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Techniques

Monte-Carlo simulations

After Wick rotation, the path integral for the partition function of QCD takes the form

Z = ∫ D U e − S [ U ] = ∫ ∏ x , μ d U μ ( x ) e − S [ U ] {\displaystyle Z=\int {\mathcal {D}}U\,e^{-S[U]}=\int \prod _{x,\mu }dU_{\mu }(x)\,e^{-S[U]}}

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

⟨ O ⟩ = 1 Z ∫ D U O ( U ) e − S [ U ] {\displaystyle \left\langle {\mathcal {O}}\right\rangle ={\frac {1}{Z}}\int {\mathcal {D}}U\,{\mathcal {O}}(U)e^{-S[U]}}

For cases where evaluating observables pertubatively is difficult or impossible, a Monte Carlo approach can be used, computing an observable O {\displaystyle {\mathcal {O}}} as

⟨ O ⟩ ≈ ∑ i = 1 N O ( U i ) {\displaystyle \left\langle {\mathcal {O}}\right\rangle \approx \sum _{i=1}^{N}{\mathcal {O}}(U_{i})}

where U 1 , … , U N {\displaystyle U_{1},\dots ,U_{N}} are i.i.d random variables distributed according to the Boltzman distribution U i ∼ e − S [ U i ] / Z {\displaystyle U_{i}\sim e^{-S[U_{i}]}/Z} . For practical calculations, the samples { U i } {\displaystyle \{U_{i}\}} are typically obtained using Markov chain Monte Carlo methods, in particular Hybrid Monte Carlo, which was invented for this purpose.¹¹

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.¹²

Lattice perturbation theory

In lattice perturbation theory physical quantities (such as the scattering matrix) are 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 (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.¹³

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.
• Computations of observables at nonzero baryon density suffer from a sign problem, preventing direct computations of thermodynamic quantities.¹⁴

See also

• Lattice field theory
• Lattice gauge theory
• Lattice model (physics)
• QCD matter
• Quantum triviality
• SU(2) color superconductivity
• QCD sum rules
• Wilson action

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 Archived 2015-02-03 at the Wayback Machine
• Flavour Lattice Averaging Group

References

1. Wilson, K. (1974). "Confinement of quarks". Physical Review D. 10 (8): 2445. Bibcode:1974PhRvD..10.2445W. doi:10.1103/PhysRevD.10.2445. /wiki/Kenneth_G._Wilson ↩

2. Davies, C. T. H.; Follana, E.; Gray, A.; Lepage, G. P.; Mason, Q.; Nobes, M.; Shigemitsu, J.; Trottier, H. D.; Wingate, M.; Aubin, C.; Bernard, C.; et al. (2004). "High-Precision Lattice QCD Confronts Experiment". Physical Review Letters. 92 (2): 022001. arXiv:hep-lat/0304004. Bibcode:2004PhRvL..92b2001D. doi:10.1103/PhysRevLett.92.022001. ISSN 0031-9007. PMID 14753930. S2CID 16205350. /wiki/Christine_Davies ↩

3. A. Bazavov; et al. (2010). "Nonperturbative QCD simulations with 2+1 flavors of improved staggered quarks". Reviews of Modern Physics. 82 (2): 1349–1417. arXiv:0903.3598. Bibcode:2010RvMP...82.1349B. doi:10.1103/RevModPhys.82.1349. S2CID 119259340. /wiki/ArXiv_(identifier) ↩

4. David J. E. Callaway and Aneesur Rahman (1982). "Microcanonical Ensemble Formulation of Lattice Gauge Theory". Physical Review Letters. 49 (9): 613–616. Bibcode:1982PhRvL..49..613C. doi:10.1103/PhysRevLett.49.613. /wiki/David_Callaway ↩

5. David J. E. Callaway and Aneesur Rahman (1983). "Lattice gauge theory in the microcanonical ensemble" (PDF). Physical Review. D28 (6): 1506–1514. Bibcode:1983PhRvD..28.1506C. doi:10.1103/PhysRevD.28.1506. /wiki/David_Callaway ↩

6. "In Memoriam: Aneesur Rahman" (PDF). cecam.org. 1987-09-01. https://www.cecam.org/themes/cecam/assets/images/history/1981-1990/In_Memoriam_Aneesur_Rahman.pdf ↩

7. S. Dürr; Z. Fodor; J. Frison; et al. (2008). "Ab Initio Determination of Light Hadron Masses". Science. 322 (5905): 1224–7. arXiv:0906.3599. Bibcode:2008Sci...322.1224D. doi:10.1126/science.1163233. PMID 19023076. S2CID 14225402. /wiki/ArXiv_(identifier) ↩

8. P. Petreczky (2012). "Lattice QCD at non-zero temperature". J. Phys. G. 39 (9): 093002. arXiv:1203.5320. Bibcode:2012JPhG...39i3002P. doi:10.1088/0954-3899/39/9/093002. S2CID 119193093. /wiki/ArXiv_(identifier) ↩

9. Rafelski, Johann (September 2015). "Melting hadrons, boiling quarks". The European Physical Journal A. 51 (9): 114. arXiv:1508.03260. Bibcode:2015EPJA...51..114R. doi:10.1140/epja/i2015-15114-0. https://doi.org/10.1140%2Fepja%2Fi2015-15114-0 ↩

10. Bennett, Ed; Lucini, Biagio; Del Debbio, Luigi; Jordan, Kirk; Patella, Agostino; Pica, Claudio; Rago, Antonio; Trottier, H. D.; Wingate, M.; Aubin, C.; Bernard, C.; Burch, T.; DeTar, C.; Gottlieb, Steven; Gregory, E. B.; Heller, U. M.; Hetrick, J. E.; Osborn, J.; Sugar, R.; Toussaint, D.; Di Pierro, M.; El-Khadra, A.; Kronfeld, A. S.; Mackenzie, P. B.; Menscher, D.; Simone, J. (2016). "BSMBench: A flexible and scalable HPC benchmark from beyond the standard model physics". 2016 International Conference on High Performance Computing & Simulation (HPCS). pp. 834–839. arXiv:1401.3733. doi:10.1109/HPCSim.2016.7568421. ISBN 978-1-5090-2088-1. S2CID 115229961. 978-1-5090-2088-1 ↩

11. Duane, Simon; Kennedy, A.D.; Pendleton, Brian J.; Roweth, Duncan (1987). "Hybrid Monte Carlo". Physics Letters B. 195 (2): 216–222. doi:10.1016/0370-2693(87)91197-X. https://doi.org/10.1016/0370-2693(87)91197-X ↩

12. A. Bazavov; et al. (2010). "Nonperturbative QCD simulations with 2+1 flavors of improved staggered quarks". Reviews of Modern Physics. 82 (2): 1349–1417. arXiv:0903.3598. Bibcode:2010RvMP...82.1349B. doi:10.1103/RevModPhys.82.1349. S2CID 119259340. /wiki/ArXiv_(identifier) ↩

13. Byrnes, Tim; Yamamoto, Yoshihisa (17 February 2006). "Simulating lattice gauge theories on a quantum computer". Physical Review A. 73 (2): 022328. arXiv:quant-ph/0510027. Bibcode:2006PhRvA..73b2328B. doi:10.1103/PhysRevA.73.022328. S2CID 6105195. /wiki/ArXiv_(identifier) ↩

14. Philipsen, O. (2008). "Lattice calculations at non-zero chemical potential: The QCD phase diagram". Proceedings of Science. 77: 011. doi:10.22323/1.077.0011. https://doi.org/10.22323%2F1.077.0011 ↩