Hidden percolation transition in kinetic replication process

2016

Phys. Rev. E

The evolution of many kinetic processes in 1+1 (space-time) dimensions results in 2d directed percolative landscapes. The active phases of these models possess numerous hidden geometric orders characterized by various types of large-scale and/or coarse-grained percolative backbones that we define. For the patterns originated in the classical directed percolation (DP) and contact process (CP) we show from the Monte-Carlo simulation data that these percolative backbones emerge at specific critical points as a result of continuous phase transitions. These geometric transitions belong to the DP universality class and their nonlocal order parameters are the capacities of corresponding backbones. The multitude of conceivable percolative backbones implies the existence of infinite cascades of such geometric transitions in the kinetic processes considered. We present simple arguments to support the conjecture that such cascades of transitions is a generic feature of percolation as well as of many other transitions with nonlocal order parameters.

“…Here the auxiliary variable of integration  is introduced to enforce the model rule   0,0 0  . For more details on this formalism, see [8].…”

Section:  mentioning

confidence: 99%

“…A variant of the kinetic replication process was introduced in [8]. Its transfer probabilities depend on the two parameters p and q.…”

Section: Replication Processmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Exploring percolative landscapes: Infinite cascades of geometric phase transitions

Timonin

2016

Phys. Rev. E

“…35 Another interesting class of models where nonlocal string-like order parameters turned out to be essential are the probabilistic cellular automata which have been experiencing a steadily growing interest during the last several decades. It was shown recently 36 that the active phases of the (1+1)-dimensional automation models possess numerous percolative nonlocal order parameters emerging as cascades of geometric phase transitions.…”

Section: Introductionmentioning

confidence: 99%

Constructing Landau framework for topological order: quantum chains and ladders

Pandey

2017

J. Stat. Mech.

We studied quantum phase transitions in the antiferromagnetic dimerized spin-1 2 XY chain and two-leg ladders. From analysis of several spin models we present our main result: the framework to deal with topological orders and hidden symmetries within the Landau paradigm. After mapping of the spin Hamiltonians onto the tight-binding models with Dirac or Majorana fermions and, when necessary, the mean-field approximation, the analysis can be done analytically. By utilizing duality transformations the calculation of nonlocal string order parameters is mapped onto the local order problem in some dual representation and done without further approximations. Calculated phase diagrams, phase boundaries, order parameters and their symmetries for each of the phases provide a comprehensive quantitative Landau description of the quantum critical properties of the models considered. Complementarily, the phases with hidden orders can also be distinguished by the Pontryagin (winding) numbers which we have calculated as well. This unified framework can be straightforwardly applied for various spin chains and ladders, topological insulators and superconductors. Applications to other systems are under way.

Infinite cascades of phase transitions in the classical Ising chain

Timonin

2017

Phys. Rev. E

We report the new exact results on one of the best studied models in statistical physics: the classical antiferromagnetic Ising chain in a magnetic field. We show that the model possesses an infinite cascade of thermal phase transitions (also known as "disorder lines" or geometric phase transitions). The phase transition is signalled by a change of asymptotic behavior of the nonlocal string-string correlation functions when their monotonous decay becomes modulated by incommensurate oscillations. The transitions occur for rarefied (m-periodic) strings with arbitrary odd m. We propose a duality transformation which maps the Ising chain onto the m-leg Ising tube with nearest-neighbor couplings along the legs and the plaquette four-spin interactions of adjacent legs. Then the m-string correlation functions of the Ising chain are mapped onto the two-point spin-spin correlation functions along the legs of the m-leg tube. We trace the origin of these cascades of phase transitions to the lines of the Lee-Yang zeros of the Ising chain in m-periodic complex magnetic field, allowing us to relate these zeros to the observable (and potentially measurable) quantities.