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arXiv2013

Interacting Particle Systems in Time-Dependent Geometries

Many complex structures and stochastic patterns emerge from simple kinetic rules and local interactions, and are governed by scale invariance properties in combination with effects of the global geometry. We consider systems that can be described effectively by space-time trajectories of interacting particles, such as domain boundaries in two-dimensional growth or river networks. We study trajectories embedded in time-dependent geometries, and the main focus is on uniformly expanding or decreasing domains for which we obtain an exact mapping to simple fixed domain systems while preserving the local scale invariance properties. This approach was recently introduced in [A. Ali et al., Phys. Rev. E 87, 020102(R) (2013)] and here we provide a detailed discussion on its applicability for self-affince Markovian models, and how it can be adapted to self-affine models with memory or explicit time dependence. The mapping corresponds to a non-linear time transformation which convergences to a finite value for a large class of trajectories, enabling an exact analysis of asymptotic properties in expanding domains. We further provide a detailed discussion of different particle interactions and generalized geometries. All our findings are based on exact computations and are illustrated numerically for various examples, including Lévy processes and fractional Brownian motion.

cond-mat.stat-mech↗

arXiv2013

Nonlocal Asymmetric exclusion process on a ring and conformal invariance

We present a one-dimensional nonlocal hopping model with exclusion on a ring. The model is related to the Raise and Peel growth model. A nonnegative parameter $u$ controls the ratio of the local backwards and nonlocal forwards hopping rates. The phase diagram and consequently the values of the current, depend on $u$ and the density of particles. In the special case of half-filling and $u = 1$ the system is conformal invariant and an exact value of the current for any size $L$ of the system is conjectured and checked for large lattice sizes in Monte Carlo simulations. For $u > 1$ the current has a non-analytic dependence on the density when the latter approaches the half-filling value.

cond-mat.stat-mech↗

arXiv2013

Binary decision making with very heterogeneous influence

We consider an extension of a binary decision model in which nodes make decisions based on influence-biased averages of their neighbors' states, similar to Ising spin glasses with on-site random fields. In the limit where these influences become very heavy-tailed, the behavior of the model dramatically changes. On complete graphs, or graphs where nodes with large influence have large degree, this model is characterized by a new "phase" with an unpredictable number of macroscopic shocks, with no associated critical phenomena. On random graphs where the degree of the most influential nodes is small compared to population size, a predictable glassy phase without phase transitions emerges. Analytic results about both of these new phases are obtainable in limiting cases. We use numerical simulations to explore the model for more general scenarios. The phases associated with very influential decision makers are easily distinguishable experimentally from a homogeneous influence phase in many circumstances, in the context of our simple model.

physics.soc-ph↗

arXiv2013

Localization and advectional spreading of convective currents under parametric disorder

We address a problem which is mathematically reminiscent of the one of Anderson localization, although it is related to a strongly dissipative dynamics. Specifically, we study thermal convection in a horizontal porous layer heated from below in the presence of a parametric disorder; physical parameters of the layer are time-independent and randomly inhomogeneous in one of the horizontal directions. Under such a frozen parametric disorder, spatially localized flow patterns appear. We focus our study on their localization properties and the effect of an imposed advection along the layer on these properties. Our interpretation of the results of the linear theory is underpinned by numerical simulation for the nonlinear problem. Weak advection is found to lead to an upstream delocalization of localized current patterns. Due to this delocalization, the transition from a set of localized patterns to an almost everywhere intense "global" flow can be observed under conditions where the disorder-free system would be not far below the instability threshold. The results presented are derived for a physical system which is mathematically described by a modified Kuramoto-Sivashinsky equation and therefore they are expected to be relevant for a broad variety of dissipative media where pattern selection occurs.

physics.flu-dyn↗

arXiv2013

Phase transitions in systems with non-additive long-range interactions

We consider spin systems with long-range interactions in nonadditive regime. When the non-additive scaling limit is employed, the energy and the entropy compete and the system exhibits some phase transitions. Such systems do not satisfy the additivity, which results in some unfamiliar properties related to phase transitions. In this paper, the concept of additivity and its consequence are explained and the recent progress on statistical mechanics of long-range interacting systems are reviewed. It is shown that the parameter space is clearly decomposed into the three regions according to the stability of the uniform state predicted by the mean-field theory. Based on this parameter space decomposition, recent results on the exactness of MF theory are explained. When the interaction is non-negative (ferromagnetic), the analysis of the mean-field theory is exact and a typical spin configuration is always uniform in the canonical ensemble. However, in the restricted canonical ensemble, i.e., the canonical ensemble with a restriction of the value of the magnetization, it is shown that the mean-field theory does not necessarily give the exact description of the system and phase transitions between the mean-field uniform states (MF phase) and the inhomogeneous states (non-MF phase) occur. A new finding is that when the interaction potential changes its sign depending on the distance, the non-MF phase appears even in the canonical ensemble.

cond-mat.stat-mech↗

arXiv2013

Bethe ansatz description of edge-localization in the open-boundary XXZ spin chain

At large values of the anisotropy Δ, the open-boundary Heisenberg spin-1/2 chain has eigenstates displaying localization at the edges. We present a Bethe ansatz description of this `edge-locking' phenomenon in the entire Δ>1 region. We focus on the simplest spin sectors, namely the highly polarized sectors with only one or two overturned spins, i.e., one-particle and two-particle sectors. Edge-locking is associated with pure imaginary solutions of the Bethe equations, which are not commonly encountered in periodic chains. In the one-particle case, at large anisotropies there are two eigenstates with imaginary Bethe momenta, related to localization at the two edges. For any finite chain size, one of the two solutions become real as the anisotropy is lowered below a certain value. For two particles, a richer scenario is observed, with eigenstates having the possibility of both particles locked on the same or different edge, one locked and the other free, and both free either as single magnons or as bound composites corresponding to `string' solutions. For finite chains, some of the edge-locked spins get delocalized at certain values of the anisotropy (`exceptional points'), corresponding to imaginary solutions becoming real. We characterize these phenomena thoroughly by providing analytic expansions of the Bethe momenta for large chains, large anisotropy, and near the exceptional points. In the large-chain limit all the exceptional points coalesce at the isotropic point (Δ=1) and edge-locking becomes stable in the whole Δ>1 region.

cond-mat.str-el↗

arXiv2013

Scaling Exponent for Incremental Records

We investigate records in a growing sequence of identical and independently distributed random variables. The record equals the largest value in the sequence, and our focus is on the increment, defined as the difference between two successive records. We investigate sequences in which all increments decrease monotonically, and find that the fraction I_N of sequences that exhibit this property decays algebraically with sequence length N, namely I_N ~ N^{-nu} as N --> infinity. We analyze the case where the random variables are drawn from a uniform distribution with compact support, and obtain the exponent nu = 0.317621... using analytic methods. We also study the record distribution and the increment distribution. Whereas the former is a narrow distribution with an exponential tail, the latter is broad and has a power-law tail characterized by the exponent nu. Empirical analysis of records in the sequence of waiting times between successive earthquakes is consistent with the theoretical results.

cond-mat.stat-mech↗

arXiv2013

A stochastic model of long-range interacting particles

We introduce a model of long-range interacting particles evolving under a stochastic Monte Carlo dynamics, in which possible increase or decrease in the values of the dynamical variables is accepted with preassigned probabilities. For symmetric increments, the system at long times settles to the Gibbs equilibrium state, while for asymmetric updates, the steady state is out of equilibrium. For the associated Fokker-Planck dynamics in the thermodynamic limit, we compute exactly the phase space distribution in the nonequilibrium steady state, and find that it has a nontrivial form that reduces to the familiar Gibbsian measure in the equilibrium limit.

cond-mat.stat-mech↗

arXiv2013

Uniqueness of the thermodynamic limit for driven disordered elastic interfaces

We study the finite size fluctuations at the depinning transition for a one-dimensional elastic interface of size $L$ displacing in a disordered medium of transverse size $M=k L^ζ$ with periodic boundary conditions, where $ζ$ is the depinning roughness exponent and $k$ is a finite aspect ratio parameter. We focus on the crossover from the infinitely narrow ($k\to 0$) to the infinitely wide ($k\to \infty$) medium. We find that at the thermodynamic limit both the value of the critical force and the precise behavior of the velocity-force characteristics are {\it unique} and $k$-independent. We also show that the finite size fluctuations of the critical force (bias and variance) as well as the global width of the interface cross over from a power-law to a logarithm as a function of $k$. Our results are relevant for understanding anisotropic size-effects in force-driven and velocity-driven interfaces.

cond-mat.stat-mech↗

arXiv2013

Experimental approaches to universal out-of-equilibrium scaling laws: turbulent liquid crystal and other developments

This is a brief survey of recent experimental studies on out-of-equilibrium scaling laws, focusing on two prominent situations where non-trivial universality classes have been identified theoretically: absorbing-state phase transitions and growing interfaces. First the article summarizes main results obtained for electrically-driven turbulent liquid crystal, which exhibited the scaling laws for the directed percolation class at the transition between two turbulent regimes, and those for the Kardar-Parisi-Zhang class in the supercritical phase where one turbulent regime invades the other. Other experimental investigations on these universality classes and related situations are then overviewed and discussed. Some remarks on analyses of these scaling laws are also given from the practical viewpoints.

cond-mat.stat-mech↗

arXiv2013

On Rényi entropies of disjoint intervals in conformal field theory

We study the Rényi entropies of N disjoint intervals in the conformal field theories given by the free compactified boson and the Ising model. They are computed as the 2N point function of twist fields, by employing the partition function of the model on a particular class of Riemann surfaces. The results are written in terms of Riemann theta functions. The prediction for the free boson in the decompactification regime is checked against exact results for the harmonic chain. For the Ising model, matrix product states computations agree with the conformal field theory result once the finite size corrections have been taken into account.

hep-th↗

arXiv2014

Stationary entanglement entropies following an interaction quench in 1D Bose gas

We analyze the entanglement properties of the asymptotic steady state after a quench from free to hard-core bosons in one dimension. The Rényi and von Neumann entanglement entropies are found to be extensive, and the latter coincides with the thermodynamic entropy of the Generalized Gibbs Ensemble (GGE). Computing the spectrum of the two-point function, we provide exact analytical results both for the leading extensive parts and the subleading terms for the entropies as well as for the cumulants of the particle number fluctuations. We also compare the extensive part of the entanglement entropy with the thermodynamic ones, showing that the GGE entropy equal the entanglement one and it is the double of the diagonal entropy.

cond-mat.quant-gas↗

arXiv2018

Generalized Riemann Hypothesis and Stochastic Time Series

Using the Dirichlet theorem on the equidistribution of residue classes modulo $q$ and the Lemke Oliver-Soundararajan conjecture on the distribution of pairs of residues on consecutive primes, we show that the domain of convergence of the infinite product of Dirichlet $L$-functions of non-principal characters can be extended from $\Re(s) > 1$ down to $\Re(s) > \half$, without encountering any zeros before reaching this critical line. The possibility of doing so can be traced back to a universal diffusive random walk behavior $C_N = {\cal O}(N^{1/2})$ of the series $C_N = \sum_{n=1}^N χ(p_n)$ over the primes $p_n$ where $χ$ is a Dirichlet character, which underlies the convergence of the infinite product of the Dirichlet functions. The series $C_N$ presents several aspects in common with stochastic time series and its control requires to address a problem similar to the Single Brownian Trajectory Problem in statistical mechanics. In the case of the Dirichlet functions of non principal characters, we show that this problem can be solved in terms of a self-averaging procedure based on an ensemble $\CE$ of block variables computed on extended intervals of primes. Those intervals, called {\em inertial intervals}, ensure the ergodicity and stationarity of the time series underlying the quantity $C_N$. The infinity of primes also ensures the absence of rare events which would have been responsible for a different scaling behavior than the universal law $C_N = {\cal O}(N^{1/2})$ of the random walks.

math.NT↗

arXiv2017

n-cluster models in a transverse magnetic field

In this paper we analize a family of one dimensional fully analytically solvable models, named the n-cluster models in a transverse magnetic field, in which a many-body cluster interaction competes with a uniform transverse magnetic field. These models, independently by the cluster size n + 2, exibit a quantum phase transition, that separates a paramagnetic phase from a cluster one, that corresponds to a nematic ordered phase or a symmetry-protected topological ordered phase for even or odd n respectively. Due to the symmetries of the spin correlation functions, we prove that these models have no genuine n+2-partite entanglement. On the contrary, a non vanishing concurrence arises between spins at the endpoints of the cluster, for a magnetic field strong enough. Due to their analyticity and peculiar entanglement properties, the n-cluster models in a transverse magnetic field serve as a prototype for studying non trivial-spin orderings and as a potential reference system for the applications of quantum information tasks.

quant-ph↗

arXiv2018

Entanglement and diagonal entropies after a quench with no pair structure

A typical working condition in the study of quantum quenches is that the initial state produces a distribution of quasiparticle excitations with an opposite-momentum-pair structure. In this work we investigate the dynamical and stationary properties of the entanglement entropy after a quench from initial states which do not have such structure: instead of pairs of excitations they generate $ν$-plets of correlated excitations with $ν>2$. Our study is carried out focusing on a system of non-interacting fermions on the lattice. We study the time evolution of the entanglement entropy showing that the standard semiclassical formula is not applicable. We propose a suitable generalisation which correctly describes the entanglement entropy evolution and perfectly matches numerical data. We finally consider the relation between the thermodynamic entropy of the stationary state and the diagonal entropy, showing that when there is no pair structure their ratio depends on the details of the initial state and lies generically between $1/2$ and $1$.

cond-mat.stat-mech↗

arXiv2018

Kardar-Parisi-Zhang Universality in First-Passage Percolation: the Role of Geodesic Degeneracy

We have characterized the scaling behavior of the first-passage percolation (FPP) model on two types of discrete networks, the regular square lattice and the disordered Delaunay lattice, thereby addressing the effect of the underlying topology. Several distribution functions for the link-times were considered. The asymptotic behavior of the fluctuations for both the minimal arrival time and the lateral deviation of the geodesic path are in perfect agreement with the Kardar-Parisi-Zhang (KPZ) universality class regardless of the type of the link-time distribution and of the lattice topology. Pre-asymptotic behavior, on the other hand, is found to depend on the uniqueness of geodesics in absence of disorder in the local crossing times, a topological property of lattice directions that we term geodesic degeneracy. This property has important consequences on the model, as for example the well-known anisotropic growth in regular lattices. In this work we provide a framework to understand its effect as well as to characterize its extent.

cond-mat.stat-mech↗

arXiv2021

Inhomogeneous asymmetric exclusion processes between two reservoirs : large deviations for the local empirical observables in the Mean-Field approximation

For a given inhomogeneous exclusion processes on $N$ sites between two reservoirs, the trajectories probabilities allow to identify the relevant local empirical observables and to obtain the corresponding rate function at Level 2.5. In order to close the hierarchy of the empirical dynamics that appear in the stationarity constraints, we consider the simplest approximation, namely the Mean-Field approximation for the empirical density of two consecutive sites, in direct correspondence with the previously studied Mean-Field approximation for the steady state. For a given inhomogeneous Totally Asymmetric model (TASEP), this Mean-Field approximation yields the large deviations for the joint distribution of the empirical density profile and of the empirical current around the mean-field steady state; the further explicit contraction over the current allows to obtain the large deviations of the empirical density profile alone. For a given inhomogeneous Asymmetric model (ASEP), the local empirical observables also involve the empirical activities of the links and of the reservoirs; the further explicit contraction over these activities yields the large deviations for the joint distribution of the empirical density profile and of the empirical current. The consequences for the large deviations properties of time-additive space-local observables are also discussed in both cases.

cond-mat.stat-mech↗

arXiv2009

Random Hamiltonian in thermal equilibrium

A framework for the investigation of disordered quantum systems in thermal equilibrium is proposed. The approach is based on a dynamical model--which consists of a combination of a double-bracket gradient flow and a uniform Brownian fluctuation--that `equilibrates' the Hamiltonian into a canonical distribution. The resulting equilibrium state is used to calculate quenched and annealed averages of quantum observables.

quant-ph↗