arXiv subjects
Publications and source records attributed to Alexander Moroz.
The applicability of the so-called isotropic and anisotropic complete photonic-band-gap (CPBG) models [S. John and J. Wang, Phys. Rev. Lett. {\bf 64}, 2418 (1990)] to capture essential features of the spontaneous emission (SE) of a fluorescent atom or molecule near a band-gap-edge of a CPBG structure is discussed.
The width $\triangle_L$ of the first Bragg's scattering peak in the (111) direction of a face-centered-cubic lattice of air spheres can be well approximated by a simple formula which only involves the volume averaged $ε$ and $ε^2$ over the lattice unit cell, $ε$ being the (position dependent) dielectric constant of the medium, and the effective dielectric constant $ε_{eff}$ in the long-wavelength limit approximated by Maxwell-Garnett's formula. Apparently, our formula describes the asymptotic behaviour of the absolute gap width $\triangle_L$ for high dielectric contrast $δ$ exactly. The standard deviation $σ$ steadily decreases well below 1% as $δ$ increases. For example $σ< 0.1%$ for the sphere filling fraction $f=0.2$ and $δ\geq 20$. On the interval $δ\in(1,100)$, our formula still approximates the absolute gap width $\triangle_L$ (the relative gap width $\triangle_L^r$) with a reasonable precision, namely with a standard deviation 3% (4.2%) for low filling fractions up to 6.5% (8%) for the close-packed case. Differences between the case of air spheres in a dielectric and dielectric spheres in air are briefly discussed.
We show that the violation of rotational symmetry for differential cross section for Aharonov-Bohm effect with nonstandard boundary conditions has been known for some time. Moreover, the results were applied to discuss the Hall effect and persistent currents of fermions in a plane pierced by a flux tube.
For the case of a simple face-centered-cubic photonic crystal of homogeneous dielectric spheres, we examine to what extent single-sphere Mie resonance frequencies are related to band gaps and whether the width of a gap can be enlarged due to nearby resonances. Contrary to some suggestions, no spectacular effects may be expected. When the dielectric constant of the spheres $ε_s$ is greater than the dielectric constant $ε_b$ of the background medium, then for any filling fraction $f$ there exists a critical $ε_c$ above which the lowest lying Mie resonance frequency falls inside the lowest stop gap in the (111) crystal direction, close to its midgap frequency. If $ε_s <ε_b$, the correspondence between Mie resonances and both the (111) stop gap and a full gap does not follow such a regular pattern. If the Mie resonance frequency is close to a gap edge, one can observe a resonance-induced widening of a relative gap width by $\approx 5%$.
We show that the photonic analogue of the Korringa-Kohn-Rostocker method is a viable alternative to the plane-wave method to analyze the spectrum of electromagnetic waves in a three-dimensional periodic dielectric lattice. Firstly, in the case of an fcc lattice of homogeneous dielectric spheres, we reproduce the main features of the spectrum obtained by the plane wave method, namely that for a sufficiently high dielectric contrast a full gap opens in the spectrum between the eights and ninth bands if the dielectric constant $ε_s$ of spheres is lower than the dielectric constant $ε_b$ of the background medium. If $ε_s> ε_b$, no gap is found in the spectrum. The maximal value of the relative band-gap width approaches 14% in the close-packed case and decreases monotonically as the filling fraction decreases. The lowest dielectric contrast $ε_b/ε_s$ for which a full gap opens in the spectrum is determined to be 8.13. Eventually, in the case of an fcc lattice of coated spheres, we demonstrate that a suitable coating can enhance gap widths by as much as 50%.
The transition operator T for the scattering of a particle from N potentials V_j can be expanded into a series featuring the transition operators t_j associated with the individual potentials. For V_j(x) both absolutely and square integrable in x, we show, using an analytic continuation argument, that if T is on-shell, i.e. in < k|T(sigma^{2}+ i0)|k' >, |k|=|k'|=σ, then each t_j is also on-shell.
Characteristics of the initial condensate in the recent experiment on Bose-Einstein condensation (BEC) of ${}^{87}$Rb atoms in an anisotropic magnetic trap is discussed. Given the aspect ratio $R$, the quality of BEC is estimated. A simple analytical Ansatz for the initial condensate wave function is proposed as a function of the aspect ratio which, in contrast to the Baym-Pethick trial wave function, reproduces both the weak and the strong intaraction limits and which is in better agreement with numerical results than the latter.
Using unitary equivalence of magnetic translation operators, explicit upper and lower convex bounds on the partition function of the Hofstadter model are given for any rational ``flux" and any value of Bloch momenta. These bounds (i) generalize straightforwardly to the case of a general asymmetric hopping and to the case of hopping of the form $t_{jn}(S_j^n+S_j^{-n})$ with $n$ arbitrary integer larger than or equal $2$, and (ii) allow to derive bounds on the derivatives of the partition function.
The Dirac Hamiltonian with the Aharonov-Bohm potential provides an example of a non-Fredholm operator for which all spectral asymmetry comes entirely from the continuous spectrum. In this case one finds that the use of standard definitions of the resolvent regularized, the heat kernel regularized, and the Witten indices misses the contribution coming from the continuous spectrum and gives vanishing spectral asymmetry and axial anomaly. This behaviour in the case of the continuous spectrum seems to be general and its origin is discussed.
A generic physical situation is considered where Im $Π$, the imaginary part of polarization operator (generalized susceptibility), can be measured on a finite interval and the high frequency asymptotics (up to a few orders) of $Π$ can be calculated theoretically. In such a case, it is desirable to derive an equivalent form of the Kramers-Kronig dispersion relation, the so-called sum rule, in which both the high-frequency part of Im $Π$ in the dispersion integral and the high-order contribution to $Π$ are suppressed. We provide a general framework for derivation of such sum rules, without any recourse to an infinite-order differential operator. We derive sum rules for a wide set of weight functions and show that any departure from the $e^{-t}$ behaviour of the weight function in sum rules leads to modulating factors on the theoretical side of sum rules, providing its low frequency regularization. We argue that by including modulating factors one can extend the domain of validity of sum rules further to an intermediate region of frequencies and can account for ``bumps" which were observed numerically on the phenomenological side of sum rules at ``intermediate'' frequencies.
A new summability method was tested to calculate the critical exponent $ν$ of the localization length for the symplectic case derived from the non-linear $σ$-model. Although we used the same series as Hikami and others, unlike them we were able to resum the series in two-dimensions (2D) and obtain the result $ν\sim 1$. Values of $ν$ in $2+\varepsilon$ dimensions seem to saturate the Harris inequality up to $\varepsilon=0.2$.
Both the nonrelativistic scattering and the spectrum in the presence of the Aharonov-Bohm potential are analyzed. The single-particle density of states (DOS) for different self-adjoint extensions is calculated. The DOS provides a link between different physical quantities and is a natural starting point for their calculation. The consequences of an asymmetry of the S matrix for the generic self-adjoint extension are examined. I. Introduction II. Impenetrable flux tube and the density of states III. Penetrable flux tube and self-adjoint extensions IV. The S matrix and scattering cross sections V. The Krein-Friedel formula and the resonance VI. Regularization VII. The R --> 0 limit and the interpretation of self-adjoint extensions VIII. Energy calculations IX. The Hall effect in the dilute vortex limit X. Persistent current of free electrons in the plane pierced by a flux tube XI. The 2nd virial coefficient of nonrelativistic interacting anyons XII. Discussion of the results and open questions
The single-particle densitity of states (DOS) for the Pauli and the Schrödinger Hamiltonians in the presence of an Aharonov-Bohm potential is calculated for different values of the particle magnetic moment. The DOS is a symmetric and periodic function of the flux. The Krein-Friedel formula can be applied to this long-ranged potential when regularized with the zeta function. We have found that whenever a bound state is present in the spectrum it is always accompanied by a resonance. The shape of the resonance is not of the Breit-Wigner type. The differential scattering cross section is asymmetric if a bound state is present and gives rise to the Hall effect. As an application, propagation of electrons in a dilute vortex limit is considered and the Hall resistivity is calculated.
The spectral properties of the Dirac Hamiltonian in the the Aharonov-Bohm potential are discussed. By using the Krein-Friedel formula, the density of states (DOS) for different self-adjoint extensions is calculated. As in the nonrelativistic case, whenever a bound state is present in the spectrum it is always accompanied by a (anti)resonance at the energy. The Aharonov-Casher theorem must be corrected for singular field configurations. There are no zero (threshold) modes in the Aharonov-Bohm potential. For our choice of the 2d Dirac Hamiltonian, the phase-shift flip is shown to occur at only positive energies. This flip gives rise to a surplus of the DOS at the lower threshold coming entirely from the continuous part of the spectrum. The results are applied to several physical quantities: the total energy, induced fermion-number, and the axial anomaly.
The density of states for a finite or an infinite cluster of scatterers in the case of both electrons and photons can be represented in a general form as the sum over all Krein-Friedel contributions of individual scatterers and a contribution due to the presence of multiple scatterers. The latter is given by the sum over all periodic orbits between different scatterers. General three dimensional multiple-scattering theory for electromagnetic waves in the presence of scatterers of arbitrary shape is presented. Vector structure constants are calculated and general rules for obtaining them from known scalar structure constants are given. The analog of the Korringa-Kohn-Rostocker equations for photons is explicitly written down. PACS numbers: 41.20.Jb, 41.20.Bt, 05.40.+j, 05.45.+b
Predictions of Akkermans et al. are essentially changed when the Krein spectral displacement operator is regularized by means of zeta function. Instead of piecewise constant persistent current of free electrons on the plane one has a current which varies linearly with the flux and is antisymmetric with regard to all time preserving values of $α$ including $1/2$. Different self-adjoint extensions of the problem and role of the resonance are discussed.
In the case of electromagnetic waves it is necessary to distinguish between inward and outward on-shell integral equations. Both kinds of equation are derived. A correct implementation of the photonic KKR method then requires the inward equations and it follows directly from them. A derivation of the KKR method from a variational principle is also outlined. Rather surprisingly, the variational KKR method cannot be entirely written in terms of surface integrals unless permeabilities are piecewise constant. Both kinds of photonic KKR method use the standard structure constants of the electronic KKR method and hence allow for a direct numerical application. As a by-product, matching rules are obtained for derivatives of fields on different sides of the discontinuity of permeabilities. Key words: The Maxwell equations, photonic band gap calculations
Recently it has been suggested by A. M. Tsvelik that quantum S=1/2 antiferromagnet can be described by the Majorana fermions in an irreducible way and without any constraint. In contrast to this claim we shall show that this representation is highly reducible. It is a direct sum of four irreducible fundamental representations of $su(2)$ algebra.