Hikaru Kawamura

Ordering of the frustrated classical Heisenberg model on the triangular lattice with an incommensurate spiral structure is studied under magnetic fields by means of a mean-field analysis and a Monte Carlo simulation. Several types of multiple-q states including the Skyrmion-lattice state is observed in addition to the standard single-q state. In contrast to the Dzyaloshinskii-Moriya interaction driven system, the present model allows both Skyrmions and anti-Skyrmions, together with a new thermodynamic phase where Skyrmion and anti-Skyrmion lattices form a domain state.

Ordering process of the antiferromagnetic Heisenberg model on the two-dimensional triangular lattice is studied both by topological analysis of defects and by Monte Carlo simulation. It is shown that the order parameter space of this model is isomorphic to the three-dimensional rotation group SO(3) due to the inherent frustration effect. Homotopy analysis shows that the system bears a topologically stable point defect characterized by a two-valued topological quantum number and exhibits a phase transition driven by the dissociation of the vortices. A Monte Carlo study on the specific heat and the behavior of vortices strongly suggests the occurence of a Kosterlitz-Thouless-type phase transition. It is, however, argued that in contrast to the two-dimensional X Y model, the spin-correlation function decays exponentially even in the low-temperature phase. In order to distinguish the high- and low-temperature phases qualitatively, we introduce a “vorticity function” analogously to the Wilson loop in the quark-confinement problem in the lattice gauge theory. A discussion is made on possible interpretations of the experimental data for triangular lattice Heisenberg antiferromagnets.

Recent theoretical and experimental studies on the critical properties of frustrated antiferromagnets with the noncollinear spin order, including stacked-triangular antiferromagnets and helimagnets, are reviewed. Particular emphasis is put on the novel critical and multicritical behaviors exhibited by these magnets, together with an important role played by the ‘chirality’. Phase transitions and critical phenomena have been a central issue of statistical physics for many years. In particular, phase transitions of magnets or of ‘spin systems ’ have attracted special interest. Thanks to extensive theoretical and experimental studies, we now have rather good understanding of the nature of phase

Chiral phase transitions are analyzed by renormalization-group techniques on the basis of a standard Ginzburg-Landau-Wilson Hamiltonian for two real n-component fields a and b with quartic couplings u${(\mathrm{a}}^{2}$${+\mathrm{b}}^{2}$${)}^{2}$ and v[(a\ensuremath{\cdot}${\mathrm{b})}^{2}$${\mathrm{\ensuremath{-}}\mathrm{a}}^{2}$${\mathrm{b}}^{2}$]. This model is a natural extension of the usual ${\mathrm{cphi}}^{4}$ O(n) model: For v>0, it represents triangular antiferromagnets, helical magnets, and the A phase of superfluid $^{3}\mathrm{He}$. An \ensuremath{\epsilon}=4-d expansion reveals a new v>0, chiral fixed point which is stable for n>21.8-23.4\ensuremath{\epsilon}+O(${\ensuremath{\epsilon}}^{2}$). An antisymmetric chirality tensor, \ensuremath{\kappa}=${a}_{\ensuremath{\lambda}}$${b}_{\ensuremath{\mu}}$-${a}_{\ensuremath{\mu}}$${b}_{\ensuremath{\lambda}}$, is a new relevant operator at this fixed point. The associated exponents \ensuremath{\gamma} and \ensuremath{\nu} are smaller than the usual O(n) exponents. A 1/n expansion yields a continuous chiral transition for 2<d<4, the exponents \ensuremath{\gamma} and \ensuremath{\nu} again being smaller than in the O(n) case. The chiral crossover exponent, ${\ensuremath{\varphi}}_{\ensuremath{\kappa}}$, exceeds \ensuremath{\gamma} in both \ensuremath{\epsilon} and 1/n expansions. The spectrum of other leading scaling operators and their exponents is obtained. On the basis of comparisons with recent Monte Carlo and experimental results, it is argued that the chiral fixed point probably remains stable down to the physically relevant cases n=2 and 3 at d=3.

The present status of research and understanding regarding the dynamics and the statistical properties of earthquakes is reviewed, mainly from a statistical physical viewpoint. Emphasis is put both on the physics of friction and fracture, which provides a microscopic basis for our understanding of an earthquake instability, and on the statistical physical modelling of earthquakes, which provides macroscopic aspects of such phenomena. Recent numerical results from several representative models are reviewed, with attention to both their critical and their characteristic properties. Some of the relevant notions and related issues are highlighted, including the origin of power laws often observed in statistical properties of earthquakes, apparently contrasting features of characteristic earthquakes or asperities, the nature of precursory phenomena and nucleation processes, and the origin of slow earthquakes, etc.