Numerical sign problem: Difference between revisions

Revision as of 03:41, 21 April 2013

Revision as of 19:34, 22 April 2013

Line 34:

=== Reweighting procedure ===

A field theory with a non-positive weight can be transformed to one with a positive weight, by incorporating the non-positive part (sign or complex phase) of the weight in tointo the observable. For example, one could decompose the weighting function in tointo its modulus and phase,

:<math>\rho[\sigma] = p[\sigma]\, \exp(i\theta[\sigma])</math>

where <math>p[\sigma]</math> is real and positive, so

Line 46:

where <math>V</math> is the volume of the system, <math>T</math> is the temperature, and <math>f</math> is an energy density. The number of Monte-Carlo sampling points needed to obtain an accurate result therefore rises exponentially as the volume of the system becomes large, and as the temperature goes to zero.

The decomposition of the weighting function in tointo modulus and phase is just one example (although it has been advocated as the optimal choice since it minimizes the variance of the denominator <ref name='Kieu'>T. D. Kieu and C. J. Griffin, "Monte Carlo simulations with indefinite and complex-valued measures", [http://pre.aps.org/abstract/PRE/v49/i5/p3855_1 Phys. Rev. E 49, 3855–3859 (1994)]</ref>). In general one could write

:<math>\rho[\sigma] = p[\sigma] \frac{\rho[\sigma]}{p[\sigma]}</math>

where <math>p[\sigma]</math> can be any positive weighting function (for example, the weighting function of the <math>\mu=0</math> theory.)<ref>I. Barbour et al, "Results on finite density QCD", Nucl. Phys. Proc. Suppl. 60A 220-234 (1998), [http://arxiv.org/abs/hep-lat/9705042 arXiv:hep-lat/9705042], presented at International Workshop on Lattice QCD on Parallel Computers, Tsukuba, Japan</ref> The badness of the sign problem is then measured by