Sector Of Fock Space Research Articles

Many-body excitations in trapped Bose gas: A non-Hermitian approach

We study a physically motivated model for a trapped dilute gas of Bosons with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states of this system by accounting for the scattering of atoms in pairs from the macroscopic state. We start with an approximate many-body Hamiltonian, H a p p \mathcal {H}_{\mathrm {app}} , in the Bosonic Fock space. This H a p p \mathcal {H}_{\mathrm {app}} conserves the total number of atoms. Inspired by Wu [J. Math. Phys. 2 (1961), 105–123], we apply a non-unitary transformation to H a p p \mathcal {H}_{\mathrm {app}} . Key in this procedure is the pair-excitation kernel, which obeys a nonlinear integro-partial differential equation. In the stationary case, we develop an existence theory for solutions to this equation by a variational principle. We connect this theory to a system of partial differential equations for one-particle excitation (“quasiparticle”-) wave functions derived by Fetter [Ann. Phys. 70 (1972), 67–101], and prove existence of solutions for this system. These wave functions solve an eigenvalue problem for a J J -self-adjoint operator. From the non-Hermitian Hamiltonian, we derive a one-particle nonlocal equation for low-lying excitations, describe its solutions, and recover Fetter’s energy spectrum. We also analytically provide an explicit construction of the excited eigenstates of the reduced Hamiltonian in the N N -particle sector of Fock space.

Potential energy curves for electronic states of the sodium dimer with multireference coupled cluster calculations

Accurate potential energy curves (PECs) were obtained for the 26 lowest lying electronic states of the sodium dimer. The current approach is based on the first principle size-extensive multireference coupled cluster method formulated in the (2,0) sector of the Fock space (FS). The (2,0) sector of FS provides the description of the states obtained by the attachment of two electrons to the reference system. This makes it possible to adopt the doubly ionised sodium dimer as a reference with its very concrete advantage in the calculations of the PECs, i.e. it dissociates into closed shell fragments (i.e. sodium cations) hence the restricted Hartree–Fock reference can be used in the whole range of interatomic distances. This results in the correct description of the interatomic potentials for all bond lengths. The theoretical PECs stay very close to the experimental curves wherever the latter are available and the computed spectroscopic parameters reproduce the experiment with very good accuracy. Relativistic corrections included at the second-order Douglass–Kroll level have a non-negligible effect on the accuracy of computed excitation and dissociation energies.

Equation-of-motion cavity quantum electrodynamics coupled-cluster theory for electron attachment.

The electron attachment variant of equation-of-motion coupled-cluster theory (EOM-EA-CC) is generalized to the case of strong light-matter coupling within the framework of cavity quantum electrodynamics (QED). The resulting EOM-EA-QED-CC formalism provides an ab initio, correlated, and non-perturbative description of cavity-induced effects in many-electron systems that complements other recently proposed cavity-QED-based extensions of CC theory. Importantly, this work demonstrates that QED generalizations of EOM-CC theory are useful frameworks for exploring particle-non-conserving sectors of Fock space, thereby establishing a path forward for the simultaneous description of both strong electron-electron and electron-photon correlation effects.

An efficient Fock space multi-reference coupled cluster method based on natural orbitals: Theory, implementation, and benchmark.

We present a natural orbital-based implementation of the intermediate Hamiltonian Fock space coupled-cluster method for the (1, 1) sector of Fock space. The use of natural orbitals significantly reduces the computational cost and can automatically choose an appropriate set of active orbitals. The new method retains the charge transfer separability of the original intermediate Hamiltonian Fock space coupled-cluster method and gives excellent performance for valence, Rydberg, and charge-transfer excited states. It offers significant computational advantages over the popular equation of motion coupled cluster method for excited states dominated by single excitations.

Finite-Field Calculations of Transition Properties by the Fock Space Relativistic Coupled Cluster Method: Transitions between Different Fock Space Sectors

Reliable information on transition matrix elements of various property operators between molecular electronic states is of crucial importance for predicting spectroscopic, electric, magnetic and radiative properties of molecules. The finite-field technique is a simple and rather accurate tool for evaluating transition matrix elements of first-order properties in the frames of the Fock space relativistic coupled cluster approach. We formulate and discuss the extension of this technique to the case of transitions between the electronic states associated with different sectors of the Fock space. Pilot applications to the evaluation of transition dipole moments between the closed-shell-like states (vacuum sector) and those dominated by single excitations of the Fermi vacuum (the 1h1p sector) in heavy atoms (Xe and Hg) and simple molecules of heavy element compounds (I2 and TlF) are reported.

Multi-time formulation of particle creation and annihilation via interior-boundary conditions

Interior-boundary conditions (IBCs) have been suggested as a possibility to circumvent the problem of ultraviolet divergences in quantum field theories. In the IBC approach, particle creation and annihilation is described with the help of linear conditions that relate the wave functions of two sectors of Fock space: [Formula: see text] at an interior point [Formula: see text] and [Formula: see text] at a boundary point [Formula: see text], typically a collision configuration. Here, we extend IBCs to the relativistic case. To do this, we make use of Dirac’s concept of multi-time wave functions, i.e. wave functions [Formula: see text] depending on [Formula: see text] space-time coordinates [Formula: see text] for [Formula: see text] particles. This provides the manifestly covariant particle-position representation that is required in the IBC approach. In order to obtain rigorous results, we construct a model for Dirac particles in 1+1 dimensions that can create or annihilate each other when they meet. Our main results are an existence and uniqueness theorem for that model, and the identification of a class of IBCs ensuring local probability conservation on all Cauchy surfaces. Furthermore, we explain how these IBCs relate to the usual formulation with creation and annihilation operators. The Lorentz invariance is discussed and it is found that, apart from a constant matrix (which is required to transform in a certain way), the model is manifestly Lorentz invariant. This makes it clear that the IBC approach can be made compatible with relativity.

Quantum jumps, superpositions, and the continuous evolution of quantum states

The apparent dichotomy between quantum jumps on the one hand, and continuous time evolution according to wave equations on the other hand, provided a challenge to Bohr's proposal of quantum jumps in atoms. Furthermore, Schrödinger's time-dependent equation also seemed to require a modification of the explanation for the origin of line spectra due to the apparent possibility of superpositions of energy eigenstates for different energy levels. Indeed, Schrödinger himself proposed a quantum beat mechanism for the generation of discrete line spectra from superpositions of eigenstates with different energies.However, these issues between old quantum theory and Schrödinger's wave mechanics were correctly resolved only after the development and full implementation of photon quantization. The second quantized scattering matrix formalism reconciles quantum jumps with continuous time evolution through the identification of quantum jumps with transitions between different sectors of Fock space. The continuous evolution of quantum states is then recognized as a sum over continually evolving jump amplitudes between different sectors in Fock space.In today's terminology, this suggests that linear combinations of scattering matrix elements are epistemic sums over ontic states. Insights from the resolution of the dichotomy between quantum jumps and continuous time evolution therefore hold important lessons for modern research both on interpretations of quantum mechanics and on the foundations of quantum computing. They demonstrate that discussions of interpretations of quantum theory necessarily need to take into account field quantization. They also demonstrate the limitations of the role of wave equations in quantum theory, and caution us that superpositions of quantum states for the formation of qubits may be more limited than usually expected.

Coupled cluster calculations in the (0,2) and (2,0) sectors of the Fock space for the lowest electronic states of the O2 molecule

The Fock space (FS) multireference coupled cluster (CC) method in the (0,2) and (2,0) sectors has been applied to study the ground and excited states of the oxygen molecule O2. The considered FS sectors – when used for the neutral molecule – yield double ionisation potential (DIP) or double electron affinity (DEA) values. Once they are applied to the doubly negative/doubly positive ions the results provide description of the neutral molecule. In the current case the FS(0,2)-CC variant was used to generate potential energy curves (PEC) for the ground X3Σ−g and two lowest excited states, a1Δg and b1Σ+g, of the O2 molecule. Due to a closed shell nature of the reference function, the PECs could be easily extended to the ca. 6 Å which is not very common for the system as complex as the oxygen molecule. A similar attempt has been made to use the FS(2,0)-CC scheme by doing calculations for the ion (isoelectronic with the N2 molecule). The energy values obtained for the equilibrium bond length could be considered as acceptable; however, the PECs are limited to the short range of interatomic distances.

A Lagrange multiplier approach for excited state properties through intermediate Hamiltonian formulation of Fock space multireference coupled-cluster theory

In this paper, we present a formulation based on Lagrange multiplier approach for efficient evaluation of excited state energy derivatives in Fock space coupled cluster theory within the intermediate Hamiltonian framework. The formulation is applied to derive the explicit generic expressions up to second order energy derivatives for [1, 1] sector of Fock space with singles and doubles approximation. Its advantage, efficiency, and interconnection in comparison to the Lagrange multiplier approach in traditional formulation of Fock space, which is built on the concept of Bloch equation based effective Hamiltonian, has been discussed. Computational strategy for their implementation has also been discussed in some detail.

Multi-reference Fock space coupled-cluster method in the intermediate Hamiltonian formulation for potential energy surfaces

The effective and intermediate Hamiltonian multi-reference coupled-cluster (CC) method with singles and doubles for the doubly ionized (0,2) sector of Fock space (FS) is formulated and implemented. The intermediate Hamiltonian realization of the (0,2) FS problem provides a robust computational scheme for solving the FS-CC equations free from the intruder state problem. By introducing an efficient factorization strategy, we obtain a very efficient tool that can be used for computing double ionization potentials but more significantly to describe multi-reference problems in CC theory, illustrated by twisted ethylene and the potential energy curve for F(2). The latter separates smoothly to two F atoms, while the former avoids the cusp behavior at the 90° dihedral. We also explore the double ionization potentials for several small molecules, H(2)O, CO, C(2)H(2), and C(2)H(4).

Intermediate Hamiltonian Fock-space multireference coupled-cluster method with full triples for calculation of excitation energies

The intermediate Hamiltonian multireference coupled-cluster (CC) method with singles, doubles, and triples within the excited (1,1) sector of Fock space (FS) is implemented and formulated to calculate excitation energies (EEs). Due to the intermediate Hamiltonian formulation, which provides a robust computational scheme for solving the FS-CC equations, coupled to an efficient factorization strategy, relatively large basis sets and model spaces are employed permitting basis set converged comparisons of the calculated vertical EEs, which can be compared to the experimental data for the N(2) and CO molecules. The issue of charge-transfer separability is also addressed.

Relation between light cone distribution amplitudes and shape function inBmesons

The Bakamjian-Thomas relativistic quark model provides a Poincare representation of bound states with a fixed number of constituents and, in the heavy quark limit, form factors of currents satisfy covariance and Isgur-Wise scaling. We compute the light cone distribution amplitudes (LCDA) of B mesons {phi}{sub {+-}}{sup B}({omega}) as well as the shape function S({omega}), that enters in the decay B{yields}X{sub s}{gamma}, that are also covariant in this class of models. The LCDA and the shape function are related through the quark model wave function. The former satisfy, in the limit of vanishing constituent light quark mass, the integral relation given by QCD in the valence sector of Fock space. Using a Gaussian wave function, the obtained S({omega}) is identical to the so-called roman shape function. From the parameters for the latter that fit the B{yields}X{sub s}{gamma} spectrum we predict the behavior of {phi}{sub {+-}}{sup B}({omega}). We discuss the important role played by the constituent light quark mass. In particular, although {phi}{sub -}{sup B}(0){ne}0 for vanishing light quark mass, a nonvanishing mass implies the unfamiliar result {phi}{sub -}{sup B}(0)=0. Moreover, we incorporate the short distance behavior of QCD to {phi}{sub +}{sup B}({omega}), which has sizeable effects at large {omega}. We obtainmore » the values for the parameters {lambda} congruent with 0.35 GeV and {lambda}{sub B}{sup -1} congruent with 1.43 GeV{sup -1}. We compare with other theoretical approaches and illustrate the great variety of models found in the literature for the functions {phi}{sub {+-}}{sup B}({omega}); hence the necessity of imposing further constraints as in the present paper. We briefly review also the different phenomena that are sensitive to the LCDA. The value that we find for {lambda}{sub B}{sup -1} fulfills the upper bound recently measured by BABAR.« less

Mixed-sector intermediate Hamiltonian Fock-space coupled cluster approach.

An alternative formulation of the intermediate Hamiltonian Fock-space coupled cluster scheme developed before is presented. The methodological and computational advantages of the new formulation include the possibility of using a model space with determinants belonging to different Fock-space sectors. This extends the scope of application of the multireference coupled cluster method, and makes possible the use of quasiclosed shells (e.g., p2, d4) as reference states. Representative applications are described, including electron affinities of group-14 atoms, ionization potentials of group-15 elements, and ionization potentials and excitation energies of silver and gold. Excellent agreement with experiment (a few hundredths of an electronvolt) is obtained, with significant improvement (by a factor of 5-10 for p3 states) over Fock-space coupled cluster results. Many states not reachable by the Fock-space approach can now be studied.

A constrained variational approach for energy derivatives in Fock-space multireference coupled-cluster theory.

In this paper, we present a formulation based on constrained variational approach to enable efficient computation of energy derivatives using Fock-space multireference coupled-cluster theory. Adopting conventional normal ordered exponential with Bloch projection approach, we present a method of deriving equations when general incomplete model spaces are used. Essential simplifications arise when effective Hamiltonian definition becomes explicit as in the case of complete model spaces or some special quasicomplete model spaces. We apply the method to derive explicit generic expressions upto third-order energy derivatives for [0,1], [1,0], and [1,1] Fock-space sectors. Specific diagrammatic expressions for zeroth-order Lagrange multiplier equations for [0,1], [1,0], and [1,1] sectors are presented.

Brillouin–Wigner coupled cluster theory. Fock-space approach

The generalization of coupled cluster (CC) theory is far from being a standard method to account for correlation effects of ubiquitous open-shell systems. This dilemma is largely due to three problems. The first concerns the incorporation of multiple reference “guess” wave functions into nonvariational theories. Next is the size-extensivity issue. Finally, and perhaps most importantly is the notorious intruder state problem. Brillouin–Wigner (BW) and generalized Brillouin–Wigner (gBW) perturbation theories are used to aid in the development of new Fock-space coupled cluster theories in an attempt to alleviate some of these problems. Bloch equations are derived which can be used to formulate BWCC and gBWCC theories of arbitrary dimension in all sectors of Fock space. Since this is our first study of Brillouin–Wigner coupled cluster theories in Fock space we have chosen to keep our approach very simple. Explicit effective Hamiltonian and amplitude equations for up to two-body S-amplitudes in the (0,1) and (1,0) sectors are given for the single reference case. Aspects concerning the connectivity of the amplitude equations are addressed.

Glueballs in a Hamiltonian light-front approach to pure-glue QCD

We calculate a renormalized Hamiltonian for pure-glue QCD and diagonalize it. The renormalization procedure is designed to produce a Hamiltonian that will yield physical states that rapidly converge in an expansion in free-particle Fock-space sectors. To make this possible, we use light-front field theory to isolate vacuum effects, and we place a smooth cutoff on the Hamiltonian to force its free-state matrix elements to quickly decrease as the difference of the free masses of the states increases. The cutoff violates a number of physical principles of light-front pure-glue QCD, including Lorentz covariance and gauge covariance. This means that the operators in the Hamiltonian are not required to respect these physical principles. However, by requiring the Hamiltonian to produce cutoff-independent physical quantities and by requiring it to respect the unviolated physical principles of pure-glue QCD, we are able to derive recursion relations that define the Hamiltonian to all orders in perturbation theory in terms of the running coupling. We approximate all physical states as two-gluon states, and use our recursion relations to calculate to second order the part of the Hamiltonian that is required to compute the spectrum. We diagonalize the Hamiltonian using basis-function expansions for the gluons' color, spin, and momentum degrees of freedom. We examine the sensitivity of our results to the cutoff and use them to analyze the nonperturbative scale dependence of the coupling. We investigate the effect of the dynamical rotational symmetry of light-front field theory on the rotational degeneracies of the spectrum and compare the spectrum to recent lattice results. Finally, we examine our wave functions and analyze the various sources of error in our calculation.

Systematic renormalization in Hamiltonian light-front field theory

We develop a systematic method for computing a renormalized light-front field theory Hamiltonian that can lead to bound states that rapidly converge in an expansion in free-particle Fock-space sectors. To accomplish this without dropping any Fock sectors from the theory, and to regulate the Hamiltonian, we suppress the matrix elements of the Hamiltonian between free-particle Fock-space states that differ in free mass by more than a cutoff. The cutoff violates a number of physical principles of the theory, and thus the Hamiltonian is not just the canonical Hamiltonian with masses and couplings redefined by renormalization. Instead, the Hamiltonian must be allowed to contain all operators that are consistent with the unviolated physical principles of the theory. We show that if we require the Hamiltonian to produce cutoff-independent physical quantities and we require it to respect the unviolated physical principles of the theory, then its matrix elements are uniquely determined in terms of the fundamental parameters of the theory. This method is designed to be applied to QCD, but for simplicity, we illustrate our method by computing and analyzing second- and third-order matrix elements of the Hamiltonian in massless phi-cubed theory in six dimensions.

Analytic energy derivatives for ionized states described by the equation-of-motion coupled cluster method

The theory for analytic energy derivatives of excited electronic states described by the equation-of-motion coupled cluster (EOM-CC) method has been generalized to treat cases in which reference and final states differ in the number of electrons. While this work specializes to the sector of Fock space that corresponds to ionization of the reference, the approach can be trivially modified for electron attached final states. Unlike traditional coupled cluster methods that are based on single determinant reference functions, several electronic configurations are treated in a balanced way by EOM-CC. Therefore, this quantum chemical approach is appropriate for problems that involve important nondynamic electron correlation effects. Furthermore, a fully spin adapted treatment of doublet electronic states is guaranteed when a spin restricted closed shell reference state is used—a desirable feature that is not easily achieved in standard coupled cluster approaches. The efficient implementation of analytic gradients reported here allows this variant of EOM-CC theory to be routinely applied to multidimensional potential energy surfaces for the first time. Use of the method is illustrated by an investigation of the formyloxyl radical (HCOO), which suffers from notorious symmetry breaking effects.

Light-front QCD. III. Coupling constant renormalization.

In this third part in a series of papers on light-front QCD, we continue the study of regularization and renormalization in old-fashioned perturbative light-front formalism. We calculate lowest order radiative corrections to the quark-gluon coupling constant in light-front QCD. An attempt is made to understand the origin of antiscreening in a formulation of QCD with only physical degrees of freedom in terms of contributions from distinct Fock space sectors. The relevance of our results to bound state calculations in QCD with a truncated Fock space are discussed.

Special example of relativistic Hamiltonian field theory.

We study fermion-boson bound and scattering states in a (3+1)-dimensional Yukawa theory, using a Tamm-Dancoff truncation of light-front field theory. We retain only two sectors of Fock space, the sector with one fermion and the sector with one fermion and one boson. Such truncations violate Lorentz covariance if one employs the canonical Hamiltonian. In order to restore covariance we modify the Hamiltonian, introducing new terms and allowing all terms to depend on the Fock-space sectors within which or between which they act. In this special example, which is closely related to the Lee model, simple sector-dependent mass and vertex counterterms are sufficient to yield finite observables that are exactly covariant for states whose invariant mass is less than a critical value determined by the cutoffs. Triviality prevents the complete removal of cutoffs from the Hamiltonian that we consider. This calculation illuminates several basic features of the renormalization program required by the nonperturbative light-front Tamm-Dancoff approach.