Discrete scale invariance of the quasi-bound states at atomic vacancies in a topological material.

Recently, log-periodic quantum oscillations have been detected in the topological materials zirconium pentatelluride (ZrTe5) and hafnium pentatelluride (HfTe5), displaying an intriguing discrete scale invariance (DSI) characteristic. In condensed materials, the DSI is considered to be related to the quasi-bound states formed by massless Dirac fermions with strong Coulomb attraction, offering a feasible platform to study the long-pursued atomic-collapse phenomenon. Here, we demonstrate that a variety of atomic vacancies in the topological material HfTe5 can host the geometric quasi-bound states with a DSI feature, resembling an artificial supercritical atom collapse. The density of states of these quasi-bound states is enhanced, and the quasi-bound states are spatially distributed in the "orbitals" surrounding the vacancy sites, which are detected and visualized by low-temperature scanning tunneling microscope/spectroscopy. By applying the perpendicular magnetic fields, the quasi-bound states at lower energies become wider and eventually invisible; meanwhile, the energies of quasi-bound states move gradually toward the Fermi energy (EF). These features are consistent with the theoretical prediction of a magnetic field-induced transition from supercritical to subcritical states. The direct observation of geometric quasi-bound states sheds light on the deep understanding of the DSI in quantum materials.

A simplex path integral and a simplex renormalization group for high-order interactions.

Modern theories of phase transitions and scale invariance are rooted in path integral formulation and renormalization groups (RGs). Despite the applicability of these approaches in simple systems with only pairwise interactions, they are less effective in complex systems with undecomposable high-order interactions (i.e. interactions among arbitrary sets of units). To precisely characterize the universality of high-order interacting systems, we propose a simplex path integral and a simplex RG (SRG) as the generalizations of classic approaches to arbitrary high-order and heterogeneous interactions. We first formalize the trajectories of units governed by high-order interactions to define path integrals on corresponding simplices based on a high-order propagator. Then, we develop a method to integrate out short-range high-order interactions in the momentum space, accompanied by a coarse graining procedure functioning on the simplex structure generated by high-order interactions. The proposed SRG, equipped with a divide-and-conquer framework, can deal with the absence of ergodicity arising from the sparse distribution of high-order interactions and can renormalize a system with intertwined high-order interactions at thep-order according to its properties at theq-order (p⩽q). The associated scaling relation and its corollaries provide support to differentiate among scale-invariant, weakly scale-invariant, and scale-dependent systems across different orders. We validate our theory in multi-order scale-invariance verification, topological invariance discovery, organizational structure identification, and information bottleneck analysis. These experiments demonstrate the capability of our theory to identify intrinsic statistical and topological properties of high-order interacting systems during system reduction.

Scaling features in high-concentrations PM2.5 evolution: the Ignored factor affecting scarlet fever incidence.

As an acute respiratory disease, scarlet fever has great harm to public health. Some evidence indicates that the time distribution pattern of heavy PM2.5 pollution occurrence may have an impact on health risks. This study aims to reveal the relation between scaling features in high-concentrations PM2.5 (HC-PM2.5) evolution and scarlet fever incidence (SFI). Based on the data of Hong Kong from 2012 to 2019, fractal box-counting dimension (D) is introduced to capture the scaling features of HC-PM2.5. It has been found that index D can quantify the time distribution of HC-PM2.5, and lower D values indicate more cluster distribution of HC-PM2.5. Moreover, scale-invariance in HC-PM2.5 at different time scales has been discovered, which indicates that HC-PM2.5 occurrence is not random but follows a typical power-law distribution. Next, the exposure-response relationship between SFI and scale-invariance in HC-PM2.5 is explored by Distributed lag non-linear model, in conjunction with meteorological factors. It has been discovered that scale-invariance in HC-PM2.5 has a nonlinear effect on SFI. Low and moderate D values of HC-PM2.5 are identified as risk factors for SFI at small time-scale. Moreover, relative risk shows a decreasing trend with the increase of exposure time. These results suggest that exposure to short-term clustered HC-PM2.5 makes individual more prone to SFI than exposure to long-term uniform HC-PM2.5. This means that individuals in slightly-polluted regions may face a greater risk of SFI, once the PM2.5 concentration keeps rising. In the future, it is expected that the relative risk of scarlet fever for a specific region can be estimated based on the quantitative analysis of scaling features in high-concentrations PM2.5 evolution.

A Maximum Entropy Resolution to the Wine/Water Paradox.

The Principle of Indifference ('PI': the simplest non-informative prior in Bayesian probability) has been shown to lead to paradoxes since Bertrand (1889). Von Mises (1928) introduced the 'Wine/Water Paradox' as a resonant example of a 'Bertrand paradox', which has been presented as demonstrating that the PI must be rejected. We now resolve these paradoxes using a Maximum Entropy (MaxEnt) treatment of the PI that also includes information provided by Benford's 'Law of Anomalous Numbers' (1938). We show that the PI should be understood to represent a family of informationally identical MaxEnt solutions, each solution being identified with its own explicitly justified boundary condition. In particular, our solution to the Wine/Water Paradox exploits Benford's Law to construct a non-uniform distribution representing the universal constraint of scale invariance, which is a physical consequence of the Second Law of Thermodynamics.

First Digits' Shannon Entropy.

Related to the letters of an alphabet, entropy means the average number of binary digits required for the transmission of one character. Checking tables of statistical data, one finds that, in the first position of the numbers, the digits 1 to 9 occur with different frequencies. Correspondingly, from these probabilities, a value for the Shannon entropy H can be determined as well. Although in many cases, the Newcomb-Benford Law applies, distributions have been found where the 1 in the first position occurs up to more than 40 times as frequently as the 9. In this case, the probability of the occurrence of a particular first digit can be derived from a power function with a negative exponent p > 1. While the entropy of the first digits following an NB distribution amounts to H = 2.88, for other data distributions (diameters of craters on Venus or the weight of fragments of crushed minerals), entropy values of 2.76 and 2.04 bits per digit have been found.

Fractal biomarker of activity in patients with bipolar disorder.

BACKGROUND: The output of many healthy physiological systems displays fractal fluctuations with self-similar temporal structures. Altered fractal patterns are associated with pathological conditions. There is evidence that patients with bipolar disorder have altered daily behaviors. METHODS: To test whether fractal patterns in motor activity are altered in patients with bipolar disorder, we analyzed 2-week actigraphy data collected from 106 patients with bipolar disorder type I in a euthymic state, 73 unaffected siblings of patients, and 76 controls. To examine the link between fractal patterns and symptoms, we analyzed 180-day actigraphy and mood symptom data that were simultaneously collected from 14 patients. RESULTS: Compared to controls, patients showed excessive regularity in motor activity fluctuations at small time scales (<1.5 h) as quantified by a larger scaling exponent (α1 > 1), indicating a more rigid motor control system. α1 values of siblings were between those of patients and controls. Further examinations revealed that the group differences in α1 were only significant in females. Sex also affected the group differences in fractal patterns at larger time scales (>2 h) as quantified by scaling exponent α2. Specifically, female patients and siblings had a smaller α2 compared to female controls, indicating more random activity fluctuations; while male patients had a larger α2 compared to male controls. Interestingly, a higher weekly depression score was associated with a lower α1 in the subsequent week. CONCLUSIONS: Our results show sex- and scale-dependent alterations in fractal activity regulation in patients with bipolar disorder. The mechanisms underlying the alterations are yet to be determined.

Isomorph Invariance of Higher-Order Structural Measures in Four Lennard-Jones Systems.

In the condensed liquid phase, both single- and multicomponent Lennard-Jones (LJ) systems obey the "hidden-scale-invariance" symmetry to a good approximation. Defining an isomorph as a line of constant excess entropy in the thermodynamic phase diagram, the consequent approximate isomorph invariance of structure and dynamics in appropriate units is well documented. However, although all measures of the structure are predicted to be isomorph invariant, with few exceptions only the radial distribution function (RDF) has been investigated. This paper studies the variation along isomorphs of the nearest-neighbor geometry quantified by the occurrence of Voronoi structures, Frank-Kasper bonds, icosahedral local order, and bond-orientational order. Data are presented for the standard LJ system and for three binary LJ mixtures (Kob-Andersen, Wahnström, NiY2). We find that, while the nearest-neighbor geometry generally varies significantly throughout the phase diagram, good invariance is observed along the isomorphs. We conclude that higher-order structural correlations are no less isomorph invariant than is the RDF.

Distinguishing cognitive effort and working memory load using scale-invariance and alpha suppression in EEG.

Despite being intuitive, cognitive effort has proven difficult to define quantitatively. Here, we proposed to study cognitive effort by investigating the degree to which the brain deviates from its default state, where brain activity is scale-invariant. Specifically, we measured such deviations by examining changes in scale-invariance of brain activity as a function of task difficulty and posited suppression of scale-invariance as a proxy for exertion of cognitive effort. While there is some fMRI evidence supporting this proposition, EEG investigations on the matter are scant, despite the EEG signal being more suitable for analysis of scale invariance (i.e., having a much broader frequency range). In the current study we validated the correspondence between scale-invariance (H) of cortical activity recorded by EEG and task load during two working memory (WM) experiments with varying set sizes. Then, we used this neural signature to disentangle cognitive effort from the number of items stored in WM within participants. Our results showed monotonic decreases in H with increased set size, even after set size exceeded WM capacity. This behavior of H contrasted with behavioral performance and an oscillatory indicator of WM load (i.e., alpha-band desynchronization), both of which showed a plateau at difficulty levels surpassing WM capacity. This is the first reported evidence for the suppression of scale-invariance in EEG due to task difficulty, and our work suggests that H suppression may be used to quantify changes in cognitive effort even when working memory load is at maximum capacity.

Nonadiabatic Energy Fluctuations of Scale-Invariant Quantum Systems in a Time-Dependent Trap.

We consider the nonadiabatic energy fluctuations of a many-body system in a time-dependent harmonic trap. In the presence of scale-invariance, the dynamics becomes self-similar and the nondiabatic energy fluctuations can be found in terms of the initial expectation values of the second moments of the Hamiltonian, square position, and squeezing operators. Nonadiabatic features are expressed in terms of the scaling factor governing the size of the atomic cloud, which can be extracted from time-of-flight images. We apply this exact relation to a number of examples: the single-particle harmonic oscillator, the one-dimensional Calogero-Sutherland model, describing bosons with inverse-square interactions that includes the non-interacting Bose gas and the Tonks-Girdardeau gas as limiting cases, and the unitary Fermi gas. We illustrate these results for various expansion protocols involving sudden quenches of the trap frequency, linear ramps and shortcuts to adiabaticity. Our results pave the way to the experimental study of nonadiabatic energy fluctuations in driven quantum fluids.

A neural microcircuit model for a scalable scale-invariant representation of time.

Scale-invariant timing has been observed in a wide range of behavioral experiments. The firing properties of recently described time cells provide a possible neural substrate for scale-invariant behavior. Earlier neural circuit models do not produce scale-invariant neural sequences. In this article, we present a biologically detailed network model based on an earlier mathematical algorithm. The simulations incorporate exponentially decaying persistent firing maintained by the calcium-activated nonspecific (CAN) cationic current and a network structure given by the inverse Laplace transform to generate time cells with scale-invariant firing rates. This model provides the first biologically detailed neural circuit for generating scale-invariant time cells. The circuit that implements the inverse Laplace transform merely consists of off-center/on-surround receptive fields. Critically, rescaling temporal sequences can be accomplished simply via cortical gain control (changing the slope of the f-I curve).