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arXiv:1406.3200 · Journal of Statistical Mechanics Theory and Experiment

Infinite-disorder critical points of models with stretched exponential interactions

Róbert Juhász

Published Sep 22, 2014• 6 citations
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Abstract

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.

DOI
10.1088/1742-5468/2014/09/p09027

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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 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.
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.
We consider disordered ladders of the transverse-field Ising model and study their critical properties and entanglement entropy for varying width, $w\ensuremath{\le}20$, by numerical application of the strong disorder renormalization group method. We demonstrate that the critical properties of the ladders for any finite www are controlled by the infinite disorder fixed point of the random chain and the correction to scaling exponents contain information about the two-dimensional model. We calculate sample dependent pseudocritical points and study the shift of the mean values as well as scaling of the width of the distributions and show that both are characterized by the same exponent, $\ensuremath{\nu}(2d)$. We also study scaling of the critical magnetization, investigate critical dynamical scaling as well as the behavior of the critical entanglement entropy. Analyzing the www dependence of the results we have obtained accurate estimates for the critical exponents of the two-dimensional model: $\ensuremath{\nu}(2d)=1.25(8)$, x(2d)=0.996(15)x(2d)=0.996(15)x(2d)=0.996(15), and $\ensuremath{\psi}(2d)=0.51(3)$.
The infinite-disorder fixed point of the random transverse-field Ising model is expected to control the critical behavior of a large class of random quantum and stochastic systems having an order parameter with discrete symmetry. Here we study the model on the square lattice with a very efficient numerical implementation of the strong disorder renormalization group method, which makes us possible to treat finite samples of linear size up to L=2048L=2048L=2048. We have calculated sample dependent pseudocritical points and studied their distribution, which is found to be characterized by the same shift and width exponent: $\ensuremath{\nu}=1.24(2)$. For different types of disorder the infinite-disorder fixed point is shown to be characterized by the same set of critical exponents, for which we have obtained improved estimates: x=0.982(15)x=0.982(15)x=0.982(15) and $\ensuremath{\psi}=0.48(2)$. We have also studied the scaling behavior of the magnetization in the vicinity of the critical point as well as dynamical scaling in the ordered and disordered Griffiths phases.
We apply a real-space block renormalization group approach to study the critical properties of the random transverse-field Ising spin chain with multispin interactions. First, we recover the known properties of the traditional model with two-spin interactions by applying the renormalization approach for the arbitrary size of the block. For the model with three-spin couplings, we calculate the critical point and demonstrate that the phase transition is controlled by an infinite disorder fixed point. We have determined the typical correlation-length critical exponent, which seems to be different from that of the random transverse Ising chain with nearest-neighbor couplings. Thus, this model represents a new infinite disorder universality class.