Constrained Path Monte Carlo method for strongly paired fermions

Shiwei Zhang

Shiwei Zhang

Siu A. Chin

Ethan Ethan, Jonathan L. DuBois, David M. Ceperley

Jodie Roberts, Beau A. Thompson, Rudolf Torsten Clay

Ground state properties of the Hubbard model are of fundamental importance to understand the mechanism of unconventional superconductivity in the high-T_c cuprates and other materials. One of the most powerful numerical methods for strongly interacting models is quantum Monte Carlo, which however faces a fundamental limitation, the Fermion sign problem. The sign problem can be mitigated using approximate methods such as Constrained Path Monte Carlo, where the sign of the overlap with a trial wavefunction is constrained to remain positive. Beyond the constraint, additional approximations must be made in order to measure observables for operators that do not commute with the Hamiltonian. We examine critically the most commonly used approximation, back propagation, as well as a recently proposed constraint release measurement technique. In comparisons with a variety of systems that can be solved numerically exactly by other methods, we find that back propagation tends to underestimate long-range superconducting pair-pair correlations. Except in special cases such as one dimension, back propagation introduces an uncontrolled approximation even when the exact trial wavefunction is used. The constraint release technique can provide accurate results, with the disadvantages that it much more computationally expensive and reintroduces the sign problem.

Joseph Carlson, James E. Gubernatis, Gerardo Ortíz +1 more

We report several important observations that underscore the distinctions between the constrained-path Monte Carlo method and the continuum and lattice versions of the fixed-node method. The main distinctions stem from the differences in the state space in which the random walk occurs and in the manner in which the random walkers are constrained. One consequence is that in the constrained-path method the so-called mixed estimator for the energy is not an upper bound to the exact energy, as previously claimed. Several ways of producing an energy upper bound are given, and relevant methodological aspects are illustrated with simple examples.