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E. Kogan

Publications and source records attributed to E. Kogan.

At least 19 recordsLinked to original sources
arXiv2026

A short proof of the Patak-Tancer theorem on non-embeddability of $k$-complexes in $2k$-manifolds

In 2019 P. Patak and M. Tancer obtained the following higher-dimensional generalization of the Heawood inequality on embeddings of graphs into surfaces. We present a short well-structured proof accessible to non-specialists in the field. Let $Δ_n^k$ be the union of $k$-dimensional faces of the $n$-dimensional simplex. Theorem. (a) If $Δ_n^k$ PL embeds into the connected sum of $g$ copies of the Cartesian product $S^k\times S^k$ of two $k$-dimensional spheres, then $g\ge\dfrac{n-2k-1}{k+2}$. (b) If $Δ_n^k$ PL embeds into a closed $(k-1)$-connected PL $2k$-manifold $M$, then $(-1)^k(χ(M)-2)\ge\dfrac{n-2k-1}{k+1}$.

math.GT↗
arXiv2021

Linking of three triangles in 3-space

Two triples of triangles having pairwise disjoint outlines in 3-space are called combinatorially isotopic if one triple can be obtained from the other by a continuous motion during which the outlines of the triangles remain pairwise disjoint. We conjecture that it can be algorithmically checked if an (ordered or unordered) triple of triangles is combinatorially isotopic to a triple of triangles having pairwise disjoint convex hulls. We also conjecture that any unordered triple of pairwise disjoint triangles in 3-space belongs to one of the 5 types of such triples listed in the paper. We present an elementary proof that triples of different types are not combinatorially isotopic.

math.GT↗
arXiv2019

Poor man's scaling and Lie algebras

We consider a general model, describing a quantum impurity with degenerate energy levels, interacting with a gas of itinerant electrons, derive general scaling equation for the model, and analyse the connection between its particular forms and the symmetry of interaction. On the basis of this analysis we write down scaling equations for the Hamiltonians which are the direct products of $su(3)$ Lie algebras and have either $SU(2)\times U(1)$ or $SU(2)$ symmetry. We also put into a new context anisotropic Coqblin -- Schrieffer models proposed by us earlier.

cond-mat.mes-hall↗
arXiv2019

Irradiation-induced broadening of the Raman spectra in monolayer graphene

Broadening of the Raman scattering (RS) spectra was studied in monolayer graphene samples irradiated with various dose of ions followed by annealing of radiation damage at different temperatures. It is shown that the width Γ (full width at half maximum, FWHM) of three main RS lines (G-, D-, and 2D) increases linearly with increase of the density of irradiation-induced point defects N d as ΔΓ = m N d . The slope m of the linear dependencies is the same for one-phonon emitting G-line and D-line, and almost double for two-phonon emitting 2D-line. It is also shown that the width of D-line Γ D for all samples is larger than one half of the width of 2D-line Γ 2D , which shows that in the case of D-line, elastic electron scattering on point defects leads to an additional decreasing the lifetime of the emitted phonon. Theoretical model of the width of D-line in disordered graphene is developed which explains the experimental observations and allows to determine the numerical coefficient in the in-plane transverse optic phonon dispersion in graphene.

cond-mat.mes-hall↗
arXiv2019

Irradiation-induced metal-insulator transition in monolayer graphene

A brief review of experiments directed to study a gradual localization of charge carriers and metal-insulator transition in samples of disordered monolayer graphene is presented. Disorder was induced by irradiation with different doses of heavy and light ions. Degree of disorder was controlled by measurements of the Raman scattering spectra. The temperature dependences of conductivity and magnetoresistance (MR) showed that at low disorder, conductivity is governed by the weak localization and antilocalization regime. Further increase of disorder leads to strong localization of charge carriers, when the conductivity is described by the variable-range-hopping (VRH) mechanism. It was observed that MR in the VRH regime is negative in perpendicular fields and is positive in parallel magnetic fields which allowed to reveal different mechanisms of hopping MR. Theoretical analysis is in a good agreement with experimental data.

cond-mat.mes-hall↗
arXiv2019

RKKY interaction in graphene at finite temperature

In our publication from 8 years ago (Phys. Rev. B {\bf 84}, 115119 (2011)) we calculated RKKY interaction between two magnetic impurities adsorbed on graphene at zero temperature. We show in this short paper that the approach based on Matsubara formalism and perturbation theory for the thermodynamic potential in the imaginary time and coordinate representation which was used then, can be easily generalized, and calculate RKKY interaction between the magnetic impurities at finite temperature.

cond-mat.mes-hall↗
arXiv2019

Decay of discrete state resonantly coupled to a continuum of finite width

A simple quantum mechanical model consisting of a discrete level resonantly coupled to a continuum of finite width, where the coupling can be varied from perturbative to strong (Fano-Anderson model), is considered. The particle is initially localized at the discrete level, and the time dependence of the amplitude to find the particle at the discrete level is calculated without resorting to perturbation theory. The deviations from the exponential decay law, predicted by the Fermi's Golden Rule, are discussed. We also study analytic structure of the Green's function (GF) for the model. We analyze the GF poles, branch points and Riemann surface, and show how the Fermi's Golden Rule, valid in perturbative regime for not to large time, appears in this context. The knowledge of analytic structure of the GF in frequency representation opens opportunities for obtaining easy for numerical calculations formulas for the GF in time representation, alternative to those using the spectral density.

quant-ph↗
arXiv2018

Poor man's scaling: anisotropic Kondo and Coqblin--Schrieffer models

We discuss Kondo effect for a general model, describing a quantum impurity with degenerate energy levels, interacting with a gas of itinerant electrons, and derive scaling equation to the second order for such a model. We show how the scaling equation for the spin-anisotropic Kondo model with the power law density of states (DOS) for itinerant electrons follows from the general scaling equation. We introduce the anisotropic Coqblin--Schrieffer model, apply the general method to derive scaling equation for that model for the power law DOS, and integrate the derived equation analytically.

cond-mat.mes-hall↗
arXiv2017

Electronic structure of graphene: (nearly) free electrons bands vs. tight-binding bands

In our previous paper (Phys. Rev. B {\bf 89}, 165430 (2014)) we have found that in graphene, in distinction to the four occupied bands, which can be described by the simple tight-binding model (TBM) with four atomic orbitals per atom, the two lowest lying at the $Γ$-point unoccupied bands (one of them of a $σ$ type and the other of a $π$ type) can not be described by such model. In the present work we suggest a minimalistic model for these two bands, based on (nearly) free electrons model (FEM), which correctly describes the symmetry of these bands, their dispersion law and their localization with respect to the graphene plane.

cond-mat.mes-hall↗
arXiv2016

Influence of ageing on Raman spectra and the conductivity of monolayer graphene samples irradiated by heavy and light ions

The influence of long-term ageing (about one year) on the Raman scattering (RS) spectra and the temperature dependence of conductivity has been studied in two series of monolayer graphene samples irradiated by different doses of C$^{+}$ and Xe$^{+}$ ions. It is shown that the main result of ageing consists of changes in the intensity and position of D- and G- and 2D-lines in RS spectra and in an increase of the conductivity. The observed effects are explained in terms of an increase of the radius of the \textquotedblleft activated\textquotedblright{} area around structural defects.

cond-mat.mes-hall↗
arXiv2016

Effect of annealing on Raman scattering spectra of monolayer graphene samples gradually disordered by ion irradiation

The Raman scattering spectra (RS) of two series of monolayer graphene samples irradiated with various doses of C$^{+}$ and Xe$^{+}$ ions were measured after annealing in high vacuum, and in forming gas (95\%Ar+5\%H$_{2}$). It was found that these methods of annealing have dramatically different influence on the RS lines. Annealing in vacuum below 500$^{\circ}$C leads to significant decrease of both D-line, associated with defects, and 2D-line, associated with the intact lattice structure, which can be explained by annealing-induced enhanced doping. Further annealing in vacuum up to 1000$^{\circ}$C leads to significant increase of 2D-line together with continuous decrease of D-line, which gives evidence of partial removal of defects and recovery of the damaged lattice. Annealing in forming gas is less effective in this sense. The blue shift of all lines is observed after annealing. It is shown that below 500$^{\circ}$C, the unintentional doping is the main mechanism of shift, while at higher annealing temperatures, the lattice strain dominates due to mismatch of the thermal expansion coefficient of graphene and the SiO$_{2}$ substrate. Inhomogeneous distribution of stress and doping across the samples leads to the correlated variation of the amplitude and the peak position of RS lines.

cond-mat.mes-hall↗
arXiv2016

Replica symmetry breaking for anisotropic magnets with quenched disorder

We study critical behaviour of a magnet with cubic anisotropy and quenched scalar disorder which is taken into account by replica method. We derive to first order in $ε$ approximation the renormalization group equations taking into account possible replica symmetry breaking. We study the stability of the replica symmetric fixed points with respect to perturbations without (in general case) replica symmetry. However, we find that if a fixed point is stable with respect to replica symmetric deviations, it is also stable with respect to deviations without replica symmetry.

cond-mat.stat-mech↗
arXiv2016

Lift force due to odd (Hall) viscosity

We study the problem of flow of a neutral gas past an infinite cylinder at right angle to its axis at low Reynolds number when the fluid is characterized by broken time-reversal invariance, and hence by odd viscosity in addition to the normal even one. We solve the Oseen approximation to Navier-Stokes equation and calculate the lift force which appears due to the odd viscosity.

physics.flu-dyn↗
arXiv2016

Critical behaviour of anisotropic magnets with quenched disorder: replica symmetry breaking studied by operator product expansion

We study critical behaviour of disordered magnets near four dimensions. We consider the system with explicit cubic anisotropy and scalar disorder and that with random direction of anisotropy axis. The quenched disorder is taken into account by replica method. Using the method of operator product expansion, we derive in the first order to $ε$ approximation the renormalization group equations taking into account possible replica symmetry breaking.

cond-mat.stat-mech↗
arXiv2015

Hopping magnetoresistance in ion irradiated monolayer graphene

Magnetoresistance (MR) of ion irradiated monolayer graphene samples with variable-range hopping (VRH) mechanism of conductivity was measured at temperatures down to $T = 1.8$ K in magnetic fields up to $B = 8$ T. It was observed that in perpendicular magnetic fields, hopping resistivity $R$ decreases, which corresponds to negative MR (NMR), while parallel magnetic field results in positive MR (PMR) at low temperatures. NMR is explained on the basis of the "orbital" model in which perpendicular magnetic field suppresses the destructive interference of many paths through the intermediate sites in the total probability of the long-distance tunneling in the VRH regime. At low fields, a quadratic dependence ($|ΔR/R|\sim B^2$) of NMR is observed, while at $B > B^*$, the quadratic dependence is replaced by the linear one. It was found that all NMR curves for different samples and different temperatures could be merged into common dependence when plotted as a function of $B/B^*$. It is shown that $B^*\sim T^{1/2}$ in agreement with predictions of the "orbital" model. The obtained values of $B^*$ allowed also to estimate the localization radius $ξ$ of charge carriers for samples with different degree of disorder. PMR in parallel magnetic fields is explained by suppression of hopping transitions via double occupied states due to alignment of electron spins.

cond-mat.mes-hall↗
arXiv2015

Localization of charge carriers in monolayer graphene gradually disordered by ion irradiation

Gradual localization of charge carriers was studied in a series of micro-size samples of monolayer graphene fabricated on the common large scale film and irradiated by different doses of C$^+$ ions with energy 35 keV. Measurements of the temperature dependence of conductivity and magnetoresistance in fields up to 4 T showed that at low disorder, the samples are in the regime of weak localization and antilocalization. Further increase of disorder leads to strong localization regime, when conductivity is described by the variable-range-hopping (VRH) mechanism. A crossover from the Mott regime to the Efros-Shklovskii regime of VRH is observed with decreasing temperature. Theoretical analysis of conductivity in both regimes showed a remarkably good agreement with experimental data.

cond-mat.mes-hall↗
arXiv2015

Raman scattering and electrical resistance of highly disordered graphene

Raman scattering (RS) spectra and current-voltage characteristics at room temperature were measured in six series of small samples fabricated by means of electron-beam lithography on the surface of a large size (5x5 mm) industrial monolayer graphene film. Samples were irradiated by different doses of C${}^+$ ion beam up to $10^{15}$ cm${}^{-2}$. It was observed that at the utmost degree of disorder, the Raman spectra lines disappear which is accompanied by the exponential increase of resistance and change in the current-voltage characteristics.These effects are explained by suggestion that highly disordered graphene film ceases to be a continuous and splits into separate fragments. The relationship between structure (intensity of RS lines) and sample resistance is defined. It is shown that the maximal resistance of the continuous film is of order of reciprocal value of the minimal graphene conductivity $πh/4e^2\approx 20$ kOhm.

cond-mat.mes-hall↗