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arXiv:2304.10139 · Physical Review Research

Renormalization theory of disordered contact processes with heavy-tailed dispersal

Róbert Juhász

Published Sep 5, 2023• 1 citation
arXivHTMLPDF

Abstract

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 nonzero 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 faster-vanishing densities as the critical point is approached. The persistence, however, shows an opposite tendency: the broadening of the dispersal makes its decline sharper 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 that model.

DOI
10.1103/physrevresearch.5.033157

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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 nonzero 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 faster-vanishing densities as the critical point is approached. The persistence, however, shows an opposite tendency: the broadening of the dispersal makes its decline sharper 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 that model.
Abstract We study the local persistence probability during non-stationary time evolutions in disordered contact processes with long-range interactions by a combination of the strong-disorder renormalization group method, a phenomenological theory of rare regions, and numerical simulations. We find that, for interactions decaying as an inverse power of the distance, the persistence probability tends to a non-zero limit not only in the inactive phase but also in the critical point. Thus, unlike in the contact process with short-range interactions, the persistence in the limit t → ∞ is a discontinuous function of the control parameter. For stretched exponentially decaying interactions, the limiting value of the persistence is found to remain continuous, similar to the model with short-range interactions.
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.
We study the time dependence of the local persistence probability during a nonstationary time evolution in the disordered contact process in d=1, 2, and 3 dimensions. We present a method for calculating the persistence with the strong-disorder renormalization group (SDRG) technique, which we then apply at the critical point analytically for d=1 and numerically for d=2,3. According to the results, the average persistence decays at late times as an inverse power of the logarithm of time, with a universal dimension-dependent generalized exponent. For d=1, the distribution of sample-dependent local persistence is shown to be characterized by a universal limit distribution of effective persistence exponents. Using a phenomenological approach of rare-region effects in the active phase, we obtain a nonuniversal algebraic decay of the average persistence for d=1 and enhanced power laws for d>1. As an exception, for randomly diluted lattices, the algebraic decay remains valid for d>1, which is explained by the contribution of dangling ends. Results on the time dependence of average persistence are confirmed by Monte Carlo simulations. We also prove the equivalence of the persistence with a return probability, a valuable tool for the argumentations.
We investigate the nonequilibrium phase transition in the disordered contact process in the presence of long-range spatial disorder correlations. These correlations greatly increase the probability for finding rare regions that are locally in the active phase while the bulk system is still in the inactive phase. Specifically, if the correlations decay as a power of the distance, the rare-region probability is a stretched exponential of the rare-region size rather than a simple exponential as is the case for uncorrelated disorder. As a result, the Griffiths singularities are enhanced and take a non-power-law form. The critical point itself is of infinite-randomness type but with critical exponent values that differ from the uncorrelated case. We report large-scale Monte Carlo simulations that verify and illustrate our theory. We also discuss generalizations to higher dimensions and applications to other systems such as the random transverse-field Ising model, itinerant magnets, and the superconductor-metal transition.
We study the low-energy properties of the long-range random transverse-field Ising chain with ferromagnetic interactions decaying as a power of the distance. Using variants of the strong-disorder renormalization group method, the critical behavior is found to be controlled by a strong-disorder fixed point with a finite dynamical exponent . Approaching the critical point, the correlation length diverges exponentially. In the critical point, the magnetization shows an -independent logarithmic finite-size scaling and the entanglement entropy satisfies the area law. These observations are argued to hold for other systems with long-range interactions, even in higher dimensions.
We formulate a strong-disorder renormalization-group (SDRG) approach to study the $\ensuremath{\beta}$ function of the tight-binding model in one dimension with both diagonal and off-diagonal disorder for states at the band center. We show that the SDRG method, when used to compute transport properties, yields exact results since it is identical to the transfer matrix method. The $\ensuremath{\beta}$ function is shown to be universal when only off-diagonal disorder is present even though single-parameter scaling is known to be violated. A different single-parameter scaling theory is formulated for this particular (particle-hole symmetric) case. Upon breaking particle-hole symmetry (by adding diagonal disorder), the $\ensuremath{\beta}$ function is shown to crossover from the universal behavior of the particle-hole symmetric case to the conventional nonuniversal one in agreement with the two-parameter scaling theory. We finally draw an analogy with the random transverse-field Ising chain in the paramagnetic phase. The particle-hole symmetric case corresponds to the critical point of the quantum Ising model, while the generic case corresponds to the Griffiths paramagnetic phase.