Analytical and numerical studies of disordered spin-1 Heisenberg chains with aperiodic couplings

We investigate the low-temperature properties of the one-dimensional spin-1 Heisenberg model with geometric fluctuations induced by aperiodic but deterministic coupling distributions, involving two parameters. We focus on two aperiodic sequences, the Fibonacci sequence and the 6-3 sequence. Our goal is to understand how these geometric fluctuations modify the physics of the (gapped) Haldane phase, which corresponds to the ground state of the uniform spin-1 chain. We make use of different adaptations of the strong-disorder renormalization-group (SDRG) scheme of Ma, Dasgupta and Hu, widely employed in the study of random spin chains, supplemented by quantum Monte Carlo and density-matrix renormalization-group numerical calculations, to study the nature of the ground state as the coupling modulation is increased. We find no phase transition for the Fibonacci chain, while we show that the 6-3 chain exhibits a phase transition to a gapless, aperiodicity-dominated phase similar to the one found for the aperiodic spin-1/2 XXZ chain. Contrary to what is verified for random spin-1 chains, we show that different adaptations of the SDRG scheme may lead to different qualitative conclusions about the nature of the ground state in the presence of aperiodic coupling modulations.Comment: Accepted for publication in Physical Review

Optimal Renormalization Group Transformation from Information Theory

Recently a novel real-space RG algorithm was introduced, identifying the relevant degrees of freedom of a system by maximizing an information-theoretic quantity, the real-space mutual information (RSMI), with machine learning methods. Motivated by this, we investigate the information theoretic properties of coarse-graining procedures, for both translationally invariant and disordered systems. We prove that a perfect RSMI coarse-graining does not increase the range of interactions in the renormalized Hamiltonian, and, for disordered systems, suppresses generation of correlations in the renormalized disorder distribution, being in this sense optimal. We empirically verify decay of those measures of complexity, as a function of information retained by the RG, on the examples of arbitrary coarse-grainings of the clean and random Ising chain. The results establish a direct and quantifiable connection between properties of RG viewed as a compression scheme, and those of physical objects i.e. Hamiltonians and disorder distributions. We also study the effect of constraints on the number and type of coarse-grained degrees of freedom on a generic RG procedure.Comment: Updated manuscript with new results on disordered system

Universality in the one-dimensional chain of phase-coupled oscillators

We apply a recently developed renormalization group (RG) method to study synchronization in a one-dimensional chain of phase-coupled oscillators in the regime of weak randomness. The RG predicts how oscillators with randomly distributed frequencies and couplings form frequency-synchronized clusters. Although the RG was originally intended for strong randomness, i.e. for distributions with long tails, we find good agreement with numerical simulations even in the regime of weak randomness. We use the RG flow to derive how the correlation length scales with the width of the coupling distribution in the limit of large coupling. This leads to the identification of a universality class of distributions with the same critical exponent $\nu$. We also find universal scaling for small coupling. Finally, we show that the RG flow is characterized by a universal approach to the unsynchronized fixed point, which provides physical insight into low-frequency clusters.Comment: 14 pages, 10 figure

The Low-Energy Fixed Points of Random Quantum Spin Chains

The one-dimensional isotropic quantum Heisenberg spin systems with random couplings and random spin sizes are investigated using a real-space renormalization group scheme. It is demonstrated that these systems belong to a universality class of disordered spin systems, characterized by weakly coupled large effective spins. In this large-spin phase the uniform magnetic susceptibility diverges as 1/T with a non-universal Curie constant at low temperatures T, while the specific heat vanishes as T^delta |ln T| for T->0. For broad range of initial distributions of couplings and spin sizes the distribution functions approach a single fixed-point form, where delta \approx 0.44. For some singular initial distributions, however, fixed-point distributions have non-universal values of delta, suggesting that there is a line of fixed points.Comment: 19 pages, REVTeX, 13 figure

3D mesh processing using GAMer 2 to enable reaction-diffusion simulations in realistic cellular geometries

Recent advances in electron microscopy have enabled the imaging of single cells in 3D at nanometer length scale resolutions. An uncharted frontier for in silico biology is the ability to simulate cellular processes using these observed geometries. Enabling such simulations requires watertight meshing of electron micrograph images into 3D volume meshes, which can then form the basis of computer simulations of such processes using numerical techniques such as the Finite Element Method. In this paper, we describe the use of our recently rewritten mesh processing software, GAMer 2, to bridge the gap between poorly conditioned meshes generated from segmented micrographs and boundary marked tetrahedral meshes which are compatible with simulation. We demonstrate the application of a workflow using GAMer 2 to a series of electron micrographs of neuronal dendrite morphology explored at three different length scales and show that the resulting meshes are suitable for finite element simulations. This work is an important step towards making physical simulations of biological processes in realistic geometries routine. Innovations in algorithms to reconstruct and simulate cellular length scale phenomena based on emerging structural data will enable realistic physical models and advance discovery at the interface of geometry and cellular processes. We posit that a new frontier at the intersection of computational technologies and single cell biology is now open.Comment: 39 pages, 14 figures. High resolution figures and supplemental movies available upon reques

Reaction Diffusion Models in One Dimension with Disorder

We study a large class of 1D reaction diffusion models with quenched disorder using a real space renormalization group method (RSRG) which yields exact results at large time. Particles (e.g. of several species) undergo diffusion with random local bias (Sinai model) and react upon meeting. We obtain the large time decay of the density of each specie, their associated universal amplitudes, and the spatial distribution of particles. We also derive the spectrum of exponents which characterize the convergence towards the asymptotic states. For reactions with several asymptotic states, we analyze the dynamical phase diagram and obtain the critical exponents at the transitions. We also study persistence properties for single particles and for patterns. We compute the decay exponents for the probability of no crossing of a given point by, respectively, the single particle trajectories ($\theta$) or the thermally averaged packets ($\bar{\theta}$). The generalized persistence exponents associated to n crossings are also obtained. Specifying to the process $A+A \to \emptyset$ or A with probabilities $(r,1-r)$, we compute exactly the exponents $\delta(r)$ and $\psi(r)$ characterizing the survival up to time t of a domain without any merging or with mergings respectively, and $\delta_A(r)$ and $\psi_A(r)$ characterizing the survival up to time t of a particle A without any coalescence or with coalescences respectively. $\bar{\theta}, \psi, \delta$ obey hypergeometric equations and are numerically surprisingly close to pure system exponents (though associated to a completely different diffusion length). Additional disorder in the reaction rates, as well as some open questions, are also discussed.Comment: 54 pages, Late