Lattice QCD
Encyclopedia
Lattice QCD is a well-established non-perturbative
approach to solving the quantum chromodynamics
(QCD) theory of quark
s and gluon
s. It is a lattice gauge theory
formulated on a grid or lattice
of points in space and time.
Analytic or perturbative solutions in low-energy QCD are hard or impossible due to the highly nonlinear nature of the strong force. This formulation of QCD in discrete rather than continuous spacetime naturally introduces a momentum cut off at the order 1/a, where a is the lattice spacing, which regularizes the theory. As a result lattice QCD is mathematically well-defined. Most importantly, lattice QCD provides a framework for investigation of non-perturbative phenomena such as confinement
and quark-gluon plasma
formation, which are intractable by means of analytic field theories.
In lattice QCD, fields representing quarks are defined at lattice sites (which leads to fermion doubling
), while the gluon fields are defined on the links connecting neighboring sites. This approximation approaches continuum QCD as the spacing between lattice sites is reduced to zero. Because the computational cost of numerical simulations can increase dramatically as the lattice spacing decreases, results are often extrapolated
to a = 0 by repeated calculations at different lattice spacings a that are large enough to be tractable.
Numerical lattice QCD calculations using Monte Carlo method
s can be extremely computationally intensive, requiring the use of the largest available supercomputer
s. To reduce the computational burden, the so-called quenched approximation
can be used, in which the quark fields are treated as non-dynamic "frozen" variables. While this was common in early lattice QCD calculations, "dynamical" fermions are now standard. These simulations typically utilize algorithms based upon molecular dynamics
or microcanonical ensemble
algorithms.
At present, lattice QCD is primarily applicable at low densities where the numerical sign problem
does not interfere with calculations. Lattice QCD predicts that confined quarks will become released to quark-gluon plasma around energies of 170 MeV. Monte Carlo method
s are free from the sign problem when applied to the case of QCD with gauge group SU(2) (QC2D).
Lattice QCD has already made successful contact with many experiments. For example the mass of the proton
has been determined theoretically with an error of less than 2 percent.
Lattice QCD has also been used as a benchmark for high-performance computing, an approach originally developed in the context of the IBM Blue Gene
supercomputer.
is a method to pseudo-randomly sample a large space of variables.
The importance sampling technique used to select the gauge configurations in the Monte-Carlo simulation imposes the use of Euclidean time
, by a Wick rotation
of space-time.
In lattice Monte-Carlo simulations the aim is to calculate correlation function
s. This is done by explicitly calculating the action
, using field configurations which are chosen according to the distribution function
, which depends on the action and the fields. Usually one starts with the gauge boson
s part and gauge-fermion
interaction part of the action to calculate the gauge configurations, and then uses the simulated gauge configurations to calculate hadron
ic propagator
s and correlation functions.
In order to compensate for the errors one improves the lattice action in various ways, to minimize mainly finite spacing errors.
small, while it fails completely when the coupling is large and higher order corrections are larger than lower orders in the perturbative series. In this region non-perturbative methods, such as Monte-Carlo sampling of the correlation function,
are necessary.
Lattice perturbation theory can also provide results for condensed matter
theory. One can use the lattice to represent the real atomic crystal
. In this case the lattice spacing is a real physical value, and not an artifact of the calculation which has to be removed, and a quantum field theory can be formulated and solved on the physical lattice.
Perturbation theory (quantum mechanics)
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...
approach to solving the quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
(QCD) theory of quark
Quark
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
s and gluon
Gluon
Gluons are elementary particles which act as the exchange particles for the color force between quarks, analogous to the exchange of photons in the electromagnetic force between two charged particles....
s. It is a lattice gauge theory
Lattice gauge theory
In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice. Gauge theories are important in particle physics, and include the prevailing theories of elementary particles: quantum electrodynamics, quantum chromodynamics and the Standard...
formulated on a grid or lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
of points in space and time.
Analytic or perturbative solutions in low-energy QCD are hard or impossible due to the highly nonlinear nature of the strong force. This formulation of QCD in discrete rather than continuous spacetime naturally introduces a momentum cut off at the order 1/a, where a is the lattice spacing, which regularizes the theory. As a result lattice QCD is mathematically well-defined. Most importantly, lattice QCD provides a framework for investigation of non-perturbative phenomena such as confinement
Colour confinement
Color confinement, often simply called confinement, is the physics phenomenon that color charged particles cannot be isolated singularly, and therefore cannot be directly observed. Quarks, by default, clump together to form groups, or hadrons. The two types of hadrons are the mesons and the baryons...
and quark-gluon plasma
Quark-gluon plasma
A quark–gluon plasma or quark soup is a phase of quantum chromodynamics which exists at extremely high temperature and/or density. This phase consists of asymptotically free quarks and gluons, which are several of the basic building blocks of matter...
formation, which are intractable by means of analytic field theories.
In lattice QCD, fields representing quarks are defined at lattice sites (which leads to fermion doubling
Fermion doubling
The fermion doubling problem is a problem that is encountered when naively trying to put fermionic fields on a lattice. It consists in the appearance of spurious states, such that one ends up having 2d fermionic particles for each original fermion...
), while the gluon fields are defined on the links connecting neighboring sites. This approximation approaches continuum QCD as the spacing between lattice sites is reduced to zero. Because the computational cost of numerical simulations can increase dramatically as the lattice spacing decreases, results are often extrapolated
Extrapolation
In mathematics, extrapolation is the process of constructing new data points. It is similar to the process of interpolation, which constructs new points between known points, but the results of extrapolations are often less meaningful, and are subject to greater uncertainty. It may also mean...
to a = 0 by repeated calculations at different lattice spacings a that are large enough to be tractable.
Numerical lattice QCD calculations using Monte Carlo method
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
s can be extremely computationally intensive, requiring the use of the largest available supercomputer
Supercomputer
A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...
s. To reduce the computational burden, the so-called quenched approximation
Quenched approximation
In particle physics, the quenched approximation is an approximation often used in lattice gauge theory in which the quantum loops of fermions in Feynman diagrams are neglected. Equivalently, the corresponding one-loop determinants are set to one...
can be used, in which the quark fields are treated as non-dynamic "frozen" variables. While this was common in early lattice QCD calculations, "dynamical" fermions are now standard. These simulations typically utilize algorithms based upon molecular dynamics
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
or microcanonical ensemble
Microcanonical ensemble
In statistical physics, the microcanonical ensemble is a theoretical tool used to describe the thermodynamic properties of an isolated system. In such a system, the possible macrostates of the system all have the same energy and the probability for the system to be in any given microstate is the same...
algorithms.
At present, lattice QCD is primarily applicable at low densities where the numerical sign problem
Numerical sign problem
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...
does not interfere with calculations. Lattice QCD predicts that confined quarks will become released to quark-gluon plasma around energies of 170 MeV. Monte Carlo method
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
s are free from the sign problem when applied to the case of QCD with gauge group SU(2) (QC2D).
Lattice QCD has already made successful contact with many experiments. For example the mass of the proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
has been determined theoretically with an error of less than 2 percent.
Lattice QCD has also been used as a benchmark for high-performance computing, an approach originally developed in the context of the IBM Blue Gene
Blue Gene
Blue Gene is a computer architecture project to produce several supercomputers, designed to reach operating speeds in the PFLOPS range, and currently reaching sustained speeds of nearly 500 TFLOPS . It is a cooperative project among IBM Blue Gene is a computer architecture project to produce...
supercomputer.
Monte-Carlo simulations
Monte-CarloMonte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
is a method to pseudo-randomly sample a large space of variables.
The importance sampling technique used to select the gauge configurations in the Monte-Carlo simulation imposes the use of Euclidean time
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, by a Wick rotation
Wick rotation
In physics, Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable...
of space-time.
In lattice Monte-Carlo simulations the aim is to calculate correlation function
Correlation function (quantum field theory)
In quantum field theory, the matrix element computed by inserting a product of operators between two states, usually the vacuum states, is called a correlation function....
s. This is done by explicitly calculating the action
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
, using field configurations which are chosen according to the distribution function
Distribution function
In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...
, which depends on the action and the fields. Usually one starts with the gauge boson
Gauge boson
In particle physics, gauge bosons are bosonic particles that act as carriers of the fundamental forces of nature. More specifically, elementary particles whose interactions are described by gauge theory exert forces on each other by the exchange of gauge bosons, usually as virtual particles.-...
s part and gauge-fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
interaction part of the action to calculate the gauge configurations, and then uses the simulated gauge configurations to calculate hadron
Hadron
In particle physics, a hadron is a composite particle made of quarks held together by the strong force...
ic propagator
Propagator
In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...
s and correlation functions.
Fermions on the lattice
Lattice QCD is a way to solve the theory exactly from first principles, without any assumptions, to the desired precision. However, in practice the calculation power is limited, which requires a smart use of the available resources. One needs to choose an action which gives the best physical description of the system, with minimum errors, using the available computational power. The limited computer resources force one to use physical constants which are different from their true physical values:- The lattice discretization means a finite lattice spacing and size, which do not exist in the continuous and infinite space-time. In addition to the automatic error introduced by this, the limited resources force the use of smaller physical lattices and larger lattice spacing than wanted in order to minimize errors.
- Another unphysical quantity is the quark masses. Quark masses are steadily going down, but to-date (2010) they are typically too high with respect to the real value.
In order to compensate for the errors one improves the lattice action in various ways, to minimize mainly finite spacing errors.
Lattice perturbation theory
The lattice was initially introduced by Wilson as a framework for studying strongly coupled theories, such as QCD, non-perturbatively. it was found to be a regularization also suitable for perturbative calculations. Perturbation theory involves an expansion in the coupling constant, and is well-justified in high-energy QCD where the coupling constant issmall, while it fails completely when the coupling is large and higher order corrections are larger than lower orders in the perturbative series. In this region non-perturbative methods, such as Monte-Carlo sampling of the correlation function,
are necessary.
Lattice perturbation theory can also provide results for condensed matter
Condensed Matter
Condensed matter may refer to several things*Condensed matter physics, the study of the physical properties of condensed phases of matter*European Physical Journal B: Condensed Matter and Complex Systems, a scientific journal published by EDP sciences...
theory. One can use the lattice to represent the real atomic crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
. In this case the lattice spacing is a real physical value, and not an artifact of the calculation which has to be removed, and a quantum field theory can be formulated and solved on the physical lattice.
See also
- Lattice field theoryLattice field theoryIn physics, lattice field theory is the study of lattice models of quantum field theory, that is, of field theory on a spacetime that has been discretized onto a lattice. Although most lattice field theories are not exactly solvable, they are of tremendous appeal because they can be studied by...
- Lattice gauge theoryLattice gauge theoryIn physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice. Gauge theories are important in particle physics, and include the prevailing theories of elementary particles: quantum electrodynamics, quantum chromodynamics and the Standard...
- QCD matterQCD matterQuark matter or QCD matter refers to any of a number of theorized phases of matter whose degrees of freedom include quarks and gluons. These theoretical phases would occur at extremely high temperatures and densities, billions of times higher than can be produced in equilibrium in laboratories...
- QCD sum rulesQCD sum rulesIn quantum chromodynamics, the confining and strong coupling nature of the theory means that conventional perturbative techniques often fail to apply. The QCD sum rules are a way of dealing with this. The idea is to work with gauge invariant operators and operator product expansions of them...
Further reading
- M. Creutz, Quarks, gluons and lattices, Cambridge University Press 1985.
- I. Montvay and G. Münster, Quantum Fields on a Lattice, Cambridge University Press 1997.
- J. Smit, Introduction to Quantum Fields on a Lattice, Cambridge University Press 2002.
- H. Rothe, Lattice Gauge Theories, An Introduction, World Scientific 2005.
- T. DeGrand and C. DeTar, Lattice Methods for Quantum Chromodynamics, World Scientific 2006.
- C. Gattringer and C. B. Lang, Quantum Chromodynamics on the Lattice, Springer 2010.