Quantum Belief Propagation

We present an accurate numerical algorithm, called quantum belief propagation (QBP), for simulation of one-dimensional quantum systems at non-zero temperature. The algorithm exploits the fact that quantum effects are short-range in these systems at non-zero temperature, decaying on a length scale inversely proportional to the temperature. We compare to exact results on a spin-1/2 Heisenberg chain. Even a very modest calculation, requiring diagonalizing only 10-by-10 matrices, reproduces the peak susceptibility with a relative error of less than $10^{-5}$, while more elaborate calculations further reduce the error.Comment: 4 pages, 1 figure; revised time estimates due to improved implementation. Typographical corrections to Eq. 7 made; thanks to David Poulin for pointing out the mistak

Disordered Topological Insulators via C∗C^*-Algebras

The theory of almost commuting matrices can be used to quantify topological obstructions to the existence of localized Wannier functions with time-reversal symmetry in systems with time-reversal symmetry and strong spin-orbit coupling. We present a numerical procedure that calculates a Z_2 invariant using these techniques, and apply it to a model of HgTe. This numerical procedure allows us to access sizes significantly larger than procedures based on studying twisted boundary conditions. Our numerical results indicate the existence of a metallic phase in the presence of scattering between up and down spin components, while there is a sharp transition when the system decouples into two copies of the quantum Hall effect. In addition to the Z_2 invariant calculation in the case when up and down components are coupled, we also present a simple method of evaluating the integer invariant in the quantum Hall case where they are decoupled.Comment: Added detail regarding the mapping of almost commuting unitary matrices to almost commuting Hermitian matrices that form an approximate representation of the sphere. 6 pages, 6 figure

Fractal to Nonfractal Phase Transition in the Dielectric Breakdown Model

A fast method is presented for simulating the dielectric-breakdown model using iterated conformal mappings. Numerical results for the dimension and for corrections to scaling are in good agreement with the recent RG prediction of an upper critical $\eta_c=4$, at which a transition occurs between branching fractal clusters and one-dimensional nonfractal clusters.Comment: 5 pages, 7 figures; corrections to scaling include

EAGLE ISS - A modular twin-channel integral-field near-IR spectrograph

The ISS (Integral-field Spectrograph System) has been designed as part of the EAGLE Phase A Instrument Study for the E-ELT. It consists of two input channels of 1.65x1.65 arcsec field-of-view, each reconfigured spatially by an image-slicing integral-field unit to feed a single near-IR spectrograph using cryogenic volume-phase-holographic (VPH) gratings to disperse the image spectrally. A 4k x 4k array detector array records the dispersed images. The optical design employs anamorphic magnification, image slicing, VPH gratings scanned with a novel cryo-mechanism and a three-lens camera. The mechanical implementation features IFU optics in Zerodur, a modular bench structure and a number of high-precision cryo-mechanisms.Comment: 12 pages, to be published in Proc SPIE 7735: Ground-based & Airborne Instrumentation for Astronomy II

Anomalies in the Entanglement Properties of the Square Lattice Heisenberg Model

We compute the bipartite entanglement properties of the spin-half square-lattice Heisenberg model by a variety of numerical techniques that include valence bond quantum Monte Carlo (QMC), stochastic series expansion QMC, high temperature series expansions and zero temperature coupling constant expansions around the Ising limit. We find that the area law is always satisfied, but in addition to the entanglement entropy per unit boundary length, there are other terms that depend logarithmically on the subregion size, arising from broken symmetry in the bulk and from the existence of corners at the boundary. We find that the numerical results are anomalous in several ways. First, the bulk term arising from broken symmetry deviates from an exact calculation that can be done for a mean-field Neel state. Second, the corner logs do not agree with the known results for non-interacting Boson modes. And, third, even the finite temperature mutual information shows an anomalous behavior as T goes to zero, suggesting that T->0 and L->infinity limits do not commute. These calculations show that entanglement entropy demonstrates a very rich behavior in d>1, which deserves further attention.Comment: 12 pages, 7 figures, 2 tables. Numerical values in Table I correcte

Nonlinear dynamics, rectification, and phase locking for particles on symmetrical two-dimensional periodic substrates with dc and circular ac drives

We investigate the dynamical motion of particles on a two-dimensional symmetric periodic substrate in the presence of both a dc drive along a symmetry direction of the periodic substrate and an additional circular ac drive. For large enough ac drives, the particle orbit encircles one or more potential maxima of the periodic substrate. In this case, when an additional increasing dc drive is applied in the longitudinal direction, the longitudinal velocity increases in a series of discrete steps that are integer multiples of the lattice constant of the substrate times the frequency. Fractional steps can also occur. These integer and fractional steps correspond to distinct stable dynamical orbits. A number of these phases also show a rectification in the positive or negative transverse direction where a non-zero transverse velocity occurs in the absence of a dc transverse drive. We map out the phase diagrams of the regions of rectification as a function of ac amplitude, and find a series of tongues. Most of the features, including the steps in the longitudinal velocity and the transverse rectification, can be captured with a simple toy model and by arguments from nonlinear maps. We have also investigated the effects of thermal disorder and incommensuration on the rectification phenomena, and find that for increasing disorder, the rectification regions are gradually smeared and the longitudinal velocity steps are no longer flat but show a linearly increasing velocity.Comment: 14 pages, 17 postscript figure

Random Vibrational Networks and Renormalization Group

We consider the properties of vibrational dynamics on random networks, with random masses and spring constants. The localization properties of the eigenstates contrast greatly with the Laplacian case on these networks. We introduce several real-space renormalization techniques which can be used to describe this dynamics on general networks, drawing on strong disorder techniques developed for regular lattices. The renormalization group is capable of elucidating the localization properties, and provides, even for specific network instances, a fast approximation technique for determining the spectra which compares well with exact results.Comment: 4 pages, 3 figure