Standard Perturbation Expansion Research Articles

Interaction of a finite quantum system with an infinite quantum system that contains a single one-parameter eigenvalue band

Interaction of a finite quantum system $${\textsf{\textbf{S}}_\rho^a}$$ that contains ρ eigenvalues and eigenstates with an infinite quantum system $${\textsf{\textbf{S}}_\infty^{\rm b}}$$ that contains a single one-parameter eigenvalue band is considered. A new approach for the treatment of the combined system $${\textsf{\textbf{S}}_\infty \equiv \textsf{\textbf{S}}_\rho^a \oplus \textsf{\textbf{S}}_\infty^b}$$ is developed. This system contains embedded eigenstates $${\left| {\psi (\epsilon)} \right\rangle}$$ with continuous eigenvalues $${\varepsilon}$$ , and, in addition, it may contain isolated eigenstates $${\left| {\psi_s } \right\rangle}$$ with discrete eigenvalues $${\varepsilon_s}$$ . Two ρ × ρ eigenvalue equations, a generic eigenvalue equation and a fractional shift eigenvalue equation are derived. It is shown that all properties of the system $${\textsf{\textbf{S}}_\rho^a}$$ that interacts with the system $${\textsf{\textbf{S}}_\infty^{\rm b}}$$ can be expressed in terms of the solutions to those two equations. The suggested method produces correct results, however strong the interaction between quantum systems $${\textsf{\textbf{S}}_\rho^a}$$ and $${\textsf{\textbf{S}}_\infty^{\rm b}}$$ . In the case of the weak interaction this method reproduces results that are usually obtained within the formalism of the perturbation expansion approach. However, if the interaction is strong one may encounter new phenomena with much more complex behavior. This is also the region where standard perturbation expansion fails. The method is illustrated with an example of a two-dimensional system $${\textsf{\textbf{S}}_2^a}$$ that interacts with the infinite system $${\textsf{\textbf{S}}_\infty^{\rm b}}$$ that contains a single one-parameter eigenvalue band. It is shown that all relevant completeness relations are satisfied, however strong the interaction between those two systems. This provides a strong verification of the suggested method.

Interaction of an isolated state with an infinite quantum system containing several one-parameter eigenvalue bands: II. time-dependent case

A new method for the exact solution of the interaction of an isolated state $$\left| \Theta \right\rangle $$ with an infinite dimensional quantum system § ∞ containing several one-parameter eigenvalue bands $$\lambda _\nu (k)\in I_\nu \equiv \left[ {a_\nu ,b_\nu } \right]$$ is developed. Unlike standard perturbation expansion approach, this method produces correct results however strong the interaction between the state $$\left| \Theta \right\rangle $$ and the system $$\S_\infty ^{\,b} $$ . It is shown that in the case of the weak interaction this method correctly reproduces standard results obtained within the formalism of the perturbation expansion method. In particular, due to the interaction with the system $$\S_\infty ^{\,b} $$ , eigenvalue E of the state $$\left| \Theta \right\rangle $$ shifts to a new position. In addition, if this eigenvalue is embedded inside the range $$D=\bigcup\nolimits_\nu {I_\nu } $$ of the unperturbed eigenvalues, this shifted eigenvalue broadens and spectral distribution of the state $$\left| \Theta \right\rangle $$ has the shape of the universal resonance curve. However, if the interaction is strong, one finds much more complex and much more complicated behavior

Matching-pursuit∕split-operator Fourier-transform simulations of nonadiabatic quantum dynamics

A rigorous and practical approach for simulations of nonadiabatic quantum dynamics is introduced. The algorithm involves a natural extension of the matching-pursuitsplit-operator Fourier-transform (MPSOFT) method [Y. Wu and V. S. Batista, J. Chem. Phys. 121, 1676 (2004)] recently developed for simulations of adiabatic quantum dynamics in multidimensional systems. The MPSOFT propagation scheme, extended to nonadiabatic dynamics, recursively applies the time-evolution operator as defined by the standard perturbation expansion to first-, or second-order, accuracy. The expansion is implemented in dynamically adaptive coherent-state representations, generated by an approach that combines the matching-pursuit algorithm with a gradient-based optimization method. The accuracy and efficiency of the resulting propagation method are demonstrated as applied to the canonical model systems introduced by Tully for testing simulations of dual curve-crossing nonadiabatic dynamics.

Derivation of nonlinear susceptibility coefficients in antiferroelectrics

We use a perturbation expansion of polarization as a function of an applied electric field and a Landau-type theory to derive expressions for the nonlinear susceptibility coefficients of antiferroelectrics, which is modeled as two collinear sublattices oppositely polarized in the antiferroelectric phase. The dynamics are incorporated via the Landau–Khalatnikov equations resulting in nonlinear equations of motion, which are solved using a standard perturbation expansion technique. The case of a superposition of two monochromatic waves as input fields is considered and the expansion is taken up to third order in nonlinearity. Such an analysis results in various combinations of the two input wave frequencies, second- and third-order in the expansion. In the calculation, we move between time and frequency domains; and show explicitly how the equations of motion in each domain are related by a Fourier transform integral pair. A generalization of the method to fa superposition of any number of monochromatic waves is indicated.

The Physical Content of the Leading Orders of Principal Component Analysis of Spectral Profiles

We consider the principal component analysis (PCA) method of expanding Stokes intensity and net polarization profiles in terms of eigenfunctions (or principal components) of the spectral covariance matrix. The expansion is ordered by the magnitude of the relevant eigenvalue from largest to smallest. We find that the ordering represents a perturbation expansion. This allows us to examine the physical content of the first few orders of the basis set for 40,000 profiles for each Stokes parameter for a solar active region. For the intensity profile, we find that the expansion represents a Taylor series with the highest ranked, or first, eigenfunction being the zeroth order term, the second as the (scaled) first derivative of the zeroth term, and the third as the (scaled) second derivative term. Thus, we can derive a velocity from the coefficients of the first derivative term and a magnetic splitting parameter from those of the second using the standard velocity perturbation and weak-field expansion. For the net polarization profiles, we find that the zeroth order terms yield, using the weak-field expansion, the vector magnetic field. A comparison with a Stokes profile fitting inversion shows that the thus-estimated velocity and magnetic parameters are in good agreement with the more time-consuming profile fitting values, but do show a roll-off, or saturation, for sufficiently large values. We also find that the bright quiet-Sun points have an upflow signature, while the dark regions have a downflow—one in good agreement with that derived by traditional analysis.

Resummed hadronic spectra of inclusiveBdecays

In this paper we investigate the hadronic mass spectra of inclusive $B$ decays. Specifically, we study how an upper cut on the invariant mass spectrum, which is necessary to extract ${V}_{\mathrm{ub}},$ results in the breakdown of the standard perturbative expansion due to the existence of large infrared logs. We first show how the decay rate factorizes at the level of the double differential distribution. Then, we present closed form expressions for the resummed cut rate for the inclusive decays $\stackrel{\ensuremath{\rightarrow}}{B}{X}_{s}\ensuremath{\gamma}$ and $\stackrel{\ensuremath{\rightarrow}}{B}{X}_{u}e\ensuremath{\nu}$ at next-to-leading order in the infrared logs. Using these results, we determine the range of cuts for which resummation is necessary, as well as the range for which the resummed expansion itself breaks down. We also use our results to extract the leading and next to leading infrared log contribution to the two loop differential rate. We find that for the phenomenologically interesting cut values, there is only a small region where the calculation is under control. Furthermore, the size of this region is sensitive to the parameter $\overline{\ensuremath{\Lambda}}.$ We discuss the viability of extracting ${V}_{\mathrm{ub}}$ from the hadronic mass spectrum.

Effective action for superconductors and BCS-Bose crossover

A standard perturbative expansion around the mean-field solution is used to derive the low-energy effective action for superconductors at $T=0.$ Taking into account the density fluctuations at the outset we get the effective action where the density $\ensuremath{\rho}$ is the conjugated momentum to the phase $\ensuremath{\theta}$ of the order parameter. In the hydrodynamic regime, the dynamics of the superconductor is described by a time-dependent nonlinear Schr\"odinger (TDNLS) equation for the field $\ensuremath{\Psi}(x)=\sqrt{\ensuremath{\rho}/2}{e}^{i\ensuremath{\theta}}.$ The evolution of the density fluctuations in the crossover from weak-coupling (BCS) to strong-coupling (Bose condensation of localized pairs) superconductivity is discussed for the attractive Hubbard model. In the bosonic limit, the TDNLS equation reduces to the Gross-Pitaevskii equation for the order parameter, as in the standard description of superfluidity. The conditions under which a phase-only action can be derived in the presence of a long-range interaction to describe the physics of the superconductivity of ``bad metals'' are discussed.

An efficient probabilistic finite element method for stochastic groundwater flow

We present an efficient numerical method for solving stochastic porous media flow problems. Single-phase flow with a random conductivity field is considered in a standard first-order perturbation expansion framework. The numerical scheme, based on finite element techniques, is computationally more efficient than traditional approaches because one can work with a much coarser finite element mesh. This is achieved by avoiding the common finite element representation of the conductivity field. Computations with the random conductivity field only arise in integrals of the log conductivity covariance function. The method is demonstrated in several two- and three-dimensional flow situations and compared to analytical solutions and Monte Carlo simulations. Provided that the integrals involving the covariance of the log conductivity are computed by higher-order Gaussian quadrature rules, excellent results can be obtained with characteristic element sizes equal to about five correlation lengths of the log conductivity field. Investigations of the validity of the proposed first-order method are performed by comparing nonlinear Monte Carlo results with linear solutions. In box-shaped domains the log conductivity standard deviation σ Y may be as large as 1.5, while the head variance is considerably influenced by nonlinear effects as σ Y approaches unity in more general domains.

Quantum Monte Carlo in the interaction representation: Application to a spin-Peierls model

A quantum Monte Carlo algorithm is constructed starting from the standard perturbation expansion in the interaction representation. The resulting configuration space is strongly related to that of the Stochastic Series Expansion (SSE) method, which is based on a direct power series expansion of exp(-beta*H). Sampling procedures previously developed for the SSE method can therefore be used also in the interaction representation formulation. The new method is first tested on the S=1/2 Heisenberg chain. Then, as an application to a model of great current interest, a Heisenberg chain including phonon degrees of freedom is studied. Einstein phonons are coupled to the spins via a linear modulation of the nearest-neighbor exchange. The simulation algorithm is implemented in the phonon occupation number basis, without Hilbert space truncations, and is exact. Results are presented for the magnetic properties of the system in a wide temperature regime, including the T-->0 limit where the chain undergoes a spin-Peierls transition. Some aspects of the phonon dynamics are also discussed. The results suggest that the effects of dynamic phonons in spin-Peierls compounds such as GeCuO3 and NaV2O5 must be included in order to obtain a correct quantitative description of their magnetic properties, both above and below the dimerization temperature.

Mesoscopic conductance and its fluctuations at a nonzero Hall angle

We consider the bilocal conductivity tensor, the two-probe conductance and its fluctuations for a disordered phase-coherent two-dimensional system of non-interacting electrons in the presence of a magnetic field, including correctly the edge effects. Analytical results are obtained by perturbation theory in the limit $\sigma_{xx} \gg 1$. For mesoscopic systems the conduction process is dominated by diffusion but we show that, due to the lack of time-reversal symmetry, the boundary condition for diffusion is altered at the reflecting edges. Instead of the usual condition, that the derivative along the direction normal to the wall of the diffusing variable vanishes, the derivative at the Hall angle to the normal vanishes. We demonstrate the origin of this boundary condition from different starting points, using (i) a simplified Chalker-Coddington network model, (ii) the standard diagrammatic perturbation expansion, and (iii) the nonlinear sigma-model with the topological term, thus establishing connections between the different approaches. Further boundary effects are found in quantum interference phenomena. We evaluate the mean bilocal conductivity tensor $\sigma_{\mu\nu}(r,r')$, and the mean and variance of the conductance, to leading order in $1/\sigma_{xx}$ and to order $(\sigma_{xy}/\sigma_{xx})^2$, and find that the variance of the conductance increases with the Hall ratio. Thus the conductance fluctuations are no longer simply described by the unitary universality class of the $\sigma_{xy}=0$ case, but instead there is a one-parameter family of probability distributions. In the quasi-one-dimensional limit, the usual universal result for the conductance fluctuations of the unitary ensemble is recovered, in contrast to results of previous authors. Also, a long discussion of current conservation.

Exact resummations in the theory of hydrodynamic turbulence. I. The ball of locality and normal scaling.

This paper is the first in a series of papers that aim at understanding the scaling behavior of hydrodynamic turbulence. We present in this paper a perturbative theory for the structure functions and the response functions of the hydrodynamic velocity field in real space and time. Starting from the Navier-Stokes equations (at high Reynolds number Re) we show that the standard perturbative expansions that suffer from infrared divergences can be exactly resummed using the Belinicher-L'vov transformation. After this exact (partial) resummation it is proven that the resulting perturbation theory is free of divergences, both in large and in small spatial separations. The hydrodynamic response and the correlations have contributions that arise from mediated interactions which take place at some space-time coordinates. It is shown that the main contribution arises when these coordinates lie within a shell of a ``ball of locality'' that is defined and discussed. We argue that the real space-time formalism that is developed here offers a clear and intuitive understanding of every diagram in the theory, and of every element in the diagrams. One major consequence of this theory is that none of the familiar perturbative mechanisms may ruin the classical 1941 Kolmogorov (K41) scaling solution for the structure functions. Accordingly, corrections to the K41 solutions should be sought in nonperturbative effects. These effects are the subjects of paper II (the following paper) and a future paper in this series that will propose a mechanism for anomalous scaling in turbulence, which in particular allows a multiscaling of the structure functions.

Wannier and Bloch orbital computation of the nonlinear susceptibility

We present a method to compute high-order derivatives of the total energy of a periodic solid with respect to a uniform electric field. We apply the 2n+1 theorem to a recently introduced total energy functional which uses a Wannier representation for the electronic orbitals and we find an expression for the static nonlinear susceptibility which is much simpler than the one obtained by standard perturbative expansions. We show that the zero-field expression of the nonlinear susceptibility can be rewritten in a Bloch representation. We test numerically the validity of our approach with a 1D model Hamiltonian.

Diagrammatic expansion of a Φ4 theory and lattice models with local interactions up to eighth order

Abstract Four-degree-vertices-type diagrams are the most frequently used in solid-state and statistical physics. These are applicable in the following fields: (i) standard perturbation expansion of the Green functions; (ii) variational description of models with local interaction; (iii) Hugenholtz-type diagrams for two-body potentials; (iv) renormalization group; (v) standard Φ4 theory, including four point and two point functions, which is applied to localized spin systems, amplitude functions and expansions; (vi) exact solution of the non-interacting, two-dimensional (sixteen, thirty-two, one hundred and twenty-eight) vertex models. We present the complete topology of the four-degree-vertices diagrams up to eighth order, in a condensed manner, characterizing a total number of contributing graphs of order 109, and we determine the topologically different contributions for every order. The result is a simplification in calculating the contributions of the different orders, e.g. in eighth order instead of ...

A computationally efficient method for evaluating the gradient of 2D NOESY intensities

A computationally efficient method for calculating the derivative of NOE intensities with respect to any parameter is presented. This method is based on an integral expression representing the gradient. We will derive this expression from first principles using standard perturbation expansion techniques, and show it to be equivalent to an analytical expression [Yip, P. and Case, D.A. (1989) J. Magn. Reson., 83, 643] Implementation of this method in a refinement scheme (NOE-MD) is also briefly mentioned.

On the Second Quantization of Gauge Systems

Within the stochastic quantization scheme the second quantization of constrained Hamiltonian systems is exemplified for the case of the relativistic point particle. In addition, the related quantization of the abelian Chern-Simons field theory is discussed. 389 One of the most appealing features of stochastic quantization lies in the quantiza­ tion of gauge field theories where, remarkably, no gauge fixing terms are needed.l) As a consequence stochastic quantization preserves the full gauge symmetry throughout and constitutes a manifestly gauge covariant quantization scheme. In the first part of my contribution I would like to summarize some generaliza­ tions2>.a> of the stochastic quantization scheme to study the second quantization of classical Hamiltonian systems with constraints. 4 >- 6 > A key ingredient for this approach is the use of the ERST formalism 7 >- 9 > which constitutes a powerful and elegant tool for dealing with constrained systems. I discuss the method for the simple case of the interacting relativistic point particle; it will be seen upon second quantiza­ tion that the standard perturbation expansion of a self interacting scalar field is obtained. In the second part I would like to point out some similarities of the above method to the stochastic quantization of the Chern-Simons field theory. 10 >-Iz> Here I restrict myself to the abelian case only. A classical system with phase space variables (x, p) is constrained or possesses a gauge invariance, if the Legendre transformation between the velocities and the momenta is singular. As a consequence not all the velocities can be expressed by the momenta and some of the Euler-Lagrange equations reduce to constraints Gi(x, p)=O, i=l, ···, m.

Correlation diagrams and perturbation theory in thermal coherent states

Thermal quantum theories employ a free parameter αϵ [0,1]. It has been demonstrated recently that a standard perturbation expansion exists only for α=0 or α=1. In this work it is shown how this limitation can be overcome in thermal coherent states at the cost of increased complexity.

Lie Group Calculation of the Green Function of Disordered Systems

ABSTRACTWithin the framework of the muffin-tin multiple-scattering theory, the scattering path operators are given by the inverse of a matrix consisting of atomic t-matrices and a structural matrix. The influence of the displacement of an atomic centre on the structural matrix can be described analytically using Lie group techniques. From this analytical expression and the standard perturbation expansion of the Lippmann-Schwinger equation, it is possible to write the Green function of a disordered system as a series of terms whichare averages over configurations. These averages can be calculated analytically from themoments of the interatomic distances. Special terms of this series are then summed up toinfinity using Dyson equation. This formalism is computationally very effective to calculate electronic properties of systems with thermal or structural disorder. In this paper, the theoretical basis of this approach is briefly described and the convergence properties of the expansions are investigated.

Renormalized expansions for functional integrals: Generalized coherent potential approach to lattice models

An expansion for generating functionals (partition sums) of models expressed as lattice functional integrals with local (on-site) interactions is presented. This expansion renormalizes the standard perturbative expansion in such a way that certain its terms are summed up non-perturbatively. A non-self-consistent and a self-consistent versions of the expansion are formulated and criteria for an estimation of validity of approximations resulting from the both expansions are given. The simplest approximation being the first term of this expansion is applied to two lattice models: classicalN-component spin model and the model of non-interacting electrons in a disordered crystal. In the former model the critical temperature is calculated within 10% accuracy and in the latter, the coherent potential approximation is obtained.

Discrete sudden perturbation theory for inelastic scattering. I. Quantum and semiclassical treatment

A double perturbation theory is constructed to treat rotationally and vibrationally inelastic scattering. It uses both the elastic scattering from the spherically averaged potential and the infinite-order sudden (IOS) approximation as the unperturbed solutions. First, a standard perturbation expansion is done to express the radial wave functions in terms of the elastic wave functions. The resulting coupled equations are transformed to the discrete-variable representation where the IOS equations are diagonal. Then, the IOS solutions are removed from the equations which are solved by an exponential perturbation approximation. The results for Ar+N2 are very much more accurate than the IOS and somewhat more accurate than a straight first-order exponential perturbation theory. The theory is then converted into a semiclassical, time-dependent form by using the WKB approximation. The result is an integral of the potential times a slowly oscillating factor over the classical trajectory. A method of interpolating the result is given so that the calculation is done at the average velocity for a given transition. With this procedure, the semiclassical version of the theory is more accurate than the quantum version and very much faster. Calculations on Ar+N2 show the theory to be much more accurate than the infinite-order sudden (IOS) approximation and the exponential time-dependent perturbation theory.

An alternative to the magnus expansion in time-dependent perturbation theory

An alternative to the Magnus expansion in time-dependent quantum mechanical perturbation theory is presented. An exponential form of the time-development operator, given as the exponential of a sum of products of integrals well known in time-dependent quantum mechanics, is derived from the standard perturbation expansion. The derivation is simple and the form of the terms in the exponential is so straightforward that the Nth order term can be written by inspection. The standard Magnus terms can be obtained by rearrangement of the terms of the new expansion probably easier than they can be obtained by iteration of the original equations. A proof by induction is presented to establish the form of the Nth term and it is shown how the new expansion terms relate to the Magnus terms. Some apparent differences between Magnus terms in the literature are resolved.