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The anisotropic potential energy surface of the (H2)2 dimer represents a challenging problem for many-body methods. Here, we determine the potential energy curves of five different dimer configurations (T, Z, X, H, and L) using the lattice regularized diffusion Monte Carlo method and a number of approximate functionals within density functional theory (DFT), including advanced orbital-dependent functionals based on the random phase approximation (RPA). We assess their performance in describing the potential wells, bond distances, and relative energies. The repulsive potential wall is studied by looking at the relative stability of the different dimer configurations as a function of an applied force acting along the intermolecular axis. It is shown that most functionals within DFT break down at finite compression, even those that give an accurate description around the potential well minima. Only by including exchange within RPA, a qualitatively correct description along the entire potential energy curve is obtained. Finally, we discuss these results in the context of solid molecular hydrogen at finite pressures. [ABSTRACT FROM AUTHOR]

Quantum computers hold immense potential in the field of chemistry, ushering new frontiers to solve complex many-body problems that are beyond the reach of classical computers. However, noise in the current quantum hardware limits their applicability to large chemical systems. This work encompasses the development of a projective formalism that aims to compute ground-state energies of molecular systems accurately using noisy intermediate scale quantum (NISQ) hardware in a resource-efficient manner. Our approach is reliant upon the formulation of a bipartitely decoupled parameterized ansatz within the disentangled unitary coupled cluster framework based on the principles of nonlinear dynamics and synergetics. Such decoupling emulates total parameter optimization in a lower dimensional manifold, while a mutual synergistic relationship among the parameters is exploited to ensure characteristic accuracy via a non-iterative energy correction. Without any pre-circuit measurements, our method leads to a highly compact fixed-depth ansatz with shallower circuits and fewer expectation value evaluations. Through analytical and numerical demonstrations, we establish the method's superior performance under noise while concurrently ensuring requisite accuracy in future fault-tolerant systems. This approach enables rapid exploration of emerging chemical spaces by the efficient utilization of near-term quantum hardware resources. [ABSTRACT FROM AUTHOR]

This article introduces HODLR3D, a class of hierarchical matrices arising out of N-body problems in three dimensions. HODLR3D relies on the fact that certain off-diagonal matrix sub-blocks arising out of the N-body problems in three dimensions are numerically low rank. For the Laplace kernel in 3D, which is widely encountered, we prove that all the off-diagonal matrix sub-blocks are rank deficient in finite precision. We also obtain the growth of the rank as a function of the size of these matrix sub-blocks. For other kernels in three dimensions, we numerically illustrate a similar scaling in rank for the different off-diagonal sub-blocks. We leverage this hierarchical low-rank structure to construct HODLR3D representation, with which we accelerate matrix-vector products. The storage and computational complexity of the HODLR3D matrix-vector product scales almost linearly with system size. We demonstrate the computational performance of HODLR3D representation through various numerical experiments. Further, we explore the performance of the HODLR3D representation on distributed memory systems. HODLR3D, described in this article, is based on a weak admissibility condition. Among the hierarchical matrices with different weak admissibility conditions in 3D, only in HODLR3D did the rank of the admissible off-diagonal blocks not scale with any power of the system size. Thus, the storage and the computational complexity of the HODLR3D matrix-vector product remain tractable for N-body problems with large system sizes. [ABSTRACT FROM AUTHOR]

We study the class of symmetric Dziobek configurations where there are p + q bodies with non-zero masses at the vertices of a regular (p + q - 1) -simplex and one more body with zero mass on the symmetric line going through the centers of a subsimplex with p vertices and its complementary subsimplex with q vertices. All bodies from the same subsimplex have equal masses and let μ ≠ 0 be the ratio of masses from different subsimplices. We consider such central configurations for homogeneous force laws with exponent a ≠ 0 . The Newtonian case of a = - 3 2 with μ > 0 is studied more intensively. Using elementary methods, we obtain results for general values of the parameters a, p, q, and μ . For a = - 3 2 , we count the real roots of a polynomial system in two variables with three parameters. We are able to reduce the number of parameters required in our symbolic computation to two or one and use some techniques to count roots rigorously for all parameter values. Our results extend many previously known cases. For example, the known three positions of the zero mass in the case of a = - 3 2 , p = q = 1 , μ > 0 are at the Lagrange points L 1 , L 2 , and L 3 discovered by Euler. [ABSTRACT FROM AUTHOR]

In the $ N $-body problem, a motion is called hyperbolic, when the mutual distances between the bodies go to infinity with non-zero limiting velocities as time goes to infinity. For Newtonian potential, in [8] Maderna and Venturelli proved that starting from any initial position there is a hyperbolic motion with any prescribed limiting velocities at infinity.Recently based on a different approach, Liu, Yan and Zhou [6] generalized this result to a larger class of $ N $-body problem. As the proof in [6] is quite long and technical, we give a simplified proof for homogeneous potentials following the approach given in the latter paper. [ABSTRACT FROM AUTHOR]

We deal, for the classical N -body problem, with the existence of action minimizing half entire expansive solutions with prescribed asymptotic direction and initial configuration of the bodies. We tackle the cases of hyperbolic, hyperbolic-parabolic and parabolic arcs in a unified manner. Our approach is based on the minimization of a renormalized Lagrangian action on a suitable functional space. With this new strategy, we are able to confirm the already-known results of the existence of both hyperbolic and parabolic solutions, and we prove for the first time the existence of hyperbolic-parabolic solutions for any prescribed asymptotic expansion in a suitable class. Associated with each element of this class we find a viscosity solution of the Hamilton-Jacobi equation as a linear correction of the value function. Besides, we also manage to give a precise description of the growth of parabolic and hyperbolic-parabolic solutions. [ABSTRACT FROM AUTHOR]

The continued development of computational approaches to many-body ground-state problems in physics and chemistry calls for a consistent way to assess its overall progress. In this work, we introduce a metric of variational accuracy, theV-score, obtained from the variational energy and its variance. We provide an extensive curated dataset of variational calculations of many-body quantum systems, identifying cases where state-ofthe-art numerical approaches show limited accuracy and future algorithms or computational platforms, such as quantum computing, could provide improved accuracy. The V-score can be used as a metric to assess the progress of quantum variational methods toward a quantum advantage for ground-state problems, especially in regimes where classical verifiability is impossible. [ABSTRACT FROM AUTHOR]

The infinite spin problem is an old problem concerning the rotational behavior of total collision orbits in the n -body problem. It has long been known that when a solution tends to total collision then its normalized configuration curve must converge to the set of normalized central configurations. In the planar n-body problem every normalized configuration determines a circle of rotationally equivalent normalized configurations and, in particular, there are circles of normalized central configurations. It's conceivable that by means of an infinite spin , a total collision solution could converge to such a circle instead of to a particular point on it. Here we prove that this is not possible, at least if the limiting circle of central configurations is isolated from other circles of central configurations. (It is believed that all central configurations are isolated, but this is not known in general.) Our proof relies on combining the center manifold theorem with the Łojasiewicz gradient inequality. [ABSTRACT FROM AUTHOR]

This Topical Collection of the European Physics Journal A is devoted to recent progress in the nuclear many-body problem. In particular, it aims at a comprehensive compilation of developments related to the work of a pioneer in that field, Peter Schuck, who passed away in 2022. Together with Peter Ring, he co-authored the book on "The Nuclear Many-Body Problem". Different concepts presented in this seminal book have been elaborated further within a broad international collaboration. For instance, the quasi-particle approaches in connection with nuclear superfluidity and cluster formation in nuclear systems, in particular alpha-particle condensation and quartetting at subsaturation densities, have been put forward inspired by Peter Schuck. These advances obtained in the nuclear many-body problem can also be applied to other systems, for instance solid state physics. This Topical Collection is considered as addendum and continuation of the textbook of P. Ring and P. Schuck. [ABSTRACT FROM AUTHOR]

The aim of this work is to explain how, starting from the orthogonality expression of two polynomials, we deduce the Schrödinger equation and the solution of the N-body problem including two-body correlations as well as the existence of shells. Generated by the behaviour of kinetic energy for a two-body interaction. The quantification of matters is obtained by the application of the weight function algorithm to the statement that two states are independents when their product integrated over the whole space is null leading to a two variables second order differential equation. The Nuclear Shell Model is a consequence of the kinetic energy behaviour for increasing number of nucleons in ground state. It leaves the mean field theory useless. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a new, analytical, method for computing the exact number of relative equilibria in the planar, circular, restricted 4-body problem of celestial mechanics. The new approach allows for a very efficient computer-aided proof, and opens a potential pathway to proving harder instances of the n-body problem. [ABSTRACT FROM AUTHOR]

In the N-body problem, it is classical that there are conserved quantities of center of mass, linear momentum, angular momentum and energy. The level sets M (c , h) of these conserved quantities are parameterized by the angular momentum c and the energy h, and are known as the integral manifolds. A long-standing goal has been to identify the bifurcation values, especially the bifurcation values of energy for fixed non-zero angular momentum, and to describe the integral manifolds at the regular values. Alain Albouy identified two categories of singular values of energy: those corresponding to bifurcations at relative equilibria; and those corresponding to "bifurcations at infinity", and demonstrated that these are the only possible bifurcation values. This work examines the bifurcations for the four body problem with equal masses. There are four singular values corresponding to bifurcations at infinity. To establish that the topology of the integral manifolds changes at each of these values, and to describe the manifolds at the regular values of energy, the homology groups of the integral manifolds are computed for the five energy regions on either side of the singular values. The homology group calculations establish that all four energy levels are indeed bifurcation values, and allows some of the global properties of the integral manifolds to be explored. A companion paper will provide the corresponding analysis of the bifurcations at relative equilibria for the four-body problem with equal masses. [ABSTRACT FROM AUTHOR]

Advancements in the implementation of quantum hardware have enabled the acquisition of data that are intractable for emulation with classical computers. The integration of classical machine learning (ML) algorithms with these data holds potential for unveiling obscure patterns. Although this hybrid approach extends the class of efficiently solvable problems compared to using only classical computers, this approach has been only realized for solving restricted problems because of the prevalence of noise in current quantum computers. Here, we extend the applicability of the hybrid approach to problems of interest in many-body physics, such as predicting the properties of the ground state of a given Hamiltonian and classifying quantum phases. By performing experiments with various error-reducing procedures on superconducting quantum hardware with 127 qubits, we managed to acquire refined data from the quantum computer. This enabled us to demonstrate the successful implementation of theoretically suggested classical ML algorithms for systems with up to 44 qubits. Our results verify the scalability and effectiveness of the classical ML algorithms for processing quantum experimental data. Classical machine learning methods trained on quantum simulation data can efficiently solve quantum many-body problems, but this has been mostly shown for numerically generated data. Here the authors successfully apply such methods to data acquired on a superconducting quantum processor with error mitigation. [ABSTRACT FROM AUTHOR]

Phenomenological relations linking thermodynamics and kinetic theory in condensed matter have been empirically verified in numerous systems, yet their theoretical derivation from first principles remains an open question. One such relation is the so-called "excess-entropy scaling". Do N-body gravitational systems exhibit a similar relation? We provide an affirmative response to this question, albeit with some limitations. Our analysis relies on a well-established thermodynamical quasi-equilibrium model for the cosmological N-body problem, along with an appropriate diffusion model for gravitational interactions. By identifying a scaling region, we were able to estimate diffusion coefficients directly from observational two-particle correlation functions or counts-in-cells distributions in large-scale structures. Intriguingly, the phenomenon of excess-entropy scaling manifests primarily during the nonlinear, asymptotic clustering phase preceding a potential thermodynamic phase transition. [ABSTRACT FROM AUTHOR]

For given k bodies of collinear central configuration of Newtonian k-body problem, we ask whether one can add other l bodies at the same time on the line without changing the configuration and motion of the initial bodies so that the total k + l bodies provide a central configuration. We call it k+l-Moulton configuration. We find the following. When l < k + 1, there exist only zero-mass solutions, masses of added bodies are all zero that means infinitesimal mass. When l = k + 1, we show the existence of k+l-Moulton configuration where masses are non-negative given as a one parameter family, m B = m B 0 t, t ≥ 0. Then there exist not only zero-mass but also positive-mass solutions whose masses are all positive. Moreover when l > k + 1, there is not zero-mass solution because one cannot put more than one body in an interval which is separated by initial k bodies. Then maximum number of added bodies is k + 1 at once in zero-mass solutions. [ABSTRACT FROM AUTHOR]

Several quantum hardware platforms, while being unable to perform fully fault-tolerant quantum computation, can still be operated as analogue quantum simulators for addressing many-body problems. However, due to the presence of errors, it is not clear to what extent those devices can provide us with an advantage with respect to classical computers. In this work, we make progress on this problem for noisy analogue quantum simulators computing physically relevant properties of many-body systems both in equilibrium and undergoing dynamics. We first formulate a system-size independent notion of stability against extensive errors, which we prove for Gaussian fermion models, as well as for a restricted class of spin systems. Remarkably, for the Gaussian fermion models, our analysis shows the stability of critical models which have long-range correlations. Furthermore, we analyze how this stability may lead to a quantum advantage, for the problem of computing the thermodynamic limit of many-body models, in the presence of a constant error rate and without any explicit error correction. Analogue quantum simulators have looser requirements than digital ones, but rigorous results on their usefulness in the noisy case are few. Here, the authors conclude that analogue quantum simulators are robust to errors and can provide superpolynomial to exponential quantum advantage when used to compute relevant many-body observables. [ABSTRACT FROM AUTHOR]

The Variational Monte Carlo (VMC) method has recently seen important advances through the use of neural network quantum states. While more and more sophisticated ansatze have been designed to tackle a wide variety of quantum many-body problems, modest progress has been made on the associated optimization algorithms. In this work, we revisit the Kronecker-Factored Approximate Curvature (KFAC), an optimizer that has been used extensively in a variety of simulations. We suggest improvements in the scaling and the direction of this optimizer and find that they substantially increase its performance at a negligible additional cost. We also reformulate the VMC approach in a game theory framework, to propose a novel optimizer based on decision geometry. We find that on a practical test case for continuous systems, this new optimizer consistently outperforms any of the KFAC improvements in terms of stability, accuracy and speed of convergence. Beyond VMC, the versatility of this approach suggests that decision geometry could provide a solid foundation for accelerating a broad class of machine learning algorithms. This article is part of the theme issue 'The liminal position of Nuclear Physics: from hadrons to neutron stars'. [ABSTRACT FROM AUTHOR]

Reduced mass value and vector are well known for the two-body problem, but the many-body reduced vector problem is not solved yet. The study of many-body problems and their applications (such as vibrational spectroscopy) is one of the more important physical problems. Vibrational spectroscopy provides a powerful tool to perceive the molecular structures and atom motions of molecules. The water molecule is a threebody system stretching vibration that has been previously quantized; their frequencies were defined and showed the infrared (IR) absorption spectrum based on Morse potential. In this work, the reduced mass of the many-body problem is being solved and then used to study the intensity of the stretching vibration modes and show the ratio is in agreement with experiments. The molecule was studied in classical and quantum mechanics to determine its absorption intensity as an example of a reduced mass problem. The results show molecular atomic motions and changes in dipole and reduced mass vector. A Morse-like model for bending was predicted based on the spectroscopic vibration frequency and intensity, defining the bending potential depth of 93.5 kJ/mol. [ABSTRACT FROM AUTHOR]

One of the oldest problems in physics is that of calculating the motion of N particles under a specified mutual force: the N-body problem. Much is known about this problem if the specified force is non-relativistic gravity, and considerable progress has been made by considering the problem in one spatial dimension. Here, I review what is known about the relativistic gravitational N-body problem. Reduction to one spatial dimension has the feature of the absence of gravitational radiation, thereby allowing for a clear comparison between the physics of one-dimensional relativistic and non-relativistic self-gravitating systems. After describing how to obtain a relativistic theory of gravity coupled to N point particles, I discuss in turn the two-body, three-body, four-body, and N-body problems. Quite general exact solutions can be obtained for the two-body problem, unlike the situation in general relativity in three spatial dimensions for which only highly specified solutions exist. The three-body problem exhibits mild forms of chaos, and provides one of the first theoretical settings in which relativistic chaos can be studied. For N ≥ 4 , other interesting features emerge. Relativistic self-gravitating systems have a number of interesting problems awaiting further investigation, providing us with a new frontier for exploring relativistic many-body systems. [ABSTRACT FROM AUTHOR]

This Perspective presents an overview of the development of the hierarchy of Davydov's Ansätze and a few of their applications in many-body problems in computational chemical physics. Davydov's solitons originated in the investigation of vibrational energy transport in proteins in the 1970s. Momentum-space projection of these solitary waves turned up to be accurate variational ground-state wave functions for the extended Holstein molecular crystal model, lending unambiguous evidence to the absence of formal quantum phase transitions in Holstein systems. The multiple Davydov Ansätze have been proposed, with increasing Ansatz multiplicity, as incremental improvements of their single-Ansatz parents. For a given Hamiltonian, the time-dependent variational formalism is utilized to extract accurate dynamic and spectroscopic properties using Davydov's Ansätze as its trial states. A quantity proven to disappear for large multiplicities, the Ansatz relative deviation is introduced to quantify how closely the Schrödinger equation is obeyed. Three finite-temperature extensions to the time-dependent variation scheme are elaborated, i.e., the Monte Carlo importance sampling, the method of thermofield dynamics, and the method of displaced number states. To demonstrate the versatility of the methodology, this Perspective provides applications of Davydov's Ansätze to the generalized Holstein Hamiltonian, variants of the spin-boson model, and systems of cavity-assisted singlet fission, where accurate dynamic and spectroscopic properties of the many-body systems are given by the Davydov trial states. [ABSTRACT FROM AUTHOR]

The pressure tensor (equivalent to the negative stress tensor) at both microscopic and macroscopic levels is fundamental to many aspects of engineering and science, including fluid dynamics, solid mechanics, biophysics, and thermodynamics. In this Perspective, we review methods to calculate the microscopic pressure tensor. Connections between different pressure forms for equilibrium and nonequilibrium systems are established. We also point out several challenges in the field, including the historical controversies over the definition of the microscopic pressure tensor; the difficulties with many-body and long-range potentials; the insufficiency of software and computational tools; and the lack of experimental routes to probe the pressure tensor at the nanoscale. Possible future directions are suggested. [ABSTRACT FROM AUTHOR]

KAM theory owes most of its success to its initial motivation: the application to problems of celestial mechanics. The masterly application was offered by Arnold in the 60s who worked out a theorem, that he named the "Fundamental Theorem" (FT), especially designed for the planetary problem. However, FT could be really used at that purpose only when, about 50 years later, a set of coordinates constructively taking the invariance by rotation and close-to-integrability into account was used. Since then, some progress has been done in the symplectic assessment of the problem, and here we review such results. [ABSTRACT FROM AUTHOR]

In this paper, the importance of isospin symmetry and its breaking in elucidating the properties of atomic nuclei is reviewed. The quark mass splitting and the electromagnetic origin of the isospin symmetry breaking (ISB) for the nuclear many-body problem is discussed. The experimental data on isobaric analogue states cannot be described only with the Coulomb interaction, and ISB terms in the nucleon–nucleon interaction are needed to discern the observed properties. In the present work, the ISB terms are explicitly considered in nuclear energy density functional and spherical shell model approaches, and a detailed investigation of the analogue states and other properties of nuclei is performed. It is observed that isospin mixing is largest for the N = Z system in the density functional approach [ABSTRACT FROM AUTHOR]

We investigate the secular evolution of a four-body planetary system, where two planets move around a binary star configuration on quasi-elliptic orbits. It is assumed that the masses of all bodies can change isotropically at different rates and the bodies attract each other according to Newton's law of universal gravitation. The purpose of this work is to investigate an influence of variability of the masses of the binary stars and their planets on the dynamical evolution of the planetary system. We consider the case of small eccentricities and inclinations of the bodies orbits and assume that their orbits do not intersect during evolution. Differential equations of the perturbed motion in the osculating analogues of canonical Poincaré elements were obtained in a general case of the many-body problem with variable masses in our previous work. Here we solve these equations numerically and investigate the secular evolution of some fictitious circumbinary 2-planet system under assumption that the two stars of the binary system lose their masses independently at different rates. In order to demonstrate the dynamical features of the equations we use the known parameters of the TOI-1338 system. Comparing the results of calculations in the cases of constant and variable masses, we show that the isotropic variability of the masses of bodies can influence substantially the secular perturbation of the orbital elements. [ABSTRACT FROM AUTHOR]

Entanglement constitutes one of the key concepts in quantum mechanics and serves as an indispensable tool in the understanding of quantum many-body systems. In this work, we perform extensive numerical investigations of extensive entanglement properties of coupled quantum spin chains. This setup has proven useful for e.g. extending the Lieb–Schultz–Mattis theorem to open systems, and contrasts the majority of previous research where the entanglement cut has one lower dimension than the system. We focus on the cases where the entanglement Hamiltonian is either gapless or exhibits spontaneous symmetry breaking behavior. We further employ conformal field theoretical formulae to identify the universal behavior in the former case. The results in our work can serve as a paradigmatic starting point for more systematic exploration of the largely uncharted physics of extensive entanglement, both analytical and numerical. [ABSTRACT FROM AUTHOR]

The channel width-dependent behaviors of viscosity (μ) and slip length (ls) in nanoslits are investigated using many-body dissipative particle dynamics simulation in both Poiseuille and Couette flow systems. In both systems, the viscosity and slip length increase as the channel width (w) grows in smaller channels, while they reach bulk values in larger channels. Moreover, as the surface wettability decreases, the slip length is found to increase, while the viscosity remains the same. The channel width-dependent behavior in nanoslits can be explained by the unique structure of the confined fluid. As the channel width narrows, the uniform density profile in the central region diminishes, and an oscillation pattern appears throughout the system. The change in the microstructure with the channel width alters friction between layers of fluid in laminar flow and fluid-solid friction, leading to a w-dependent μ and ls. Nonetheless, the alteration of surface wettability influences only fluid–solid interactions but not the friction between layers of fluid. [ABSTRACT FROM AUTHOR]

We consider the two- and n-body problems on the two-dimensional conformal sphere M R 2 , with a radius R > 0 . We employ an alternative potential free of singularities at antipodal points. We study the limit of relative equilibria under the SO(2) symmetry; we examine the specific conditions under which a pair of positive-mass particles, situated at antipodal points, can maintain a state of relative equilibrium as they traverse along a geodesic. It is identified that, under an appropriate radius–mass relationship, these particles experience an unrestricted and free movement in alignment with the geodesic of the canonical Killing vector field in M R 2 . An even number of bodies with pairwise conjugated positions, arranged in a regular n-gon, all with the same mass m, move freely on a geodesic with suitable velocities, where this geodesic motion behaves like a relative equilibrium. Also, a center of mass formula is included. A relation is found for the relative equilibrium in the two-body problem in the sphere similar to the Snell law. [ABSTRACT FROM AUTHOR]

The rigorous description of correlated quantum many-body systems constitutes one of the most challenging tasks in contemporary physics and related disciplines. In this context, a particularly useful tool is the concept of effective pair potentials that take into account the effects of the complex many-body medium consistently. In this work, we present extensive, highly accurate ab initio path integral Monte Carlo (PIMC) results for the effective interaction and the effective force between two electrons in the presence of the uniform electron gas. This gives us a direct insight into finite-size effects, thereby, opening up the possibility for novel domain decompositions and methodological advances. In addition, we present unassailable numerical proof for an effective attraction between two electrons under moderate coupling conditions, without the mediation of an underlying ionic structure. Finally, we compare our exact PIMC results to effective potentials from linear-response theory, and we demonstrate their usefulness for the description of the dynamic structure factor. All PIMC results are made freely available online and can be used as a thorough benchmark for new developments and approximations. [ABSTRACT FROM AUTHOR]

The study of Brownian ratchets has taught how time-periodic driving supports a time-periodic steady state that generates nonequilibrium transport. When a single particle is transported in one dimension, it is possible to rationalize the current in terms of the potential, but experimental efforts have ventured beyond that single-body case to systems with many interacting carriers. Working with a lattice model of volume-excluding particles in one dimension, we analyze the impact of interactions on a flashing ratchet's current. To surmount the many-body problem, we employ the time-dependent variational principle applied to binary tree tensor networks. Rather than propagating individual trajectories, the tensor network approach propagates a distribution over many-body configurations via a controllable variational approximation. The calculations, which reproduce Gillespie trajectory sampling, identify and explain a shift in the frequency of maximum current to higher driving frequency as the lattice occupancy increases. [ABSTRACT FROM AUTHOR]

Downfolding coupled cluster techniques have recently been introduced into quantum chemistry as a tool for the dimensionality reduction of the many-body quantum problem. As opposed to earlier formulations in physics and chemistry based on the concept of effective Hamiltonians, the appearance of the downfolded Hamiltonians is a natural consequence of the single-reference exponential parameterization of the wave function. In this paper, we discuss the impact of higher-order terms originating in double commutators. In analogy to previous studies, we consider the case when only one- and two-body interactions are included in the downfolded Hamiltonians. We demonstrate the efficiency of the many-body expansions involving single and double commutators for the unitary extension of the downfolded Hamiltonians on the example of the beryllium atom, and bond-breaking processes in the Li2 and H2O molecules. For the H2O system, we also analyze energies obtained with downfolding procedures as functions of the active space size. [ABSTRACT FROM AUTHOR]

Based on the idea of adiabatic symmetry, we present a novel basis set expansion method—the kinetic energy partition (KEP) method—for solving quantum eigenvalue problems. Broken symmetry is responsible for quantum entanglement in many-body systems via parametric non-adiabatic corrections. Starting from simple one-particle-in-one-dimension problems, we gradually increase the complexity in the number of particles and the interaction patterns. Our goal in the mini-review is to advocate for the utility of the KEP method in front-line research, in particular for research beginners in quantum many-body problems. [ABSTRACT FROM AUTHOR]

We consider the problem: given a configuration, find the masses which make it central. This is the inverse problem for central configurations. The main result is a symmetry theorem (see Theorem 1) that allows simplifying the equations for symmetric central configurations, with any symmetry group, at any dimension, using linear representations of finite groups. Using this theorem, one can study the existence of central configurations with a given symmetry group and also study the masses allowed in this central configuration. As an application, we determine the masses when the configurations are the Platonic solids. The simplifications significantly reduce the equations allowing to use of a computer algebra system to make exact computations. In this case, we prove that a central configuration is possible if and only if the masses are equal (see Theorem 5). Except for the Tetrahedron that is known to be a central configuration for any masses. [ABSTRACT FROM AUTHOR]

The focus here is on a thin solid body passing through a channel flow and interacting with the flow. Unsteady two-dimensional interactive properties from modelling, analysis and computation are presented along with comparisons. These include the effects of a finite dilation or constriction, as the body travels through, and the effects of a continuing expansion of the vessel. Finite-time clashing of the body with the channel walls is investigated as well as the means to avoid clashing. Sustained oscillations are found to be possible. Wake properties behind the body are obtained, and broad agreement in trends between full-system and reduced-system responses is found for increased body mass. [ABSTRACT FROM AUTHOR]

In this paper, we study the n -body problem in the Poincaré upper half-plane ℍ R 2 , where the radius R of the Poincaré disk is fixed. We introduce a new potential to derive the condition for hyperbolic relative equilibria on ℍ R 2 . We analyze the relative equilibrium of positive masses moving along geodesics under the SL (2 , ℝ) group. This result is utilized to establish the existence of relative equilibria for the n -body problem on ℍ R 2 for n = 2 and n = 3. We revisit previously known results and uncover new qualitative findings on relative equilibria that are not evident in an extrinsic context. Additionally, we provide a simple expression for the center of mass of a system of point particles on a two-dimensional surface with negative constant Gaussian curvature. [ABSTRACT FROM AUTHOR]

The present paper is the second of a series of publications that aim at investigating relevant directions to turn the nuclear energy density functional (EDF) method as an effective field theory (EFT). The EDF approach has known numerous successes in nuclear theory over the past decades [1] and is currently the only microscopic technique that can be applied to all atomic nuclei. However, the phenomenological character of the EDF method also comes with important limitations, such as the lack of an explicit connection with quantum chromodynamics (QCD). As was argued in the first paper of this series [2], reformulating the EDF framework as an EFT would enable us to overcome these limitations. In particular, path-integral (PI) techniques are suited to achieve such a purpose as they allow us to design numerous non-perturbative approximations and can take Lagrangians possibly derived from EFTs of QCD as inputs. In our previous paper [2], we have illustrated such technical features for diagrammatic PI techniques in a study of the (0+0)-D O(N)-symmetric φ 4 -theory. In the present work, we consider another class of PI techniques, i.e. functional renormalization group (FRG) approaches, that we apply to the same toy model. Despite our explicit interest for the nuclear many-body problem, the presented study is also directed towards FRG practitioners from various fields: technical details are provided for FRG techniques based on 1-particle-irreducible (1PI), 2-particle-irreducible (2PI) and 2-particle-point-irreducible (2PPI) effective actions, coined respectively as 1PI-, 2PI- and 2PPI-FRGs, and the treatment of the O(N) symmetry is also addressed thoroughly. Connections between these various FRG methods are identified as well. [ABSTRACT FROM AUTHOR]

Explicit symplectic integrators are powerful and widely used for Hamiltonian systems. However, once the post-Newtonian (PN) effect is considered to provide more precise modeling for the N-body problem, explicit symplectic methods cannot be constructed due to the nonseparability of the Hamiltonian. Thus, the available symplectic method is either fully implicit or semi-implicit, which decreases the efficiency because of the implicit iteration used during the evolution. In this paper, we aim to explore efficient explicit methods whose performance is mostly like symplectic methods for PN Hamiltonian systems. Taking the small parameter ε appearing in PN terms into consideration, we replace the implicit symplectic solver with explicit solvers in the mixed symplectic method to solve the PN term and then derive three explicit methods. It is theoretically shown that the proposed methods are respectively second-order, fourth-order, and pseudo-fourth-order, and that their closeness to the corresponding symplectic methods are O (ε 3 h 3) , O (ε 5 h 5) , and O (ε 3 h 3). That is, they are explicit near-symplectic methods with the presence of the small parameter ε. Numerical experiments with the Hamiltonian problem of spinning compact binaries show that the energy errors and orbital errors of the proposed explicit near-symplectic methods are indistinguishable from the corresponding mixed semi-implicit symplectic methods. The very small magnitude of the difference between the proposed explicit near-symplectic methods and the mixed symplectic methods confirms our theoretical analysis of their closeness to symplecticity. Finally, the much less CPU time consumed by the proposed methods highlights their most important advantage of high efficiency over the mixed symplectic methods. [ABSTRACT FROM AUTHOR]

Usually, expected values for various physical quantities, such as the number of electrons occupying certain states or the Coulomb interaction between different states of electrons, can be expressed in terms of integrals. In contrast, our method, based on differential forms, shows that expected values can be obtained by averaging over time. To confirm the validity of our method, we prepare the two cases: one is a very simple case with no many-body interaction, and the other is the case where the many-body term is included (the simplest Anderson Hamiltonian). Regarding the simple case without inclusion of many-body term, we prove analytically that the number of electrons occupying any state derived from our method is equivalent to the analytical one evaluated from the Greenʼs function method. When the many-body term is included, our results show good numerical agreement with the analytical ones derived from the Greenʼs function method. By the two cases, the calculation of expected values based on our method is considered valid. [ABSTRACT FROM AUTHOR]

The Kondo effect, deriving from a local magnetic impurity mediating electron-electron interactions, constitutes a flourishing basis for understanding a large variety of intricate many-body problems. Its experimental implementation in tunable circuits has made possible important advances through well-controlled investigations. However, these have mostly concerned transport properties, whereas thermodynamic observations - notably the fundamental measurement of the spin of the Kondo impurity - remain elusive in test-bed circuits. Here, with a novel combination of a 'charge' Kondo circuit with a charge sensor, we directly observe the state of the impurity and its progressive screening. We establish the universal renormalization flow from a single free spin to a screened singlet, the associated reduction in the magnetization, and the relationship between scaling Kondo temperature and microscopic parameters. In our device, a Kondo pseudospin is realized by two degenerate charge states of a metallic island, which we measure with a non-invasive, capacitively coupled charge sensor. Such pseudospin probe of an engineered Kondo system opens the way to the thermodynamic investigation of many exotic quantum states, including the clear observation of Majorana zero modes through their fractional entropy. Previous work on charge Kondo circuits, in which a spin is formed by two degenerate charge states of a metallic island, has been limited to transport measurements of multi-channel Kondo problems. Piquard et al. use thermodynamic measurements via a charge sensor to study the evolution of a single Kondo impurity. [ABSTRACT FROM AUTHOR]

The paper is devoted to the problem of the origin of the Moon. The discussed modern scenarios for the formation of the Earth–Moon system are: simultaneous formation of the Earth and the Moon in the circumsolar gas-dust disk; impact partial destruction of the Earth by a massive asteroid; gravitational capture of the Moon by the Earth; and destruction of the double Moon at the beginning when approaching the Earth with possible subsequent absorption components of smaller mass by the Earth. We offer two-stage scenario of gravitational capture of the Moon by the Earth in the early stages of Solar System. In the first stage, using a hybrid numerical model in the formulation of the three-body problem (Sun, Earth and Moon) and N-bodies, the search and selection of temporary orbits of the Moon around the Earth is carried out. Using the backward integration method in the N-body problem formulation, the influence of tidal forces on pumping of orbital moment of the Moon () relative to the Earth at its own moment is estimated. The simulation shows that actions tidal forces alone are not enough to capture the Moon by the Earth in a short time scale of ~100 years (). At the second stage, the factor is considered viscous-dissipative environment leading to additional "slowing down" of the Moon, due, for example, to collisions with asteroids and the transition of tidal energy into heat, which helps the Moon get rid of excess kinetic energy and gain constant orbit around the Earth. [ABSTRACT FROM AUTHOR]

We study the importance of few-nucleon forces in chiral effective field theory for describing many-nucleon systems. A combinatorial argument suggests that three-nucleon forces-which are conventionally regarded as next-to-next-to-leading order-should accompany the two-nucleon force already at leading order (LO) starting with mass number A ≃ 10–20. We find that this promotion enables the first realistic description of the 16 O ground state based on a renormalization-group-invariant LO interaction. We also performed coupled-cluster calculations of the equation of state for symmetric nuclear matter and our results indicate that LO four-nucleon forces could play a crucial role for describing heavy-mass nuclei. The enhancement mechanism we found is very general and could be important also in other many-body problems. [ABSTRACT FROM AUTHOR]

Wave functions based on electron-pair states provide inexpensive and reliable models to describe quantum many-body problems containing strongly correlated electrons, given that broken-pair states have been appropriately accounted for by, for instance, a posteriori corrections. In this article, we analyze the performance of electron-pair methods in predicting orbital-based correlation spectra. We focus on the (orbital-optimized) pair-coupled cluster doubles (pCCD) ansatz with a linearized coupled-cluster (LCC) correction. Specifically, we scrutinize how orbital-based entanglement and correlation measures can be determined from a pCCD-tailored CC wave function. Furthermore, we employ the single-orbital entropy, the orbital-pair mutual information, and the eigenvalue spectra of the two-orbital reduced density matrices to benchmark the performance of the LCC correction for the one-dimensional Hubbard model with the periodic boundary condition as well as the N2 and F2 molecules against density matrix renormalization group reference calculations. Our study indicates that pCCD-LCC accurately reproduces the orbital-pair correlation patterns in the weak correlation limit and for molecules close to their equilibrium structure. Hence, we can conclude that pCCD-LCC predicts reliable wave functions in this regime. [ABSTRACT FROM AUTHOR]

Finite temperature auxiliary field-based quantum Monte Carlo methods, including determinant quantum Monte Carlo and Auxiliary Field Quantum Monte Carlo (AFQMC), have historically assumed pivotal roles in the investigation of the finite temperature phase diagrams of a wide variety of multidimensional lattice models and materials. Despite their utility, however, these techniques are typically formulated in the grand canonical ensemble, which makes them difficult to apply to condensates such as superfluids and difficult to benchmark against alternative methods that are formulated in the canonical ensemble. Working in the grand canonical ensemble is furthermore accompanied by the increased overhead associated with having to determine the chemical potentials that produce desired fillings. Given this backdrop, in this work, we present a new recursive approach for performing AFQMC simulations in the canonical ensemble that does not require knowledge of chemical potentials. To derive this approach, we exploit the convenient fact that AFQMC solves the many-body problem by decoupling many-body propagators into integrals over one-body problems to which non-interacting theories can be applied. We benchmark the accuracy of our technique on illustrative Bose and Fermi--Hubbard models and demonstrate that it can converge more quickly to the ground state than grand canonical AFQMC simulations. We believe that our novel use of HS-transformed operators to implement algorithms originally derived for non-interacting systems will motivate the development of a variety of other methods and anticipate that our technique will enable direct performance comparisons against other many-body approaches formulated in the canonical ensemble. [ABSTRACT FROM AUTHOR]

The exact canonical partition function of a hard disk system in a narrow quasi-one-dimensional pore of given length and width is derived analytically in the thermodynamic limit. As a result, the many body problem is reduced to solving the single transcendental equation. The pressures along and across the pore, distributions of contact distances along the pore, and disks' transverse coordinates are found analytically and presented in the whole density range for three different pore widths. The transition from the solidlike zigzag to the liquidlike state is found to be quite sharp in the density scale but shows no genuine singularity. This transition is quantitatively described by the distribution of zigzag's windows through which disks exchange their positions across the pore. The windowlike defects vanish only in the densely packed zigzag, which is in line with a continuous Kosterlitz–Thouless transition. [ABSTRACT FROM AUTHOR]

We study the sensitivity and practicality of Henderson's theorem in classical statistical mechanics, which states that the pair potential v (r) that gives rise to a given pair correlation function g2(r) [or equivalently, the structure factor S(k)] in a classical many-body system at number density ρ and temperature T is unique up to an additive constant. While widely invoked in inverse-problem studies, the utility of the theorem has not been quantitatively scrutinized to any large degree. We show that Henderson's theorem has practical shortcomings for disordered and ordered phases for certain densities and temperatures. Using proposed sensitivity metrics, we identify illustrative cases in which distinctly different potential functions give very similar pair correlation functions and/or structure factors up to their corresponding correlation lengths. Our results reveal that due to a limited range and precision of pair information in either direct or reciprocal space, there is effective ambiguity of solutions to inverse problems that utilize pair information only, and more caution must be exercised when one claims the uniqueness of any resulting effective pair potential found in practice. We have also identified systems that possess virtually identical pair statistics but have distinctly different higher-order correlations. Such differences should be reflected in their individually distinct dynamics (e.g., glassy behaviors). Finally, we prove a more general version of Henderson's theorem that extends the uniqueness statement to include potentials that involve two- and higher-body interactions. [ABSTRACT FROM AUTHOR]

The quantum many-body problem in condensed phases is often simplified using a quasiparticle description, such as effective mass theory for electron motion in a periodic solid. These approaches are often the basis for understanding many fundamental condensed phase processes, including the molecular mechanisms underlying solar energy harvesting and photocatalysis. Despite the importance of these effective particles, there is still a need for computational methods that can explore their behavior on chemically relevant length and time scales. This is especially true when the interactions between the particles and their environment are important. We introduce an approach for studying quasiparticles in condensed phases by combining effective mass theory with the path integral treatment of quantum particles. This framework incorporates the generally anisotropic electronic band structure of materials into path integral simulation schemes to enable modeling of quasiparticles in quantum confinement, for example. We demonstrate the utility of effective mass path integral simulations by modeling an exciton in solid potassium chloride and electron trapping by a sulfur vacancy in monolayer molybdenum disulfide. [ABSTRACT FROM AUTHOR]

Effective Hamiltonians, which are commonly used for fitting experimental observables, provide a coarse-grained representation of exact many-electron states obtained in quantum chemistry calculations; however, the mapping between the two is not trivial. In this contribution, we apply Bloch's formalism to equation-of-motion coupled-cluster wave functions to rigorously derive effective Hamiltonians in Bloch's and des Cloizeaux's forms. We report the key equations and illustrate the theory by application to systems with two or three unpaired electrons, which give rise to electronic states of covalent and ionic characters. We show that Hubbard's and Heisenberg's Hamiltonians can be extracted directly from the so-obtained effective Hamiltonians. By establishing a quantitative connection between many-body states and simple models, the approach facilitates the analysis of the correlated wave functions. We propose a simple diagnostic for assessing the validity of the model space choice based on the overlaps between the target- and model-space states. Artifacts affecting the quality of electronic structure calculations such as spin contamination are also discussed. [ABSTRACT FROM AUTHOR]