Scaling Symmetry Research Articles

Scaling symmetry reductions of coupled KdV systems

Abstract In this paper we discuss the Painlevé reductions of coupled KdV systems. We start by comparing the procedure with that of stationary reductions. Indeed, we see that exactly the same construction can be used at each step and parallel results obtained. For simplicity, we restrict attention to the t2 flow of the KdV and DWW hierarchies and derive respectively 2 and 3 compatible Poisson brackets, which have identical structure to those of their stationary counterparts. In the KdV case, we derive a discrete version, which is a non-autonomous generalisation of the well known Darboux transformation of the stationary case.

Using Symmetries to Investigate the Complete Integrability, Solitary Wave Solutions and Solitons of the Gardner Equation

In this paper, using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations, thereby establishing their complete integrability. The Gardner equation is chosen as the key example, as it comprises both the Korteweg–de Vries and modified Korteweg–de Vries equations. The Gardner and Miura transformations, which connect these equations, are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota’s method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case), whereas the focusing Gardner equation has standard elastically colliding solitons. This paper’s aim is to provide a review of the integrability properties and solutions of the Gardner equation and to illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica, but can be adapted for major computer algebra systems.

On AdS3/ICFT2 with a dynamical scalar field located on the brane

We exploit the holographic duality to study the system of a one-dimensional interface contacting two semi-infinite two-dimensional CFTs. Central to our investigation is the introduction of a dynamical scalar field located on the bulk interface brane which breaks the scaling symmetry of the dual interface field theory, along with its consequential backreaction on the system. We define an interface entropy from holographic entanglement entropy, to construct a g-function. At zero temperature we construct several illustrative examples and consistently observe that the g-theorem is always satisfied. These examples also reveal distinct features of the interface entropy that are intricately linked to the scalar potential profiles. At finite temperature we find that the dynamical scalar field enables the bulk theory to have new configurations which would be infeasible solely with a tension term on the interface brane.

Fermion mass generation without symmetry breaking

We study the generation of fermion mass in a context where interactions break a discrete chiral symmetry. Then, fermion mass is not protected by a symmetry, no symmetry is broken by the generation of mass, and a vanishing mass no longer enhances a symmetry. We elaborate these scenarios for template fermionic and Yukawa theories in three dimensions where mass can be generated either by fluctuations, strong dynamics, or the vacuum expectation value of a scalar field. We find that fluctuation-induced contributions to fermion mass are parametrically suppressed in the number of fermion flavors N. The generation of fermion mass then takes the form of a rapid crossover which turns into a second order quantum phase transition for large N, much like in settings with fundamental chiral symmetry. We further discuss theories where fermion mass can be generated spontaneously without breaking any symmetry other than scale symmetry. Implications of our findings are discussed. Published by the American Physical Society 2024

Continuation of Bianchi spacetimes through the big bang

In this paper we present a framework in which the relational description of general relativity can be used to smoothly continue cosmological dynamical systems through the big bang without invoking quantum gravity effects. Cosmological spacetimes contain as a key dynamical variable a notion of scale through the volume factor ν. However, no cosmological observer is ever able to separate their measuring apparatus from the system they are measuring, in that sense every measurement is a relative one, and measurable dynamical variables are, in fact, dimensionless ratios. This is manifest in the identification of a scaling symmetry or “dynamical similarity” in the Einstein-Hilbert action associated with the volume factor. By quotienting out this scaling symmetry, we form a relational system defined on a contact manifold whose dynamical variables are decoupled from scale. When the phase space is reduced to shape space, we show that there exists unique solutions to the equations of motion that pass smoothly through the initial cosmological singularity in flat Friedmann-Lemaître-Robertson-Walker, Bianchi I, and Quiescent Bianchi IX cosmologies. Published by the American Physical Society 2024

Hamiltonian Cycles on Ammann-Beenker Tilings

We provide a simple algorithm for constructing Hamiltonian graph cycles (visiting every vertex exactly once) on a set of arbitrarily large finite subgraphs of aperiodic two-dimensional Ammann-Beenker (AB) tilings. Using this result, and the discrete scale symmetry of AB tilings, we find exact solutions to a range of other problems which lie in the complexity class NP-complete for general graphs. These include the equal-weight traveling salesperson problem, providing, for example, the most efficient route a scanning tunneling microscope tip could take to image the atoms of physical quasicrystals with AB symmetries; the longest path problem, whose solution demonstrates that collections of flexible molecules of any length can adsorb onto AB quasicrystal surfaces at density one, with possible applications to catalysis; and the three-coloring problem, giving ground states for the q-state Potts model (q≥3) of magnetic interactions defined on the planar dual to AB, which may provide useful models for protein folding. Published by the American Physical Society 2024

Scale symmetry breaking and generation of mass at quantum critical points

We study an asymptotically free theory of N relativistic Dirac fermions and a real scalar field coupled by Yukawa and scalar self-interactions in three dimensions using functional renormalisation. In the limit of many fermion flavours, the cubic scalar coupling becomes exactly marginal due to quantum fluctuations, leading to a line of strongly-coupled infrared fixed points. Fermion mass can be generated through a quantum phase transition even if chiral symmetry is absent. The line of fixed points terminates at a critical endpoint due to the loss of vacuum stability. Exactly at the endpoint, scale symmetry is broken spontaneously, leading to the generation of fermion mass. Intriguingly, the absence of chiral symmetry is a prerequisite for the spontaneous generation of fermion mass, and not a consequence thereof. We also highlight close similarities between Gross-Neveu and Gross-Neveu-Yukawa theories at and away from critical points, and establish the large-N equivalence of their functional RG flows and quantum effective actions. Further implications including for conformal field theories are indicated.

Interlinking Rayleigh–Taylor/Richtmyer–Meshkov interfacial mixing with variable acceleration and canonical Kolmogorov turbulence

Canonical turbulence and Rayleigh–Taylor/Richtmyer–Meshkov mixing with variable acceleration are paradigmatic complexities in science, mathematics, and engineering, with broadly ranging applications in nature, technology, and industry. We employ scaling symmetries and invariant forms to represent these challenging processes and to assess their very different properties. We directly link—for the first time to our knowledge—the attributes of Rayleigh–Taylor/Richtmyer–Meshkov interfacial mixing with variable acceleration to those of canonical turbulence, including scaling laws, spectral shapes, and characteristic scales. We explore the role of control dimensional parameters in quantifying these processes. The theory results compare well with available observations, the chart perspectives for future experiments and simulations, and for better understanding realistic complexity.

A Bridge between Trace Anomalies and Deconfinement Phase Transitions

Inspired by the fact that both the dilaton potential encoding the trace anomalies of QCD and the Polyakov loop potential measuring the deconfinement phase transition can be expressed in the logarithmic forms, as well as the fact that the scale symmetry is expected to be restoring and colors are deconfined in extreme conditions such as high temperatures and/or densities, we conjecture a relation between the dilaton potential and the Polyakov loop potential. Explicitly, we start from the Coleman–Weinberg type potential of a real scalar field—a dilaton or conformal compensator—and make an ansatz of the relation between this scalar field and the Polyakov loop to obtain the Polyakov loop potential, which can be parameterized in Lattice QCD (LQCD) in the pure glue sector. We find that the coefficients of Polyakov potential fitted from Lattice data are automatically satisfied in this ansatz, the locations of deconfinement and scale restoration are locked to each other, and the first-order phase transition can be realized. Extensions to the low-energy effective quark models are also discussed. The conjectured relation may deepen our understanding of the evolution of the universe, the mechanism of electroweak symmetry breaking, the phase diagram of QCD matter, and the properties of neutron stars.

Noether symmetries of F(T,X,ϕ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(T,X,\\phi )$$\\end{document} cosmology

This article explores an analysis via the Noether symmetry approach to investigate an extended teleparallel F(T,X,ϕ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(T,X,\\phi )$$\\end{document} gravity model in the context of a Friedmann–Robertson–Walker spacetime. The model incorporates the torsion scalar T, scalar field ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document} and the kinetic term X of the scalar field. This approach allows us to select physically interesting models by restricting arbitrariness in the F(T,X,ϕ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(T,X,\\phi )$$\\end{document} function. Thus we focus on two types of the function F(T,X,ϕ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(T,X,\\phi )$$\\end{document}, namely models involving the generalized teleparallel dark energy and the nonlinear function of the kinetic term and analyze the Noether symmetry properties and demonstrate the presence of non-vanishing Noether vectors. In the first model, by specifically considering the X4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf{X_{4}}$$\\end{document} Noether symmetry, representing a scaling symmetry, we simplify the field equations and introduce a cyclic variable to enable a more workable formulation. Furthermore, we present a comprehensive graphical analysis, showcasing the time evolution of the scale factor and the scalar field by using the solutions obtained through these new variables. Additionally, we examine some fundamental cosmological parameters. For the second model, which includes the nonlinear form of the kinetic term, Noether symmetry results in obtaining simple power-law solutions for the dynamic variables. Our findings clearly show that the considered models represent an accelerating expansion of the Universe, which is consistent with observations. This study highlights the efficacy of the Noether symmetry approach in defining the functional form of F(T,X,ϕ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(T,X,\\phi )$$\\end{document} and obtaining precise cosmological solutions. The analytical solutions and graphical analysis offer valuable insights into the dynamic behavior of the models, contributing to understanding the evolving nature of the Universe.

Massless Lifshitz field theory for arbitrary z

By using the notion of fractional derivatives, we introduce a class of massless Lifshitz scalar field theory in (1+1)-dimension with an arbitrary anisotropy index z. The Lifshitz scale invariant ground state of the theory is constructed explicitly and takes the form of Rokhsar-Kivelson (RK). We show that there is a continuous family of ground states with degeneracy parameterized by the choice of solution to the equation of motion of an auxiliary classical system. The quantum mechanical path integral establishes a 2d/1d correspondence with the equal time correlation functions of the Lifshitz scalar field theory. We study the entanglement properties of the Lifshitz theory for arbitrary z using the path integral representation. The entanglement measures are expressed in terms of certain cross ratio functions we specify, and satisfy the c-function monotonicity theorems. We also consider the holographic description of the Lifshitz theory. In order to match with the field theory result for the entanglement entropy, we propose a z-dependent radius scale for the Lifshitz background. This relation is consistent with the z-dependent scaling symmetry respected by the Lifshitz vacuum. Furthermore, the time-like entanglement entropy is determined using holography. Our result suggests that there should exist a fundamental definition of time-like entanglement other than employing analytic continuation as performed in relativistic field theory.

Aging properties of the voter model with long-range interactions

We investigate the aging properties of the one-dimensional voter model with long-range interactions in its ordering kinetics. In this system, an agent, Si=±1 , positioned at a lattice vertex i, copies the state of another one located at a distance r, selected randomly with a probability P(r)∝r−α . Employing both analytical and numerical methods, we compute the two-time correlation function G(r;t,s) ( t⩾s ) between the state of a variable Si at time s and that of another one, at distance r, at time t. At time t, the memory of an agent of its former state at time s, expressed by the autocorrelation function A(t,s)=G(r=0;t,s) , decays algebraically for α > 1 as [L(t)/L(s)]−λ , where L is a time-increasing coherence length and λ is the Fisher–Huse exponent. We find λ = 1 for α > 2, and λ=1/(α−1) for 1<α⩽2 . For α⩽1 , instead, there is an exponential decay, as in the mean field. Then, in contrast with what is known for the related Ising model, here we find that λ increases upon decreasing α. The space-dependent correlation G(r;t,s) obeys a scaling symmetry G(r;t,s)=g[r/L(s);L(t)/L(s)] for α > 2. Similarly, for 1<α⩽2 , one has G(r;t,s)=g[r/L(t);L(t)/L(s)] , where the length L regulating two-time correlations now differs from the coherence length as L∝Lδ , with δ=1+2(2−α) .

Holographic renormalization of Horndeski gravity

We study the renormalization of a particular sector of Horndeski theory. In particular, we focus on the nonminimal coupling of a scalar field to the Gauss-Bonnet term and its kinetic coupling to the Einstein tensor. Adopting a power expansion on the scalar function that couples the Gauss-Bonnet term, we find specific conditions on their coefficients such that the action and charges are finite. To accomplish the latter, we add a finite set of intrinsic boundary terms. The contribution of the nonminimal coupling generates an effective scalar mass, allowing us to recover a modified Breitenlohner-Freedman bound. Furthermore, we compute the holographic 1-point functions and Ward identities associated with the scalar field and the metric. We constrain the parameter space of the theory by taking into account the preservation of scaling symmetry at the boundary.

N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document}=4 Supergravity with Local Scaling Symmetry in Four Dimensions

We construct the most general four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supergravity coupled to an arbitrary number n of vector multiplets in which the global scaling symmetry is gauged, in addition to a subgroup of SL(2, ℝ) × SO(6, n). The various gaugings are parametrized by an embedding tensor built out of 2n+63+4n+6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ 2\\left(\\begin{array}{c}n+6\\\\ {}3\\end{array}\\right)+4\\left(n+6\\right) $$\\end{document} parameters that satisfy a specific set of quadratic consistency constraints, to which we provide explicit solutions. We also derive the local supersymmetry transformation rules and the equations of motion for the four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 matter-coupled supergravity with local scaling symmetry. Such supergravity theories do not possess an action, since the scaling symmetry is only an on-shell symmetry of the corresponding ungauged theories.

Dilaton in a multicritical 3+epsilon-D parity violating field theory

The multi-critical behavior of an approximately scale and conformal invariant quantum field theory, which can be regarded as the deformation of the critical Gross-Neveu model in 3+epsilon dimensions by a nearly marginal parity violating operator, is studied using a large N expansion. When epsilon is greater than a number of order 1/N, the deformation is marginally relevant and it is found to exhibit spontaneous breaking of the approximate scale symmetry accompanied by the appearance of a light scalar in its spectrum. The scalar mass is parametrically small, of order epsilon times the fermion mass and it can be identified with a light dilaton. When the dimension is reduced to 3 the deformation of the Gross-Neveu model becomes marginally irrelevant, what was a minimum of the potential becomes a maximum and the theory has a non-perturbative global instability. There is a metastable perturbative phase where the scalar does not condense and the fermions are massless separated by an energy barrier with height of order one (rather than N) from an energetically favored phase with a runaway condensate.

Duality between the quantum inverted harmonic oscillator and inverse square potentials

In this paper we show how the quantum mechanics of the inverted harmonic oscillator (IHO) can be mapped to the quantum mechanics of a particle in a super-critical inverse square potential (ISP). We demonstrate this by relating both of these systems to the Berry–Keating system with Hamiltonian H=(xp+px)/2 . It has long been appreciated that the quantum mechanics of the ISP has an ambiguity in choosing a boundary condition near the origin and we show how this ambiguity is mapped to the IHO system. Imposing a boundary condition requires specifying a distance scale where it is applied and changes to this scale come with a renormalisation group (RG) evolution of the boundary condition that ensures observables do not directly depend on the scale (which is arbitrary). Physical scales instead emerge as RG invariants of this evolution. The RG flow for the ISP is known to follow limit cycles describing the discrete breaking of classical scale invariance in a simple example of a quantum anomaly, and we find that limit cycles also occur for the IHO. However, unlike the ISP where the continuous scaling symmetry is explicit, in the case of the IHO it is hidden and occurs because the Hamiltonian is part of a larger su(1,1) spectrum generating algebra. Our map does not require the boundary condition to be self-adjoint, as can be appropriate for systems that involve the absorption or emission of particles.

Scaling symmetry, Smarr relation, and the extended first law in lower-dimensional Lovelock gravity

Recently, it was discovered that lower-dimensional versions of Lovelock gravity exist as scalar-tensor theories that are examples of Horndeski gravity. We study the thermodynamics of the static black hole solutions in these theories up to cubic order through Euclidean methods. Considering solutions with spherical, planar and hyperbolic event horizons (k=+1,0,−1), we show that the universality of the thermodynamics for planar black holes (k=0) and the extended first law that include the variation of the couplings together with their associated potentials hold also in lower dimensions. We find that in D=4,6 where the 2nd- and the 3rd-order Lovelock Lagrangians are boundary terms respectively, the Smarr relation is modified since the entropy is not a homogenous function in these dimensions. We also present a derivation of the Smarr relation and its modified version based on the global scaling properties of the reduced action that is used to obtain the solutions consistently. Unlike the other hairy black hole solutions that have been analyzed before, despite the terms in the reduced action that break the scaling symmetry, the derivation still follows from a conserved Noether charge.

Modelling ductile damage in metals and alloys through Weyl condition exploiting local gauge symmetries

Local translational and scaling symmetries in space–time is exploited for modelling ductile damage in metals and alloys over wide ranges of strain rate and temperature. The invariant energy density corresponding to the ductile deformation is constructed through the gauge invariant curvature tensor by imposing the Weyl like condition. In contrast, the energetics of the plastic deformation is brought in through the gauge compensating field emerged due to local translation and attempted to explore the geometric interpretation of certain internal variables often used in classical viscoplasticity models. Invariance of the energy density under the local action of translation and scaling is preserved through minimally replaced space–time gauge covariant operators. Minimal replacement introduces two non-trivial gauge compensating fields pertaining to local translation and scaling. These are used to describe ductile damage, including plastic flow and micro-crack evolution in the material. A space–time pseudo-Riemannian metric is used to lay out the kinematics in a finite-deformation setting. Recognizing the available insights in classical theories of viscoplasticity, we also establish a correspondence of the gauge compensating field due to spatial translation with Kröner’s multiplicative decomposition of the deformation gradient. Thermodynamically consistent coupling between viscoplasticity and ductile damage is ensured through an appropriate degradation function. Non-ordinary state-based (NOSB) peridynamics (PD) discretization of the model is used for numerical implementation. We conduct simulations of uniaxial deformation to validate the model against available experimental evidence and to assess its predictive features. The model’s viability is tested in reproducing a few experimentally known facts, viz., strain rate locking in the stress–strain response, whose origin is traced to a nonlinear microscopic inertia term arising out of the space–time translation symmetry. Finally, we solved 2D and axisymmetric deformation problems for qualitatively validating the model’s viability. NOSB peridynamics axisymmetric formulation in finite deformation setup is also presented.

Stability against fluctuations: a two-dimensional study of scaling, bifurcations and spontaneous symmetry breaking in stochastic models of synaptic plasticity

Stochastic models of synaptic plasticity must confront the corrosive influence of fluctuations in synaptic strength on patterns of synaptic connectivity. To solve this problem, we have proposed that synapses act as filters, integrating plasticity induction signals and expressing changes in synaptic strength only upon reaching filter threshold. Our earlier analytical study calculated the lifetimes of quasi-stable patterns of synaptic connectivity with synaptic filtering. We showed that the plasticity step size in a stochastic model of spike-timing-dependent plasticity (STDP) acts as a temperature-like parameter, exhibiting a critical value below which neuronal structure formation occurs. The filter threshold scales this temperature-like parameter downwards, cooling the dynamics and enhancing stability. A key step in this calculation was a resetting approximation, essentially reducing the dynamics to one-dimensional processes. Here, we revisit our earlier study to examine this resetting approximation, with the aim of understanding in detail why it works so well by comparing it, and a simpler approximation, to the system’s full dynamics consisting of various embedded two-dimensional processes without resetting. Comparing the full system to the simpler approximation, to our original resetting approximation, and to a one-afferent system, we show that their equilibrium distributions of synaptic strengths and critical plasticity step sizes are all qualitatively similar, and increasingly quantitatively similar as the filter threshold increases. This increasing similarity is due to the decorrelation in changes in synaptic strength between different afferents caused by our STDP model, and the amplification of this decorrelation with larger synaptic filters.

Kirillov structures and reduction of Hamiltonian systems by scaling and standard symmetries

AbstractIn this paper, we discuss the reduction of symplectic Hamiltonian systems by scaling and standard symmetries which commute. We prove that such a reduction process produces a so‐called Kirillov Hamiltonian system. Moreover, we show that if we reduce first by the scaling symmetries and then by the standard ones or in the opposite order, we obtain equivalent Kirillov Hamiltonian systems. In the particular case when the configuration space of the symplectic Hamiltonian system is a Lie group , which coincides with the symmetry group, the reduced structure is an interesting Kirillov version of the Lie–Poisson structure on the dual space of the Lie algebra of . We also discuss a reconstruction process for symplectic Hamiltonian systems which admit a scaling symmetry. All the previous results are illustrated in detail with some interesting examples.