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Lattice QCD is a well-established non-
perturbative 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 w ...
approach to solving the
quantum chromodynamics 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 ...
(QCD) theory of
quark A quark () is a type of 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. All commonly ...
s and
gluon A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind ...
s. It is a
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 ...
formulated on a grid or
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
of points in space and time. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum QCD is recovered. Analytic or perturbative solutions in low-energy QCD are hard or impossible to obtain due to the highly
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
nature of the
strong force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
and the large
coupling constant In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
at low energies. 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 Confinement may refer to * With respect to humans: ** An old-fashioned or archaic synonym for childbirth ** Postpartum confinement (or postnatal confinement), a system of recovery after childbirth, involving rest and special foods ** Civil confi ...
and
quark–gluon plasma Quark–gluon plasma (QGP) or quark soup is an interacting localized assembly of quarks and gluons at thermal (local kinetic) and (close to) chemical (abundance) equilibrium. The word ''plasma'' signals that free color charges are allowed. In a ...
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 Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deter ...
s can be extremely computationally intensive, requiring the use of the largest available
supercomputer A supercomputer is a computer with a high level of performance as compared to a general-purpose computer. The performance of a supercomputer is commonly measured in floating-point operations per second ( FLOPS) instead of million instructio ...
s. To reduce the computational burden, the so-called
quenched approximation In lattice field theory, 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 t ...
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 (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of th ...
or
microcanonical ensemble In statistical mechanics, the microcanonical ensemble is a statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assumed to be isolated in the sense that it canno ...
algorithms. At present, lattice QCD is primarily applicable at low densities where the
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 n ...
does not interfere with calculations.
Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deter ...
s are free from the sign problem when applied to the case of QCD with gauge group SU(2) (QC2D). Lattice QCD has already successfully agreed with many experiments. For example, the mass of the
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
has been determined theoretically with an error of less than 2 percent. Lattice QCD predicts that the transition from confined quarks to
quark–gluon plasma Quark–gluon plasma (QGP) or quark soup is an interacting localized assembly of quarks and gluons at thermal (local kinetic) and (close to) chemical (abundance) equilibrium. The word ''plasma'' signals that free color charges are allowed. In a ...
occurs around a temperature of (), within the range of experimental measurements. 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 is an IBM project aimed at designing supercomputers that can reach operating speeds in the petaFLOPS (PFLOPS) range, with low power consumption. The project created three generations of supercomputers, Blue Gene/L, Blue Gene/P, ...
supercomputer.


Techniques


Monte-Carlo simulations

Monte-Carlo Monte Carlo (; ; french: Monte-Carlo , or colloquially ''Monte-Carl'' ; lij, Munte Carlu ; ) is officially an administrative area of the Principality of Monaco, specifically the ward of Monte Carlo/Spélugues, where the Monte Carlo Casino is ...
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 In physics, Wick rotation, named after Italian physicist 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 s ...
of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
. In lattice Monte-Carlo simulations the aim is to calculate
correlation function A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables r ...
s. This is done by explicitly calculating 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 ...
, 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 In particle physics, a gauge boson is a bosonic elementary particle that acts as the force carrier for elementary fermions. Elementary particles, whose interactions are described by a gauge theory, interact with each other by the exchange of ga ...
s part and gauge-
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 ...
interaction part of the action to calculate the gauge configurations, and then uses the simulated gauge configurations to calculate
hadron In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the ...
ic
propagator In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. ...
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 approximate physical constants which are different from their true physical values: * The lattice discretization means approximating continuous and infinite space-time by a finite lattice spacing and size. The smaller the lattice, and the bigger the gap between nodes, the bigger the error. Limited resources commonly force the use of smaller physical lattices and larger lattice spacing than wanted, leading to larger errors than wanted. * The quark masses are also approximated. Quark masses are larger than experimentally measured. These have been steadily approaching their physical values, and within the past few years a few collaborations have used nearly physical values to extrapolate down to physical values. In order to compensate for the errors one improves the lattice action in various ways, to minimize mainly finite spacing errors.


Lattice perturbation theory

In lattice perturbation theory the
scattering matrix In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More forma ...
is expanded in powers of the lattice spacing, ''a''. The results are used primarily to renormalize Lattice QCD Monte-Carlo calculations. In perturbative calculations both the operators of the action and the propagators are calculated on the lattice and expanded in powers of ''a''. When renormalizing a calculation, the coefficients of the expansion need to be matched with a common continuum scheme, such as the MS-bar scheme, otherwise the results cannot be compared. The expansion has to be carried out to the same order in the continuum scheme and the lattice one. The lattice regularization was initially introduced by
Wilson Wilson may refer to: People *Wilson (name) ** List of people with given name Wilson ** List of people with surname Wilson * Wilson (footballer, 1927–1998), Brazilian manager and defender * Wilson (footballer, born 1984), full name Wilson R ...
as a framework for studying strongly coupled theories non-perturbatively. However, it was found to be a regularization suitable also for perturbative calculations. Perturbation theory involves an expansion in the coupling constant, and is well-justified in high-energy QCD where the coupling constant is 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 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 su ...
theory. One can use the lattice to represent the real atomic
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
. In this case the lattice spacing is a real physical value, and not an artifact of the calculation which has to be removed (a UV regulator), and a quantum field theory can be formulated and solved on the physical lattice.


Quantum computing

In 2005 researchers of the
National Institute of Informatics The is a Japanese research institute located in Chiyoda, Tokyo, Japan. NII was established in April 2000 for the purpose of advancing the study of informatics. This institute also works on creating systems to facilitate the spread of scienti ...
reformulated the U(1), SU(2), and SU(3) lattice gauge theories into a form that can be simulated using "spin qubit manipulations" on a
universal quantum computer A quantum Turing machine (QTM) or universal quantum computer is an abstract machine used to model the effects of a quantum computer. It provides a simple model that captures all of the power of quantum computation—that is, any quantum algorith ...
.


Limitations

The method suffers from a few limitations: * Currently there is no formulation of lattice QCD that allows us to simulate the real-time dynamics of a quark-gluon system such as quark–gluon plasma. * It is computationally intensive, with the bottleneck not being
flops In computing, floating point operations per second (FLOPS, flops or flop/s) is a measure of computer performance, useful in fields of scientific computations that require floating-point calculations. For such cases, it is a more accurate me ...
but the bandwidth of memory access. * It provides reliable predictions only for hadrons containing heavy quarks, such as
hyperon In particle physics, a hyperon is any baryon containing one or more strange quarks, but no charm, bottom, or top quark. This form of matter may exist in a stable form within the core of some neutron stars. Hyperons are sometimes generically rep ...
s, which have one or more
strange quark The strange quark or s quark (from its symbol, s) is the third lightest of all quarks, a type of elementary particle. Strange quarks are found in subatomic particles called hadrons. Examples of hadrons containing strange quarks include kaons ( ...
s.


See also

*
Lattice model (physics) In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of c ...
*
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 sol ...
*
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 ...
* QCD matter * SU(2) color superconductivity * QCD sum rules *
Wilson action In lattice field theory, the Wilson action is a discrete formulation of the Yang–Mills action, forming the foundation of lattice gauge theory. Rather than using Lie algebra valued gauge fields as the fundamental parameters of the theory, grou ...


References


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.


External links


Gupta - Introduction to Lattice QCD

Lombardo - Lattice QCD at Finite Temperature and Density

Chandrasekharan, Wiese - An Introduction to Chiral Symmetry on the Lattice

Kuti, Julius - Lattice QCD and String Theory

The FermiQCD Library for Lattice Field theory

Flavour Lattice Averaging Group
{{States of matter Lattice field theory Quantum chromodynamics