Hypercontact structures and Floer homology

We introduce a new Floer theory associated to a pair consisting of a Cartan hypercontact 3-manifold M and a hyperkaehler manifold X.Comment: 74 page

The three dimensional Fueter equation and divergence free frames

A divergence free frame on a closed three manifold is called regular if every solution of the linear Fueter equation is constant and is called singular otherwise. The set of singular divergence free frames is an analogue of the Maslov cycle. Regular divergence free frames satisfy an analogue of the Arnold conjecture for flat hyperkaehler target manifolds. The Seiberg-Witten equations can be viewed as gauged versions of the Fueter equation, and so can the Donaldson-Thomas equations on certain seven dimensional product manifolds.Comment: Final version, 34 pages. Abhandlungen aus dem Mathematischen Seminar der Universitaet Hambur

Percolation on uniform infinite planar maps

We construct the uniform infinite planar map (UIPM), obtained as the n \to \infty local limit of planar maps with n edges, chosen uniformly at random. We then describe how the UIPM can be sampled using a "peeling" process, in a similar way as for uniform triangulations. This process allows us to prove that for bond and site percolation on the UIPM, the percolation thresholds are p_c^bond=1/2 and p_c^site=2/3 respectively. This method also works for other classes of random infinite planar maps, and we show in particular that for bond percolation on the uniform infinite planar quadrangulation, the percolation threshold is p_c^bond=1/3.Comment: 26 pages, 9 figure

Poor Outlook for 4d and 5d Kitaev Candidate Materials: Density Functional Theory Calculations Point to Small Kitaev Terms

Na2IrO3 and other “Kitaev candidate” materials are of much scientific and technological interest. In such materials, proposed anisotropic, Ising-like interactions promote magnetic frustration, potentially leading to a quantum spin liquid (QSL) ground state with excitations (non-Abelian anyons) that could enable topological quantum computing – a form of quantum computing particularly robust to decoherence. Unfortunately, experimental work indicates that Na2IrO3 and other Kitaev candidates do not exhibit a QSL ground state. While there are several proposals for manipulating Kitaev candidates into a QSL state, there is no consensus over whether the effective-spin interactions of these materials are proximate to the values necessary for such a state. Here, we use density functional theory (DFT) and mathematical techniques to investigate the electronic properties and effective-spin interactions of Kitaev candidate materials. Our approach for determining spin interaction terms, Compressive Sensing Spin Dynamics (CSSD), involves two main steps: 1) by appending the Kohn-Sham energy functional with a Lagrangian term that allows for quickly finding the lowest energy solution of a material given fixed magnetic moments, we perform many calculations of the material with the spins slightly perturbed from the equilibrium state; 2) we perform compressive sensing on these data (on the fixed spins and magnetic fields required to stabilize the desired spin arrangement) to yield the interaction terms with far fewer data than otherwise required. Performing this procedure on the Kitaev candidates Na2IrO3 and α-RuCl3 indicates neither is close to the Kitaev QSL regime. We further investigate manipulated versions of Kitaev candidate materials and find all have problems: epitaxial and single layers of Na2IrO3 are conductors (implying Na2IrO3 cannot be cleanly separated into individual layers without changing the band structure), while straining Na2IrO3 in-plane introduces different problems if under tension (the Ir-O-Ir angles increase, leading to less destructive interference of undesirable interactions) and compression (the Ir-Ir distances decrease, leading to larger isotropic Heisenberg interaction, which competes with the Kitaev interaction). We additionally use CSSD to evaluate the interaction terms of idealized versions of Na2IrO3 and α-RuCl3 (90� Ir-O-Ir and Ru-Cl-Ru angles), and our results indicate that these, too, are far from the Kitaev QSL state. Finally, we investigate if YbBr3, as a lanthanide material, may circumvent the problems with 4d and 5d candidates, due to smaller f-f orbital overlap compared to d-d overlap. While the results with YbBr3 are somewhat ambiguous, we conclude f-orbital Kitaev candidate materials warrant further investigation

High-temperature ab initio calculations on FeSi and NiSi

The Fe–Ni–Si system is potentially a very important component of terrestrial planetary cores. However, at present, even the behaviour of the FeSi and NiSi end members is poorly understood, especially at low to moderate pressures—the data for FeSi are contradictory and NiSi has been little studied. For FeSi, there is general agreement that there is a phase transition from the ε-FeSi to the CsCl structure with increasing pressure, but, in experiments, there is disagreement as to the position and slope of the phase boundary and the range of coexistence of the two phases. In this paper we have used ab initio lattice dynamics calculations to determine the phase boundary between the ε-FeSi and CsCl structures as a function of pressure and temperature in both FeSi and NiSi. For FeSi, we find that the transition pressure at zero Kelvin is ~11 GPa and that the boundary between the ε-FeSi and CsCl phases varies little with temperature, having a slight negative Clapeyron slope, going from ~11 GPa at 300 K to ~3 GPa at 2000 K. For NiSi, there is much greater variation of the transition pressure with temperature, with a much shallower negative Clapeyron slope, going from ~156 GPa at 300 K to ~94 GPa at 2000 K

The fate of high redshift massive compact galaxies in dense environments

Massive compact galaxies seem to be more common at high redshift than in the local universe, especially in denser environments. To investigate the fate of such massive galaxies identified at z~2 we analyse the evolution of their properties in three cosmological hydrodynamical simulations that form virialised galaxy groups of mass ~10^13 Msun hosting a central massive elliptical/S0 galaxy by redshift zero. We find that at redshift ~2 the population of galaxies with M_*> 2 10^10 Msun is diverse in terms of mass, velocity dispersion, star formation and effective radius, containing both very compact and relatively extended objects. In each simulation all the compact satellite galaxies have merged into the central galaxy by redshift 0 (with the exception of one simulation where one of such satellite galaxy survives). Satellites of similar mass at z = 0 are all less compact than their high redshift counterparts. They form later than the galaxies in the z = 2 sample and enter the group potential at z < 1, when dynamical friction times are longer than the Hubble time. Also, by z = 0 the central galaxies have increased substantially their characteristic radius via a combination of in situ star formation and mergers. Hence in a group environment descendants of compact galaxies either evolve towards larger sizes or they disappear before the present time as a result of the environment in which they evolve. Since the group-sized halos that we consider are representative of dense environments in the LambdaCDM cosmology, we conclude that the majority of high redshift compact massive galaxies do not survive until today as a result of the environment.Comment: 10 pages, 4 figures, submitted to MNRA