Cancellation time for correlated random walkers

Abstract

We have studied the dynamics of the correlated diffusion of pairs of random walkers of opposite signs. The use of populations of such pairs has been proposed for the Monte Carlo treatment of many-fermion systems, where the possibility of their cancellation might prevent the characteristic decay of the signal-to-noise ratio. For four model systems – free fermions, the harmonic oscillator, an N-body system of attractive and repulsive harmonic forces, and an extensive system interacting by Pöschl–Teller potentials – we have explored analytically and by computation the behavior of the time to cancellation as a function of initial conditions and, equally important, as a function of system size. We find that for these systems the computational efficiency does not decay either with large imaginary time or with large N.

Keywords:

• Quantum Monte Carlo
• Fermion Monte Carlo
• Diffusion Monte Carlo
• correlated walkers
• harmonic oscillator

Acknowledgments

We are honored to dedicate this paper to Luciano Reatto in recognition of his decades of contributions to condensed matter physics, for the deep theoretical insights he has derived from Quantum Monte Carlo, and for his friendship. We are grateful to Kevin Schmidt who called our attention to the need to investigate the time to cancellation in a systematic way. Joseph Carlson suggested the importance of studying the free-fermion problem as a basic challenge and as a source of insight. This paper has benefited greatly from critical comments by David Hardin. F.A.S. acknowledges financial support from the Spanish Ministerio de Educación during his stay at the Lawrence Livermore National Laboratory and from the project FIS2009-07390. F.A.S. and F.P. thank Eric Schwegler and the Quantum Simulation Group for their hospitality during visits to the Lawrence Livermore Laboratory. This work was performed under the auspices of the Lawrence Livermore National Security, LLC (LLNS) under Contract No. DE-AC52-07NA27344.