High orders of perturbation theory: are renormalons significant?

According to Lipatov, the high orders of perturbation theory are determined by saddle-point configurations (instantons) of the corresponding functional integrals. According to t'Hooft, some individual large diagrams, renormalons, are also significant and they are not contained in the Lipatov contribution. The history of the conception of renormalons is presented, and the arguments in favor of and against their significance are discussed. The analytic properties of the Borel transforms of functional integrals, Green functions, vertex parts, and scaling functions are investigated in the case of \phi^4 theory. Their analyticity in a complex plane with a cut from the first instanton singularity to infinity (the Le Guillou - Zinn-Justin hypothesis) is proved. It rules out the existence of the renormalon singularities pointed out by t'Hooft and demonstrates the nonconstructiveness of the conception of renormalons as a whole. The results can be interpreted as an indication of the internal consistency of \phi^4 theory.Comment: 28 pages, 8 figures include

Summing Divergent Perturbative Series in a Strong Coupling Limit. The Gell-Mann - Low Function of the \phi^4 Theory

An algorithm is proposed for determining asymptotics of the sum of a perturbative series in the strong coupling limit using given values of the expansion coefficients. Operation of the algorithm is illustrated by test examples, method for estimating errors is developed, and an optimization procedure is described. Application of the algorithm to the $\phi^4$ theory gives a behavior $\beta(g)\approx 7.4 g^{0.96}$ at large $g$ for its Gell-Mann -- Low function. The fact that the exponent is close to unity can be interpreted as a manifestation of the logarithmic branching of the type $\beta(g)\sim g (\ln g)^{-\gamma}$ (with $\gamma\approx 0.14$), which is confirmed by independent evidence. In any case, the $\phi^4$ theory is internally consistent. The procedure of summing perturbartive series with arbitrary values of expansion parameter is discussed.Comment: 23 pages, PD

Divergent Perturbation Series

Various perturbation series are factorially divergent. The behavior of their high-order terms can be found by Lipatov's method, according to which they are determined by the saddle-point configurations (instantons) of appropriate functional integrals. When the Lipatov asymptotics is known and several lowest order terms of the perturbation series are found by direct calculation of diagrams, one can gain insight into the behavior of the remaining terms of the series. Summing it, one can solve (in a certain approximation) various strong-coupling problems. This approach is demonstrated by determining the Gell-Mann - Low functions in \phi^4 theory, QED, and QCD for arbitrary coupling constants. An overview of the mathematical theory of divergent series is presented, and interpretation of perturbation series is discussed. Explicit derivations of the Lipatov asymptotic forms are presented for some basic problems in theoretical physics. A solution is proposed to the problem of renormalon contributions, which hampered progress in this field in the late 1970s. Practical schemes for summation of perturbation series are described for a coupling constant of order unity and in the strong-coupling limit. An interpretation of the Borel integral is given for 'non-Borel-summable' series. High-order corrections to the Lipatov asymptotics are discussed.Comment: Review article, 45 pages, PD

Structure of High Order Corrections to Lipatov's Asymptotics

High orders of perturbation theory can be calculated by the Lipatov method, whereby they are determined by saddle-point configurations (instantons) of the corresponding functional integrals. For most field theories, the Lipatov asymptotics has the functional form c a^N \Gamma(N+b) (N is the order of perturbation theory), and the relative corrections to it have a form of a series in powers of 1/N. It is shown that this series diverges factorially and its high-order coefficients can be calculated using a procedure similar to Lipatov's one. The K-th expansion coefficient has the form const(\ln(S1/S0))^{-K}\Gamma(K+(r1-r0)/2), where S0 and S1 are the values of the action for the first and the second instanton of the field theory under consideration, while r0 and r1 are the corresponding numbers of zero modes. The instantons satisfy the same equation as in the Lipatov method and are assumed to be renumbered in order of increasing of their action. This result has the universal character and is valid in any field theory for which the Lipatov asymptotic form is as specified above.Comment: 8 pages, p

Thermoelectric properties of Cu-dispersed bi0.5sb1.5te3

A novel and simple approach was used to disperse Cu nanoparticles uniformly in the Bi0.5Sb1.5Te3 matrix, and the thermoelectric properties were evaluated for the Cu-dispersed Bi0.5Sb1.5Te3. Polycrystalline Bi0.5Sb1.5Te3 powder prepared by encapsulated melting and grinding was dry-mixed with Cu(OAc)2 powder. After Cu(OAc)2 decomposition, the Cu-dispersed Bi0.5Sb1.5Te3 was hot-pressed. Cu nanoparticles were well-dispersed in the Bi0.5Sb1.5Te3 matrix and acted as effective phonon scattering centers. The electrical conductivity increased systematically with increasing level of Cu nanoparticle dispersion. All specimens had a positive Seebeck coefficient, which confirmed that the electrical charge was transported mainly by holes. The thermoelectric figure of merit was enhanced remarkably over a wide temperature range of 323-523 K

Gell-Mann - Low Function for QCD in the strong-coupling limit

The Gell-Mann - Low function \beta(g) in QCD (g=g0^2/16\pi^2 where g0 is the coupling constant in the Lagrangian) is shown to behave in the strong-coupling region as \beta_\infty g^\alpha with \alpha\approx -13, \beta_\infty\sim 10^5.Comment: 5 pages, PD

The Role of Bound States in Time-Dependent Quantum Transport

Charge transport through a nanoscale junction coupled to two macroscopic electrodes is investigated for the situation when bound states are present. We provide numerical evidence that bound states give rise to persistent, non-decaying current oscillations in the junction. We also show that the amplitude of these oscillations can exhibit a strong dependence on the history of the applied potential as well as on the initial equilibrium configuration. Our simulations allow for a quantitative investigation of several transient features. We also discuss the existence of different time-scales and address their microscopic origin.Comment: 10 pages, 8 figure

Dynamical mean-field approach to materials with strong electronic correlations

We review recent results on the properties of materials with correlated electrons obtained within the LDA+DMFT approach, a combination of a conventional band structure approach based on the local density approximation (LDA) and the dynamical mean-field theory (DMFT). The application to four outstanding problems in this field is discussed: (i) we compute the full valence band structure of the charge-transfer insulator NiO by explicitly including the p-d hybridization, (ii) we explain the origin for the simultaneously occuring metal-insulator transition and collapse of the magnetic moment in MnO and Fe2O3, (iii) we describe a novel GGA+DMFT scheme in terms of plane-wave pseudopotentials which allows us to compute the orbital order and cooperative Jahn-Teller distortion in KCuF3 and LaMnO3, and (iv) we provide a general explanation for the appearance of kinks in the effective dispersion of correlated electrons in systems with a pronounced three-peak spectral function without having to resort to the coupling of electrons to bosonic excitations. These results provide a considerable progress in the fully microscopic investigations of correlated electron materials.Comment: 24 pages, 14 figures, final version, submitted to Eur. Phys. J. for publication in the Special Topics volume "Cooperative Phenomena in Solids: Metal-Insulator Transitions and Ordering of Microscopic Degrees of Freedom

Electronic Structure Calculation by First Principles for Strongly Correlated Electron Systems

Recent trends of ab initio studies and progress in methodologies for electronic structure calculations of strongly correlated electron systems are discussed. The interest for developing efficient methods is motivated by recent discoveries and characterizations of strongly correlated electron materials and by requirements for understanding mechanisms of intriguing phenomena beyond a single-particle picture. A three-stage scheme is developed as renormalized multi-scale solvers (RMS) utilizing the hierarchical electronic structure in the energy space. It provides us with an ab initio downfolding of the global band structure into low-energy effective models followed by low-energy solvers for the models. The RMS method is illustrated with examples of several materials. In particular, we overview cases such as dynamics of semiconductors, transition metals and its compounds including iron-based superconductors and perovskite oxides, as well as organic conductors of kappa-ET type.Comment: 44 pages including 38 figures, to appear in J. Phys. Soc. Jpn. as an invited review pape