Symmetry of the charge density wave in cuprates

We derive and analyze an effective Ginzburg-Landau (GL) functional for a charge density wave (CDW) for a model of electrons on a tight binding square lattice with density-density interactions. We show, using realistic electronic dispersions for the cuprates, that for the simplest GL theory, the preferred symmetry is of the uni-directional (stripe) type, but upon inclusion of third-order terms, the CDW may exhibit a transition from a uni-directional to a checkerboard pattern depending on the strength and range of the interaction.Comment: 8 pages, 8 figure

Index theoretic characterization of d-wave superconductors in the vortex state

We employ index theoretic methods to study analytically the low energy spectrum of a lattice d-wave superconductor in the vortex lattice state. This allows us to compare singly quantized $hc/2e$ and doubly quantized $hc/e$ vortices, the first of which must always be accompanied by $Z_2$ branch cuts. For an inversion symmetric vortex lattice and in the presence of particle-hole symmetry we prove an index theorem that imposes a lower bound on the number of zero energy modes. Generic cases are constructed in which this bound exceeds the number of zero modes of an equivalent lattice of doubly quantized vortices, despite the identical point group symmetries. The quasiparticle spectrum around the zero modes is doubly degenerate and exhibits a Dirac-like dispersion, with velocities that become universal functions of $\Delta_0/t$ in the limit of low magnetic field. For weak particle-hole symmetry breaking, the gapped state can be characterized by a topological quantum number, related to spin Hall conductivity, which generally differs in the cases of the $hc/2e$ and $hc/e$ vortex lattices.Comment: 4 pages, 2 figures, 1 table (accepted for publication in PRL; substantially rewritten for presentation clarity; references to quantum order and visons omitted on referee's demand

Mixed state of a lattice d-wave superconductor

We study the mixed state in an extreme type-II lattice d-wave superconductor in the regime of intermediate magnetic fields H_{c1} << H << H_{c2}. We analyze the low energy spectrum of the problem dominated by nodal Dirac-like quasiparticles with momenta near k_F=(\pm k_D,\pm k_D) and find that the spectrum exhibits characteristic oscillatory behavior with respect to the product of k_D and magnetic length l. The Simon-Lee scaling, predicted in this regime, is satisfied only on average, with the magnitude of the oscillatory part of the spectrum displaying the same 1/l dependence as its monotonous ``envelope'' part. The oscillatory behavior of the spectrum is due to the inter-nodal interference enhanced by the singular nature of the low energy eigenfunctions near vortices. We also study a separate problem of a single vortex piercing an isolated superconducting grain of size L by L. Here we find that the periodicity of the quasiparticle energy oscillations with respect to k_D L is doubled relative to the case where the field is zero and the vortex is absent, both such oscillatory behaviors being present at the leading order in 1/L. Finally, we review the overall features of the tunneling conductance experiments in YBCO and BSCCO, and suggest an interpretation of the peaks at 5-20 meV observed in the tunneling local density of states in these materials.Comment: 16 pages, 11 figure

Dopant-modulated pair interaction in cuprate superconductors

Comparison of recent experimental STM data with single-impurity and many-impurity Bogoliubov-de Gennes calculations strongly suggests that random out-of-plane dopant atoms in cuprates modulate the pair interaction locally. This type of disorder is crucial to understanding the nanoscale electronic structure inhomogeneity observed in BSCCO-2212, and can reproduce observed correlations between the positions of impurity atoms and various aspects of the local density of states such as the gap magnitude and the height of the coherence peaks. Our results imply that each dopant atom modulates the pair interaction on a length scale of order one lattice constant.Comment: 5 pages, 4 figure

Andreev states near short-ranged pairing potential impurities

We study Andreev states near atomic scale modulations in the pairing potential in both $s$- and d-wave superconductors with short coherence lengths. For a moderate reduction of the local gap, the states exist only close to the gap edge. If one allows for local sign changes of the order parameter, however, resonances can occur at energies close to the Fermi level. The local density of states (LDOS) around such pairing potential defects strongly resembles the patterns observed by tunneling measurements around Zn impurities in Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ (BSCCO). We discuss how this phase impurity model of the Zn LDOS pattern can be distinguished from other proposals experimentally.Comment: 4 pages, 4 figure

Thermodynamic transitions in inhomogeneous d-wave superconductors

We study the spectral and thermodynamic properties of inhomogeneous d-wave superconductors within a model where the inhomogeneity originates from atomic scale pair disorder. This assumption has been shown to be consistent with the small charge and large gap modulations observed by scanning tunnelling spectroscopy (STS) on BSCCO. Here we calculate the specific heat within the same model, and show that it gives a semi-quantitative description of the transition width in this material. This model therefore provides a consistent picture of both surface sensitive spectroscopy and bulk thermodynamic properties.Comment: 4 pages, 4 figure

Fourier transform spectroscopy of d-wave quasiparticles in the presence of atomic scale pairing disorder

The local density of states power spectrum of optimally doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ (BSCCO) has been interpreted in terms of quasiparticle interference peaks corresponding to an "octet'' of scattering wave vectors connecting k-points where the density of states is maximal. Until now, theoretical treatments have not been able to reproduce the experimentally observed weights and widths of these "octet'' peaks; in particular, the predominance of the dispersing "q$_1$'' peak parallel to the Cu-O bond directions has remained a mystery. In addition, such theories predict "background'' features which are not observed experimentally. Here, we show that most of the discrepancies can be resolved when a realistic model for the out-of-plane disorder in BSCCO is used. Weak extended potential scatterers, which are assumed to represent cation disorder, suppress large-momentum features and broaden the low-energy "q$_7$''-peaks, whereas scattering at order parameter variations, possibly caused by a dopant-modulated pair interaction around interstitial oxygens, strongly enhances the dispersing "q$_1$''-peaks.Comment: 7 pages, 3 figure

The genetic history of the Southern Arc: a bridge between West Asia and Europe

By sequencing 727 ancient individuals from the Southern Arc (Anatolia and its neighbors in Southeastern Europe and West Asia) over 10,000 years, we contextualize its Chalcolithic period and Bronze Age (about 5000 to 1000 BCE), when extensive gene flow entangled it with the Eurasian steppe. Two streams of migration transmitted Caucasus and Anatolian/Levantine ancestry northward, and the Yamnaya pastoralists, formed on the steppe, then spread southward into the Balkans and across the Caucasus into Armenia, where they left numerous patrilineal descendants. Anatolia was transformed by intra–West Asian gene flow, with negligible impact of the later Yamnaya migrations. This contrasts with all other regions where Indo-European languages were spoken, suggesting that the homeland of the Indo-Anatolian language family was in West Asia, with only secondary dispersals of non-Anatolian Indo-Europeans from the steppe

Pattern-Based Approach to Current Density Verification

Methodology of static verification of current density based on layout patterns common in IC designs is proposed. The methodology is based on pre-calculation of current density distribution for common layout patterns. Then using the obtained data to calculate current densities of large circuits by partitioning them to selected patterns. Presented experimental results show the effectiveness of the approach