Critical Scaling Regime Research Articles

High‐dimensional limit theorems for SGD: Effective dynamics and critical scaling

AbstractWe study the scaling limits of stochastic gradient descent (SGD) with constant step‐size in the high‐dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite‐dimensional functions) of SGD as the dimension goes to infinity. Our approach allows one to choose the summary statistics that are tracked, the initialization, and the step‐size. It yields both ballistic (ODE) and diffusive (SDE) limits, with the limit depending dramatically on the former choices. We show a critical scaling regime for the step‐size, below which the effective ballistic dynamics matches gradient flow for the population loss, but at which, a new correction term appears which changes the phase diagram. About the fixed points of this effective dynamics, the corresponding diffusive limits can be quite complex and even degenerate. We demonstrate our approach on popular examples including estimation for spiked matrix and tensor models and classification via two‐layer networks for binary and XOR‐type Gaussian mixture models. These examples exhibit surprising phenomena including multimodal timescales to convergence as well as convergence to sub‐optimal solutions with probability bounded away from zero from random (e.g., Gaussian) initializations. At the same time, we demonstrate the benefit of overparametrization by showing that the latter probability goes to zero as the second layer width grows.

Isobaric heat capacity of classical and quantum fluids: extending experimental data sets into the critical scaling regime

Isobaric heat capacity of classical and quantum fluids: extending experimental data sets into the critical scaling regime

Equilibrium fluctuations in mean-field disordered models.

Mean-field models of glasses that present a random first order transition exhibit highly nontrivial fluctuations. Building on previous studies that focused on the critical scaling regime, we here obtain a fully quantitative framework for all equilibrium conditions. By means of the replica method we evaluate Gaussian fluctuations of the overlaps around the thermodynamic limit, decomposing them in thermal fluctuations inside each state and heterogeneous fluctuations between different states. We first test and compare our analytical results with numerical simulation results for the p-spin spherical model and the random orthogonal model, and then analyze the random Lorentz gas. In all cases, a strong quantitative agreement is obtained. Our analysis thus provides a robust scheme for identifying the key finite-size (or finite-dimensional) corrections to the mean-field treatment of these paradigmatic glass models.

Universality Class of Ising Critical States with Long-Range Losses.

We show that spatial resolved dissipation can act on d-dimensional spin systems in the Ising universality class by qualitatively modifying the nature of their critical points. We consider power-law decaying spin losses with a Lindbladian spectrum closing at small momenta as ∝q^{α}, with α a positive tunable exponent directly related to the power-law decay of the spatial profile of losses at long distances, 1/r^{(α+d)}. This yields a class of soft modes asymptotically decoupled from dissipation at small momenta, which are responsible for the emergence of a critical scaling regime ascribable to the nonunitary counterpart of the universality class of long-range interacting Ising models. For α<1 we find a nonequilibrium critical point ruled by a dynamical field theory described by a Langevin model with coexisting inertial (∼∂_{t}^{2}) and frictional (∼∂_{t}) kinetic coefficients, and driven by a gapless Markovian noise with variance ∝q^{α} at small momenta. This effective field theory is beyond the Halperin-Hohenberg description of dynamical criticality, and its critical exponents differ from their unitary long-range counterparts. Our Letter lays out perspectives for a revision of universality in driven open systems by employing dark states tailored by programmable dissipation.

Gaussian and critical scalings in the magnetoconductivity fluctuations of Y3Ba5Cu8O18 superconductor

We have studied the superconducting transition and the magnetoconductivity fluctuations in the polycrystalline Y3Ba5Cu8O[Formula: see text] (Y358) superconductor under magnetic fields upto 1 T. A two-step superconducting transition could be observed as a consequence of the granular structure of the sample, which is strongly affected by the applied magnetic field. Gaussian and genuine critical 3D-XY-E fluctuation regimes were identified. A critical scaling regime beyond 3D-XY was identified for magnetic fields upto 0.25 T, corresponding to the averaged exponent 0.19 and suggesting the occurrence of the weak first-order character of the superconducting transition. In the approximation to the zero resistance a power law regime could be observed, corresponding to the averaged exponent 2.37, which are smaller than previously reported for the Y358 system. Our results are discussed in terms of the Y358 and YBa2Cu3O[Formula: see text](Y123) results in the literature.

Aging dynamics in quenched noisy long-range quantum Ising models

We consider the $d$-dimensional transverse-field Ising model with power-law interactions $J/r^{d+\sigma}$ in the presence of a noisy longitudinal field with zero average. We study the longitudinal-magnetization dynamics of an initial paramagnetic state after a sudden switch-on of both the interactions and the noisy field. While the system eventually relaxes to an infinite-temperature state with vanishing magnetization correlations, we find that two-time correlation functions show aging at intermediate times. Moreover, for times shorter than the inverse noise strength $\kappa$ and distances longer than $a(J/\kappa)^{2/\sigma}$ with $a$ being the lattice spacing, we find a critical scaling regime of correlation and response functions consistent with the model A dynamical universality class with an initial-slip exponent $\theta=1$ and dynamical critical exponent $z=\sigma/2$. We obtain our results analytically by deriving an effective action for the magnetization field including the noise in a non-perturbative way. The above scaling regime is governed by a non-equilibrium fixed point dominated by the noise fluctuations.

Weyl Semimetal to Metal Phase Transitions Driven by Quasiperiodic Potentials.

We explore the stability of three-dimensional Weyl and Dirac semimetals subject to quasiperiodic potentials. We present numerical evidence that the semimetal is stable for weak quasiperiodic potentials, despite being unstable for weak random potentials. As the quasiperiodic potential strength increases, the semimetal transitions to a metal, then to an "inverted" semimetal, and then finally to a metal again. The semimetal and metal are distinguished by the density of states at the Weyl point, as well as by level statistics, transport, and the momentum-space structure of eigenstates near the Weyl point. The critical properties of the transitions in quasiperiodic systems differ from those in random systems: we do not find a clear critical scaling regime in energy; instead, at the quasiperiodic transitions, the density of states appears to jump abruptly (and discontinuously to within our resolution).

Metamagnetic Anomalies near Dynamic Phase Transitions.

We report the existence of anomalous metamagnetic fluctuations in the vicinity of the dynamic phase transition (DPT) that do not occur for the corresponding thermodynamic behavior of simple ferromagnets. Our results demonstrate that key characteristics associated with the DPT are qualitatively different from conventional thermodynamic phase transitions. We also provide evidence that these differences are tunable by showing that the presence of metamagnetic fluctuations and the size of the critical scaling regime depend strongly on the amplitude of the oscillating field that is driving the DPT in the first place.

Smoothed Analysis for the Conjugate Gradient Algorithm

The purpose of this paper is to establish bounds on the rate of convergence of the conjugate gradient algorithm when the underlying matrix is a random positive definite perturbation of a deterministic positive definite matrix. We estimate all finite moments of a natural halting time when the random perturbation is drawn from the Laguerre unitary ensemble in a critical scaling regime explored in Deift et al. (2016). These estimates are used to analyze the expected iteration count in the framework of smoothed analysis, introduced by Spielman and Teng (2001). The rigorous results are compared with numerical calculations in several cases of interest.

Critical behaviour of the O(n)-ϕ4 model with an antisymmetric tensor order parameter

Critical behaviour of the O(n)-symmetric ϕ4 model with an antisymmetric tensor order parameter is studied by means of the field theoretic renormalization group (RG) in the leading order of the ε = 4 − d expansion (one-loop approximation). For n = 2 and 3 the model is equivalent to the scalar and the O(3)-symmetric vector models; for n ⩾ 4 it involves two independent interaction terms and two coupling constants. It is shown that for n > 4 the RG equations have no infrared (IR) attractive fixed points and their solutions (RG flows) leave the stability region of the model. This means that fluctuations of the order parameter change the nature of the phase transition from the second-order type (suggested by the mean-field theory) to the first-order one. For n = 4, the IR attractive fixed point exists and the IR behaviour is non-universal: if the coupling constants belong to the basin of attraction for the IR point, the phase transition is of the second order and the IR critical scaling regime is realized. The corresponding critical exponents ν and η are presented in the order ε and ε2, respectively. Otherwise the RG flows pass outside the stability region and the first-order transition takes place.

Solitonic Asymptotics for the Korteweg–de Vries Equation in the Small Dispersion Limit

We study the small dispersion limit for the Korteweg-de Vries (KdV) equation u(t) + 6uu(x) + epsilon(2)u(xxx) = 0 in a critical scaling regime where x approaches the trailing edge of the region where the KdV solution shows oscillatory behavior. Using the Riemann-Hilbert approach, we obtain an asymptotic expansion for the KdV solution in a double scaling limit, which shows that the oscillations degenerate to sharp pulses near the trailing edge. Locally those pulses resemble soliton solutions of the KdV equation.

Extended scaling in high dimensions

We apply and test the recently proposed ‘extended scaling’ scheme in an analysis of themagnetic susceptibility of Ising systems above the upper critical dimension. The data areobtained by Monte Carlo simulations using both the conventional Wolff cluster algorithmand the Prokof’ev–Svistunov worm algorithm. As already observed for other models,extended scaling is shown to extend the high-temperature critical scaling regime over arange of temperatures much wider than that achieved conventionally. It allows for anaccurate determination of leading and sub-leading scaling indices, critical temperatures andamplitudes of the confluent corrections.

T−Jmodel for a bilayer: Coexisting superconductivity and planar-flux phases

We study the normal state and the superconducting transition in the Attractive Hubbard Model in three dimensions, using self-consistent diagrammatics. Our results for the self-consistent $T$-matrix approximation are consistent with 3D-XY power-law critical scaling and finite-size scaling. This is in contrast to the exponential 2D-XY scaling the method was able to capture in our previous 2D calculation. We find the 3D transition temperature at quarter-filling and $U=-4t$ to be $T_c=0.207t$. The 3D critical regime is much narrower than in 2D and the ratio of the mean-field transition to $T_c$ is about 5 times smaller than in 2D. We also find that, for the parameters we consider, the pseudogap regime in 3D (as in 2D) coincides with the critical scaling regime.

Topological phase fluctuations, amplitude fluctuations, and criticality in extreme type-II superconductors

We study the effect of critical fluctuations on the $(B,T)$ phase diagram in extreme type-II superconductors in zero and finite magnetic field using large-scale Monte Carlo simulations on the Ginzburg-Landau model in a frozen gauge approximation. We show that a vortex-loop unbinding gives a correct picture of the zero field superconducting-normal transition even in the presence of amplitude fluctuations, which are far from being critical at $T_c$. We extract critical exponents of the dual model by studying the topological excitations of the original model. From the vortex-loop distribution function we extract the anomalous dimension of the dual field $\eta \simeq -0.18$, and conclude that the charged Ginzburg-Landau model and the neutral 3DXY model belong to different universality classes. We find are two distinct scaling regimes for the vortex-line lattice melting line: a high-field scaling regime and a distinct low-field 3DXY critical scaling regime. We also find indications of an abrupt change in the connectivity of the vortex-tangle in the vortex liquid along a line $T_L \geq T_M$. This is the finite field counter-part of the zero-field vortex-loop blowout. Which at low enough fields appears to coincide with $T_M$. Here, a description of the vortex system only in terms of field induced vortex lines is inadequate at and above the VLL melting temperature.