Numerical sign problem


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.
The sign problem is one of the major unsolved problems in the physics of many-particle systems. 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.

Overview

In physics the sign problem is typically 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. Because the particles are fermions, their wavefunction changes sign when any two fermions are interchanged. 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:
In a field theory approach to multi-particle systems, the fermion density is controlled by the value of the fermion chemical potential. One evaluates the partition function by summing over all classical field configurations, weighted by where is the action of the configuration. The sum over fermion fields can be performed analytically, and one is left with a sum over the bosonic fields
where represents the measure for the sum over all configurations of the bosonic fields, weighted by
where is now the action of the bosonic fields, and is a matrix that encodes how the fermions were coupled to the bosons. The expectation value of an observable is therefore an average over all configurations weighted by
If is positive, then it can be interpreted as a probability measure, and can be calculated by performing the sum over field configurations numerically, using standard techniques such as Monte Carlo importance sampling.
The sign problem arises when is non-positive. This typically occurs in theories of fermions when the fermion chemical potential is nonzero, i.e. when there is a nonzero background density of fermions. If there is no particle-antiparticle symmetry, and, and hence the weight, 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 of the weight into the observable. For example, one could decompose the weighting function into its modulus and phase,
where 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,. However, the phase 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 : if it is much less than 1 then the sign problem is severe.
It can be shown that
where is the volume of the system, is the temperature, and 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. In general one could write
where can be any positive weighting function The badness of the sign problem is then measured by
which again goes to zero exponentially in the large-volume limit.

Methods for reducing the sign problem

The sign problem is NP-hard, implying that a full and generic solution of the sign problem would also solve all problems in the complexity class NP in polynomial time. If there are no polynomial-time solutions to NP problems, then there is no generic solution to the sign problem. 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 to real, or Taylor expansion in powers of.
There are various proposals for solving systems with a severe sign problem: