Critical Scaling Behavior Research Articles

A classical density functional theory for solvation across length scales.

A central aim of multiscale modeling is to use results from the Schrödinger equation to predict phenomenology on length scales that far exceed those of typical molecular correlations. In this work, we present a new approach rooted in classical density functional theory (cDFT) that allows us to accurately describe the solvation of apolar solutes across length scales. Our approach builds on the Lum-Chandler-Weeks (LCW) theory of hydrophobicity [K. Lum et al., J. Phys. Chem. B 103, 4570 (1999)] by constructing a free energy functional that uses a slowly varying component of the density field as a reference. From a practical viewpoint, the theory we present is numerically simpler and generalizes to solutes with soft-core repulsion more easily than LCW theory. Furthermore, by assessing the local compressibility and its critical scaling behavior, we demonstrate that our LCW-style cDFT approach contains the physics of critical drying, which has been emphasized as an essential aspect of hydrophobicity by recent theories. As our approach is parameterized on the two-body direct correlation function of the uniform fluid and the liquid-vapor surface tension, it straightforwardly captures the temperature dependence of solvation. Moreover, we use our theory to describe solvation at a first-principles level on length scales that vastly exceed what is accessible to molecular simulations.

Critical surface adsorption of confined binary liquids with locally conserved mass and composition

Close to a solid surface, the properties of a fluid deviate significantly from their bulk values. In this context, we study the surface adsorption profiles of a symmetric binary liquid confined to a slit pore by means of molecular dynamics simulations; the latter naturally entails that mass and concentration are locally conserved. Near a bulk consolute point, where the liquid exhibits a demixing transition with the local concentration as the order parameter, we determine the order parameter profiles and characterise the relevant critical scaling behaviour, in the regime of strong surface attraction, for a range of pore widths and temperatures. The obtained order parameter profiles decay monotonically near the surfaces, also in the presence of a pronounced layering in the number density. Overall, our results agree qualitatively with recent theoretical predictions from a mesoscopic field-theoretical approach for the canonical ensemble.

Scaling towards the critical point in the combined reaction/Gibbs ensemble

We explore the impact of force field parameters and reaction equilibrium on the scaling behavior towards the critical point in reactive binary systems, focusing on NO2/N2O4. This system can be considered as a special single-component system since NO2 and N2O4 are in chemical equilibrium via the chemical reaction 2NO2⇌N2O4. We simplify the system by representing both components as single LJ particles, achieving excellent agreement with densities computed using molecular simulations in which all-atom force fields were used. We investigate the effect of force field parameters (ɛ and σ) on phase behavior and show that the critical exponent β remains constant, which means that intermolecular interactions do not affect the scaling to the critical point when the chemical reaction takes place. We also investigate the sensitivity of the reaction equilibrium constant and show that even small changes in isolated molecule partition functions lead to large differences in chemical equilibria. We show that the critical exponent β is different for systems with different reaction equilibrium constants, so a careful parameterization of β is needed for an accurate computation of critical temperatures of reactive mixtures. We perform a screening of reactive binary mixtures for a wide range of ideal gas reaction equilibrium constants, revealing key insights into the thermodynamic behavior and critical properties. Thereby we facilitate the efficient screening of reactive binary mixtures for various applications. Our results emphasize the importance of accurately parameterizing β and provide valuable insights into the critical scaling behavior of complex reactive systems.

Sweeny dynamics for the random-cluster model with small Q.

The Sweeny algorithm for the Q-state random-cluster model in two dimensions is shown to exhibit a rich mixture of critical dynamical scaling behaviors. As Q decreases, the so-called critical speeding-up for nonlocal quantities becomes more and more pronounced. However, for some quantity of a specific local pattern, e.g., the number of half faces on the square lattice, we observe that, as Q→0, the integrated autocorrelation time τ diverges as Q^{-ζ}, with ζ≃1/2, leading to the nonergodicity of the Sweeny method for Q→0. Such Q-dependent critical slowing-down, attributed to the peculiar form of the critical bond weight v=sqrt[Q], can be eliminated by a combination of the Sweeny and the Kawasaki algorithm. Moreover, by classifying the occupied bonds into bridge bonds and backbone bonds, and the empty bonds into internal-perimeter bonds and external-perimeter bonds, one can formulate an improved version of the Sweeny-Kawasaki method such that the autocorrelation time for any quantity is of order O(1).

Charge fluctuation and charge-resolved entanglement in a monitored quantum circuit with U(1) symmetry

We study a ($1+1$)-dimensional quantum circuit consisting of Haar-random unitary gates and projective measurements, both of which conserve a total $U(1)$ charge and thus have $U(1)$ symmetry. In addition to a measurement-induced entanglement transition between a volume-law and an area-law entangled phase, we find a phase transition between two phases characterized by bipartite charge fluctuation growing with the subsystem size or staying constant. At this charge-fluctuation transition, steady-state quantities obtained by evolving an initial state with a definitive total charge exhibit critical scaling behaviors akin to Tomonaga-Luttinger-liquid theory for equilibrium critical quantum systems with $U(1)$ symmetry, such as logarithmic scaling of bipartite charge fluctuation, power-law decay of charge correlation functions, and logarithmic scaling of charge-resolved entanglement whose coefficient becomes a universal quadratic function in a flux parameter. These critical features, however, do not persist below the transition, in contrast to a recent prediction based on replica field theory and mapping to a classical statistical mechanical model.

Connecting Complex Electronic Pattern Formation to Critical Exponents

Scanning probes reveal complex, inhomogeneous patterns on the surface of many condensed matter systems. In some cases, the patterns form self-similar, fractal geometric clusters. In this paper, we advance the theory of criticality as it pertains to those geometric clusters (defined as connected sets of nearest-neighbor aligned spins) in the context of Ising models. We show how data from surface probes can be used to distinguish whether electronic patterns observed at the surface of a material are confined to the surface, or whether the patterns originate in the bulk. Whereas thermodynamic critical exponents are derived from the behavior of Fortuin–Kasteleyn (FK) clusters, critical exponents can be similarly defined for geometric clusters. We find that these geometric critical exponents are not only distinct numerically from the thermodynamic and uncorrelated percolation exponents, but that they separately satisfy scaling relations at the critical fixed points discussed in the text. We furthermore find that the two-dimensional (2D) cross-sections of geometric clusters in the three-dimensional (3D) Ising model display critical scaling behavior at the bulk phase transition temperature. In particular, we show that when considered on a 2D slice of a 3D system, the pair connectivity function familiar from percolation theory displays more robust critical behavior than the spin-spin correlation function, and we calculate the corresponding critical exponent. We discuss the implications of these two distinct length scales in Ising models. We also calculate the pair connectivity exponent in the clean 2D case. These results extend the theory of geometric criticality in the clean Ising universality classes, and facilitate the broad application of geometric cluster analysis techniques to maximize the information that can be extracted from scanning image probe data in condensed matter systems.

Modifying Critical Exponents of Magnetic Phase Transitions via Nanoscale Materials Design.

We demonstrate a nanoscale materials design path that allows us to bypass universality in thin ferromagnetic films and enables us to tune the critical exponents of ferromagnetic phase transitions in a very wide parameter range, while at the same time preserving scaling in an extended phase space near the Curie temperature. Our detailed magnetometry results reveal that single crystal CoRu alloy films, in which the predefined depth dependent exchange coupling strength follows a V-shaped profile, exhibit critical scaling behavior over many orders of magnitude. Their critical exponents, however, can be designed and controlled by modifying their specific nanoscale structures, thus demonstrating full tunability of critical behavior. The reason for this tunability and the disappearance of universality is shown to be the competing relevance of collective versus interface propagating progression of ferromagnetic phase transitions, whose balance we find to be dependent on the specifics of the underlying exchange coupling strength profile.

Phase transition in the binary mixture of jammed particles with large size dispersity

It has been well established that particulate systems show the jamming transition and critical scaling behaviors associated with it. However, our knowledge is limited to (nearly) monodisperse systems. Recently, a binary mixture of jammed particles with large size dispersity was studied, and it was suggested that two distinct jammed phases appeared. Here, we conduct a thorough numerical study on this system with a special focus on the statistics of and finite-size effects on the fraction of small particles that participate in the rigid network. We present strong evidence that two distinct jammed phases appear depending on the pressure and composition of two species, which are separated by the first-order phase transition. In one of two phases, only large particles are jammed, whereas both small and large particles are jammed in the other phase. We also describe the phase diagram in the pressure-composition plane, where the first-order phase transition line terminates at a critical point. In addition, we investigate the mechanical properties in terms of the elastic moduli over the phase diagram and find that discontinuous changes in elastic moduli emerge across the phase transition. Remarkably, despite the discontinuities, the elastic moduli in each jammed phase exhibit identical scaling laws to those in the monodisperse systems.

From the transverse field Ising chain to the quantum <i>E</i><sub>8</sub> integrable model

This review reports a series of theoretical and experimental progress on researches of the transverse field Ising chain (TFIC) and the quantum <i>E</i><sub>8</sub> integrable model. For the TFIC, on one hand, a unique exotic quantum critical behavior of Grüneisen ratio (a ratio from magnetic or thermal expansion coefficient to specific heat) is theoretically established; on the other hand microscopic models can accommodate the TFIC universality class are substantially expanded. These progresses successfully promote a series of experiments collaborations to first-time realize the TFIC universality class in quasi one-dimensional anti-ferromagnetic material BaCo<sub>2</sub>V<sub>2</sub>O<sub>8</sub> and SrCo<sub>2</sub>V<sub>2</sub>O<sub>8</sub>. For the quantum <i>E</i><sub>8</sub> integrable model, the low temperature local dynamics and the dynamical structure factor with zero transfer momentum of this system are analytically determined, where a cascade of edge singularities with power-law divergences are obtained in the continuum region of the dynamical structure factor. After combining with detailed quantum critical scaling behaviors analysis and large scale iTEBD calculation, it successfully facilitates a series of experiments, including THz spectrum measurements, inelastic neutron scattering and NMR experiments, to realize the quantum <i>E</i><sub>8</sub> integrable model in BaCo<sub>2</sub>V<sub>2</sub>O<sub>8</sub> for the first time. The experimental realization of the quantum <i>E</i><sub>8</sub> integrable model substantially extends the frontiers of studying quantum integrable models in real materials. The series of progress and developments on the TFIC and the quantum <i>E</i><sub>8</sub> integrable model lay down a concrete ground to go beyond quantum integrability, and can inspire studies in condensed matter systems, cold atom systems, statistical field theory and conformal field theory.

Critical dynamics of the nonconserved strongly anisotropic permutation symmetric three-vector model

We explore, employing the renormalization-group theory, the critical scaling behavior of the permutation symmetric three-vector model that obeys nonconserving dynamics and has a relevant anisotropic perturbation which drives the system into a nonequilibrium steady state. We explicitly find the independent critical exponents with corrections up to two loops. They include the static exponents ν and η, the off equilibrium exponent η[over ̃], the dynamic exponent z, and the strong anisotropy exponent Δ. We also express the other anisotropy exponents in terms of these.

Superuniversality from disorder at two-dimensional topological phase transitions

We investigate the effects of quenched randomness on topological quantum phase transitions in strongly interacting two-dimensional systems. We focus first on transitions driven by the condensation of a subset of fractionalized quasiparticles (`anyons') identified with `electric charge' excitations of a phase with intrinsic topological order. All other anyons have nontrivial mutual statistics with the condensed subset and hence become confined at the anyon condensation transition. Using a combination of microscopically exact duality transformations and asymptotically exact real-space renormalization group techniques applied to these two-dimensional disordered gauge theories, we argue that the resulting critical scaling behavior is `superuniversal' across a wide range of such condensation transitions, and is controlled by the same infinite-randomness fixed point as that of the 2D random transverse-field Ising model. We validate this claim using large-scale quantum Monte Carlo simulations that allow us to extract zero-temperature critical exponents and correlation functions in (2+1)D disordered interacting systems. We discuss generalizations of these results to a large class of ground-state and excited-state topological transitions in systems with intrinsic topological order as well as those where topological order is either protected or enriched by global symmetries. When the underlying topological order and the symmetry group are Abelian, our results provide prototypes for topological phase transitions between distinct many-body localized phases.

Evidence of the Berezinskii-Kosterlitz-Thouless phase in a frustrated magnet

The Berezinskii-Kosterlitz-Thouless (BKT) mechanism, building upon proliferation of topological defects in 2D systems, is the first example of phase transition beyond the Landau-Ginzburg paradigm of symmetry breaking. Such a topological phase transition has long been sought yet undiscovered directly in magnetic materials. Here, we pin down two transitions that bound a BKT phase in an ideal 2D frustrated magnet TmMgGaO4, via nuclear magnetic resonance under in-plane magnetic fields, which do not disturb the low-energy electronic states and allow BKT fluctuations to be detected sensitively. Moreover, by applying out-of-plane fields, we find a critical scaling behavior of the magnetic susceptibility expected for the BKT transition. The experimental findings can be explained by quantum Monte Carlo simulations applied on an accurate triangular-lattice Ising model of the compound which hosts a BKT phase. These results provide a concrete example for the BKT phase and offer an ideal platform for future investigations on the BKT physics in magnetic materials.

Mechanical and Vibrational Properties of Three-Dimensional Dimer Packings Near the Jamming Transition

We comprehensively study mechanical and vibrational properties of dimer packings in three-dimensional space with particular attention on critical scaling behaviors near the jamming transition. First, we confirm the dependence of the packing fraction at the transition on the aspect ratio, the isostatic contact number at the transition, and the scaling dependence of the excess contact number on the excess density. Second, we study the elastic moduli, bulk and shear moduli, and establish power-law scaling of them. Finally, we study the vibrational density of states and its characteristic frequency. The vibrational density of states shows two plateaus in the lower- and higher-frequency regions, which are characterized by rotational and translational vibrational modes, respectively. The onset frequency of the lower-frequency plateau scales linearly to the square root of excess density. The scaling laws in the mechanical and vibrational properties are consistent between two- and three-dimensional dimers, and they are identical to those in spheres.

LiZn2V3O8: a new geometrically frustrated cluster spin-glass

We have investigated the structural and magnetic properties of a new cubic spinel LiZn2V3O8 (LZVO) through x-ray diffraction, dc and ac susceptibility, magnetic relaxation, aging, memory effect, heat capacity and 7Li nuclear magnetic resonance (NMR) measurements. A Curie–Weiss fit of the dc susceptibility () yields a Curie–Weiss temperature K. This suggests strong antiferromagnetic (AFM) interactions among the magnetic vanadium ions. The dc and ac susceptibility data indicate the spin-glass behavior below a freezing temperature Tf 3 K. The frequency dependence of the Tf is characterized by the Vogel–Fulcher law and critical dynamic scaling behavior or power law. From both fitting, we obtained the value of the characteristic angular frequency 3.56 106 Hz, the dynamic exponent 2.65, and the critical time constant 1.82 10−6 s, which falls in the conventional range for typical cluster spin-glass (CSG) systems. The value of relative shift in freezing temperature 0.039 supports a CSG ground states. We also found aging phenomena and memory effects in LZVO. The asymmetric response of the magnetic relaxation below Tf supports the hierarchical model. Heat capacity data show no long-range or short-range ordering down to 2 K. Only about 25% magnetic entropy change signifies the presence of strong frustration in the system. The 7Li NMR spectra show a shift and broadening with decreasing temperature. The spin-lattice and spin–spin relaxation rates show anomalies due to spin freezing around 3 K as the bulk magnetization.

Critical Scaling Behaviors of Entanglement Spectra**Supported by the start-up funding from Westlake University, and the National Natural Science Foundation of China under Grant No 11974288.

We investigate the evolution of entanglement spectra under a global quantum quench from a short-range correlated state to the quantum critical point. Motivated by the conformal mapping, we find that the dynamical entanglement spectra demonstrate distinct finite-size scaling behaviors from the static case. As a prototypical example, we compute real-time dynamics of the entanglement spectra of a one-dimensional transverse-field Ising chain. Numerical simulation confirms that the entanglement spectra scale with the subsystem size l as ∼l−1 for the dynamical equilibrium state, much faster than ∝ ln−1 l for the critical ground state. In particular, as a byproduct, the entanglement spectra at the long time limit faithfully gives universal tower structure of underlying Ising criticality, which shows the emergence of operator-state correspondence in the quantum dynamics.

Itinerant quantum critical point with fermion pockets and hotspots

Metallic quantum criticality is among the central themes in the understanding of correlated electronic systems, and converging results between analytical and numerical approaches are still under review. In this work, we develop a state-of-the-art large-scale quantum Monte Carlo simulation technique and systematically investigate the itinerant quantum critical point on a 2D square lattice with antiferromagnetic spin fluctuations at wavevector [Formula: see text]-a problem that resembles the Fermi surface setup and low-energy antiferromagnetic fluctuations in high-Tc cuprates and other critical metals, which might be relevant to their non-Fermi-liquid behaviors. System sizes of [Formula: see text] ([Formula: see text]) are comfortably accessed, and the quantum critical scaling behaviors are revealed with unprecedented high precision. We found that the antiferromagnetic spin fluctuations introduce effective interactions among fermions and the fermions in return render the bare bosonic critical point into a different universality, different from both the bare Ising universality class and the Hertz-Mills-Moriya RPA prediction. At the quantum critical point, a finite anomalous dimension [Formula: see text] is observed in the bosonic propagator, and fermions at hotspots evolve into a non-Fermi liquid. In the antiferromagnetically ordered metallic phase, fermion pockets are observed as the energy gap opens up at the hotspots. These results bridge the recent theoretical and numerical developments in metallic quantum criticality and can serve as the stepping stone toward final understanding of the 2D correlated fermions interacting with gapless critical excitations.

On the viscosity of adhesive hard sphere dispersions: Critical scaling and the role of rigid contacts

The viscosity of colloidal dispersions is a highly sensitive function of the microstructure of the dispersion and the nature of the interactions among the particles. While the viscosity of solutions of colloidal hard spheres is well understood, the viscosity in a broader array of colloidal solutions remains poorly studied. Here, we use immersed boundary simulations to investigate the viscosity of a prototypical colloidal solution, adhesive hard spheres, which under the right conditions can percolate to form sample spanning networks. We examine how rigidity of the particle–particle contacts in these dispersions can enhance the dispersion viscosity. By rigidly constraining a fraction of the adhered particles, we establish a reaction coordinate, the extent of rigidity, along which we can track the development of a gelled state characterized by a diverging viscosity. The fraction of rigid bonds is analogous to the extent of reaction for cross-linking in chemical gels, which suggests a close connection between chemical gelation with polymers and physical gelation with adhesive hard spheres. A critical gel point is identified when particles connected by rigid bonds begin to percolate the sample. In the vicinity of critical gel point, we observe a diverging viscosity and critical scaling behavior in agreement with previous studies on adhesive hard spheres. We use these results to formulate empirical equations for the viscosity across a broad range of volume fractions as a function of the extent of rigidity. Most importantly, through these simulations, we show that rigid constraints among the bonded particles are essential to producing the large viscosities measured experimentally in dispersions of adhesive hard spheres. Hydrodynamic interactions alone are insufficient for reproducing experimental observations.

Numerical simulations study of a spin-1 Blume–Emery–Griffiths model on a square lattice

We use the Monte Carlo simulation technique to study the critical behavior of a three-state spin model, with bilinear and biquadratic nearest-neighbor interactions, known as the Blume–Emery–Griffiths model (BEG), in a square lattice. In order to characterize this model, we study the phase diagram, in which we identify three different phases: ferromagnetic, paramagnetic and quadrupolar, the later with one sublattice filled with spins and the other with vacancies. We perform our studies by using two algorithms: Metropolis update (MU) and Wang–Landau (WL). The critical scaling behavior of the model is complementary studied by applying results obtained by using both algorithms, while tricritical points and the tricritical scaling behavior is analyzed by means of WL measuring the joint density of states and using the method of field mixing in conjunction with finite-size scaling. Furthermore, motivated by the decoupling between spins observed within the quadrupolar phase, we further generalize the BEG model in order to study the behavior of the system by adding a next nearest neighbour (NNN) interaction between spins. We found that by increasing the strength of the (ferromagnetic) NNN interaction, a new ferromagnetic phase takes over that contains both quadrupolar and ferromagnetic order.

Universality and scaling in human and social systems

The objective of statistical physics is to understand macroscopic behavior of a many-body system from the interactions of the constituents of that system. When many-body systems reach critical states, simple universal and scaling behaviors appear. In this talk, I first introduce the concepts of universality and scaling in critical physical systems, I then briefly review some examples of universal and scaling behaviors in human and social systems, e.g. universal crossover behavior of stock returns, universality and scaling in the statistical data of literary works, universal trend in the evolution of states or countries etc. Finally, I mention some interesting problems for further studies.

Crossovers and critical scaling in the one-dimensional transverse-field Ising model

We consider the scaling behavior of thermodynamic quantities in the one-dimensional transverse-field Ising model near its quantum critical point (QCP). Our study has been motivated by the question about the thermodynamical signatures of this paradigmatic quantum critical system and, more generally, by the issue of how quantum criticality accumulates entropy. We find that the crossovers in the phase diagram of temperature and (the non-thermal control parameter) transverse field obey a general scaling ansatz, and so does the critical scaling behavior of the specific heat and magnetic expansion coefficient. Furthermore, the Gr\"{u}neisen ratio diverges in a power-law way when the QCP is accessed as a function of the transverse field at zero temperature, which follows the prediction of quantum critical scaling. However, at the critical field, upon decreasing the temperature, the Gr\"uneisen ratio approaches a constant instead of showing the expected divergence. We are able to understand this unusual result in terms of a peculiar form of the quantum critical scaling function for the free energy; the contribution to the Gr\"uneisen ratio vanishes at the linear order in a suitable Taylor expansion of the scaling function. In spite of this special form of the scaling function, we show that the entropy is still maximized near the QCP, as expected from the general scaling argument. Our results establish the telltale thermodynamic signature of a transverse-field Ising chain, and will thus facilitate the experimental identification of this model quantum-critical system in real materials.