantifies the amount of both classical and quantum correlations, is computed exactly in the two limiting cases of vanishing magnetic field and vanishing temperature. For all other situations, numerical results provide evidence of a finite mutual information at all temperatures except at criticality. There, it diverges as the logarithm of the system size, with a prefactor that can take only two values, depending on whether the critical temperature vanishes or not. Our work provides a simple example in which the mutual information appears as a powerful tool to detect finite-temperature phase transitions, contrary to entanglement measures such as the concurrence.

Disentangling multi-level systems: averaging, correlations and memory

Abstract: We consider two weakly coupled systems and adopt a perturbative approach based on the Ruelle response theory to study their interaction. We propose a systematic way of parameterizing the effect of the coupling as a function of only the variables of a system of interest. Our focus is on describing the impacts of the coupling on the long term statistics rather than on the finite-time behavior. By direct calculation, we find that, at first order, the coupling can be surrogated by adding a deterministic perturbation to the autonomous dynamics of the system of interest. At second order, there are additionally two separate and very different contributions. One is a term taking into account the second-order contributions of the fluctuations in the coupling, which can be parameterized as a stochastic forcing with given spectral properties. The other one is a memory term, coupling the system of interest to its previous history, through the correlations of the second system. If these correlations are known, this effect can be implemented as a perturbation with memory on the single system. In order to treat this case, we present an extension to Ruelle's response theory able to deal with integral operators. We discuss our results in the context of other methods previously proposed for disentangling the dynamics of two coupled systems. We emphasize that our results do not rely on assuming a time scale separation, and, if such a separation exists, can be used equally well to study the statistics of the slow variables and that of the fast variables. By recursively applying the technique proposed here, we can treat the general case of multi-level systems.

Efficient rare event simulation by optimal nonequilibrium forcing

Abstract: Rare event simulation and estimation for systems in equilibrium are among the most challenging topics in molecular dynamics. As was shown by Jarzynski and others, nonequilibrium forcing can theoretically be used to obtain equilibrium rare event statistics. The advantage seems to be that the external force can speed up the sampling of the rare events by biasing the equilibrium distribution towards a distribution under which the rare events is no longer rare. Yet algorithmic methods based on Jarzynski's and related results often fail to be efficient because they are based on sampling in path space. We present a new method that replaces the path sampling problem by minimization of a cross-entropy-like functional which boils down to finding the optimal nonequilibrium forcing. We show how to solve the related optimization problem in an efficient way by using an iterative strategy based on milestoning.

Beyond Itô versus Stratonovich

Abstract: Recently, a novel framework to handle stochastic processes has emerged from a series of studies in biology, showing situations beyond ?It? versus Stratonovich?. Its internal consistency can be demonstrated via the zero mass limit of a generalized Klein?Kramers equation. Moreover, the connection to other integrations becomes evident: the obtained Fokker?Planck equation defines a new type of stochastic calculus that in general differs from the ?-type interpretation. A unique advantage of this new approach is a natural correspondence between stochastic and deterministic dynamics, which is useful or may even be essential in practice. The core of the framework is a transformation from the usual Langevin equation to a form that contains a potential function with two additional dynamical matrices, which reveals an underlying symplectic structure. The framework has a direct physical meaning and a straightforward experimental realization. A recent experiment has offered a first empirical validation of this new stochastic integration.

How glassy are neural networks

Abstract: In this paper we continue our investigation on the high storage regime of a neural network with Gaussian patterns. Through an exact mapping between its partition function and one of a bipartite spin glass (whose parties consist of Ising and Gaussian spins respectively), we give a complete control of the whole annealed region. The strategy explored is based on an interpolation between the bipartite system and two independent spin glasses built respectively by dichotomic and Gaussian spins: critical line, behavior of the principal thermodynamic observables and their fluctuations as well as overlap fluctuations are obtained and discussed. Then, we move further, extending such an equivalence beyond the critical line, to explore the broken ergodicity phase under the assumption of replica symmetry and show that the quenched free energy of this (analogical) Hopfield model can be described as a linear combination of the two quenched spin glass free energies even in the replica symmetric framework.

On quantum mean-field models and their quantum annealing

Abstract: This paper deals with fully connected mean-field models of quantum spins with p-body ferromagnetic interactions and a transverse field. For p = 2 this corresponds to the quantum Curie–Weiss model (a special case of the Lipkin–Meshkov–Glick model) which exhibits a second-order phase transition, while for p > 2 the transition is first order. We provide a refined analytical description both of the static and of the dynamic properties of these models. In particular we obtain analytically the exponential rate of decay of the gap at the first-order transition. We also study the slow annealing from the pure transverse field to the pure ferromagnet (and vice versa) and discuss the effect of the first-order transition and of the spinodal limit of metastability on the residual excitation energy, both for finite and exponentially divergent annealing times. In the quantum computation perspective this quantity would assess the efficiency of the quantum adiabatic procedure as an approximation algorithm.

Reconstruction of financial networks for robust estimation of systemic risk

Abstract: In this paper we estimate the propagation of liquidity shocks through interbank markets when the information about the underlying credit network is incomplete. We show that techniques such as maximum entropy currently used to reconstruct credit networks severely underestimate the risk of contagion by assuming a trivial (fully connected) topology, a type of network structure which can be very different from the one empirically observed. We propose an efficient message-passing algorithm to explore the space of possible network structures and show that a correct estimation of the network degree of connectedness leads to more reliable estimations for systemic risk. Such an algorithm is also able to produce maximally fragile structures, providing a practical upper bound for the risk of contagion when the actual network structure is unknown. We test our algorithm on ensembles of synthetic data encoding some features of real financial networks (sparsity and heterogeneity), finding that more accurate estimations of risk can be achieved. Finally we find that this algorithm can be used to control the amount of information that regulators need to require from banks in order to sufficiently constrain the reconstruction of financial networks.

An empirical study on human mobility and its agent-based modeling

Abstract: This paper aims to analyze the GPS traces of 258 volunteers in order to obtain a better understanding of both the human mobility patterns and the mechanism. We report the regular and scaling properties of human mobility for several aspects, and importantly we identify its Levy flight characteristic, which is consistent with those from previous studies. We further assume two factors that may govern the Levy flight property: (1) the scaling and hierarchical properties of the purpose clusters which serve as the underlying spatial structure, and (2) the individual preferential behaviors. To verify the assumptions, we implement an agent-based model with the two factors, and the simulated results do indeed capture the same Levy flight pattern as is observed. In order to enable the model to reproduce more mobility patterns, we add to the model a third factor: the jumping factor, which is the probability that one person may cancel their regular mobility schedule and explore a random place. With this factor, our model can cover a relatively wide range of human mobility patterns with scaling exponent values from 1.55 to 2.05.

Logarithmic observables in critical percolation

Abstract: Although it has long been known that the proper quantum field theory description of critical percolation involves a logarithmic conformal field theory (LCFT), no direct consequence of this has been observed so far. Representing critical bond percolation as the Q → 1 limit of the Q-state Potts model, and analyzing the underlying SQ symmetry of the Potts spins, we identify a class of simple observables whose two-point functions scale logarithmically for Q → 1. The logarithm originates from the mixing of the energy operator with a logarithmic partner that we identify as the field that creates two propagating clusters. In d = 2 dimensions this agrees with general LCFT results, and in particular the universal prefactor of the logarithm can be computed exactly. We confirm its numerical value by carrying out extensive Monte Carlo simulations.

Surface free energy of the open XXZ spin-1/2 chain

Abstract: We study the boundary free energy of the XXZ spin-1/2 chain subject to diagonal boundary fields. We first show that the representation for its finite Trotter number approximant obtained by Gohmann, Bortz and Frahm is related to the partition function of the six-vertex model with reflecting ends. Building on the Tsuchiya determinant representation for the latter quantity we are able to take the infinite Trotter number limit. This yields a representation for the surface free energy which involves the solution of the non-linear integral equation that governs the thermodynamics of the XXZ spin-1/2 chain subject to periodic boundary conditions. We show that this integral representation allows one to extract the low-T asymptotic behavior of the boundary magnetization at finite external magnetic field on the one hand and numerically plot this function on the other hand.

Thermostatistics with minimal length uncertainty relation

Abstract: The existence of minimal length is suggested in any quantum theory of gravity such as string theory, double special relativity and black hole physics. One way to impose minimal length is by deforming Heisenberg algebra in a phase space which is called the generalized uncertainty principle (GUP). In this paper, we develop statistical mechanics in the GUP framework. Our method is quite general and does not need to fix the generalized coordinates and momenta. We define a general transformation in phase space which transforms the usual Heisenberg algebra to a deformed one. In this method, quantum gravity effects only act on the structure of phase space and we relate these effects to the density of states. We find an interesting phenomenon in Maxwell–Boltzmann statistics which has no classical analogy. We show that there is an upper bound for the number of excited particles in the limit of high temperature which implies condensation. Also we study modification of Bose–Einstein condensation and the completely degenerate gas.