In applied mathematics, the numerical sign problem is the problem of numerically evaluating the integral of a highly oscillatory function of a large number of variables.
Numerical methods fail because of the near-cancellation of the positive and negative contributions to the integral.
Each has to be integrated to very high precision in order for their difference to be obtained with useful accuracy.
It often arises in calculations of the properties of a quantum mechanical system with large number of strongly interacting fermions, or in field theories involving a non-zero density of strongly interacting fermions.
In physics the sign problem is typically (but not exclusively) encountered in calculations of the properties of a quantum mechanical system with large number of strongly interacting fermions, or in field theories involving a non-zero density of strongly interacting fermions.
Because the particles are strongly interacting, perturbation theory is inapplicable, and one is forced to use brute-force numerical methods.
So unless there are cancellations arising from some symmetry of the system, the quantum-mechanical sum over all multi-particle states involves an integral over a function that is highly oscillatory, hence hard to evaluate numerically, particularly in high dimension.
Since the dimension of the integral is given by the number of particles, the sign problem becomes severe in the thermodynamic limit.
The field-theoretic manifestation of the sign problem is discussed below.
The sign problem is one of the major unsolved problems in the physics of many-particle systems, impeding progress in many areas: [a]In a field-theory approach to multi-particle systems, the fermion density is controlled by the value of the fermion chemical potential
by summing over all classical field configurations, weighted by
(which may have been originally part of the theory, or have been produced by a Hubbard–Stratonovich transformation to make the fermion action quadratic) where
is a matrix that encodes how the fermions were coupled to the bosons.
can be calculated by performing the sum over field configurations numerically, using standard techniques such as Monte Carlo importance sampling.
, is in general a complex number, so Monte Carlo importance sampling cannot be used to evaluate the integral.
is real and positive, so 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
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
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[6]).
[7] The badness of the sign problem is then measured by which again goes to zero exponentially in the large-volume limit.
This leaves open the possibility that there may be solutions that work in specific cases, where the oscillations of the integrand have a structure that can be exploited to reduce the numerical errors.
In systems with a moderate sign problem, such as field theories at a sufficiently high temperature or in a sufficiently small volume, the sign problem is not too severe and useful results can be obtained by various methods, such as more carefully tuned reweighting, analytic continuation from imaginary
[3][9] There are various proposals for solving systems with a severe sign problem: