3,391 research outputs found
A nested sequence of projectors (2): Multiparameter multistate statistical models, Hamiltonians, S-matrices
Our starting point is a class of braid matrices, presented in a previous
paper, constructed on a basis of a nested sequence of projectors. Statistical
models associated to such matrices for odd are studied
here. Presence of free parameters is the crucial feature
of our models, setting them apart from other well-known ones. There are
possible states at each site. The trace of the transfer matrix is shown to
depend on parameters. For order , eigenvalues consitute
the trace and the remaining eigenvalues involving the full range of
parameters come in zero-sum multiplets formed by the -th roots of unity, or
lower dimensional multiplets corresponding to factors of the order when
is not a prime number. The modulus of any eigenvalue is of the form
, where is a linear combination of the free parameters,
being the spectral parameter. For a prime number an amusing
relation of the number of multiplets with a theorem of Fermat is pointed out.
Chain Hamiltonians and potentials corresponding to factorizable -matrices
are constructed starting from our braid matrices. Perspectives are discussed.Comment: 32 pages, no figure, few mistakes are correcte
Connected component identification and cluster update on GPU
Cluster identification tasks occur in a multitude of contexts in physics and
engineering such as, for instance, cluster algorithms for simulating spin
models, percolation simulations, segmentation problems in image processing, or
network analysis. While it has been shown that graphics processing units (GPUs)
can result in speedups of two to three orders of magnitude as compared to
serial codes on CPUs for the case of local and thus naturally parallelized
problems such as single-spin flip update simulations of spin models, the
situation is considerably more complicated for the non-local problem of cluster
or connected component identification. I discuss the suitability of different
approaches of parallelization of cluster labeling and cluster update algorithms
for calculations on GPU and compare to the performance of serial
implementations.Comment: 15 pages, 14 figures, one table, submitted to PR
The Statistics of the Number of Minima in a Random Energy Landscape
We consider random energy landscapes constructed from d-dimensional lattices
or trees. The distribution of the number of local minima in such landscapes
follows a large deviation principle and we derive the associated law exactly
for dimension 1. Also of interest is the probability of the maximum possible
number of minima; this probability scales exponentially with the number of
sites. We calculate analytically the corresponding exponent for the Cayley tree
and the two-leg ladder; for 2 to 5 dimensional hypercubic lattices, we compute
the exponent numerically and compare to the Cayley tree case.Comment: 18 pages, 8 figures, added background on landscapes and reference
Multicolored Temperley-Lieb lattice models. The ground state
Using inversion relation, we calculate the ground state energy for the
lattice integrable models, based on a recently obtained baxterization of non
trivial multicolored generalization of Temperley-Lieb algebras. The simplest
vertex and IRF models are analyzed and found to have a mass gap.Comment: 15 pages 2 figure
Algebras in Higher Dimensional Statistical Mechanics - the Exceptional Partition (MEAN Field) Algebras
We determine the structure of the partition algebra (a generalized
Temperley-Lieb algebra) for specific values of Q \in \C, focusing on the
quotient which gives rise to the partition function of site -state Potts
models (in the continuous formulation) in arbitrarily high lattice
dimensions (the mean field case). The algebra is non-semi-simple iff is a
non-negative integer less than . We determine the dimension of the key
irreducible representation in every specialization.Comment: 4 page
A hard-sphere model on generalized Bethe lattices: Statics
We analyze the phase diagram of a model of hard spheres of chemical radius
one, which is defined over a generalized Bethe lattice containing short loops.
We find a liquid, two different crystalline, a glassy and an unusual
crystalline glassy phase. Special attention is also paid to the close-packing
limit in the glassy phase. All analytical results are cross-checked by
numerical Monte-Carlo simulations.Comment: 24 pages, revised versio
Critical properties of joint spin and Fortuin-Kasteleyn observables in the two-dimensional Potts model
The two-dimensional Potts model can be studied either in terms of the
original Q-component spins, or in the geometrical reformulation via
Fortuin-Kasteleyn (FK) clusters. While the FK representation makes sense for
arbitrary real values of Q by construction, it was only shown very recently
that the spin representation can be promoted to the same level of generality.
In this paper we show how to define the Potts model in terms of observables
that simultaneously keep track of the spin and FK degrees of freedom. This is
first done algebraically in terms of a transfer matrix that couples three
different representations of a partition algebra. Using this, one can study
correlation functions involving any given number of propagating spin clusters
with prescribed colours, each of which contains any given number of distinct FK
clusters. For 0 <= Q <= 4 the corresponding critical exponents are all of the
Kac form h_{r,s}, with integer indices r,s that we determine exactly both in
the bulk and in the boundary versions of the problem. In particular, we find
that the set of points where an FK cluster touches the hull of its surrounding
spin cluster has fractal dimension d_{2,1} = 2 - 2 h_{2,1}. If one constrains
this set to points where the neighbouring spin cluster extends to infinity, we
show that the dimension becomes d_{1,3} = 2 - 2 h_{1,3}. Our results are
supported by extensive transfer matrix and Monte Carlo computations.Comment: 15 pages, 3 figures, 2 table
Statistical Mechanical Models and Topological Color Codes
We find that the overlapping of a topological quantum color code state,
representing a quantum memory, with a factorized state of qubits can be written
as the partition function of a 3-body classical Ising model on triangular or
Union Jack lattices. This mapping allows us to test that different
computational capabilities of color codes correspond to qualitatively different
universality classes of their associated classical spin models. By generalizing
these statistical mechanical models for arbitrary inhomogeneous and complex
couplings, it is possible to study a measurement-based quantum computation with
a color code state and we find that their classical simulatability remains an
open problem. We complement the meaurement-based computation with the
construction of a cluster state that yields the topological color code and this
also gives the possibility to represent statistical models with external
magnetic fields.Comment: Revtex4, color figures, submitted for publicatio
The Blob Algebra and the Periodic Temperley-Lieb Algebra
We determine the structure of two variations on the Temperley-Lieb algebra,
both used for dealing with special kinds of boundary conditions in statistical
mechanics models.
The first is a new algebra, the `blob' algebra (the reason for the name will
become obvious shortly!). We determine both the generic and all the exceptional
structures for this two parameter algebra. The second is the periodic
Temperley-Lieb algebra. The generic structure and part of the exceptional
structure of this algebra have already been studied. Here we complete the
analysis, using results from the study of the blob algebra.Comment: 12 page
On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain with integrable boundary
Motivated by a study of the crossing symmetry of the `gemini' representation
of the affine Hecke algebra we give a construction for crossing tensor space
representations of ordinary Hecke algebras. These representations build
solutions to the Yang--Baxter equation satisfying the crossing condition (that
is, integrable quantum spin chains). We show that every crossing representation
of the Temperley--Lieb algebra appears in this construction, and in particular
that this construction builds new representations. We extend these to new
representations of the blob algebra, which build new solutions to the Boundary
Yang--Baxter equation (i.e. open spin chains with integrable boundary
conditions).
We prove that the open spin chain Hamiltonian derived from Sklyanin's
commuting transfer matrix using such a solution can always be expressed as the
representation of an element of the blob algebra, and determine this element.
We determine the representation theory (irreducible content) of the new
representations and hence show that all such Hamiltonians have the same
spectrum up to multiplicity, for any given value of the algebraic boundary
parameter. (A corollary is that our models have the same spectrum as the open
XXZ chain with nondiagonal boundary -- despite differing from this model in
having reference states.) Using this multiplicity data, and other ideas, we
investigate the underlying quantum group symmetry of the new Hamiltonians. We
derive the form of the spectrum and the Bethe ansatz equations.Comment: 43 pages, multiple figure
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