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 whi ...
approach to solving the
quantum chromodynamics (QCD) theory of
quarks and
gluons. 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 elec ...
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 othe ...
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 n ...
and the large
coupling constant 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
In lattice field theory, fermion doubling occurs when naively putting fermionic fields on a lattice, resulting in more fermionic states than expected. For the naively discretized Dirac fermions in d Euclidean dimensions, each fermionic field res ...
), 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
In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between kn ...
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 determi ...
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 instructions ...
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 to ...
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 the ...
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 determi ...
s are free from the sign problem when applied to the case of QCD with gauge group SU(2) (QC
2D).
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 differen ...
.
In lattice Monte-Carlo simulations the aim is to calculate
correlation functions. 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 bosons 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 an ...
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 e ...
ic
propagators 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
In quantum field theory, the minimal subtraction scheme, or MS scheme, is a particular renormalization scheme used to absorb the infinities that arise in perturbative calculations beyond leading-order, leading order, introduced independently by Ge ...
, 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 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 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 re ...
s, which have one or more
strange quarks.
See also
*
Lattice model (physics)
*
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 elec ...
*
QCD 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 conferenc ...
*
SU(2) color superconductivity
*
QCD sum rules
In quantum chromodynamics, the confining and strong coupling nature of the theory means that conventional perturbative techniques often fail to apply. The QCD sum rules (or Shifman– Vainshtein–Zakharov sum rules) are a way of dealing with t ...
*
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 QCDLombardo - Lattice QCD at Finite Temperature and DensityChandrasekharan, Wiese - An Introduction to Chiral Symmetry on the LatticeKuti, Julius - Lattice QCD and String TheoryThe FermiQCD Library for Lattice Field theoryFlavour Lattice Averaging Group
{{States of matter
Lattice field theory
Quantum chromodynamics