A New Method to Calculate the Spin-Glass Order Parameter of the Two-Dimensional +/-J Ising Model

A new method to numerically calculate the $n$th moment of the spin overlap of the two-dimensional $\pm J$ Ising model is developed using the identity derived by one of the authors (HK) several years ago. By using the method, the $n$th moment of the spin overlap can be calculated as a simple average of the $n$th moment of the total spins with a modified bond probability distribution. The values of the Binder parameter etc have been extensively calculated with the linear size, $L$, up to L=23. The accuracy of the calculations in the present method is similar to that in the conventional transfer matrix method with about $10^{5}$ bond samples. The simple scaling plots of the Binder parameter and the spin-glass susceptibility indicate the existence of a finite-temperature spin-glass phase transition. We find, however, that the estimation of $T_{\rm c}$ is strongly affected by the corrections to scaling within the present data ($L\leq 23$). Thus, there still remains the possibility that $T_{\rm c}=0$, contrary to the recent results which suggest the existence of a finite-temperature spin-glass phase transition.Comment: 10 pages,8 figures: final version to appear in J. Phys.

Crystal engineering rescues a solution organic synthesis in a cocrystallization that confirms the configuration of a molecular ladder

Treatment of an achiral molecular ladder of C2h symmetry composed of five edge-sharing cyclobutane rings, or a [5]-ladderane, with acid results in cis- to trans-isomerization of end pyridyl groups. Solution NMR spectroscopy and quantum chemical calculations support the isomerization to generate two diastereomers. The NMR data, however, could not lead to unambiguous configurational assignments of the two isomers. Single-crystal X-ray diffraction was employed to determine each configuration. One isomer readily crystallized as a pure form and X-ray diffraction revealed the molecule as being achiral based on Ci symmetry. The second isomer resisted crystallization under a variety of conditions. Consequently, a strategy based on a cocrystallization was developed to generate single crystals of the second isomer. Cocrystallization of the isomer with a carboxylic acid readily afforded single crystals that confirmed a chiral ladderane based on C2 symmetry. The chiral ladderane and acid self-assembled to generate a five-component hydrogen-bonded complex that packs to form large solvent-filled homochiral channels of nanometer-scale dimensions. Whereas cocrystallizations are frequently applied to structure determinations of proteins, our study represents the first application of a cocrystallization to confirm the relative configuration of a small-molecule diastereomer generated in a solution-phase organic synthesis