Finite-time Scaling Analysis Research Articles

Universal dynamical scaling laws in three-state quantum walks.

We perform a finite-time scaling analysis over the detrapping point of a three-state quantum walk on the line. The coin operator is parametrized by ρ that controls the wave packet spreading velocity. The input state prepared at the origin is set as a symmetric linear combination of two eigenstates of the coin operator with a characteristic mixing angle θ, one of them being the component that results in full spreading occurring at θ_{c}(ρ) for which no fraction of the wave packet remains trapped near the initial position. We show that relevant quantities, such as the survival probability and the participation ratio assume single parameter scaling forms at the vicinity of the detrapping angle θ_{c}. In particular, we show that the participation ratio grows linearly in time with a logarithmic correction, thus, shedding light on previous reports of sublinear behavior.

Short-time Monte Carlo simulation of the majority-vote model on cubic lattices

We perform short-time Monte Carlo simulations to study the criticality of the isotropic two-state majority-vote model on cubic lattices of volume N=L3, with L up to 2048. We obtain the precise location of the critical point by examining the scaling properties of a new auxiliary function Ψ. We perform finite-time scaling analysis to accurately calculate the whole set of critical exponents, including the dynamical critical exponent z=2.027(9), and the initial slip exponent θ=0.1081(1). Our results indicate that the majority-vote model in three dimensions belongs to the same universality class of the three-dimensional Ising model.

One-dimensional Absorbing phase transition in the fermionic parity-conserving particle process with second neighbors branching

By means of Monte Carlo procedure this article analyzes a stochastic process in one-dimensional model. The parity-conserving particle process with infinite annihilation and branching of the two offspring to the second neighbors, each one at each side. The model was firstly considered in Takayasu and Tretyakov (1991) and Jensen (1994) but branching of four offspring to four first neighbors. It has also been solved in Zhong and ben-Avraham (1995) with branching of the two particles to the first neighbors, each one at each side, but with finite annihilation. Now the model has been studied with infinite annihilation in presence of only second neighbors branching of two particles, one to left and the other to the right. Simulations were performed on a regular one dimension lattice in order to determine the threshold of absorbing phase transition. From steady state simulations and finite time scaling analysis, it is shown that the transition exists and critical exponents are obtained. The exponents are found to differ from those of the directed percolation (DP) universality class and are the same found in Takayasu and Tretyakov (1991), Jensen (1994) and Zhong and ben-Avraham (1995). It is also calculated the short-time relaxation dynamical critical exponents and checked the consistence of the related scaling relation.

Absorbing phase transition in a parity-conserving particle process on a Sierpinski carpet fractal

We study a stochastic lattice model with parity-conserving particle process using a Monte Carlo procedure. We perform simulations on a Sierpinski carpet fractal with dimension . We calculate the critical exponents at the threshold of the absorbing phase transition at the known value for the critical diffusion p c = 1 (Cardy and Tauber 1996 Phys. Rev. Lett. 77 4780). Using finite-size and finite-time scaling analysis we calculate the critical exponents at p c = 1 and below, where a finite density of particles is found in the long-time limit. From dynamic simulations we calculate the dynamical exponents Z, , , and , and they are found to differ from the mean-field values, as well as the stationary exponent . We check the consistence of the results with the hyperscaling relation.

Dynamical Ginzburg criterion for the quantum-classical crossover of the Kibble-Zurek mechanism

We introduce a simple criterion for lattice models to predict quantitatively the crossover between the classical and the quantum scaling of the Kibble-Zurek mechanism, as the one observed in a quantum $\phi^4$-model on a 1D lattice [Phys. Rev. Lett. 116, 225701 (2016)]. We corroborate that the crossover is a general feature of critical models on a lattice, by testing our paradigm on the quantum Ising model in transverse field for arbitrary spin-$s$ ($s \geq 1/2$) in one spatial dimension. By means of tensor network methods, we fully characterize the equilibrium properties of this model, and locate the quantum critical regions via our dynamical Ginzburg criterion. We numerically simulate the Kibble-Zurek quench dynamics and show the validity of our picture, also according to finite-time scaling analysis.

Absorbing phase transition in the three dimensional fermionic parity-conserving particle process

By use of a Monte Carlo procedure we analyze a stochastic lattice model with parity-conserving particle process, previously considered in Takayasu and Tretyakov (1991). We perform simulations on a regular three-dimensional (3D) lattice in order to determine the threshold of absorbing phase transition. A finite-time scaling analysis is employed to calculate the critical exponents at the critical diffusion probability pc, below which a finite density of particles is developed in the long-time limit. From steady state simulations and finite-time scaling analysis we obtained these critical exponents and found them to differ from those of the 3D directed percolation (DP) universality class. We also calculated the short-time relaxation dynamical critical exponents and checked the consistence with the hyperscaling relation.

Hamiltonian short-time critical dynamics of the three-dimensional XY model.

The short-time relaxation critical behavior of the XY model on a simple-cubic lattice is investigated within the scope of deterministic Hamiltonian dynamics. The Hamiltonian includes a first-neighbor interaction between planar vectors and a rotational kinetic term from which the motion equations are derived. The dynamical evolution from a fully ordered initial state is followed by employing a symplectic algorithm based on a high-order Trotter-Suzuki decomposition of the time-evolution operator. A finite-time scaling analysis is performed to provide accurate estimates of the critical energy density, the order-parameter relaxation exponent, and the dynamical critical exponent. The estimated critical exponents are consistent with prior theoretical and experimental values reported for the superfluid ^{4}He, extreme type-II superconducting, and Bose-Einstein condensation transitions.

Critical spreading dynamics of parity conserving annihilating random walks with power-law branching

We investigate the critical spreading of the parity conserving annihilating random walks model with Lévy-like branching. The random walks are considered to perform normal diffusion with probability p on the sites of a one-dimensional lattice, annihilating in pairs by contact. With probability 1−p, each particle can also produce two offspring which are placed at a distance r from the original site following a power-law Lévy-like distribution P(r)∝1∕rα. We perform numerical simulations starting from a single particle. A finite-time scaling analysis is employed to locate the critical diffusion probability pc below which a finite density of particles is developed in the long-time limit. Further, we estimate the spreading dynamical exponents related to the increase of the average number of particles at the critical point and its respective fluctuations. The critical exponents deviate from those of the counterpart model with short-range branching for small values of α. The numerical data suggest that continuously varying spreading exponents sets up while the branching process still results in a diffusive-like spreading.

Finite-time scaling at the Anderson transition for vibrations in solids

A model in which a three-dimensional elastic medium is represented by a network of identical masses connected by springs of random strengths and allowed to vibrate only along a selected axis of the reference frame, exhibits an Anderson localization transition. To study this transition, we assume that the dynamical matrix of the network is given by a product of a sparse random matrix with real, independent, Gaussian-distributed non-zero entries and its transpose. A finite-time scaling analysis of system's response to an initial excitation allows us to estimate the critical parameters of the localization transition. The critical exponent is found to be $\nu = 1.57 \pm 0.02$ in agreement with previous studies of Anderson transition belonging to the three-dimensional orthogonal universality class.

Coherent Backscattering Reveals the Anderson Transition.

We develop an accurate finite-time scaling analysis of the angular width of the coherent backscattering (CBS) peak for waves propagating in 3D random media. Applying this method to ultracold atoms in optical speckle potentials, we show how to determine both the mobility edge and the critical exponent of the Anderson transition from the temporal behavior of the CBS width. Our method could be used in experiments to fully characterize the 3D Anderson transition.

Nonequilibrium Relaxation Analysis of the Spin-Glass Correlation Length

Relaxation functions of the spin-glass correlation length are investigated in the three-dimensional ±J XY model. The algebraic divergence is observed at the spin-glass transition temperature, T sg ≃ 0.45. The finite-time scaling analysis provides results consistent with our previous scaling analysis of the spin-glass susceptibility. The relaxation functions in the low-temperature phase exhibit the same behavior with the Ising spin glass: the exponent is linearly dependent on the temperature as 1/z(T) ≃ 0.21T/T sg . Relaxation functions of the chiral-glass correlation length do not exhibit such behavior as the spin-glass ones do. All the findings suggest that the transition is driven by the spin degrees of freedom.

Nonvanishing spin-glass transition temperature of the±JXYmodel in three dimensions

A three-dimensional $\pm J$ XY spin-glass model is investigated by a nonequilibrium relaxation method. We have introduced a new criterion for the finite-time scaling analysis. A transition temperature is obtained by a crossing point of obtained data. The scaling analysis on the relaxation functions of the spin-glass susceptibility and the chiral-glass susceptibility shows that both transitions occur simultaneously. The result is checked by relaxation functions of the Binder parameters and the glass correlation lengths of the spin and the chirality. Every result is consistent if we consider that the transition is driven by the spin degrees of freedom.

Critical exponents of isotropic-hexatic phase transition in the hard-disk system.

The hard-disk system is studied by observing the nonequilibrium relaxation behavior of a bond-orientational order parameter. The density dependence of characteristic relaxation time tau is estimated from the finite-time scaling analysis. The critical point between the fluid and the hexatic phase is refined to be 0.899 (1) by assuming the divergence behavior of the Kosterlitz-Thouless transition. The value of the critical exponent eta is also studied by analyzing the fluctuation of the order parameter at the criticality and estimated as eta=0.25 (2). These results are consistent with the prediction by the Kosterlitz-Thouless-Halperin-Nelson-Young theory.

Nonequilibrium relaxation analysis of two-dimensional melting.

The phase diagram of a hard-disk system is studied by observing nonequilibrium relaxation functions of a bond-orientational order parameter using particle dynamics simulations. From a finite-time scaling analysis, two Kosterlitz-Thouless transitions can be observed when the density is increased from the isotropic fluid phase to closest packing. The transition densities are estimated to be 0.901(2) and 0.910(2), where the density denotes the fraction of area occupied by the particles, the density is normalized to one for the quadratic packing configuration. These observations are consistent with the predictions of the Kosterlitz-Thouless-Halperin-Nelson-Young theory.

A Spin-Glass and Chiral-Glass Transition in a ±JHeisenberg Spin-Glass Model in Three Dimensions

The three-dimensional $\pm J$ Heisenberg spin-glass model is investigated by the non-equilibrium relaxation method from the paramagnetic state. Finite-size effects in the non-equilibrium relaxation are analyzed, and the relaxation functions of the spin-glass susceptibility and the chiral-glass susceptibility in the infinite-size system are obtained. The finite-time scaling analysis gives the spin-glass transition at $T_{\rm sg}/J=0.21_{-0.02}^{+0.01}$ and the chiral-glass transition at $T_{\rm cg}/J=0.22_{-0.03}^{+0.01}$. The results suggest that both transitions occur simultaneously. The critical exponent of the spin-glass susceptibility is estimated as $\gamma_{\rm sg}= 1.7 \pm 0.3$, which makes an agreement with the experiments of the insulating and the canonical spin-glass materials.