  In a number of classical statistical-physical models, there exists a
characteristic dimensionality called the upper critical dimension above which
one observes the mean-field critical behavior. Instead of constructing
high-dimensional lattices, however, one can also consider infinite-dimensional
structures, and the question is whether this mean-field character extends to
quantum-mechanical cases as well. We therefore investigate the transverse-field
quantum Ising model on the globally coupled network and the Watts-Strogatz
small-world network by means of quantum Monte Carlo simulations and the
finite-size scaling analysis. We confirm that both the structures exhibit
critical behavior consistent with the mean-field description. In particular, we
show that the existing cumulant method has a difficulty in estimating the
correct dynamic critical exponent and suggest that an order parameter based on
the quantum-mechanical expectation value can be a practically useful numerical
observable to determine critical behavior when there is no well-defined
dimensionality.
