272 research outputs found
Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices
Let S be a flat surface of genus g with cone type singularities. Given a
bipartite graph G isoradially embedded in S, we define discrete analogs of the
2^{2g} Dirac operators on S. These discrete objects are then shown to converge
to the continuous ones, in some appropriate sense. Finally, we obtain necessary
and sufficient conditions on the pair (S,G) for these discrete Dirac operators
to be Kasteleyn matrices of the graph G. As a consequence, if these conditions
are met, the partition function of the dimer model on G can be explicitly
written as an alternating sum of the determinants of these 2^{2g} discrete
Dirac operators.Comment: 39 pages, minor change
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
Exact Ground States of Large Two-Dimensional Planar Ising Spin Glasses
Studying spin-glass physics through analyzing their ground-state properties
has a long history. Although there exist polynomial-time algorithms for the
two-dimensional planar case, where the problem of finding ground states is
transformed to a minimum-weight perfect matching problem, the reachable system
sizes have been limited both by the needed CPU time and by memory requirements.
In this work, we present an algorithm for the calculation of exact ground
states for two-dimensional Ising spin glasses with free boundary conditions in
at least one direction. The algorithmic foundations of the method date back to
the work of Kasteleyn from the 1960s for computing the complete partition
function of the Ising model. Using Kasteleyn cities, we calculate exact ground
states for huge two-dimensional planar Ising spin-glass lattices (up to
3000x3000 spins) within reasonable time. According to our knowledge, these are
the largest sizes currently available. Kasteleyn cities were recently also used
by Thomas and Middleton in the context of extended ground states on the torus.
Moreover, they show that the method can also be used for computing ground
states of planar graphs. Furthermore, we point out that the correctness of
heuristically computed ground states can easily be verified. Finally, we
evaluate the solution quality of heuristic variants of the Bieche et al.
approach.Comment: 11 pages, 5 figures; shortened introduction, extended results; to
appear in Physical Review E 7
Non-Abelian Anyons and Topological Quantum Computation
Topological quantum computation has recently emerged as one of the most
exciting approaches to constructing a fault-tolerant quantum computer. The
proposal relies on the existence of topological states of matter whose
quasiparticle excitations are neither bosons nor fermions, but are particles
known as {\it Non-Abelian anyons}, meaning that they obey {\it non-Abelian
braiding statistics}. Quantum information is stored in states with multiple
quasiparticles, which have a topological degeneracy. The unitary gate
operations which are necessary for quantum computation are carried out by
braiding quasiparticles, and then measuring the multi-quasiparticle states. The
fault-tolerance of a topological quantum computer arises from the non-local
encoding of the states of the quasiparticles, which makes them immune to errors
caused by local perturbations. To date, the only such topological states
thought to have been found in nature are fractional quantum Hall states, most
prominently the \nu=5/2 state, although several other prospective candidates
have been proposed in systems as disparate as ultra-cold atoms in optical
lattices and thin film superconductors. In this review article, we describe
current research in this field, focusing on the general theoretical concepts of
non-Abelian statistics as it relates to topological quantum computation, on
understanding non-Abelian quantum Hall states, on proposed experiments to
detect non-Abelian anyons, and on proposed architectures for a topological
quantum computer. We address both the mathematical underpinnings of topological
quantum computation and the physics of the subject using the \nu=5/2 fractional
quantum Hall state as the archetype of a non-Abelian topological state enabling
fault-tolerant quantum computation.Comment: Final Accepted form for RM
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