Universal Scaling Properties Research Articles

Kardar-Parisi-Zhang universality in two-component driven diffusive models: Symmetry and renormalization group perspectives.

We elucidate the universal spatiotemporal scaling properties of the time-dependent correlation functions in a class of two-component one-dimensional (1D) driven diffusive system that consists of two coupled asymmetric exclusion processes. By using a perturbative renormalization group framework, we show that the relevant scaling exponents have values same as those for the 1D Kardar-Parisi-Zhang (KPZ) equation. We connect these universal scaling exponents with the symmetries of the model equations. We thus establish that these models belong to the 1D KPZ universality class.

Reconciling scaling of the optical conductivity of cuprate superconductors with Planckian resistivity and specific heat

Materials tuned to a quantum critical point display universal scaling properties as a function of temperature T and frequency ω. A long-standing puzzle regarding cuprate superconductors has been the observed power-law dependence of optical conductivity with an exponent smaller than one, in contrast to T-linear dependence of the resistivity and ω-linear dependence of the optical scattering rate. Here, we present and analyze resistivity and optical conductivity of La2−xSrxCuO4 with x = 0.24. We demonstrate ℏω/kBT scaling of the optical data over a wide range of frequency and temperature, T-linear resistivity, and optical effective mass proportional to sim ln T corroborating previous specific heat experiments. We show that a T, ω-linear scaling Ansatz for the inelastic scattering rate leads to a unified theoretical description of the experimental data, including the power-law of the optical conductivity. This theoretical framework provides new opportunities for describing the unique properties of quantum critical matter.

Experimental Test of Sinai's Model in DNA Unzipping.

The experimental measurement of correlation functions and critical exponents in disordered systems is key to testing renormalization group (RG) predictions. We mechanically unzip single DNA hairpins with optical tweezers, an experimental realization of the diffusive motion of a particle in a one-dimensional random force field, known as the Sinai model. We measure the unzipping forces F_{w} as a function of the trap position w in equilibrium and calculate the force-force correlator Δ_{m}(w), its amplitude, and correlation length, finding agreement with theoretical predictions. We study the universal scaling properties since the effective trap stiffness m^{2} decreases upon unzipping. Fluctuations of the position of the base pair at the unzipping junction u scales as u∼m^{-ζ}, with a roughness exponent ζ=1.34±0.06, in agreement with the analytical prediction ζ=4/3. Our study provides a single-molecule test of the functional RG approach for disordered elastic systems in equilibrium.

Complex scaling flows in the quench dynamics of interacting particles

Many-body systems driven out of equilibrium can exhibit scaling flows of the quantum state. For a sudden quench to resonant interactions between particles we construct a new class of analytical scaling solutions for the time evolved wave function with a complex scale parameter. These solutions determine the exact dynamical scaling of observables such as the pair correlation function, the contact and the fidelity. We give explicit examples of the nonequilibrium dynamics for two trapped fermions or bosons quenched to unitarity, for ideal Bose polarons, and for resonantly interacting, Borromean three-body systems. These solutions reveal universal scaling properties of interacting many-body systems that arise from the buildup of correlations at short times after the quench.

Dimensional crossover for universal scaling far from equilibrium

We perform a dynamical finite-size scaling analysis of a nonequilibrium Bose gas which is confined in the transverse plane. Varying the transverse size, we establish a dimensional crossover for universal scaling properties far from equilibrium. Our results suggest that some aspects of the dynamical universal behavior of anisotropic systems can be classified in terms of fractional spatial dimensions. We discuss our findings in view of recent experimental results with quasi one-dimensional setups of quenched ultracold quantum gases.

Universal properties of the Kardar-Parisi-Zhang equation with quenched columnar disorders.

Inspired by the recent results on totally asymmetric simple exclusion processes on a periodic lattice with short-ranged quenched hopping rates [A. Haldar and A. Basu, Phys. Rev. Research 2, 043073 (2020)2643-156410.1103/PhysRevResearch.2.043073], we study the universal scaling properties of the Kardar-Parisi-Zhang (KPZ) equation with short-ranged quenched columnar disorder in general d dimensions. We show that there are generic propagating modes in the system that have their origin in the quenched disorder and make the system anisotropic. We argue that the presence of the propagating modes actually make the effects of the quenched disorder irrelevant, making the universal long wavelength scaling property belong to the well-known KPZ universality class. On the other hand, when these waves vanish in a special limit of the model, new universality class emerges with dimension d=4 as the lower critical dimension, above which the system is speculated to admit a disorder-induced roughening transition to a perturbatively inaccessible rough phase.

Universality in a class of the modified Villain–Lai–Das Sarma equation

The universal scaling properties of the original and modified versions of the Villain–Lai–Das Sarma (VLDS) growth system are investigated numerically in both (1 + 1) and (2 + 1) dimensions. The modified VLDS equation with instability suppression by an exponentially decreasing function is equivalent to the VLDS with infinitely many weakly relevant nonlinear terms (VLDS∞). The growth instability and scaling properties are discussed based on the modification of the VLDS growth system. Our results show that the existence of infinitely many weakly relevant nonlinear terms in the modified VLDS system could: (i) lead to nontrivial scaling behavior in a generic way, such as anomalous scaling; (ii) be partially effective at suppressing numerical instabilities in the normal VLDS equation.

Universal Scaling Properties of QCD Close to the Chiral Limit

We present a lattice QCD based determination of the chiral phase transition temperature in QCD with two massless (up and down) and one strange quark having its physical mass. We propose and calculate two novel estimators for the chiral transition temperature for several values of the light quark masses, corresponding to Goldstone pion masses in the range of $58~{\rm MeV}\lesssim m_\pi\lesssim 163~{\rm MeV}$. The chiral phase transition temperature is determined by extrapolating to vanishing pion mass using universal scaling relations. After thermodynamic, continuum and chiral extrapolations we find the chiral phase transition temperature $T_c^0=132^{+3}_{-6}$ MeV. We also show some preliminary calculations that use the conventional estimator for the pseudo-critical temperature and compare with the new estimators for $T_c^0$. Furthermore, we show results for the ratio of the chiral order parameter and its susceptibility and argue that this ratio can be used to differentiate between $O(N)$ and $Z_2$ universality classes in a non-parametric manner.

Dynamic scaling in the quenched disordered classicalN-vector model

We revisit the effects of short-ranged random quenched disorder on the universal scaling properties of the classical $N$-vector model with cubic anisotropy. We set up the nonconserved relaxational dynamics of the model, and study the universal dynamic scaling near the second order phase transition. We extract the critical exponents and the dynamic exponent in a one-loop dynamic renormalisation group calculation with short-ranged isotropic disorder. We show that the dynamics near a critical point is generically slower when the quenched disorder is relevant than when it is not, independent of whether the pure model is isotropic or cubic anisotropic. We demonstrate the surprising thresholdless instability of the associated universality class due to perturbations from rotational invariance breaking quenched disorder-order parameter coupling, indicating breakdown of dynamic scaling. We speculate that this may imply a novel first order transition in the model, induced by a symmetry-breaking disorder.

Quantum Hall Network Models as Floquet Topological Insulators.

Network models for equilibrium integer quantum Hall (IQH) transitions are described by unitary scattering matrices that can also be viewed as representing nonequilibrium Floquet systems. The resulting Floquet bands have zero Chern number, and are instead characterized by a chiral Floquet winding number. This begs the question, How can a model without Chern number describe IQH systems? We resolve this puzzle by showing that nonzero Chern number is recovered from the network model via the energy dependence of network model scattering parameters. This relationship shows that, despite their topologically distinct origins, IQH and chiral Floquet topology-changing transitions share identical universal scaling properties.

Bifurcation analysis and structural stability of simplicial oscillator populations

We present an analytical description for the collective dynamics of oscillator ensembles with higher-order coupling encoded by simplicial structure, which serves as an illustrative and insightful paradigm for brain function and information storage. The novel dynamics of the system, including abrupt desynchronization and multistability, are rigorously characterized and the critical points that correspond to a continuum of first-order phase transitions are found to satisfy universal scaling properties. More importantly, the underlying bifurcation mechanism giving rise to multiple clusters with arbitrary ensemble size is characterized using a rigorous spectral analysis of the stable cluster states. As a consequence of $SO_2$ group symmetry, we show that the continuum of abrupt desynchronization transitions result from the instability of a collective mode under the nontrivial antisymmetric manifold in the high dimensional phase space.

Supercritical Fluid Gaseous and Liquid States: A Review of Experimental Results.

We review the experimental evidence, from both historic and modern literature of thermodynamic properties, for the non-existence of a critical-point singularity on Gibbs density surface, for the existence of a critical density hiatus line between 2-phase coexistence, for a supercritical mesophase with the colloidal characteristics of a one-component 2-state phase, and for the percolation loci that bound the existence of gaseous and liquid states. An absence of any critical-point singularity is supported by an overwhelming body of experimental evidence dating back to the original pressure-volume-temperature (p-V-T) equation-of-state measurements of CO2 by Andrews in 1863, and extending to the present NIST-2019 Thermo-physical Properties data bank of more than 200 fluids. Historic heat capacity measurements in the 1960s that gave rise to the concept of “universality” are revisited. The only experimental evidence cited by the original protagonists of the van der Waals hypothesis, and universality theorists, is a misinterpretation of the isochoric heat capacity Cv. We conclude that the body of extensive scientific experimental evidence has never supported the Andrews–van der Waals theory of continuity of liquid and gas, or the existence of a singular critical point with universal scaling properties. All available thermodynamic experimental data, including modern computer experiments, are compatible with a critical divide at Tc, defined by the intersection of two percolation loci at gaseous and liquid phase bounds, and the existence of a colloid-like supercritical mesophase comprising both gaseous and liquid states.

Universal scaling of active nematic turbulence

A landmark of turbulence is the emergence of universal scaling laws, such as Kolmogorov's $E(q)\sim q^{-5/3}$ scaling of the kinetic energy spectrum of inertial turbulence with the wave vector $q$. In recent years, active fluids have been shown to exhibit turbulent-like flows at low Reynolds number. However, the existence of universal scaling properties in these flows has remained unclear. To address this issue, here we propose a minimal defect-free hydrodynamic theory for two-dimensional active nematic fluids at vanishing Reynolds number. By means of large-scale simulations and analytical arguments, we show that the kinetic energy spectrum exhibits a universal scaling $E(q)\sim q^{-1}$ at long wavelengths. We find that the energy injection due to activity has a peak at a characteristic length scale, which is selected by a nonlinear mechanism. In contrast to inertial turbulence, energy is entirely dissipated at the scale where it is injected, thus precluding energy cascades. Nevertheless, the non-local character of the Stokes flow establishes long-ranged velocity correlations, which lead to the scaling behavior. We conclude that active nematic fluids define a distinct universality class of turbulence at low Reynolds number.

Characterizing nonaffinity upon decompression of soft-sphere packings.

Athermal elastic moduli of soft-sphere packings are known to exhibit universal scaling properties near the unjamming point, most notably the vanishing of the shear-to-bulk moduli ratio G/B upon decompression. Interestingly, the smallness of G/B stems from the large nonaffinity of deformation-induced displacements under shear strains, compared to insignificant nonaffinity of displacements under compressive strains. In this work, we show using numerical simulations that the relative weights of the affine and nonaffine contributions to the bulk modulus, and their dependence on the proximity to the unjamming point, can differ qualitatively between different models that feature the same generic unjamming phenomenology. In canonical models of unjamming, we observe that the ratio of the nonaffine to total bulk moduli B_{na}/B approaches a constant upon decompression, while in other, less well-studied models, it vanishes. We show that the vanishing of B_{na}/B in noncanonical models stems from the emergence of an invariance of net (zero) forces on the constituent particles to compressive strains at the onset of unjamming. We provide a theoretical scaling analysis that fully explains our numerical observations, and allows us to predict the scaling behavior of B_{na}/B upon unjamming, given the functional form of the pairwise interaction potential.

Percolated Strain Networks and Universal Scaling Properties of Strain Glasses.

Strain glass is being established as a conceptually new state of matter in highly doped alloys, yet the understanding of its microscopic formation mechanism remains elusive. Here, we use a combined numerical and experimental approach to establish, for the first time, that the formation of strain glasses actually proceeds via the gradual percolation of strain clusters, namely, localized strain clusters that expand to reach the percolating state. Furthermore, our simulation studies of a wide variety of specific materials systems unambiguously reveal the existence of distinct scaling properties and universal behavior in the physical observables characterizing the glass transition, as obeyed by many existing experimental findings. The present work effectively enriches our understanding of the underlying physical principles governing glassy disordered materials.

Scaling Properties of Out-of-equilibrium Hydrodynamic Fluctuations near the QCD Critical Point

In a heavy ion experiment the system travels along trajectories of constant n/s in absence of dissipation. Ideally one can tune the beam energy so that the system goes exactly through the QCD critical point (CP), in which case the equilibrium two point correlator 〈δ(n/s)δ(n/s)〉eq [5] of the thermal fluctuations δ(n/s) diverges at the CP while the actual value of this correlator 〈δ(n/s)δ(n/s)〉 falls out of equilibrium and consequently remains finite. In this paper we first show that this out-of-equilibrium dynamics exhibits universal scaling properties known as Kibble-Zurek scaling. The relevant physical scale is called Kibble-Zurek scale and it is a function of a small dimensionless parameter ϵ≡τo/τQ, where τo is the microscopic mean-free-time and τQ is the macroscopic total expansion time. Realistically the trajectory always misses the CP, the equilibrium correlator is then cut off by a new scale which we call the crossing scale. The crossing scale is a function of another small dimensionless parameter Δs≡(nc/sc)(s/n−sc/nc) that's controlled by the beam energy. We then show from the scaling properties of Ising EOS that the equilibrium correlator scales universally with the crossing scale. The paper is concluded by a quantitative analysis on how the out-of-equilibrium KZ dynamics competes with missing the CP since they both limit the growth of the correlator 〈δ(n/s)δ(n/s)〉 near the CP.

Dynamic nuclear polarization from topological insulator helical edge states

Topological insulators are promising for spintronics and related technologies due to their spin-momentum-locked edge states, which are protected by time-reversal symmetry. In addition to the unique fundamental physics that arises in these systems, the potential technological applications of these protected states has also been driving TI research over the past decade. However, most known topological insulator materials naturally contain spinful nuclei, and their hyperfine coupling to helical edge states intrinsically breaks time-reversal symmetry, removing the topological protection and enabling the buildup of dynamic nuclear spin polarization through hyperfine-assisted backscattering. Here, we calculate scattering probabilities and nuclear polarization for edge channels containing up to $34$ nuclear spins using a numerically exact analysis that exploits the symmetries of the problem to drastically reduce the computational complexity. We then show the emergence of universal scaling properties that allow us to extrapolate our findings to vastly larger and experimentally relevant system sizes. We find that significant nuclear polarization can result from relatively weak helical edge currents, suggesting that it may be an important factor affecting spin transport in topological insulator devices.

Inverse finite-size scaling for high-dimensional significance analysis.

We propose an efficient procedure for significance determination in high-dimensional dependence learning based on surrogate data testing, termed inverse finite-size scaling (IFSS). The IFSS method is based on our discovery of a universal scaling property of random matrices which enables inference about signal behavior from much smaller scale surrogate data than the dimensionality of the original data. As a motivating example, we demonstrate the procedure for ultra-high-dimensional Potts models with order of 10^{10} parameters. IFSS reduces the computational effort of the data-testing procedure by several orders of magnitude, making it very efficient for practical purposes. This approach thus holds considerable potential for generalization to other types of complex models.

Symmetries and scaling in generalised coupled conserved Kardar–Parisi–Zhang equations

We study the noisy nonequilibrium dynamics of a conserved density that is driven by a fluctuating surface governed by the conserved Kardar–Parisi–Zhang equation. We uncover the universal scaling properties of the conserved density. We consider two separate minimal models where the surface fluctuations couple (i) with the spatial variation of the conserved density, and (ii) directly with the magnitude of the conserved density. Both these two models conserve the density, but differ from a symmetry stand point. We use our result to highlight the dependence of nonequilibrium universality classes on the interplay between symmetries and conservation laws.

Nonstationary dynamics of encounters: Mean valuable territory covered by a random searcher.

Inspired by recent experiments on the organism Caenorhabditis elegans we present a stochastic problem to capture the adaptive dynamics of search in living beings, which involves the exploration-exploitation dilemma between remaining in a previously preferred area and relocating to new places. We assess the question of search efficiency by introducing a new magnitude, the mean valuable territory covered by a Browinan searcher, for the case where each site in the domain becomes valuable only after a random time controlled by a nonhomogeneous rate which expands from the origin outwards. We explore analytically this magnitude for domains of dimensions 1, 2, and 3 and discuss the theoretical and applied (biological) interest of our approach. As the main results here, we (i) report the existence of some universal scaling properties for the mean valuable territory covered as a function of time and (ii) reveal the emergence of an optimal diffusivity which appears only for domains in two and higher dimensions.