General Perturbation Research Articles

Systematic Investigation of Electronic States and Bond Properties of LnO, LnO+, LnS, and LnS+ (Ln = La-Lu) by Spin-Orbit Multiconfiguration Perturbation Theory.

The electronic structures of lanthanide monoxides (LnO/LnO+) and monosulfides (LnS/LnS+) for all lanthanide series elements (Ln = La-Lu) have been systematically analyzed with sophisticated quantum chemical calculations. The ground electronic configuration has been determined to be Ln 4fn6s1 or 4fn+1 for the neutral molecules and Ln 4fn for the cations. The low-lying energy states resulting from spin-orbit coupling and ligand field effects have been resolved using spin-orbit multiconfiguration quasi-degenerate second-order perturbation theory calculations. The ionization energies of LnO (5.20-7.06 eV) are about 0.3-2.2 eV lower than those of LnS (5.54-9.22 eV) due to the difference in the Ln 6s and 4f orbital energies from which an electron is removed during the ionization process. The bond dissociation energies (BDEs) have been computed by the state-averaged general multiconfigurational perturbation theory and the completely renormalized coupled-cluster [CR-CC(2,3)] methods. The BDEs are highly dependent on the lanthanide elements as several factors of the lanthanides affect the bond dissociation. The calculated bond lengths and energies agree well with available experimental values and are systematically predicted for the series of lanthanide monoxides and monosulfides where experimental values are not available. Furthermore, the LS terms of low-lying energy states and their corresponding bond properties have been clarified in detail to systematize the similarities and differences of the lanthanide compounds.

Adversarial Training With Anti-Adversaries.

Adversarial training is effective in improving the robustness of deep neural networks. However, existing studies still exhibit significant drawbacks in terms of the robustness, generalization, and fairness of models. In this study, we validate the importance of different perturbation directions (i.e., adversarial and anti-adversarial) and bounds from both theoretical and practical perspectives. The influence of adversarial training on deep learning models in terms of fairness, robustness, and generalization is theoretically investigated under a more general perturbation scope that different samples can have different perturbation directions and varied perturbation bounds. Our theoretical explorations suggest that combining adversaries and anti-adversaries with varied bounds in training can be more effective in achieving better fairness among classes and a better tradeoff among robustness, accuracy, and fairness in some typical learning scenarios compared with standard adversarial training. Inspired by our theoretical findings, a more general learning objective that combines adversaries and anti-adversaries with varied bounds on each training sample is presented. To solve this objective, two adversarial training frameworks based on meta-learning and reinforcement learning are proposed, in which the perturbation direction and bound for each sample are determined by its training characteristics. Furthermore, the role of the combination strategy with varied bounds is explained from a regularization perspective. Extensive experiments under different learning scenarios verify our theoretical findings and the effectiveness of the proposed methodology.

Topologically ordered time crystals

Time crystals are a dynamical phase of periodically driven quantum many-body systems where discrete time-translation symmetry is broken spontaneously. Time-crystallinity however subtly requires also spatial order, ordinarily related to further symmetries, such as spin-flip symmetry when the spatial order is ferromagnetic. Here we define topologically ordered time crystals, a time-crystalline phase borne out of intrinsic topological order—a particularly robust form of spatial order that requires no symmetry. We show that many-body localization can stabilize this phase against generic perturbations and establish some of its key features and signatures, including a dynamical, time-crystal form of the perimeter law for topological order. We link topologically ordered and ordinary time crystals through three complementary perspectives: higher-form symmetries, quantum error-correcting codes, and a holographic correspondence. Topologically ordered time crystals may be realized in programmable quantum devices, as we illustrate for the Google Sycamore processor.

Gauge invariant perturbations of static spatially compact LRS II spacetimes

Abstract We present a framework to describe completely general first-order perturbations of static, spatially compact, and locally rotationally symmetric class II spacetimes within the theory of general relativity. The perturbation variables are by construction covariant and identification gauge invariant and encompass the geometry and the thermodynamics of the fluid sources. The new equations are then applied to the study of isotropic, adiabatic perturbations. We discuss how the choice of frame in which perturbations are described can significantly simplify the mathematical analysis of the problem and show that it is possible to change frames directly from the linear level equations. We find explicitly that the case of isotropic, adiabatic perturbations can be reduced to a singular Sturm–Liouville eigenvalue problem, and lower bounds for the values of the eigenfrequencies can be derived. These results lay the theoretical groundwork to analytically describe linear, isotropic, and adiabatic perturbations of static, spherically symmetric spacetimes.

Lower Stackelberg equilibria: from bilevel optimization to Stackelberg games

Both pessimistic and optimistic bilevel optimization problems may be not stable under perturbation when the lower-level problem has not a unique solution, meaning that the limit of sequences of solutions (resp. equilibria) to perturbed bilevel problems is not necessarily a solution (resp. an equilibrium) to the original problem. In this paper, we investigate the notion of lower Stackelberg equilibrium, an equilibrium concept arising as a limit point of pessimistic equilibria and of optimistic equilibria of perturbed bilevel problems. First, connections with pessimistic equilibria and optimistic equilibria are obtained in a general setting, together with existence and closure results. Secondly, the problem of finding a lower Stackelberg equilibrium is shown to be stable under general perturbation, differently from what happens for pessimistic and optimistic bilevel problems. Then, moving to the game theory viewpoint, the set of lower Stackelberg equilibria is proved to coincide with the set of subgame perfect Nash equilibrium outcomes of the associated Stackelberg game. These results allow to achieve a comprehensive look on various equilibrium concepts in bilevel optimization and in Stackelberg games as well as to add a new interpretation in terms of game theory to the previous limit results on pessimistic equilibria and optimistic equilibria under perturbation.

Dynamics of U(1) gauged Q -balls in three spatial dimensions

We investigate the dynamics of U(1) gauged Q-balls using fully three-dimensional numerical simulations. We consider two different scenarios: first, the classical stability of gauged Q-balls with respect to generic three-dimensional perturbations, and second, the behavior of gauged Q-balls during head-on and off-axis collisions at relativistic velocities. With regard to stability, we find that there exist gauged Q-ball configurations which are classically stable in both logarithmic and polynomial scalar field models. With regard to relativistic collisions, we find that the dynamics can depend on many different parameters such as the collision velocity, relative phase, relative charge, and impact parameter of the colliding Q-balls. Published by the American Physical Society 2024

Astrometric search for ultralight dark matter

Precision astrometry offers a way to probe new physics. By measuring the angular position of light sources at unprecedented precision, astrometry could probe minuscule fluctuations of underlying spacetime. This work explores the possibility of probing ultralight dark matter candidates using precision astrometry. Through the coherent and stochastic density fluctuations over the scale of its wavelength, ultralight dark matter perturbs the propagation of light and the geodesics of the observer and source, leading to unique time-dependent signatures in the angular position of background light sources. With detector specifications similar to the current and future astrometry observations, such as Gaia and the Roman Space Telescope, it is shown that the ultralight scalar dark matter of mass 10−18–10−16 eV could be probed when its density near the Solar System is about a few thousand times larger than the nominal dark matter density measured on a much larger kpc scale. This sensitivity is comparable to current pulsar timing array observations at a similar mass range. Explicit expressions for the angular deflection induced by most generic metric perturbations are derived and its gauge invariance is explicitly checked at the linear order. Published by the American Physical Society 2024

Finite-size corrections to the energy spectra of gapless one-dimensional systems in the presence of boundaries

We present the finite-size scaling theory of one-dimensional quantum critical systems in the presence of boundaries. While the finite-size spectrum in the conformal limit, namely of a conformal field theory with conformally invariant boundary conditions, is related to the dimensions of boundary operators by Cardy, the actual spectra of lattice models are affected by both bulk and boundary perturbations and contain non-universal boundary energies. We obtain a general expression of the finite-size energy levels in the presence of bulk and boundary perturbations. In particular, a generic boundary perturbation related to the energy-momentum tensor gives rise to a renormalization of the effective system size. We verify our field-theory formulation by comparing the results with the exact solution of the critical transverse-field Ising chain and with accurate numerical results on the critical three-state Potts chain obtained by Density-Matrix Renormalization Group.

Inter-Satellite Link Prediction with Supervised Learning Based on Kepler and SGP4 Orbits

Distributed Space Systems (DSS) are gaining prominence in the space industry due to their ability to increase mission performance by allowing cooperation and resource sharing between multiple satellites. In DSS where communication between heterogeneous satellites is necessary, achieving autonomous cooperation while minimizing energy consumption is a critical requirement, particularly in sparse constellations with nano-satellites. In order to minimize the functioning time and energy consumed by the Inter-Satellite Links established for satellite-to-satellite communication, their temporal encounters must be anticipated. This work proposes an autonomous solution based on Supervised Learning that allows heterogeneous satellites in circular polar Low-Earth Orbits to predict their close-approach encounters given the Orbital Elements. The model performance is evaluated and compared in two different scenarios: (1) a simplified scenario assuming Kepler orbits and (2) a realistic scenario assuming Simplified General Perturbations 4 orbital model. The obtained results demonstrate a Balanced Accuracy exceeding 95% when compared to realistic data from an available database. This work represents a promising initial stage in developing an alternative approach within the field of DSS.

Nonlinear stability of planar viscous shock wave to three-dimensional compressible Navier–Stokes equations

We prove the nonlinear stability of the planar viscous shock up to a time-dependent shift for the three-dimensional (3D) compressible Navier–Stokes equations under the generic perturbations, in particular, without zero mass conditions. Moreover, the time-dependent shift function keeps the shock profile shape time-asymptotically. Our stability result is unconditional for the weak planar Navier–Stokes shock. Our proof is motivated by the a -contraction method (a kind of weighted L^{2} -relative entropy method) with time-dependent shift for the stability of viscous shock in the one-dimensional (1D) case. Instead of the classical anti-derivative techniques, we perform the stability analysis of the planar Navier–Stokes shock in the original H^{2} -perturbation framework and therefore zero mass conditions are not necessarily needed, which, in turn, brings out the essential difficulties due to the compressibility of viscous shock. Furthermore, compared with 1D case, there are additional difficulties coming from the wave propagations along the multi-dimensional transverse directions and their interactions with the viscous shock. To overcome these difficulties, a multi-dimensional version of the sharp weighted Poincaré inequality, a -contraction techniques with time-dependent shift, and some essential physical structures of the multi-dimen­sional Navier–Stokes system are fully used.

Hadamard variation of eigenvalues with respect to general domain perturbations

Hadamard variation of eigenvalues with respect to general domain perturbations

Resolvent expansions of 3D magnetic Schrödinger operators and Pauli operators

We obtain asymptotic resolvent expansions at the threshold of the essential spectrum for magnetic Schrödinger and Pauli operators in dimension three. These operators are treated as perturbations of the Laplace operator in L2(R3) and L2(R3;C2), respectively. The main novelty of our approach is to show that the relative perturbations, which are first order differential operators, can be factorized in suitably chosen auxiliary spaces. This allows us to derive the desired asymptotic expansions of the resolvents around zero. We then calculate their leading and sub-leading terms explicitly. Analogous factorization schemes for more general perturbations, including e.g. finite rank perturbations, are discussed as well.

General perturbations in D+1 standard and brane cosmology revisited

In this work, we generalize the theory of perturbations in a ([Formula: see text])-dimensional space–time with cosmological constant, studying scalar, vector and tensor perturbations, as well as its structure in Newtonian and Synchronous gauge. We also show the theory of perturbations in the context of brane cosmology, where branes are embedded in a set of D-spatial dimensions, a temporal dimension, and an additional spatial dimension. In both standard and brane cosmology, an unperturbed space–time is provided with a Friedmann–Lemaitre–Robertson–Walker metric and arbitrary sectional curvature, the matter content has the shape of a perfect fluid. In addition, we consider the arbitrary sectional curvature, obtaining the respective equations in the Newtonian and Synchronous gauge. We highlight that the results presented in this paper can be used to treat brane cosmology with two concentric branes or tackle the [Formula: see text] tension with a braneworld approach. Finally, as an example of the utility of all the technology presented in this paper, we show an application to the energy flux and the repercussions in the framework of the braneworlds.

The Arsenal of Perturbation Bounds for Finite Continuous-Time Markov Chains: A Perspective

Perturbation bounds are powerful tools for investigating the phenomenon of insensitivity to perturbations, also referred to as stability, for stochastic and deterministic systems. This perspective article presents a focused account of some of the main concepts and results in inequality-based perturbation theory for finite state-space, time-homogeneous, continuous-time Markov chains. The diversity of perturbation bounds and the logical relationships between them highlight the essential stability properties and factors for this class of stochastic processes. We discuss the linear time dependence of general perturbation bounds for Markov chains, as well as time-independent (i.e., time-uniform) perturbation bounds for chains whose stationary distribution is unique. Moreover, we prove some new results characterizing the absolute and relative tightness of time-uniform perturbation bounds. Specifically, we show that, in some of them, an equality is achieved. Furthermore, we analytically compare two types of time-uniform bounds known from the literature. Possibilities for generalizing Markov-chain stability results, as well as connections with stability analysis for other systems and processes, are also discussed.

Features, Paradoxes and Amendments of Perturbative Non-Hermitian Quantum Mechanics

Quantum mechanics of unitary systems is considered in quasi-Hermitian representation and in the dynamical regime in which one has to take into account the ubiquitous presence of perturbations, random or specific. In this paper, it is shown that multiple technical obstacles encountered in such a context can be circumvented via just a mild amendment of the so-called Rayleigh–Schrödinger perturbation–expansion approach. In particular, the quasi-Hermitian formalism characterized by an enhancement of flexibility is shown to remain mathematically tractable while, on the phenomenological side, opening several new model-building horizons. It is emphasized that they include, i.a., the study of generic random perturbations and/or of multiple specific non-Hermitian toy models. In parallel, several paradoxes and open questions are shown to survive.

A note on the persistence of multiplicity of eigenvalues of fractional Laplacian under perturbations

We consider the eigenvalue problem for the fractional Laplacian (−Δ)s, s∈(0,1), in a bounded domain Ω with Dirichlet boundary condition. A recent result (see Fall et al., 2023) states that, under generic small perturbations of the coefficient of the equation or of the domain Ω, all the eigenvalues are simple. In this paper we give a condition for which a perturbation of the coefficient or of the domain preserves the multiplicity of a given eigenvalue. Also, in the case of an eigenvalue of multiplicity ν=2 we prove that the set of perturbations of the coefficients which preserve the multiplicity is a smooth manifold of codimension 2 in C1(Rn).

Gauge-invariant cosmological perturbations in general teleparallel gravity

We study linear cosmological perturbations in the most general teleparallel gravity setting, where gravity is mediated by the torsion and nonmetricity of a flat connection alongside the metric. For a general linear perturbation of this geometry around a homogeneous and isotropic background geometry, we derive the irreducible decomposition of the perturbation variables, as well as their behavior under gauge transformations, i.e., infinitesimal diffeomorphisms generated by a vector field. In addition, we also study these properties for the most general set of matter variables and gravitational field equations. We then make use of these result to construct gauge-invariant perturbation variables, using a general approach based on gauge conditions. We further calculate these quantities also in the metric and symmetric teleparallel geometries, where nonmetricity or torsion is imposed to vanish. To illustrate our results, we derive the energy-momentum–hypermomentum conservation equations for both the cosmological background and the linear perturbations. As another example, we study the propagation of tensor perturbations in the f(G), f(T) and f(Q) class of theories.

ExROPPP: Fast, accurate, and spin-pure calculation of the electronically excited states of organic hydrocarbon radicals.

Recent years have seen an explosion of interest in organic radicals due to their promise for highly efficient organic light-emitting diodes and molecular qubits. However, accurately and inexpensively computing their electronic structure has been challenging, especially for excited states, due to the spin-contamination problem. Furthermore, while alternacy or "pseudoparity" rules have guided the interpretation and prediction of the excited states of closed-shell hydrocarbons since the 1950s, similar general rules for hydrocarbon radicals have not to our knowledge been found yet. In this article, we present solutions to both of these challenges. First, we combine the extended configuration interaction singles method with Pariser-Parr-Pople (PPP) theory to obtain a method that we call ExROPPP (Extended Restricted Open-shell PPP) theory. We find that ExROPPP computes spin-pure excited states of hydrocarbon radicals with comparable accuracy to experiment as high-level general multi-configurational quasi-degenerate perturbation theory calculations but at a computational cost that is at least two orders of magnitude lower. We then use ExROPPP to derive widely applicable rules for the spectra of alternant hydrocarbon radicals, which are completely consistent with our computed results. These findings pave the way for highly accurate and efficient computation and prediction of the excited states of organic radicals.

Parameter dependence of entanglement spectra in quantum field theories

In this paper, we explore the characteristics of reduced density matrix spectra in quantum field theories. Previous studies mainly focus on the function P(λ)≔∑iδ(λ−λi), where λi denote the eigenvalues of the reduced density matrix. We introduce a series of functions designed to capture the parameter dependencies of these spectra. These functions encompass information regarding the derivatives of eigenvalues concerning the parameters, notably including the function PαJ(λ)≔∑i∂λi∂αJδ(λ−λi), where αJ denotes the specific parameter. Computation of these functions is achievable through the utilization of Rényi entropy. Intriguingly, we uncover compelling relationships among these functions and demonstrate their utility in constructing the eigenvalues of reduced density matrices for select cases. We perform computations of these functions across several illustrative examples. Especially, we conducted a detailed study of the variations of P(λ) and PαJ(λ) under general perturbation, elucidating their physical implications. In the context of holographic theory, we ascertain that the zero point of the function PαJ(λ) possesses universality, determined as λ0=e−S, where S denotes the entanglement entropy of the reduced density matrix. Furthermore, we exhibit potential applications of these functions in analyzing the properties of entanglement entropy. Published by the American Physical Society 2024

Asymptotic stability of the fourth-order $\phi^{4}$ kink for general perturbations in the energy space

The fourth-order \phi^{4} model extends the classical \phi^{4} model of quantum field theory to the fourth-order case, but sharing the same kink solution. It is also the dispersive counterpart of the well-known parabolic Cahn–Hilliard equation. Mathematically speaking, the kink is characterized by a fourth-order nonnegative linear operator with a simple kernel at the origin but no spectral gap. In this paper, we consider the kink of this theory, and prove orbital and asymptotic stability for any perturbation in the energy space.