5,913 research outputs found
Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices
We present an efficient algorithm for computing the partition function of the
q-colouring problem (chromatic polynomial) on regular two-dimensional lattice
strips. Our construction involves writing the transfer matrix as a product of
sparse matrices, each of dimension ~ 3^m, where m is the number of lattice
spacings across the strip. As a specific application, we obtain the large-q
series of the bulk, surface and corner free energies of the chromatic
polynomial. This extends the existing series for the square lattice by 32
terms, to order q^{-79}. On the triangular lattice, we verify Baxter's
analytical expression for the bulk free energy (to order q^{-40}), and we are
able to conjecture exact product formulae for the surface and corner free
energies.Comment: 17 pages. Version 2: added 4 further term to the serie
Extended Scaling for the high dimension and square lattice Ising Ferromagnets
In the high dimension (mean field) limit the susceptibility and the second
moment correlation length of the Ising ferromagnet depend on temperature as
chi(T)=tau^{-1} and xi(T)=T^{-1/2}tau^{-1/2} exactly over the entire
temperature range above the critical temperature T_c, with the scaling variable
tau=(T-T_c)/T. For finite dimension ferromagnets temperature dependent
effective exponents can be defined over all T using the same expressions. For
the canonical two dimensional square lattice Ising ferromagnet it is shown that
compact "extended scaling" expressions analogous to the high dimensional limit
forms give accurate approximations to the true temperature dependencies, again
over the entire temperature range from T_c to infinity. Within this approach
there is no cross-over temperature in finite dimensions above which
mean-field-like behavior sets in.Comment: 6 pages, 6 figure
Construction of some missing eigenvectors of the XYZ spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix
We discuss an algebraic method for constructing eigenvectors of the transfer
matrix of the eight vertex model at the discrete coupling parameters. We
consider the algebraic Bethe ansatz of the elliptic quantum group for the case where the parameter satisfies for arbitrary integers , and . When or
is odd, the eigenvectors thus obtained have not been discussed previously.
Furthermore, we construct a family of degenerate eigenvectors of the XYZ spin
chain, some of which are shown to be related to the loop algebra
symmetry of the XXZ spin chain. We show that the dimension of some degenerate
eigenspace of the XYZ spin chain on sites is given by , if
is an even integer. The construction of eigenvectors of the transfer matrices
of some related IRF models is also discussed.Comment: 19 pages, no figure (revisd version with three appendices
Comment on `Series expansions from the corner transfer matrix renormalization group method: the hard-squares model'
Earlier this year Chan extended the low-density series for the hard-squares
partition function to 92 terms. Here we analyse this extended
series focusing on the behaviour at the dominant singularity which lies
on on the negative fugacity axis. We find that the series has a confluent
singularity of order 2 at with exponents and
. We thus confirm that the exponent has the exact
value as observed by Dhar.Comment: 5 pages, 1 figure, IoP macros. Expanded second and final versio
Information requirements for enterprise systems
In this paper, we discuss an approach to system requirements engineering, which is based on using models of the responsibilities assigned to agents in a multi-agency system of systems. The responsibility models serve as a basis for identifying the stakeholders that should be considered in establishing the requirements and provide a basis for a structured approach, described here, for information requirements elicitation. We illustrate this approach using a case study drawn from civil emergency management
Free Energy of the Eight Vertex Model with an Odd Number of Lattice Sites
We calculate the bulk contribution for the doubly degenerated largest
eigenvalue of the transfer matrix of the eight vertex model with an odd number
of lattice sites N in the disordered regime using the generic equation for
roots proposed by Fabricius and McCoy. We show as expected that in the
thermodynamic limit the result coincides with the one in the N even case.Comment: 11 pages LaTeX New introduction, Method change
Critical Point of a Symmetric Vertex Model
We study a symmetric vertex model, that allows 10 vertex configurations, by
use of the corner transfer matrix renormalization group (CTMRG), a variant of
DMRG. The model has a critical point that belongs to the Ising universality
class.Comment: 2 pages, 6 figures, short not
Dislocation-mediated melting of one-dimensional Rydberg crystals
We consider cold Rydberg atoms in a one-dimensional optical lattice in the
Mott regime with a single atom per site at zero temperature. An external laser
drive with Rabi frequency \Omega and laser detuning \Delta, creates Rydberg
excitations whose dynamics is governed by an effective spin-chain model with
(quasi) long-range interactions. This system possesses intrinsically a large
degree of frustration resulting in a ground-state phase diagram in the
(\Delta,\Omega) plane with a rich topology. As a function of \Delta, the
Rydberg blockade effect gives rise to a series of crystalline phases
commensurate with the optical lattice that form a so-called devil's staircase.
The Rabi frequency, \Omega, on the other hand, creates quantum fluctuations
that eventually lead to a quantum melting of the crystalline states. Upon
increasing \Omega, we find that generically a commensurate-incommensurate
transition to a floating Rydberg crystal occurs first, that supports gapless
phonon excitations. For even larger \Omega, dislocations within the floating
Rydberg crystal start to proliferate and a second,
Kosterlitz-Thouless-Nelson-Halperin-Young dislocation-mediated melting
transition finally destroys the crystalline arrangement of Rydberg excitations.
This latter melting transition is generic for one-dimensional Rydberg crystals
and persists even in the absence of an optical lattice. The floating phase and
the concomitant transitions can, in principle, be detected by Bragg scattering
of light.Comment: 21 pages, 9 figures; minor changes, published versio
Critical and Tricritical Hard Objects on Bicolorable Random Lattices: Exact Solutions
We address the general problem of hard objects on random lattices, and
emphasize the crucial role played by the colorability of the lattices to ensure
the existence of a crystallization transition. We first solve explicitly the
naive (colorless) random-lattice version of the hard-square model and find that
the only matter critical point is the non-unitary Lee-Yang edge singularity. We
then show how to restore the crystallization transition of the hard-square
model by considering the same model on bicolored random lattices. Solving this
model exactly, we show moreover that the crystallization transition point lies
in the universality class of the Ising model coupled to 2D quantum gravity. We
finally extend our analysis to a new two-particle exclusion model, whose
regular lattice version involves hard squares of two different sizes. The exact
solution of this model on bicolorable random lattices displays a phase diagram
with two (continuous and discontinuous) crystallization transition lines
meeting at a higher order critical point, in the universality class of the
tricritical Ising model coupled to 2D quantum gravity.Comment: 48 pages, 13 figures, tex, harvmac, eps
Study of the Potts Model on the Honeycomb and Triangular Lattices: Low-Temperature Series and Partition Function Zeros
We present and analyze low-temperature series and complex-temperature
partition function zeros for the -state Potts model with on the
honeycomb lattice and on the triangular lattice. A discussion is given
as to how the locations of the singularities obtained from the series analysis
correlate with the complex-temperature phase boundary. Extending our earlier
work, we include a similar discussion for the Potts model with on the
honeycomb lattice and with on the kagom\'e lattice.Comment: 33 pages, Latex, 9 encapsulated postscript figures, J. Phys. A, in
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