Scale Invariant Systems Research Articles

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 the p-order according to its properties at the q-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.

Observation of the Fermionic Joule-Thomson Effect.

We report the first observation of the quantum Joule-Thomson (JT) effect in ideal and unitary Fermi gases. We study the temperature dynamics of these systems while they undergo an energy-per-particle conserving rarefaction. For scale-invariant systems, whose equations of state satisfy the relation U∝PV, this rarefaction conserves the specific enthalpy, which makes it thermodynamically equivalent to a JT throttling process. We observe JT heating in an ideal Fermi gas, a direct consequence of Pauli blocking. In a unitary Fermi gas, we observe that the JT heating is marginal in the temperature range 0.2≲T/T_{F}≲0.8 as the repulsive quantum-statistical effect is lessened by the attractive interparticle interactions.

An Operational Approach to Fractional Scale-Invariant Linear Systems

The fractional scale-invariant systems are introduced and studied, using an operational formalism. It is shown that the impulse and step responses of such systems belong to the vector space generated by some special functions here introduced. For these functions, the fractional scale derivative is a decremental index operator, allowing the construction of an algebraic framework that enables to compute the impulse and step responses of such systems. The effectiveness and accuracy of the method are demonstrated through various numerical simulations.

Fractional Scale Calculus: Hadamard vs. Liouville

A general fractional scale derivative is introduced and studied. Its relation with the Hadamard derivatives is established and reformulated. A new derivative similar to the Grünwald–Letnikov’s is deduced. Tempered versions are also introduced. Scale-invariant systems are described and exemplified. For solving the corresponding differential equations, a new logarithmic Mittag-Leffler series is proposed.

The mechanism of scale-invariance

A system is invariant with respect to an input transformation if we can transform any dynamic input by this function and obtain the same output dynamics after adjusting the initial conditions appropriately. Often, the set of all such input transformations forms a Lie group, the most prominent examples being scale-invariant (u↦epu, p∈R) and translational-invariant (u↦u+p) systems, the latter comprising linear systems with transfer function zeros at the origin. Here, we derive a necessary and sufficient normal form for invariant systems and, by analyzing this normal form, provide a complete characterization of the mechanism by which invariance can be achieved. In this normal form, all invariant systems (i) estimate the applied input transformation by means of an integral feedback, and (ii) then apply the inverse of this estimate to the input before processing it in any other way. We demonstrate our results based on three examples: a scale-invariant “feed-forward loop”, a bistable switch, and a system resembling the core of the mammalian circadian network.

Ideal-Gas Approach to Hydrodynamics

Transport is one of the most important physical processes in all energy and length scales. Ideal gases and hydrodynamics are, respectively, two opposite limits of transport. Here, we present an unexpected mathematical connection between these two limits; that is, there exist situations that the solution to a class of interacting hydrodynamic equations with certain initial conditions can be exactly constructed from the dynamics of noninteracting ideal gases. We analytically provide three such examples. The first two examples focus on scale-invariant systems, which generalize fermionization to the hydrodynamics of strongly interacting systems, and determine specific initial conditions for perfect density oscillations in a harmonic trap. The third example recovers the dark soliton solution in a one-dimensional Bose condensate. The results can explain a recent puzzling experimental observation in ultracold atomic gases by the Paris group and make further predictions for future experiments. We envision that extensive examples of such an ideal-gas approach to hydrodynamics can be found by systematical numerical search, which can find broad applications in different problems in various subfields of physics.

Feedback control of surface roughness in a one-dimensional Kardar-Parisi-Zhang growth process.

Control of generically scale-invariant systems, i.e., targeting specific cooperative features in nonlinear stochastic interacting systems with many degrees of freedom subject to strong fluctuations and correlations that are characterized by power laws, remains an important open problem. We study the control of surface roughness during a growth process described by the Kardar-Parisi-Zhang (KPZ) equation in (1+1) dimensions. We achieve the saturation of the mean surface roughness to a prescribed value using nonlinear feedback control. Numerical integration is performed by means of the pseudospectral method, and the results are used to investigate the coupling effects of controlled (linear) and uncontrolled (nonlinear) KPZ dynamics during the control process. While the intermediate time kinetics is governed by KPZ scaling, at later times a linear regime prevails, namely the relaxation toward the desired surface roughness. The temporal crossover region between these two distinct regimes displays intriguing scaling behavior that is characterized by nontrivial exponents and involves the number of controlled Fourier modes. Due to the control, the height probability distribution becomes negatively skewed, which affects the value of the saturation width.

Quantum many-body conformal dynamics: Symmetries, geometry, conformal tower states, and entropy production

In this article we study the quench dynamics of Galilean and scale invariant many-body systems which can be prepared using interacting atomic gases. The far-away from equilibrium dynamics are investigated by employing $m$-body density matrices, which are most conveniently defined in terms of a special basis - the conformal tower states. We explicitly illustrate that, although during the initial stage of the dynamics all symmetries can be broken and absent in the unitary evolution because of the initialization of the state, there is always an emergent conformal symmetry in the long time limit. The emergence of this dynamic conformal symmetry is robust, and always occurs; it uniquely defines the characteristics of the asymptotic dynamics at a scale invariant fixed point. As an immediate application of the asymptotic dynamics of the microscopic density matrices, we have focused on the effects of this emergent conformal symmetry on two observables: the moment of inertia tensor, $I_{ij}(t)$, $i,j=x,y,z$, and the entropy density field, $S({\bf r}, t)$, in the hydrodynamic flow of strongly interacting particles. We show that the long time behaviour of these observables is completely set by conformal symmetry, while the leading long time corrections depend on interference effects between different conformal tower states. The emergent conformal symmetry naturally leads to entropy conservation, and {\em conformal cooling}, an energy conserving cooling of a strongly interacting gas during free expansion. When the interaction Hamiltonian breaks the scale symmetry, we further demonstrate that there is a direct cause-effect relation between conformal symmetry breaking in the long time limit, and a non-vanishing entropy production. This suggests that the entropy production rate is a natural {\em parameter} for categorizing the breaking of conformal symmetry.

Multipartite entanglement and quantum Fisher information in conformal field theories

Bipartite entanglement entropy of a segment with the length $l$ in $1+1$ dimensional conformal field theories (CFT) follows the formula $S=\frac{c}{3}\ln l+\gamma$, where $c$ is the central charge of the CFT and $\gamma$ is a cut-off dependent constant which diverges in the absence of an ultraviolet cutoff. According to this formula, systems with larger central charges have {\it{more}} bipartite entanglement entropy. Using quantum Fisher information (QFI), we argue that systems with bigger central charges not only have larger bipartite entanglement entropy but also have more multipartite entanglement content. In particular, we argue that since systems with smaller {\it{smallest scaling dimension}} have bigger QFI, the multipartite entanglement content of a CFT is dependent on the value of the smallest scaling dimension present in the spectrum of the system. We show that our argument seems to be consistent with some of the existing results regarding the von Neumann entropy, negativity, and localizable entanglement in $1+1$ dimensions. Furthermore, we also argue that the QFI decays under renormalization group (RG) flow between two unitary CFTs. Finally, we also comment on the non-conformal but scale invariant systems.

Scale invariance in natural and artificial collective systems: a review.

Self-organized collective coordinated behaviour is an impressive phenomenon, observed in a variety of natural and artificial systems, in which coherent global structures or dynamics emerge from local interactions between individual parts. If the degree of collective integration of a system does not depend on size, its level of robustness and adaptivity is typically increased and we refer to it as scale-invariant. In this review, we first identify three main types of self-organized scale-invariant systems: scale-invariant spatial structures, scale-invariant topologies and scale-invariant dynamics. We then provide examples of scale invariance from different domains in science, describe their origins and main features and discuss potential challenges and approaches for designing and engineering artificial systems with scale-invariant properties.

Avalanching Systems with Longer Range Connectivity: Occurrence of a Crossover Phenomenon and Multifractal Finite Size Scaling

Many out-of-equilibrium systems respond to external driving with nonlinear and self-similar dynamics. This near scale-invariant behavior of relaxation events has been modeled through sand pile cellular automata. However, a common feature of these models is the assumption of a local connectivity, while in many real systems, we have evidence for longer range connectivity and a complex topology of the interacting structures. Here, we investigate the role that longer range connectivity might play in near scale-invariant systems, by analyzing the results of a sand pile cellular automaton model on a Newman–Watts network. The analysis clearly indicates the occurrence of a crossover phenomenon in the statistics of the relaxation events as a function of the percentage of longer range links and the breaking of the simple Finite Size Scaling (FSS). The more complex nature of the dynamics in the presence of long-range connectivity is investigated in terms of multi-scaling features and analyzed by the Rank-Ordered Multifractal Analysis (ROMA).

Cornering Gapless Quantum States via Their Torus Entanglement.

The entanglement entropy (EE) has emerged as an important window into the structure of complex quantum states of matter. We analyze the universal part of the EE for gapless systems on tori in 2D and 3D, denoted by χ. Focusing on scale-invariant systems, we derive general nonperturbative properties for the shape dependence of χ and reveal surprising relations to the EE associated with corners in the entangling surface. We obtain closed-form expressions for χ in 2D and 3D within a model that arises in the study of conformal field theories (CFTs), and we use them to obtain Ansätze without fitting parameters for the 2D and 3D free boson CFTs. Our numerical lattice calculations show that the Ansätze are highly accurate. Finally, we discuss how the torus EE can act as a fingerprint of exotic states such as gapless quantum spin liquids, e.g., Kitaev's honeycomb model.

Affine‐scale invariant feature transform and two‐dimensional principal component analysis: a novel framework for affine and scale invariant face recognition

Face recognition (FR) is one of the most effervescent fields of research with extensive applications that span numerous domains, and it stands resolutely as one of the most challenging problems in computer vision. The accuracy of FR systems is severely affected when two images under consideration for a match, vary in their scale and/or affine angles. The prevalent affine and scale invariant recognition systems have been predominantly developed only for objects, and hence in this study, the authors propose a novel approach for faces based on the affine‐SIFT (ASIFT) and two‐dimensional principal component analysis (2DPCA) techniques, to accomplish the formidable task of facial image recognition, invariant of scale and affine angles, i.e. the ability to simulate with enough accuracy, all the distortions caused by the differences in resolution and the variation of the camera optical axis direction. In the formulation of ASIFT‐2DPCA, they investigate three different variants of 2DPCA: classical 2DPCA, quaternion 2DPCA and sparse 2DPCA to gauge as to which is more effective. The authors'experimentations will demonstrate that the proposed approach can robustly handle affine and scale variations, and hence provide better accuracy and matching performance than the state‐of‐the‐art methodologies.

Energy–pressure relation for low-dimensional gases

A particularly simple relation of proportionality between internal energy and pressure holds for scale-invariant thermodynamic systems (with Hamiltonians homogeneous functions of the coordinates), including classical and quantum – Bose and Fermi – ideal gases. One can quantify the deviation from such a relation by introducing the internal energy shift as the difference between the internal energy of the system and the corresponding value for scale-invariant (including ideal) gases. After discussing some general thermodynamic properties associated with the scale-invariance, we provide criteria for which the internal energy shift density of an imperfect (classical or quantum) gas is a bounded function of temperature. We then study the internal energy shift and deviations from the energy–pressure proportionality in low-dimensional models of gases interpolating between the ideal Bose and the ideal Fermi gases, focusing on the Lieb–Liniger model in 1d and on the anyonic gas in 2d. In 1d the internal energy shift is determined from the thermodynamic Bethe ansatz integral equations and an explicit relation for it is given at high temperature. Our results show that the internal energy shift is positive, it vanishes in the two limits of zero and infinite coupling (respectively the ideal Bose and the Tonks–Girardeau gas) and it has a maximum at a finite, temperature-depending, value of the coupling. Remarkably, at fixed coupling the energy shift density saturates to a finite value for infinite temperature. In 2d we consider systems of Abelian anyons and non-Abelian Chern–Simons particles: as it can be seen also directly from a study of the virial coefficients, in the usually considered hard-core limit the internal energy shift vanishes and the energy is just proportional to the pressure, with the proportionality constant being simply the area of the system. Soft-core boundary conditions at coincident points for the two-body wavefunction introduce a length scale, and induce a non-vanishing internal energy shift: the soft-core thermodynamics is considered in the dilute regime for both the families of anyonic models and in that limit we can show that the energy–pressure ratio does not match the area of the system, opposed to what happens for hard-core (and in particular 2d Bose and Fermi) ideal anyonic gases.

Circular Kardar-Parisi-Zhang equation as an inflating, self-avoiding ring polymer

We consider the Kardar-Parisi-Zhang equation for a circular interface in two dimensions, unconstrained by the standard small-slope and no-overhang approximations. Numerical simulations using an adaptive scheme allow us to elucidate the complete time evolution as a crossover between a short-time regime with the interface fluctuations of a self-avoiding ring or two-dimensional vesicle, and a long-time regime governed by the Tracy-Widom distribution expected for this geometry. For small-noise amplitudes, scaling behavior is only of the latter type. Large noise is also seen to renormalize the bare physical parameters of the ring, akin to analogous parameter renormalization for equilibrium three-dimensional membranes. Our results bear particular importance on the relation between relevant universality classes of scale-invariant systems in two dimensions.

Low-frequency optical conductivity in graphene and in other scale-invariant two-band systems

We investigate optical transitions of non-interacting electron systems consisting of two symmetric energy bands touching each other at the Fermi energy (e.g. graphene at half filling). Optical conductivity is obtained using Kubo formula at zero temperature. We show that for particles whose pseudospin direction is determined solely by the direction of their momentum, the optical conductivity has power law frequency dependence with the exponent $(d-2)/z$ where $d$ is the dimension of the system and $z$ is the dynamical exponent. According to our result two-dimensional systems with the above pseudospin characteristics always exhibit frequency-independent optical conductivity.

Quantification of anisotropic scale invariance of geochemical anomalies associated with Sn-Cu mineralization in Gejiu, Yunan Province, China

In order to understand various anisotropic scale invariant systems, two different models are combined into a single method: the scale invariant generator (SIG) model quantifies anisotropies by estimating the generalized scale invariance (GSI) in frequency domain, which is a form of scale transformation in GSI representing how a scaling field is stratified and how it rotates; and the spectrum–area (S–A) model quantifies anisotropies by estimating the anisotropic scaling exponent in 2D frequency domain and decomposes multi-component data into separate patterns based on their distinct anisotropic scaling properties. The application of the combined model of S–A and SIG to stream sediment geochemical data from the Gejiu mineral district has shown that the new method can be used not only for separation of anomaly from background but also for characterization of anisotropic scale invariance properties such as differential compression and rotational scaling. Anomalies obtained by means of the S–A model can be utilized to delineate target areas for mineral exploration. The parameters obtained by means of the SIG model for the decomposed anomalies are useful for characterization of their orientation and compression distributions, which may provide insights to controls on mineralization or anomaly formation.

Variational principle underlying scale invariant social systems

MaxEnt's variational principle, in conjunction with Shannon's logarithmic information measure, yields only exponential functional forms in straightforward fashion. In this communication we show how to overcome this limitation via the incorporation, into the variational process, of suitable dynamical information. As a consequence, we are able to formulate a somewhat generalized Shannonian Maximum Entropy approach which provides a unifying "thermodynamic-like" explanation for the scale-invariant phenomena observed in social contexts, as city-population distributions. We confirm the MaxEnt predictions by means of numerical experiments with random walkers, and compare them with some empirical data.

On the Fractional Linear Scale Invariant Systems

The linear scale invariant systems are introduced for both integer and fractional orders. They are defined by the generalized Euler-Cauchy differential equation. The quantum fractional derivatives are suitable for dealing with this kind of systems, allowing us to define impulse response and transfer function with the help of the Mellin transform. It is shown how to compute the impulse responses corresponding to the two half plane regions of convergence of the transfer function.

On conjectured local generalizations of anisotropic scale invariance and their implications

The theory of generalized local scale invariance of strongly anisotropic scale invariant systems proposed some time ago by Henkel [M. Henkel, Phenomenology of local scale invariance: from conformal invariance to dynamical scaling, Nucl. Phys. B 641 (3) (2002) 405–486] is examined. The case of so-called type-I systems is considered. This was conjectured to be realized by systems at m-axial Lifshitz points; in support of this claim, scaling functions of two-point cumulants at the uniaxial Lifshitz point of the three-dimensional ANNNI model were predicted on the basis of this theory and found to be in excellent agreement with Monte Carlo results [M. Pleimling, M. Henkel, Anisotropic scaling and generalized conformal invariance at Lifshitz points, Phys. Rev. Lett. 87 (12) (2001) 125702]. The consequences of the conjectured invariance equations are investigated, with emphasis put on this case. It is shown that fewer solutions than anticipated by Henkel generally exist and contribute to the scaling functions if these equations are assumed to hold for all (positive and negative) values of the d-dimensional space (or space time) coordinates ( t , r ) ∈ R × R d − 1 . Specifically, a single rather than two independent physically acceptable solutions exists in the case relevant for the mentioned fit of Monte Carlo data for the ANNNI model. Renormalization-group improved perturbation theory in 4 + m / 2 − ϵ dimensions is used to determine the scaling functions of the order-parameter and energy-density two-point cumulants in momentum space to two-loop order. The results are mathematically incompatible with Henkel's predictions except in free-field-theory cases. However, the scaling function of the energy-density cumulant we obtain for m = 1 upon extrapolation of our two-loop RG results to d = 3 differs numerically little from that of an effective free field theory.