550 research outputs found
Exact Algorithm for Sampling the 2D Ising Spin Glass
A sampling algorithm is presented that generates spin glass configurations of
the 2D Edwards-Anderson Ising spin glass at finite temperature, with
probabilities proportional to their Boltzmann weights. Such an algorithm
overcomes the slow dynamics of direct simulation and can be used to study
long-range correlation functions and coarse-grained dynamics. The algorithm
uses a correspondence between spin configurations on a regular lattice and
dimer (edge) coverings of a related graph: Wilson's algorithm [D. B. Wilson,
Proc. 8th Symp. Discrete Algorithms 258, (1997)] for sampling dimer coverings
on a planar lattice is adapted to generate samplings for the dimer problem
corresponding to both planar and toroidal spin glass samples. This algorithm is
recursive: it computes probabilities for spins along a "separator" that divides
the sample in half. Given the spins on the separator, sample configurations for
the two separated halves are generated by further division and assignment. The
algorithm is simplified by using Pfaffian elimination, rather than Gaussian
elimination, for sampling dimer configurations. For n spins and given floating
point precision, the algorithm has an asymptotic run-time of O(n^{3/2}); it is
found that the required precision scales as inverse temperature and grows only
slowly with system size. Sample applications and benchmarking results are
presented for samples of size up to n=128^2, with fixed and periodic boundary
conditions.Comment: 18 pages, 10 figures, 1 table; minor clarification
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
Matching Kasteleyn Cities for Spin Glass Ground States
As spin glass materials have extremely slow dynamics, devious numerical
methods are needed to study low-temperature states. A simple and fast
optimization version of the classical Kasteleyn treatment of the Ising model is
described and applied to two-dimensional Ising spin glasses. The algorithm
combines the Pfaffian and matching approaches to directly strip droplet
excitations from an excited state. Extended ground states in Ising spin glasses
on a torus, which are optimized over all boundary conditions, are used to
compute precise values for ground state energy densities.Comment: 4 pages, 2 figures; minor clarification
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
Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions
The aim of the paper is to formulate a discrete analogue of the claim made by
Alvarez-Gaume et al., realizing the partition function of the free fermion on a
closed Riemann surface of genus g as a linear combination of 2^{2g} Pfaffians
of Dirac operators. Let G=(V,E) be a finite graph embedded in a closed Riemann
surface X of genus g, x_e the collection of independent variables associated
with each edge e of G (collected in one vector variable x) and S the set of all
2^{2g} Spin-structures on X. We introduce 2^{2g} rotations rot_s and (2|E|
times 2|E|) matrices D(s)(x), s in S, of the transitions between the oriented
edges of G determined by rotations rot_s. We show that the generating function
for the even subsets of edges of G, i.e., the Ising partition function, is a
linear combination of the square roots of 2^{2g} Ihara-Selberg functions
I(D(s)(x)) also called Feynman functions. By a result of Foata--Zeilberger
holds I(D(s)(x))= det(I-D'(s)(x)), where D'(s)(x) is obtained from D(s)(x) by
replacing some entries by 0. Thus each Feynman function is computable in
polynomial time. We suggest that in the case of critical embedding of a
bipartite graph G, the Feynman functions provide suitable discrete analogues
for the Pfaffians of discrete Dirac operators
The 3D Dimer and Ising Problems Revisited
We express the finite 3D Dimer partition function as a linear combination of
determinants of oriented adjacency matrices, and the finite 3D Ising partition
sum as a linear combination of products over aperiodic closed walks. The
methodology we use is embedding of cubic lattice on 2D surfaces of large genus
Planar Ising model at criticality: state-of-the-art and perspectives
In this essay, we briefly discuss recent developments, started a decade ago
in the seminal work of Smirnov and continued by a number of authors, centered
around the conformal invariance of the critical planar Ising model on
and, more generally, of the critical Z-invariant Ising model on
isoradial graphs (rhombic lattices). We also introduce a new class of
embeddings of general weighted planar graphs (s-embeddings), which might, in
particular, pave the way to true universality results for the planar Ising
model.Comment: 19 pages (+ references), prepared for the Proceedings of ICM2018.
Second version: two references added, a few misprints fixe
Lateral critical Casimir force in two-dimensional inhomogeneous Ising strip. Exact results
We consider two-dimensional Ising strip bounded by two planar, inhomogeneous
walls. The inhomogeneity of each wall is modeled by a magnetic field acting on
surface spins. It is equal to except for a group of sites where it
is equal to . The inhomogeneities of the upper and lower wall are shifted
with respect to each other by a lateral distance . Using exact
diagonalization of the transfer matrix, we study both the lateral and normal
critical Casimir forces as well as magnetization profiles for a wide range of
temperatures and system parameters. The lateral critical Casimir force tends to
reduce the shift between the inhomogeneities, and the excess normal force is
attractive. Upon increasing the shift we observe, depending on the
temperature, three different scenarios of breaking of the capillary bridge of
negative magnetization connecting the inhomogeneities of the walls across the
strip. As long as there exists a capillary bridge in the system, the magnitude
of the excess total critical Casimir force is almost constant, with its
direction depending on . By investigating the bridge morphologies we have
found a relation between the point at which the bridge breaks and the
inflection point of the force. We provide a simple argument that some of the
properties reported here should also hold for a whole range of different models
of the strip with the same type of inhomogeneity
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