Algorithms for entanglement renormalization

We describe an iterative method to optimize the multi-scale entanglement renormalization ansatz (MERA) for the low-energy subspace of local Hamiltonians on a D-dimensional lattice. For translation invariant systems the cost of this optimization is logarithmic in the linear system size. Specialized algorithms for the treatment of infinite systems are also described. Benchmark simulation results are presented for a variety of 1D systems, namely Ising, Potts, XX and Heisenberg models. The potential to compute expected values of local observables, energy gaps and correlators is investigated.Comment: 23 pages, 28 figure

Entropy of Folding of the Triangular Lattice

The problem of counting the different ways of folding the planar triangular lattice is shown to be equivalent to that of counting the possible 3-colorings of its bonds, a dual version of the 3-coloring problem of the hexagonal lattice solved by Baxter. The folding entropy Log q per triangle is thus given by Baxter's formula q=sqrt(3)(Gamma[1/3])^(3/2)/2pi =1.2087...Comment: 9 pages, harvmac, epsf, uuencoded, 5 figures included, Saclay preprint T/9401

The Initial Conditions of Clustered Star Formation. II. N2H+ Observations of the Ophiuchus B Core

We present a Nobeyama 45 m Radio Telescope map and Australia Telescope Compact Array pointed observations of N2H+ 1-0 emission towards the clustered, low mass star forming Oph B Core within the Ophiuchus molecular cloud. We compare these data with previously published results of high resolution NH3 (1,1) and (2,2) observations in Oph B. We use 3D Clumpfind to identify emission features in the single-dish N2H+ map, and find that the N2H+ `clumps' match well similar features previously identified in NH3 (1,1) emission, but are frequently offset to clumps identified at similar resolution in 850 micron continuum emission. Wide line widths in the Oph B2 sub-Core indicate non-thermal motions dominate the Core kinematics, and remain transonic at densities n ~ 3 x 10^5 cm^-3 with large scatter and no trend with N(H2). Non-thermal motions in Oph B1 and B3 are subsonic with little variation, but also show no trend with H2 column density. Over all Oph B, non-thermal N2H+ line widths are substantially narrower than those traced by NH3, making it unlikely NH3 and N2H+ trace the same material, but the v_LSR of both species agree well. We find evidence for accretion in Oph B1 from the surrounding ambient gas. The NH3/N2H+ abundance ratio is larger towards starless Oph B1 than towards protostellar Oph B2, similar to recent observational results in other star-forming regions. Small-scale structure is found in the ATCA N2H+ 1-0 emission, where emission peaks are again offset from continuum emission. In particular, the ~1 M_Sun B2-MM8 clump is associated with a N2H+ emission minimum and surrounded by a broken ring-like N2H+ emission structure, suggestive of N2H+ depletion. We find a strong general trend of decreasing N2H+ abundance with increasing N(H2) in Oph B which matches that found for NH3.Comment: 55 pages (manuscript), 15 figures, ApJ accepte

Critical and Multicritical Semi-Random (1+d)-Dimensional Lattices and Hard Objects in d Dimensions

We investigate models of (1+d)-D Lorentzian semi-random lattices with one random (space-like) direction and d regular (time-like) ones. We prove a general inversion formula expressing the partition function of these models as the inverse of that of hard objects in d dimensions. This allows for an exact solution of a variety of new models including critical and multicritical generalized (1+1)-D Lorentzian surfaces, with fractal dimensions $d_F=k+1$, k=1,2,3,..., as well as a new model of (1+2)-D critical tetrahedral complexes, with fractal dimension $d_F=12/5$. Critical exponents and universal scaling functions follow from this solution. We finally establish a general connection between (1+d)-D Lorentzian lattices and directed-site lattice animals in (1+d) dimensions.Comment: 44 pages, 15 figures, tex, harvmac, epsf, references adde

Abundant cyanopolyynes as a probe of infall in the Serpens South cluster-forming region

We have detected bright HC7N J = 21-20 emission toward multiple locations in the Serpens South cluster-forming region using the K-Band Focal Plane Array at the Robert C. Byrd Green Bank Telescope. HC7N is seen primarily toward cold filamentary structures that have yet to form stars, largely avoiding the dense gas associated with small protostellar groups and the main central cluster of Serpens South. Where detected, the HC7N abundances are similar to those found in other nearby star forming regions. Toward some HC7N `clumps', we find consistent variations in the line centroids relative to NH3 (1,1) emission, as well as systematic increases in the HC7N non-thermal line widths, which we argue reveal infall motions onto dense filaments within Serpens South with minimum mass accretion rates of M ~ 2-5 M_sun Myr^-1. The relative abundance of NH3 to HC7N suggests that the HC7N is tracing gas that has been at densities n ~ 10^4 cm^-3, for timescales t < 1-2 x 10^5 yr. Since HC7N emission peaks are rarely co-located with those of either NH3 or continuum, it is likely that Serpens South is not particularly remarkable in its abundance of HC7N, but instead the serendipitous mapping of HC7N simultaneously with NH3 has allowed us to detect HC7N at low abundances in regions where it otherwise may not have been looked for. This result extends the known star-forming regions containing significant HC7N emission from typically quiescent regions, like the Taurus molecular cloud, to more complex, active environments.Comment: 19 pages, 13 figures, accepted to MNRAS. Version with full resolution figures available at http://www.dunlap.utoronto.ca/~friesen/Friesen_HC7N.pd

Thermopower in the Coulomb blockade regime for Laughlin quantum dots

Using the conformal field theory partition function of a Coulomb-blockaded quantum dot, constructed by two quantum point contacts in a Laughlin quantum Hall bar, we derive the finite-temperature thermodynamic expression for the thermopower in the linear-response regime. The low-temperature results for the thermopower are compared to those for the conductance and their capability to reveal the structure of the single-electron spectrum in the quantum dot is analyzed.Comment: 11 pages, 3 figures, Proceedings of the 10-th International Workshop "Lie Theory and Its Applications in Physics", 17-23 June 2013, Varna, Bulgari

A transfer matrix approach to the enumeration of plane meanders

A closed plane meander of order $n$ is a closed self-avoiding curve intersecting an infinite line $2n$ times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm, based on transfer matrix methods, for the enumeration of plane meanders. While the algorithm has exponential complexity, its rate of growth is much smaller than that of previous algorithms. The algorithm is easily modified to enumerate various systems of closed meanders, semi-meanders, open meanders and many other geometries.Comment: 13 pages, 9 eps figures, to appear in J. Phys.

A path model for Whittaker vectors

40 pages, 2 figuresIn this paper we construct weighted path models to compute Whittaker vectors in the completion of Verma modules, as well as Whittaker functions of fundamental type, for all finite-dimensional simple Lie algebras, affine Lie algebras, and the quantum algebra $U_q(\mathfrak{sl}_{r+1})$. This leads to series expressions for the Whittaker functions. We show how this construction leads directly to the quantum Toda equations satisfied by these functions, and to the $q$-difference equations in the quantum case. We investigate the critical limit of affine Whittaker functions computed in this way