Exponent Nu Research Articles

Machine learning of pair-contact process with diffusion

The pair-contact process with diffusion (PCPD), a generalized model of the ordinary pair-contact process (PCP) without diffusion, exhibits a continuous absorbing phase transition. Unlike the PCP, whose nature of phase transition is clearly classified into the directed percolation (DP) universality class, the model of PCPD has been controversially discussed since its infancy. To our best knowledge, there is so far no consensus on whether the phase transition of the PCPD falls into the unknown university classes or else conveys a new kind of non-equilibrium phase transition. In this paper, both unsupervised and supervised learning are employed to study the PCPD with scrutiny. Firstly, two unsupervised learning methods, principal component analysis (PCA) and autoencoder, are taken. Our results show that both methods can cluster the original configurations of the model and provide reasonable estimates of thresholds. Therefore, no matter whether the non-equilibrium lattice model is a random process of unitary (for instance the DP) or binary (for instance the PCP), or whether it contains the diffusion motion of particles, unsupervised learning can capture the essential, hidden information. Beyond that, supervised learning is also applied to learning the PCPD at different diffusion rates. We proposed a more accurate numerical method to determine the spatial correlation exponent nu _{perp }, which, to a large degree, avoids the uncertainty of data collapses through naked eyes.

Anisotropic scaling for 3D topological models

A proposal to study topological models beyond the standard topological classification and that exhibit breakdown of Lorentz invariance is presented. The focus of the investigation relies on their anisotropic quantum critical behavior. We study anisotropic effects on three-dimensional (3D) topological models, computing their anisotropic correlation length critical exponent nu obtained from numerical calculations of the penetration length of the zero-energy surface states as a function of the distance to the topological quantum critical point. A generalized Weyl semimetal model with broken time-reversal symmetry is introduced and studied using a modified Dirac equation. An approach to characterize topological surface states in topological insulators when applied to Fermi arcs allows to capture the anisotropic critical exponent theta =nu _{x}/nu _{z}. We also consider the Hopf insulator model, for which the study of the topological surface states yields unusual values for nu and for the dynamic critical exponent z. From an analysis of the energy dispersions, we propose a scaling relation nu _{bar{alpha }}z_{bar{alpha }}=2q and theta =nu _{x}/nu _{z}=z_{z}/z_{x} for nu and z that only depends on the Hopf insulator Hamiltonian parameters p and q and the axis direction bar{alpha }. An anisotropic quantum hyperscaling relation is also obtained.

Critical exponents of the O(N)-symmetric phi ^4 model from the varepsilon ^7 hypergeometric-Meijer resummation

We extract the varepsilon -expansion from the recently obtained seven-loop g-expansion for the renormalization group functions of the O(N)-symmetric model. The different series obtained for the critical exponents nu , omega and eta have been resummed using our recently introduced hypergeometric-Meijer resummation algorithm. In three dimensions, very precise results have been obtained for all the critical exponents for N=0,1,2,3 and 4. To shed light on the obvious improvement of the predictions at this order, we obtained the divergence of the specific heat critical exponent alpha for the XY model. We found the result -0.0123(11) which is compatible with the famous experimental result of -0.0127(3) from the specific heat of zero gravity liquid helium superfluid transition while the six-loop Borel with conformal mapping resummation result in literature gives the value -0.007(3). For the challenging case of resummation of the varepsilon -expansion series in two dimensions, we showed that our resummation results reflect a significant improvement to the previous six-loop resummation predictions.

Real-space renormalization-group approach to the random transverse-field Ising model in finite dimensions

The transverse-field Ising models with random exchange interactions in finite dimensions are investigated by means of a real-space renormalization-group method. The scheme yields the exact values of the critical point and critical exponent nu in one dimension and some previous results in the case of random ferromagnetic interactions are reproduced in two and three dimensions. We apply the scheme to spin glasses in transverse fields in two and three dimensions, which have not been analyzed very extensively. The phase diagrams and the critical exponent nu are obtained and evidence for the existence of an infinite-randomness fixed point in these models is found.

Accurate exponents from approximate tensor renormalizations

We explain the recent numerical successes obtained by Tao Xiang's group, who developed and applied Tensor Renormalization Group methods for the Ising model on square and cubic lattices, by the fact that their new truncation method sharply singles out a surprisingly small subspace of dimension two. We show that in the two-state approximation, their transformation can be handled analytically yielding a value 0.964 for the critical exponent nu much closer to the exact value 1 than 1.338 obtained in the Migdal-Kadanoff approximation. We propose two alternative blocking procedures that preserve the isotropy and improve the accuracy to nu=0.987 and 0.993 respectively. We discuss applications to other classical lattice models, including models with fermions, and suggest that it could become a competitor for Monte Carlo methods suitable to calculate accurately critical exponents, take continuum limits and study near-conformal systems in arbitrarily large volumes.

Thermal metal-insulator transition in a helical topological superconductor

Two-dimensional superconductors with time-reversal symmetry have a Z_2 topological invariant, that distinguishes phases with and without helical Majorana edge states. We study the topological phase transition in a class-DIII network model, and show that it is associated with a metal-insulator transition for the thermal conductance of the helical superconductor. The localization length diverges at the transition with critical exponent nu approx 2.0, about twice the known value in a chiral superconductor.

Multifractal finite-size scaling and universality at the Anderson transition

We describe a new multifractal finite size scaling (MFSS) procedure and its application to the Anderson localization-delocalization transition. MFSS permits the simultaneous estimation of the critical parameters and the multifractal exponents. Simulations of system sizes up to L^3=120^3 and involving nearly 10^6 independent wavefunctions have yielded unprecedented precision for the critical disorder W_c=16.530 (16.524,16.536) and the critical exponent nu=1.590 (1.579,1.602). We find that the multifractal exponents Delta_q exhibit a previously predicted symmetry relation and we confirm the non-parabolic nature of their spectrum. We explain in detail the MFSS procedure first introduced in our Letter [Phys. Rev. Lett. 105, 046403 (2010)] and, in addition, we show how to take account of correlations in the simulation data. The MFSS procedure is applicable to any continuous phase transition exhibiting multifractal fluctuations in the vicinity of the critical point.

Numerical study of critical properties and hidden orders in dimerized spin ladders

Dimerized antiferromagnetic spin-1/2 ladders are known to exhibit a quantum critical phase transition in the ground state, the existence or absence of which is dependent on the dimerization pattern of the ladder. The gapped phases cannot be distinguished by the conventional Landau long-range order parameter. However, they possess a non-local (hidden) string order parameter, which is non-zero in one phase and vanishes in the other. We use an exact diagonalization technique to calculate ground state energies, energy gaps and string order parameters of dimerized two- and three-leg Heisenberg ladders, as well as a critical scaling analysis to yield estimates of the critical exponents nu and beta.

Scaling behavior of explosive percolation on the square lattice

Clusters generated by the product-rule growth model of Achlioptas, D'Souza, and Spencer on a two-dimensional square lattice are shown to obey qualitatively different scaling behavior than standard (random growth) percolation. The threshold with unrestricted bond placement (allowing loops) is found precisely using several different criteria based on both moments and wrapping probabilities, yielding p(c)=0.526 565 ± 0.000005, consistent with the recent result of Radicchi and Fortunato. The correlation-length exponent ν is found to be close to 1. The qualitative difference from regular percolation is shown dramatically in the behavior of the percolation probability P∞ (size of largest cluster), of the susceptibility, and of the second moment of finite clusters, where discontinuities appear at the threshold. The critical cluster-size distribution does not follow a consistent power law for the range of system sizes we study (L ≤ 8192) but may approach a power law with τ>2 for larger L .

Generalized Lifshitz-Kosevich scaling at quantum criticality from the holographic correspondence

We characterize quantum oscillations in the magnetic susceptibility of a quantum critical non-Fermi liquid. The computation is performed in a strongly interacting regime using the nonperturbative holographic correspondence. The temperature dependence of the amplitude of the oscillations is shown to depend on a critical exponent nu. For general nu the temperature scaling is distinct from the textbook Lifshitz-Kosevich formula. At the `marginal' value nu = 1/2, the Lifshitz-Kosevich formula is recovered despite strong interactions. As a by-product of our analysis we present a formalism for computing the amplitude of quantum oscillations for general fermionic theories very efficiently.

Vortex glass scaling in Pb-doped Bi-2223 single crystal

We report on study of the vortex liquid in Pb-doped Bi-2223 single crystal using the in-plane resistivity measurements as a function of temperature and magnetic field up to 6T applied perpendicular to CuO planes. Below Tc at the upper part of superconducting transition we found Arrhenius-like resistivity behavior. With further temperature decrease close to resistivity onset resistivity shows power law dependence on temperature signaling approaching vortex-glass transition. The critical exponents nu(z-1)=4.6 plus-minus 0.5 are found to be field independent within experimental errors. We also present magnetic phase diagram defining region of nonzero critical current for Pb-doped Bi-2223 single crystal.

Individual bottle brush molecules in dense 2D layers restoring high degree of extension after collapse-decollapse cycle: Directly measured scaling exponent

We prepared dense films of adsorbed brush-like macromolecules on mica substrate by transfer of compressed Langmuir monolayers from water subphase. The main macromolecular contours in the dense films were clearly resolved by SFM. The films were subjected to successive treatments by ethanol and water vapours. In accordance with previous results for isolated macromolecules, the films underwent collapse and subsequent decollapse morphological transformations in the changing vapour environment. Statistical analysis of the macromolecular dimensions in the films allowed us to measure the values of the scaling exponent nu determining the correlation between mean lateral and linear dimensions of the macromolecules. The analysis showed that the macromolecular conformations in the film as transferred were similar to the previously described conformations of the same macromolecules deposited directly on mica as isolated chains at much lower surface densities. The determined nu was close to the 0.75 value corresponding to the 2D SAW statistics. We assumed that the molecules retained the high degree of extension during the compression step due to suppressed reorganisation of the side chains. Differently from previous observations for isolated macromolecules, the restored conformations in the dense films after collapse-decollapse cycle were more extended with the nu of about 0.73 value. A theoretical explanation of the high degree of re-extension is proposed.

Percolation threshold parameters of fluids

Extensive Monte Carlo simulations on three qualitatively different model supercritical fluids (square-well fluid, Lennard-Jonesium, and primitive water) have been performed to examine percolation threshold parameters for continuum (correlated) models and their relation to general results valid for random lattice models; random-site percolation simple-cubic lattice has therefore been considered as well. Two different bond criteria, the configurational and self-bound ones, defining a cluster have been used. In addition to the percolation threshold occupation probability pc and the percolation threshold fluid density rhoc, the correlation length exponent nu and the wrapping probability at the percolation threshold Rw,c have also been evaluated. It is found that parameters nu and Rw,c exhibit not only strong temperature dependence but also, unlike the case of lattice systems, dependence on the nature of the system considered and the employed definition of the cluster.

Structure, Dynamic Properties, and Phase Transitions of Tethered Membranes

A coarse-grained model of a self-avoiding tethered membrane with hexagonal coordination, embedded in three-dimensional space, is studied by means of extensive Monte Carlo computer simulations. The simulations are performed at various temperatures for membranes with linear size 5 < or =L< or = 30. We find that the membrane undergoes several folding transitions from a high-temperature flat phase to multiple-folded structure as the temperature is steadily decreased. Using a suitable order parameter and finite size scaling analysis, these phase transitions are shown to be of first order. The equilibrium shape of the membranes is analyzed by calculating the eigenvalues lambda(2) (max)> or =lambda(2) (med)> or =lambda(2) (min) of the inertia tensor. We present a systematic finite size scaling analysis of the radius of gyration and the eigenvalues of the inertia tensor at different phases of the observed folding transitions. In the high temperature flat phase, the radius of gyration R(g) grows with the linear size of the membrane L as R(g) proportional, variant L(nu), where the exponent nu approximately 1.0. The eigenvalues of the inertia tensor scale as lambda(max) proportional, variant lambda(med) proportional, variant L(nu) and lambda(min) proportional variant L(nu min) whereby the roughness exponent nu(min) approximately 0.7. We also find that the Rouse relaxation time tau(R) of a self-avoiding membrane scales as tau(R) proportional, variant L(2nu+2), in good agreement with the theoretical predictions.

Critical dynamics of the two-dimensional random-bond Potts model with nonequilibrium Monte Carlo simulations

We study two-dimensional q -state random-bond Potts models for both q=8 and q=5 with a linearly varying temperature. By applying a successive Monte Carlo renormalization group procedure, both the static and dynamic critical exponents are obtained for randomness amplitudes (the strong to weak coupling ratio) of r_{0}=3 , 10, 15, and 20. The correlation length exponent nu increases with disorder from less than to larger than unity and this variation is justified by the good collapse of the specific heat near the critical region. The specific heat exponent is obtained by the usual hyperscaling relation alpha=2-dnu and thus indicates no possibility of the activated dynamic scaling. Both r_{0} and q have effects on the critical dynamics of the disordered systems, which can be seen from variations of the rate exponent, the hysteresis exponent, and the dynamic critical exponent. Implications of these results are discussed.

Conformation of Circular DNA in Two Dimensions

The conformation of circular DNA molecules of various lengths adsorbed in a 2D conformation on a mica surface is studied. The results confirm the conjecture that the critical exponent nu is topologically invariant and equal to the self-avoiding walk value (in the present case nu=3/4), and that the topology and dimensionality of the system strongly influence the crossover between the rigid regime and the self-avoiding regime at a scale L approximately 7l{p}. Additionally, the bond correlation function scales with the molecular length L as predicted. For molecular lengths L<or=5l{p}, circular DNA behaves like a stiff molecule with an approximately elliptic shape.

Thermal fluctuations and phase diagrams of the phase-field crystal model with pinning

We study the influence of thermal fluctuations in the phase diagram of a recently introduced two-dimensional phase field crystal model with an external pinning potential. The model provides a continuum description of pinned lattice systems allowing for both elastic deformations and topological defects. We introduce a nonconserved version of the model and determine the ground-state phase diagram as a function of lattice mismatch and strength of the pinning potential. Monte Carlo simulations are used to determine the phase diagram as a function of temperature near commensurate phases. The results show a rich phase diagram with commensurate, incommensurate, and liquidlike phases with a topology strongly dependent on the type of ordered structure. A finite-size scaling analysis of the melting transition for the c(2x2) commensurate phase shows that the thermal correlation length exponent nu and specific heat behavior are consistent with the Ising universality class as expected from analytical arguments.

Topological effect in ring polymers investigated with Monte Carlo simulation

We studied equilibrium conformations of ring polymers in the melt over the wide range of segment number up to 1000 by the Monte Carlo simulations and the bond fluctuation model, and estimated Flory's scaling exponent nu. The radial distribution function of segments for the ring polymers in the melt is obtained. We have found that nu for ring polymers is decreased with increasing segment number N, and nu goes down to 0.365 when N reaches 1000, whose value is apparently smaller than the theoretically predicted one, i.e., 25. Those values are in contrast to the well established nu value of 0.5 for linear polymers in the melt. This is because ring polymer chains in the melt are squeezed both by their own topological effect and the compression effect by the neighboring ring polymer coils which are also squeezed at bulk state. The difference in our result and the theory may be due to the fact that the estimation of topological entropy loss was ignored in the theoretical prediction, while it has been taken into consideration in the present study. If polymer coils repel each other in melt at N --> infinity, we have the limiting nu value of 13, so we conclude that nu is in the range of 13 < or = nu << 0.365 when the molecular weight of a ring polymer is high enough.

Defect Production in Nonlinear Quench across a Quantum Critical Point

We show that the defect density n, for a slow nonlinear power-law quench with a rate tau(-1) and an exponent alpha>0, which takes the system through a critical point characterized by correlation length and dynamical critical exponents nu and z, scales as n approximately tau(-alphanud/(alphaznu+1)) [n approximately (alphag((alpha-1)/alpha)/tau)(nud/(znu+1))] if the quench takes the system across the critical point at time t=0 [t=t(0) not = 0], where g is a nonuniversal constant and d is the system dimension. These scaling laws constitute the first theoretical results for defect production in nonlinear quenches across quantum critical points and reproduce their well-known counterpart for a linear quench (alpha=1) as a special case. We supplement our results with numerical studies of well-known models and suggest experiments to test our theory.

Duffing oscillators: Control and memory effects

In the first part of this article we study the hysteretic bistable response of Duffing oscillators and show ways to control the switching between stable branches of this nonlinear response. The control mechanism is either applied through a pulse that can be in phase or out of phase with the periodic driving force or through a frequency-modulated driving force. In the second part we show how memory effects in dissipation qualitatively and quantitatively alter the dynamics of Duffing oscillators. We show how memory functions corresponding to different dissipative regimes (diffusion, subdiffusion, and superdiffusion) affect the oscillator. In particular, we obtain universal power laws for the absoption when the driving frequency omega-->0 . For subdiffusive memories the power law exponents nu<2 , for diffusive memories nu=2 , and for superdiffusive memories nu>2 .