Many-particle Hamiltonian Research Articles

Approximation of the ground and excited states of atoms and molecules with a single determinant by means of the exact many-particle Hamiltonian

In the present work, we search for the maximum of the functional | ⟨ Φ ′ | H ^ | Φ ⟩ | , where | Φ ′ ⟩ is a Slater Determinant (SlDet) and H ^ is the exact Hamiltonian of an atom or a molecule, starting from a SlDet | Φ ⟩ , with its spin orbitals calculated by the standard Hartree-Fock (HF) equation or other approximation or any determinant. The element | ⟨ Φ 1 | H ^ | Φ ⟩ | with | Φ 1 ⟩ the maximising | Φ ′ ⟩ gives a value greater or equal to | ⟨ Φ | H ^ | Φ ⟩ | . Next we calculate the corresponding maximum overlap | ⟨ Φ 2 | H ^ | Φ 1 ⟩ | and finally | ⟨ Φ n + 1 | H ^ | Φ n ⟩ | until | ⟨ Φ n + 1 | H ^ | Φ n ⟩ − ⟨ Φ n | H ^ | Φ n − 1 ⟩ | ≤ ε , where ε determines the desired numerical accuracy. We show that the sequence ⟨ Φ n + 1 | H ^ | Φ n ⟩ converges to an extremum ⟨ Φ | H ^ | Φ ⟩ . Having found the SlDet with the lowest energy we repeat the procedure in the orthogonal subspace for finding the first and the higher excited states. We applied this method in order to determine the energies of several configurations of H 3 , the Lithium atom, LiH and Be.

On the microscopic shell-model version of the Bohr-Mottelson collective model

The recently proposed microscopic shell-model version of the Bohr-Mottelson (BM) collective model is considered in more detail in the coordinate representation. The latter possesses a clear and transparent physical meaning, which reveals several features of the new version of the collective model missed in the previous formulation. The relationship to the original BM model is considered, along with the relationships between the different limiting submodels of the microscopic version of the BM model, which closely resemble the relationships of the original Wilets-Jean and rotor models. The kinematically correct many-particle wave functions of the microscopic version of the BM model, conserving the experimentally observed integrals of motion, are shown to consist of collective irrotational-flow and intrinsic components in accordance with the original BM unified model. The general BM Hamiltonian is obtained as a contraction limit of the microscopic many-particle nuclear Hamiltonian, or, alternatively, by restricting the latter to the scalar irreducible collective space.

A new microscopic look into pressure fields and local buoyancy

Defining local pressures is pivotal to understanding phenomena such as buoyancy and atmospheric physics, which always require space-varying pressure fields to be considered. Such fields have precise definitions within the frameworks of hydro-thermodynamics and the kinetic theory of gases, but a pedagogical approach based on standard statistical mechanics techniques is still lacking. In this paper, we propose a new microscopic definition of local pressure inside a classical fluid, relying upon a local barometer potential that is built into the many-particle Hamiltonian. This setup allows pressure to be locally defined as an ensemble average of the radial force exerted by the barometer potential on the gas' particles. Moreover, this same framework is also employed in a microscopic derivation of buoyancy of immersed bodies in the presence of arbitrary external fields. As instructive examples, buoyant forces are calculated for ideal fluids in the presence of a uniform force field, in the presence of a spherical harmonic confinement field and in rotation with a constant angular velocity.

Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double-Well.

We examine the spectral structure and many-body dynamics of two and three repulsively interacting bosons trapped in a one-dimensional double-well, for variable barrier height, inter-particle interaction strength, and initial conditions. By exact diagonalization of the many-particle Hamiltonian, we specifically explore the dynamical behavior of the particles launched either at the single-particle ground state or saddle-point energy, in a time-independent potential. We complement these results by a characterization of the cross-over from diabatic to quasi-adiabatic evolution under finite-time switching of the potential barrier, via the associated time evolution of a single particle’s von Neumann entropy. This is achieved with the help of the multiconfigurational time-dependent Hartree method for indistinguishable particles (MCTDH-X)—which also allows us to extrapolate our results for increasing particle numbers.

Theory of second-order excitonic nonlinearities in transition metal dichalcogenides

Single layers of transition metal dichalcogenides exhibit tightly bound excitons with large oscillator strength making them to promising materials for future optoelectronic applications. Despite that the linear response of these ultrathin materials is extensively studied, a profound understanding of the excitonic second-order nonlinear optical response, including trigonal warping and Berry curvature effects, is lacking so far. To bridge this gap, in this paper we report a microscopic model for the second-order nonlinear susceptibility tensor of these materials. Based on a $kp$ description of the underlying many-particle Hamiltonian we are able to address excitonic features as well as trigonal warping effects, spin-valley coupling, and also Berry curvature to the susceptibility tensor. We present explicit calculation of the latter for the exemplary materials ${\mathrm{MoS}}_{2}$ and ${\mathrm{WS}}_{2}$ on silicon substrate and encapsulated in hBN. For instance we predict resonant susceptibility for second-harmonic generation $0.5--1.3\phantom{\rule{4pt}{0ex}}\mathrm{nm}/\mathrm{V}$ and double-resonant susceptibility $10--30\phantom{\rule{4pt}{0ex}}\mathrm{n}\mathrm{m}/\mathrm{V}$ for the difference and sum-frequency generation processes. Our results are in good agreement with available experimental findings in the literature.

High density limit of the Fermi polaron with infinite mass

We analyze the ground state energy for $N$ identical fermions in a two-dimensional box of volume $L^2$ interacting with an external point scatterer. Since the point scatterer can be considered as an impurity particle of infinite mass, this system is a limit case of the Fermi polaron. We prove that its ground state energy in the limit of high density $N/L^2 \gg 1$ is given by the polaron energy. The polaron energy is an energy estimate based on trial states up to first order in particle-hole expansion, which was proposed by F. Chevy in the physics literature. The relative error in our result is shown to be small uniformly in $L$. Hence, we do not require a gap of fixed size in the spectrum of the Laplacian on the box. The strategy of our proof relies on a twofold Birman-Schwinger type argument applied to the many-particle Hamiltonian of the system.

Explicitly correlated Gaussian functions with shifted-center and projection techniques in pre-Born-Oppenheimer calculations.

Numerical projection methods are elaborated for the calculation of eigenstates of the non-relativistic many-particle Coulomb Hamiltonian with selected rotational and parity quantum numbers employing shifted explicitly correlated Gaussian functions, which are, in general, not eigenfunctions of the total angular momentum and parity operators. The increased computational cost of numerically projecting the basis functions onto the irreducible representations of the three dimensional rotation-inversion group is the price to pay for the increased flexibility of the basis functions. This increased flexibility allowed us to achieve a substantial improvement for the variational upper bound to the Pauli-allowed ground-state energy of the molecular ion treated as an explicit five-particle system. We compare our pre-Born-Oppenheimer result obtained for this molecular ion with rotational-vibrational calculations carried out on a potential energy surface.

Dot-ring nanostructure: Rigorous analysis of many-electron effects.

We discuss the quantum dot-ring nanostructure (DRN) as canonical example of a nanosystem, for which the interelectronic interactions can be evaluated exactly. The system has been selected due to its tunability, i.e., its electron wave functions can be modified much easier than in, e.g., quantum dots. We determine many-particle states for Ne = 2 and 3 electrons and calculate the 3- and 4-state interaction parameters, and discuss their importance. For that purpose, we combine the first- and second-quantization schemes and hence are able to single out the component single-particle contributions to the resultant many-particle state. The method provides both the ground- and the first-excited-state energies, as the exact diagonalization of the many-particle Hamiltonian is carried out. DRN provides one of the few examples for which one can determine theoretically all interaction microscopic parameters to a high accuracy. Thus the evolution of the single-particle vs. many-particle contributions to each state and its energy can be determined and tested with the increasing system size. In this manner, we contribute to the wave-function engineering with the interactions included for those few-electron systems.

Classical-quantum correspondence in bosonic two-mode conversion systems: Polynomial algebras and Kummer shapes

Bosonic quantum conversion systems can be modeled by many-particle single-mode Hamiltonians describing a conversion of $n$ molecules of type A into $m$ molecules of type B and vice versa. These Hamiltonians are analyzed in terms of generators of a polynomially deformed $su(2)$ algebra. In the mean-field limit of large particle numbers, these systems become classical and their Hamiltonian dynamics can again be described by polynomial deformations of a Lie algebra, where quantum commutators are replaced by Poisson brackets. The Casimir operator restricts the motion to Kummer shapes, deformed Bloch spheres with cusp singularities depending on $m$ and $n$. It is demonstrated that the many-particle eigenvalues can be recovered from the mean-field dynamics using a WKB type quantization condition. The many-particle state densities can be semiclassically approximated by the time-periods of periodic orbits, which show characteristic steps and singularities related to the fixed points, whose bifurcation properties are analyzed.

Self-ordering dynamics of ultracold atoms in multicolored cavity fields

We study light-induced spatial crystallization of ultracold quantum particles confined along the axis of a high-$Q$ linear cavity via a transverse multicolor pump using numerical simulations. Whenever a pump frequency is tuned close to resonance with a longitudinal cavity mode, the dynamics favors bistable spatial particle ordering into a Bragg grating at a wavelength distance. Simultaneous pumping at several resonant frequencies fosters competition between the different spatial lattice orders, exhibiting complex nonlinear field dynamics involving several metastable atom-field states. For few particles, even superpositions of different spatial orders entangled with different light-mode amplitudes appear. By a proper choice of trap geometry and pump frequencies, a broad variety of many-particle Hamiltonians with a nontrivial long-range coupling can be emulated in such a setup. When applying quantum Monte Carlo wave-function simulations to study time evolution, we find simultaneous superradiant scattering into several light modes and the buildup of strong nonclassical atom-field correlations.

Quantum Codes Give Counterexamples to the Unique Preimage Conjecture of theN-Representability Problem

It is well known that the ground state energy of many-particle Hamiltonians involving only 2-body interactions can be obtained using constrained optimizations over density matrices which arise from reducing an N-particle state. While determining which 2-particle density matrices are "N-representable" is a computationally hard problem, all known extreme N-representable 2-particle reduced density matrices arise from a unique N-particle preimage, satisfying a conjecture established in 1972. We present explicit counterexamples to this conjecture through giving Hamiltonians with 2-body interactions which have degenerate ground states that cannot be distinguished by any 2-body operator. We relate the existence of such counterexamples to quantum error correction codes and topologically ordered spin systems.

Matrix elements of a many-particle hamiltonian evaluated using hyperspherical coordinates

Two formulas derived in previous articles are used to evaluate the matrix elements of a general many-particle Hamiltonian. The first formula is an expansion of the Coulomb potential of the system in terms of hyperspherical harmonics, while the second is a general formula for the evaluation of angular integrals in many-dimensional spaces. These two formulas lead to explicit expressions for the Hamiltonian matrix elements when the basis functions are mononomials in the 3N coordinates multiplied by functions of the hyperradius.

Many-particle Hamiltonian for open systems with full Coulomb interaction: Application to classical and quantum time-dependent simulations of nanoscale electron devices

A many-particle Hamiltonian for a set of particles with Coulomb interaction inside an open system is described without any perturbative or mean-field approximation. The boundary conditions of the Hamiltonian on the borders of the open system [in the real three-dimensional (3D) space representation] are discussed in detail to include the Coulomb interaction between particles inside and outside of the open system. The many-particle Hamiltonian provides the same electrostatic description obtained from the image-charge method, but it has the fundamental advantage that it can be directly implemented into realistic (classical or quantum) electron device simulators via a 3D Poisson solver. Classically, the solution of this many-particle Hamiltonian is obtained via a coupled system of Newton-type equations with a different electric field for each particle. The quantum-mechanical solution of this many-particle Hamiltonian is achieved using the quantum (Bohm) trajectory algorithm [X. Oriols, Phys. Rev. Lett. 98, 066803 (2007)]. The computational viability of the many-particle algorithms to build powerful nanoscale device simulators is explicitly demonstrated for a (classical) double-gate field-effect transistor and a (quantum) resonant tunneling diode. The numerical results are compared with those computed from time-dependent mean-field algorithms showing important quantitative differences.

A many-particle approach to the gyro-kinetic theory

AbstractA systematic first-principles approach to the many-particle formulation of the gyro-kinetic theory is suggested. The gyro-kinetic many-particle Hamiltonian is derived using the Lie transform technique. The generalized gyro-kinetic equation is obtained following the Born–Bogoliubov–Green–Kirkwood–Yvon approach. The microscopic expression for the self-consistent potential and the polarization density is obtained. It is shown that new terms appear in the gyro-kinetic polarization that can not be derived in the conventional approach. An expression for the collision term is obtained in the Landau approximation.

Static and dynamic properties of the spinless Falicov-Kimball model

The spinless Falicov-Kimball model is studied by the use of a recently developed projector-based renormalization method for many-particle Hamiltonians. The method is used to evaluate static and dynamic quantities of the one-dimensional model at half-filling. These include the quasiparticle excitation energy ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{k}$, the momentum distribution ${n}_{k}$ of the conduction electrons, and the spatial correlation functions of the localized electrons. One of the most remarkable results is the appearance of a gap in ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{k}$ at the Fermi level of the order of the Coulomb repulsion $U$, which is accompanied by a smooth behavior for ${n}_{k}$. The density of states for the conduction electrons and the one-particle spectral functions for the localized electrons are also discussed. In both quantities, a gap opens with increasing $U$.

Expectation value constraints for theN-representability problem

It has long been known that reduced density matrices (RDM) can be used to obtain the exact ground state energy of many-particle Hamiltonians by performing a constrained minimization over the set of all $N$-representable reduced density matrices. In this paper, we present a set of $N$-representability constraints for both bosons and fermions based on expectation values. These expectation value constraints are derived from a simple yet nontrivial recursive formula, and can easily be included in RDM-based variational schemes. The expectation value constraints are shown to provide additional $N$-representability information when compared to existing $N$-representability constraints.

On the Essential Spectrum of Many-Particle Pseudorelativistic Hamiltonians with Permutational Symmetry Account

In this paper we formulate our results on the essential spectrum of many-particle pseudorelativistic Hamiltonians without magnetic and external potential fields in the spaces of functions, having arbitrary type of the permutational symmetry. We discover location of the essential spectrum for all and for some cases we establish new properties of the lower bound of this spectrum, which are useful for study of the discrete spectrum. In this paper we formulate our results on the essential spectrum of many-particle pseudore- lativistic Hamiltonians without magnetic and external potential fields in spaces of functions, having arbitrary type of the permutational symmetry. We discover the location of the essential spectrum for all (Theorem 1) and for some cases we establish new properties of the lower bound

Renormalization of the periodic Anderson model: An alternative analytical approach to heavy-fermion behavior

In this paper a recently developed projector-based renormalization method (PRM) for many-particle Hamiltonians is applied to the periodic Anderson model (PAM) with the aim to describe heavy Fermion behavior. In this method high-energetic excitation operators instead of high energetic states are eliminated. We arrive at an effective Hamiltonian for a quasi-free system which consists of two non-interacting heavy-quasiparticle bands. The resulting renormalization equations for the parameters of the Hamiltonian are valid for large as well as small degeneracy $\nu_f$ of the angular momentum. An expansion in $1/\nu_f$ is avoided. Within an additional approximation which adapts the idea of a fixed renormalized \textit{f} level $\tilde{\epsilon}_{f}$, we obtain coupled equations for $\tilde{\epsilon}_{f}$ and the averaged \textit{f} occupation $<n_{f}>$. These equations resemble to a certain extent those of the usual slave boson mean-field (SB) treatment. In particular, for large $\nu_f$ the results for the PRM and the SB approach agree perfectly whereas considerable differences are found for small $\nu_f$.

Nonlinear Dynamics and Chaos in Many-Particle Hamiltonian Systems

We report the results of studies of nonlinear dynamics and dynamical chaos in Hamiltonian systems composed of many interacting particles. The importance of the Lyapunov exponents and the Kolmogorov-Sinai entropy is discussed in the context of ergodic theory and nonequilibrium statistical mechanics. Two types of systems are studied: hard-ball models for the motion of a tracer or Brownian particle interacting with the particles of a surrounding fluid and microplasmas which are composed of positively charged ions confined in a Penning electromagnetic trap. Lyapunov exponents are studied for both classes of systems. In microplasmas, transitions between different regimes of nonlinear behavior and chaos are reported.

The generalized Douglas–Kroll transformation

We derive the most general parametrization of the unitary matrices in the Douglas–Kroll (DK) transformation sequence for relativistic electronic structure calculations. It is applied for a detailed analysis of the generalized DK transformation up to fifth order in the external potential. While DKH2–DKH4 are independent of the parametrization of the unitary matrices, DKH5 turns out to be dependent on the third expansion coefficient of the innermost unitary transformation which is carried out after the initial free-particle Foldy–Wouthuysen transformation. The freedom in the choice of this expansion coefficient vanishes consistently if the optimum unitary transformation is sought for. Since the standard protocol of the DK method is the application of unitary transformations to the one-electron Dirac operator, we analyze the DKH procedure up to fifth order for hydrogenlike atoms. We find remarkable accuracy of the higher-order DK corrections as compared to the exact Dirac ground state energy. In the case of many-electron atomic systems, we investigate the order of magnitude of the higher-order corrections in the light of the neglect of the DK transformation of the two-electron terms of the many-particle Hamiltonian. A careful analysis of the silver and gold atoms demonstrates that both the fourth- and fifth-order one-electron DK transformation yield a smaller contribution to the total electronic energy than the DK transformation of the two-electron terms. In order to improve significantly on the third-order correction DKH3, it is thus mandatory to include the DK transformation of the two-electron terms as well as the spin-dependent terms before proceeding to higher orders in the transformation of the one-electron terms. However, an analysis of the ionization energies of these atoms indicates that already DKH3 yields a highly accurate treatment of the scalar-relativistic effects on properties.