use for the model is to apply the original Hubbard model to bosons instead of fermions. In that case the interaction *U* suppresses multiple occupancy of sites and in the infinite *U* limit gives the “hard-core” boson mode. Such models have been used to model superfluid helium-four. Furthermore, the addition of disorder leads to the so-called dirty boson model used to describe, for example, the phenomenon where one class of cuprate superconductors becomes either superconducting or insulating as the temperature is lowered and depending on the degree of disorder.

A principal method to attack all such Hamiltonians is the quantum Monte Carlo (QMC) method, which is based on a stochastic approach. Unfortunately, the effort to construct a probability runs into a considerable obstacle in the Hubbard model; for some moves in the stochastic walk, the probability is negative, thereby precluding a probability interpretation. This so-called fermion sign problem has limited the application of QMC calculations, especially since it gets exponentially worse as the temperature is lowered. So far, every attempt to solve the fermion sign problem for lattice models has been either unsuccessful or so difficult to implement that it has not been attempted.

Another critical problem is a first-principles computation of the model parameters such as the onsite Coulomb interaction, let alone accurate estimates of what is being omitted in such a simple model. There have been pioneering studies using both quantum chemistry methods and local-density approximations, but these have clearly demonstrated the limitations of both. The connection from quantum chemical techniques to the construction of accurate models is an open question.

In addition to the lattice models described above, Monte Carlo methods (either variational or fixed-node diffusion) have been successfully applied to the treatment of real materials with long-range Coulomb interaction, such as the covalent semiconductor and metals. Extension of these methods to highly correlated systems is currently an active area of research. Another direction is a hybrid approach combining a Hubbard-type interaction with a standard local spin density functional Hamiltonian. This semiempirical approach allows interpretation of spectroscopic experiments.

Monte Carlo methods are intrinsically well suited to take advantage of the unprecedented increases in computing power afforded by emerging MPP environments. The computation time for path-integral Monte Carlo calculations scales as *N*^{3}*L*, where *N* is the number of sites and *L* is the number of “time slices” in the computations. In going to lower temperatures, *L* will increase and necessitate larger *N* in order to see longer-range correlations. But, in fact, the fermion sign problem has prevented large-scale application to low-temperature systems.

At present, there is essentially no agreement on any features of the ground state or the phase diagram of even the simplest Hubbard model as a function of the concentration of electrons. At low temperatures, the ill-conditioned nature of the matrices requires extensive computation, growing exponentially on small data sets. These potential complexities are reminiscent of quantum chromodynamics on a lattice problem, which has spawned both the extensive use of parallel computers and the development of special computer architectures to achieve multiteraflop speeds. The variational and fixed-node methods do not suffer from the sign problems but provide only a variational solution. In contrast with path integral methods, these calculations scale as *N*^{3}, where *N* is the number of electrons, and simulations with *N*~1,000 electrons have been performed.