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arXiv:1312.3771 · Physical Review E

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

Róbert Juhász

Published Mar 10, 2014• 4 citations
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Abstract

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 disordered contact process, the survival probability P(t) is found to show multiscaling in the critical point, meaning that P(t)=t-δ, where the (environment and time-dependent) exponent δ has a universal limit distribution when t→∞. 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 δ, which are based on numerical results, are formulated. By the same method, the survival probability in the problem of random walks in random environments is also shown to exhibit multiscaling with an exponential limit distribution. In addition to this, the (imaginary-time) spin-spin autocorrelation function of the 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.

DOI
10.1103/physreve.89.032108

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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 disordered contact process, the survival probability P(t) is found to show multiscaling in the critical point, meaning that P(t)=t-δ, where the (environment and time-dependent) exponent δ has a universal limit distribution when t→∞. 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 δ, which are based on numerical results, are formulated. By the same method, the survival probability in the problem of random walks in random environments is also shown to exhibit multiscaling with an exponential limit distribution. In addition to this, the (imaginary-time) spin-spin autocorrelation function of the 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.
We study the nonequilibrium phase transition in the two-dimensional contact process on a randomly diluted lattice by means of large-scale Monte Carlo simulations for times up to 10;{10} and system sizes up to 8000x8000 sites. Our data provide strong evidence for the transition being controlled by an exotic infinite-randomness critical point with activated (exponential) dynamical scaling. We calculate the critical exponents of the transition and find them to be universal, i.e., independent of disorder strength. The Griffiths region between the clean and the dirty critical points exhibits power-law dynamical scaling with continuously varying exponents. We discuss the generality of our findings and relate them to a broader theory of rare region effects at phase transitions with quenched disorder. Our results are of importance beyond absorbing state transitions because, according to a strong-disorder renormalization group analysis, our transition belongs to the universality class of the two-dimensional random transverse-field Ising model.
diffusion in disordered environments, random quantum spin chains Abstract. — We study the first passage time properties of an integrated Brownian curve both in homogeneous and disordered environments. In a disordered medium we relate the scaling properties of this center of mass persistence of a random walker to the average persistence, the latter being the probability P pr(t) that the expectation value 〈x(t) 〉 of the walker’s position after time t has not returned to the initial value. The average persistence is then connected to the statistics of extreme events of homogeneous random walks which can be computed exactly for moderate system sizes. As a result we obtain a logarithmic dependence P pr(t) ∼ ln(t) −θ with a new exponent θ = 0.191 ± 0.002. We note on a complete correspondence between the average persistence of random walks and the magnetization autocorrelation function of the transverse-field Ising chain, in the homogeneous and disordered case. First passage time or persistence problems have a long history in the physical literature [1]. Recently they gained a lot of interest, since the persistence exponents, describing the asymptotic behavior of first passage time probabilities, are shown to be independent dynamical critical exponents, which have been calculated for various models exactly[2, 3]. Not much is known about analogous quantities in systems with quenched disorder, for instance the random walk (or diffusion) in a disordered environment [4], which is in the one-dimensional case the Sinai-model[5]. For this model the first passage / persistence exponent for a single walker has been determined by us in a previous work [6]. In this letter we introduce and study the concept of average persistence of random walks both in homogeneous and random environments. We consider a random walk with nearest neighbor hopping in one dimension defined by the Master equation pi(t = 0) = δi,1,
Using exact expressions for the persistence probability and for the leading eigenvalue of the Fokker-Planck operator of a random walk in a random environment, we establish a fundamental relation between the statistical properties of anomalous diffusion and the critical and off-critical behavior of random quantum spin chains. Many exact results are obtained from this correspondence, including the space and time correlations of surviving random walks and the distribution of the gaps of the corresponding Fokker-Planck operator. In turn we derive analytically the dynamical exponent of the random transverse-field Ising spin chain in the Griffiths-McCoy region.
We show that an interaction decaying as a stretched exponential function of distance, , 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 find that the critical behavior of the model is controlled by infinite-disorder fixed points different from those of the short-range model 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 at the critical point but, unlike the short-range model, the prefactor is dependent on disorder 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.
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We consider cooperative processes (quantum spin chains and random walks) in one-dimensional fluctuating random and aperiodic environments characterized by fluctuating exponents #>0. At the critical point the random and aperiodic systems scale essentially anisotropically in a similar fashion: length (L) and time (t) scales are related as L # (ln t) 1/# . Also some critical exponents, characterizing the singularities of average quantities, are found to be universal functions of #, whereas some others do depend on details of the distribution of the disorder. In the off-critical region there is an important di#erence between the two types of environments: in aperiodic systems there are no extra (Griffiths)-singularities.