Spin stiffness and topological defects in two-dimensional frustrated spin systems

Abstract

Using a collective Monte Carlo algorithm we study the low-temperature and long-distance properties of two systems of two-dimensional classical tops. Both systems have the same spin-wave dynamics (low-temperature behavior) as a large class of Heisenberg frustrated spin systems. They are constructed so that to differ only by their topological properties. The spin-stiffnesses for the two systems of tops are calculated for different temperatures and different sizes of the sample. This allows to investigate the role of topological defects in frustrated spin systems. Comparisons with Renormalization Group results based on a Non Linear Sigma model approach and with the predictions of some simple phenomenological model taking into account the topological excitations are done.

Nonperturbative renormalization-group approach to frustrated magnets

“…This was, in particular, the position advocated by Azaria et al [151,161]. The outstanding fact is that although the SO(4) behavior has indeed been seen numerically in d = 2 [74,78], it actually does not exist far from two dimensions. This is clear since no such fixed point is found in d = 4 − ǫ and since, as already emphasized, the SO(4) behavior is not seen in any numerical or experimental data in d = 3.…”

“…It corresponds to the RP 3 = SO(4)/(SO(3) × Z Z 2 ) model. Note that, had we kept the microscopical coupling constants: P = diag(J, J, 0), the Hamiltonian (20) would be supplemented by terms breaking the SO(4) global symmetry and leaving untouched the Z Z 2 local symmetry which is the important point for our purpose (see [78] for details).…”

First-order transition induced by topological defects in theO(3)principal chiral model

“…However, there are arguments against the existence of a phase transition at a finite temperature based on the σ-model approach [20][21][22] . The σ-model is the effective theory describing low-energy (weak) fluctuations, so it also describes a low-temperature behavior.…”

Homotopy in statistical physics

“…In the nematic n = 3 case, evidence was presented for a transition described by a diverging correlation length and susceptibility but a cusp (as opposed to a divergence) in the specific heat was reported. Similar to the two-dimensional XY case, both the correlation length and the susceptibility appeared [56,60] (n → ∞); BKT or 2 nd -order transition for n = 3 [57,59,61,62]; No transition for n = 4 [65] O(n) 3 n = 2: Z, vortices 2 nd -order transitions (m = 1); n = 3: Z, (see [42] and references therein) monopoles (m = 0); no defects for n 4 RP n−1 3 n = 3 only: Z, 2 nd -order transition [51] (from for n 3 monopoles (m = 0); perturbation theory); n 3: Z2, vortices/ 1 st -order transition [56] (n → ∞); disclinations (m = 1) 1 st -order transition [48,52,53] (n = 3); 1 st -order transition [54,55] (n = 4)…”

“…However, while the work of [57,59,61,62] favours a second-order or BKT-type transition with divergent correlation length in the n = 3 case, and (see also [64]), another study [65] of the RP n−1 defects in a d = 2 model of tops favours the absence of a true phase transition there (at least for n = 4). Instead it is claimed that there is a crossover in the correlation length.…”