We present a stochastic method able to deal with complex Hermitian Hamiltonians where time reversal invariance is broken explicitly. We fix the phase of the wave function and show that the equation for the modulus can be solved by quantum Monte Carlo techniques. Then, any choice for its phase provides a variational upper bound for the ground state energy of the system. We apply the fixed-phase method to the 2D electron gas in the presence of a magnetic field with generalized periodic boundary conditions, where we study the transition between an incompressible \ensuremath{\nu}=1/m Laughlin liquid and a Wigner crystal.
We study the ground-state structural properties of a hydrogen molecule in superstrong magnetic fields (B\ensuremath{\approxeq}
1012 G) using quantum Monte Carlo (fixed-phase and variational) approaches. We determine that the ground state (spin-triplet) belongs to the sector of total (z-component) angular momentum M=-1
(3${\mathrm{\ensuremath{\Pi}}}_{0}$), meaning that paramagnetic contributions to the total energy cannot be neglected. This non-time-reversal invariant ground state has a strong interatomic interaction, suggesting that a hydrogen gas under the same physical conditions has a tendency to form strong bonded molecules.