Many-particle Systems Research Articles

The Scott conjecture for large Coulomb systems: a review

We review some older and more recent results concerning the energy and particle distribution in ground states of heavy Coulomb systems. The reviewed results are asymptotic in nature: they describe properties of many-particle systems in the limit of a large number of particles. Particular emphasis is put on models that take relativistic kinematics into account. While non-relativistic models are typically rather well understood, this is generally not the case for relativistic ones and leads to a variety of open questions.

Symmetry breaking/symmetry preserving circuits and symmetry restoration on quantum computers

We discuss here some aspects related to the symmetries of a quantum many-body problem when trying to treat it on a quantum computer. Several features related to symmetry conservation, symmetry breaking, and possible symmetry restoration are reviewed. After briefly discussing some of the standard symmetries relevant for many-particle systems, we discuss the advantage of encoding some symmetries directly in quantum ansätze, especially to reduce the quantum register size. It is, however, well-known that the use of symmetry-breaking states can also be a unique way to incorporate specific internal correlations when a spontaneous symmetry breaking occurs. These aspects are discussed in the quantum computing context. Ultimately, an accurate description of quantum systems can be achieved only when the initially broken symmetries are properly restored. We review several methods explored previously to perform symmetry restoration on a quantum computer, for instance, the ones based on symmetry filtering by quantum phase estimation and by an iterative independent set of Hadamard tests. We propose novel methods that pave the new directions to perform symmetry restoration, like those based on the purification of the state employing the linear combination of unitaries (LCU) approach.

Energetics of a self-gravitating quantum system of charged particles

Extending our model of Newtonian Quantum Gravity (NQG) in a self-gravitating system (Lotte B. K. and Mishra S., Mod. Phys. Lett. A, 35 (2020) 2050081), we study here the energetics of a charged self-gravitating quantum many-particle system. The model is based on the use of the uncertainty principle and the incorporation of necessary relativistic corrections. From the resulting ground state energy we have derived the effective radius of the system of charged particles, after it gets collapsed under its own gravity. Additional results like the Hawking temperature and the Buchdahl-Andréasson (BA) limit for this system are also explored. We further show a possible link of this collapsed system of self-gravitating charge particles, with the notion of a charged black hole.

Block-encoding dense and full-rank kernels using hierarchical matrices: applications in quantum numerical linear algebra

Many quantum algorithms for numerical linear algebra assume black-box access to a block-encoding of the matrix of interest, which is a strong assumption when the matrix is not sparse. Kernel matrices, which arise from discretizing a kernel function k(x,x′), have a variety of applications in mathematics and engineering. They are generally dense and full-rank. Classically, the celebrated fast multipole method performs matrix multiplication on kernel matrices of dimension N in time almost linear in N by using the linear algebraic framework of hierarchical matrices. In light of this success, we propose a block-encoding scheme of the hierarchical matrix structure on a quantum computer. When applied to many physical kernel matrices, our method can improve the runtime of solving quantum linear systems of dimension N to O(κpolylog⁡(Nε)), where κ and ε are the condition number and error bound of the matrix operation. This runtime is near-optimal and, in terms of N, exponentially improves over prior quantum linear systems algorithms in the case of dense and full-rank kernel matrices. We discuss possible applications of our methodology in solving integral equations and accelerating computations in N-body problems.

Magnetic-field-induced Wigner crystallization of charged interlayer excitons in van der Waals heterostructures

Ever since its inception, coherent excited states of semiconductors have been the focus of semiconductor materials research to evolve into a vibrant field of low-dimensional solid-state physics. The field is gaining new momentum these days due to emerging transdimensional semiconductors such as van der Waals bound layers of transition metal dichalcogenides (TMDs) of controlled thickness. Here, we develop the theory of magnetic-field-induced Wigner crystallization for charged interlayer excitons (CIE) discovered recently in TMD heterobilayers. We derive the ratio of the potential interaction energy to the kinetic energy for the many-particle CIE system in the perpendicular magnetostatic field of an arbitrary strength and predict the crystallization effect in the strong field regime. We show that magnetic-field-induced Wigner crystallization and melting of CIEs can be observed in magneto-photoluminescence experiments with TMD bilayers of systematically varied electron-hole doping concentrations. Our results advance the capabilities of this new generation of transdimensional quantum materials.

Homogeneous generalized master equation for an arbitrary initial state of a many-particle classical system

A special projection operator is introduced, which exactly transforms the Nakajima–Zwanzig inhomogeneous Generalized Master Equation (GME) with an irrelevant initial condition term (a source) into a completely closed homogeneous GME (with no initial correlations term) for the relevant part of an N-particle distribution function FN(t) for an arbitrary initial distribution function FN(0) of a system of classical particles. It resolves the problem of the derivation of a closed linear evolution equation from the Liouville equation with no “molecular chaos” type approximation. The obtained equation is equivalent to the completely closed equation for s-particle (s≤N) distribution function Fs(t). The initial correlations are accounted for in this equation in the modified by the introduced projection operator kernel governing the evolution of Fs(t). This equation is further simplified by presenting its kernel in the linear in the particles’ density n approximation. In this approximation the equations for one-particle F1(t) and two-particle F2(t) distribution functions are derived. Special attention is given to analyzing the equation for F1(t) and showed that this equation in the Markov approximation can be written as the linear Boltzmann or Landau (for a small interparticle interaction) equations but with additional terms caused by initial correlations.

Derivation and Properties of the Angular Momentum Relative Amended Potential

AbstractThe paper concerns the determination of the Angular Momentum Relative Amended Potential (AMR potential) in the framework of the Full n-Body problem and some of its basic properties are discussed. The AMR potential is derived using two different approaches : first using a Routh reduction of the system relative to rotation about the total angular momentum, second as a variation of the Sundman Inequality, using the Cauchy inequality.

The Number of Relative Equilibria in the PCR4BP

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.

Average-fluctuation separation in energy levels in many-particle quantum systems with k-body interactions using q-Hermite polynomials

Separation between average and fluctuation parts in the state density in many-particle quantum systems with $k$-body interactions, modeled by the $k$-body embedded Gaussian orthogonal random matrices (EGOE($k$)), is demonstrated using the method of normal mode decomposition of the spectra and also verified through power spectrum analysis, for both fermions and bosons. The smoothed state density is represented by the $q$-normal distribution ($f_{qN}$) (with corrections) which is the weight function for $q$-Hermite polynomials. As the rank of interaction $k$ increases, the fluctuations set in with smaller order of corrections in the smooth state density. They are found to be of GOE type, for all $k$ values, for both fermion and boson systems.

COMPACT—a new complex plasma facility for the ISS

Complex plasma is a state of soft matter where micrometer-sized particles are immersed in a weakly ionized gas. The particles acquire negative charges of the order of several thousand elementary charges in the plasma, and they can form gaseous, liquid and crystalline states. Direct optical observation of individual particles allows to study their dynamics on the kinetic level even in large many-particle systems. Gravity is the dominant force in ground-based experiments, restricting the research to vertically compressed, inhomogeneous clouds, or two-dimensional systems, and masking dynamical processes mediated by weaker forces. An environment with reduced gravity, such as provided on the International Space Station (ISS), is therefore essential to overcome this limitations. We will present the research goals for the next generation complex plasma facility COMPACT to be operated onboard the ISS. COMPACT is envisaged as an international multi-purpose and multi-user facility that gives access to the full three-dimensional kinetic properties of the particles.

Is there a one-to-one correspondence between interparticle interactions and physical properties of liquid?

In this study, we present the original method for reconstructing the potential of interparticle interaction from statistically averaged structural data, namely, the radial distribution function of particles in many-particle system. This method belongs to a family of machine learning methods and is implemented through the differential evolution algorithm. As demonstrated for the case of the Lennard-Jones liquid taken as an example, there is no one-to-one correspondence between structure and potential of interparticle interaction of a many-particle disordered system at a certain thermodynamic state. Namely, a whole family of the Mie potentials determined by two parameters p1 and p2 related to each other according to a certain rule can reproduce properly a unique structure of the Lennard-Jones liquid at a given thermodynamic state. It is noteworthy that this family of the potentials quite correctly reproduces for the Lennard-Jones liquid the transport properties (in particular, the self-diffusion coefficient) over a temperature range as well as the dynamic structure factor, which is one of the key characteristics of the collective dynamics of particles.

Production of hot Jupiter candidates from high-eccentricity mechanisms for different initial planetary mass configurations

ABSTRACT Hot Jupiters (HJs) are giant planets with orbital periods of the order of a few days with semimajor axis within ∼0.1 au. Several theories have been invoked in order to explain the origin of this type of planets, one of them being the high-eccentricity migration. This migration can occur through different high-eccentricity mechanisms. Our investigation focused on six different kinds of high-eccentricity mechanisms, namely, direct dispersion, coplanar, Kozai–Lidov, secular chaos, E1 and E2 mechanisms. We investigated the efficiency of these mechanisms for the production of HJ candidates in multiplanet systems initially tightly-packed in the semimajor axis, considering a large set of numerical simulations of the exact equations of motion in the context of the N-body problem. In particular, we analyzed the sensitivity of our results to the initial number of planets, the initial semimajor axis of the innermost planetary orbit, the initial configuration of planetary masses, and to the inclusion of general relativity (GR) effects. We found that the E1 mechanism is the most efficient in producing HJ candidates both in simulations with and without the contribution of GR, followed by the Kozai–Lidov and E2 mechanisms. Our results also revealed that, except for the initial equal planetary mass configuration, the E1 mechanism was notably efficient in the other initial planetary mass configurations considered in this work. Finally, we investigated the production of HJ candidates with prograde, retrograde, and alternating orbits. According to our statistical analysis, the Kozai–Lidov mechanism has the highest probability of significantly exciting the orbital inclinations of the HJ candidates.

Phonon and Optical-roton Branches of Excitations of the Bose System

For a system of a large number of Bose particles, a chain of coupled equations for the averages of field operators is obtained. In the approximation where only the averages of one field operator and the averages of products of two operators at zero temperature are taken into account, there is derived a closed system of dynamic equations. Taking into account the finite range of the interaction potential between particles, the spectrum of elementary excitations of a many-particle Bose system is calculated, and it is shown that it has two branches: a sound branch and an optical branch with an energy gap at zero momentum. At high density, both branches are nonmonotonic and have the roton-like minima. The dispersion of the phonon part of the spectrum is considered. The performed calculations and analysis of experiments on neutron scattering allow to make a statement about the complex structure of the Landau dispersion curve in the superfluid $$^4$$ He.

Artificial neural network states for nonadditive systems

Methods inspired from machine learning have recently attracted great interest in the computational study of quantum many-particle systems. So far, however, it has proven challenging to deal with microscopic models in which the total number of particles is not conserved. To address this issue, we propose a variant of neural network states, which we term neural coherent states. Taking the Fr\"ohlich impurity model as a case study, we show that neural coherent states can learn the ground state of nonadditive systems very well. In particular, we recover exact diagonalization in all regimes tested and observe substantial improvement over the standard coherent state estimates in the most challenging intermediate-coupling regime. Our approach is generic and does not assume specific details of the system, suggesting wide applications.

Critical points at infinity in charged [formula omitted]-body systems

We define the notion of critical points at infinity for the charged N-body problem, following the approach of Albouy (1993). We give a characterisation of such points and show how they can be found in the charged 3-body problem. The symmetry group of the N-body problem and accompanying integrals play a key role. In fact critical points at infinity are indispensible in understanding the bifurcations of the integral map. Together with the critical points at infinity in the charged 3-body problem, we present the bifurcation values.

Non-equilibrium dynamics of bacterial colonies—growth, active fluctuations, segregation, adhesion, and invasion

Colonies of bacteria endowed with a pili-based self-propulsion machinery are ideal models for investigating the structure and dynamics of active many-particle systems. We study Neisseria gonorrhoeae colonies with a molecular-dynamics-based approach. A generic, adaptable simulation method for particle systems with fluctuating bond-like interactions is devised. The simulations are employed to investigate growth of bacterial colonies and the dependence of the colony structure on cell-cell interactions. In colonies, pilus retraction enhances local ordering. For colonies consisting of different types of cells, the simulations show a segregation depending on the pili-mediated interactions among different cells. These results agree with experimental observations. Next, we quantify the power-spectral density of colony-shape fluctuations in silico. Simulations predict a strong violation of the equilibrium fluctuation-response relation. Furthermore, we show that active force generation enables colonies to spread on surfaces and to invade narrow channels. The methodology can serve as a foundation for future studies of active many-particle systems at boundaries with complex shape.

Complexity calculation for an amorphous metastable solid

We use a field-theoretic model to study the partition function of a many-particle system without quenched-disorder. The model is formulated in terms of the coarse-grained density field ρ(x) used in classical density functional theory (DFT). A corresponding free energy landscape (FEL) is depicted in the function space of the continuum field. The number of local minima in the FEL is defined as the Complexity Sc. Following the standard trick of mapping into a composite system of m identical replicas, and finally taking the m→1 limit, Sc is calculated. The two-point structural correlations of the density fluctuations are needed as an input. A vital step of the present work is using the static correlations of the inhomogeneous metastable state obtained in classical DFT. The inhomogeneous state is described in terms of overlapping Gaussian density profiles centred on a set of random lattice points {Ri}. Our results for Sc(η) on extrapolation to higher packing fractions (η)tends to zero, obtaining the corresponding Kauzmann point ηK. We obtain the dependence of the ηK on the properties of the inhomogeneous structure for the structure {Ri}.

Variational Adiabatic Gauge Transformation on Real Quantum Hardware for Effective Low-Energy Hamiltonians and Accurate Diagonalization

Effective low-energy theories represent powerful theoretical tools to reduce the complexity in modeling interacting quantum many-particle systems. However, common theoretical methods rely on perturbation theory, which limits their applicability to weak interactions. Here we introduce the Variational Adiabatic Gauge Transformation (VAGT), a nonperturbative hybrid quantum algorithm that can use nowadays quantum computers to learn the variational parameters of the unitary circuit that brings the Hamiltonian to either its block-diagonal or full-diagonal form. If a Hamiltonian can be diagonalized via a shallow quantum circuit, then VAGT can learn the optimal parameters using a polynomial number of runs. The accuracy of VAGT is tested through numerical simulations, as well as simulations on Rigetti and IonQ quantum computers.

Analysis and mean-field derivation of a porous-medium equation with fractional diffusion

A mean-field-type limit from stochastic moderately interacting many-particle systems with singular Riesz potential is performed, leading to nonlocal porous-medium equations in the whole space. The nonlocality is given by the inverse of a fractional Laplacian, and the limit equation can be interpreted as a transport equation with a fractional pressure. The proof is based on Oelschläger’s approach and a priori estimates for the associated diffusion equations, coming from energy-type and entropy inequalities as well as parabolic regularity. An existence analysis of the fractional porous-medium equation is also provided, based on a careful regularization procedure, new variants of fractional Gagliardo–Nirenberg inequalities, and the div-curl lemma. A consequence of the mean-field limit estimates is the propagation of chaos property.

Many-particle covalency, ionicity, and atomicity revisited for a few simple example molecules

We analyze two-particle binding factors of H2, LiH, and HeH+ molecules/ions with the help of our original exact diagonalization ab initio approach. The interelectronic correlations are taken into account rigorously within the second quantization scheme for restricted basis of renormalized single-particle wave functions, i.e., with their size readjusted in the correlated state. This allows us to determine the many-particle covalency and ionicity factors in a natural and intuitive manner in terms of the microscopic single-particle and interaction parameters, also determined within our method. We discuss the limitations of those basic characteristics and introduce the concept of atomicity, corresponding to the Mott and Hubbard criterion concerning localization threshold in many-particle systems. This addition introduces an atomic ingredient into the electron states and thus removes a spurious behavior of covalency with the increasing interatomic distance, as well as provides a more complete physical interpretation of bonding.