The quantum XY model was first proposed
[Matsubara:56a] in 1956 to study the lattice quantum fluids.
Later, high-temperature series studies raised the possibility of a
divergent susceptibility for the two-dimensional model. For the
classical planar model, the remarkable theory of Kosterlitz and
Thouless [Kosterlitz:73a] provided a clear physical picture and
correctly predicted a number of important properties. However, much
less is known about the quantum model. In fact, it has been
controversial. Using a large-order high-temperature expansion,
Rogiers, et al. [Rogiers:79a] suggested a second-order
transition at 
for spin-1/2. Later, real-space
renormalization group analysis was applied to the model with
contradictory and inconclusive results. DeRaedt, et al.
[DeRaedt:84a] then presented an exact solution and Monte Carlo
simulation, both based on the Suzuki-Trotter transformation with small
Trotter number m. Their results, both analytical and numerical,
supported an Ising-like (second-order) transition at the Ising
point 
, with a logarithmically
divergent specific heat. Loh, et al. [Loh:85a] simulated
the system with an improved technique. They found that specific peak
remains finite and argued that a phase transition occurs at
-0.5 by measuring the change of the ``twist energy'' from the
lattice to the 
lattice. The dispute between
DeRaedt, et al., and Loh, et al., centered on the
importance of using a large Trotter number m and the global updates
in small-size systems, which move the system from one subspace to
another. Recent attempts to solve this problem still add fuel to the
controversy.