Degenerate Perturbation Theory Research Articles

Global Symmetry is Unnecessary for Fast Quantum Search

Grover's quantum search algorithm can be formulated as a quantum particle randomly walking on the (highly symmetric) complete graph, with one vertex marked by a nonzero potential. From an initial equal superposition, the state evolves in a two-dimensional subspace. Strongly regular graphs have a local symmetry that ensures that the state evolves in a \emph{three}-dimensional subspace, but most have no \emph{global} symmetry. Using degenerate perturbation theory, we show that quantum random walk search on known families of strongly regular graphs nevertheless achieves the full quantum speedup of $\Theta(\sqrt{N})$, disproving the intuition that fast quantum search requires global symmetry.

The impact of highly correlated potential energy surfaces on the anharmonically corrected IR spectrum of acetonitrile

This paper discusses the quality and feasibility of highly correlated ab initio techniques in a vibrational self-consistent field (VSCF) approach using acetonitrile as a model system. The topical renormalized coupled-cluster technique exploiting the similarity-transformed Hamiltonian’s left eigenstates (i.e. CR-CC(2,3)) is investigated alongside the well-known Hartree–Fock (HF), Møller–Plesset second-order perturbation theory (MP2) and coupled cluster (CCSD(T)) methods. The inclusion of mode triple interactions is discussed and it is found that the use of an effective core potential (ECP) serves as a viable compromise during the highly demanding task of computing such contributions, thus enabling a grid-based evaluation of three mode interaction terms with coupled cluster techniques also for larger molecules. In this context, a previously proposed reduced coupling scheme [1] is investigated, confirming the applicability of this technique to a system exhibiting a rather complex electronic structure. A combination of Ahlrichs’ triple-ζ valence polarized (TZVP) basis set with Dunning’s set of core-valence correlation functions is found to deliver results in good agreement with experiment while being computationally very feasible. Since CH3CN exhibits four degenerate vibrational degrees of freedom, it serves as an ideal model system for critically assessing the qualities of the degenerate second-order perturbation theory corrected (DPT2) VSCF technique. Besides fundamental vibrations, a thorough investigation of overtone transitions and combination bands is conducted by means of comparing the results to both available and newly recorded experimental data.

Fine structures in the spectrum of the open-boundary Heisenberg chain at large anisotropies

At large anisotropies, the spectrum of the Heisenberg XXZ spin chain separates into ``bands'' with energies largely determined by the number of domain walls. The band structure is richer with open-boundary conditions: there are more bands and the bands develop intricate fine structures. We characterize and explain these structures and substructures in the open-boundary chain. The fine structures are explained using degenerate perturbation theory. We also present some dynamical consequences of these subband structures, through explicit time evolution of the wave function from initial states motivated by the fine-structure analysis.

Multi-shell effective interactions

Background: Effective interactions, either derived from microscopic theories or based on fitting selected properties of nuclei in specific mass regions, are widely used inputs to shell-model studies of nuclei. The commonly used unperturbed basis functions are given by the harmonic oscillator. Until recently, most shell-model calculations have been confined to a single oscillator shell like the $sd$ shell or the $pf$ shell. Recent interest in nuclei away from the stability line requires, however, larger shell-model spaces. Because the derivation of microscopic effective interactions has been limited to degenerate models spaces, there are both conceptual and practical limits to present shell-model calculations that utilize such interactions.Purpose: The aim of this work is to present a novel microscopic method to calculate effective nucleon-nucleon interactions for the nuclear shell model. Its main difference from existing theories is that it can be applied not only to degenerate model spaces but also to nondegenerate model spaces. This has important consequences, in particular for intershell matrix elements of effective interactions.Methods: The formalism is presented in the form of a many-body perturbation theory based on the recently developed extended Kuo-Krenciglowa method. Our method enables us to microscopically construct effective interactions not only in one oscillator shell but also for several oscillator shells.Results: We present numerical results using effective interactions within (i) a single oscillator shell (a so-called degenerate model space) like the $sd$ shell or the $pf$ shell and (ii) two major shells (nondegenerate model space) like the $sd{f}_{7}{p}_{3}$ shell or the $pf{g}_{9}$ shell. We also present energy levels of several nuclei that have two valence nucleons on top of a given closed-shell core.Conclusions: Our results show that the present method works excellently in shell-model spaces that comprise several oscillator shells, as well as in a single oscillator shell. We show, in particular, that the microscopic intershell interactions are much more attractive than has been expected by degenerate perturbation theory. The consequences for shell-model studies are discussed.

How to Compute the Fukui Matrix and Function for Systems with (Quasi-)Degenerate States.

A system in a spatially (quasi-)degenerate ground state responds in a qualitatively different way to a change in the external potential. Consequently, the usual method for computing the Fukui function, namely, taking the difference between the electron densities of the N- and N ± 1 electron systems, cannot be applied directly. It is shown how the Fukui matrix, and thus also the Fukui function, depends on the nature of the perturbation. One thus needs to use degenerate perturbation theory for the given perturbing potential to generate the density matrix whose change with respect to a change in the number of electrons equals the Fukui matrix. Accounting for the degeneracy in the case of nitrous oxide reveals that an average over the degenerate states differs significantly from using the proper density matrix. We further show the differences in Fukui functions depending on whether a Dirac delta perturbation is used or an interaction with a true point charge (leading to the Fukui potential).

PHOTONIC DIRAC CONES REALIZED BY ACCIDENTAL DEGENERACY ON THE BRILLOUIN-ZONE BOUNDARY

The condition for the formation of Dirac cones at arbitrary points in the Brillouin zone by the accidental degeneracy of two photonic bands was examined by the degenerate perturbation theory and group theory based on the spatial symmetry of two modes. The analysis was applied to a two-dimensional square-lattice photonic crystal with C4v symmetry and the dispersion relation in the vicinity of the M point was examined. Exact agreement between the analytical and numerical calculation was obtained.

Small two-component Fermi gases in a cubic box with periodic boundary conditions

The properties of two-component Fermi gases become universal if the interspecies $s$-wave scattering length ${a}_{s}$ and the average interparticle spacing are much larger than the range of the underlying two-body potential. Using an explicitly correlated Gaussian basis set expansion approach, we determine the eigenenergies of two-component Fermi gases in a cubic box with periodic boundary conditions as functions of the interspecies $s$-wave scattering length and the effective range of the two-body potential. The universal properties of systems consisting of up to four particles are determined by extrapolating the finite-range energies to the zero-range limit. We determine the eigenenergies of states with vanishing and finite momenta. In the weakly attractive BCS regime, we analyze the energy spectra and degeneracies using first-order degenerate perturbation theory. Excellent agreement between the perturbative energy shifts and the numerically determined energies is obtained. For the infinitely large scattering length case, we compare our results---where available---with those presented in the literature.

Correlated four-component EPR g-tensors for doublet molecules

The first correlated ab initio four-component calculations of electron paramagnetic resonance (EPR) g-tensors for doublet radicals are reported. We have implemented a first-order degenerate perturbation theory approach based on the four-component Dirac-Coulomb Hamiltonian and fully relativistic configuration interaction wave functions in the DIRAC program package. We find that the correlation effects on the g-tensors can be sufficiently well described with manageable basis sets of triple-zeta quality and manageable configuration spaces. The new fully relativistic EPR module in DIRAC should be useful for benchmarking density functional theory approaches, however, with future optimization of the code we believe it will also be useful for applications.

Boundary term contribution to the volume of a small causal diamond

In his calculation of the spacetime volume of a small Alexandrov interval in four dimensions, Myrheim introduced a term which he referred to as a surface integral (Myrheim 1978 CERN preprint TH-2538). The evaluation of this term has remained opaque and led subsequent authors to obtain a formula for the volume using other techniques (Gibbons and Solodukhin 2007 Phys. Lett. B 649 317). It is the purpose of this work to explicitly calculate this integral and in the process complete the proof for the volume formula in arbitrary dimensions. We point out that it arises from the difference in the flat spacetime volumes of the curved and flat spacetime intervals. We use first-order degenerate perturbation theory to evaluate this difference and find that it adds a dimension-independent factor to the volume of the flat spacetime interval as the lowest order correction. Our analysis admits a simple extension to a more general class of integrals over the same domain. Using a combination of techniques we also find that the next-order correction to the volume vanishes.

Anomalous behavior of the spin gap of a spin‐1/2 two‐leg antiferromagnetic ladder with Ising‐like rung interactions

AbstractUsing mainly numerical methods, we investigate the width of the spin gap of a spin‐1/2 two‐leg ladder described by ${\cal H} = J_{{\rm l}} \sum\nolimits_{j = 1}^{N/2} {} [{\bf S}_{j,a} \cdot {\bf S}_{j + 1,a} + {\bf S}_{j,b} \cdot {\bf S}_{j + 1,b} ] + J_{{\rm r}} \sum\nolimits_{j = 1}^{N/2} {} [\lambda (S_{j,a}^{x} S_{j,b}^{x} + S_{j,a}^{y} S_{j,b}^{y} ) + S_{j,a}^{z} S_{j,b}^{z} ]$, where $S_{j,a(b)}^{\alpha } $ denotes the α‐component of the spin‐1/2 operator at the j‐th site of the a (b) chain. We mainly focus on the $J_{{\rm r}}$ ≫ $J_{{\rm l}} > 0$ and $|\lambda |\ll 1$ case. The width of the spin gap between the M = 0 and 1 subspaces (M is the total magnetization) as a function of λ anomalously increases near λ = 0; for instance, for $- 0.1 $ ≲ $\lambda$ ≲ $0.1$ when Jl/Jr = 0.1. The gap formation mechanism is thought to be different for the λ < 0 and λ > 0 cases. Since, in usual cases, the width of the gap becomes zero or small at the point where the gap formation mechanism changes, the above gap‐increasing phenomenon in the present case is anomalous. We explain the origin of this anomalous phenomenon by use of the degenerate perturbation theory. We also draw the ground‐state phase diagram.

Spectra of open-shell nuclei with Padé-resummed degenerate perturbation theory

We apply degenerate many-body perturbation theory at high orders for the ab-initio description of ground states and excitation spectra of open-shell nuclei using soft realistic nucleon-nucleon interactions. We derive a recursive formulation of standard degenerate many-body perturbation theory that enables us to evaluate order-by-order perturbative energy and state corrections up to the 30th order. We study 6,7-Li as test cases using a similarity renormalization group (SRG) evolved nucleon-nucleon interaction from chiral effective field theory. The simple perturbation series exhibits a strong, often oscillatory divergence, as was observed previously for ground states of closed-shell nuclei. Even for very soft interactions resulting from SRG evolutions up to large flow parameter, i.e. low momentum scales, the perturbation series still diverges. However, a resummation of the perturbation series via Pad\'e approximants yields very stable and converged ground and excited-state energies in very good agreement with exact no-core shell-model calculations for the same model space.

Proof of the universality of mode symmetries in creating photonic Dirac cones

We formulate a degenerate perturbation theory for the vector electromagnetic field of periodic structures and apply it to the problem of the creation of Dirac cones in the Brillouin-zone center by accidental degeneracy of two modes. We derive a necessary condition by which we can easily select candidates of mode combinations that enable the creation of the Dirac cone. We analyze the structure of a matrix that determines the first-order correction to eigen frequencies by examining its transformation by symmetry operations. Thus, we can obtain the analytical solution of dispersion curves in the vicinity of the zone center and can judge the presence of the Dirac cone. All these findings clearly show that the presence or absence of the Dirac cone in the zone center is solely determined by the spatial symmetry of the two modes.

Quantum breathers in a finite Heisenberg spin chain with antisymmetric interactions

A study of the likelihood of quantum breathers in a quantum Heisenberg spin system including a Dzyaloshinsky-Moriya interaction (DMI) is done through an extended Bose-Hubbard model while using the scheme of few body physics. The energy spectrum of the resulting Bose-Hubbard Hamiltonian, on a periodic one-dimensional lattice containing more than two quanta shows interesting detailed band structures. From a non degenerate, and a degenerate perturbation theory in addition to a numerical diagonalization, a careful investigation of these fine structures is set up. The attention is focussed on the effects of various interactions that are; the DMI, the Heisenberg in-plane (X, Y) as well as the out of plane exchange interaction on the energy spectrum of such a system. The outcome displays a possibility of an energy self-compensation in the system. We also computed the weight function of the eigenstates in direct space and in the space of normal modes. From a perturbation theory it is shown that the interaction between the quanta leads to an algebraic localization of the modified extended states in the normal-mode space of the non-interacting system that are coined quantum q-breathers excitations.

Microscopic approach to the shell model: study of nuclei north-east of 48Ca

We report on shell-model calculations for nuclei with valence nucleons outside 48Ca wherein a fully microscopic approach is adopted. Namely, the single-particle energies and the matrix elements of the two-body interaction are derived theoretically starting from the high-precision CD-Bonn nucleon-nucleon potential, renormalized by way of the Vlow−k approach. This has been done within the framework of the time-dependent degenerate linked-diagram perturbation theory. Our results compare satisfactorily with a large amount of experimental data.

Multiphoton ionization of a metastable hydrogen atom by electron and positron impact and charge asymmetry

Charge asymmetry is studied theoretically for multiphoton ionization (nγ e±, e e±) of a hydrogen atom from its metastable (2S) state using coplanar geometry. The external laser field is chosen to be single-mode, spatially homogeneous, linearly polarized with laser intensity that is quite high by laboratory standards. The continuum states of the impinging electron (e) or positron (e+) are represented by plane wave Volkov states, while the wavefunctions for the ejected electron and the scattered electron/positron are chosen to be of Coulomb–Volkov type. The dressed wavefunction for the metastable (2S) hydrogen is constructed in the framework of degenerate perturbation theory. Charge asymmetry is noted in the triple differential cross sections (TDCS) of the two projectiles both in the field-free (FF) and laser-assisted situations. The positron binary peak intensity is much higher than that of the electron, while the reverse is true for the recoil peak. As for the laser modifications, the single-photon TDCS exhibits a distinct four lobed structure in contrast to the FF (two lobed) and is highly suppressed w.r.t. the latter, while the multiphoton TDCS usually tends to approach the FF (with some exceptions), obeying the famous Kroll–Watson (KW) sum rule for both projectiles. For e+ impact, the electron capture to the continuum (ECC) effect is also studied (for the symmetric geometry), where enhancement is noted in both the single and multiphoton ECC peaks thereby deviating from the KW sum rules.

Fully microscopic shell-model calculations with realistic effective hamiltonians

The advent of nucleon-nucleon potentials derived from chiral perturbation theory, as well as the so-called Vlow-k approach to the renormalization of the strong short-range repulsion contained in the potentials, have brought renewed interest in realistic shell-model calculations. Here we focus on calculations where a fully microscopic approach is adopted. No phenomenological input is needed in these calculations, because single-particle energies, matrix elements of the two-body interaction, and matrix elements of the electromagnetic multipole operators are derived theoretically. This has been done within the framework of the time-dependent degenerate linked-diagram perturbation theory. We present results for some nuclei in different mass regions. These evidence the ability of realistic effective hamiltonians to provide an accurate description of nuclear structure properties.

Theoretical Analysis of the Spin Hamiltonian Parameters in Co(II)S4 Complexes, Using Density Functional Theory and Correlated ab initio Methods

A systematic Density Functional Theory (DFT) and multiconfigurational ab initio computational analysis of the Spin Hamiltonian (SH) parameters of tetracoordinate S = 3/2 Co((II))S(4)-containing complexes has been performed. The complexes under study bear either arylthiolato, ArS(-), or dithioimidodiphosphinato, [R(2)P(S)NP(S)R'(2)](-) ligands. These complexes were chosen because accurate structural and spectroscopic data are available, including extensive Electron Paramagnetic Resonance (EPR)/Electron Nuclear Double Resonance (ENDOR) studies. For comparison purposes, the [Co(PPh(3))(2)Cl(2)] complex, which was thoroughly studied in the past by High-Field and Frequency EPR and Variable Temperature, Variable Field Magnetic Circular Dichroism (MCD) spectroscopies, was included in the studied set. The magnitude of the computed axial zero-field splitting parameter D (ZFS), of the Co((II))S(4) systems, was found to be within ~10% of the experimental values, provided that the property calculation is taken beyond the accuracy obtained with a second-order treatment of the spin-orbit coupling interaction. This is achieved by quasi degenerate perturbation theory (QDPT), in conjunction with complete active space configuration interaction (CAS-CI). The accuracy was increased upon recovering dynamic correlation with multiconfigurational ab initio methods. Specifically, spectroscopy oriented configuration interaction (SORCI), and difference dedicated configuration interaction (DDCI) were employed for the calculation of the D-tensor. The sign and magnitude of parameter D was analyzed in the framework of Ligand Field Theory, to reveal the differences in the electronic structures of the investigated Co((II))S(4) systems. For the axial complexes, accurate effective g'-tensors were obtained in the QDPT studies. These provide a diagnostic tool for the adopted ground state configuration (±3/2 or ±1/2) and are hence indicative of the sign of D. On the other hand, for the rhombic complexes, the determination of the sign of D required the SH parameters to be derived along suitably constructed symmetry interconversion pathways. This procedure, which introduces a dynamic perspective into the theoretical investigation, helped to shed some light on unresolved issues of the corresponding experimental studies. The metal hyperfine and ligand super-hyperfine A-tensors of the C(2) [Co{(SPPh(2))(SP(i)Pr(2))N}(2)] complex were estimated by DFT calculations. The theoretical data were shown to be in good agreement with the available experimental data. Decomposition of the metal A-tensor into individual contributions revealed that, despite the large ZFS, the observed significant anisotropy should be largely attributed to spin-dipolar contributions. The analysis of both, metal and ligand A-tensors, is consistent with a highly covalent character of the Co-S bonds.

EXPLORATION OF THE MEMORY EFFECT ON THE PHOTON-ASSISTED TUNNELING VIA A SINGLE QUANTUM DOT: A GENERALIZED FLOQUET THEORETICAL APPROACH

The generalized Floquet approach is developed to study memory effect on electron transport phenomena through a periodically driven single quantum dot in an electrode–multi-level dot–electrode nanoscale quantum device. The memory effect is treated using a multi-function Lorentzian spectral density (LSD) model that mimics the spectral density of each electrode in terms of multiple Lorentzian functions. For the symmetric single-function LSD model involving a single-level dot, the underlying single-particle propagator is shown to be related to a 2×2 effective time-dependent Hamiltonian that includes both the periodic external field and the electrode memory effect. By invoking the generalized Van Vleck (GVV) nearly degenerate perturbation theory, an analytical Tien-Gordon-like expression is derived for arbitrary order multi-photon resonance d.c. tunneling current. Numerically converged simulations and the GVV analytical results are in good agreement, revealing the origin of multi-photon coherent destruction of tunneling and accounting for the suppression of the staircase jumps of d.c. current due to the memory effect. Specially, a novel blockade phenomenon is observed, showing distinctive oscillations in the field-induced current in the large bias voltage limit.

Edge states of bilayer graphene in the quantum Hall regime

We study the low-energy edge states of bilayer graphene in a strong perpendicular magnetic field. Several possible simple boundary geometries related to zigzag edges are considered. Tight-binding calculations reveal three types of edge-state behaviors: weakly, strongly, and nondispersive edge states. These three behaviors may all be understood within a continuum model, and related by nonlinear transformations to the spectra of quantum Hall edge states in a conventional two-dimensional electron system. In all cases, the edge states closest to zero energy include a holelike edge state of one valley and a particlelike state of the other on the same edge, which may or may not cross, depending on the boundary condition. Edge states with the same spin generically have anticrossings that complicate the spectra, but which may be understood within degenerate perturbation theory. The results demonstrate that the number of edge states crossing the Fermi level in clean, undoped bilayer graphene depends both on boundary conditions and the energies of the bulk states.

Schrieffer–Wolff transformation for quantum many-body systems

The Schrieffer–Wolff (SW) method is a version of degenerate perturbation theory in which the low-energy effective Hamiltonian H eff is obtained from the exact Hamiltonian by a unitary transformation decoupling the low-energy and high-energy subspaces. We give a self-contained summary of the SW method with a focus on rigorous results. We begin with an exact definition of the SW transformation in terms of the so-called direct rotation between linear subspaces. From this we obtain elementary proofs of several important properties of H eff such as the linked cluster theorem. We then study the perturbative version of the SW transformation obtained from a Taylor series representation of the direct rotation. Our perturbative approach provides a systematic diagram technique for computing high-order corrections to H eff . We then specialize the SW method to quantum spin lattices with short-range interactions. We establish unitary equivalence between effective low-energy Hamiltonians obtained using two different versions of the SW method studied in the literature. Finally, we derive an upper bound on the precision up to which the ground state energy of the n th-order effective Hamiltonian approximates the exact ground state energy.