5,913 research outputs found

    Bulk, surface and corner free energy series for the chromatic polynomial on the square and triangular lattices

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    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

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    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

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    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 Eτ,η(sl2)E_{\tau, \eta}(sl_2) for the case where the parameter η\eta satisfies 2Nη=m1+m2τ2 N \eta = m_1 + m_2 \tau for arbitrary integers NN, m1m_1 and m2m_2. When m1m_1 or m2m_2 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 sl2sl_2 loop algebra symmetry of the XXZ spin chain. We show that the dimension of some degenerate eigenspace of the XYZ spin chain on LL sites is given by N2L/NN 2^{L/N}, if L/NL/N 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'

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    Earlier this year Chan extended the low-density series for the hard-squares partition function Îș(z)\kappa(z) to 92 terms. Here we analyse this extended series focusing on the behaviour at the dominant singularity zdz_d which lies on on the negative fugacity axis. We find that the series has a confluent singularity of order 2 at zdz_d with exponents Ξ=0.83333(2)\theta=0.83333(2) and Ξâ€Č=1.6676(3)\theta'= 1.6676(3). We thus confirm that the exponent Ξ\theta has the exact value 56\frac56 as observed by Dhar.Comment: 5 pages, 1 figure, IoP macros. Expanded second and final versio

    Information requirements for enterprise systems

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    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

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    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

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    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

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    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

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    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

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    We present and analyze low-temperature series and complex-temperature partition function zeros for the qq-state Potts model with q=4q=4 on the honeycomb lattice and q=3,4q=3,4 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 q=3q=3 on the honeycomb lattice and with q=3,4q=3,4 on the kagom\'e lattice.Comment: 33 pages, Latex, 9 encapsulated postscript figures, J. Phys. A, in pres
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