Many-particle Systems Research Articles

Quantum-Field Multiloop Calculations in Critical Dynamics

The quantum-field renormalization group method is one of the most efficient and powerful tools for studying critical and scaling phenomena in interacting many-particle systems. The multiloop Feynman diagrams underpin the specific implementation of the renormalization group program. In recent years, multiloop computation has had a significant breakthrough in both static and dynamic models of critical behavior. In the paper, we focus on the state-of-the-art computational techniques for critical dynamic diagrams and the results obtained with their help. The generic nature of the evaluated physical observables in a wide class of field models is manifested in the asymptotic character of perturbation expansions. Thus, the Borel resummation of series is required to process multiloop results. Such a procedure also enables one to take high-order contributions into consideration properly. The paper outlines the resummation framework in dynamic models and the circumstances in which it can be useful. An important resummation criterion is the properties of the higher-order asymptotics of the perturbation theory. In static theories, these properties are determined by the method of instanton analysis. A similar approach is applicable in critical dynamics models. We describe the calculation of these asymptotics in dynamical models and present the results of the corresponding resummation.

Accelerating nonequilibrium Green function simulations with embedding self-energies

Real-time nonequilibrium Green functions (NEGFs) have been very successfully used to simulate the dynamics of correlated many-particle systems far from equilibrium. However, NEGF simulations are computationally expensive since the effort scales cubically with the simulation duration. Recently, we introduced the G1--G2 scheme that allows for a dramatic reduction to time-linear scaling [N. Schl\"unzen et al., Phys. Rev. Lett. 124, 076601 (2020); J.-P. Joost et al., Phys. Rev. B 101, 245101 (2020)]. Here we tackle another problem: the rapid growth of the computational effort with the system size. In many situations where the system of interest is coupled to a bath, to electric contacts, or to similar macroscopic systems for which a microscopic resolution of the electronic properties is not necessary, efficient simplifications are possible. This is achieved by the introduction of an embedding self-energy---a concept that has been successful in standard NEGF simulations. Here, we demonstrate how the embedding concept can be introduced into the G1--G2 scheme, allowing us to drastically accelerate NEGF embedding simulations. The approach is compatible with all advanced self-energies that can be represented by the G1--G2 scheme [as described in J.-P. Joost et al., Phys. Rev. B 105, 165155 (2022)] and retains the memoryless structure of the equations and their time-linear scaling. As a numerical illustration, we investigate the charge transfer between a Hubbard nanocluster and an additional site which is of relevance for the neutralization of ions in matter.

Quantum Phase Recognition via Quantum Kernel Methods

The application of quantum computation to accelerate machine learning algorithms is one of the most promising areas of research in quantum algorithms. In this paper, we explore the power of quantum learning algorithms in solving an important class of Quantum Phase Recognition (QPR) problems, which are crucially important in understanding many-particle quantum systems. We prove that, under widely believed complexity theory assumptions, there exists a wide range of QPR problems that cannot be efficiently solved by classical learning algorithms with classical resources. Whereas using a quantum computer, we prove the efficiency and robustness of quantum kernel methods in solving QPR problems through Linear order parameter Observables. We numerically benchmark our algorithm for a variety of problems, including recognizing symmetry-protected topological phases and symmetry-broken phases. Our results highlight the capability of quantum machine learning in predicting such quantum phase transitions in many-particle systems.

Entanglement of particles versus entanglement of fields: Independent quantum resources

Nature allows one to explore a manifold of remarkable quantum effects. Most prominently, quantum entanglement can be observed in many-particle systems, between multiple quantized fields, and hybrid combinations thereof. This diversity, however, also leads to contradicting conclusions about what truly constitutes entanglement in any given physical scenario. By explicitly allowing various perspectives, we rigorously consider different notions of entanglement in the context of first and second quantization. By providing instructive examples, we show that particle entanglement and field entanglement are actually distinct phenomena that can occur and be observed independently of each other. This finding not only affects our fundamental understanding but has direct implications on quantum technology which can harness those independent forms of entanglement in practical scenarios.

Dependence of the probability of escape on the Jacobi constant in the N-body ring problem without central body

In this work, we perform a numerical exploration of the escape in the N-body ring problem in absence of a central body, for 4le Nle 9 and ten values of the Jacobi constant. We show how the probability of escape per interval of time varies as a function of the Jacobi constant, finding that, for values of the Jacobi constant smaller than a certain limit value, the probability of escape from the system tends to decrease with time. However, if we consider values of the Jacobi constant larger than this limit value, the probability of escape grows with time, for times of escape smaller than 100 units of time.

Significance of the nature of disorder on the universal features of the spatio-temporal correlations of two-dimensional Coulomb-clusters

We comprehend the role of imperfections in materials consisting of interacting particles, arising from different origins on their universal features. Specifically, we report the static and dynamic responses in a cluster of Coulomb interacting particles in two dimensions. Confined systems with pinned impurities are studied, and results are compared with those from irregularly trapped system of particles. While the disorder of first type leads to diffusive single-particle dynamics, the motion of a single particle in an irregular trap is chaotic but ballistic. The many-particle system does not differentiate between these two models of disorder insofar as their qualitative properties are concerned -- particularly for describing the thermal melting of the underlying Coulomb-"solid". However, quantitative differences persist -- the relaxation time scales differ significantly by tuning impurity concentration.

A new method for identifying dynamical transitions in rubble-pile asteroid scenarios

Context. Evidence supports the idea that asteroids are rubble piles, that is, gravitational aggregates of loosely consolidated material. This makes their dynamics subject not only to the complex N-body gravitational interactions between its constituents, but also to the laws of granular mechanics, which is one of the main unsolved problems in physics. Aims. We aim to develop a new method to identify dynamical transitions and predict qualitative behavior in the granular N-body problem, in which the dynamics of individual bodies are driven both by mutual gravity, contact and collision interactions. Methods. The method has its foundation in the combination of two elements: a granular N-body simulation code that can resolve the dynamics of granular fragments to particle-scale precision, and a theoretical framework that can decode the nature of particle-scale dynamics and their transitions by means of ad hoc indicators. Results. We present here a proof-of-concept of the method, with application to the spinning rubble-pile asteroid problem. We investigate the density-spin parameter space and demonstrate that the approach can identify the breakup limit and reshape region for spinning rubble-pile aggregates. Conclusions. We provide the performance of several ad hoc indicators and discuss whether they are suitable for identifying and predicting the features of the dynamical problem.

Revealing the dynamics of ultrarelativistic non-equilibrium many-electron systems with phase space tomography

The description of physical processes with many-particle systems is a key approach to the modeling of numerous physical systems. For example in storage rings, where ultrarelativistic particles are agglomerated in dense bunches, the modeling and measurement of their phase-space distribution is of paramount importance: at any time the phase-space distribution not only determines the complete space-time evolution but also provides fundamental performance characteristics for storage ring operation. Here, we demonstrate a non-destructive tomographic imaging technique for the 2D longitudinal phase-space distribution of ultrarelativistic electron bunches. For this purpose, we utilize a unique setup, which streams turn-by-turn near-field measurements of bunch profiles at MHz repetition rates. To demonstrate the feasibility of our method, we induce a non-equilibrium state and show that the phase-space distribution microstructuring as well as the phase-space distribution dynamics can be observed in great detail. Our approach offers a pathway to control ultrashort bunches and supports, as one example, the development of compact accelerators with low energy footprints.

On the restricted (N + 1)-body problem on surfaces of constant curvature

In this paper, we consider a restricted (N+1)-body problem on surfaces Mκ2, where the constant κ≠0 is the Gaussian curvature, which by means of a rescaling can be reduced to κ=±1. This problem consists in the study of the dynamics of an infinitesimal mass particle attracted by N primaries of identical masses describing elliptic relative equilibria of the N-body problem on Mκ2, i.e., a solution where the primaries are rotating uniformly with constant angular velocity ω on a fixed parallel of S2 or H2 and placed at the vertices of a regular polygon. In a rotating frame, this problem yields a two degrees of freedom Hamiltonian system. The goal of this paper is to describe analytically some dynamics features for κ=±1. Precisely, we study the relative location of equilibria, obtaining, in particular, that the poles of S2 and vertex of H2 represent equilibrium points for any value of the parameters. Thus, analysis of the nonlinear stability of these equilibria is carried out, as well as two types of bifurcations are detected: Hamiltonian-Hopf and N-bifurcation. Additionally, we prove the existence of a family of Hill's orbits and a family of periodic orbits when the primaries are located near the poles of S2 or the vertex of H2. Finally we prove the existence of KAM 2-tori related to these periodic orbits.

On the centered co-circular central configurations for the n-body problem

For the power-law potential n-body problem, we study a special kind of central configurations where all the masses lie on a circle and the center of mass coincides with the center of the circle. It is also called the centered co-circular central configuration. We get some symmetry results for such central configurations. We show that for positive numbers α>0 and integers n≥3 satisfying 1n∑j=1n−1cscα⁡jπn≤1+α4, the regular n-gon with equal masses is the unique centered co-circular central configuration for the n-body problem with power-law potential Uα. It quickly follows that for the Newtonian n-body problem (in the case α=1) and n≤6, the regular n-gon is the unique centered co-circular central configuration.

On the stability of ring relative equilibria in the N-body problem on with Hodge potential

AbstractIn this paper, we study the stability of the ring solution of the N-body problem in the entire sphere $\mathbb {S}^2$ by using the logarithmic potential proposed in Boatto et al. (2016, Proceedings of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 472, 20160020) and Dritschel (2019, Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 377, 20180349), derived through a definition of central force and Hodge decomposition theorem for 1-forms in manifolds. First, we characterize the ring solution and study its spectral stability, obtaining regions (spherical caps) where the ring solution is spectrally stable for $2\leq N\leq 6$ , while, for $N\geq 7$ , the ring is spectrally unstable. The nonlinear stability is studied by reducing the system to the homographic regular polygonal solutions, obtaining a 2-d.o.f. Hamiltonian system, and therefore some classic results on stability for 2-d.o.f. Hamiltonian systems are applied to prove that the ring solution is unstable at any parallel where it is placed. Additionally, this system can be reduced to 1-d.o.f. by using the angular momentum integral, which enables us to describe the phase portraits and use them to find periodic ring solutions to the full system. Some of those solutions are numerically approximated.

Classification of positive elliptic-elliptic rotopulsators on Clifford tori

We prove that positive elliptic-elliptic rotopulsator solutions of the n-body problem in spaces of constant Gaussian curvature that move on Clifford tori of nonconstant size either lie on great circles, or project onto regular polygons. We additionally prove for the case that the configurations project onto regular polygons that all masses are equal and show that all these different types of positive elliptic-elliptic rotopulsator exist.

A Directional Equispaced Interpolation-Based Fast Multipole Method for Oscillatory Kernels

Fast multipole methods (FMMs) based on the oscillatory Helmholtz kernel can reduce the cost of solving N-body problems arising from boundary integral equations (BIEs) in acoustics or electromagnetics. However, their cost strongly increases in the high-frequency regime. This paper introduces a new directional FMM for oscillatory kernels (defmm: directional equispaced interpolation-based fmm), whose precomputation and application are FFT-accelerated due to polynomial interpolations on equispaced grids. We demonstrate the consistency of our FFT approach and show how symmetries can be exploited in the Fourier domain. We also describe the algorithmic design of defmm, well-suited for the BIE nonuniform particle distributions, and present performance optimizations on one CPU core. Finally, we exhibit important performance gains on all test cases for defmm over a state-of-the-art FMM library for oscillatory kernels.

Determination of the doubly symmetric periodic orbits in the restricted three-body problem and Hill’s lunar problem

We review some recent progress on the research of the periodic orbits of the N-body problem, and numerically study the spatial doubly symmetric periodic orbits (SDSPs for short). Both comet- and lunar-type SDSPs in the circular restricted three-body problem are computed, as well as the Hill-type SDSPs in Hill’s lunar problem. Double symmetries are exploited so that the SDSPs can be computed efficiently. The monodromy matrix can be calculated by the information of one fourth period. The periodicity conditions are solved by Broyden’s method with a line-search, and some numerical examples show that the scheme is very efficient. For a fixed period ratio and a given acute angle, there exist sixteen cases of initial values. For the restricted three-body problem, the cases of “Copenhagen problem” and the Sun–Jupiter–asteroid model are considered. New SDSPs are also numerically found in Hill’s lunar problem. Though the period ratio should be small theoretically, some new periodic orbits are found when the ratio is not too small, and the linear stability of the searched SDSPs is numerically determined.

The Metastable State of Fermi–Pasta–Ulam–Tsingou Models

Classical statistical mechanics has long relied on assumptions such as the equipartition theorem to understand the behavior of the complicated systems of many particles. The successes of this approach are well known, but there are also many well-known issues with classical theories. For some of these, the introduction of quantum mechanics is necessary, e.g., the ultraviolet catastrophe. However, more recently, the validity of assumptions such as the equipartition of energy in classical systems was called into question. For instance, a detailed analysis of a simplified model for blackbody radiation was apparently able to deduce the Stefan-Boltzmann law using purely classical statistical mechanics. This novel approach involved a careful analysis of a "metastable" state which greatly delays the approach to equilibrium. In this paper, we perform a broad analysis of such a metastable state in the classical Fermi-Pasta-Ulam-Tsingou (FPUT) models. We treat both the α-FPUT and β-FPUT models, exploring both quantitative and qualitative behavior. After introducing the models, we validate our methodology by reproducing the well-known FPUT recurrences in both models and confirming earlier results on how the strength of the recurrences depends on a single system parameter. We establish that the metastable state in the FPUT models can be defined by using a single degree-of-freedom measure-the spectral entropy (η)-and show that this measure has the power to quantify the distance from equipartition. For the α-FPUT model, a comparison to the integrable Toda lattice allows us to define rather clearly the lifetime of the metastable state for the standard initial conditions. We next devise a method to measure the lifetime of the metastable state tm in the α-FPUT model that reduces the sensitivity to the exact initial conditions. Our procedure involves averaging over random initial phases in the plane of initial conditions, the P1-Q1 plane. Applying this procedure gives us a power-law scaling for tm, with the important result that the power laws for different system sizes collapse down to the same exponent as Eα2→0. We examine the energy spectrum E(k) over time in the α-FPUT model and again compare the results to those of the Toda model. This analysis tentatively supports a method for an irreversible energy dissipation process suggested by Onorato et al.: four-wave and six-wave resonances as described by the "wave turbulence" theory. We next apply a similar approach to the β-FPUT model. Here, we explore in particular the different behavior for the two different signs of β. Finally, we describe a procedure for calculating tm in the β-FPUT model, a very different task than for the α-FPUT model, because the β-FPUT model is not a truncation of an integrable nonlinear model.

First post-Newtonian N-body problem in Einstein–Cartan theory with the Weyssenhoff fluid: Lagrangian and first integrals

The rotational dynamics of an N-body system at the first post-Newtonian order in Einstein–Cartan theory is derived. This result is achieved by performing the point-particle limit of the equations of motion of the Weyssenhoff fluid, which models the quantum spin effects residing inside the bodies. For the special case of binary systems, we determine the Lagrangian function and the resulting first integrals underlying the translational dynamics and the spin precession.

Degree of Atomicity in the Chemical Bonding: Why Return to the H2 Molecule?

We analyze two-particle binding factors for the case of \ch{H2} molecule with the help of our original Exact Diagonalization \textit{Ab Intio} (EDABI) approach. Explicitly, we redefine the many-particle covalency and ionicity factors as a function of interatomic distance. Insufficiency of those basic characteristics is stressed and the concept of \textit{atomicity} is introduced and corresponds to the Mott and Hubbard criteria concerning the localization in many-particle systems. This additional characteristic introduces atomic ingredient into the essentially molecular states and thus eliminates a spurious behavior of the standard covalency factor with the increasing interatomic distance, as well as provides a physical reinterpretation of the chemical bond's nature.

Implementation and study of a drag model for an N-body problem simulator

Implementation and study of a drag model for an N-body problem simulator

Old perturbative methods for a new problem in Celestial Mechanics: the space debris dynamics

Perturbative methods have been developed and widely used in the XVIII and XIX century to study the behavior of N-body problems in Celestial Mechanics. Such methods apply to nearly-integrable Hamiltonian systems and they have the remarkable property to be constructive. A well-known application of perturbative techniques is represented by the construction of the so-called proper elements, which are quasi-invariants of the dynamics, obtained by removing the perturbing function to higher orders. They have been used to identify families of asteroids; more recently, they have been used in the context of space debris, which is the main core of this work. We describe the dynamics of space debris, considering a model including the Earth’s gravitational attraction, the influence of Sun and Moon, and the Solar radiation pressure. We construct a Lie series normalization procedure and we compute the proper elements associated to the orbital elements. To provide a concrete example, we analyze three different break-up events with nearby initial orbital elements. We use the information coming from proper elements to successfully group the fragments; the clusterization is supported by statistical data analysis and by machine learning methods. These results show that perturbative methods still play an important role in the study of the dynamics of space objects.

Modular Approach to Creating Functionalized Surface Arrays of Molecular Qubits.

The quest for developing quantum technologies is driven by the promise of exponentially faster computations, ultrahigh performance sensing, and achieving thorough understanding of many-particle quantum systems. Molecular spins are excellent qubit candidates because they feature long coherence times, are widely tunable through chemical synthesis, and can be interfaced with other quantum platforms such as superconducting qubits. A present challenge for molecular spin qubits is their integration in quantum devices, which requires arranging them in thin films or monolayers on surfaces. However, clear proof of the survival of quantum properties of molecular qubits on surfaces has not been reported so far. Furthermore, little is known about the change in spin dynamics of molecular qubits going from the bulk to monolayers. Here, a versatile bottom-up method is reported to arrange molecular qubits as functional groups of self-assembled monolayers (SAMs) on surfaces, combining molecular self-organization and click chemistry. Coherence times of up to 13 µs demonstrate that qubit properties are maintained or even enhanced in the monolayer.