8,372 research outputs found
Exact enumeration of Hamiltonian circuits, walks, and chains in two and three dimensions
We present an algorithm for enumerating exactly the number of Hamiltonian
chains on regular lattices in low dimensions. By definition, these are sets of
k disjoint paths whose union visits each lattice vertex exactly once. The
well-known Hamiltonian circuits and walks appear as the special cases k=0 and
k=1 respectively. In two dimensions, we enumerate chains on L x L square
lattices up to L=12, walks up to L=17, and circuits up to L=20. Some results
for three dimensions are also given. Using our data we extract several
quantities of physical interest
A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings
Combining tree decomposition and transfer matrix techniques provides a very
general algorithm for computing exact partition functions of statistical models
defined on arbitrary graphs. The algorithm is particularly efficient in the
case of planar graphs. We illustrate it by computing the Potts model partition
functions and chromatic polynomials (the number of proper vertex colourings
using Q colours) for large samples of random planar graphs with up to N=100
vertices. In the latter case, our algorithm yields a sub-exponential average
running time of ~ exp(1.516 sqrt(N)), a substantial improvement over the
exponential running time ~ exp(0.245 N) provided by the hitherto best known
algorithm. We study the statistics of chromatic roots of random planar graphs
in some detail, comparing the findings with results for finite pieces of a
regular lattice.Comment: 5 pages, 3 figures. Version 2 has been substantially expanded.
Version 3 shows that the worst-case running time is sub-exponential in the
number of vertice
Logarithmic observables in critical percolation
Although it has long been known that the proper quantum field theory
description of critical percolation involves a logarithmic conformal field
theory (LCFT), no direct consequence of this has been observed so far.
Representing critical bond percolation as the Q = 1 limit of the Q-state Potts
model, and analyzing the underlying S_Q symmetry of the Potts spins, we
identify a class of simple observables whose two-point functions scale
logarithmically for Q = 1. The logarithm originates from the mixing of the
energy operator with a logarithmic partner that we identify as the field that
creates two propagating clusters. In d=2 dimensions this agrees with general
LCFT results, and in particular the universal prefactor of the logarithm can be
computed exactly. We confirm its numerical value by extensive Monte-Carlo
simulations.Comment: 11 pages, 2 figures. V2: as publishe
An integrable spin chain for the SL(2,R)/U(1) black hole sigma model
We introduce a spin chain based on finite-dimensional spin-1/2 SU(2)
representations but with a non-hermitian `Hamiltonian' and show, using mostly
analytical techniques, that it is described at low energies by the SL(2,R)/U(1)
Euclidian black hole Conformal Field Theory. This identification goes beyond
the appearance of a non-compact spectrum: we are also able to determine the
density of states, and show that it agrees with the formulas in [J. Math. Phys.
42, 2961 (2001)] and [JHEP 04, 014 (2002)], hence providing a direct `physical
measurement' of the associated reflection amplitude.Comment: 6 pages, 3 figures, in RevTeX. Corrected some typo
Dense loops, supersymmetry, and Goldstone phases in two dimensions
Loop models in two dimensions can be related to O(N) models. The
low-temperature dense-loops phase of such a model, or of its reformulation
using a supergroup as symmetry, can have a Goldstone broken-symmetry phase for
N<2. We argue that this phase is generic for -2< N <2 when crossings of loops
are allowed, and distinct from the model of non-crossing dense loops first
studied by Nienhuis [Phys. Rev. Lett. 49, 1062 (1982)]. Our arguments are
supported by our numerical results, and by a lattice model solved exactly by
Martins et al. [Phys. Rev. Lett. 81, 504 (1998)].Comment: RevTeX, 5 pages, 3 postscript figure
Tetromino tilings and the Tutte polynomial
We consider tiling rectangles of size 4m x 4n by T-shaped tetrominoes. Each
tile is assigned a weight that depends on its orientation and position on the
lattice. For a particular choice of the weights, the generating function of
tilings is shown to be the evaluation of the multivariate Tutte polynomial
Z\_G(Q,v) (known also to physicists as the partition function of the Q-state
Potts model) on an (m-1) x (n-1) rectangle G, where the parameter Q and the
edge weights v can take arbitrary values depending on the tile weights.Comment: 8 pages, 6 figure
Simulations of energetic beam deposition: from picoseconds to seconds
We present a new method for simulating crystal growth by energetic beam
deposition. The method combines a Kinetic Monte-Carlo simulation for the
thermal surface diffusion with a small scale molecular dynamics simulation of
every single deposition event. We have implemented the method using the
effective medium theory as a model potential for the atomic interactions, and
present simulations for Ag/Ag(111) and Pt/Pt(111) for incoming energies up to
35 eV. The method is capable of following the growth of several monolayers at
realistic growth rates of 1 monolayer per second, correctly accounting for both
energy-induced atomic mobility and thermal surface diffusion. We find that the
energy influences island and step densities and can induce layer-by-layer
growth. We find an optimal energy for layer-by-layer growth (25 eV for Ag),
which correlates with where the net impact-induced downward interlayer
transport is at a maximum. A high step density is needed for energy induced
layer-by-layer growth, hence the effect dies away at increased temperatures,
where thermal surface diffusion reduces the step density. As part of the
development of the method, we present molecular dynamics simulations of single
atom-surface collisions on flat parts of the surface and near straight steps,
we identify microscopic mechanisms by which the energy influences the growth,
and we discuss the nature of the energy-induced atomic mobility
The staggered vertex model and its applications
New solvable vertex models can be easily obtained by staggering the spectral
parameter in already known ones. This simple construction reveals some
surprises: for appropriate values of the staggering, highly non-trivial
continuum limits can be obtained. The simplest case of staggering with period
two (the case) for the six-vertex model was shown to be related, in one
regime of the spectral parameter, to the critical antiferromagnetic Potts model
on the square lattice, and has a non-compact continuum limit. Here, we study
the other regime: in the very anisotropic limit, it can be viewed as a zig-zag
spin chain with spin anisotropy, or as an anyonic chain with a generic
(non-integer) number of species. From the Bethe-Ansatz solution, we obtain the
central charge , the conformal spectrum, and the continuum partition
function, corresponding to one free boson and two Majorana fermions. Finally,
we obtain a massive integrable deformation of the model on the lattice.
Interestingly, its scattering theory is a massive version of the one for the
flow between minimal models. The corresponding field theory is argued to be a
complex version of the Toda theory.Comment: 38 pages, 14 figures, 3 appendice
Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices
We present an efficient algorithm for computing the partition function of the
q-colouring problem (chromatic polynomial) on regular two-dimensional lattice
strips. Our construction involves writing the transfer matrix as a product of
sparse matrices, each of dimension ~ 3^m, where m is the number of lattice
spacings across the strip. As a specific application, we obtain the large-q
series of the bulk, surface and corner free energies of the chromatic
polynomial. This extends the existing series for the square lattice by 32
terms, to order q^{-79}. On the triangular lattice, we verify Baxter's
analytical expression for the bulk free energy (to order q^{-40}), and we are
able to conjecture exact product formulae for the surface and corner free
energies.Comment: 17 pages. Version 2: added 4 further term to the serie
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