Dynamical Exponent Research Articles

Role of finite probe size in measuring growth exponent in film deposition

We use computer simulations to investigate the effects of the tip diameter of an electrostatic force microscope (EFM) operating at a constant force on the extraction of the growth exponent β during film growing in a one-dimensional substrate. Laplace’s equation is solved in the EFM simulation using the finite element method to determine the electrostatic force between the tip and the film interface. Importantly, for EFM tips with sufficiently large apex diameters, the topographies calculated with EFM and those computed with the transformed mean height profile (TMHP) method, where the interface is divided into bins of the same tip diameter size and the average height within each bin is used to transform the original interface, are almost identical. This was shown in the context of lattice models of the Kardar–Parisi–Zhang (KPZ) and Villain–Lai–Das–Sarma (VLDS) classes. The global roughness of the film surface, W, scales with the diameter of the EFM tip, ε, as W/a=(ε/a)αg[Ψ], where a is the lattice parameter, α is the KPZ/VLDS roughness exponent, and g is a universal scaling function of the argument Ψ≡t/(ε/a)z, where t and z are the reduced time of deposition and the KPZ/VLDS dynamic exponent, respectively. These results provide a limit for ε from which a KPZ/VLDS growth exponent can be reliably determined with EFM at a constant force. When the EFM tip diameter is larger than the surface correlation length, a misleading effective growth exponent consistent with uncorrelated growth is found.

Bulk-Boundary Correspondence and Singularity-Filling in Long-Range Free-Fermion Chains

The bulk-boundary correspondence relates topologically protected edge modes to bulk topological invariants and is well understood for short-range free-fermion chains. Although case studies have considered long-range Hamiltonians whose couplings decay with a power-law exponent $\ensuremath{\alpha}$, there has been no systematic study for a free-fermion symmetry class. We introduce a technique for solving gapped, translationally invariant models in the 1D BDI and AIII symmetry classes with $\ensuremath{\alpha}>1$, linking together the quantized winding invariant, bulk topological string-order parameters, and a complete solution of the edge modes. The physics of these chains is elucidated by studying a complex function determined by the couplings of the Hamiltonian: in contrast to the short-range case where edge modes are associated to roots of this function, we find that they are now associated to singularities. A remarkable consequence is that the finite-size splitting of the edge modes depends on the topological winding number, which can be used as a probe of the latter. We furthermore generalize these results by (i) identifying a family of BDI chains with $\ensuremath{\alpha}<1$ where our results still hold and (ii) showing that gapless symmetry-protected topological chains can have topological invariants and edge modes when $\ensuremath{\alpha}\ensuremath{-}1$ exceeds the dynamical critical exponent.

Nonsteady dynamics at the dynamic depinning transition in the two-dimensional Gaussian random-field Ising model.

With large-scale Monte Carlo simulations, we investigate the nonsteady relaxation at the dynamic depinning transition in the two-dimensional Gaussian random-field Ising model. The dynamic scaling behavior is carefully analyzed, and the transition fields as well as static and dynamic exponents are accurately determined based on the short-time dynamic scaling form. Different from the usual assumption, two distinguished growth processes of spatial correlation lengths for the velocity and height of the domain wall are found. Thus, the universality class of the depinning transition is established, which significantly differs from that of the quenched disorder equationbut agrees with that of the recent experiment as well as other simulations works. Under the influence of the mesoscopic time regime, the crossover from the second-order phase transition to the first-order one is confirmed in the weak-disorder regime, yielding an abnormal disorder-dependent nature of the criticality.

Discrete Laplacian thermostat for spin systems with conserved dynamics

A well-established numerical technique to study the dynamics of spin systems in which symmetries and conservation laws play an important role is to microcanonically integrate their reversible equations of motion, obtaining thermalization through initial conditions drawn with the canonical distribution. In order to achieve a more realistic relaxation of the magnetic energy, numerically expensive methods that explicitly couple the spins to the underlying lattice are normally employed. Here we introduce a stochastic conservative thermostat that relaxes the magnetic energy while preserving the constant of motions, thus turning microcanonical spin dynamics into a conservative canonical dynamics, without actually simulating the lattice. We test the thermostat on the Heisenberg antiferromagnet in $d=3$ and show that the method reproduces the exact values of the static and dynamic critical exponents, while in the low-temperature phase it yields the correct spin wave phenomenology. Finally, we demonstrate that the relaxation coefficient of the new thermostat is quantitatively connected to the microscopic parameters of the spin-lattice coupling.

Anchored advected interfaces, Oslo model, and roughness at depinning

There is a plethora of one-dimensional advected systems with an absorbing boundary: the Toom model of anchored interfaces, the directed exclusion process where in addition to diffusion particles and holes can jump over their right neighbour, simple diffusion with advection, and Oslo sandpiles. All these models share a roughness exponent of , while the dynamic exponent z varies, depending on the observable. We show that for the first three models z = 1, z = 2, and are realized, depending on the observable. The Oslo model is apart with a conjectured dynamic exponent of . Since the height in the latter is the gradient of the position of a disordered elastic string, this shows that for a driven elastic string at depinning.

Traversing the nucleation-growth landscape through heterogeneous random walks.

The nucleation-growth process is a crucial component of crystallization. While previous theoretical models have focused on nucleation events and postnucleation growth, such as the classical nucleation theory and Lifshitz-Slyozov-Wagner model, recent advancements in experiments and simulations have highlighted the inability of classical models to explain the transient dynamics during the early development of nanocrystals. To address these shortcomings, we present a model that describes the nucleation-growth dynamics of individual nanocrystals as a series of reversible chain reactions, with the free energy landscape extended to include activation-adsorption-relaxation reaction pathways. By using the Monte Carlo method based on the transition state theory, we simulate the crystallization dynamics. We derive a Fokker-Planck formalism from the master equation to describe the nucleation-growth process as a heterogeneous random walk on the extended free energy landscape with activated states. Our results reveal the transient quasiequilibrium of the prenucleation stage before nucleation starts, and we identify a postnucleation crossover regime where the dynamic growth exponents asymptotically converge towards classical limits. Additionally, we generalize the power laws to address the dimension and scale effects for the growth of large crystals.

Entanglement wedge cross section growth during thermalization

Motivated by exploring the thermalization process in relativistic and non-relativistic holographic field theories after a non-local quench, we investigate some features in the time evolution of the entanglement wedge cross section (EWCS). This quantity is a possible holographic dual to some non-local information measures such as entanglement of purification. In particular, we focus on the time dependence of EWCS during black hole formation in D+2 dimensional AdS spacetime as well as geometries with Lifshitz and hyperscaling violating exponents. A combination of analytic and numerical results for large symmetric strip shaped boundary subregions shows that the scaling of EWCS at early times only depends on the Lifshitz exponent. In addition, this early growth regime is followed by a linear growth regime whose velocity depends on the dimensions of spacetime, the Lifshitz exponent, and the hyperscaling parameter. This velocity is the same as the entanglement velocity and for nontrivial dynamical exponent depends on the temperature of the final equilibrium state.

Quantum phase transition in NbN superconducting thin films

We systematically investigated the low-temperature transport properties of a series of NbN epitaxial films with thickness $t$ ranging from $\sim$2.0 to $\sim$4.0 nm. The films undergo a superconductor-insulator transition (SIT) with decreasing film thickness, and the critical sheet resistance for the SIT is close to the quantum resistance of Cooper pairs $h/4e^2$ (6.45 k$\Omega$). Besides the Berezinski-Koterlitz-Thouless transition, a magnetic-field-driven SIT is observed in those two-dimensional (2D) superconducting films (2.6 nm $\lesssim t \lesssim 4.0$ nm). Interestingly, it is found that the low-temperature magnetoresistance isotherms do not cross at a single fixed point but at a well-distinguished region for these superconducting films. The dynamical critical exponent obtained by analyzing these magnetoresistance isotherms is divergent as the quantum critical point is being approached. The behavior of the dynamical critical exponent, originating from quenched disorder at ultralow temperatures, provides direct evidence for the occurrence of quantum Griffiths singularity in the quantum phase transition process of the films. The field-driven anomalous metal (quantum metal) state does not appear in these films. Our results suggest that the quantum Griffiths singularity not only occurs in the highly crystalline 2D superconductors with superconductor-metal transition but also in those with SIT.

Loop current fluctuations and quantum critical transport

We study electrical transport at quantum critical points (QCPs) associated with loop current ordering in a metal, focusing specifically on models of the “Hertz-Millis” type. At the infrared (IR) fixed point and in the absence of disorder, the simplest such models have infinite DC conductivity and zero incoherent conductivity at nonzero frequencies. However, we find that a particular deformation, involving NN species of bosons and fermions with random couplings in flavor space, admits a finite incoherent, frequency-dependent conductivity at the IR fixed point, \sigma(\omega>0)\sim\omega^{-2/z}σ(ω>0)∼ω−2/z, where zz is the boson dynamical exponent. Leveraging the non-perturbative structure of quantum anomalies, we develop a powerful calculational method for transport. The resulting "anomaly-assisted large NN expansion" allows us to extract the conductivity systematically. Although our results imply that such random-flavor models are problematic as a description of the physical N = 1N=1 system, they serve to illustrate some general conditions for quantum critical transport as well as the anomaly-assisted calculational methods. In addition, we revisit an old result that irrelevant operators generate a frequency-dependent conductivity, \sigma(\omega>0) \sim \omega^{-2(z-2)/z}σ(ω>0)∼ω−2(z−2)/z, in problems of this kind. We show explicitly, within the scope of the original calculation, that this result does not hold for any order parameter.

Observation of magnetic structural universality and jamming transition with NMR

Nuclear magnetic resonance (NMR) has been instrumental in deciphering the structure of proteins. Here we show that transverse NMR relaxation, through its time-dependent relaxation rate, is distinctly sensitive to the structure of complex materials or biological tissues at the mesoscopic scale, from micrometers to tens of micrometers. Based on the ideas of universality, we show analytically and numerically that the time-dependent transverse relaxation rate approaches its long-time limit in a power-law fashion, with the dynamical exponent reflecting the universality class of mesoscopic magnetic structure. The spectral line shape acquires the corresponding non-analytic power law singularity at zero frequency. We experimentally detect the change in the dynamical exponent as a result of the transition into maximally random jammed state characterized by hyperuniform correlations. The relation between relaxational dynamics and magnetic structure opens the way for noninvasive characterization of porous media, complex materials and biological tissues

Dynamic multiscaling in stochastically forced Burgers turbulence

We carry out a detailed study of dynamic multiscaling in the turbulent nonequilibrium, but statistically steady, state of the stochastically forced one-dimensional Burgers equation. We introduce the concept of interval collapse time, which we define as the time taken for a spatial interval, demarcated by a pair of Lagrangian tracers, to collapse at a shock. By calculating the dynamic scaling exponents of the moments of various orders of these interval collapse times, we show that (a) there is not one but an infinity of characteristic time scales and (b) the probability distribution function of the interval collapse times is non-Gaussian and has a power-law tail. Our study is based on (a) a theoretical framework that allows us to obtain dynamic-multiscaling exponents analytically, (b) extensive direct numerical simulations, and (c) a careful comparison of the results of (a) and (b). We discuss possible generalizations of our work to higher dimensions, for the stochastically forced Burgers equation, and to other compressible flows that exhibit turbulence with shocks.

Wetting transition in the transverse-field spin- 12 XY model with boundary fields

The wetting transition in the transverse-field spin-$\frac{1}{2}$ XY model with opposite boundary fields ${h}_{L}^{x}{h}_{R}^{x}<0$ is studied analytically and numerically. We find that the phase diagram is complex and that the wetting transition is of three types: first, second, and fourth order. The energy gap is obtained analytically, and the magnetization profile, correlation functions, and wetting layer thickness are obtained numerically. For $|{h}_{L}^{x}|,|{h}_{R}^{x}|<{h}_{w}$, a first-order phase transition occurs at ${h}_{L}^{x}=\ensuremath{-}{h}_{R}^{x}$, where ${h}_{w}$ is the continuous wetting transition point. For $|{h}_{R}^{x}|$ larger than ${h}_{w}$, the continuous wetting transition occurs at ${h}_{L}^{x}={h}_{w}$, and vice versa. For $g\ensuremath{\ne}1\ensuremath{-}{\ensuremath{\gamma}}^{2}$, the wetting transition is second order, and commensurate and incommensurate phases occur for $g<1\ensuremath{-}{\ensuremath{\gamma}}^{2}$ and $g>1\ensuremath{-}{\ensuremath{\gamma}}^{2}$, respectively. For $g=1\ensuremath{-}{\ensuremath{\gamma}}^{2}$, the wetting transition is fourth order. For this fourth-order phase transition, the third derivative of the surface magnetization oscillates and diverges near the transition point. The correlation length exponent is $\ensuremath{\nu}=2$, and the dynamic exponent is $z=2$. Thus, this fourth-order transition belongs to a new universality class. The wetting behavior is induced by asymmetric boundary fields ${h}_{L}^{x}=\ensuremath{-}{h}_{R}^{x}$ and its finite-size scaling is discussed.

Experimental critical dynamics of 3-methyl pyridine/D2O mixtures without and with antagonistic salt

We observed criticality in the structure and dynamics of a three-dimensional (3D) and two-dimensional (2D) Ising system consisting of 3-methyl pyridine $\text{(3MP)}/{\mathrm{D}}_{2}\mathrm{O}$ without and with antagonistic salt. We could describe both dynamic criticalities by the Kawasaki crossover function. The dynamic critical exponent was $z=0.063\ifmmode\pm\else\textpm\fi{}0.020$ and $0.005\ifmmode\pm\else\textpm\fi{}0.019$ for three and two dimensions, which confirms earlier observations in the 3D case and confirms expectations in the 2D case. The amplitudes of the critical dynamics are governed by the bare viscosities experimentally, and by the coefficient $R$ theoretically [the latter is proportional to ${(4\ensuremath{-}d)}^{\ensuremath{-}1}$ with the dimensionality $d$]. This finding is in accordance with the lubrication effect [N. Gov, A. G. Zilman, and S. Safran, Phys. Rev. E 70, 011104 (2004)], which is also connected to lamellar systems of the Brazovskii criticality [M. Gvaramia et al., Colloid Polym. Sci. 297, 1507 (2019)]. This lubrication effect is tightly connected to a laminar flow enforced by the domain structure, and it also holds for our 2D Ising system. The experimental techniques employed were small-angle neutron and x-rays scattering for the static criticality, and dynamic light scattering and neutron spin echo spectroscopy for the critical dynamics. Furthermore, the criticality of the viscosities was measured.

Meson excitation time as a probe of holographic critical point

We study the time evolution of expectation value of Wilson loop as a non-local observable in a strongly coupled field theory with a critical point at finite temperature and nonzero chemical potential, which is dual to an asymptotically AdS charged black hole via gauge/gravity duality. Due to inject of energy into the plasma, the temperature and a chemical potential increase to finite values and the plasma experiences an out-of-equilibrium process. By defining meson excitation time t_{ex} as a time at which the meson falls into the final excited state, we investigate the behavior of t_{ex} near the critical point as the system evolves towards the critical point. We observe that by increasing the interquark distance the dynamical critical exponent increases smoothly. Also, we obtain for slow quenches different values of the dynamical critical exponent, although for fast quenches our result for the dynamical critical exponent is in agreement with the one that is reported for studying the quasi-normal modes. Consequently, this indicates that in this model for fast quenches and small values of interquark distances the gauge invariant Wilson loop is a good non-local observable to probe the critical point.

Quasiparticle dynamics in a quasiperiodic Ising model with temporally fluctuating transverse fields

We study quasiparticle dynamics in a quasiperiodic Ising model with temporally fluctuating transverse fields. Specifically, we calculate the dynamical exponents of the standard deviation of a quasiparticle spreading under a field chosen randomly from binary values $\ifmmode\pm\else\textpm\fi{}h$ at every time interval. We find that the short-time behavior of the dynamical exponents depends on the interval of the temporally fluctuating fields. We also reveal how the quasiparticle dynamics affects the relaxation of spin-spin correlation functions. The dynamics can be explained via the overlap between the eigenvectors of a Hamiltonian with $\ifmmode\pm\else\textpm\fi{}h$.

Thermalization and prethermalization in the soft-wall AdS/QCD model

The real-time dynamics of chiral phase transition is investigated in a two-flavor ($N_f=2$) soft-wall AdS/QCD model. To understand the dynamics of thermalization, we quench the system from initial states deviating from the equilibrium states. Then, we solve the nonequilibrium evolution of the order parameter (chiral condensate $\langle \sigma\equiv\bar{q}q\rangle$). It is shown that the system undergoes an exponential relaxation at temperatures away from the critical temperature $T_c$. The relaxation time diverges at $T_c$, presenting a typical behavior of critical slowing down. Numerically, we extract the dynamic critical exponent $z$, and get $z\approx 2$ by fitting the scaling behavior $\sigma\propto t^{-\beta/(\nu z)}$, where the mean-field static critical exponents (order parameter critical exponent $\beta=1/2$, correlation length critical exponent $\nu=1/2$ ) have been applied. More interestingly, it is remarked that, for a large class of initial states, the system would linger over a quasi-steady state for a certain period of time before the thermalization. It is suggested that the interesting phenomenon, known as prethermalization, has been observed in the framework of holographic models. In such prethermal stage, we verify that the system is characterized by a universal dynamical scaling law and described by the initial-slip exponent $\theta=0$.

Disentangling growth and decay of domains during phase ordering.

Using Monte Carlo simulations we study phase-ordering dynamics of a multispecies system modeled via the prototype q-state Potts model. In such a multispecies system, we identify a spin state or species as the winner if it has survived as the majority in the final state, otherwise, we mark them as loser. We disentangle the time (t) dependence of the domain length of the winner from losers, rather than monitoring the average domain length obtained by treating all spin states or species alike. The kinetics of domain growth of the winner at a finite temperature in space dimension d=2 reveal that the expected Lifshitz-Cahn-Allen scaling law ∼t^{1/2} can be realized with no early-time corrections, even for system sizes much smaller than what is traditionally used. Up to a certain period, all others species, i.e., the losers, also show a growth that, however, is dependent on the total number of species, and slower than the expected ∼t^{1/2} growth. Afterwards, the domains of the losers start decaying with time, for which our numerical data appear to be consistent with a ∼t^{-2} behavior. We also demonstrate that this approach of looking into the kinetics even provides new insights for the special case of phase ordering at zero temperature, both in d=2 and d=3.

Quantum critical dynamics in the two-dimensional transverse Ising model

In the vicinity of the quantum critical point (QCP), thermodynamic properties diverge toward zero temperature governed by universal exponents. Although this fact is well known, how it is reflected in quantum dynamics has not been addressed. The QCP of the transverse Ising model on a triangular lattice is an ideal platform to test the issue, since it has an experimental realization, the dielectrics being realized in an organic dimer Mott insulator, $\ensuremath{\kappa}\text{\ensuremath{-}}{\mathrm{ET}}_{2}X$, where a quantum electric dipole represents the Ising degrees of freedom. We track the Glauber-type dynamics of the model by constructing a kinetic protocol based on the quantum Monte Carlo method. The dynamical susceptibility takes the form of the Debye function and shows a significant peak narrowing in approaching a QCP due to the divergence of the relaxation timescale. It explains the anomaly of dielectric constants observed in the organic materials, indicating that the material is very near the ferroelectric QCP. We disclose how the dynamical and other critical exponents develop near QCP beyond the simple field theory.

Robust quantum Griffiths singularity above 1.5 K in nitride thin films

Quantum Griffiths singularity (QGS), which is closely correlated with the quenched disorder, is characterized by the divergence of the dynamical critical exponent and the presence of activated scaling behavior. Typically such a quantum phenomenon is rather rare and only observed in extremely low temperatures. Here we report the experimental observation of a robust QGS in nitride thin films, NbN, which survives in a rather high temperature range. The electrical transport propertied were measured under the magnetic field up to 12 T. The field induced superconductor-metal transitions were observed with the continuously changed transition points with the decrease of temperature. The dynamical critical exponent based on the conventional power-law scaling reveals a divergent trend when approaching the the low temperature limit. Moreover, the temperature and field dependence of sheet resistance can be described by the activated scaling analysis in the temperature region up to 1.6 K $\leq T \leq$ 4.0 K. We argue that the robustness of QGS in the present system originates from the Pauli paramagnetic effect due to the strong coupling between the magnetic field and the spin degree of freedom.

Mechanical excitation and marginal triggering during avalanches in sheared amorphous solids.

We study plastic strain during individual avalanches in overdamped particle-scale molecular dynamics (MD) and mesoscale elastoplastic models (EPM) for amorphous solids sheared in the athermal quasistatic limit. We show that the spatial correlations in plastic activity exhibit a short length scale that grows as t^{3/4} in MD and ballistically in EPM, which is generated by mechanical excitation of nearby sites not necessarily close to their stability thresholds, and a longer lengthscale that grows diffusively for both models and is associated with remote marginally stable sites. These similarities in spatial correlations explain why simple EPMs accurately capture the size distribution of avalanches observed in MD, though the temporal profiles and dynamical critical exponents are quite different.