Universality Classes Of Phase Transitions Research Articles

Formalizing the Use of the Activation Function in Neural Inference

We investigate how the activation function can be used to describe neural firing in an abstract way, and in turn, why it works well in artificial neural networks. We discuss how a spike in a biological neuron belongs to a particular universality class of phase transitions in statistical physics. We then show that the artificial neuron is, mathematically, a mean-field model of biological neural membrane dynamics, which arises from modeling spiking as a phase transition. This allows us to treat selective neural firing in an abstract way and formalize the role of the activation function in perceptron learning. The resultant statistical physical model allows us to recover the expressions for some known activation functions as various special cases. Along with deriving this model and specifying the analogous neural case, we analyze the phase transition to understand the physics of neural network learning. Together, it is shown that there is not only a biological meaning but a physical justification for the emergence and performance of typical activation functions; implications for neural learning and inference are also discussed.

Tuning the universality class of phase transitions by feedback: Open quantum systems beyond dissipation

We shift the paradigm of feedback control from the control of quantum states to the control of phase transitions in quantum systems. We show that feedback allows tuning the universality class of phase transitions via modifying its critical exponent. We expand our previous treatment [D. A. Ivanov, T. Yu. Ivanova, S. F. Caballero-Benitez, and I. B. Mekhov, Phys. Rev. Lett. 124, 010603 (2020)] of Dicke model and go beyond the approximation of adiabatically eliminated light field. Both linearized and nonlinear models of spin ensembles are considered. The tunability of quantum fluctuations near the critical point by the feedbacks of nontrivial shapes is explained by considering the fluctuation spectra and the system behavior at single quantum trajectories.

Probing Yang–Lee edge singularity by central spin decoherence

Yang–Lee edge singularities are the branch point of free energy on the complex plane of physical parameters and have been shown to be the simplest universality class of phase transitions. However, Yang–Lee edge singularities have not been regarded as experimentally observable since they occur at complex physical parameters which are unphysical. A recent discovery regarding the relation between partition functions and probe spin coherence makes it experimentally feasible to access the complex plane of physical parameters. However, how to extract the critical point and the critical exponent of Yang–Lee edge singularities in many-body systems, which occurs only at the thermodynamic limit, has still been elusive. Here we show that the quantum coherence of a probe spin coupled to finite-size Ising-type spin systems presents universal scaling behavior near the Yang–Lee edge singularity. The finite-size scaling behavior of the quantum coherence of the probe spin predicts that one can extract the critical point and the critical exponent of the Yang–Lee edge singularity of an Ising-type spin system in the thermodynamic limit from the spin coherence measurement of the probe spin coupled to finite-size Ising-type spin systems. This finding provides a practical approach to studying the nature of Yang–Lee edge singularities of many-body systems.

Conformational transitions of single polymer adsorption in poor solvent: Wetting transition due to molecular confinement induced line tension.

We report a theory capable of describing conformational transitions for single polymer adsorption in a poor solvent. We show that an additional molecular confinement effect near the contact line can act exactly like line tension, playing a critical role in the behavior of an absorbed polymer chain. Using this theory, distinct conformational states: desorbed globule (DG), surface attached cap (SAC), and adsorbed lens (AL), can be vividly revealed, resembling the drying-wetting transition of a nanodroplet. But the transitions between these states can behave rather differently from those in the usual wetting transitions. The DG-SAC transition is discrete, occurring at the adsorption threshold when the globule size at the desorbed state is equal to the adsorption blob. The SAC-AL transition is smooth for finite chain lengths, but can change to discontinuous in the infinite chain limit, characterized by the different end-to-end exponent 3/8 and the unique crossover exponent 1/4. Distinctive critical exponents near this transition are also determined, indicating that it is an additional universality class of phase transitions. This work also sheds light on nanodrop spreading, wherein the important role played by line tension might simply be a manifestation of the local molecular confinement near the contact line.

Frequency-dependent selection at rough expanding fronts

Microbial colonies are experimental model systems for studying the colonization of new territory by biological species through range expansion. We study a generalization of the two-species Eden model, which incorporates local frequency-dependent selection, in order to analyze how social interactions between two species influence surface roughness of growing microbial colonies. The model includes several classical scenarios from game theory. We then concentrate on an expanding public goods game, where either cooperators or defectors take over the front depending on the system parameters. We analyze in detail the critical behavior of the nonequilibrium phase transition between global cooperation and defection and thereby identify a new universality class of phase transitions dealing with absorbing states. At the transition, the number of boundaries separating sectors decays with a novel power law in time and their superdiffusive motion crosses over from Eden scaling to a nearly ballistic regime. In parallel, the width of the front initially obeys Eden roughening and, at later times, passes over to selective roughening.

Phase transition properties of the Bell-Lavis model.

Using Monte Carlo calculations we analyze the order and the universality class of phase transitions into a low-density (honeycomb) phase of a triangular antiferromagnetic three-state Bell-Lavis model. The results are obtained in a whole interval of chemical potential μ corresponding to the honeycomb phase. Our results demonstrate that the phase transitions might be attributed to the three-state Potts universality class for all μ values except for the edges of the honeycomb phase existence. At the honeycomb phase and the low-density gas phase boundary the transitions become of the first order. At another, honeycomb-to-frustrated phase boundary, we observe the approach to the crossover from the three-state Potts to the Ising model universality class. We also obtain the Schottky anomaly in the specific heat close to this edge. We show that the intermediate planar phase, found in a very similar antiferromagnetic triangular Blume-Capel model, does not occur in the Bell-Lavis model.

Continuous percolation phase transitions of random networks under a generalized Achlioptas process

Using finite-size scaling, we have investigated the percolation phase transitions of evolving random networks under a generalized Achlioptas process (GAP). During this GAP, the edge with a minimum product of two connecting cluster sizes is taken with a probability p from two randomly chosen edges. This model becomes the Erdös-Rényi network at p=0.5 and the random network under the Achlioptas process at p=1. Using both the fixed point of the size ratio s{2}/s{1} and the straight line of lns{1}, where s{1} and s{2} are the reduced sizes of the largest and the second-largest cluster, we demonstrate that the phase transitions of this model are continuous for 0.5 ≤ p ≤ 1. From the slopes of lns{1} and ln(s{2}/s{1})' at the critical point, we get critical exponents β and ν of the phase transitions. At 0.5 ≤ p ≤ 0.8, it is found that β, ν, and s{2}/s{1} at critical point are unchanged and the phase transitions belong to the same universality class. When p ≥ 0.9, β, ν, and s{2}/s{1} at critical point vary with p and the universality class of phase transitions depends on p.

Low-dimensional Yang-Mills theories: Matrix models and emergent geometry

In a simple example of a bosonic three-matrix model, we show how a background geometry can condense as the temperature or coupling constant passes through a critical value. We show that this example belongs to a new universality class of phase transitions where the background geometry is itself emergent.

Development of the critical exponent at the antiferromagnetic phase transition of YbRh2Si2 under chemical pressure

AbstractWe present specific‐heat measurements in the vicinity of the antiferromagnetic phase transition on single crystals of the alloy Yb(Rh1−xCox)2Si2 for x ≤ 0.38. This study was motivated by the violation of critical universality in the undoped YbRh2Si2 [Krellner et al., Phys. Rev. Lett. 102, 196402, 2009] where we have found a large critical exponent α = 0.38. For Co‐doped samples we observe a drastic change of the critical fluctuations resulting in a negative α, explainable within the universality classes of phase transitions. The development of α under chemical pressure gives strong indication that the violation of critical universality in YbRh2Si2 is due to the nearby quantum critical point.

Percolation transition at growing spatiotemporal fractal patterns in models of mesoscopic neural networks

Spike packet propagation is modeled in mesoscopic-scale networks, composed of locally and recurrently coupled neural pools, and embedded in a two-dimensional lattice. Site dynamics is governed by three key parameters--pool connectedness probability, synaptic strength (following the steady-state distribution of some realizations of spike-timing-dependent plasticity learning rule), and the neuron refractoriness. Formation of spatiotemporal patterns in our model, synfire chains, exhibits critical behavior, with the emerging percolation phase transition controlled by the probability for nonzero synaptic strength value. Applying the finite-size scaling method, we infer the critical probability dependence on synaptic strength and refractoriness and determine the effects of connection topology on the pertaining percolation clusters fractal dimensions. We find that the directed percolation and the pair contact process with diffusion constitute the relevant universality classes of phase transitions observed in a class of mesoscopic-scale network models, which may be related to recently reported data on in vitro cultures.

Polarized neutron scattering studies of chiral criticality, and new universality classes of phase transitions

Using a novel polarised neutron scattering technique, the critical exponents for the spin chirality and chiral susceptibility are determined for the triangular lattice antiferromagnet (TLA) CsMnBr 3 in the ranges of reduced temperature τ>10 −3 and τ>7×10 −3, respectively. Their values, β C=0.44(2) and γ C=0.85(3), together with the scaling relation α + 2 β C + γ C = 2.13 ( 9 ) , including the critical exponent where α for the specific heat, prove that the spin-ordering transition belongs to the XY chiral universality class. In the case of helimagnet Ho, it is found that β C - 2 β = 0.14 ( 4 ) , where β is the staggered magnetisation exponent. The scaling relation α + 2 β + γ = 2 could be fulfilled with a reasonable α=0.23(4), although for the chiral critical exponents β C =0.90(2) and γ C =0.69(5) one needs α=−0.49(5) in contradiction with any experimental data. As the scaling relation always holds, we assume that the spin-ordering transition in Ho is of the first order. In the quantum antiferromagnet CsCuCl 3, a triangular spin order coexists with a long-period Dzyaloshinskii helix. The Dzyaloshinskii axial vector should remove the helix chiral degeneracy, which has not been observed in reality. The critical exponent β=0.22(2) is found to be in agreement with the XY chiral scenario for a TLA. Chiral scattering above T N is very weak, probably being masked by zero-point quantum fluctuations. A modulation of the crystal structure with the periodicity of the helix is observed, indicating strong coupling of the Dzyaloshinskii–Moriya interaction with the lattice.

The effect of dislocations on the roughening transition in the weak coupling approximation

The analysis of the roughening transition proposed by Nozieres and Gallet is extended to interfaces threaded with either randomly placed screw dislocations or dislocation loops. In both cases we consider a low density of dislocations which intersect normally with the surface leading to a local deformation of the crystal planes and to the formation of small steps in the interface. We show that dislocation loops do not change the roughening transition significantly: it still remains within the Kosterlitz-Thouless universality class of phase transitions. In contrast, randomly placed screw dislocations modify the phase transition: the Kosterlitz-Thouless transition is rounded on a length scale of order the spacing of dislocations. However there is a transition at a slightly lower temperature for randomly placed screw dislocations: the superroughening transition. This conclusion is only valid if the dislocations are randomly placed, not if they are distributed as closely spaced pairs of opposite sign.

Universal finite-size scaling amplitudes of interfacial free energies in Monte Carlo simulations

The study of the universal finite-size scaling amplitudes of interfacial free energies in Monte Carlo simulations provides an accurate method of determining the critical temperature and the universality class of phase transitions. These universal amplitudes can replace the critical exponents to identify universality classes. The authors use Bennett's optimisation scheme (1976) for the calculation of the interfacial free energies in Monte Carlo simulations. Tests for the Ising and three-state Potts model confirm the effectiveness of this method.