Numerical sign problem: Difference between revisions

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The '''numerical sign problem''' refers to the difficulty 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 [[fermion]]s, or in field theories involving a non-zero density of strongly- interacting fermions.

==The sign problem in physics==

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 [[fermion]]s, 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 (due to the symmetry of the wave function, see [[Pauli principle]]). 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, and 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:

* Condensed matter physics. It prevents the numerical solution of systems with a high density of strongly- correlated electrons, such as the [[Hubbard model]].<ref>E. Loh et al., "Sign problem in the numerical simulation of many-electron systems" [http://prb.aps.org/abstract/PRB/v41/i13/p9301_1 Phys. Rev. B 41, 9301–9307 (1990)]</ref>

* Nuclear physics. It prevents the ab-initio calculation of properties of [[nuclear matter]] and hence limits our understanding of [[atomic nucleus|nuclei]] and [[neutron star]]s.

* Particle physics. It prevents the use of [[Lattice QCD]] to predict the phases and properties of [[quark matter]].<ref name='Philipsen'>O. Philipsen, "Lattice calculations at non-zero chemical potential: The QCD phase diagram", [http://pos.sissa.it//archive/conferences/077/011/Confinement8_011.pdf PoS Confinement8 011 (2008)], Plenary talk at Quark Confinement and the Hadron Spectrum 8, Mainz, Germany, Sept 2008</ref>