Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial

We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as the solution of a special one-dimensional polymer model. We also obtain the large-q expansion of the bulk and surface (resp. corner) free energies for the zero-temperature antiferromagnet (= chromatic polynomial) through order q^{-47} (resp. q^{-46}). Finally, we compute chromatic roots for strips of widths 9 <= m <= 12 with free boundary conditions and locate roughly the limiting curves.Comment: 111 pages (LaTeX2e). Includes tex file, three sty files, and 19 Postscript figures. Also included are Mathematica files data_CYL.m and data_FREE.m. Many changes from version 1: new material on series expansions and their analysis, and several proofs of previously conjectured results. Final version to be published in J. Stat. Phy

Symmetries and Elasticity of Nematic Gels

A nematic liquid-crystal gel is a macroscopically homogeneous elastic medium with the rotational symmetry of a nematic liquid crystal. In this paper, we develop a general approach to the study of these gels that incorporates all underlying symmetries. After reviewing traditional elasticity and clarifying the role of broken rotational symmetries in both the reference space of points in the undistorted medium and the target space into which these points are mapped, we explore the unusual properties of nematic gels from a number of perspectives. We show how symmetries of nematic gels formed via spontaneous symmetry breaking from an isotropic gel enforce soft elastic response characterized by the vanishing of a shear modulus and the vanishing of stress up to a critical value of strain along certain directions. We also study the phase transition from isotropic to nematic gels. In addition to being fully consistent with approaches to nematic gels based on rubber elasticity, our description has the important advantages of being independent of a microscopic model, of emphasizing and clarifying the role of broken symmetries in determining elastic response, and of permitting easy incorporation of spatial variations, thermal fluctuations, and gel heterogeneity, thereby allowing a full statistical-mechanical treatment of these novel materials.Comment: 21 pages, 4 eps figure

Transfer matrices and partition-function zeros for antiferromagnetic Potts models. VI. Square lattice with special boundary conditions

We study, using transfer-matrix methods, the partition-function zeros of the square-lattice q-state Potts antiferromagnet at zero temperature (= square-lattice chromatic polynomial) for the special boundary conditions that are obtained from an m x n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column. We provide numerical evidence that the partition-function zeros are becoming dense everywhere in the complex q-plane outside the limiting curve B_\infty(sq) for this model with ordinary (e.g. free or cylindrical) boundary conditions. Despite this, the infinite-volume free energy is perfectly analytic in this region.Comment: 114 pages (LaTeX2e). Includes tex file, three sty files, and 23 Postscript figures. Also included are Mathematica files data_Eq.m, data_Neq.m,and data_Diff.m. Many changes from version 1, including several proofs of previously conjectured results. Final version to be published in J. Stat. Phy

Potts model on recursive lattices: some new exact results

We compute the partition function of the Potts model with arbitrary values of $q$ and temperature on some strip lattices. We consider strips of width $L_y=2$, for three different lattices: square, diced and `shortest-path' (to be defined in the text). We also get the exact solution for strips of the Kagome lattice for widths $L_y=2,3,4,5$. As further examples we consider two lattices with different type of regular symmetry: a strip with alternating layers of width $L_y=3$ and $L_y=m+2$, and a strip with variable width. Finally we make some remarks on the Fisher zeros for the Kagome lattice and their large q-limit.Comment: 17 pages, 19 figures. v2 typos corrected, title changed and references, acknowledgements and two further original examples added. v3 one further example added. v4 final versio

Limit on Bs0B^0_s oscillation using a jet charge method

A lower limit is set on the B_{s}^{0} meson oscillation parameter \Delta m_{s} using data collected from 1991 to 1994 by the ALEPH detector. Events with a high transverse momentum lepton and a reconstructed secondary vertex are used. The high transverse momentum leptons are produced mainly by b hadron decays, and the sign of the lepton indicates the particle/antiparticle final state in decays of neutral B mesons. The initial state is determined by a jet charge technique using both sides of the event. A maximum likelihood method is used to set a lower limit of \, \Delta m_{s}. The 95\% confidence level lower limit on \Delta m_s ranges between 5.2 and 6.5(\hbar/c^{2})~ps^{-1} when the fraction of b quarks from Z^0 decays that form B_{s}^{0} mesons is varied from 8\% to 16\%. Assuming that the B_{s}^{0} fraction is 12\%, the lower limit would be \Delta m_{s} 6.1(\hbar/c^{2})~ps^{-1} at 95\% confidence level. For x_s = \Delta m_s \, \tau_{B_s}, this limit also gives x_s 8.8 using the B_{s}^{0} lifetime of \tau_{B_s} = 1.55 \pm 0.11~ps and shifting the central value of \tau_{B_s} down by 1\sigma

Measurement of the Bs0^0_s lifetime and production rate with Ds−l+^-_s l^+ combinations in Z decays

The lifetime of the \bs meson is measured in approximately 3 million hadronic Z decays accumulated using the ALEPH detector at LEP from 1991 to 1994. Seven different \ds decay modes were reconstructed and combined with an opposite sign lepton as evidence of semileptonic \bs decays. Two hundred and eight \dsl candidates satisfy selection criteria designed to ensure precise proper time reconstruction and yield a measured \bs lifetime of \mbox{\result .} Using a larger, less constrained sample of events, the product branching ratio is measured to be \mbox{\pbrresult