The challenge of the chiral Potts model

The chiral Potts model continues to pose particular challenges in statistical mechanics: it is ``exactly solvable'' in the sense that it satisfies the Yang-Baxter relation, but actually obtaining the solution is not easy. Its free energy was calculated in 1988 and the order parameter was conjectured in full generality a year later. However, a derivation of that conjecture had to wait until 2005. Here we discuss that derivation.Comment: 22 pages, 3 figures, 29 reference

The structure of the C4 cluster radical

The first infrared spectrum of gas phase, jet-cooled C4 has been measured by high resolution diode laser absorption spectroscopy. Twelve rovibrational transitions are assigned to the nu3(sigmau) antisymmetric stretch of linear 3Sigma - g C4. No evidence is observed for the bent structure of triplet C4 recently observed in a matrix study by Cheung and Graham [J. Chem. Phys. 91, 6664 (1989)]. Indeed, the measured band origin (1548.9368(21) cm^–1) and effective ground state C–C bond length [1.304 31(21)A] are consistent with several ab initio predictions of a rigid, linear, cumulenic structure for this cluster radical

The Utility of Coherent States and other Mathematical Methods in the Foundations of Affine Quantum Gravity

Affine quantum gravity involves (i) affine commutation relations to ensure metric positivity, (ii) a regularized projection operator procedure to accomodate first- and second-class quantum constraints, and (iii) a hard-core interpretation of nonlinear interactions to understand and potentially overcome nonrenormalizability. In this program, some of the less traditional mathematical methods employed are (i) coherent state representations, (ii) reproducing kernel Hilbert spaces, and (iii) functional integral representations involving a continuous-time regularization. Of special importance is the profoundly different integration measure used for the Lagrange multiplier (shift and lapse) functions. These various concepts are first introduced on elementary systems to help motivate their application to affine quantum gravity.Comment: 15 pages, Presented at the X-International Conference on Symmetry Methods in Physic

Derivation of the order parameter of the chiral Potts model

We derive the order parameter of the chiral Potts model, using the method of Jimbo et al. The result agrees with previous conjectures.Comment: Version 2 submitted 21 Feb 2005. It has 7 pages, 2 figures. The introduction has been expanded and a significant typographical error in eqn 23 has been correcte

Theory of size-dependent resonance Raman intensities in InP nanocrystals

The resonance Raman spectrum of InP nanocrystals is characterized by features ascribable to both longitudinal (LO) and transverse (TO) optical modes. The intensity ratio of these modes exhibits a strong size dependence. To calculate the size dependence of the LO and TO Raman cross sections, we combine existing models of Raman scattering, the size dependence of electronic and vibrational structure, and electron vibration coupling in solids. For nanocrystals with a radius >10 Å, both the LO and TO coupling strengths increase with increasing radius. This, together with an experimentally observed increase in the electronic dephasing rate with decreasing size, allows us to account for the observed ratio of LO/TO Raman intensities

The order parameter of the chiral Potts model

An outstanding problem in statistical mechanics is the order parameter of the chiral Potts model. An elegant conjecture for this was made in 1983. It has since been successfully tested against series expansions, but as far as the author is aware there is as yet no proof of the conjecture. Here we show that if one makes a certain analyticity assumption similar to that used to derive the free energy, then one can indeed verify the conjecture. The method is based on the ``broken rapidity line'' approach pioneered by Jimbo, Miwa and Nakayashiki.Comment: 29 pages, 7 figures. Citations made more explicit and some typos correcte