Scaling Property of the F-AF Spin Chain Near the Exactly Solvable Point

We investigate the ground state of the $J_1$-$J_2$ spin-1/2 chain with $J_10$ in the case that the nearest-neighbor $J_1$ interaction in the $z$-direction has a weak anisotropy as $J_1 (1-\alpha)$. We perform a perturbational analysis for small $\alpha$ and $\lambda \equiv J_2- |J_1|/4$ with the exact solution of the unperturbed ground state for $\alpha = \lambda = 0$. The scaling property of the ground state energy is examined in detail. By the numerical diagonalization analysis of finite size systems, we found the phase boundary equation between the spin fluid and dimer phases as $\alpha_{\rm c} = 14 \lambda_{\rm c}^{4/3}$.Comment: 15 pages, 10 figure

Existence and regularity of mean curvature flow with transport term in higher dimensions

Given an initial $C^1$ hypersurface and a time-dependent vector field in a Sobolev space, we prove a time-global existence of a family of hypersurfaces which start from the given hypersurface and which move by the velocity equal to the mean curvature plus the given vector field. We show that the hypersurfaces are $C^1$ for a short time and, even after some singularities occur, almost everywhere $C^1$ away from higher multiplicity region.Comment: 60 page

Inversion Phenomena of the Anisotropies of the Hamiltonian and the Wave-Function in the Distorted Diamond Type Spin Chain

We investigate the ground-sate phase diagram of the XXZ version of the S=1/2 distorted diamond chain by use of the degenerate perturbation theory near the truncation point. In case of the XY-like interaction anisotropy, the phase diagram consists of the Neel phase and the spin-fluid phase. For the Ising-like interaction anisotropy case, it consists of three phases: the ferrimagnetic phase, the Neel phase and the spin-fluid phase. The magnetization in the ferrimagnetic phase is 1/3 of the saturation magnetization. The remarkable nature of the phase diagram is the existence of the Neel phase, although the interaction anisotropy is XY-like. And also, the spin-fluid phase appears in spite of the Ising-like interaction anisotropy. We call these regions "inversion regions".Comment: 7 pages, 4 figure

Convergence of phase-field approximations to the Gibbs-Thomson law

We prove the convergence of phase-field approximations of the Gibbs-Thomson law. This establishes a relation between the first variation of the Van-der-Waals-Cahn-Hilliard energy and the first variation of the area functional. We allow for folding of diffuse interfaces in the limit and the occurrence of higher-multiplicities of the limit energy measures. We show that the multiplicity does not affect the Gibbs-Thomson law and that the mean curvature vanishes where diffuse interfaces have collided. We apply our results to prove the convergence of stationary points of the Cahn-Hilliard equation to constant mean curvature surfaces and the convergence of stationary points of an energy functional that was proposed by Ohta-Kawasaki as a model for micro-phase separation in block-copolymers.Comment: 25 page

Convergence of the Allen-Cahn equation with Neumann boundary conditions

We study a singular limit problem of the Allen-Cahn equation with Neumann boundary conditions and general initial data of uniformly bounded energy. We prove that the time-parametrized family of limit energy measures is Brakke's mean curvature flow with a generalized right angle condition on the boundary.Comment: 26 pages, 1 figur