Editing Lattice QCD

{{short description|Quantum chromodynamics on a lattice}}
{{Quantum field theory}}
'''Lattice QCD''' is a well-established non-[[Perturbation theory (quantum mechanics)|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 (group)|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.<ref name="wilson">{{cite journal | authorlink=Kenneth G. Wilson | first=K. | last= Wilson | journal=[[Physical Review D]]| volume=10 | issue=8 | page=2445 | title=Confinement of quarks | year= 1974 | doi=10.1103/PhysRevD.10.2445|bibcode = 1974PhRvD..10.2445W }}</ref><ref name="DaviesFollana2004">{{cite journal|last1=Davies|first1=C. T. H.|authorlink1=Christine Davies|last2=Follana|first2=E.|last3=Gray|first3=A.|last4=Lepage|first4=G. P.|last5=Mason|first5=Q.|last6=Nobes|first6=M.|last7=Shigemitsu|first7=J.|author7-link= Junko Shigemitsu |last8=Trottier|first8=H. D.|last9=Wingate|first9=M.|last10=Aubin|first10=C.|last11=Bernard|first11=C.|last12=Burch|first12=T.|last13=DeTar|first13=C.|last14=Gottlieb|first14=Steven|last15=Gregory|first15=E. B.|last16=Heller|first16=U. M.|last17=Hetrick|first17=J. E.|last18=Osborn|first18=J.|last19=Sugar|first19=R.|last20=Toussaint|first20=D.|last21=Pierro|first21=M. Di|last22=El-Khadra|first22=A.|last23=Kronfeld|first23=A. S.|last24=Mackenzie|first24=P. B.|last25=Menscher|first25=D.|last26=Simone|first26=J.|title=High-Precision Lattice QCD Confronts Experiment|display-authors=11|journal=[[Physical Review Letters]]|volume=92|issue=2|pages=022001|year=2004|issn=0031-9007|doi=10.1103/PhysRevLett.92.022001|pmid=14753930|arxiv=hep-lat/0304004|bibcode=2004PhRvL..92b2001D|s2cid=16205350}}</ref>
 
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#QCD and asymptotic freedom|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 [[colour confinement|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 increases as the lattice spacing decreases, results must be [[extrapolation|extrapolated]] to ''a = 0'' (the [[continuum limit]]) by repeated calculations at different lattice spacings ''a''.

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.<ref name="Bazavov">{{cite journal | author=A. Bazavov| title=Nonperturbative QCD simulations with 2+1 flavors of improved staggered quarks | journal=Reviews of Modern Physics | volume=82 | issue=2 | year=2010 | pages=1349–1417 | doi=10.1103/RevModPhys.82.1349 | arxiv=0903.3598 | bibcode=2010RvMP...82.1349B| s2cid=119259340 |display-authors=etal}}</ref>  These simulations typically utilize algorithms based upon [[molecular dynamics]] or [[microcanonical ensemble]] algorithms.<ref>{{cite journal | author=[[David Callaway|David J. E. Callaway]] and [[Aneesur Rahman]] | title=Microcanonical Ensemble Formulation of Lattice Gauge Theory | journal=Physical Review Letters | volume=49 | year=1982 | issue=9 |pages=613–616 | doi=10.1103/PhysRevLett.49.613 | bibcode=1982PhRvL..49..613C}}</ref><ref>{{cite journal | author=[[David Callaway|David J. E. Callaway]] and [[Aneesur Rahman]] | title=Lattice gauge theory in the microcanonical ensemble | journal=Physical Review | volume=D28 |year=1983 | issue=6 | pages=1506–1514 | doi=10.1103/PhysRevD.28.1506|bibcode = 1983PhRvD..28.1506C | url=https://cds.cern.ch/record/144746/files/PhysRevD.28.1506.pdf }}</ref>

At present, lattice QCD is primarily applicable at low densities where the [[numerical sign problem]] does not interfere with calculations. [[Monte Carlo method]]s are free from the sign problem when applied to the case of QCD with gauge group SU(2) (QC<sub>2</sub>D).

Lattice QCD has already successfully agreed with many experiments. For example, the mass of the [[proton]] has been determined theoretically with an error of less than 2 percent.<ref>{{cite journal | journal=Science | volume=322 | issue=5905 | pages=1224–7 |author1=S. Dürr |author2=Z. Fodor |author3=J. Frison | title=Ab Initio Determination of Light Hadron Masses | year=2008 | doi=10.1126/science.1163233 | pmid=19023076 | arxiv=0906.3599|bibcode = 2008Sci...322.1224D | s2cid=14225402 |display-authors=etal}}</ref> Lattice QCD predicts that the transition from confined quarks to [[quark–gluon plasma]] occurs around a temperature of {{val|150|ul=MeV}} ({{val|1.7e12|ul=K}}), within the range of experimental measurements.<ref>{{cite journal | author=P. Petreczky | title=Lattice QCD at non-zero temperature | journal=J. Phys. G | volume=39  | issue=9 | year=2012 | pages= 093002 | doi=10.1088/0954-3899/39/9/093002 | arxiv=1203.5320 |bibcode = 2012JPhG...39i3002P | s2cid=119193093 }}</ref><ref>{{cite journal |last1=Rafelski |first1=Johann |title=Melting hadrons, boiling quarks |journal=The European Physical Journal A |date=September 2015 |volume=51 |issue=9 |pages=114 |doi=10.1140/epja/i2015-15114-0 |arxiv=1508.03260 |bibcode=2015EPJA...51..114R |doi-access=free }}</ref>

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.<ref>{{Cite book |arxiv = 1401.3733|doi = 10.1109/HPCSim.2016.7568421|chapter = BSMBench: A flexible and scalable HPC benchmark from beyond the standard model physics|title = 2016 International Conference on High Performance Computing & Simulation (HPCS)|pages = 834–839|year = 2016|last1 = Bennett|first1 = Ed|last2 = Lucini|first2 = Biagio|last3 = Del Debbio|first3 = Luigi|last4 = Jordan|first4 = Kirk|last5 = Patella|first5 = Agostino|last6 = Pica|first6 = Claudio|last7 = Rago|first7 = Antonio|last8 = Trottier|first8 = H. D.|last9 = Wingate|first9 = M.|last10 = Aubin|first10 = C.|last11 = Bernard|first11 = C.|last12 = Burch|first12 = T.|last13 = DeTar|first13 = C.|last14 = Gottlieb|first14 = Steven|last15 = Gregory|first15 = E. B.|last16 = Heller|first16 = U. M.|last17 = Hetrick|first17 = J. E.|last18 = Osborn|first18 = J.|last19 = Sugar|first19 = R.|last20 = Toussaint|first20 = D.|last21 = Di Pierro|first21 = M.|last22 = El-Khadra|first22 = A.|last23 = Kronfeld|first23 = A. S.|last24 = Mackenzie|first24 = P. B.|last25 = Menscher|first25 = D.|last26 = Simone|first26 = J.|isbn = 978-1-5090-2088-1|s2cid = 115229961}}</ref>

==Techniques==

===Monte-Carlo simulations===

After [[Wick rotation]], the [[Path integral formulation|path integral]] for the [[Partition function (quantum field theory)|partition function]] of QCD takes the form

<math> Z = \int \mathcal{D} U \, e^{-S[U]} 
= \int \prod_{x, \mu} dU_\mu(x) \, e^{-S[U]} </math>

where the gauge links <math>U_\mu(x) \in \mathrm{SU}(3)</math> range over all the sites <math>x</math> and space-time directions <math>\mu</math> in a 4-dimensional space-time lattice, <math>S[U]</math> denotes the (Euclidean) [[Action (physics)|action]] and <math>dU_\mu(x)</math>  denotes the [[Haar measure]] on <math>\mathrm{SU}(3)</math>. Physical information is obtained by computing observables

<math> \left\langle \mathcal{O} \right\rangle
= \frac{1}{Z} \int \mathcal{D} U \, \mathcal{O}(U)
e^{-S[U]} </math>

For cases where evaluating observables pertubatively is difficult or impossible, a [[Monte Carlo method|Monte Carlo]] approach can be used, computing an observable <math> \mathcal{O} </math> as

<math> \left\langle \mathcal{O} \right\rangle
\approx \sum_{i=1}^{N} \mathcal{O}(U_i)
</math>

where <math>U_1, \dots, U_{N}</math> are [[Independent and identically distributed random variables|i.i.d random variables]] distributed according to the [[Boltzmann distribution|Boltzman distribution]] <math> U_i \sim e^{-S[U_i]}/Z </math>. For practical calculations, the samples <math>\{U_i\}</math> are typically obtained using [[Markov chain Monte Carlo]] methods, in particular [[Hybrid Monte Carlo]], which was invented for this purpose.<ref>{{cite journal | url=https://doi.org/10.1016/0370-2693(87)91197-X | doi=10.1016/0370-2693(87)91197-X | title=Hybrid Monte Carlo | date=1987 | last1=Duane | first1=Simon | last2=Kennedy | first2=A.D. | last3=Pendleton | first3=Brian J. | last4=Roweth | first4=Duncan | journal=Physics Letters B | volume=195 | issue=2 | pages=216–222 }}</ref>

===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.<ref name="Bazavov" />

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 physical quantities (such as the [[scattering matrix]]) are [[taylor expansion|expanded]] in powers of the lattice spacing, ''a''. The results are used primarily to [[renormalization|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 [[Kenneth G. Wilson|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 (a UV regulator), and a quantum field theory can be formulated and solved on the physical lattice.

===Quantum computing===
The U(1), SU(2), and SU(3) lattice gauge theories can be reformulated into a form that can be simulated using "spin qubit manipulations" on a [[universal quantum computer]].<ref>{{Cite journal|last1=Byrnes|first1=Tim|last2=Yamamoto|first2=Yoshihisa|title=Simulating lattice gauge theories on a quantum computer|journal=Physical Review A|volume=73|issue=2|pages=022328|doi=10.1103/PhysRevA.73.022328|date=17 February 2006|arxiv=quant-ph/0510027|bibcode=2006PhRvA..73b2328B|s2cid=6105195}}</ref>

==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|flops]] but the bandwidth of memory access.
* Computations of observables at nonzero baryon density suffer from a [[sign problem]], preventing direct computations of thermodynamic quantities.<ref name='Philipsen'>{{cite journal |last=Philipsen |first=O. |year=2008 |title=Lattice calculations at non-zero chemical potential: The QCD phase diagram |journal=Proceedings of Science |volume=77 |pages=011 |doi=10.22323/1.077.0011|doi-access=free }}</ref>

==See also==
* [[Lattice model (physics)]]
* [[Lattice field theory]]
* [[Lattice gauge theory]]
* [[QCD matter]]
* [[SU(2) color superconductivity]]
* [[QCD sum rules]]
* [[Wilson action]]

==References==
{{reflist|35em}}

==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.
* [[Jan Smit (physicist)|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==
* [https://arxiv.org/abs/hep-lat/9807028 Gupta - Introduction to Lattice QCD]
* [https://arxiv.org/abs/hep-lat/0509180 Lombardo - Lattice QCD at Finite Temperature and Density]
* [https://arxiv.org/abs/hep-lat/0405024 Chandrasekharan, Wiese - An Introduction to Chiral Symmetry on the Lattice]
* [http://pos.sissa.it/archive/conferences/020/001/LAT2005_001.pdf Kuti, Julius - Lattice QCD and String Theory]
* [http://fermiqcd.net The FermiQCD Library for Lattice Field theory] {{Webarchive|url=https://web.archive.org/web/20150203181754/http://fermiqcd.net/ |date=2015-02-03 }}
* [http://flag.unibe.ch Flavour Lattice Averaging Group]
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{{States of matter}}

[[Category:Lattice field theory]]
[[Category:Quantum chromodynamics]]