Fast Algorithms for Simulating Chiral Fermions in U(1) Lattice Gauge Theory

Dafina. Xhako; Artan. Boriçi

doi:doi:10.11648/j.ajpa.20140202.15

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Fast Algorithms for Simulating Chiral Fermions in U(1) Lattice Gauge Theory

Dafina. Xhako, Artan. Boriçi

Published in American Journal of Physics and Applications (Volume 2, Issue 2)

Received: 6 March 2014 Accepted: 11 April 2014 Published: 20 April 2014

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Abstract

Lattice QCD with chiral fermions are extremely computationally expensive, but on the other hand provides an accurate tool for studying the physics of strong interactions. Since the truncated overlap variant of domain wall fermions are equivalent to overlap fermions in four dimensions at any lattice spacing, in this paper we have used domain wall fermions for our simulations. The physical information of lattice QCD theory is contained in quark propagators. In practice computing quark propagator in lattice is an inversion problem of the Dirac operator matrix representing this quarks. In order to develop fast inversion algorithms we have used overlap solvers in two dimensions. Lattice QED theory with U(1) group symmetry in two dimensional space-times dimensions has always been a testing ground for algorithms. By the other side, motivated by our previews work that the two-grid algorithm converge faster than the standard iterative methods for overlap inversion but not for all quark masses, we thought to test this idea in less dimensions such as U(1) gauge theory. Our main objective of this paper it is to implement and develop the idea of a two level algorithm in a new algorithm coded in QCDLAB. This implementation is presented in the preconditioned GMRESR algorithm, as our new contribution in QCDLAB package. The preconditioned part of our algorithm, different from the one of [18], is the approximation of the overlap operator with the truncated overlap operator with finite N3 dimension. We have tested it for 100 statistically independent configurations on 32 x 32 lattice background U(1) field at coupling constant β=1 and for different bare quark masses mq = [0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1]. We have compared the convergence history of the preconditioned GMRESR residual norm with another overlap inverter of QCDLAB as an optimal one, such as SHUMR. We have shown that our algorithm converges faster than SHUMR for different quark masses. Also, we have demonstrated that it saves more time for light quarks compared to SHUMR algorithm. Our algorithm is approximately independent from the quark mass. This is a key result in simulations with chiral fermions in lattice theories. By the other side, if we compare the results of [18] for quark mass 0.1 in SU(3), results that our chosen preconditioned saves a factor of 2 but in U(1). Our next step is to test this algorithm in SU(3) and to adopt it in parallel

Published in	American Journal of Physics and Applications (Volume 2, Issue 2)
DOI	10.11648/j.ajpa.20140202.15
Page(s)	67-72
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

Chiral Fermions, Quark Propagator, Inverting Algorithms, U(1) Gauge Theory

References

[1]	M. Lüscher, “Lattice QCD — from quark conﬁnement to asymptotic freedom”, Annales Henri Poincare 4, S197-S210, 2003. arXiv:hep-ph/0211220
[2]	D. B. Kaplan, “A Method for Simulating Chiral Fermions on the Lattice”, Phys. Lett. B 228, 1992, pp 342.
[3]	V. Furman, Y. Shamir, “Axial symmetries in lattice QCD with Kaplan fermions”, Nucl. Phys. B439, 1995, pp 54-78.
[4]	R. Narayanan, H. Neuberger, “Infinitely many regulator fields for chiral fermions”, Phys. Lett. B 302, 1993, pp 62.
[5]	R. Narayanan, H. Neuberger, “A construction of lattice chiral gauge theories”, Nucl. Phys. B 443, 1995, pp 305.
[6]	A. Boriçi, “Computational methods for the fermion determinant and the link between overlap and domain wall fermions”, in QCD and Numerical Analysis III, ed. Boriçi et al, Springer 2005.
[7]	A. Boriçi, “Truncated Overlap Fermions”, Nucl. Phys. Proc. Suppl. 83, 2000, pp 771-773.
[8]	A. Boriçi, “Truncated Overlap Fermions: the link between Overlap and Domain Wall Fermions”, in V. Mitrjushkin and G. Schierholz (edts.), Lattice Fermions and Structure of the Vacuum, Kluwer Academic Publishers, 2000.6
[9]	A. Boriçi, “QCDLAB: Designing Lattice QCD Algorithms with MATLAB”, High Energy Physics - Lattice (hep-lat), October 2006, arXiv:hep-lat/0610054.
[10]	A. Boriçi, “Speeding up Domain Wall Fermion Algorithms using QCDLAB”, Invited talk given at the 'Domain Wall Fermions at Ten Years', Brookhaven National Laboratory, 15-17 March 2007, arXiv:hep-lat/0703021.
[11]	D. Xhako, A. Boriçi, “Invertimi i operatorit të mbulimit me anë të algoritmit me dy rrjeta në QCD-në rrjetore”, AKTET, Revistë Shkencore e Institutit Alb-Shkenca, ISSN 2073-2244, 2011, pp 384 – 391.
[12]	A. Boriçi, “The two-grid algorithm confronts a shifted unitary orthogonal method”, Nucl.Phys.Proc.Suppl. 140 , 2005, pp 850-852, hep-lat/0409078.
[13]	H. Neuberger, Exactly massless quarks on the lattice, Phys. Lett. B 417, 1998, pp 141.
[14]	A. Boriçi , for a review of these methods in lattice QCD see A. Borici, “Krylov Subspace Methods in Lattice QCD”, PhD thesis, CSCS TR-96-27, ETH Zurich 1996.
[15]	A. Borici, “Lanczos approach to the inverse square root of a large and sparse matrix”, J. Comp. Phys. 162, 2000, pp 123-131.
[16]	A. D. Kennedy, “Fast Evaluation of Zolotarev Coefficients”, QCD and Numerical Analysis III, ISBN: 978-3-540-21257-7, Springer 2005, proceedings of the Third International Workshop on Numerical Analysis and Lattice QCD, Edinburgh June-July 2003.
[17]	A. Borici, A. Allkoci, “A fast minimal residual solver for overlap fermions”, High Energy Physics - Lattice (hep-lat), February 2006, hep-lat/0602015 .
[18]	N. Cundy, A. Frommer, J. van den Eshof, Th. Lippert, S. Krieg, K. Schäfer, “Numerical Methods for the QCD Overlap Operator: III. Nested Iterations”, Comput. Phys. Commun. 165 , 2005, pp 221-242.
[19]	Y. Saad, “Iterative Methods for Sparse Linear Systems”, Second Edition, Copyright, 2003 by the Society for Industrial and Applied Mathematics.

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APA Style

Dafina. Xhako, Artan. Boriçi. (2014). Fast Algorithms for Simulating Chiral Fermions in U(1) Lattice Gauge Theory. American Journal of Physics and Applications, 2(2), 67-72. https://doi.org/10.11648/j.ajpa.20140202.15

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ACS Style

Dafina. Xhako; Artan. Boriçi. Fast Algorithms for Simulating Chiral Fermions in U(1) Lattice Gauge Theory. Am. J. Phys. Appl. 2014, 2(2), 67-72. doi: 10.11648/j.ajpa.20140202.15

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AMA Style

Dafina. Xhako, Artan. Boriçi. Fast Algorithms for Simulating Chiral Fermions in U(1) Lattice Gauge Theory. Am J Phys Appl. 2014;2(2):67-72. doi: 10.11648/j.ajpa.20140202.15

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@article{10.11648/j.ajpa.20140202.15,
  author = {Dafina. Xhako and Artan. Boriçi},
  title = {Fast Algorithms for Simulating Chiral Fermions in U(1) Lattice Gauge Theory},
  journal = {American Journal of Physics and Applications},
  volume = {2},
  number = {2},
  pages = {67-72},
  doi = {10.11648/j.ajpa.20140202.15},
  url = {https://doi.org/10.11648/j.ajpa.20140202.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpa.20140202.15},
  abstract = {Lattice QCD with chiral fermions are extremely computationally expensive, but on the other hand provides an accurate tool for studying the physics of strong interactions. Since the truncated overlap variant of domain wall fermions are equivalent to overlap fermions in four dimensions at any lattice spacing, in this paper we have used domain wall fermions for our simulations. The physical information of lattice QCD theory is contained in quark propagators. In practice computing quark propagator in lattice is an inversion problem of the Dirac operator matrix representing this quarks. In order to develop fast inversion algorithms we have used overlap solvers in two dimensions. Lattice QED theory with U(1) group symmetry in two dimensional space-times dimensions has always been a testing ground for algorithms. By the other side, motivated by our previews work that the two-grid algorithm converge faster than the standard iterative methods for overlap inversion but not for all quark masses, we thought to test this idea in less dimensions such as U(1) gauge theory. Our main objective of this paper it is to implement and develop the idea of a two level algorithm in a new algorithm coded in QCDLAB. This implementation is presented in the preconditioned GMRESR algorithm, as our new contribution in QCDLAB package. The preconditioned part of our algorithm, different from the one of [18], is the approximation of the overlap operator with the truncated overlap operator with finite N3 dimension. We have tested it for 100 statistically independent configurations on 32 x 32 lattice background U(1) field at coupling constant β=1 and for different bare quark masses mq = [0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1]. We have compared the convergence history of the preconditioned GMRESR residual norm with another overlap inverter of QCDLAB as an optimal one, such as SHUMR. We have shown that our algorithm converges faster than SHUMR for different quark masses. Also, we have demonstrated that it saves more time for light quarks compared to SHUMR algorithm.  Our algorithm is approximately independent from the quark mass. This is a key result in simulations with chiral fermions in lattice theories. By the other side, if we compare the results of [18] for quark mass 0.1 in SU(3), results that our chosen preconditioned saves a factor of 2 but in U(1). Our next step is to test this algorithm in SU(3) and to adopt it in parallel},
 year = {2014}
}

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TY - JOUR
T1 - Fast Algorithms for Simulating Chiral Fermions in U(1) Lattice Gauge Theory
AU - Dafina. Xhako
AU - Artan. Boriçi
Y1 - 2014/04/20
PY - 2014
N1 - https://doi.org/10.11648/j.ajpa.20140202.15
DO - 10.11648/j.ajpa.20140202.15
T2 - American Journal of Physics and Applications
JF - American Journal of Physics and Applications
JO - American Journal of Physics and Applications
SP - 67
EP - 72
PB - Science Publishing Group
SN - 2330-4308
UR - https://doi.org/10.11648/j.ajpa.20140202.15
AB - Lattice QCD with chiral fermions are extremely computationally expensive, but on the other hand provides an accurate tool for studying the physics of strong interactions. Since the truncated overlap variant of domain wall fermions are equivalent to overlap fermions in four dimensions at any lattice spacing, in this paper we have used domain wall fermions for our simulations. The physical information of lattice QCD theory is contained in quark propagators. In practice computing quark propagator in lattice is an inversion problem of the Dirac operator matrix representing this quarks. In order to develop fast inversion algorithms we have used overlap solvers in two dimensions. Lattice QED theory with U(1) group symmetry in two dimensional space-times dimensions has always been a testing ground for algorithms. By the other side, motivated by our previews work that the two-grid algorithm converge faster than the standard iterative methods for overlap inversion but not for all quark masses, we thought to test this idea in less dimensions such as U(1) gauge theory. Our main objective of this paper it is to implement and develop the idea of a two level algorithm in a new algorithm coded in QCDLAB. This implementation is presented in the preconditioned GMRESR algorithm, as our new contribution in QCDLAB package. The preconditioned part of our algorithm, different from the one of [18], is the approximation of the overlap operator with the truncated overlap operator with finite N3 dimension. We have tested it for 100 statistically independent configurations on 32 x 32 lattice background U(1) field at coupling constant β=1 and for different bare quark masses mq = [0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1]. We have compared the convergence history of the preconditioned GMRESR residual norm with another overlap inverter of QCDLAB as an optimal one, such as SHUMR. We have shown that our algorithm converges faster than SHUMR for different quark masses. Also, we have demonstrated that it saves more time for light quarks compared to SHUMR algorithm. Our algorithm is approximately independent from the quark mass. This is a key result in simulations with chiral fermions in lattice theories. By the other side, if we compare the results of [18] for quark mass 0.1 in SU(3), results that our chosen preconditioned saves a factor of 2 but in U(1). Our next step is to test this algorithm in SU(3) and to adopt it in parallel
VL - 2
IS - 2
ER -

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Author Information

Dafina. Xhako

Department of Physics, Faculty of Natural Sciences, University of Tirana, Tirana, Albania
Artan. Boriçi

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APA Style

Dafina. Xhako, Artan. Boriçi. (2014). Fast Algorithms for Simulating Chiral Fermions in U(1) Lattice Gauge Theory. American Journal of Physics and Applications, 2(2), 67-72. https://doi.org/10.11648/j.ajpa.20140202.15

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ACS Style

Dafina. Xhako; Artan. Boriçi. Fast Algorithms for Simulating Chiral Fermions in U(1) Lattice Gauge Theory. Am. J. Phys. Appl. 2014, 2(2), 67-72. doi: 10.11648/j.ajpa.20140202.15

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AMA Style

Dafina. Xhako, Artan. Boriçi. Fast Algorithms for Simulating Chiral Fermions in U(1) Lattice Gauge Theory. Am J Phys Appl. 2014;2(2):67-72. doi: 10.11648/j.ajpa.20140202.15

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@article{10.11648/j.ajpa.20140202.15,
  author = {Dafina. Xhako and Artan. Boriçi},
  title = {Fast Algorithms for Simulating Chiral Fermions in U(1) Lattice Gauge Theory},
  journal = {American Journal of Physics and Applications},
  volume = {2},
  number = {2},
  pages = {67-72},
  doi = {10.11648/j.ajpa.20140202.15},
  url = {https://doi.org/10.11648/j.ajpa.20140202.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpa.20140202.15},
  abstract = {Lattice QCD with chiral fermions are extremely computationally expensive, but on the other hand provides an accurate tool for studying the physics of strong interactions. Since the truncated overlap variant of domain wall fermions are equivalent to overlap fermions in four dimensions at any lattice spacing, in this paper we have used domain wall fermions for our simulations. The physical information of lattice QCD theory is contained in quark propagators. In practice computing quark propagator in lattice is an inversion problem of the Dirac operator matrix representing this quarks. In order to develop fast inversion algorithms we have used overlap solvers in two dimensions. Lattice QED theory with U(1) group symmetry in two dimensional space-times dimensions has always been a testing ground for algorithms. By the other side, motivated by our previews work that the two-grid algorithm converge faster than the standard iterative methods for overlap inversion but not for all quark masses, we thought to test this idea in less dimensions such as U(1) gauge theory. Our main objective of this paper it is to implement and develop the idea of a two level algorithm in a new algorithm coded in QCDLAB. This implementation is presented in the preconditioned GMRESR algorithm, as our new contribution in QCDLAB package. The preconditioned part of our algorithm, different from the one of [18], is the approximation of the overlap operator with the truncated overlap operator with finite N3 dimension. We have tested it for 100 statistically independent configurations on 32 x 32 lattice background U(1) field at coupling constant β=1 and for different bare quark masses mq = [0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1]. We have compared the convergence history of the preconditioned GMRESR residual norm with another overlap inverter of QCDLAB as an optimal one, such as SHUMR. We have shown that our algorithm converges faster than SHUMR for different quark masses. Also, we have demonstrated that it saves more time for light quarks compared to SHUMR algorithm.  Our algorithm is approximately independent from the quark mass. This is a key result in simulations with chiral fermions in lattice theories. By the other side, if we compare the results of [18] for quark mass 0.1 in SU(3), results that our chosen preconditioned saves a factor of 2 but in U(1). Our next step is to test this algorithm in SU(3) and to adopt it in parallel},
 year = {2014}
}

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