Stable and unstable attractors in Boolean networks

Boolean networks at the critical point have been a matter of debate for many years as, e.g., scaling of number of attractor with system size. Recently it was found that this number scales superpolynomially with system size, contrary to a common earlier expectation of sublinear scaling. We here point to the fact that these results are obtained using deterministic parallel update, where a large fraction of attractors in fact are an artifact of the updating scheme. This limits the significance of these results for biological systems where noise is omnipresent. We here take a fresh look at attractors in Boolean networks with the original motivation of simplified models for biological systems in mind. We test stability of attractors w.r.t. infinitesimal deviations from synchronous update and find that most attractors found under parallel update are artifacts arising from the synchronous clocking mode. The remaining fraction of attractors are stable against fluctuating response delays. For this subset of stable attractors we observe sublinear scaling of the number of attractors with system size.Comment: extended version, additional figur

Causality Violation and Naked Time Machines in AdS_5

We study supersymmetric charged rotating black holes in AdS$_5$, and show that closed timelike curves occur outside the event horizon. Also upon lifting to rotating D3 brane solutions of type IIB supergravity in ten dimensions, closed timelike curves are still present. We believe that these causal anomalies correspond to loss of unitarity in the dual ${\cal N}=4$, D=4 super Yang-Mills theory, i.e. the chronology protection conjecture in the AdS bulk is related to unitarity bounds in the boundary CFT. We show that no charged or uncharged geodesic can penetrate the horizon, so that the exterior region is geodesically complete. These results still hold true in the quantum case, i.~e.~the total absorption cross section for Klein-Gordon scalars propagating in the black hole background is zero. This suggests that the effective temperature is zero instead of assuming the naively found imaginary value.Comment: 22 pages, Latex, uses JHEP.cls, 1 figure. v3: comments on unitarity in CFT and 2 references added. v4: changes in final remarks, final version to appear in JHE

Theory of Coherent cc-Axis Josephson Tunneling between Layered Superconductors

We calculate exactly the Josephson current for $c$-axis coherent tunneling between two layered superconductors, each with internal coherent tight-binding intra- and interlayer quasiparticle dispersions. Our results also apply when one or both of the superconductors is a bulk material, and include the usually neglected effects of surface states. For weak tunneling, our results reduce to our previous results derived using the tunneling Hamiltonian. Our results are also correct for strong tunneling. However, the $c$-axis tunneling results of Tanaka and Kashiwaya are shown to be incorrect in any limit. In addition, we consider the $c$-axis coherent critical current between two identical layered superconductors twisted an angle $\phi_0$ about the $c$-axis with respect to each other. Regardless of the order parameter symmetry, our coherent tunneling results using a tight-binding intralayer quasiparticle dispersion are inconsistent with the recent $c$-axis twist bicrystal Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ twist junction experiments of Li {\it et al.}Comment: 11 pages, 13 figures, submitted to Physical Review

Holomorphic Anomaly in Gauge Theories and Matrix Models

We use the holomorphic anomaly equation to solve the gravitational corrections to Seiberg-Witten theory and a two-cut matrix model, which is related by the Dijkgraaf-Vafa conjecture to the topological B-model on a local Calabi-Yau manifold. In both cases we construct propagators that give a recursive solution in the genus modulo a holomorphic ambiguity. In the case of Seiberg-Witten theory the gravitational corrections can be expressed in closed form as quasimodular functions of Gamma(2). In the matrix model we fix the holomorphic ambiguity up to genus two. The latter result establishes the Dijkgraaf-Vafa conjecture at that genus and yields a new method for solving the matrix model at fixed genus in closed form in terms of generalized hypergeometric functions.Comment: 34 pages, 2 eps figures, expansion at the monopole point corrected and interpreted, and references adde

Epidemic threshold in structured scale-free networks

We analyze the spreading of viruses in scale-free networks with high clustering and degree correlations, as found in the Internet graph. For the Suscetible-Infected-Susceptible model of epidemics the prevalence undergoes a phase transition at a finite threshold of the transmission probability. Comparing with the absence of a finite threshold in networks with purely random wiring, our result suggests that high clustering and degree correlations protect scale-free networks against the spreading of viruses. We introduce and verify a quantitative description of the epidemic threshold based on the connectivity of the neighborhoods of the hubs.Comment: 4 pages, 4 figure

Single-ion and exchange anisotropy effects and multiferroic behavior in high-symmetry tetramer single molecule magnets

We study single-ion and exchange anisotropy effects in equal-spin $s_1$ tetramer single molecule magnets exhibiting $T_d$, $D_{4h}$, $D_{2d}$, $C_{4h}$, $C_{4v}$, or $S_4$ ionic point group symmetry. We first write the group-invariant quadratic single-ion and symmetric anisotropic exchange Hamiltonians in the appropriate local coordinates. We then rewrite these local Hamiltonians in the molecular or laboratory representation, along with the Dzyaloshinskii-Moriay (DM) and isotropic Heisenberg, biquadratic, and three-center quartic Hamiltonians. Using our exact, compact forms for the single-ion spin matrix elements, we evaluate the eigenstate energies analytically to first order in the microscopic anisotropy interactions, corresponding to the strong exchange limit, and provide tables of simple formulas for the energies of the lowest four eigenstate manifolds of ferromagnetic (FM) and anitiferromagnetic (AFM) tetramers with arbitrary $s_1$. For AFM tetramers, we illustrate the first-order level-crossing inductions for $s_1=1/2,1,3/2$, and obtain a preliminary estimate of the microscopic parameters in a Ni$_4$ from a fit to magnetization data. Accurate analytic expressions for the thermodynamics, electron paramagnetic resonance absorption and inelastic neutron scattering cross-section are given, allowing for a determination of three of the microscopic anisotropy interactions from the second excited state manifold of FM tetramers. We also predict that tetramers with symmetries $S_4$ and $D_{2d}$ should exhibit both DM interactions and multiferroic states, and illustrate our predictions for $s_1=1/2, 1$.Comment: 30 pages, 14 figures, submitted to Phys. Rev.

Theory of Bi2_2Sr2_2CaCu2_2O8+δ_{8+\delta} Cross-Whisker Josephson Junctions

Takano {\it et al.} [Phys. Rev. B {\bf 65}, 140513 (2002) and unpublished] made Josephson junctions from single crystal whiskers of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ crossed an angle $\phi_0$ about the $c$ axis. From the mesa structures that formed at the cross-whisker interface, they inferred a critical current density $J_c(\phi_0)$. Like the single crystal results of Li {\it et al.} [Phys. Rev. Lett. {\bf 83}, 4160 (1999)], we show that the whisker data are unlikely to result from a predominantly d-wave order parameter. However, unlike the single crystals, these results, if correct, require the whisker c-axis transport to be coherent.Comment: 5 pages, 4 figures, accepted for publication in Physical Review

Heat capacity of the quantum magnet TiOCl

Measurements of the heat capacity C(T,H) of the one-dimensional quantum magnet TiOCl are presented for temperatures 2K < T < 300K and magnetic fields up to 5T. Distinct anomalies at 91K and 67K signal two subsequent phase transitions. The lower of these transitions clearly is of first order and seems to be related to the spin degrees of freedom. The transition at 92K probably involves the lattice and/or orbital moments. A detailed analysis of the data reveals that the entropy change through both transitions is surprisingly small (~ 0.1R), pointing to the existence strong fluctuations well into the non-ordered high-temperature phase. No significant magnetic field dependence was detected.Comment: 4 pages, 2 figure

Topological Strings and (Almost) Modular Forms

The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Gamma, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H^3(X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Gamma. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi-Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local P_2 and P_1 x P_1. As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold C^3/Z_3.Comment: 62 pages, 1 figure; v2: minor correction