Self-consistent Perturbation Expansion Research Articles

Non-perturbative intertwining between spin and charge correlations: A ``smoking gun'' single-boson-exchange result

We study the microscopic mechanism controlling the interplay between local charge and local spin fluctuations in correlated electron systems via a thorough investigation of the generalized on-site charge susceptibility of several fundamental many-electron models, such as the Hubbard atom, the Anderson impurity model, and the Hubbard model. By decomposing the numerically determined generalized susceptibility in terms of physically transparent single-boson exchange processes, we unveil the microscopic mechanisms responsible for the breakdown of the self-consistent many-electron perturbation expansion. In particular, we unambiguously identify the origin of the significant suppression of its diagonal entries in (Matsubara) frequency space and the slight increase of the off-diagonal ones which cause the breakdown. The suppression effect on the diagonal elements originates directly from the electronic scattering on local magnetic moments, reflecting their increasingly longer lifetime as well as their enhanced effective coupling with the electrons. Instead, the slight and diffuse enhancement of the off-diagonal terms can be mostly ascribed to multiboson scattering processes. The strong intertwining between spin and charge sectors is partly weakened at the Kondo temperature due to a progressive reduction of the effective spin-fermion coupling of local magnetic fluctuations in the low frequency regime. Our analysis, thus, clarifies the precise mechanism through which the physical information is transferred between different scattering channels of interacting electron problems and highlights the pivotal role played by such an intertwining in the physics of correlated electrons beyond the perturbative regime.

Perturbation theory for an Anderson quantum dot asymmetrically attached to two superconducting leads

Self-consistent perturbation expansion up to the second order in the interaction strength is used to study a single-level quantum dot with local Coulomb repulsion attached asymmetrically to two generally different superconducting leads. At zero temperature and wide range of other parameters the spin-symmetric version of the expansion yields excellent results for the position of the $0-\pi$ impurity quantum phase transition boundary and Josephson current together with the energy of Andreev bound states in the $0$-phase as confirmed by numerical calculations using the Numerical Renormalisation Group method. We analytically prove that the method is charge-conserving as well as thermodynamically consistent. Explicit formulas for the position of the $0-\pi$ phase boundary are presented for the Hartree-Fock approximation as well as for its variant called Generalized Atomic Limit. It is shown that the Generalized Atomic Limit can be used as a quick estimate for the position of the phase boundary at half-filling in a broad range of parameters. We apply our second order perturbation method to the interpretation of the existing experimental data on the phase boundary with very satisfactory outcome suggesting that the so far employed heavy numerical tools such as Numerical Renormalization Group and/or Quantum Monte Carlo are not necessary in a class of generic situations and can be safely replaced by a perturbative approach.

Numerical study of long-time dynamics and ergodic-nonergodic transitions in dense simple fluids.

Since the mid-1980s, mode-coupling theory (MCT) has been the de facto theoretic description of dense fluids and the transition from the fluid state to the glassy state. MCT, however, is limited by the approximations used in its construction and lacks an unambiguous mechanism to institute corrections. We use recent results from a new theoretical framework--developed from first principles via a self-consistent perturbation expansion in terms of an effective two-body potential--to numerically explore the kinetics of systems of classical particles, specifically hard spheres governed by Smoluchowski dynamics. We present here a full solution for such a system to the kinetic equation governing the density-density time correlation function and show that the function exhibits the characteristic two-step decay of supercooled fluids and an ergodic-nonergodic transition to a dynamically arrested state. Unlike many previous numerical studies--and in stark contrast to experiment--we have access to the full time and wave-number range of the correlation function with great precision and are able to track the solution unprecedentedly close to the transition, covering nearly 15 decades in scaled time. Using asymptotic approximation techniques analogous to those developed for MCT, we fit the solution to predicted forms and extract critical parameters. We find complete qualitative agreement with known glassy behavior (e.g. power-law divergence of the α-relaxation time scale in the ergodic phase and square-root growth of the glass form factors in the nonergodic phase), as well as some limited quantitative agreement [e.g. the transition at packing fraction η*=0.60149761(10)], consistent with previous static solutions under this theory and with comparable colloidal suspension experiments. However, most importantly, we establish that this new theory is able to reproduce the salient features seen in other theories, experiments, and simulations but has the advantages of being derived from first principles and possessing a clear mechanism for making systematic corrections.

Perturbation theory of a superconducting 0 - π impurity quantum phase transition.

A single-level quantum dot with Coulomb repulsion attached to two superconducting leads is studied via the perturbation expansion in the interaction strength. We use the Nambu formalism and the standard many-body diagrammatic representation of the impurity Green functions to formulate the Matsubara self-consistent perturbation expansion. We show that at zero temperature second order of the expansion in its spin-symmetric version yields a nearly perfect agreement with the numerically exact calculations for the position of the 0 − π phase boundary at which the Andreev bound states reach the Fermi energy as well as for the values of single-particle quantities in the 0-phase. We present results for phase diagrams, level occupation, induced local superconducting gap, Josephson current, and energy of the Andreev bound states with the precision surpassing any (semi)analytical approaches employed thus far.

Effective Action for Bose–Einstein Condensates

We clarify basic properties of an effective action (i.e., self-consistent perturbation expansion) for interacting Bose-Einstein condensates, where field $\psi$ itself acquires a finite thermodynamic average $\langle \psi\rangle$ besides two-point Green's function $\hat G$ to form an off-diagonal long-range order. It is shown that the action can be expressed concisely order by order in terms of the interaction vertex and a special combination of $\langle\psi\rangle$ and $\hat G$ so as to satisfy both Noether's theorem and Goldstone's theorem (I) corresponding to the first proof. The self-energy is predicted to have a one-particle-reducible structure due to $\langle \psi\rangle\neq 0$ to transform the Bogoliubov mode into a bubbling mode with a substantial decay rate.

Lifetime of Single-Particle Excitations in a Dilute Bose–Einstein Condensate at Zero Temperature

We study the lifetime of single-particle excitations in a dilute homogeneous Bose-Einstein condensate at zero temperature based on a self-consistent perturbation expansion of satisfying Goldstone's theorem and conservation laws simultaneously.It is shown that every excitation for each momentum ${\bf p}$ should have a finite lifetime proportional to the inverse $a^{-1}$ of the $s$-wave scattering length $a$, instead of $a^{-2}$ for the normal state, due to a new class of Feynman diagrams for the self-energy that emerges upon condensation. We calculate the lifetime as a function of $|{\bf p}|$ approximately.

Ground-State Energy and Condensate Density of a Dilute Bose Gas Revisited

The ground-state energy per particle \(E/N\) and condensate density \(n_{0}\) of a dilute Bose gas are studied with a self-consistent perturbation expansion satisfying the Hugenholtz–Pines theorem and conservation laws simultaneously. A class of Feynman diagrams for the self-energy, which has escaped consideration so far, is shown to add an extra constant \(c_{\text{ip}}\sim O(1)\) to the expressions reported by Lee et al. [Phys. Rev. 106 (1957) 1135] as \(E/N=(2\pi\hbar^{2}an/m)[1+(128/15\sqrt{\pi}+16c_{\text{ip}}/5)\sqrt{a^{3}n}]\) and \(n_{0}/n=1-(8/3\sqrt{\pi}+c_{\text{ip}})\sqrt{a^{3}n}\), where \(a\), \(n\), and \(m\) are are the \(s\)-wave scattering length, particle density, and particle mass, respectively. We present a couple of estimates for \(c_{\text{ip}}\); the third-order perturbation expansion yields \(c_{\text{ip}}=0.412\).

Self-Consistent Approximations for Superconductivity beyond the Bardeen–Cooper–Schrieffer Theory

We develop a concise self-consistent perturbation expansion for superconductivity where all the pair processes are naturally incorporated without drawing "anomalous" Feynman diagrams. This simplification results from introducing an interaction vertex that is symmetric in the particle-hole indices besides the ordinary space-spin coordinates. The formalism automatically satisfies conservation laws, includes the Luttinger-Ward theory as the normal-state limit, and reproduces the Bardeen-Cooper-Schrieffer theory as the lowest-order approximation. It enables us to study the thermodynamic, single-particle, two-particle, and dynamical properties of superconductors with competing fluctuations based on a single functional $\Phi[\hat{G}]$ of Green's function $\hat{G}$ in the Nambu space. Specifically, we derive closed equations in the FLEX-S approximation, i.e., the fluctuation exchange approximation for superconductivity with all the pair processes, which contains extra terms besides those in the standard FLEX approximation.

Nonequilibrium dynamical mean-field calculations based on the noncrossing approximation and its generalizations

We solve the impurity problem which arises within nonequilibrium dynamical mean-field theory for the Hubbard model by means of a self-consistent perturbation expansion around the atomic limit. While the lowest order, known as the non-crossing approximation (NCA), is reliable only when the interaction U is much larger than the bandwidth, low-order corrections to the NCA turn out to be sufficient to reproduce numerically exact Monte Carlo results in a wide parameter range that covers the insulating phase and the metal-insulator crossover regime at not too low temperatures. As an application of the perturbative strong-coupling impurity solver we investigate the response of the double occupancy in the Mott insulating phase of the Hubbard model to a dynamical change of the interaction or the hopping, a technique which has been used as a probe of the Mott insulating state in ultracold fermionic gases.

Introduction to Nonequilibrium Statistical Mechanics with Quantum Field Theory

In this article, we present a concise and self-contained introduction to nonequilibrium statistical mechanics with quantum field theory by considering an ensemble of interacting identical bosons or fermions as an example. Readers are assumed to be familiar with the Matsubara formalism of equilibrium statistical mechanics such as Feynman diagrams, the proper self-energy, and Dyson's equation. The aims are threefold: (i) to explain the fundamentals of nonequilibrium quantum field theory as simple as possible on the basis of the knowledge of the equilibrium counterpart; (ii) to elucidate the hierarchy in describing nonequilibrium systems from Dyson's equation on the Keldysh contour to the Navier-Stokes equation in fluid mechanics via quantum transport equations and the Boltzmann equation; (iii) to derive an expression of nonequilibrium entropy that evolves with time. In stage (i), we introduce nonequilibrium Green's function and the self-energy uniquely on the round-trip Keldysh contour, thereby avoiding possible confusions that may arise from defining multiple Green's functions at the very beginning. We try to present the Feynman rules for the perturbation expansion as simple as possible. In particular, we focus on the self-consistent perturbation expansion with the Luttinger-Ward thermodynamic functional, i.e., Baym's Phi-derivable approximation, which has a crucial property for nonequilibrium systems of obeying various conservation laws automatically. We also show how the two-particle correlations can be calculated within the Phi-derivable approximation, i.e., an issue of how to handle the "Bogoliubov-Born-Green-Kirkwood-Yvons (BBGKY) hierarchy".

Self-consistent perturbation expansion for Bose-Einstein condensates satisfying Goldstone’s theorem and conservation laws

Quantum-field-theoretic descriptions of interacting condensed bosons have suffered from the lack of self-consistent approximation schemes satisfying Goldstone's theorem and dynamical conservation laws simultaneously. We present a procedure to construct such approximations systematically by using either an exact relation for the interaction energy or the Hugenholtz-Pines relation to express the thermodynamic potential in a Luttinger-Ward form. Inspection of the self-consistent perturbation expansion up to the third order with respect to the interaction shows that the two relations yield a unique identical result at each order, reproducing the conserving-gapless mean-field theory [T. Kita, J. Phys. Soc. Jpn. 74, 1891 (2005)] as the lowest-order approximation. The uniqueness implies that the series becomes exact when infinite terms are retained. We also derive useful expressions for the entropy and superfluid density in terms of Green's function and a set of real-time dynamical equations to describe thermalization of the condensate.

What really enhances the adsorption of polymers onto chemically nonuniform surfaces: Surface randomness or its heterogeneity?

We theoretically perform a comparative analysis of the adsorption of polymers onto the regularly and randomly nonuniform surfaces. By developing and making use of the self-consistent perturbation expansion we calculate the surface excesses of the polymers adsorbed on the random and periodically patterned surfaces. In both cases the enhancement of the polymer adsorption is indicated, as compared to the adsorption onto the homogeneous surface that has the same average affinity for polymers. Moreover, the results obtained for the randomly nonuniform and periodically patterned adsorbing surfaces show striking quantitative similarity, when compared at the same characteristic sizes of inhomogeneities of these surfaces. This finding leads to the conclusion that the adsorption ability of the nonuniform surface primarily depends on the characteristic size of the surface inhomogeneity, rather than on the spatial distribution of the inhomogeneities on this surface. In all cases, the calculated total surface excess is found to be a decaying function of the ratio of the radius of gyration of polymers to the characteristic size of the surface inhomogeneity. The effect of the excluded volume is found to reduce the polymer adsorption.

Parity Effects in Nanoscale Metallic Grain Including a Magnetic Impurity

In ultrasmall systems such as nanoscale metallic grains, the discrete energy spectrum together with electron correlations may give dramatic properties at low temperatures. Among others, the Kondo effect has attracted considerable attention recently. In particular, parity effects should emerge depending on the number of total electrons: e.g., the Kondo effect appears (disappears) at low temperatures for an even (odd) number of electrons, yielding a finite (divergent) susceptibility when the Kondo temperature is smaller than the level spacing. More remarkable parity effects were pointed out by Thimm et al. in dynamical as well as transport properties when the mean level spacing is nearly equal to (or even smaller than) the Kondo temperature. They found that the Kondo resonance is discretized into a series of subpeaks with one main peak (two split peaks) at the Fermi level for an even (odd) number of electrons. They claimed that this behavior characterizes the Kondo effect in a metallic grain, which also leads to the oscillation behavior in the conductance. Although such parity effects are naturally expected, it is not clear how a single (double) peak structure is developed in the Kondo resonance depending on whether the electron count is even or odd. More generally we ask which aspect is universal in the parity effects. In this note we revisit this question about the Kondo effect in a metallic grain. Using the self-consistent second-order perturbation expansion, we demonstrate that although the parity effect indeed appears generically, the shape of the spectrum exhibits nonuniversal properties depending on a model system employed, in contrast to the claim of Thimm et al. In the nonequilibrium case, we find the oscillation behavior possessing the parity effect in the differential conductance. We consider a small metallic grain including a magnetic impurity described by the Anderson Hamiltonian,

Damped orbital excitations in the titanates

A possible mechanism for the removal of the orbital degeneracy in RTiO3 (where R=La, Y, ...) is considered. The calculation is based on the Kugel-Khomskii Hamiltonian for electrons residing in the t2g orbitals of the Ti ions, and uses a self-consistent pe rturbation expansion in the interaction between the orbital and the spin degrees of freedom. The latter are assumed to be ordered in a Neel state, brought about by delicate interactions that are not included in the Kugel-Khomskii Hamiltonian. Within our model calculations, each of the t2g bands is found to acquire a finite, temperature-dependent dispersion, that lifts the orbital degeneracy. The orbital excitations are found to be heavily damped over a rather wide band. Consequently, they do not participate as a separate branch of excitations in the low-temperature thermodynamics.e

Self-consistent density-functional approach to the correlated ground states and an unrestricted many-body perturbation theory

Abstract We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and in...

Anderson model: Lattice effects in high dimensions

The periodic Anderson model with infinite Coulomb interaction U on a D-dimensional lattice is studied by mapping the lattice problem onto an impurity model with a self-consistency condition, using a systematic 1/D-expansion. The mapping becomes exact in the limit of infinite dimension. The remaining impurity model with a renormalized band is investigated within the self-consistent perturbation expansion in the hybridization V. For low temperatures the imaginary part of the f-electron self-energy vanishes quadratically at the Fermi level and thus provides the expected Fermi-liquid behavior.

Neural networks based on peano curves and hairy neurons

Neural Intelligence (NI) can automatically extract pattern features given an Artificial Intelligence (AI) performance rule. For example, neurocomputing searches for features that satisfy a rule-based criterion: intraclass-minimum-interclass-maximum cost function among features. The features representing images so discovered by NI can be passed to AI for further processing to increase efficiency and reliability, for AI can follow any formulated algorithm exactly, and provide (an imagery context knowledge base as) a constraint on further NI neurocomputing. Such integration, with a dialogue between the two, is believed actually to happen in the human visual system in tracking of moving objects, for instance. Consequently, we propose that both AI and NI are two sides of an intelligence coin that can together solve the pattern recognition problem and unify an intelligent machine. The rule-based minimax cost function used for feature search can, furthermore, give us a top-down architectural design of neural networks by means of Taylor series expansion of the cost function. A typical minimax cost function consists of the sample variance of each class in the numerator, and separation of the centers of classes in the denominator. Thus, when the total cost energy is minimized, the conflicting goals of intraclass clustering and interclass segregation are achieved simultaneously. This Taylor expansion variable must be a one-dimensional array that traces along a space-filling curve which, as Peano proved, preserves two-dimensional neighborhood relationships. Therefore, this one-dimensional array can support Taylor series expansion in terms of neighborhood derivatives. An adaptive space-filling capability is postulated for useful neuronic representations by using a top layer neural network, similar to Adaptive Resonance Theory, when more detailed spatial resolution becomes desirable at the place in the picture where an interesting change occurs. A self-consistent perturbation expansion can speed up the training procedure. A hairy neuron model that has two internal degrees of freedom is useful to determine a dynamically self-reconfigurable architecture. The convergence theorem of such a morphology is given for a hairy neural network under arbitrary time scales.

Consistent perturbation expansion for phase-changing material in a finite domain

A self-consistent perturbation expansion is developed for the case of a finite slab of phase-changing material with constant boundary conditions corresponding to a ‘‘two-phase’’ problem. Consistency is achieved by a power-series expansion in functions of both solid and liquid Stefan numbers. Terms through third order are calculated.

Renormalised perturbation theory of normal systems

Employing an infinite hierarchy of equations for the Green functions of a many particle system in its ground state mod psi N), the well known non-relativistic, time-dependent, ground-state perturbation theory of a normal system of interacting fermions is derived, without introducing the hypothesis that mod psi N) is the adiabatic transform of a non-interacting state. This new formulation of the problem allows us to renormalise conventional perturbation theory by introducing an effective interaction Gamma and the final result is a highly summed and manifestly self-consistent perturbation expansion in powers of Gamma . The authors calculate Gamma to second order and demonstrate that many of the well known results of time-dependent perturbation theory may be obtained by choosing the simplest approximations for Gamma . The results are readily generalised to finite temperatures and the formalism provides a generalised form of Hartree-Fock theory which may have important applications in many areas of physics.

Microscopic description of static quadrupole moments of 2 1 + states by time-dependent HF equations

Large quadrupole moments of 1-phonon states of sperical nuclei are shown to follow in a simple and direct way from time-dependent HF equations using the self-consistent perturbation expansion. Whereas the first order describes harmonic quadrupole oscillations, anharmonic terms arise in second order. The scattering of quasiparticles by quasi-particle pairs, which is neglected in RPA calculations, but included in this paper, leads to a second order response of the system and is shown to produce a strong enhancement of the quadrupole moments. Explicit results are obtained with a pairing-plus-quadrupole interaction and discussed within a schematic two level model.