Tumbling of a rigid rod in a shear flow

The tumbling of a rigid rod in a shear flow is analyzed in the high viscosity limit. Following Burgers, the Master Equation is derived for the probability distribution of the orientation of the rod. The equation contains one dimensionless number, the Weissenberg number, which is the ratio of the shear rate and the orientational diffusion constant. The equation is solved for the stationary state distribution for arbitrary Weissenberg numbers, in particular for the limit of high Weissenberg numbers. The stationary state gives an interesting flow pattern for the orientation of the rod, showing the interplay between flow due to the driving shear force and diffusion due to the random thermal forces of the fluid. The average tumbling time and tumbling frequency are calculated as a function of the Weissenberg number. A simple cross-over function is proposed which covers the whole regime from small to large Weissenberg numbers.Comment: 22 pages, 9 figure

Scaling in the vicinity of the four-state Potts fixed point

We study a self-dual generalization of the Baxter-Wu model, employing results obtained by transfer matrix calculations of the magnetic scaling dimension and the free energy. While the pure critical Baxter-Wu model displays the critical behavior of the four-state Potts fixed point in two dimensions, in the sense that logarithmic corrections are absent, the introduction of different couplings in the up- and down triangles moves the model away from this fixed point, so that logarithmic corrections appear. Real couplings move the model into the first-order range, away from the behavior displayed by the nearest-neighbor, four-state Potts model. We also use complex couplings, which bring the model in the opposite direction characterized by the same type of logarithmic corrections as present in the four-state Potts model. Our finite-size analysis confirms in detail the existing renormalization theory describing the immediate vicinity of the four-state Potts fixed point.Comment: 19 pages, 7 figure

d=2 transverse-field Ising model under the screw-boundary condition: An optimization of the screw pitch

A length-N spin chain with the \sqrt{N}(=v)-th neighbor interaction is identical to a two-dimensional (d=2) model under the screw-boundary (SB) condition. The SB condition provides a flexible scheme to construct a d\ge2 cluster from an arbitrary number of spins; the numerical diagonalization combined with the SB condition admits a potential applicability to a class of systems intractable with the quantum Monte Carlo method due to the negative-sign problem. However, the simulation results suffer from characteristic finite-size corrections inherent in SB. In order to suppress these corrections, we adjust the screw pitch v(N) so as to minimize the excitation gap for each N. This idea is adapted to the transverse-field Ising model on the triangular lattice with N\le32 spins. As a demonstration, the correlation-length critical exponent $\nu$ is analyzed in some detail

Effective Field Theory of the Zero-Temperature Triangular-Lattice Antiferromagnet: A Monte Carlo Study

Using a Monte Carlo coarse-graining technique introduced by Binder et al., we have explicitly constructed the continuum field theory for the zero-temperature triangular Ising antiferromagnet. We verify the conjecture that this is a gaussian theory of the height variable in the interface representation of the spin model. We also measure the height-height correlation function and deduce the stiffness constant. In addition, we investigate the nature of defect-defect interactions at finite temperatures, and find that the two-dimensional Coulomb gas scenario applies at low temperatures.Comment: 26 pages, 9 figure

Extended surface disorder in the quantum Ising chain

We consider random extended surface perturbations in the transverse field Ising model decaying as a power of the distance from the surface towards a pure bulk system. The decay may be linked either to the evolution of the couplings or to their probabilities. Using scaling arguments, we develop a relevance-irrelevance criterion for such perturbations. We study the probability distribution of the surface magnetization, its average and typical critical behaviour for marginal and relevant perturbations. According to analytical results, the surface magnetization follows a log-normal distribution and both the average and typical critical behaviours are characterized by power-law singularities with continuously varying exponents in the marginal case and essential singularities in the relevant case. For enhanced average local couplings, the transition becomes first order with a nonvanishing critical surface magnetization. This occurs above a positive threshold value of the perturbation amplitude in the marginal case.Comment: 15 pages, 10 figures, Plain TeX. J. Phys. A (accepted

The Magnetization of the 3D Ising Model

We present highly accurate Monte Carlo results for simple cubic Ising lattices containing up to $256^3$ spins. These results were obtained by means of the Cluster Processor, a newly built special-purpose computer for the Wolff cluster simulation of the 3D Ising model. We find that the magnetization $M(t)$ is perfectly described by $M(t)=(a_0-a_1 t^{\theta} - a_2 t) t^{\beta}$, where $t=(T_{\rm c}-T)/T_{\rm c}$, in a wide temperature range $0.0005 < t < 0.26$. If there exist corrections to scaling with higher powers of $t$, they are very small. The magnetization exponent is determined as $\beta=0.3269$ (6). An analysis of the magnetization distribution near criticality yields a new determination of the critical point: $K_{\rm c}=J/k_B T_{\rm c}=0.2216544$, with a standard deviation of $3\cdot 10^{-7}$.Comment: 7 pages, 5 Postscript figure

Geometric properties of two-dimensional O(n) loop configurations

We study the fractal geometry of O($n$) loop configurations in two dimensions by means of scaling and a Monte Carlo method, and compare the results with predictions based on the Coulomb gas technique. The Monte Carlo algorithm is applicable to models with noninteger $n$ and uses local updates. Although these updates typically lead to nonlocal modifications of loop connectivities, the number of operations required per update is only of order one. The Monte Carlo algorithm is applied to the O($n$) model for several values of $n$, including noninteger ones. We thus determine scaling exponents that describe the fractal nature of O($n$) loops at criticality. The results of the numerical analysis agree with the theoretical predictions.Comment: 18 pages, 6 figure

Monte Carlo Renormalization of the 3-D Ising model: Analyticity and Convergence

We review the assumptions on which the Monte Carlo renormalization technique is based, in particular the analyticity of the block spin transformations. On this basis, we select an optimized Kadanoff blocking rule in combination with the simulation of a d=3 Ising model with reduced corrections to scaling. This is achieved by including interactions with second and third neighbors. As a consequence of the improved analyticity properties, this Monte Carlo renormalization method yields a fast convergence and a high accuracy. The results for the critical exponents are y_H=2.481(1) and y_T=1.585(3).Comment: RevTeX, 4 PostScript file

Finite-size scaling and conformal anomaly of the Ising model in curved space

We study the finite-size scaling of the free energy of the Ising model on lattices with the topology of the tetrahedron and the octahedron. Our construction allows to perform changes in the length scale of the model without altering the distribution of the curvature in the space. We show that the subleading contribution to the free energy follows a logarithmic dependence, in agreement with the conformal field theory prediction. The conformal anomaly is given by the sum of the contributions computed at each of the conical singularities of the space, except when perfect order of the spins is precluded by frustration in the model.Comment: 4 pages, 4 Postscript figure