Twisted Dirac Operators over Quantum Spheres

We construct new families of spectral triples over quantum spheres, with a particular attention focused on the standard Podles quantum sphere and twisted Dirac operators.Comment: 17 page

Acquaintance time of random graphs near connectivity threshold

Benjamini, Shinkar, and Tsur stated the following conjecture on the acquaintance time: asymptotically almost surely $AC(G) \le p^{-1} \log^{O(1)} n$ for a random graph $G \in G(n,p)$, provided that $G$ is connected. Recently, Kinnersley, Mitsche, and the second author made a major step towards this conjecture by showing that asymptotically almost surely $AC(G) = O(\log n / p)$, provided that $G$ has a Hamiltonian cycle. In this paper, we finish the task by showing that the conjecture holds in the strongest possible sense, that is, it holds right at the time the random graph process creates a connected graph. Moreover, we generalize and investigate the problem for random hypergraphs

Spin susceptibility of interacting two-dimensional electrons in the presence of spin-orbit coupling

A long-range interaction via virtual particle-hole pairs between Fermi-liquid quasiparticles leads to a nonanalytic behavior of the spin susceptibility $\chi$ as a function of the temperature ($T$), magnetic field ($\mathbf{B}$), and wavenumber. In this paper, we study the effect of the Rashba spin-orbit interaction (SOI) on the nonanalytic behavior of $\chi$ for a two-dimensional electron liquid. Although the SOI breaks the SU(2) symmetry, it does not eliminate nonanalyticity but rather makes it anisotropic: while the linear scaling of $\chi_{zz}$ with $T$ and $|\mathbf{B}|$ saturates at the energy scale set by the SOI, that of $\chi_{xx}$ ($=\chi_{yy}$) continues through this energy scale, until renormalization of the electron-electron interaction in the Cooper channel becomes important. We show that the Renormalization Group flow in the Cooper channel has a non-trivial fixed point, and study the consequences of this fixed point for the nonanalytic behavior of $\chi$. An immediate consequence of SOI-induced anisotropy in the nonanalytic behavior of $\chi$ is a possible instability of a second-order ferromagnetic quantum phase transition with respect to a first-order transition to an XY ferromagnetic state.Comment: 34 pages, 12 figure

Ferromagnetic order of nuclear spins coupled to conduction electrons: a combined effect of the electron-electron and spin-orbit interactions

We analyze the ordered state of nuclear spins embedded in an interacting two-dimensional electron gas (2DEG) with Rashba spin-orbit interaction (SOI). Stability of the ferromagnetic nuclear-spin phase is governed by nonanalytic dependences of the electron spin susceptibility $\chi^{ij}$ on the momentum ($\tilde{\mathbf{q}}$) and on the SOI coupling constant ($\alpha$). The uniform (\tq=0) spin susceptibility is anisotropic (with the out-of-plane component, $\chi^{zz}$, being larger than the in-plane one, $\chi^{xx}$, by a term proportional to $U^2(2k_F)|\alpha|$, where $U(q)$ is the electron-electron interaction). For \tq \leq 2m^*|\alpha|, corrections to the leading, $U^2(2k_F)|\alpha|$, term scale linearly with \tq for $\chi^{xx}$ and are absent for $\chi^{zz}$. This anisotropy has important consequences for the ferromagnetic nuclear-spin phase: $(i)$ the ordered state--if achieved--is of an Ising type and $(ii)$ the spin-wave dispersion is gapped at \tq=0. To second order in $U(q)$, the dispersion a decreasing function of \tq, and anisotropy is not sufficient to stabilize long-range order. However, renormalization in the Cooper channel for \tq\ll2m^*|\alpha| is capable of reversing the sign of the \tq-dependence of $\chi^{xx}$ and thus stabilizing the ordered state. We also show that a combination of the electron-electron and SO interactions leads to a new effect: long-wavelength Friedel oscillations in the spin (but not charge) electron density induced by local magnetic moments. The period of these oscillations is given by the SO length $\pi/m^*|\alpha|$.Comment: 22 pages, 15 figure

A hierarchical approach to the prediction of the quaternary structure of GCN4 and its mutants

First published in DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 23 (1996) published by the American Mathematical Society.Presented at DIMACS Workshop on Global Minimization of Nonconvex Energy Functions: Molecular Conformation and Protein Folding, March 20-21, 1995.A hierarchical approach to protein folding is employed to examine the folding pathway and predict the quaternary structure of the GCN4 leucine zipper. Structures comparable in quality to experiment have been predicted. In addition, the equilibrium between dimers, trimers and tetramers of a number of GCN4 mutants has been examined. In five out of eight cases, the simulation results are in accordance with the experimental studies of Harbury, et al

Twisted Hochschild Homology of Quantum Hyperplanes

We calculate the Hochschild dimension of quantum hyperplanes using the twisted Hochschild homology.Comment: 12 pages, LaTe