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arXiv:2008.10807

Universal spectrum structure at nonequilibrium critical points in the (1+1)-dimensional directed percolation

Kenji Harada

Published Aug 25, 2020• 0 citations
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Abstract

Using a tensor renormalization group method with oblique projectors for an anisotropic tensor network, we confirm that the rescaled spectrum of transfer matrices at nonequilibrium critical points in the (1+1)-dimensional directed percolation, a canonical model of nonequilibrium critical phenomena, is scale-invariant and its structure is universal.

DOI
10.48550/arxiv.2008.10807

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Using a tensor renormalization group method with oblique projectors for an anisotropic tensor network, we confirm that the rescaled spectrum of transfer matrices at nonequilibrium critical points in the (1+1)-dimensional directed percolation, a canonical model of nonequilibrium critical phenomena, is scale-invariant and its structure is universal.
Universal behavior is a typical emergent feature of critical systems. A paramount model of the nonequilibrium critical behavior is the directed bond percolation process that exhibits an active-to-absorbing state phase transition in the vicinity of a percolation threshold. Fluctuations of the ambient environment might affect or destroy the universality properties completely. In this work, we assume that the random environment can be described by means of compressible velocity fluctuations. Using field-theoretic models and renormalization group methods, we investigate large-scale and long-time behavior. Altogether, 11 universality classes are found, out of which 4 are stable in the infrared limit and thus macroscopically accessible. In contrast to the model without velocity fluctuations, a possible candidate for a realistic three-dimensional case, a regime with relevant short-range noise, is identified. Depending on the dimensionality of space and the structure of the turbulent flow, we calculate critical exponents of the directed percolation process. In the limit of the purely transversal random force, critical exponents comply with the incompressible results obtained by previous authors. We have found intriguing nonuniversal behavior related to the mutual effect of compressibility and advection.
Recently, considerable progress has been made in understanding finite-size scaling in equilibrium systems. Here, we study finite-size scaling in nonequilibrium systems at the instance of directed percolation (DP), which has become the paradigm of nonequilibrium phase transitions into absorbing states, above, at, and below the upper critical dimension. We investigate the finite-size scaling behavior of DP analytically and numerically by considering its steady state generated by a homogeneous constant external source on a d-dimensional hypercube of finite edge length L with periodic boundary conditions near the bulk critical point. In particular, we study the order parameter and its higher moments using renormalized field theory. We derive finite-size scaling forms of the moments in a one-loop calculation. Moreover, we introduce and calculate a ratio of the order parameter moments that plays a similar role in the analysis of finite size scaling in absorbing nonequilibrium processes as the famous Binder cumulant in equilibrium systems and that, in particular, provides a signature of the DP universality class. To complement our analytical work, we perform Monte Carlo simulations which confirm our analytical results.
We generalize nonperturbative renormalization group methods to nonequilibrium critical phenomena. Within this formalism, reaction-diffusion processes are described by a scale-dependent effective action, the flow of which is derived. We investigate branching and annihilating random walks with an odd number of offspring. Along with recovering their universal physics (described by the directed percolation universality class), we determine their phase diagrams and predict that a transition occurs even in three dimensions, contrarily to what perturbation theory suggests.
Using transfer-matrix extended phenomenological renormalization-group methods, we study the critical properties of the spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths J→=(J′,J′,J)\mathop J\limits^ \to = (J',J',J)J→​=(J′,J′,J) . The universality of both fundamental critical exponents y t and y h is confirmed. It is shown that the critical finite-size scaling amplitude ratios U=Aχ(4)Aκ/Aχ2,Y1=Aκ′′/AχU = A_{\chi ^{(4)} } A_\kappa /A_\chi ^2 ,Y_1 = A_{\kappa ''} /A_\chiU=Aχ(4)​Aκ​/Aχ2​,Y1​=Aκ′′​/Aχ​ , and Y2=Aκ(4)/Aχ(4)Y_2 = A_{\kappa ^{(4)} } /A_{\chi ^{(4)} }Y2​=Aκ(4)​/Aχ(4)​ are independent of the lattice anisotropy parameter Δ=J′/J. For the Y 2 invariant of the three-dimensional Ising universality class, we give the first quantitative estimate Y 2≈2.013 (shape L×L×∞, periodic boundary conditions in both transverse directions).
It is shown that the universal critical properties of two recently introduced coupled directed percolation processes can be described by a single rapidity reversal invariant stochastic reaction-diffusion model. It is demonstrated that all renormalizations needed for the calculation of the universal scaling behavior near the multicritical point can be gained from the Gribov process (Reggeon field theory). Consequently the crossover exponent describing the scaling of the linear coupling parameter is given by Phi = 1 to all orders of the perturbation expansion.