Universal Critical Exponents Research Articles

Quantum tricriticality in a generalized quantum Rabi system

Quantum tricriticality, a unique form of high-order criticality, is expected to exhibit fascinating features including unconventional critical exponents and universal scaling laws. However, a quantum tricritical point (QTCP) is much harder to access, and the corresponding phenomena at tricriticality have rarely been investigated. In this study, we explore a tricritical quantum Rabi model, which incorporates a non-trivial parameter to adjust the coupling ratio between a cavity and a three-level atom. The QTCP emerges at the intersection of first- and second-order superradiant phase transitions according to Landau theory. By using finite-frequency scaling analysis on quantum fluctuations and the average photon number, universal critical exponents differentiate the QTCP from the second-order critical point. Our results indicate that the phase transition at the tricritical point goes beyond the conventional second-order phase transition. Our work explores an interesting direction in the generalization of the well-known Rabi model for the study of higher-order critical points due to its high control and tunability.

Thermalization slowing down in multidimensional Josephson junction networks.

We characterize thermalization slowing down of Josephson junction networks in one, two, and three spatial dimensions for systems with hundreds of sites by computing their entire Lyapunov spectra. The ratio of Josephson coupling E_{J} to energy density h controls two different universality classes of thermalization slowing down, namely, the weak-coupling regime, E_{J}/h→0, and the strong-coupling regime, E_{J}/h→∞. We analyze the Lyapunov spectrum by measuring the largest Lyapunov exponent and by fitting the rescaled spectrum with a general ansatz. We then extract two scales: the Lyapunov time (inverse of the largest exponent) and the exponent for the decay of the rescaled spectrum. The two universality classes, which exist irrespective of network dimension, are characterized by different ways the extracted scales diverge. The universality class corresponding to the weak-coupling regime allows for the coexistence of chaos with a large number of near-conserved quantities and is shown to be characterized by universal critical exponents, in contrast with the strong-coupling regime. We expect our findings, which we explain using perturbation theory arguments, to be a general feature of diverse Hamiltonian systems.

Critical Scaling through Gini Index.

In the systems showing critical behavior, various response functions have a singularity at the critical point. Therefore, as the driving field is tuned toward its critical value, the response functions change drastically, typically diverging with universal critical exponents. In this Letter, we quantify the inequality of response functions with measures traditionally used in economics, namely by constructing a Lorenz curve and calculating the corresponding Gini index. The scaling of such a response function, when written in terms of the Gini index, shows singularity at a point that is at least as universal as the corresponding critical exponent. The critical scaling, therefore, becomes a single parameter fit, which is a considerable simplification from the usual form where the critical point and critical exponents are independent. We also show that another measure of inequality, the Kolkata index, crosses the Gini index at a point just prior to the critical point. Therefore, monitoring these two inequality indices for a system where the critical point is not known can produce a precursory signal for the imminent criticality. This could be useful in many systems, including that in condensed matter, bio- and geophysics to atmospheric physics. The generality and numerical validity of the calculations are shown with the MonteCarlo simulations of the two dimensional Ising model, site percolation on square lattice, and the fiber bundle model of fracture.

Hamiltonian cycles on bicolored random planar maps

We study the statistics of Hamiltonian cycles on various families of bicolored random planar maps (with the spherical topology). These families fall into two groups corresponding to two distinct universality classes with respective central charges c=−1 and c=−2. The first group includes generic p-regular maps with vertices of fixed valency p≥3, whereas the second group comprises maps with vertices of mixed valencies, and the so-called rigid case of 2q-regular maps (q≥2) for which, at each vertex, the unvisited edges are equally distributed on both sides of the cycle. We predict for each class its universal configuration exponent γ, as well as a new universal critical exponent ν characterizing the number of long-distance contacts along the Hamiltonian cycle. These exponents are theoretically obtained by using the Knizhnik, Polyakov and Zamolodchikov (KPZ) relations, with the appropriate values of the central charge, applied, in the case of ν, to the corresponding critical exponent on regular (hexagonal or square) lattices. These predictions are numerically confirmed by analyzing exact enumeration results for p-regular maps with p=3,4,…,7, and for maps with mixed valencies (2,3), (2,4) and (3,4).

Irrelevant Corrections at the Quantum Hall Transition

The quantum Hall effect is one of the most extensively studied topological effects in solid‐state physics. The transitions between different quantum Hall states exhibit critical phenomena described by universal critical exponents. Numerous numerical finite size scaling studies have focused on the critical exponent ν for the correlation length. In such studies, it is important to take proper account of irrelevant corrections to scaling. Recently, Dresselhaus et al. proposed a new scaling ansatz, which they applied to the two terminal conductance of the Chalker–Coddington model. Herein, their proposal is applied to the previously reported data for the Lyapunov exponents of that model using both polynomial fitting and Gaussian process fitting.

Unified modeling and experimental realization of electrical and thermal percolation in polymer composites

Correlations between electrical and thermal conduction in polymer composites are blurred due to the complex contribution of charge and heat carriers at the nanoscale junctions of filler particles. Conflicting reports on the lack or existence of thermal percolation in polymer composites have made it the subject of great controversy for decades. Here, we develop a generalized percolation framework that describes both electrical and thermal conductivity within a remarkably wide range of filler-to-matrix conductivity ratios (Yf/Ym), covering 20 orders of magnitude. Our unified theory provides a genuine classification of electrical conductivity with typical Yf/Ym≥1010 as insulator–conductor percolation with the standard power-law behavior and of thermal conductivity with 102≤Yf/Ym≤104 as poor–good conductor percolation characterized by two universal critical exponents. Experimental verification of the universal and unified features of our theoretical framework is conducted by constructing a 3D segregated and well-extended network of multiwalled carbon nanotubes in polypropylene as a model polymer matrix under a carefully designed fabrication method. We study the evolution of the electrical and thermal conductivity in our fabricated composites at different loading levels up to 5 vol. %. Significantly, we find an ultralow electrical percolation threshold at 0.02 vol. % and a record-low thermal percolation threshold at 1.5 vol. %. We also apply our theoretical model to a number of 23 independent experimental and numerical datasets reported in the literature, including more than 350 data points, for systems with different microscopic details, and show that all collapse onto our proposed universal scaling function, which depends only on dimensionality.

Kardar–Parisi–Zhang universality in a one-dimensional polariton condensate

Revealing universal behaviours is a hallmark of statistical physics. Phenomena such as the stochastic growth of crystalline surfaces1 and of interfaces in bacterial colonies2, and spin transport in quantum magnets3-6 all belong to the same universality class, despite the great plurality of physical mechanisms they involve at the microscopic level. More specifically, in all these systems, space-time correlations show power-law scalings characterized by universal critical exponents. This universality stems from a common underlying effective dynamics governed by the nonlinear stochastic Kardar-Parisi-Zhang (KPZ) equation7. Recent theoretical works have suggested that this dynamics also emerges in the phase of out-of-equilibrium systems showing macroscopic spontaneous coherence8-17. Here we experimentally demonstrate that the evolution of the phase in a driven-dissipative one-dimensional polariton condensate falls in the KPZ universality class. Our demonstration relies on a direct measurement of KPZ space-time scaling laws18,19, combined with a theoretical analysis that reveals other key signatures of this universality class. Our results highlight fundamental physical differences between out-of-equilibrium condensates and their equilibrium counterparts, and open a paradigm for exploring universal behaviours in driven open quantumsystems.

Squeezing as a probe of the universality hypothesis

We compute analytically the radiative quantum corrections, up to next-to-leading loop order, to the universal critical exponents for both massless and massive O(N) [Formula: see text] scalar squeezed field theories for probing the universality hypothesis. For that, we employ six distinct and independent methods. The outcomes for the universal squeezed critical exponents obtained through these methods are identical among them and reduce to the conventional ones where squeezing is absent. Although the squeezing mechanism modifies the internal properties of the field, the squeezed critical indices are not affected by the squeezing effect, thus implying the validity of the universality hypothesis, at least at the loop level considered. At the end, we present the corresponding physical interpretation for the results in terms of the geometric symmetry properties of the squeezed field.

Nonequilibrium dynamics of the localization-delocalization transition in the non-Hermitian Aubry-André model

In this paper, we investigate the driven dynamics of the localization transition in the non-Hermitian Aubry-Andr\'e model with the periodic boundary condition. Depending on the strength of the quasiperiodic potential $\ensuremath{\lambda}$, this model undergoes a localization-delocalization phase transition. We find that the localization length $\ensuremath{\xi}$ satisfies $\ensuremath{\xi}\ensuremath{\sim}{\ensuremath{\varepsilon}}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\varepsilon}$ being the distance from the critical point and $\ensuremath{\nu}=1$ being a universal critical exponent independent of the non-Hermitian parameter. In addition, from the finite-size scaling of the energy gap between the ground state and the first excited state, we determine the dynamic exponent $z$ as $z=2$. The critical exponent of the inverse participation ratio for the $n\mathrm{th}$ eigenstate is also determined as $s=0.1197$. By changing $\ensuremath{\varepsilon}$ linearly to cross the critical point, we find that the driven dynamics can be described by the Kibble-Zurek scaling (KZS). Moreover, we show that the KZS with the same set of the exponents can be generalized to the localization phase transitions in the excited states.

Nonextensive statistical field theory

We introduce a field-theoretic approach for describing the critical behavior of nonextensive systems, systems displaying global correlations among their degrees of freedom, encoded by the nonextensive parameter q. As some applications, we report, to our knowledge, the first analytical computation of both universal static and dynamic q-dependent nonextensive critical exponents for O(N) vector models, valid for all loop orders and |q−1|<1. Then emerges the new nonextensive O(N)q universality class. Both exponents ηq and zq reduce to those when q=1 for N→∞. We employ six independent methods which furnish identical results. Particularly, the results for nonextensive 2d Ising systems, exact within the referred approximation, agree with that obtained from computer simulations, within the margin of error, as better as q is closer to 1. We argue that the present approach can be applied to all models described by extensive statistical field theory as well. The results show an interplay between global correlations and fluctuations.

Harnessing nonadiabatic excitations promoted by a quantum critical point: Quantum battery and spin squeezing

Crossing a quantum critical point in finite time challenges the adiabatic condition due to the closing of the energy gap, which ultimately results in the formation of excitations. Such nonadiabatic excitations are typically deemed detrimental in many scenarios, and consequently several strategies have been put forward to circumvent their formation. Here, however, we show how these nonadiabatic excitations—originated from the failure to meet the adiabatic condition due to the presence of a quantum critical point—can be controlled and thus harnessed to perform certain tasks advantageously. We focus on closed cycles reaching the quantum critical point of fully connected models analyzing two examples. First, a quantum battery that is loaded by approaching a quantum critical point, whose stored and extractable work increases exponentially via repeating cycles. Second, a scheme for the fast preparation of spin squeezed states containing multipartite entanglement that offer a metrological advantage, analogous to a two-axis twisting scheme. The corresponding figure of merit in both examples crucially depends on the universal critical exponents and the scaling of the protocol in the vicinity of the transition. Our results highlight the rich interplay between quantum thermodynamics and metrology with critical nonequilibrium dynamics.Received 11 May 2021Accepted 11 March 2022DOI:https://doi.org/10.1103/PhysRevResearch.4.L022017Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCoherent controlQuantum criticalityQuantum phase transitionsQuantum quenchQuantum statistical mechanicsQuantum thermodynamicsPhysical SystemsQuantum spin modelsTechniquesCritical phenomenaGeneral PhysicsAtomic, Molecular & OpticalStatistical PhysicsQuantum InformationCondensed Matter, Materials & Applied Physics

Financial markets and the phase transition between water and steam

A lattice gas model of financial markets is presented that has the potential to explain previous empirical observations of the interplay of trends and reversion in the markets, including a Langevin equation for the time evolution of trends. In this model, the shares of an asset correspond to gas molecules that are distributed across a hidden social network of investors. Neighbors in the network tend to align their positions due to herding behavior, corresponding to an attractive force of the gas molecules.This model is equivalent to the Ising model on this network, with the magnetization in the role of the deviation of the market price of an asset from its long-term value. Moreover, in efficient markets the system should naturally drive itself to its critical temperature, where it undergoes a second-order phase transition. There, it is characterized by long-range correlations and universal critical exponents, in analogy with the phase transition between water and steam.Applying scalar field theory and the renormalization group, we show that these critical exponents imply predictions for the auto-correlations of financial market returns and for Hurst exponents. For a network topology of RD, consistency with observation implies a fractal dimension of the network of D≈3, and a correlation time of at least the length of the economic cycle. However, while this simplest model agrees well with market data on very long and on short time scales, it does not explain the observed market trends on intermediate time horizons from one month to one year.In a next step, the approach should therefore be extended to other models of critical dynamics, to general network topologies, and to the neighbourhood of the critical point. It allows us to indirectly measure universal properties of the hidden social network of investors from the empirically observable scaling behaviour of financial markets.

Quantum and Classical Temporal Correlations in (1+1)D Quantum Cellular Automata.

We employ (1+1)-dimensional quantum cellular automata to study the evolution of entanglement and coherence near criticality in quantum systems that display nonequilibrium steady-state phase transitions. This construction permits direct access to the entire space-time structure of the underlying nonequilibrium dynamics, and allows for the analysis of unconventional correlations, such as entanglement in the time direction between the "present" and the "past." We show how the uniquely quantum part of these correlations-the coherence-can be isolated and that, close to criticality, its dynamics displays a universal power-law behavior on approach to stationarity. Focusing on quantum generalizations of classical nonequilibrium systems: the Domany-Kinzel cellular automaton and the Bagnoli-Boccara-Rechtman model, we estimate the universal critical exponents for both the entanglement and coherence. As these models belong to the one-dimensional directed percolation universality class, the latter provides a key new critical exponent, one that is unique to quantum systems.

Universality classes of the Anderson transition in the three-dimensional symmetry classes AIII, BDI, C, D, and CI

We clarify universal critical properties of delocalization-localization transitions in three-dimensional unitary and orthogonal classes with particle-hole and/or chiral symmetries (classes AIII, BDI, D, C, and CI). We first introduce tight-binding models on cubic lattice that belong to these five nonstandard symmetry classes, respectively. Unlike the Bogoliubov-de Gennes Hamiltonian for superconductors, all the five models have finite areas of Fermi surfaces in the momentum space in the clean limit. Thereby, the scaling theory of the Anderson transition guarantees the presence of the delocalization-localization transitions at finite disorder strength in these models. Based on this expectation, we carry out extensive transfer matrix calculations of the Lyapunov exponents for zero-energy eigenstates of the disordered tight-binding models with quasi-one-dimensional geometry. Near the Anderson transition point, the correlation length diverges with a universal critical exponent $\ensuremath{\nu}$. Using finite-size scaling analysis of the localization length, we determine the critical exponent of the transitions and scaling function of the (normalized) localization length in the three nonstandard unitary symmetry classes: ${\ensuremath{\nu}}_{\text{AIII}}=1.06\ifmmode\pm\else\textpm\fi{}0.02$, ${\ensuremath{\nu}}_{\text{D}}=0.87\ifmmode\pm\else\textpm\fi{}0.03$, and ${\ensuremath{\nu}}_{\text{C}}=0.996\ifmmode\pm\else\textpm\fi{}0.012$. Our result of the class C is consistent with a previous study of classical network model [M. Ortu\~no et al., Phys. Rev. Lett. 102, 070603 (2009)]. The critical exponents of the two nonstandard orthogonal classes are estimated as ${\ensuremath{\nu}}_{\text{CI}}=1.17\ifmmode\pm\else\textpm\fi{}0.02$ and ${\ensuremath{\nu}}_{\text{BDI}}=1.12\ifmmode\pm\else\textpm\fi{}0.06$. Our result of the class CI is consistent with another previous study of a lattice model, while the exponent of the class BDI is at variance with the previous evaluation of nodal Dirac ring model [X. Luo et al., Phys. Rev. B 101, 020202(R) (2020)].

Planar random-cluster model: fractal properties of the critical phase

This paper is studying the critical regime of the planar random-cluster model on {mathbb {Z}}^2 with cluster-weight qin [1,4). More precisely, we prove crossing estimates in quads which are uniform in their boundary conditions and depend only on their extremal lengths. They imply in particular that any fractal boundary is touched by macroscopic clusters, uniformly in its roughness or the configuration on the boundary. Additionally, they imply that any sub-sequential scaling limit of the collection of interfaces between primal and dual clusters is made of loops that are non-simple. We also obtain a number of properties of so-called arm-events: three universal critical exponents (two arms in the half-plane, three arms in the half-plane and five arms in the bulk), quasi-multiplicativity and well-separation properties (even when arms are not alternating between primal and dual), and the fact that the four-arm exponent is strictly smaller than 2. These results were previously known only for Bernoulli percolation (q = 1) and the FK-Ising model (q = 2). Finally, we prove new bounds on the one, two and four-arm exponents for qin [1,2], as well as the one-arm exponent in the half-plane. These improve the previously known bounds, even for Bernoulli percolation.

Liquid–Liquid Demonstrations: Phase Equilibria and the Lever Rule

The lever rule applies to any two-component system existing in a state of two-phase equilibrium. Here, several classroom demonstrations are described to help students understand liquid–liquid coexistence and the lever rule using a set of isobutyric acid + water + dye samples ranging in composition from water + dye to isobutyric acid + dye. The critical temperature of the partially miscible isobutyric acid + water system is just above room temperature, so students can perceive the general shape of the coexistence curve by watching a progression of phase separations take place over time as the set of prewarmed single-phase samples are allowed to cool. Qualitative visualization of the lever rule is demonstrated by showing students the set of samples after equilibration at a temperature well below the critical temperature. An interactive Excel spreadsheet is provided to help students connect their observations of the samples with the phase diagram. Quantitative and intuitive interpretation of the lever rule can be achieved through height measurements of the biphasic sample layers, leading to immediate graphical determination of the compositions of the isothermal equilibrated phases in the classroom. The set of samples may also be used in an extended classroom or laboratory project to map out the coexistence curve, and advanced students can use the coexistence data to estimate the critical composition of the system, as well as the universal critical exponent β. Finally, preparation of a stable hexaphasic immiscible liquid mixture is described.

Liquid–Liquid Demonstrations: Critical Opalescence

In the vicinity of a critical point, the impact of a substance’s chemical identity is superseded by universal critical behavior for a number of physical properties, such as visible light scattering. In this classroom demonstration, students observe a single-phase critical mixture of partially miscible isobutyric acid + water (IAW) as the liquid–liquid sample cools down to its critical temperature. Students see the sample change from colorless and transparent, to pale blue, to white and turbid. This optical behavior is called critical opalescence. No pressure control is necessary for the demonstration since the system is a condensed phase, so the sample can be large (100 mL here) and suitable for classroom visualization. The experiment is easy to set up and highly repeatable because the critical temperature of the IAW system is just above room temperature. The demonstration can be used to show the difference in angular dependence for light scattering from a single-phase sample versus a two-phase sample containing a suspension of droplets, and also to segue to a conversation on why the sky is blue. Furthermore, the critical IAW sample may be incorporated into a quantitative classroom demonstration or laboratory project in which laser light scattering measured at 90° of incidence is used to calculate the correlation length amplitude ξ0 of the IAW system and the universal critical exponent ν.

Strain tuning of nematicity and superconductivity in single crystals of FeSe

Strain is a powerful experimental tool to explore new electronic states and understand unconventional superconductivity. Here, we investigate the effect of uniaxial strain on the nematic and superconducting phase of single crystal FeSe using magnetotransport measurements. We find that the resistivity response to the strain is strongly temperature dependent and it correlates with the sign change in the Hall coefficient being driven by scattering, coupling with the lattice and multiband phenomena. Band structure calculations suggest that under strain the electron pockets develop a large in-plane anisotropy as compared with the hole pocket. Magnetotransport studies at low temperatures indicate that the mobility of the dominant carriers increases with tensile strain. Close to the critical temperature, all resistivity curves at constant strain cross in a single point, indicating a universal critical exponent linked to a strain-induced phase transition. Our results indicate that the superconducting state is enhanced under compressive strain and suppressed under tensile strain, in agreement with the trends observed in FeSe thin films and overdoped pnictides, whereas the nematic phase seems to be affected in the opposite way by the uniaxial strain. By comparing the enhanced superconductivity under strain of different systems, our results suggest that strain on its own cannot account for the enhanced high $T_c$ superconductivity of FeSe systems.

Continued Functions and Perturbation Series: Simple Tools for Convergence of Diverging Series in O(n)-Symmetric $$\phi ^4$$ Field Theory at Weak Coupling Limit

We determine universal critical exponents that describe the continuous phase transitions in different dimensions of space. We use continued functions without any external unknown parameters to obtain analytic continuation for the recently derived 7- loop $\epsilon$ expansion from $O(n)$-symmetric $\phi^4$ field theory. Employing a new blended continued function, we obtain critical exponent $\alpha=-0.0121(22)$ for the phase transition of superfluid helium which matches closely with the most accurate experimental value. This result addresses the long-standing discrepancy between the theoretical predictions and precise experimental result of $O(2)$ $\phi^4$ model known as "$\lambda$-point specific heat experimental anomaly". Further we have also examined the applicability of such continued functions in other examples of field theories.

Order parameter for the multichannel Kondo model at quantum criticality

A multichannel Kondo model, where two or more equivalent but independent channels of electrons compete to screen a spin-1/2 impurity, shows overcompensation of the impurity spin, leading to the non-Fermi-liquid behavior in various thermodynamic and transport properties. However, when the channel symmetry is broken, an impurity quantum phase transition can occur at zero temperature. Identification of an order parameter describing the impurity quantum phase transition is very difficult since it is beyond the conventional Landau-Ginzburg-Wilson theory. By employing the natural orbitals renormalization group method, we study both two-channel and threechannel Kondo models, from the perspective of spin correlation between the impurity and electrons in electronic channels. Here we demonstrate that by introducing the spin-correlation ratio as an order parameter we can characterize impurity quantum phase transitions driven by channel asymmetry. In particular, the universal critical exponents $\beta$ of the spin-correlation ratio and $\nu$ of the correlation length are explicitly determined by finite-sizescaling analysis, namely, $\beta = 0.10(1), \nu = 2.0(1)$, and $\beta = 0.10(1), \nu = 2.5(1)$ for the two-channel and three-channel Kondo models, respectively.