In
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemat ...
, the numerical sign problem is the problem of numerically evaluating the
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of a highly
oscillatory
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
of a large number of variables.
Numerical methods
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
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
Accuracy and precision are two measures of '' observational error''.
''Accuracy'' is how close a given set of measurements (observations or readings) are to their '' true value'', while ''precision'' is how close the measurements are to each ot ...
.
The sign problem is one of the major unsolved problems in the physics of
many-particle system
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
s. It often arises in calculations of the properties of a
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qu ...
system with large number of strongly interacting
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s, or in field theories involving a non-zero density of strongly interacting fermions.
Overview
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
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
is inapplicable, and one is forced to use brute-force numerical methods. Because the particles are fermions, their
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ma ...
changes sign when any two fermions are interchanged (due to the anti-symmetry of the wave function, see
Pauli principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
). 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
In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blunde ...
. 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
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the s ...
— It prevents the numerical solution of systems with a high density of strongly correlated electrons, such as the
Hubbard model
The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems.
It is particularly useful in solid-state physics. The model is named for John Hubbard.
The Hubbard model states that each ...
.
*
Nuclear physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter.
Nuclear physics should not be confused with atomic physics, which studies the ...
— It prevents the ''
ab initio
''Ab initio'' ( ) is a Latin term meaning "from the beginning" and is derived from the Latin ''ab'' ("from") + ''initio'', ablative singular of ''initium'' ("beginning").
Etymology
Circa 1600, from Latin, literally "from the beginning", from a ...
'' calculation of properties of
nuclear matter
Nuclear matter is an idealized system of interacting nucleons (protons and neutrons) that exists in several phases of exotic matter that, as of yet, are not fully established.
It is ''not'' matter in an atomic nucleus, but a hypothetical sub ...
and hence limits our understanding of
nuclei and
neutron star
A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. w ...
s.
*
Quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
— It prevents the use of
lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the ...
to predict the phases and properties of
quark matter
Quark matter or QCD matter (quantum chromodynamic) refers to any of a number of hypothetical phases of matter whose degrees of freedom include quarks and gluons, of which the prominent example is quark-gluon plasma. Several series of conferences ...
.
(In
lattice field theory
In physics, lattice field theory is the study of lattice models of quantum field theory, that is, of field theory on a space or spacetime that has been discretised onto a lattice.
Details
Although most lattice field theories are not exactly s ...
, the problem is also known as the complex action problem.)
The sign problem in field theory
In a field-theory approach to multi-particle systems, the fermion density is controlled by the value of the fermion
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
. One evaluates the
partition function by summing over all classical field configurations, weighted by
, where
is the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of the configuration. The sum over fermion fields can be performed analytically, and one is left with a sum over the
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
ic fields
(which may have been originally part of the theory, or have been produced by a
Hubbard–Stratonovich transformation The Hubbard–Stratonovich (HS) transformation is an exact mathematical transformation invented by Russian physicist Ruslan L. Stratonovich and popularized by British physicist John Hubbard. It is used to convert a particle theory into its respect ...
to make the fermion action quadratic)
:
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
where
represents the measure for the sum over all configurations
of the bosonic fields, weighted by
:
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
= \det(M(\mu,\sigma)) \exp(-S
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
,
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
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
/math> is therefore an average over all configurations weighted by
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
/math>:
:
If
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
/math> 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
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
/math> 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 (sign or complex phase) of the weight into the observable. For example, one could decompose the weighting function into its modulus and phase:
:
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
= p
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
, \exp(i\theta
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
,
where
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
/math> 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
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
/math>. However, the phase
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
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
sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
\rangle_p: if it is much less than 1, then the sign problem is severe.
It can be shown
[ that
:]sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
\rangle_p \propto \exp(-f V/T),
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 (although it has been advocated as the optimal choice since it minimizes the variance of the denominator). In general one could write
:sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
= psigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
\frac,
where sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used ...
/math> can be any positive weighting function (for example, the weighting function of the theory). 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
In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard pr ...
, 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 (as is generally suspected) there are no polynomial-time solutions to NP problems (see P versus NP problem
The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved.
The informal term ''quickly'', used above ...
), 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:
* Meron-cluster algorithms. These achieve an exponential speed-up by decomposing the fermion world lines into clusters that contribute independently. Cluster algorithms have been developed for certain theories,] but not for the Hubbard model of electrons, nor for QCD
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
, the theory of quarks.
* Stochastic quantization In theoretical physics, stochastic quantization is a method for modelling quantum mechanics, introduced by Edward Nelson in 1966, and streamlined by Parisi and Wu.
Details
Stochastic quantization serves to quantize Euclidean field theories, and ...
. The sum over configurations is obtained as the equilibrium distribution of states explored by a complex Langevin equation
In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Lange ...
. So far, the algorithm has been found to evade the sign problem in test models that have a sign problem but do not involve fermions.
* Fixed-node method. One fixes the location of nodes (zeros) of the multiparticle wavefunction, and uses Monte Carlo methods to obtain an estimate of the energy of the ground state, subject to that constraint.
* Majorana algorithms. Using Majorana fermion representation to perform Hubbard-Stratonovich transformations can help to solve the fermion sign problem of a class of fermionic many-body models.
* Diagrammatic Monte Carlo - based on stochastically and strategically sampling Feynman diagrams
See also
* Method of stationary phase In mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to the limit as k \to \infty .
This method originates from the 19th century, and is due to George Gabriel Stokes and Lord Kelvin.
It is close ...
* Oscillatory integral
In mathematical analysis an oscillatory integral is a type of distribution. Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals. It is possible to represent approximate solution operators for ...
Footnotes
References
{{DEFAULTSORT:Numerical Sign Problem
Statistical mechanics
Numerical artefacts
Unsolved problems in physics