Ordering of the Heisenberg Spin Glass in High Dimensions

TL;DR

This study uses Monte Carlo simulations to explore the ordering behavior of Heisenberg spin glasses in four and five dimensions, revealing different transition phenomena and suggesting four dimensions are near a marginal point.

Contribution

It provides new insights into the dimensional dependence of spin-glass and chiral-glass orderings in Heisenberg models through detailed numerical analysis.

Findings

Five-dimensional model shows a combined spin-glass and chiral-glass transition at finite temperature.

Four-dimensional model exhibits a chiral-glass transition without standard spin-glass order.

The critical region for the chiral-glass transition in four dimensions is very narrow, indicating proximity to marginal dimensionality.

Abstract

Ordering of the Heisenberg spin glass with the nearest-neighbor Gaussian coupling is investigated by equilibrium Monte Carlo simulations in four and five dimensions. Ordering of the mean-field Heisenberg spin-glass is also studied for comparison. Particular attention is paid to the nature of the spin-glass and the chiral-glass orderings. Our numerical data suggest that, in five dimensions, the model exhibits a single spin-glass transition at a finite temperature, where the spin-glass order accompanying the simultaneous chiral-glass order sets in. In four dimensions, by contrast, the model exhibits a chiral-glass transition at a finite temperature, not accompanying the standard spin-glass order. The critical region associated with the chiral-glass transition, however, is very narrow, suggesting that dimension four is close to the marginal dimensionality.