Universality Arguments Research Articles

Putting theory in its place: The relationship between universality arguments and empirical constraints

Putting theory in its place: The relationship between universality arguments and empirical constraints

NoBLE for Lattice Trees and Lattice Animals

We study lattice trees (LTs) and animals (LAs) on the nearest-neighbor lattice {mathbb {Z}}^d in high dimensions. We prove that LTs and LAs display mean-field behavior above dimension 16 and 17, respectively. Such results have previously been obtained by Hara and Slade in sufficiently high dimensions. The dimension above which their results apply was not yet specified. We rely on the non-backtracking lace expansion (NoBLE) method that we have recently developed. The NoBLE makes use of an alternative lace expansion for LAs and LTs that perturbs around non-backtracking random walk rather than around simple random walk, leading to smaller corrections. The NoBLE method then provides a careful computational analysis that improves the dimension above which the result applies. Universality arguments predict that the upper critical dimension, above which our results apply, is equal to d_c=8 for both models, as is known for sufficiently spread-out models by the results of Hara and Slade mentioned earlier. The main ingredients in this paper are (a) a derivation of a non-backtracking lace expansion for the LT and LA two-point functions; (b) bounds on the non-backtracking lace-expansion coefficients, thus showing that our general NoBLE methodology can be applied; and (c) sharp numerical bounds on the coefficients. Our proof is complemented by a computer-assisted numerical analysis that verifies that the necessary bounds used in the NoBLE are satisfied.

Universal signatures of Dirac fermions in entanglement and charge fluctuations

We investigate the entanglement entropy (EE) and charge fluctuations in models where the low energy physics is governed by massless Dirac fermions. We focus on the response to flux insertion which, for the EE, is widely assumed to be universal, \emph{i.e.}, independent of the microscopic details. We provide an analytical derivation of the EE and charge fluctuations for the seminal example of graphene, using the dimensional reduction of its tight-binding model to the one-dimensional Su-Schrieffer-Heeger model. Our asymptotic expression for the EE matches the conformal field theory prediction. We show that the charge variance has the same asymptotic behavior, up to a constant prefactor. To check the validity of universality arguments, we numerically consider several models, with different geometries and number of Dirac cones, and either for strictly two-dimensional models or for gapless surface mode of three-dimensional topological insulators. We also show that the flux response does not depend on the entangling surface geometry as long as it encloses the flux. Finally we consider the universal corner contributions to the EE. We show that in the presence of corners, the Kitaev-Preskill subtraction scheme provides non-universal, geometry dependent results.

Quartic cumulant of baryon number in the presence of a QCD critical point

In the context of the ongoing search for the QCD critical point at the Relativistic Heavy-Ion Collider, we study the equation of state near the critical point in the temperature and baryon chemical potential plane. We use the parametric representation introduced in earlier literature, which maps the universal 3D Ising equation of state onto the QCD phase diagram using several non-universal parameters. We focus on the quartic cumulant of the baryon number, or baryon number susceptibility~$\chi_4^B$, which can be accessed experimentally via net-proton fluctuation kurtosis measurements. It was originally predicted, through universality arguments based on the {\em leading} singular contribution, that $\chi_4^B$ and net-proton kurtosis should show a specific non-monotonic behavior due to the critical point. In particular, when following the freeze-out curve on the phase diagram by decreasing beam energy, the kurtosis is expected to dip, and then peak, when the beam energy scan passes close to the critical point. We study the effects of the non-universal and thus far unknown parameters of the Ising-to-QCD mapping on the behavior of~$\chi_4^B$. We find that, while the peak remains a solid feature, the presence of the critical point does not necessarily cause a dip in $\chi_4^B$ on the freezeout line {\em below} the transition temperature. The critical point contribution to the dip appears only for a narrow set of mapping parameters, when subleading singular terms are sufficiently suppressed.

Estimation of low-rank matrices via approximate message passing

Consider the problem of estimating a low-rank matrix when its entries are perturbed by Gaussian noise, a setting that is also known as “spiked model” or “deformed random matrix.” If the empirical distribution of the entries of the spikes is known, optimal estimators that exploit this knowledge can substantially outperform simple spectral approaches. Recent work characterizes the asymptotic accuracy of Bayes-optimal estimators in the high-dimensional limit. In this paper, we present a practical algorithm that can achieve Bayes-optimal accuracy above the spectral threshold. A bold conjecture from statistical physics posits that no polynomial-time algorithm achieves optimal error below the same threshold (unless the best estimator is trivial). Our approach uses Approximate Message Passing (AMP) in conjunction with a spectral initialization. AMP algorithms have proved successful in a variety of statistical estimation tasks, and are amenable to exact asymptotic analysis via state evolution. Unfortunately, state evolution is uninformative when the algorithm is initialized near an unstable fixed point, as often happens in low-rank matrix estimation problems. We develop a new analysis of AMP that allows for spectral initializations, and builds on a decoupling between the outlier eigenvectors and the bulk in the spiked random matrix model. Our main theorem is general and applies beyond matrix estimation. However, we use it to derive detailed predictions for the problem of estimating a rank-one matrix in noise. Special cases of this problem are closely related—via universality arguments—to the network community detection problem for two asymmetric communities. For general rank-one models, we show that AMP can be used to construct confidence intervals and control false discovery rate. We provide illustrations of the general methodology by considering the cases of sparse low-rank matrices and of block-constant low-rank matrices with symmetric blocks (we refer to the latter as to the “Gaussian block model”).

On the Universality of Hawking Radiation

Abstract A physically consistent semi-classical treatment of black holes requires universality arguments to deal with the ‘trans-Planckian’ problem where quantum spacetime effects appear to be amplified such that they undermine the entire semi-classical modelling framework. We evaluate three families of such arguments in comparison with Wilsonian renormalization group universality arguments found in the context of condensed matter physics. Our analysis is framed by the crucial distinction between robustness and universality. Particular emphasis is placed on the quality whereby the various arguments are underpinned by ‘integrated’ notions of robustness and universality. Whereas the principal strength of Wilsonian universality arguments can be understood in terms of the presence of such integration, the principal weakness of all three universality arguments for Hawking radiation is its absence. 1Introduction2Hawking Radiation and the Trans-Planckian Problem 2.1What is Hawking radiation?2.2Red-shift and robustness2.3Ultraviolet catastrophe3Universality Arguments in Condensed Matter Systems 3.1The Wilsonian approach to critical phenomena3.2The six qualities4Universality Arguments for Hawking Radiation 4.1Arguments from the equivalence principle4.2Arguments from horizon symmetries4.3Arguments from modified dispersion relations5Conclusion

Hawking radiation and analogue experiments: A Bayesian analysis

We present a Bayesian analysis of the epistemology of analogue experiments with particular reference to Hawking radiation. First, we prove that such experiments can be confirmatory in Bayesian terms based upon appeal to ‘universality arguments’. Second, we provide a formal model for the scaling behaviour of the confirmation measure for multiple distinct realisations of the analogue system and isolate a generic saturation feature. Finally, we demonstrate that different potential analogue realisations could provide different levels of confirmation. Our results provide a basis both to formalise the epistemic value of analogue experiments that have been conducted and to advise scientists as to the respective epistemic value of future analogue experiments.

Topological defects in quantum field theory with matrix product states

Topological defects (kinks) in a relativistic $\phi^{4}$ scalar field theory in $D=(1+1)$ are studied using the matrix product state tensor network. The one kink state is approximated as a matrix product state and the kink mass is calculated. The approach used is quite general and can be applied to a variety of theories and tensor networks. Additionally, the contribution of kink-antikink excitations to the ground state is examined and a general method to estimate the scalar mass from equal time ground state observables is provided. The scalar and kink mass are compared at strong coupling and behave as expected from universality arguments. This suggests that the matrix product state can adequately capture the physics of defect-antidefect excitations and thus provides a promising technique to study challenging non-equilibrium physics such as the Kibble-Zurek mechanism of defect formation.

Predicting the evolutionary dynamic behavior of a laser with injected signal using Lyapunov exponents

We show that the role of the Lyapunov exponents can be extended beyond the customary local instability, such as limit cycle behavior, to include its use as an evolutionary predictor of the dynamics of a laser with injected signal (LIS). Numerical studies of LIS reveal that as a function of the input-signal strength the evolution of two nonzero Lyapunov exponents (generally equal) distinctively predicts the evolutionary trend of the fundamental frequency of the laser output signal (an important dynamic characteristic of the LIS) even with the presence of some noise. This globally predictive behavior of the Lyapunov exponents includes also the dynamic behavior of the individual coexisting attractors. Different coexisting attractors of LIS and configurations of Lyapunov exponents for both individual attractors and the global system are reported. Two LIS case studies are considered: (I) a high-gain system with a rich history of nonlinear behavior but not experimentally accessible, and (II) a low-gain system that has complex dynamics and is experimentally accessible for Class B lasers. Universality arguments support the thesis that these different configurations and the extended role of the Lyapunov exponents as an evolutionary predictor of the dynamics will be observed in other nonlinear, dynamic dissipative systems as well.

Nonextensive lattice gauge theories: Algorithms and methods

High-energy phenomena presenting strong dynamical correlations, long-range interactions and microscopic memory effects are well described by nonextensive versions of the canonical Boltzmann–Gibbs statistical mechanics. After a brief theoretical review, we introduce a class of generalized heat-bath algorithms that enable Monte Carlo lattice simulations of gauge fields on the nonextensive statistical ensemble of Tsallis. The algorithmic performance is evaluated as a function of the Tsallis parameter q in equilibrium and nonequilibrium setups. Then, we revisit short-time dynamic techniques, which in contrast to usual simulations in equilibrium present negligible finite-size effects and no critical slowing down. As an application, we investigate the short-time critical behaviour of the nonextensive hot Yang–Mills theory at q-values obtained from heavy-ion collision experiments. Our results imply that, when the equivalence of statistical ensembles is obeyed, the long-standing universality arguments relating gauge theories and spin systems hold also for the nonextensive framework.

Dimensional fragility of the Kardar–Parisi–Zhang universality class

We assess the dependence on substrate dimensionality of the asymptotic scaling behavior of a whole family of equations that feature the basic symmetries of the Kardar–Parisi–Zhang (KPZ) equation. Even for cases in which, as expected from universality arguments, these models display KPZ values for the critical exponents and limit distributions, their behavior deviates from KPZ scaling for increasing system dimensions. Such a fragility of KPZ universality contradicts naive expectations, and questions straightforward application of universality principles for the continuum description of experimental systems.

Phase transitions in the frustrated Ising model on the square lattice

We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions ${J}_{1}<0$ (nearest neighbor, ferromagnetic) and ${J}_{2}>0$ (second neighbor, antiferromagnetic). The striped phase breaks a ${Z}_{4}$ symmetry and is stabilized at low temperatures for $g={J}_{2}/|{J}_{1}|>1/2$. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The ${J}_{1}$-${J}_{2}$ model has a line of very weak first-order phase transitions in the whole region $1/2<g<{g}^{*}$, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from $g={g}^{*}$ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point ${g}^{*}$ in the ${J}_{1}$-${J}_{2}$ model. The continuous transitions near ${g}^{*}$ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of $g>1/2$. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.

Phase structure of the linear sigma model with the standard symmetry breaking term

The linear sigma model at finite isospin chemical potential μ and temperature T is systematically studied by means of the Cornwal–Jackiw–Tomboulis (CJT) effective potential calculated in the improved Hartree–Fock (HF) approximation, where the Goldstone theorem and the thermodynamic consistency are respected. It results that in the chiral limit, for μ=0 the chiral phase transition is second order as expected from the general universality arguments, and for μ≠0 the phase diagram for the pion condensation in the (T,μ) plane exhibits a tricritical point which is crossover from first-order to second-order phase transitions. In the physical world, where the chiral symmetry is explicitly broken, the pion condensation occurs at μ=mπ, the pion mass in vacuum, and its phase diagram is basically in agreement with those found from the chiral perturbation theory. The chiral symmetry gets restored at high values of T for fixed μ and of μ for fixed T.

Future exploration of the nucleon structure at COMPASS

Up to now, COMPASS experiment [1] essentially focussed, in it's program with muon beams, on studying aspects of the longitudinal momentum structure of the nucleon in the collinear approximation, like Δq(x) and ΔG/G(x) [2, 3, 4]. However, quarks can also have intrinsic transverse momentum in the nucleon, which give rise to a new class of Transverse Momentum Distribution (TMD) Parton Distribution Functions. As an example, Sievers function has been measured by both COMPASS and HERMES to be non zero on the proton, paving thus the way for more precise investigations[5, 6].It is precisely the goal of the new COMPASS phase II proposal [7] to investigate in more detail new transverse description of the nucleon structure. Deeply Virtual Compton Scattering (DVCS) will allow studies in the transverse space via Generalized Parton Distributions (GPDs). Transverse Mometum Dependent PDFs will essentially be studied in Drell-Yan (DY) reaction and SIDIS, and some universality arguments in QCD imply different signs for Sievers and Boer-Mulders functions in DY and SIDIS.

Soliton form factors from lattice simulations

The form factor provides a convenient way to describe properties of topological solitons in the full quantum theory, when semiclassical concepts are not applicable. It is demonstrated that the form factor can be calculated numerically using lattice Monte Carlo simulations. The approach is very general and can be applied to essentially any type of soliton. The technique is illustrated by calculating the kink form factor near the critical point in 1+1-dimensional scalar field theory. As expected from universality arguments, the result agrees with the exactly calculable scaling form factor of the two-dimensional Ising model.

SPECTRUM OF SCREENING MASSES FOR SU(2) GAUGE THEORY: UNIVERSALITY CLASS

In this work we study the spectrum of the lowest screening masses for Yang–Mills theories on the lattice. We used the SU(2) gauge group in (3 + 1) dmensions. We adopted the multiple exponential method and the so-called "variational" method, in order to detect possible excited states. The calculations were done near the critical temperature of the confinement-deconfinement phase transition. We obtained values for the ratios of the screening masses consistent with predictions from universality arguments. A Monte Carlo evolution of the screening masses in the gauge theory confirms the validity of the predictions.

Quark propagator in the Nambu–Jona-Lasinio model in a self-consistent1/Ncexpansion

The quark propagator is calculated in the Nambu-Jona-Lasinio (NJL) model in a self-consistent 1/Nc-expansion at next-to-leading order. The calculations are carried out iteratively in Euclidean space. The chiral quark condensate and its dependence on temperature and chemical potential is calculated directly and compared with the mean-field results. In the chiral limit, we find a second-order phase transition at finite temperature and zero chemical potential, in agreement with universality arguments. At zero temperature and finite chemical potential, the phase transition is first order. In comparison with the mean-field results, the critical temperature and chemical potential are slightly reduced. We determine spectral functions from the Euclidean propagators by employing the Maximum-Entropy-Method (MEM). Thereby quark and meson masses are estimated and decay channels identified. For testing this method, we also apply it to evaluate perturbative spectral functions, which can be calculated directly in Minkowski space. In most cases we find that MEM is able to reproduce the rough features of the spectral functions, but not the details.

Spectrum of screening masses in pure SU(2) gauge theory in [formula omitted] dimensions

In this work we study the spectrum of the lowest screening masses for Yang-Mills theories on the lattice. We used the SU(2) gauge group in 3 + 1 dimensions. The calculations were done near the critical temperature of the confinement-deconfinement phase transition. We obtained values for the ratios of the screening masses consistent with predictions from universality arguments.

Phase transitions in ideal and weakly interacting Bose gases with a finite number of particles confined in a box

We generalize the scheme to characterize phase transitions of finite systems in a complex temperature plane and approach the classifications of phase transitions in ideal and weakly interacting Bose gases of a finite number of particles, confined in a cubic box of volume L3 with different boundary conditions. For this finite ideal Bose system, by extending the classification parameters to all regions, we predict that the phase transition for periodic boundary conditions is of second order, while the transition in Dirichlet boundary conditions is of first order. For a weakly interacting Bose gas with periodic boundary conditions, we discuss the effects of finite particle numbers and inter-particle interactions on the nature of the phase transitions. We show that this homogenous weakly interacting Bose gas undergoes a second-order phase transition, which is in accordance with universality arguments for infinite systems. We also discuss the dependence of transition temperature on interaction strengths and particle numbers.

Linear sigma model at finite density in the1/Nexpansion to next-to-leading order

We study relativistic Bose-Einstein condensation at finite density and temperature using the linear sigma model in the one-particle-irreducible 1/N expansion. We derive the effective potential to next-to-leading (NLO) order and show that it can be renormalized in a temperature-independent manner. As a particular application, we study the thermodynamics of the pion gas in the chiral limit as well as with explicit symmetry breaking. At nonzero temperature we solve the NLO gap equation and show that the results describe the chiral-symmetry-restoring second-order phase transition in agreement with general universality arguments. However, due to nontrivial regularization issues, we are not able to extend the NLO analysis to nonzero chemical potential.