  Fractional derivatives are nonlocal differential operators of real order that
often appear in models of anomalous diffusion and a variety of nonlocal
phenomena. Recently, a version of the Schr\"odinger Equation containing a
fractional Laplacian has been proposed. In this work, we develop a Fractional
Path Integral Monte Carlo algorithm that can be used to study the finite
temperature behavior of the time-independent Fractional Schr\"odinger Equation
for a variety of potentials. In so doing, we derive an analytic form for the
finite temperature fractional free particle density matrix and demonstrate how
it can be sampled to acquire new sets of particle positions. We employ this
algorithm to simulate both the free particle and $^{4}$He (Aziz) Hamiltonians.
We find that the fractional Laplacian strongly encourages particle
delocalization, even in the presence of interactions, suggesting that
fractional Hamiltonians may manifest atypical forms of condensation. Our work
opens the door to studying fractional Hamiltonians with arbitrarily complex
potentials that escape analytical solutions.
