359 research outputs found
Quadri-tilings of the plane
We introduce {\em quadri-tilings} and show that they are in bijection with
dimer models on a {\em family} of graphs arising from rhombus
tilings. Using two height functions, we interpret a sub-family of all
quadri-tilings, called {\em triangular quadri-tilings}, as an interface model
in dimension 2+2. Assigning "critical" weights to edges of , we prove an
explicit expression, only depending on the local geometry of the graph ,
for the minimal free energy per fundamental domain Gibbs measure; this solves a
conjecture of \cite{Kenyon1}. We also show that when edges of are
asymptotically far apart, the probability of their occurrence only depends on
this set of edges. Finally, we give an expression for a Gibbs measure on the
set of {\em all} triangular quadri-tilings whose marginals are the above Gibbs
measures, and conjecture it to be that of minimal free energy per fundamental
domain.Comment: Revised version, minor changes. 30 pages, 13 figure
A Measure of data-collapse for scaling
Data-collapse is a way of establishing scaling and extracting associated
exponents in problems showing self-similar or self-affine characteristics as
e.g. in equilibrium or non-equilibrium phase transitions, in critical phases,
in dynamics of complex systems and many others. We propose a measure to
quantify the nature of data collapse. Via a minimization of this measure, the
exponents and their error-bars can be obtained. The procedure is illustrated by
considering finite-size-scaling near phase transitions and quite strikingly
recovering the exact exponents.Comment: 3 pages, revtex, 3 figures,2 in colour. Replaced by the proper
version - slightly longer and no mismatch of abstrac
A generalized Kac-Ward formula
The Kac-Ward formula allows to compute the Ising partition function on a
planar graph G with straight edges from the determinant of a matrix of size 2N,
where N denotes the number of edges of G. In this paper, we extend this formula
to any finite graph: the partition function can be written as an alternating
sum of the determinants of 2^{2g} matrices of size 2N, where g is the genus of
an orientable surface in which G embeds. We give two proofs of this generalized
formula. The first one is purely combinatorial, while the second relies on the
Fisher-Kasteleyn reduction of the Ising model to the dimer model, and on
geometric techniques. As a consequence of this second proof, we also obtain the
following fact: the Kac-Ward and the Fisher-Kasteleyn methods to solve the
Ising model are one and the same.Comment: 23 pages, 8 figures; minor corrections in v2; to appear in J. Stat.
Mech. Theory Ex
On measurement-based quantum computation with the toric code states
We study measurement-based quantum computation (MQC) using as quantum
resource the planar code state on a two-dimensional square lattice (planar
analogue of the toric code). It is shown that MQC with the planar code state
can be efficiently simulated on a classical computer if at each step of MQC the
sets of measured and unmeasured qubits correspond to connected subsets of the
lattice.Comment: 9 pages, 5 figure
Dimers and the Critical Ising Model on Lattices of genus>1
We study the partition function of both Close-Packed Dimers and the Critical
Ising Model on a square lattice embedded on a genus two surface. Using
numerical and analytical methods we show that the determinants of the Kasteleyn
adjacency matrices have a dependence on the boundary conditions that, for large
lattice size, can be expressed in terms of genus two theta functions. The
period matrix characterizing the continuum limit of the lattice is computed
using a discrete holomorphic structure. These results relate in a direct way
the lattice combinatorics with conformal field theory, providing new insight to
the lattice regularization of conformal field theories on higher genus Riemann
Surfaces.Comment: 44 pages, eps figures included; typos corrected, figure and comments
added to section
Critical frontier for the Potts and percolation models on triangular-type and kagome-type lattices II: Numerical analysis
In a recent paper (arXiv:0911.2514), one of us (FYW) considered the Potts
model and bond and site percolation on two general classes of two-dimensional
lattices, the triangular-type and kagome-type lattices, and obtained
closed-form expressions for the critical frontier with applications to various
lattice models. For the triangular-type lattices Wu's result is exact, and for
the kagome-type lattices Wu's expression is under a homogeneity assumption. The
purpose of the present paper is two-fold: First, an essential step in Wu's
analysis is the derivation of lattice-dependent constants for various
lattice models, a process which can be tedious. We present here a derivation of
these constants for subnet networks using a computer algorithm. Secondly, by
means of a finite-size scaling analysis based on numerical transfer matrix
calculations, we deduce critical properties and critical thresholds of various
models and assess the accuracy of the homogeneity assumption. Specifically, we
analyze the -state Potts model and the bond percolation on the 3-12 and
kagome-type subnet lattices , , for which the
exact solution is not known. To calibrate the accuracy of the finite-size
procedure, we apply the same numerical analysis to models for which the exact
critical frontiers are known. The comparison of numerical and exact results
shows that our numerical determination of critical thresholds is accurate to 7
or 8 significant digits. This in turn infers that the homogeneity assumption
determines critical frontiers with an accuracy of 5 decimal places or higher.
Finally, we also obtained the exact percolation thresholds for site percolation
on kagome-type subnet lattices for .Comment: 31 pages,8 figure
Crossing bonds in the random-cluster model
We derive the scaling dimension associated with crossing bonds in the
random-cluster representation of the two-dimensional Potts model, by means of a
mapping on the Coulomb gas. The scaling field associated with crossing bonds
appears to be irrelevant, on the critical as well as on the tricritical branch.
The latter result stands in a remarkable contrast with the existing result for
the tricritical O(n) model that crossing bonds are relevant. In order to obtain
independent confirmation of the Coulomb gas result for the crossing-bond
exponent, we perform a finite-size-scaling analysis based on numerical
transfer-matrix calculations.Comment: 2 figure
Disorder induced rounding of the phase transition in the large q-state Potts model
The phase transition in the q-state Potts model with homogeneous
ferromagnetic couplings is strongly first order for large q, while is rounded
in the presence of quenched disorder. Here we study this phenomenon on
different two-dimensional lattices by using the fact that the partition
function of the model is dominated by a single diagram of the high-temperature
expansion, which is calculated by an efficient combinatorial optimization
algorithm. For a given finite sample with discrete randomness the free energy
is a pice-wise linear function of the temperature, which is rounded after
averaging, however the discontinuity of the internal energy at the transition
point (i.e. the latent heat) stays finite even in the thermodynamic limit. For
a continuous disorder, instead, the latent heat vanishes. At the phase
transition point the dominant diagram percolates and the total magnetic moment
is related to the size of the percolating cluster. Its fractal dimension is
found d_f=(5+\sqrt{5})/4 and it is independent of the type of the lattice and
the form of disorder. We argue that the critical behavior is exclusively
determined by disorder and the corresponding fixed point is the isotropic
version of the so called infinite randomness fixed point, which is realized in
random quantum spin chains. From this mapping we conjecture the values of the
critical exponents as \beta=2-d_f, \beta_s=1/2 and \nu=1.Comment: 12 pages, 12 figures, version as publishe
Local Statistics of Realizable Vertex Models
We study planar "vertex" models, which are probability measures on edge
subsets of a planar graph, satisfying certain constraints at each vertex,
examples including dimer model, and 1-2 model, which we will define. We express
the local statistics of a large class of vertex models on a finite hexagonal
lattice as a linear combination of the local statistics of dimers on the
corresponding Fisher graph, with the help of a generalized holographic
algorithm. Using an torus to approximate the periodic infinite
graph, we give an explicit integral formula for the free energy and local
statistics for configurations of the vertex model on an infinite bi-periodic
graph. As an example, we simulate the 1-2 model by the technique of Glauber
dynamics
Potts and percolation models on bowtie lattices
We give the exact critical frontier of the Potts model on bowtie lattices.
For the case of , the critical frontier yields the thresholds of bond
percolation on these lattices, which are exactly consistent with the results
given by Ziff et al [J. Phys. A 39, 15083 (2006)]. For the Potts model on
the bowtie-A lattice, the critical point is in agreement with that of the Ising
model on this lattice, which has been exactly solved. Furthermore, we do
extensive Monte Carlo simulations of Potts model on the bowtie-A lattice with
noninteger . Our numerical results, which are accurate up to 7 significant
digits, are consistent with the theoretical predictions. We also simulate the
site percolation on the bowtie-A lattice, and the threshold is
. In the simulations of bond percolation and site
percolation, we find that the shape-dependent properties of the percolation
model on the bowtie-A lattice are somewhat different from those of an isotropic
lattice, which may be caused by the anisotropy of the lattice.Comment: 18 pages, 9 figures and 3 table
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