On the Coble quartic

We review and extend the known constructions relating Kummer threefolds, G¨opel systems, theta constants and their derivatives, and the GIT quotient for 7 points in P^2 to obtain an explicit expression for the Coble quartic. The Coble quartic was recently determined completely in [RSSS12], where it was computed completely explicitly, as a polynomial with 372060 monomials of bidegree (28, 4) in theta constants of the second order and theta functions of the second order, respectively. Our expression is in terms of products of theta constants with characteristics corresponding to G¨opel systems, and is a polynomial with 134 terms. Our approach is based on the beautiful geometry studied by Coble and further investigated by Dolgachev and Ortland and highlights the geometry and combinatorics of syzygetic octets of characteristics, and the corresponding representations of Sp(6, F_2). One new ingredient is the relationship of G¨opel systems and Jacobian determinants of theta functions. In genus 2, we similarly obtain a short explicit equation for the universal Kummer surface, and relate modular forms of level two to binary invariants of six points on P^

Effect of nonmagnetic dilution in honeycomb lattice iridates Na2_2IrO3_3 and Li2_2IrO3_3

We have synthesized single crystals of Na$_2$(Ir$_{1-x}$Ti$_x$)O$_3$ and polycrystals of Li$_2$(Ir$_{1-x}$Ti$_x$)O$_3$ and studied the effect of magnetic depletion on the magnetic properties by measurements of the magnetic susceptibility, specific heat and magnetocaloric effect at temperatures down to 0.1~K. In both systems, the non-magnetic substitution rapidly changes the magnetically ordered ground state into a spin glass, indicating strong frustration. While for the Li system the Weiss temperature $\Theta_{\rm W}$ remains unchanged up to $x=0.55$, a strong decrease $|\Theta_{\rm W}|$ is found for the Na system. This suggests that only for the former system magnetic exchange beyond nearest neighbors is dominating. This is also corroborated by the observation of a smeared quantum phase transition in Li$_2$(Ir$_{1-x}$Ti$_x$)O$_3$ near $x=0.5$, i.e. much beyond the site percolation threshold of the honeycomb lattice.Comment: 8 pages including supplemental, 12 figure

Investigation of unconventional reconstruction and electronic properties on the Na2IrO3 surface

Na2IrO3 is an intriguing material for which spin-orbit coupling plays a key role. Theoretical predictions, so far unverified, have been made that the surface of Na2IrO3 should exhibit a clear signature of the quantum spin Hall effect. We studied the surface of Na2IrO3 using scanning tunneling microscopy and density-functional theory calculations. We observed atomic level resolution of the surface and two types of terminations with different surface periodicity and Na content. By comparing bias-dependent experimental topographic images to simulated images, we determined the detailed atomistic structure of both observed surfaces. One of these reveals a strong relaxation to the surface of Na atoms from the subsurface region two atomic layers below. Such dramatic structural changes at the surface cast doubt on any prediction of surface properties based on bulk electronic structure. Indeed, using spatially resolved tunneling spectroscopy we found no indication of the predicted quantum spin Hall behavior

Origin of the insulating state in honeycomb iridates and rhodates

A burning question in the emerging field of spin-orbit driven insulating iridates, such as Na2IrO3 and Li2IrO3 is whether the observed insulating state should be classified as a Mott-Hubbard insulator derived from a half-filled relativistic j_eff=1/2 band or as a band insulator where the gap is assisted by spin-orbit interaction, or Coulomb correlations, or both. The difference between these two interpretations is that only for the former, strong spin-orbit coupling (lambda >~ W, where W is the band width) is essential. We have synthesized the isostructural and isoelectronic Li2RhO3 and report its electrical resistivity and magnetic susceptibility. Remarkably it shows insulating behavior together with fluctuating effective S=1/2 moments, similar to Na2IrO3 and Li2IrO3, although in Rh4+ (4d5) the spin-orbit coupling is greatly reduced. We show that this behavior has non-relativistic one-electron origin (although Coulomb correlations assist in opening the gap), and can be traced down to formation of quasi-molecular orbitals, similar to those in Na2IrO3.Comment: 7 pages, 7 figure

Relevance of the Heisenberg-Kitaev model for the honeycomb lattice iridates A_2IrO_3

Combining thermodynamic measurements with theoretical density functional and thermodynamic calculations we demonstrate that the honeycomb lattice iridates A2IrO3 (A = Na, Li) are magnetically ordered Mott insulators where the magnetism of the effective spin-orbital S = 1/2 moments can be captured by a Heisenberg-Kitaev (HK) model with Heisenberg interactions beyond nearest-neighbor exchange. Experimentally, we observe an increase of the Curie-Weiss temperature from \theta = -125 K for Na2IrO3 to \theta = -33 K for Li2IrO3, while the antiferromagnetic ordering temperature remains roughly the same T_N = 15 K for both materials. Using finite-temperature functional renormalization group calculations we show that this evolution of \theta, T_N, the frustration parameter f = \theta/T_N, and the zig-zag magnetic ordering structure suggested for both materials by density functional theory can be captured within this extended HK model. Combining our experimental and theoretical results, we estimate that Na2IrO3 is deep in the magnetically ordered regime of the HK model (\alpha \approx 0.25), while Li2IrO3 appears to be close to a spin-liquid regime (0.6 < \alpha < 0.7).Comment: Version accepted for publication in PRL. Additional DFT and thermodynamic calculations have been included. 6 pages of supplementary material include

Classical theta constants vs. lattice theta series, and super string partition functions

Recently, various possible expressions for the vacuum-to-vacuum superstring amplitudes has been proposed at genus $g=3,4,5$. To compare the different proposals, here we will present a careful analysis of the comparison between the two main technical tools adopted to realize the proposals: the classical theta constants and the lattice theta series. We compute the relevant Fourier coefficients in order to relate the two spaces. We will prove the equivalence up to genus 4. In genus five we will show that the solutions are equivalent modulo the Schottky form and coincide if we impose the vanishing of the cosmological constant.Comment: 21 page

The puzzling symbiotic X-ray system 4U1700+24

Symbiotic X-ray binaries form a subclass of low-mass X-ray binary systems consisting of a neutron star accreting material from a red giant donor star via stellar wind or Roche lobe overflow. Only a few confirmed members are currently known; 4U 1700+24 is a good candidate as it is a relatively bright X-ray object, possibly associated with the late-type star V934 Her. We analysed the archive {\it XMM}-Newton and Swift/XRT observations of 4U 1700+24 in order to have a uniform high-energy ($0.3-10$ keV) view of the source. We confirmed the existence of a red-shifted O VIII Ly-$\alpha$ transition (already observed in the 2002 {\it XMM}-Newton data) in the high-resolution spectra collected via the RGS instruments. The red-shift of the line is found in all the analysed observations and, on average, it was estimated to be $\simeq 0.009$. We also observed a modulation of the centroid energy of the line on short time scales (a few days) and discuss the observations in the framework of different scenarios. If the modulation is due to the gravitational red-shift of the neutron star, it might arise from a sudden re-organization of the emitting $X$-ray matter on the scale of a few hundreds of km. Alternatively, we are witnessing a uni-polar jet of matter (with typical velocity of $1000-4000$ km s$^{-1}$) possibly emitted by the neutron star in an almost face-on system. The second possibility seems to be required by the apparent lack of any modulation in the observed $X$-ray light curve. We also note also that the low-resolution spectra (both {\it XMM}-Newton and Swift/XRT in the $0.3-10$ keV band) show the existence of a black body radiation emitted by a region (possibly associated with the neutron star polar cap) with typical size from a few tens to hundreds of meters. The size of this spot-like region reduces as the overall luminosity of 4U 1700+24 decreases.Comment: In press on A&

Effect of isoelectronic doping on honeycomb lattice iridate A_2IrO_3

We have investigated experimentally and theoretically the series (Na$_{1-x}$Li$_{x}$)$_{2}$IrO$_{3}$. Contrary to what has been believed so far, only for $x\leq0.25$ the system forms uniform solid solutions. For larger Li content, as evidenced by powder X-ray diffraction, scanning electron microscopy and density functional theory calculations, the system shows a miscibility gap and a phase separation into an ordered Na$_{3}$LiIr$_2$O$_{6}$ phase with alternating Na$_3$ and LiIr$_2$O$_6$ planes, and a Li-rich phase close to pure Li$_{2}$IrO$_{3}$. For $x\leq 0.25$ we observe (1) an increase of $c/a$ with Li doping up to $x=0.25$, despite the fact that $c/a$ in pure Li$_{2}$IrO$_{3}$ is smaller than in Na$_{2}$IrO$_{3}$, and (2) a gradual reduction of the antiferromagnetic ordering temperature $T_{N}$ and ordered moment. The previously proposed magnetic quantum phase transition at $x\approx 0.7$ may occur in a multiphase region and its nature needs to be re-evaluated.Comment: 8 pages, 9 figures including supplemental informatio