8,125 research outputs found
Bethe Equations "on the Wrong Side of Equator"
We analyse the famous Baxter's equations for () spin chain
and show that apart from its usual polynomial (trigonometric) solution, which
provides the solution of Bethe-Ansatz equations, there exists also the second
solution which should corresponds to Bethe-Ansatz beyond . This second
solution of Baxter's equation plays essential role and together with the first
one gives rise to all fusion relations.Comment: 13 pages, original paper was spoiled during transmissio
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
General scalar products in the arbitrary six-vertex model
In this work we use the algebraic Bethe ansatz to derive the general scalar
product in the six-vertex model for generic Boltzmann weights. We performed
this calculation using only the unitarity property, the Yang-Baxter algebra and
the Yang-Baxter equation. We have derived a recurrence relation for the scalar
product. The solution of this relation was written in terms of the domain wall
partition functions. By its turn, these partition functions were also obtained
for generic Boltzmann weights, which provided us with an explicit expression
for the general scalar product.Comment: 24 page
Exact clesed form of the return probability on the Bethe lattice
An exact closed form solution for the return probability of a random walk on
the Bethe lattice is given. The long-time asymptotic form confirms a previously
known expression. It is however shown that this exact result reduces to the
proper expression when the Bethe lattice degenerates on a line, unlike the
asymptotic result which is singular. This is shown to be an artefact of the
asymptotic expansion. The density of states is also calculated.Comment: 7 pages, RevTex 3.0, 2 figures available upon request from
[email protected], to be published in J.Phys.A Let
A Potts/Ising Correspondence on Thin Graphs
We note that it is possible to construct a bond vertex model that displays
q-state Potts criticality on an ensemble of phi3 random graphs of arbitrary
topology, which we denote as ``thin'' random graphs in contrast to the fat
graphs of the planar diagram expansion.
Since the four vertex model in question also serves to describe the critical
behaviour of the Ising model in field, the formulation reveals an isomorphism
between the Potts and Ising models on thin random graphs. On planar graphs a
similar correspondence is present only for q=1, the value associated with
percolation.Comment: 6 pages, 5 figure
Avalanche Collapse of Interdependent Network
We reveal the nature of the avalanche collapse of the giant viable component
in multiplex networks under perturbations such as random damage. Specifically,
we identify latent critical clusters associated with the avalanches of random
damage. Divergence of their mean size signals the approach to the hybrid phase
transition from one side, while there are no critical precursors on the other
side. We find that this discontinuous transition occurs in scale-free multiplex
networks whenever the mean degree of at least one of the interdependent
networks does not diverge.Comment: 4 pages, 5 figure
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
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
Star-Triangle Relation for a Three Dimensional Model
The solvable -chiral Potts model can be interpreted as a
three-dimensional lattice model with local interactions. To within a minor
modification of the boundary conditions it is an Ising type model on the body
centered cubic lattice with two- and three-spin interactions. The corresponding
local Boltzmann weights obey a number of simple relations, including a
restricted star-triangle relation, which is a modified version of the
well-known star-triangle relation appearing in two-dimensional models. We show
that these relations lead to remarkable symmetry properties of the Boltzmann
weight function of an elementary cube of the lattice, related to spatial
symmetry group of the cubic lattice. These symmetry properties allow one to
prove the commutativity of the row-to-row transfer matrices, bypassing the
tetrahedron relation. The partition function per site for the infinite lattice
is calculated exactly.Comment: 20 pages, plain TeX, 3 figures, SMS-079-92/MRR-020-92. (corrupted
figures replaced
Hopping on the Bethe lattice: Exact results for densities of states and dynamical mean-field theory
We derive an operator identity which relates tight-binding Hamiltonians with
arbitrary hopping on the Bethe lattice to the Hamiltonian with nearest-neighbor
hopping. This provides an exact expression for the density of states (DOS) of a
non-interacting quantum-mechanical particle for any hopping. We present
analytic results for the DOS corresponding to hopping between nearest and
next-nearest neighbors, and also for exponentially decreasing hopping
amplitudes. Conversely it is possible to construct a hopping Hamiltonian on the
Bethe lattice for any given DOS. These methods are based only on the so-called
distance regularity of the infinite Bethe lattice, and not on the absence of
loops. Results are also obtained for the triangular Husimi cactus, a recursive
lattice with loops. Furthermore we derive the exact self-consistency equations
arising in the context of dynamical mean-field theory, which serve as a
starting point for studies of Hubbard-type models with frustration.Comment: 14 pages, 9 figures; introduction expanded, references added;
published versio
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