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arXiv:1911.05964 · Physical review. E

Tricritical directed percolation with long-range interaction in one and two dimensions

Minjae Jo, B. Kahng

Published Feb 18, 2020• 8 citations
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Abstract

Recently, the quantum contact process, in which branching and coagulation processes occur both coherently and incoherently, was theoretically and experimentally investigated in driven open quantum spin systems. In the semiclassical approach, the quantum coherence effect was regarded as a process in which two consecutive atoms are involved in the excitation of a neighboring atom from the inactive (ground) state to the active state (excited s-state). In this case, both second-order and first-order transitions occur. Therefore, a tricritical point exists at which the transition belongs to the tricritical directed percolation (TDP) class. On the other hand, when an atom is excited to the d-state, long-range interaction is induced. Here, to account for this long-range interaction, we extend the TDP model to one with long-range interaction in the form of ∼1/r^{d+σ} (denoted as LTDP), where r is the separation, d is the spatial dimension, and σ is a control parameter. In particular, we investigate the properties of the LTDP class below the upper critical dimension d_{c}=min(3,1.5σ). We numerically obtain a set of critical exponents in the LTDP class and determine the interval of σ for the LTDP class. Finally, we construct a diagram of universality classes in the space (d,σ).

DOI
10.1103/physreve.101.022121

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Recently, the quantum contact process, in which branching and coagulation processes occur both coherently and incoherently, was theoretically and experimentally investigated in driven open quantum spin systems. In the semiclassical approach, the quantum coherence effect was regarded as a process in which two consecutive atoms are involved in the excitation of a neighboring atom from the inactive (ground) state to the active state (excited s-state). In this case, both second-order and first-order transitions occur. Therefore, a tricritical point exists at which the transition belongs to the tricritical directed percolation (TDP) class. On the other hand, when an atom is excited to the d-state, long-range interaction is induced. Here, to account for this long-range interaction, we extend the TDP model to one with long-range interaction in the form of ∼1/r^{d+σ} (denoted as LTDP), where r is the separation, d is the spatial dimension, and σ is a control parameter. In particular, we investigate the properties of the LTDP class below the upper critical dimension d_{c}=min(3,1.5σ). We numerically obtain a set of critical exponents in the LTDP class and determine the interval of σ for the LTDP class. Finally, we construct a diagram of universality classes in the space (d,σ).
We investigate a nonequilibrium phase transition in a dissipative and coherent quantum spin system using the quantum Langevin equation and mean-field theory. Recently, the quantum contact process (QCP) was theoretically investigated using the Rydberg antiblockade effect, in particular, when the Rydberg atoms were excited in s states so that their interactions were regarded as being between the nearest neighbors. However, when the atoms are excited to d states, the dipole-dipole interactions become effective, and long-range interactions must be considered. Here we consider a quantum spin model with a long-range QCP, where the branching and coagulation processes are allowed not only for the nearest-neighbor pairs but also for long-distance pairs, coherently and incoherently. Using the semiclassical approach, we show that the mean-field phase diagram of our long-range model is similar to that of the nearest-neighbor QCP, where the continuous (discontinuous) transition is found in the weak (strong) quantum regime. However, at the tricritical point, we find a new universality class, which was neither that of the QCP at the tricritical point nor that of the classical directed percolation model with long-range interactions. Implementation of the long-range QCP using interacting cold gases is discussed.
We present detailed simulations of a generalization of the Domany–Kinzel model to 2+1 dimensions. It has two control parameters, p and q , which describe the probabilities P k of a site to be wetted, if exactly k of its 'upstream' neighbours are already wetted. If P k depends only weakly on k , the active/adsorbed phase transition is in the directed percolation (DP) universality class. If, however, P k increases quickly with k so that the formation of inactive holes surrounded by active sites is suppressed, the transition is first order. These two transition lines meet at a tricritical point. This point should be in the same universality class as a tricritical transition in the contact process studied recently by Lübeck. Critical exponents for it have been calculated previously by means of the field-theoretic epsilon-expansion ( = 3 − d , with d = 2 in the present case). Rather poor agreement is found with both.
Phase transitions to absorbing states are among the simplest examples of critical phenomena out of equilibrium. The characteristic feature of these models is the presence of a fluctuationless configuration that the dynamics cannot leave, which has proved a rather stringent requirement in experiments. Recently, a proposal to seek such transitions in highly tunable systems of cold atomic gases offers to probe this physics and, at the same time, to investigate the robustness of these transitions to quantum coherent effects. Here, the authors specifically focus on the interplay between classical and quantum fluctuations in a simple driven open quantum model which, in the classical limit, reproduces a contact process, which is known to undergo a continuous transition in the ``directed percolation'' universality class. The authors derive an effective long-wavelength field theory for the present class of open spin systems and show that, due to quantum fluctuations, the nature of the transition changes from second to first order, passing through a critical point which appears to belong instead to the ``tricritical directed percolation'' class.
The contact process is a paradigmatic classical stochastic system displaying critical behavior even in one dimension. It features a nonequilibrium phase transition into an absorbing state that has been widely investigated and shown to belong to the directed percolation universality class. When the same process is considered in a quantum setting, much less is known. So far, mainly semiclassical studies have been conducted and the nature of the transition in low dimensions is still a matter of debate. Also, from a numerical point of view, from which the system may look fairly simple-especially in one dimension-results are lacking. In particular, the presence of the absorbing state poses a substantial challenge, which appears to affect the reliability of algorithms targeting directly the steady state. Here we perform real-time numerical simulations of the open dynamics of the quantum contact process and shed light on the existence and on the nature of an absorbing state phase transition in one dimension. We find evidence for the transition being continuous and provide first estimates for the critical exponents. Beyond the conceptual interest, the simplicity of the quantum contact process makes it an ideal benchmark problem for scrutinizing numerical methods for open quantum nonequilibrium systems.
We investigate the influence of the range of interactions in the two-dimensional bond percolation model by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges, as expressed by the number zzz of equivalent neighbors. We also consider the $z\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, i.e., the complete graph case, where percolation bonds are allowed between each pair of sites, and the model becomes mean-field-like. All investigated models with finite zzz are found to belong to the short-range universality class. There is no evidence of a tricritical point separating the short-range and long-range behavior, such as is known to occur for q=3q=3q=3 and q=4q=4q=4 Potts models. We determine the renormalization exponent describing a finite-range perturbation at the mean-field limit as ${y}_{r}\ensuremath{\approx}2/3$. Its relevance confirms the continuous crossover from mean-field percolation universality to short-range percolation universality. For finite interaction ranges, we find approximate relations between the coordination numbers and the amplitudes of the leading correction terms as found in the finite-size scaling analysis.
Crossover behaviors from the pair contact process with diffusion (PCPD) and the driven PCPD (DPCPD) to the directed percolation (DP) are studied in one dimension by introducing a single particle annihilation and/or branching dynamics. The crossover exponents phi are estimated numerically as 1/phi approximately 0.58 +/- 0.03 for the PCPD and 1/phi approximately 0.49+/-0.02 for the DPCPD. Nontriviality of the PCPD crossover exponent strongly supports the non-DP nature of the PCPD critical scaling, which is further evidenced by the anomalous critical amplitude scaling near the PCPD point. In addition, we find that the DPCPD crossover is consistent with the mean field prediction of the tricritical DP class as expected.