Line intensity measurements of methane’s \nu _3-band using a cw-OPO

We report on absolute line strength measurements of P(1), R(0) and R(1) singlet lines in the \(3.3\,\upmu\hbox {m}\,\nu _{3}\) (C–H stretching) band of methane \(^{12}\hbox {CH}_4\) at reference temperature \(T=296\)  K. Line strength measurements are performed at low pressure \((P \le 1\hbox { Torr})\) using direct absorption spectroscopy technique based on a widely tunable continuous-wave singly resonant optical parametric oscillator. The \(1\sigma \) overall accuracy in line strength determinations ranges between 7 and 8 % mostly limited by pressure and frequency measurements. A comparison with previous reported values is made. Our results show good agreement with the HITRAN 2012 database.     

Antineutrino science in KamLAND

The primary goal of KamLAND is a search for the oscillation of \({\bar{\nu }}_\mathrm{e}\) ’s emitted from distant power reactors. The long baseline, typically 180 km, enables KamLAND to address the oscillation solution of the “solar neutrino problem” with \({\bar{\nu }}_{e} \) ’s under laboratory conditions. KamLAND found fewer reactor \({\bar{\nu }}_{e} \) events than expected from standard assumptions about \(\overline{\nu }_e\) propagation at more than 9 \(\sigma \) confidence level (C.L.). The observed energy spectrum disagrees with the expected spectral shape at more than 5 \(\sigma \) C.L., and prefers the distortion from neutrino oscillation effects. A three-flavor oscillation analysis of the data from KamLAND and KamLAND + solar neutrino experiments with CPT invariance, yields \(\Delta m_{21}^2 \) = [ \(7.54_{-0.18}^{+0.19} \times \) 10 \(^{-5}\) eV \(^{2}\) , \(7.53_{-0.18}^{+0.19} \times \) 10 \(^{-5}\) eV \(^{2}\) ], tan \(^{2}\theta _{12}\) = [ \(0.481_{-0.080}^{+0.092} \) , \(0.437_{-0.026}^{+0.029} \) ], and sin \(^{2}\theta _{13}\) = [ \(0.010_{-0.034}^{+0.033} \) , \(0.023_{-0.015}^{+0.015} \) ]. All solutions to the solar neutrino problem except for the large mixing angle region are excluded. KamLAND also demonstrated almost two cycles of the periodic feature expected from neutrino oscillation effects. KamLAND performed the first experimental study of antineutrinos from the Earth’s interior so-called geoneutrinos (geo \({\bar{\nu }}_{e} \) ’s), and succeeded in detecting geo \({\bar{\nu }}_{e} \) ’s produced by the decays of \(^{238}\) U and \(^{232}\) Th within the Earth. Assuming a chondritic Th/U mass ratio, we obtain \(116_{-27}^{+28} {\bar{\nu }}_{e}\) events from \(^{238}\) U and \(^{232}\) Th, corresponding a geo \({\bar{\nu }}_{e}\) flux of \(3.4_{-0.8}^{+0.8}\times \) 10 \(^{6}\) cm \(^{-2}\)  s \(^{-1}\) at the KamLAND location. We evaluate various bulk silicate Earth composition models using the observed geo \({\bar{\nu }}_{e} \) rate.     

Phase Transitions in Layered Systems

We consider the Ising model on \(\mathbb Z\times \mathbb Z\) where on each horizontal line \(\{(x,i), x\in \mathbb Z\}\) , called “layer”, the interaction is given by a ferromagnetic Kac potential with coupling strength \(J_{ \gamma }(x,y)={ \gamma }J({ \gamma }(x-y))\) , where \(J(\cdot )\) is smooth and has compact support; we then add a nearest neighbor ferromagnetic vertical interaction of strength \({ \gamma }^{A}\) , where \(A\ge 2\) is fixed, and prove that for any \(\beta \) larger than the mean field critical value there is a phase transition for all \({ \gamma }\) small enough.     

Investigation of conductivity of adhesive layer including indium tin oxide particles for multi-junction solar cells

We report connection conductivity ( \(C_{\rm c}\) ) of adhesive which including \(\hbox {In}_2\hbox {O}_3\) – \(\hbox {SnO}_2\) (ITO) particles developed for fabrication of stacked-type-multi-junction solar cells. The commercial 20- \(\upmu \) m sized ITO particles were heated in vacuum at temperature ranging from 800 to 1,300  \(^{\circ }{\rm C}\) for 10 min to increase \(C_{\rm c}\) . 6.2 wt% ITO particles were dispersed in commercial Cemedine adhesive gel to form 100 samples structured with n-type Si/adhesive/n-type Si (n-Si sample) and p-type Si/adhesive/p-type Si (p-Si sample). Current density as a function of voltage (J–V) characteristics gave \(C_{\rm c}\) . It ranged from 4.3 to 1.0 S/cm \(^2\) for the n-Si sample with 800 \(^{\circ }{\rm C}\) heat-treated ITO particles. Its standard deviation was 0.59 S/cm \(^2\) . On the other hand, it ranged from 2.0 to 0.6 S/cm \(^2\) for the p-Si sample with 800  \(^{\circ }{\rm C}\) heat-treated ITO particles. Its standard deviation was 0.22 S/cm \(^2\) . The distribution of \(C_{\rm c}\) mainly resulted from contact efficiency of ITO particles to substrate. We theoretically estimated that present \(C_{\rm c}\) achieved a low loss of the power conversion efficiency ( \(E_{\rm ff}\) ) lower than 0.3 % in the application of fabrication of multi-junction solar cell with an intrinsic \(E_{\rm ff}\) of 30 % and an open circuit voltage above 1.9 V.     

Optimal N-Point Configurations on the Sphere: “Magic” Numbers and Smale’s 7th Problem

This paper inquires into the concavity of the map \(N\mapsto v_s(N)\) from the integers \(N\ge 2\) into the minimal average standardized Riesz pair-energies \(v_s(N)\) of \(N\) -point configurations on the sphere \(\mathbb {S}^2\) for various \(s\in \mathbb {R}\) . The standardized Riesz pair-energy of a pair of points on \(\mathbb {S}^2\) a chordal distance \(r\) apart is \(V_s(r)= s^{-1}\left( r^{-s}-1 \right) \) , \(s \ne 0\) , which becomes \(V_0(r) = \ln \frac{1}{r}\) in the limit \(s\rightarrow 0\) . Averaging it over the \(\left( \begin{array}{c} N\\ 2\end{array}\right) \) distinct pairs in a configuration and minimizing over all possible \(N\) -point configurations defines \(v_s(N)\) . It is known that \(N\mapsto v_s(N)\) is strictly increasing for each \(s\in \mathbb {R}\) , and for \(s<2\) also bounded above, thus “overall concave.” It is (easily) proved that \(N\mapsto v_{-2}^{}(N)\) is even locally strictly concave, and that so is the map \(2n\mapsto v_s(2n)\) for \(s<-2\) . By analyzing computer-experimental data of putatively minimal average Riesz pair-energies \(v_s^x(N)\) for \(s\in \{-1,0,1,2,3\}\) and \(N\in \{2,\ldots ,200\}\) , it is found that the map \(N\mapsto {v}_{-1}^x(N)\) is locally strictly concave, while \(N\mapsto {v}_s^x(N)\) is not always locally strictly concave for \(s\in \{0,1,2,3\}\) : concavity defects occur whenever \(N\in {\mathcal {C}}^{x}_+(s)\) (an \(s\) -specific empirical set of integers). It is found that the empirical map \(s\mapsto {\mathcal {C}}^{x}_+(s),\ s\in \{-2,-1,0,1,2,3\}\) , is set-theoretically increasing; moreover, the percentage of odd numbers in \({\mathcal {C}}^{x}_+(s),\ s\in \{0,1,2,3\}\) is found to increase with \(s\) . The integers in \({\mathcal {C}}^{x}_+(0)\) are few and far between, forming a curious sequence of numbers, reminiscent of the “magic numbers” in nuclear physics. It is conjectured that these new “magic numbers” are associated with optimally symmetric optimal-log-energy \(N\) -point configurations on \(\mathbb {S}^2\) . A list of interesting open problems is extracted from the empirical findings, and some rigorous first steps toward their solutions are presented. It is emphasized how concavity can assist in the solution to Smale’s \(7\) th Problem, which asks for an efficient algorithm to find near-optimal \(N\) -point configurations on \(\mathbb {S}^2\) and higher-dimensional spheres.     

Split supersymmetry under GUT and current dark matter constraints

There are four types of two-Higgs doublet models under a discrete \(Z_2\) symmetry imposed to avoid tree-level flavor-changing neutral current, i.e. type-I, type-II, type-X, and type-Y models. We investigate the possibility to discriminate the four models in the light of the flavor physics data, including \(B_s\) – \(\bar{B}_s\) mixing, \(B_{s,d} \rightarrow \mu ^+ \mu ^-\) , \(B\rightarrow \tau \nu \) and \(\bar{B} \rightarrow X_s \gamma \) decays, the recent LHC Higgs data, the direct search for charged Higgs at LEP, and the constraints from perturbative unitarity and vacuum stability. After deriving the combined constraints on the Yukawa interaction parameters, we have shown that the correlation between the mass eigenstate rate asymmetry \(A_{\Delta \Gamma }\) of \(B_{s} \rightarrow \mu ^+ \mu ^-\) and the ratio \(R=\mathcal{B}(B_{s} \rightarrow \mu ^+ \mu ^-)_\mathrm{exp}/ \mathcal{B}(B_{s} \rightarrow \mu ^+ \mu ^-)_\mathrm{SM}\) could be a sensitive probe to discriminate the four models with future precise measurements of the observables in the \(B_{s} \rightarrow \mu ^+ \mu ^-\) decay at LHCb.     

Third-order nonlinear optical properties of an azobenzene-containing ionic liquid crystalline polymer

The nonlinear optical properties of an azobenzene-containing ionic liquid crystalline polymer were investigated by a single beam Z-scan technique employing 38 ps and 6 ns laser pulses at 532 nm. The polymer film exhibited large nonlinear refraction and negligible nonlinear absorption under both ps and ns pulses excitation. The nonlinear refractive indices of the polymer were in the order of \(10^{-11} \hbox {cm}^{2}/\hbox {W}\) under ps excitation and \(10^{-8}\hbox {cm}^{2}/\hbox {W}\) under ns excitation, respectively. The mechanism accounting for the process of nonlinear refraction was discussed in terms of photoinduced trans-cis transformation and thermal effect.     

Design of spectral crosstalk suppressing structure in two-color HgCdTe infrared focal plane arrays detector

Spectral crosstalk suppressing design of two-color HgCdTe medium-wave/long-wave (MW/LW) \(\hbox {n}^{+}\) – \(\hbox {p}_{1}\) – \(\hbox {P}_{2}\) – \(\hbox {P}_{3}\) – \(\hbox {N}^{+}\) infrared focal plane arrays (IRFPAs) detector functioning in simultaneous mode is carried out in this study, using Crosslight Technology Computer Aided Design (TCAD) software. A compositional barrier of \(\hbox {P}_{2}\) -region sandwiched between LW absorption layer of \(\hbox {p}_{1}\) -region and MW absorption layer of \(\hbox {P}_{3}\) -region is designed to suppress spectral crosstalk. MW-to-LW crosstalk can be significantly suppressed to 2.1 % while LW-to-MW crosstalk can be maintained less than 1 % by integrating an optimized compositional barrier.     

Preliminary studies of the Raman spectra of \hbox {Ag}_{2}\hbox {Te}\hbox { and }\hbox {Ag}_{5}\hbox {Te}_{3}

The theoretical calculations indicated that the monoclinic low-temperature phase of silver telluride $(\upbeta \hbox {-Ag}_{2}\hbox {Te})$ is a new binary topological insulator with highly anisotropic single Dirac cone surface. We obtained $\upbeta \hbox {-Ag}_{2}\hbox {Te}$ crystal ingots containing few grains by the Bridgman method. We also deposited thin films of tellurium, $\hbox {Ag}_{5}\hbox {Te}_{3}\hbox { and }(\hbox {Te+Ag}_{5}\hbox {Te}_{3})$ by thermal evaporation method. The Raman spectra of $\upbeta \hbox {-Ag}_{2}\hbox {Te}$ , tellurium and $\hbox {Ag}_{5}\hbox {Te}_{3}$ were measured at three excitation wave lengths: 633, 515 and 488 nm. The Raman active modes of $\upbeta \hbox {-Ag}_{2}\hbox {Te}$ , tellurium and $\hbox {Ag}_{5}\hbox {Te}_{3}$ are situated at frequencies below 300  $\hbox {cm}^{-1}$ while vibrations of other phases appear at higher frequencies.     

Is the CNMSSM more credible than the CMSSM?

“Post-sphaleron baryogenesis”, a fresh and profound mechanism of baryogenesis accounts for the matter–antimatter asymmetry of our present universe in a framework of Pati–Salam symmetry. We attempt here to embed this mechanism in a non-SUSY SO(10) grand unified theory by reviving a novel symmetry breaking chain with Pati–Salam symmetry as an intermediate symmetry breaking step and as well to address post-sphaleron baryogenesis and neutron–antineutron oscillation in a rational manner. The Pati–Salam symmetry based on the gauge group \(\mathrm{SU}(2)_L \times \mathrm{SU}(2)_{R} \times \mathrm{SU}(4)_C\) is realized in our model at \(10^{5}\) – \(10^{6}\)  GeV and the mixing time for the neutron–antineutron oscillation process having \(\Delta B=2\) is found to be \(\tau _{n-\bar{n}} \simeq 10^{8}\) – \(10^{10}\)  s with the model parameters, which is within the reach of forthcoming experiments. Other novel features of the model include low scale right-handed \(W^{\pm }_R\) , \(Z_R\) gauge bosons, explanation for neutrino oscillation data via the gauged inverse (or extended) seesaw mechanism and most importantly TeV scale color sextet scalar particles responsible for an observable \(n\) – \(\bar{n}\) oscillation which may be accessible to LHC. We also look after gauge coupling unification and an estimation of the proton lifetime with and without the addition of color sextet scalars.     

Modeling of HOT (111) HgCdTe MWIR detector for fast response operation

The paper reports on photoelectrical performance of the mid-wave infrared (MWIR) (111) HgCdTe high operating temperature detector for the fast response conditions. Detector structure was simulated with software APSYS by Crosslight Inc. The detailed analysis of the time response as a function of device architecture and applied voltage was performed pointing out optimal working conditions. The time response of the MWIR HgCdTe detector with 50 % cut-off wavelength of \(\lambda _{c} \approx 5.3\, \upmu \hbox {m}\) at \(T = 200\)  K was estimated at the level of \(\tau _{s} \approx \) 2,500 ps for \(V = 100\)  mV and series resistance \(R_{Series} = 510\,\Omega \) . The series resistance’s reduction enables to reach \(\tau _{s}\approx 60\!-\!500\)  ps.     

Functional Properties of Hörmander’s Space of Distributions Having a Specified Wavefront Set

The space \({\mathcal{D}_\Gamma^\prime}\) of distributions having their wavefront sets in a closed cone \({\Gamma}\) has become important in physics because of its role in the formulation of quantum field theory in curved spacetime. In this paper, the topological and bornological properties of \({\mathcal{D}_\Gamma^\prime}\) and its dual \({\mathcal{E}_\Lambda^\prime}\) are investigated. It is found that \({\mathcal{D}_\Gamma^\prime}\) is a nuclear, semi-reflexive and semi-Montel complete normal space of distributions. Its strong dual \({\mathcal{E}_\Lambda^\prime}\) is a nuclear, barrelled and (ultra)bornological normal space of distributions which, however, is not even sequentially complete. Concrete rules are given to determine whether a distribution belongs to \({\mathcal{D}_\Gamma^\prime}\) , whether a sequence converges in \({\mathcal{D}_\Gamma^\prime}\) and whether a set of distributions is bounded in \({\mathcal{D}_\Gamma^\prime}\) .     

Feasibility of a low-voltage discharge in pure molecular hydrogen

The feasibility of initiating a low-voltage discharge in pure (free of readily ionizable impurity) molecular hydrogen is studied theoretically. A discharge with cathode fall φ₁ = 10 V, interelectrode gap L = 2 cm, and total hydrogen concentration \(N_{H_2 }^{(0)} = 2 \times 10^{15}\) cm^?3 is considered by way of example. The plasma parameters, including concentration \(N_{H^ - }\) of negative hydrogen ions H^?, are calculated. The concentration of H^? ions is maximal in the near-anode plasma and may reach \(\left( {N_{H^ - } } \right)_{\max } = 0.5 \times 10^{12}\) cm^?3. Concentration \(N_{H^ - }\) can be increased severalfold by introducing a small amount of cesium into the discharge, \(N_{Cs}^{(0)} \leqslant 10^{13}\) cm^?3. Cesium ionizes completely and concentrates in narrow near-electrode layers, which are depleted with the plasma in the purely hydrogen discharge. The discharge modifications studied in this work may be of interest as low-voltage volume plasma sources of H^? ions under conditions when a high concentration of cesium in the source plasma is undesirable.     

Large N approach to Kaon decays and mixing 28 years later: \Delta I=1/2 rule, \hat{B}_K,and \Delta M_K

We review and update our results for $K\rightarrow \pi \pi $ decays and $K^0$ – $\bar{K}^0$ mixing obtained by us in the 1980s within an analytic approximate approach based on the dual representation of QCD as a theory of weakly interacting mesons for large $N$ , where $N$ is the number of colors. In our analytic approach the Standard Model dynamics behind the enhancement of $\hbox {Re}A_0$ and suppression of $\hbox {Re}A_2$ , the so-called $\Delta I=1/2$ rule for $K\rightarrow \pi \pi $ decays, has a simple structure: the usual octet enhancement through the long but slow quark–gluon renormalization group evolution down to the scales $\mathcal{O}(1\, {\hbox { GeV}})$ is continued as a short but fast meson evolution down to zero momentum scales at which the factorization of hadronic matrix elements is at work. The inclusion of lowest-lying vector meson contributions in addition to the pseudoscalar ones and of Wilson coefficients in a momentum scheme improves significantly the matching between quark–gluon and meson evolutions. In particular, the anomalous dimension matrix governing the meson evolution exhibits the structure of the known anomalous dimension matrix in the quark–gluon evolution. While this physical picture did not yet emerge from lattice simulations, the recent results on $\hbox {Re}A_2$ and $\hbox {Re}A_0$ from the RBC-UKQCD collaboration give support for its correctness. In particular, the signs of the two main contractions found numerically by these authors follow uniquely from our analytic approach. Though the current–current operators dominate the $\Delta I=1/2$ rule, working with matching scales $\mathcal{O}(1 \, {\hbox { GeV}})$ we find that the presence of QCD-penguin operator $Q_6$ is required to obtain satisfactory result for $\hbox {Re}A_0$ . At NLO in $1/N$ we obtain $R=\hbox {Re}A_0/\hbox {Re}A_2= 16.0\pm 1.5$ which amounts to an order of magnitude enhancement over the strict large $N$ limit value $\sqrt{2}$ . We also update our results for the parameter $\hat{B}_K$ , finding $\hat{B}_K=0.73\pm 0.02$ . The smallness of $1/N$ corrections to the large $N$ value $\hat{B}_K=3/4$ results within our approach from an approximate cancelation between pseudoscalar and vector meson one-loop contributions. We also summarize the status of $\Delta M_K$ in this approach.     

Efficient saturation of an ion in free space

We report on the demonstration of a light-matter interface coupling light to a single \(^{174}\hbox {Yb}^+\) ion in free space. The interface is realized through a parabolic mirror partially surrounding the ion. It transforms a Laguerre–Gaussian beam into a linear dipole wave converging at the mirror’s focus. By measuring the non-linear response of the atomic transition, we deduce the power required for reaching an upper-level population of \(1/4\) to be \(692\pm 20\,\hbox {pW}\) at half linewidth detuning from the atomic resonance. Performing this measurement while scanning the ion through the focus provides a map of the focal intensity distribution. From the measured power, we infer a coupling efficiency of \(7.2\pm 0.2\,\%\) on the linear dipole transition when illuminating from half solid angle, being among the best coupling efficiencies reported for a single atom in free space.     

Near-Extreme Eigenvalues and the First Gap of Hermitian Random Matrices

We study the phenomenon of “crowding” near the largest eigenvalue \(\lambda _\mathrm{max}\) of random \(N \times N\) matrices belonging to the Gaussian Unitary Ensemble of random matrix theory. We focus on two distinct quantities: (i) the density of states (DOS) near \(\lambda _\mathrm{max}\) , \(\rho _\mathrm{DOS}(r,N)\) , which is the average density of eigenvalues located at a distance \(r\) from \(\lambda _\mathrm{max}\) and (ii) the probability density function of the gap between the first two largest eigenvalues, \(p_\mathrm{GAP}(r,N)\) . In the edge scaling limit where \(r = \mathcal{O}(N^{-1/6})\) , which is described by a double scaling limit of a system of unconventional orthogonal polynomials, we show that \(\rho _\mathrm{DOS}(r,N)\) and \(p_\mathrm{GAP}(r,N)\) are characterized by scaling functions which can be expressed in terms of the solution of a Lax pair associated to the Painlevé XXXIV equation. This provides an alternative and simpler expression for the gap distribution, which was recently studied by Witte et al. in Nonlinearity 26:1799, 2013. Our expressions allow to obtain precise asymptotic behaviors of these scaling functions both for small and large arguments.     

Numerical analysis of quantum ring intersubband photodetector for far infrared detection

A quantum ring based intersubband photodetector is designed and its optical performance is studied based on a developed theoretical model. Intersubband absorption of quantum rings is numerically calculated for two transitions, from ground state to first excited state and to continuum energy and the effects of broadening mechanisms are studied. Device show responsivity about 0.2 A/W at T = 80 K and V = 2 V. Dark current of structure with and without the resonant tunneling barriers is calculated. Specific detectivity \((D^*)\) is studied at different temperatures and peak values about \(3\times 10^{12}\,\hbox {cm Hz}^{1/2}/\hbox {W}\) is obtained at T = 50 K. Results show that inclusion of resonant tunneling barriers leads to improvement in \(D^*\) about one order of magnitude.     

Dilaton gravity, (quasi-) black holes,and scalar charge

Electrically charged dust is considered in the framework of Einstein–Maxwell–dilaton gravity with a Lagrangian containing the interaction term \(P(\chi )F_{\mu \nu }F^{\mu \nu }\) , where \(P(\chi )\) is an arbitrary function of the dilaton scalar field \(\chi \) , which can be normal or phantom. Without assumption of spatial symmetry, we show that static configurations exist for arbitrary functions \(g_{00} = \exp (2\gamma (x^{i}))\) ( \(i=1,2,3\) ) and \(\chi =\chi (\gamma )\) . If \(\chi = \mathrm{const}\) , the classical Majumdar–Papapetrou (MP) system is restored. We discuss solutions that represent black holes (BHs) and quasi-black holes (QBHs), deduce some general results and confirm them by examples. In particular, we analyze configurations with spherical and cylindrical symmetries. It turns out that cylindrical BHs and QBHs cannot exist without negative energy density somewhere in space. However, in general, BHs and QBHs can be phantom-free, that is, can exist with everywhere nonnegative energy densities of matter, scalar and electromagnetic fields.     

Reunion Probabilities of N One-Dimensional Random Walkers with Mixed Boundary Conditions

In this work we extend the results of the reunion probability of \(N\) one-dimensional random walkers to include mixed boundary conditions between their trajectories. The level of the mixture is controlled by a parameter \(c\) , which can be varied from \(c=0\) (independent walkers) to \(c\rightarrow \infty \) (vicious walkers). The expressions are derived by using Quantum Mechanics formalism (QMf) which allows us to map this problem into a Lieb-Liniger gas (LLg) of \(N\) one-dimensional particles. We use Bethe ansatz and Gaudin’s conjecture to obtain the normalized wave-functions and use this information to construct the propagator. As it is well-known, depending on the boundary conditions imposed at the endpoints of a line segment, the statistics of the maximum heights of the reunited trajectories have some connections with different ensembles in Random Matrix Theory. Here we seek to extend those results and consider four models: absorbing, periodic, reflecting, and mixed. In all four cases, the probability that the maximum height is less or equal than \(L\) takes the form \(F_N(L)=A_N\sum _{\varvec{k}\in \Omega _{\text {B}}} \mathrm{e}^{-\sum _{j=1}^Nk_j^2}\mathcal {V}_N(\varvec{k})\) , where \(A_N\) is a normalization constant, \(\mathcal {V}_N(\varvec{k})\) contains a deformed and weighted Vandermonde determinant, and \(\Omega _{\text {B}}\) is the solution set of quasi-momenta \(\varvec{k}\) obeying the Bethe equations for that particular boundary condition.     

The Colored Hofstadter Butterfly for the Honeycomb Lattice

We rely on a recent method for determining edge spectra and we use it to compute the Chern numbers for Hofstadter models on the honeycomb lattice having rational magnetic flux per unit cell. Based on the bulk-edge correspondence, the Chern number \(\sigma _\mathrm{H}\) is given as the winding number of an eigenvector of a \(2 \times 2\) transfer matrix, as a function of the quasi-momentum \(k\in (0,2\pi )\) . This method is computationally efficient (of order \(\mathcal {O}(n^4)\) in the resolution of the desired image). It also shows that for the honeycomb lattice the solution for \(\sigma _\mathrm{H}\) for flux \(p/q\) in the \(r\) -th gap conforms with the Diophantine equation \(r=\sigma _\mathrm{H}\cdot p+ s\cdot q\) , which determines \(\sigma _\mathrm{H}\mod q\) . A window such as \(\sigma _\mathrm{H}\in (-q/2,q/2)\) , or possibly shifted, provides a natural further condition for \(\sigma _\mathrm{H}\) , which however turns out not to be met. Based on extensive numerical calculations, we conjecture that the solution conforms with the relaxed condition \(\sigma _\mathrm{H}\in (-q,q)\) .