M. V. Komarova

Instantons for dynamic models from B to H

Abstract Instanton analysis is applied to models B–H of critical dynamics. It is shown that the static instanton of the massless ϕ 4 model determines the large-order asymptotes of the perturbation expansion of these near-equilibrium dynamic models leading to factorial growth with the order of perturbation theory.

Asymptotic Behavior of Renormalization Constants in Higher Orders of the Perturbation Expansion for the (4−∈)-Dimensionally Regularized O(n)-Symmetric φ4 Theory

Higher-order asymptotic expansions for renormalization constants and critical exponents of the O(n)-symmetric φ4 theory are found based on the instanton approach in the minimal subtraction scheme for the (4−∈)-dimensional regularization. The exactly known expansion terms differ substantially from their asymptotic values. We find expressions that improve the asymptotic expansions for unknown expansion terms of the renormalization constants.

Large-order asymptotes for dynamic models near equilibrium

Abstract Instanton analysis is applied to model A of critical dynamics. It is shown that the static instanton of the massless ϕ 4 model determines the large-order asymptotes of the perturbation expansion of the dynamic model.

Asymptotic Behavior of Higher-Order Perturbations: Scaling Functions of the O(n)-Symmetric φ4-Theory in the (4−ε)-Expansion

We use an instantonic approach to calculate the asymptotic behavior of higher orders of the (4−ε)-expansion for the scaling function of the pair correlator of the O(n)-symmetric φ4-theory in the minimal subtraction scheme. Our results differ substantially from the known exact expression for the ε3 order of the expansion of the scaling function in the small-τ domain.

Large-order asymptotes for dynamic models

Large-order asymptotics in dynamic field theories constructed from Langevin equations with the aid of the Martin–Siggia–Rose formalism are considered. The existence of instantons in dynamic models is discussed. Specific features of the instanton approach in the dynamic models are shown in the examples of the standard dynamic 4-based models from A to H in the common classification and Kraichnan model of passive scalar advection in turbulent flow. The results obtained demonstrate that the series of the perturbation expansions for the dynamic 4-related models is—as usual—asymptotic with zero radius of convergence. Main parameters of large-order asymptotes are determined. In the Kraichnan model, however, the situation is different. Our results show that the series here has a finite radius of convergence. This radius as well as the character of the singularity of the functions investigated was determined.

Superfluid phase transition with activated velocity fluctuations: Renormalization group approach.

A quantum field model that incorporates Bose-condensed systems near their phase transition into a superfluid phase and velocity fluctuations is proposed. The stochastic Navier-Stokes equation is used for a generation of the velocity fluctuations. As such this model generalizes model F of critical dynamics. The field-theoretic action is derived using the Martin-Siggia-Rose formalism and path integral approach. The regime of equilibrium fluctuations is analyzed within the perturbative renormalization group method. The double (ε,δ)-expansion scheme is employed, where ε is a deviation from space dimension 4 and δ describes scaling of velocity fluctuations. The renormalization procedure is performed to the leading order. The main corollary gained from the analysis of the thermal equilibrium regime suggests that one-loop calculations of the presented models are not sufficient to make a definite conclusion about the stability of fixed points. We also show that critical exponents are drastically changed as a result of the turbulent background and critical fluctuations are in fact destroyed by the developed turbulence fluctuations. The scaling exponent of effective viscosity is calculated and agrees with expected value 4/3.

Influence of hydrodynamic fluctuations on the phase transition in the E and F models of critical dynamics

We use the renormalization group method to study the E model of critical dynamics in the presence of velocity fluctuations arising in accordance with the stochastic Navier-Stokes equation. Using the Martin-Siggia-Rose theorem, we obtain a field theory model that allows a perturbative renormalization group analysis. By direct power counting and an analysis of ultraviolet divergences, we show that the model is multiplicatively renormalizable, and we use a two-parameter expansion in ∈ and δ to calculate the renormalization constants. Here, ∈ is the deviation from the critical dimension four, and δ is the deviation from the Kolmogorov regime. We present the results of the one-loop approximation and part of the fixedpoint structure. We briefly discuss the possible effect of velocity fluctuations on the arge-scale behavior of the model.

Study of temperature Green’s functions of graphene-like systems in a half-space

We consider the formalism of temperature Green’s functions to study the electronic properties of a semiinfinite two-dimensional graphene lattice at a given temperature. Under most general assumptions about the graphene boundary structure, we calculate the propagator in the corresponding diagram technique. The obtained propagator survives limit transitions between physically different states of the system boundary, i.e., a zig-zag edge and a boundary condition in the “infinite mass” approximation, and also correctly describes the problem where the electron–hole symmetry is violated because of the presence of an external potential applied to the graphene boundary. We illustrate the use of the propagator, its analytic properties, and specific features of calculating with it in the example of determining the dependence of the electron density on the distance to the system boundary.