  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.
