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 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 nature of the strong force
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 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 methods can be extremely computationally
intensive, requiring the use of the largest
available supercomputers.
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 150 MeV.
Monte Carlo methods 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.
== Techniques ==
=== Monte-Carlo simulations ===
Monte-Carlo 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 spacetime.
In lattice Monte-Carlo simulations the aim
is to calculate correlation functions.
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 bosons part
and gauge-fermion interaction part of the
action to calculate the gauge configurations,
and then uses the simulated gauge configurations
to calculate hadronic 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 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, 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 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 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 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.
=== Quantum computing ===
In 2005 researchers of the National Institute
of Informatics 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.
== Limitations ==
The method suffers from a few limitations:
Currently there is no formulation of lattice
QCD that allows us to simulate real-time dynamics
of quark-gluon system such as quark-gluon
plasma.
It is computationally intensive, with the
bottleneck not being flops but the bandwidth
of memory access.
== See also ==
Lattice model (physics)
Lattice field theory
Lattice gauge theory
QCD matter
SU(2) color superconductivity
QCD sum rules
