Variational Perturbation Theory Research Articles

Superfluid excitations in rotating two-dimensional ring traps

We studied a rotating Bose–Einstein condensate confined in ring trap configurations that can be produced starting with a bubble trap confinement, approximated by a Mexican hat and shift harmonic oscillator potentials. Using a variational technique and perturbation theory, we determined the vortex configurations in this system by varying the interparticle interaction and the angular velocity of the atomic cloud. We found that the phase diagram of the system has macrovortex structures for small positive values of the interaction parameter, and the charge of the central vortex increases with rotation. Strengthening the atomic interaction makes the macrovortex unstable, and it decays into multiple singly charged vortices that arrange themselves in a lattice configuration. We also look for experimentally realizable methods to determine the vortex configuration without relying upon absorption imaging since the structures are not always visible in the latter. More specifically, we study how the vortex distribution affects the collective modes of the condensate by solving the Gross–Pitaevskii equation numerically and by analytical predictions using the sum-rule approach for the frequencies of the modes. These results reveal important signatures to characterize the macrovortices and vortex lattice transitions in the experiments.

Variational density functional perturbation theory for metals

Variational density functional perturbation theory for metals

Variational perturbation theory for dynamic polarizabilities and dispersion coefficients.

An efficient method based on the variational perturbation theory (VPT) is proposed to conveniently calculate the atomic real- and imaginary-frequency dynamic polarizabilities and the interatomic dispersion coefficients. The developed method holds the great advantage that only the system ground state wave function and corresponding radial mean values are needed. Verification of the VPT method on one- and two-electron atoms indicates that the present approximation shows good agreement with calculations based on the sophisticated sum-over-states method. We apply the VPT method to examine the approximate Z-scaling laws of polarizabilities and dispersion coefficients in the He isoelectronic sequence, and to investigate the plasma screening effect on these quantities for embedded atoms. Our calculation demonstrates very well that the VPT method is capable of producing reasonably accurate static and dynamic polarizabilities as well as two- and three-atom dispersion coefficients for plasma-embedded atoms in a wide range of screening parameters.

Theoretical Simulations of Kinetic Isotope Effects on Decarboxylation of 3-Carboxybenzisoxazole.

Kinetic isotope effect values on the decarboxylation of 3-carboxybenzisoxazole have been computed using the second-order Kleinert's variational perturbation theory in the framework of Feynman's path integrals along with the potential energy surface obtained at the MP2/6-31+G(d) level. Good agreement with the experimental data was obtained, demonstrating that this novel computational approach for computing KIE values of organic reaction is a viable alternative to the traditional method employing the Bigeleisen equation and harmonic vibrational frequencies. Compared with the experimental measurements, consideration of anharmonicity and tunneling effects can significantly improve the calculated KIE values, reducing the root-mean-square deviation from 1.19 % for traditional method to 0.20 % for path-integral method.

Analytic equation of state for the Buckingham Exp(α, m) fluids at high temperatures and densities based on the Ross variation perturbation theory

High temperatures and densities analytical thermodynamic properties are derived for the general Buckingham Exp(α, m) fluids by: a) using the Ross analytical (RA) variation perturbation theory developed by Sun [Mol. Phys. 105 (2007) 3139] for multi-Yukawa potentials; b) using a two-Yukawa potential matching the Exp(α, m), and c) introducing correcting terms which account for the discrepancies between the original Exp(α, m) potential and its two-Yukawa representation. We call this method the Ross analytical corrected theory (RAC), which leads to analytic expressions for the excess Helmholtz free energy, excess internal energy and the compressibility factor Z (i.e. the equation of state). They are valid at high temperatures and densities, since they are obtained from the Ross theory, which extends the range of validity of perturbation theories to higher densities. The matching two-Yukawa potential can also be considered as a reference potential for a first-order perturbation theory, which after further simplification leads to an analytic augmented van der Waals (avdW) theory for the above thermodynamic properties of the Exp(α, m) fluids. For the Exp(11.5, 6) and Exp(14.5, 6) fluids, the excess internal energy, compressibility factor Z and reduced pressure obtained from the RAC and avdW theories are compared with Monte Carlo (MC) simulation results corresponding to both the Exp and TY potentials. The overall accuracy of the RAC and avdW theories for the Exp(11.5, 6) fluid is also assessed in comparison to the Ross numerical (RN) theory (called also MCRSR for Mansoori, Canfield, Rasaiah, Stell and Ross) and to more complex methods such as the KLRR-T (Kang, Lee, Ree, Ree-Theostar) and the HMSA/C (hypernetted mean spherical approximation with Chebyshev approximants) theories, which require more extensive numerical calculations.

Weak- and strong-expansions of coherent states eigenfunctions for the generalized q-deformed harmonic oscillator and their completeness relations

The explicit expressions for generalized q-deformed harmonic oscillator coherent states (q-DHO CS) obtained in terms of a weak- and strong-behavior expansions are investigated. We first employ the weak-deformed version () of q-boson annihilation operator to solve Barut-Girardello’s eigenvalues coherent states equation for generalized q-deformed harmonic oscillator, using for this some lemmas directly related to the resolution of Riccati differential equation. The similarity with the perturbation theory is established. In strong-behavior limit, (), the previous results are resumed using the variational perturbation theory, and the corresponding q-DHO CS are deduced. We also construct, in both limits, their associated completeness relations which turns out to be the elliptic Jacobi -function.

Time dependent variation perturbation calculation of two-photon transition probability and hyperfine shift in hydrogen atom under plasma environment

A new theoretical approach has been adopted to calculate the non-relativistic parity conserved two-photon transition probability (TPTP) from ground to excited S and D states along with the hyperfine splitting (HFS) of ground and excited S states of H atom within principal quantum number n≤4 under classical and quantum plasma environment represented by screened potential model. The methodology involves direct calculation of TPTP via an intermediate virtual state obtained from time dependent variation perturbation formalism. The standard formula with sum over states representing the TPTP has been replaced by its variational equivalent obtained from currently established procedure for two-photon excitation via fourth order time dependent variation perturbation theory. Unlike previous methodologies the current formalism yields a compact representation of the physical nature of the intermediate virtual state having well defined profile. The HFS has been evaluated under first order perturbation theory. The plasma embedded ground state energy is obtained from standard diagonalization procedure. Finite Slater type basis sets have been used for the ground and all the excited states. The variational coefficients for the excited states are determined from optimization of an appropriate time averaged variational functional. Results for free H atom agree well with existing estimates for all the states. The results for the plasma embedded states follow smooth and interesting pattern.

Convergence of Eigenvector Continuation.

Eigenvector continuation is a computational method that finds the extremal eigenvalues and eigenvectors of a Hamiltonian matrix with one or more control parameters. It does this by projection onto a subspace of eigenvectors corresponding to selected training values of the control parameters. The method has proven to be very efficient and accurate for interpolating and extrapolating eigenvectors. However, almost nothing is known about how the method converges, and its rapid convergence properties have remained mysterious. In this Letter, we present the first study of the convergence of eigenvector continuation. In order to perform the mathematical analysis, we introduce a new variant of eigenvector continuation that we call vector continuation. We first prove that eigenvector continuation and vector continuation have identical convergence properties and then analyze the convergence of vector continuation. Our analysis shows that, in general, eigenvector continuation converges more rapidly than perturbation theory. The faster convergence is achieved by eliminating a phenomenon that we call differential folding, the interference between nonorthogonal vectors appearing at different orders in perturbation theory. From our analysis we can predict how eigenvector continuation converges both inside and outside the radius of convergence of perturbation theory. While eigenvector continuation is a nonperturbative method, we show that its rate of convergence can be deduced from power series expansions of the eigenvectors. Our results also yield new insights into the nature of divergences in perturbation theory.

Analytic equation of state for Mie(α, β) fluids based on an improved Ross variation perturbation theory

The analytic expressions of thermodynamic properties obtained by Sun for multi-Yukawa fluids from the Ross variation perturbation theory (Mol. Phys. 105 (2007) 3139) can be used to study other practical fluids over wide ranges of temperature and density by mimicking their intermolecular potentials with two-Yukawa (TY) functions. Using the Lennard-Jones LJ (12, 6) as an example, Sun noticed that the accuracy obtained is sensitive to the choice of the fitted TY potential. To remedy this drawback we propose in this work to introduce into Sun Ross analytical (RA) theory correction terms that take into account the discrepancies between the more general Mie(α, β) potential (the LJ is the Mie(12, 6)) and its two-Yukawa representations leading to an improved Ross analytical corrected (RAC) theory. By using three different TY potentials fitted on the LJ from the literature the corresponding excess Helmholtz free energy, excess internal energy and compressibility factor Z obtained from the Ross analytical corrected (RAC) theory are seen to be all in good agreement with the values obtained from the original Ross numerical (RN) theory applied directly to the LJ potential for a wide range of temperatures and densities. These results show that the introduction of these analytical correcting terms increases the accuracy of Sun Ross analytical (RA) theory, which also becomes less dependent on the choice of the TY parameters.

Theoretical Simulations of Heavy-Atom Kinetic Isotope Effects in Aliphatic Claisen Rearrangement.

The aliphatic Claisen rearrangement of allyl vinyl ether has attracted great interest for its broad applications in chemical synthesis and biosynthesis. Although it is well agreed that this reaction proceeds via a concerted, "chair-like" transition state, certain inconsistencies of kinetic isotope effect (KIE) data between experimental measurements and theoretical simulations or between independent experiments indicate that the nature and mechanism of this important reaction need to be investigated in more detail. In order to verify two independent sets of experimental data, we present theoretical calculations on heavy-atom KIE values of alipahtic Claisen rearrangement, using our recently developed path-integral method with the second-order Kleinert's variational perturbation theory, which goes beyond the traditional method for computing KIE values by employing the Bigeleisen equation. Amazingly, the results demonstrate that both sets of experimental measurements are correct, while the inconsistency originates from the fact that the aliphatic Claisen rearrangement undergoes similar but different mechanisms in gas and solution phases. Different experimental conditions will alter the actual reactant state by tuning the population distribution of reactant conformers. According to the comparison between experimental and theoretical results, a more clear reaction mechanism of aliphatic Claisen rearrangement is revealed.

A Variational Perturbative Approach to Planning in Graph-Based Markov Decision Processes

Coordinating multiple interacting agents to achieve a common goal is a difficult task with huge applicability. This problem remains hard to solve, even when limiting interactions to be mediated via a static interaction-graph. We present a novel approximate solution method for multi-agent Markov decision problems on graphs, based on variational perturbation theory. We adopt the strategy of planning via inference, which has been explored in various prior works. We employ a non-trivial extension of a novel high-order variational method that allows for approximate inference in large networks and has been shown to surpass the accuracy of existing variational methods. To compare our method to two state-of-the-art methods for multi-agent planning on graphs, we apply the method different standard GMDP problems. We show that in cases, where the goal is encoded as a non-local cost function, our method performs well, while state-of-the-art methods approach the performance of random guess. In a final experiment, we demonstrate that our method brings significant improvement for synchronization tasks.

Dipole polarizability for confined two-electron atomic systems: simple approximations

The electric dipole polarizability is analyzed for a two-electron system embedded in an impenetrable spherically symmetric cavity. Methods to find simple estimates of polarizability and corrections to them are considered. The expressions for matrix elements in Hylleraas coordinates required for energy and polarizability variational calculations are presented for the two-electron atomic system. The results of simple polarizability estimates are presented for the ground state of the helium atom and are compared with polarizabilities calculated using Hylleraas variational perturbation theory and with recently calculated polarizabilities reported in the literature for confined systems.

Frequency dependent hyperpolarizability and two photon excitations in hydrogen atom confined under classical plasma environment

Frequency dependent dipole hyperpolarizability of hydrogen atom confined under classical plasma is estimated using nonlinear response theory under a monochromatic electromagnetic field. The screened Coulomb potential is used to model the classical plasma. Time-dependent variation perturbation theory with Slater basis is applied for the first time for such an analysis. As an offshoot, two photon excitations are obtained from the poles of frequency dependent hyperpolarizability. We have specifically studied two photon excited states 2s,3s/3d and 4s/4d from the ground 1s state of H atom. The positions and radial density profiles of two photon excited states of free H atom are in excellent agreement with experimental and analytical results. The accidental degeneracy above principal quantum number n=2 is lifted systematically with respect to increased screening parameters. The results are expected to be important for different nonlinear optical processes like the Kerr effect, degenerate four wave mixing, harmonic generation etc. under external confinement.

“Quantum Equilibrium-Disequilibrium”: Asset price dynamics, symmetry breaking, and defaults as dissipative instantons

We propose a simple non-equilibrium model of a financial market as an open system with a possible exchange of money with an outside world and market frictions (trade impacts) incorporated into asset price dynamics via a feedback mechanism. Using a linear market impact model, this produces a non-linear two-parametric extension of the classical Geometric Brownian Motion (GBM) model, that we call the “Quantum Equilibrium-Disequilibrium” (QED) model.The QED model incorporates non-linear mean-reverting dynamics, broken scale invariance, and corporate defaults. In the simplest one-stock (1D) formulation, our parsimonious model has only one degree of freedom, yet calibrates to both equity returns and credit default swap spreads. Defaults and market crashes are associated with dissipative tunneling events, and correspond to instanton (saddle-point) solutions of the model. When market frictions and inflows/outflows of money are neglected altogether, “classical” GBM scale-invariant dynamics with an exponential asset growth and without defaults are formally recovered from the QED dynamics. However, we argue that this is only a formal mathematical limit, and in reality the GBM limit is non-analytic due to non-linear effects that produce both defaults and divergence of perturbation theory in a small market friction parameter.

Total energy dynamics and asymptotics of the momentum distribution following an interaction quench in a two-component Fermi gas

The absence of a characteristic momentum scale in the pseudopotential description of atomic interactions in ultracold (two-component Fermi) gases is known to lead to divergences in perturbation theory. Here we show that they also plague the calculation of the dynamics of the total energy following a quantum quench. A procedure to remove the divergences is devised, which provides finite answers for the time evolution of the total energy after a quench in which the interaction strength is ramped up according to an arbitrary protocol. An important result of this analysis is the time evolution of the asymptotic tail of the momentum distribution (related to Tan's contact) to leading order in the scattering length. Explicit expressions for the dynamics of the total energy and the contact for a linear interaction ramp are obtained, as a function of the interaction ramp time in the crossover from the sudden quench to the adiabatic limit. In the sudden quench limit, the contact, following a rapid oscillation, reaches a stationary value which is different from the equilibrium one. In the adiabatic limit, the contact grows quadratically in time and later saturates to the equilibrium value for the final value of the scattering length.

Spin-orbital excitons and their potential condensation in pentavalent iridates

We investigate magnetic excitations in iridium insulators with pentavalent Ir$^{5+}$ ($5d^4$) ions with strong spin-orbit coupling. We obtain a microscopic model based on the local Ir$^{5+}$ multiplets involving $J=0$ (singlet), $J=1$ (triplet), and $J=2$ (quintet) spin-orbital states. We get effective interactions between these multiplets on square and face-centered-cubic (fcc) structures of magnetic ions in the layered-perovskites and the double-perovkites, in particular Ba$_2$YIrO$_6$. Further, we derive an effective spin-orbital Hamiltonian in terms of bond bosons and explore possible instabilities towards magnetic and quadrupole orderings. Additionally, we study charge excitations with help of the variational cluster perturbation theory and calculate the electronic charge gap as a function of hopping and Coulomb interactions. Based on both electronic and magnetic phase diagrams, we verify the possibility of excitonic magnetism due to condensation of spin-orbital excitons in Ir$^{5+}$ iridates.

Divergence of perturbation theory in large scale structures

We make progress towards an analytical understanding of the regime of validity of perturbation theory for large scale structures and the nature of some non-perturbative corrections. We restrict ourselves to 1D gravitational collapse, for which exact solutions before shell crossing are known. We review the convergence of perturbation theory for the power spectrum, recently proven by McQuinn and White [1], and extend it to non-Gaussian initial conditions and the bispectrum. In contrast, we prove that perturbation theory diverges for the real space two-point correlation function and for the probability density function (PDF) of the density averaged in cells and all the cumulants derived from it. We attribute these divergences to the statistical averaging intrinsic to cosmological observables, which, even on very large and “perturbative” scales, gives non-vanishing weight to all extreme fluctuations. Finally, we discuss some general properties of non-perturbative effects in real space and Fourier space.

Phase transitions of titanite CaTiSiO5 from density functional perturbation theory

Phonon dispersion of titanite ${\mathrm{CaTiSiO}}_{5}$ has been calculated using the variational density functional perturbation theory. The experimentally known out-of-center distortion of the Ti atom is confirmed. The distortion is associated with a ${B}_{u}$ mode that is unstable for wave vectors normal to the octahedral chain direction of the $C2/c$ aristotype structure. The layer of wave vectors with imaginary mode frequencies also comprises the Brillouin zone boundary point $Y$ $(0,1,0)$, which is critical for the transition to the $P{2}_{1}/c$ ground-state structure. The phonon branch equivalent to the imaginary branch of the titanite aristotype is found to be stable in malayaite ${\mathrm{CaSnSiO}}_{5}$. The unstable phonon mode in titanite leads to the formation of transoriented short and long Ti-O1 bonds. The Ti as well as the connecting O1 atom exhibit strongly anomalous Born effective charges along the octahedral chain direction [001], indicative of the strong covalency in this direction. Accordingly and in contrast to malayaite, LO-TO splitting is very large in titanite. In the $C2/c$ phase of titanite, the Ti-O1-Ti distortion chain is disordered with respect to neighboring distortion chains, as all chain configurations are equally unstable along the phonon branch. This result is in agreement with diffuse x-ray scattering in layers normal to the chain direction that is observed at temperatures close to the $P{2}_{1}/c$ to $C2/c$ transition temperature and above. The resulting dynamic chains of correlated Ti displacements are expected to order in two dimensions to yield the $P{2}_{1}/c$ ground-state structure of titanite.

'Quantum Equilibrium-Disequilibrium': Asset Price Dynamics, Symmetry Breaking, and Defaults as Dissipative Instantons

We propose a simple non-equilibrium model of a financial market as an open system with a possible exchange of money with an outside world and market frictions (trade impacts) incorporated into asset price dynamics via a feedback mechanism. Using a linear market impact model, this produces a non-linear two-parametric extension of the classical Geometric Brownian Motion (GBM) model, that we call the ”Quantum Equilibrium-Disequilibrium” (QED) model. The QED model gives rise to non-linear mean-reverting dynamics, broken scale invariance, and corporate defaults. In the simplest one-stock (1D) formulation, our parsimonious model has only one degree of freedom, yet calibrates to both equity returns and credit default swap spreads. Defaults and market crashes are associated with dissipative tunneling events, and correspond to instanton (saddle-point) solutions of the model. When market frictions and inflows/outflows of money are neglected altogether, ”classical” GBM scale-invariant dynamics with an exponential asset growth and without defaults are formally recovered from the QED dynamics. However, we argue that this is only a formal mathematical limit, and in reality the GBM limit is non- analytic due to non-linear effects that produce both defaults and divergence of perturbation theory in a small market friction parameter.

Convergence from divergence

We show how to convert divergent series, which typically occur in many applications in physics, into rapidly convergent inverse factorial series. This can be interpreted physically as a novel resummation of perturbative series. Being convergent, these new series allow rigorous extrapolation from an asymptotic region with a large parameter, to the opposite region where the parameter is small. We illustrate the method with various physical examples, and discuss how these convergent series relate to standard methods such as Borel summation, and also how they incorporate the physical Stokes phenomenon. We comment on the relation of these results to Dyson’s physical argument for the divergence of perturbation theory. This approach also leads naturally to a wide class of relations between bosonic and fermionic partition functions, and Klein–Gordon and Dirac determinants.