Entanglement across extended random defects in the XX spin chain

We study the half-chain entanglement entropy in the ground state of the spin-1/2 XX chain across an extended random defect, where the strength of disorder decays with the distance from the interface algebraically as $\Delta_l\sim l^{-\kappa}$. In the whole regime $\kappa\ge 0$, the average entanglement entropy is found to increase logarithmically with the system size $L$ as $S_L\simeq\frac{c_{\rm eff}(\kappa)}{6}\ln L+const$, where the effective central charge $c_{\rm eff}(\kappa)$ depends on $\kappa$. In the regime $\kappa<1/2$, where the extended defect is a relevant perturbation, the strong-disorder renormalization group method gives $c_{\rm eff}(\kappa)=(1-2\kappa)\ln2$, while, in the regime $\kappa\ge 1/2$, where the extended defect is irrelevant in the bulk, numerical results indicate a non-zero effective central charge, which increases with $\kappa$. The variation of $c_{\rm eff}(\kappa)$ is thus found to be non-monotonic and discontinuous at $\kappa=1/2$.Comment: 16 pages, 8 figure

Distribution of dynamical quantities in the contact process, random walks, and quantum spin chains in random environments

We study the distribution of dynamical quantities in various one-dimensional, disordered models the critical behavior of which is described by an infinite randomness fixed point. In the {\it disordered contact process}, the quenched survival probability $\mathcal{P}(t)$ defined in fixed random environments is found to show multi-scaling in the critical point, meaning that $\mathcal{P}(t)=t^{-\delta}$, where the (environment and time-dependent) exponent $\delta$ has a universal limit distribution when $t\to\infty$. The limit distribution is determined by the strong disorder renormalization group method analytically in the end point of a semi-infinite lattice, where it is found to be exponential, while, in the infinite system, conjectures on its limiting behaviors for small and large $\delta$, which are based on numerical results, are formulated. By the same method, the quenched survival probability in the problem of {\it random walks in random environments} is also shown to exhibit multi-scaling with an exponential limit distribution. In addition to this, the (imaginary-time) spin-spin autocorrelation function of the {\it random transverse-field Ising chain} is found to have a form similar to that of survival probability of the contact process at the level of the renormalization approach. Consequently, a relationship between the corresponding limit distributions in the two problems can be established. Finally, the distribution of the spontaneous magnetization in this model is also discussed.Comment: 16 pages, 7 figure

Infinite-disorder critical points of models with stretched exponential interactions

We show that an interaction decaying as a stretched exponential function of the distance, $J(l)\sim e^{-cl^a}$, is able to alter the universality class of short-range systems having an infinite-disorder critical point. To do so, we study the low-energy properties of the random transverse-field Ising chain with the above form of interaction by a strong-disorder renormalization group (SDRG) approach. We obtain that the critical behavior of the model is controlled by infinite-disorder fixed points different from that of the short-range one if $0<a<1/2$. In this range, the critical exponents calculated analytically by a simplified SDRG scheme are found to vary with $a$, while, for $a>1/2$, the model belongs to the same universality class as its short-range variant. The entanglement entropy of a block of size $L$ increases logarithmically with $L$ in the critical point but, as opposed to the short-range model, the prefactor is disorder-dependent in the range $0<a<1/2$. Numerical results obtained by an improved SDRG scheme are found to be in agreement with the analytical predictions. The same fixed points are expected to describe the critical behavior of, among others, the random contact process with stretched exponentially decaying activation rates.Comment: 16 pages, 3 figure

Critical behavior of models with infinite disorder at a star junction of chains

We study two models having an infinite-disorder critical point --- the zero temperature random transverse-field Ising model and the random contact process --- on a star-like network composed of $M$ semi-infinite chains connected to a common central site. By the strong disorder renormalization group method, the scaling dimension $x_M$ of the local order parameter at the junction is calculated. It is found to decrease rapidly with the number $M$ of arms, but remains positive for any finite $M$. This means that, in contrast with the pure transverse-field Ising model, where the transition becomes of first order for $M>2$, it remains continuous in the disordered models, although, for not too small $M$, it is hardly distinguishable from a discontinuous one owing to a close-to-zero $x_M$. The scaling behavior of the order parameter in the Griffiths-McCoy phase is also analyzed.Comment: 14 page

Superdiffusion in a class of networks with marginal long-range connections

A class of cubic networks composed of a regular one-dimensional lattice and a set of long-range links is introduced. Networks parametrized by a positive integer k are constructed by starting from a one-dimensional lattice and iteratively connecting each site of degree 2 with a $k$th neighboring site of degree 2. Specifying the way pairs of sites to be connected are selected, various random and regular networks are defined, all of which have a power-law edge-length distribution of the form $P_>(l)\sim l^{-s}$ with the marginal exponent s=1. In all these networks, lengths of shortest paths grow as a power of the distance and random walk is super-diffusive. Applying a renormalization group method, the corresponding shortest-path dimensions and random-walk dimensions are calculated exactly for k=1 networks and for k=2 regular networks; in other cases, they are estimated by numerical methods. Although, s=1 holds for all representatives of this class, the above quantities are found to depend on the details of the structure of networks controlled by k and other parameters.Comment: 10 pages, 9 figure

Extinction transition in stochastic population dynamics in a random, convective environment

Motivated by modeling the dynamics of a population living in a flowing medium where the environmental factors are random in space, we have studied an asymmetric variant of the one-dimensional contact process, where the quenched random reproduction rates are systematically greater in one direction than in the opposite one. The spatial disorder turns out to be a relevant perturbation but, according to results of Monte Carlo simulations, the behavior of the model at the extinction transition is different from the (infinite randomness) critical behavior of the disordered, symmetric contact process. Depending on the strength a of the asymmetry, the critical population drifts either with a finite velocity or with an asymptotically vanishing velocity as x(t)∼tμ(a), where μ(a)<1. Dynamical quantities are non-self-averaging at the extinction transition; the survival probability, for instance, shows multiscaling, i.e. it is characterized by a broad spectrum of effective exponents. For a sufficiently weak asymmetry, a Griffiths phase appears below the extinction transition, where the survival probability decays as a non-universal power of the time while, above the transition, another extended phase emerges, where the front of the population advances anomalously with a diffusion exponent continuously varying with the control parameter

Renormalization theory of disordered contact processes with heavy-tailed dispersal

Motivated by long-range dispersal in ecological systems, we formulate and apply a general strong-disorder renormalization group (SDRG) framework to describe one-dimensional disordered contact processes with heavy-tailed, such as power law, stretched exponential, and log-normal dispersal kernels, widely used in ecology. The focus is on the close-to-critical scaling of the order parameters, including the commonly used density, as well as the less known persistence, which is non-zero in the inactive phase. Our analytic and numerical results obtained by SDRG schemes at different levels of approximation reveal that the more slowly decaying dispersal kernels lead to more smoothly vanishing densities as the critical point is approached. The persistence, however, shows an opposite tendency: the broadening of the dispersal makes its decline more singular at the critical point, becoming discontinuous for the extreme case of power-law dispersal. The SDRG schemes presented here also describe the quantum phase transition of random transverse-field Ising chains with ferromagnetic long-range interactions, the density corresponding to the magnetization of this model.Comment: 16 pages, 9 figure