Numerical sign problem: Difference between revisions

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==The sign problem in field theory==

{{efn|Sources for this section include Chandrasekharan & Wiese (1999)<ref name='Wiese-cluster'/> and Kieu & Griffin (1994),<ref name='Kieu'/> in addition to those cited.}}In a field -theory approach to multi-particle systems, the fermion density is controlled by the value of the fermion [[chemical potential]] <math>\mu</math>. One evaluates the [[Partition function (quantum field theory)|partition function]] <math>Z</math> by summing over all classical field configurations, weighted by <math>\exp(-S)</math>, where <math>S</math> is the [[Action (physics)|action]] of the configuration. The sum over fermion fields can be performed analytically, and one is left with a sum over the [[boson]]ic fields <math>\sigma</math> (which may have been originally part of the theory, or have been produced by a [[Hubbard–Stratonovich transformation]] to make the fermion action quadratic)

:<math>Z = \int D \sigma \;, \rho[\sigma],</math>

where <math>D \sigma</math> represents the measure for the sum over all configurations <math>\sigma(x)</math> of the bosonic fields, weighted by

:<math>\rho[\sigma] = \det(M(\mu,\sigma)) \exp(-S[\sigma]),</math>

where <math>S</math> is now the action of the bosonic fields, and <math>M(\mu,\sigma)</math> is a matrix that encodes how the fermions were coupled to the bosons. The expectation value of an observable <math>A[\sigma]</math> is therefore an average over all configurations weighted by <math>\rho[\sigma]</math>:

:<math>

\langle A \rangle_\rho = \frac{\int D \sigma \;, A[\sigma] \;, \rho[\sigma]}{\int D \sigma \;, \rho[\sigma]} .

</math>

If <math>\rho[\sigma]</math> is positive, then it can be interpreted as a probability measure, and <math>\langle A \rangle_\rho</math> can be calculated by performing the sum over field configurations numerically, using standard techniques such as [[Monte Carlo integration|Monte Carlo importance sampling]].

The sign problem arises when <math>\rho[\sigma]</math> is non-positive. This typically occurs in theories of fermions when the fermion chemical potential <math>\mu</math> is nonzero, i.e. when there is a nonzero background density of fermions. If <math>\mu \neq 0</math>, there is no particle-antiparticleparticle–antiparticle symmetry, and <math>\det(M(\mu,\sigma))</math>, and hence the weight <math>\rho(\sigma)</math>, is in general a complex number, so Monte Carlo importance sampling cannot be used to evaluate the integral.

=== 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 into the observable. For example, one could decompose the weighting function into its modulus and phase,:

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

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

:<math> \langle A \rangle_\rho

= \frac{ \int D\sigma A[\sigma] \exp(i\theta[\sigma])\;, p[\sigma]}{\int D\sigma \exp(i\theta[\sigma])\;, p[\sigma]}

= \frac{ \langle A[\sigma] \exp(i\theta[\sigma]) \rangle_p}{ \langle \exp(i\theta[\sigma]) \rangle_p} .</math>

Note that the desired expectation value is now a ratio where the numerator and denominator are expectation values that both use a positive weighting function, <math>p[\sigma]</math>. However, the phase <math>\exp(i\theta[\sigma])</math> is a highly oscillatory function in the configuration space, so if one uses Monte Carlo methods to evaluate the numerator and denominator, each of them will evaluate to a very small number, whose exact value is swamped by the noise inherent in the Monte Carlo sampling process. The "badness" of the sign problem is measured by the smallness of the denominator <math>\langle \exp(i\theta[\sigma]) \rangle_p</math>: if it is much less than 1, then the sign problem is severe.

It can be shown (e.g.<ref name='Wiese-cluster'/>) that

:<math>\langle \exp(i\theta[\sigma]) \rangle_p \propto \exp(-f V/T),</math>

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 into 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'>{{cite journal |doi=10.1103/PhysRevE.49.3855 |pmid=9961673 |arxiv=hep-lat/9311072 |bibcode=1994PhRvE..49.3855K |title=Monte Carlo simulations with indefinite and complex-valued measures |journal=Physical Review E |volume=49 |issue=5 |pages=3855–3859 |year=1994 |last1=Kieu |first1=T. D. |last2=Griffin |first2=C. J.|s2cid=46652412 }}</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>{{Cite journal |arxiv=hep-lat/9705042 |last1=Barbour |first1=I. M. |title=Results on Finite Density QCD |journal=Nuclear Physics B - Proceedings Supplements |volume=60 |issue=1998 |pages=220–233 |last2=Morrison |first2=S. E. |last3=Klepfish |first3=E. G. |last4=Kogut |first4=J. B. |last5=Lombardo |first5=M.-P. |doi=10.1016/S0920-5632(97)00484-2 |year=1998|bibcode=1998NuPhS..60..220B |s2cid=16172956 }}</ref> The badness of the sign problem is then measured by

:<math>\left\langle \frac{\rho[\sigma]}{p[\sigma]}\right\rangle_p \propto \exp(-f V/T),</math>

which again goes to zero exponentially in the large-volume limit.