We present certain exact analytical results for dynamical spin correlation functions in the Kitaev Model. It is the first result of its kind in non-trivial quantum spin models. The result is also novel: in spite of presence of gapless propagating Majorana fermion excitations, dynamical two spin correlation functions are identically zero beyond nearest neighbor separation. This shows existence of a gapless but short range spin liquid. An unusual, all energy scale fractionization of a spin -flip quanta, into two infinitely massive π-fluxes and a dynamical Majorana fermion, is shown to occur. As the Kitaev Model exemplifies topological quantum computation, our result presents new insights into qubit dynamics and generation of topological excitations.PACS numbers: 75.10.jm, 03.67.Lx, 71.10.Pm In the field of quantum computers and quantum communications, practical realizations of qubits that are robust and escape decoherence is a foremost challenge [1]. In this context Kitaev proposed[2] certain emergent topological excitations in strongly correlated quantum many body systems as robust qubits. In a fault tolerant quantum computation scheme [2,3,4], Kitaev constructed a non-trivial and exactly solvable 2-dimensional spin model [2] and illustrated basic ideas. In some limit it also becomes the celebrated 'toric code' Hamiltonian. The Kitaev model has come closer to reality, after recent proposals for experimental realizations [5,6] and schemes for manipulation and detection [7]. In initialisation, error correction and read out operations, it is 'spins' rather than emergent topological degrees of freedom that are directly accessed from outside. Thus an understanding of dynamic spin correlations is of paramount importance.We present certain exact analytical results for time dependent spin correlation functions, in arbitrary eigenstates of the Kitaev Model. Our results are non-trivial and novel, with possible implications for new quantum computational schemes. Further our result is unique in the sense that it is the first exact result for equilibrium dynamical spin correlation functions in a non trivial 2D quantum spin model.We show that dynamical two spin correlation functions are short ranged and vanish identically beyond nearest neighbor sites for all time t, for all values of the coupling constants J x , J y and J z , even in the domain of J's where the model is gapless. Our result shows rigorously that it is a short range quantum spin liquid and long range spin order is absent. We obtain a compact form for the time dependence, which makes the physics transparent.Kitaev Model is known to support dynamical Majorana fermions and static π-flux eigen-excitations. We show how fractionization [8, 9] of a local spin-flip quanta into a bound pair of static π-flux excitations and a free Majorana fermion occurs.In the present paper we have restricted our calculation to dynamical correlation functions for time independent Hamiltonians, in arbitrary eigen-states and thermal states. In actual quantum computations, key manipulat...

We introduce a spin-1/2 model in three dimensions which is a generalization of the well-known Kitaev model on a honeycomb lattice. Following Kitaev, we solve the model exactly by mapping it to a theory of non-interacting fermions in the background of a static Z_2 gauge field. The phase diagram consists of a gapped phase and a gapless one, similar to the two-dimensional case. Interestingly, unlike in the two-dimensional model, in the gapless phase the gap vanishes on a contour in the k space. Furthermore, we show that the flux excitations of the gauge field, due to some local constraints, form loop like structures; such loops exist on a lattice formed by the plaquettes in the original lattice and is topologically equivalent to the pyrochlore lattice. Finally, we derive a low-energy effective Hamiltonian that can be used to study the properties of the excitations in the gapped phase.Comment: 9 pages, 7 figures; published version; a new section and more references adde

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2H-Chromenes (2H-1-benzopyran derivatives) display a broad spectrum of biological activities. The 2H-chromene substructure is an important structural motif present in a variety of medicines, natural products, and materials showing unique photophysical properties. Hence, the structural importance of the benzopyran moiety has elicited a great deal of interest in the field of organic synthesis and chemical biology to develop new and improved synthesis of these molecular skeletons. This review gives an up-to-date overview of different catalytic methodologies developed for the synthesis of 2H-chromenes and is structured around the three main approaches applied in catalytic 2H-chromene synthesis: (I) catalysis with (transition) metals, (II) metal-free Brønsted and Lewis acid/base catalysis, which includes examples of nonenantioselective organocatalysis, and (III) enantioselective organo-catalysis. The section in which the metal-catalyzed reactions are discussed describes different ring-closing strategies based on (transition) metal catalysis, including a few enantioselective approaches. For most of these reactions, plausible mechanisms are delineated. Moreover, synthesis of some natural products and medicinally important drugs are included. Specific advantages and disadvantages of the several synthetic methodologies are discussed. The review focuses on catalytic 2H-chromene synthesis. However, for a complete overview, synthetic routes involving some stoichiometric steps and reactions producing ring-scaffolds that are closely related to 2H-chromenes are also included.

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