On the classification of discrete Hirota-type equations in 3D

In the series of recent publications we have proposed a novel approach to the classification of integrable differential/difference equations in 3D based on the requirement that hydrodynamic reductions of the corresponding dispersionless limits are `inherited' by the dispersive equations. In this paper we extend this to the fully discrete case. Our only constraint is that the initial ansatz possesses a non-degenerate dispersionless limit (this is the case for all known Hirota-type equations). Based on the method of deformations of hydrodynamic reductions, we classify discrete 3D integrable Hirota-type equations within various particularly interesting subclasses. Our method can be viewed as an alternative to the conventional multi-dimensional consistency approach.Comment: 29 page

Anomalous scaling in two and three dimensions for a passive vector field advected by a turbulent flow

A model of the passive vector field advected by the uncorrelated in time Gaussian velocity with power-like covariance is studied by means of the renormalization group and the operator product expansion. The structure functions of the admixture demonstrate essential power-like dependence on the external scale in the inertial range (the case of an anomalous scaling). The method of finding of independent tensor invariants in the cases of two and three dimensions is proposed to eliminate linear dependencies between the operators entering into the operator product expansions of the structure functions. The constructed operator bases, which include the powers of the dissipation operator and the enstrophy operator, provide the possibility to calculate the exponents of the anomalous scaling.Comment: 9 pages, LaTeX2e(iopart.sty), submitted to J. Phys. A: Math. Ge

Superconformal spaces and implications for superstrings

We clarify some properties of projective superspace by using a manifestly superconformal notation. In particular, we analyze the N=2 scalar multiplet in detail, including its action, and the propagator and its super-Schwinger parameters. The internal symmetry is taken to be noncompact (after Wick rotation), allowing boundary conditions that preserve it off shell. Generalization to N=4 suggests the coset superspace PSU(2,2|4)/OSp(4|4) for the AdS/CFT superstring.Comment: 19 pages, no figures; v2: fixed sign, added note & reference; v3: added note & references, version to appear in Physical Review

Temperature-dependent Drude transport in a two-dimensional electron gas

We consider transport of dilute two-dimensional electrons, with temperature between Fermi and Debye temperatures. In this regime, electrons form a nondegenerate plasma with mobility limited by potential disorder. Different kinds of impurities contribute unique signatures to the resulting temperature-dependent Drude conductivity, via energy-dependent scattering. This opens up a way to characterize sample disorder composition. In particular, neutral impurities cause a slow decrease in conductivity with temperature, whereas charged impurities result in conductivity growing as a square root of temperature. This observation serves as a precaution for literally interpreting metallic or insulating conductivity dependence, as both can be found in a classical metallic system.Comment: 5 pages, 2 figures, published versio

On the classification of scalar evolutionary integrable equations in 2+12+1 dimensions

We consider evolutionary equations of the form $u_t=F(u, w)$ where $w=D_x^{-1}D_yu$ is the nonlocality, and the right hand side $F$ is polynomial in the derivatives of $u$ and $w$. The recent paper \cite{FMN} provides a complete list of integrable third order equations of this kind. Here we extend the classification to fifth order equations. Besides the known examples of Kadomtsev-Petviashvili (KP), Veselov-Novikov (VN) and Harry Dym (HD) equations, as well as fifth order analogues and modifications thereof, our list contains a number of equations which are apparently new. We conjecture that our examples exhaust the list of scalar polynomial integrable equations with the nonlocality $w$. The classification procedure consists of two steps. First, we classify quasilinear systems which may (potentially) occur as dispersionless limits of integrable scalar evolutionary equations. After that we reconstruct dispersive terms based on the requirement of the inheritance of hydrodynamic reductions of the dispersionless limit by the full dispersive equation

Energy Anomaly and Polarizability of Carbon Nanotubes

The energy of electron Fermi sea perturbed by external potential, represented as energy anomaly which accounts for the contribution of the deep-lying states, is analyzed for massive d = 1+1 Dirac fermions on a circle. The anomaly is a universal function of the applied field, and is related to known field-theoretic anomalies. We express transverse polarizability of Carbon nanotubes via the anomaly, in a way which exhibits the universality and scale-invariance of the response dominated by pi-electrons and qualitatively different from that of dielectric and conducting shells. Electron band transformation in a strong-field effect regime is predicted.Comment: 4 pg

London's limit for the lattice superconductor

A stability problem for the current state of the strong coupling superconductor has been considered within the lattice Ginzburg-Landau model. The critical current problem for a thin superconductor film is solved within the London limit taking into account the crystal lattice symmetry. The current dependence on the order parameter modulus is computed for the superconductor film for various coupling parameter magnitudes. The field penetration problem is shown to be described in this case by the one-dimensional sine-Gordon equation. The field distribution around the vortex is described at the same time by the two-dimensional elliptic sine-Gordon equation.Comment: 7 pages, 3 figures, Revtex4, mostly technical correction; extended abstrac

Explicitly solvable cases of one-dimensional quantum chaos

We identify a set of quantum graphs with unique and precisely defined spectral properties called {\it regular quantum graphs}. Although chaotic in their classical limit with positive topological entropy, regular quantum graphs are explicitly solvable. The proof is constructive: we present exact periodic orbit expansions for individual energy levels, thus obtaining an analytical solution for the spectrum of regular quantum graphs that is complete, explicit and exact