The polarization of graphene is calculated exactly within the random phase approximation for arbitrary frequency, wave vector, and doping. At finite doping, the static susceptibility saturates to a constant value for low momenta. At q = 2k F it has a discontinuity only in the second derivative.In the presence of a charged impurity this results in Friedel oscillations which decay with the same power law as the Thomas Fermi contribution, the latter being always dominant. The spin density oscillations in the presence of a magnetic impurity are also calculated. The dynamical polarization for low q and arbitrary ω is employed to calculate the dispersion relation and the decay rate of plasmons and acoustic phonons as a function of doping. The low screening of graphene, combined with the absence of a gap, leads to a significant stiffening of the longitudinal acoustic lattice vibrations.

We study the electronic structure and the phase diagram of non-interacting fermions confined to hexagonal optical lattices. In the first part, we compare the properties of Dirac points arising in the eigenspectrum of either honeycomb or triangular lattices. Numerical results are complemented by analytical equations for weak and strong confinements. In the second part we discuss the phase diagram and the evolution of Dirac points in honeycomb lattices applying a tight-binding description with arbitrary nearest-neighbor hoppings. With increasing asymmetry between the hoppings the Dirac points approach each other. At a critical asymmetry the Dirac points merge to open an energy gap, thus changing the topology of the eigenspectrum. We analyze the trajectory of the Dirac points and study the density of states in the different phases. Manifestations of the phase transition in the temperature dependence of the specific heat and in the structure factor are discussed.

A series of spiro[[2]benzopyran-1,4'-piperidines] and spiro[[2]benzofuran-1,4'-piperidines] of general structure 10 is prepared, and the affinity for sigma(1)- and sigma(2)-receptors is investigated by means of radioligand binding assays. The synthesis of the spiropiperidines 14a and 23 proceeds from bromine/lithium exchange of the bromoacetals 11 and 21, addition to piperidin-4-one 12a, and subsequent cyclization. Systematic variations of the substituent R at the nitrogen atom, the group X in position 3, and the ring size of the oxygen heterocycle are performed. The sigma(1)- and sigma(2)-receptor affinities are determined with guinea pig brain and rat liver membrane preparations using [(3)H]-labeled (+)-pentazocine and ditolylguanidine, respectively. Test results show that a benzyl residue at the piperidine nitrogen atom and a methoxy group in position 3 are advantageous for high sigma(1)-receptor affinity. In this series the 1'-benzyl-3-methoxy-3,4-dihydrospiro[[2]benzopyran-1,4'-piperidine] (14a) and the 1'-benzyl-3-methoxy-3H-spiro[[2]benzofuran-1,4'-piperidine] (23) are among the most potent sigma(1)-ligands interacting in the low nanomolar range with sigma(1)-receptors (14a, K(i) = 1.29 nM; 23, K(i) = 1.14 nM). Variation of the nitrogen substituent R from benzyl to H, alkyl, phenyl, or omega-phenylalkyl and the group X from methoxy to hydroxy, carbonyl, or alkyloxy led to reduced sigma(1)-receptor affinity. In addition to their high sigma(1)-receptor affinity, the spiropiperidines 14a and 23 display excellent selectivity toward sigma(2)-receptors (sigma(1)/sigma(2) = 2708 and 1130) and several other receptor and reuptake systems. Introduction of a polar hydroxy group in position 3 and elongation of the distance between the piperidine nitrogen atom and the phenyl moiety result in ligands with considerable sigma(2)-receptor affinity and therefore diminished sigma(1)/sigma(2)-receptor selectivity. The hemiacetalic 1'-(3-phenylpropyl)-3,4-dihydrospiro[[2]benzopyran-1,4'-piperidin]-3-ol (15e) represents the most active sigma(2)-receptor ligand in this series with a K(i) value of 83.1 nM.