The projective translation equation and unramified 2-dimensional flows with rational vector fields

Let X=(x,y). Previously we have found all rational solutions of the 2-dimensional projective translation equation, or PrTE, (1-z)f(X)=f(f(Xz)(1-z)/z); here f(X)=(u(x,y),v(x,y)) is a pair of two (real or complex) functions. Solutions of this functional equation are called projective flows. A vector field of a rational flow is a pair of 2-homogenic rational functions. On the other hand, only special pairs of 2-homogenic rational functions give rise to rational flows. In this paper we are interested in all non-singular (satisfying the boundary condition) and unramified (without branching points, i.e. single-valued functions in C^2\{union of curves}) projective flows whose vector field is still rational. We prove that, up to conjugation with 1-homogenic birational plane transformation, these are of 6 types: 1) the identity flow; 2) one flow for each non-negative integer N - these flows are rational of level N; 3) the level 1 exponential flow, which is also conjugate to the level 1 tangent flow; 4) the level 3 flow expressable in terms of Dixonian (equianharmonic) elliptic functions; 5) the level 4 flow expressable in terms of lemniscatic elliptic functions; 6) the level 6 flow expressable in terms of Dixonian elliptic functions again. This reveals another aspect of the PrTE: in the latter four cases this equation is equivalent and provides a uniform framework to addition formulas for exponential, tangent, or special elliptic functions (also addition formulas for polynomials and the logarithm, though the latter appears only in branched flows). Moreover, the PrTE turns out to have a connection with Polya-Eggenberger urn models. Another purpose of this study is expository, and we provide the list of open problems and directions in the theory of PrTE; for example, we define the notion of quasi-rational projective flows which includes curves of arbitrary genus.Comment: 34 pages, 2 figure

Band-edge problem in the theoretical determination of defect energy levels: the O vacancy in ZnO as a benchmark case

Calculations of formation energies and charge transition levels of defects routinely rely on density functional theory (DFT) for describing the electronic structure. Since bulk band gaps of semiconductors and insulators are not well described in semilocal approximations to DFT, band-gap correction schemes or advanced theoretical models which properly describe band gaps need to be employed. However, it has become apparent that different methods that reproduce the experimental band gap can yield substantially different results regarding charge transition levels of point defects. We investigate this problem in the case of the (+2/0) charge transition level of the O vacancy in ZnO, which has attracted considerable attention as a benchmark case. For this purpose, we first perform calculations based on non-screened hybrid density functionals, and then compare our results with those of other methods. While our results agree very well with those obtained with screened hybrid functionals, they are strikingly different compared to those obtained with other band-gap corrected schemes. Nevertheless, we show that all the different methods agree well with each other and with our calculations when a suitable alignment procedure is adopted. The proposed procedure consists in aligning the electron band structure through an external potential, such as the vacuum level. When the electron densities are well reproduced, this procedure is equivalent to an alignment through the average electrostatic potential in a calculation subject to periodic boundary conditions. We stress that, in order to give accurate defect levels, a theoretical scheme is required to yield not only band gaps in agreement with experiment, but also band edges correctly positioned with respect to such a reference potential

Asymptotic formula for the moments of Minkowski question mark function in the interval [0,1]

In this paper we prove the asymptotic formula for the moments of Minkowski question mark function, which describes the distribution of rationals in the Farey tree. The main idea is to demonstrate that certain a variation of a Laplace method is applicable in this problem, hence the task reduces to a number of technical calculations.Comment: 11 pages, 1 figure (final version). Lithuanian Math. J. (to appear

Optical Signatures of Quantum Emitters in Suspended Hexagonal Boron Nitride

Hexagonal boron nitride (h-BN) is a tantalizing material for solid-state quantum engineering. Analogously to three-dimensional wide-bandgap semiconductors like diamond, h-BN hosts isolated defects exhibiting visible fluorescence, and the ability to position such quantum emitters within a two-dimensional material promises breakthrough advances in quantum sensing, photonics, and other quantum technologies. Critical to such applications, however, is an understanding of the physics underlying h-BN's quantum emission. We report the creation and characterization of visible single-photon sources in suspended, single-crystal, h-BN films. The emitters are bright and stable over timescales of several months in ambient conditions. With substrate interactions eliminated, we study the spectral, temporal, and spatial characteristics of the defects' optical emission, which offer several clues about their electronic and chemical structure. Analysis of the defects' spectra reveals similarities in vibronic coupling despite widely-varying fluorescence wavelengths, and a statistical analysis of their polarized emission patterns indicates a correlation between the optical dipole orientations of some defects and the primitive crystallographic axes of the single-crystal h-BN film. These measurements constrain possible defect models, and, moreover, suggest that several classes of emitters can exist simultaneously in free-standing h-BN, whether they be different defects, different charge states of the same defect, or the result of strong local perturbations

Measurement and Control of Single Nitrogen-Vacancy Center Spins above 600 K

We study the spin and orbital dynamics of single nitrogen-vacancy (NV) centers in diamond between room temperature and 700 K. We find that the ability to optically address and coherently control single spins above room temperature is limited by nonradiative processes that quench the NV center's fluorescence-based spin readout between 550 and 700 K. Combined with electronic structure calculations, our measurements indicate that the energy difference between the 3E and 1A1 electronic states is approximately 0.8 eV. We also demonstrate that the inhomogeneous spin lifetime (T2*) is temperature independent up to at least 625 K, suggesting that single NV centers could be applied as nanoscale thermometers over a broad temperature range.Comment: 8 pages, 5 figures, and 14 pages of supplemental material with additional figures. Title change and minor revisions from previous version. DMT and DJC contributed equally to this wor

Multi-variable translation equation which arises from homothety

In many regular cases, there exists a (properly defined) limit of iterations of a function in several real variables, and this limit satisfies the functional equation (1-z)f(x)=f(f(xz)(1-z)/z); here z is a scalar and x is a vector. This is a special case of a well-known translation equation. In this paper we present a complete solution to this functional equation in case f is a continuous function on a single point compactification of a 2-dimensional real vector space. It appears that, up to conjugation by a homogeneous continuous function, there are exactly four solutions. Further, in a 1-dimensional case we present a solution with no regularity assumptions on f.Comment: 15 page