Asymptotic scaling in the two-dimensional SU(3) <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi></math></span> model at correlation length 4 × <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>5</mn></mrow></msup></mrow></math></span>

Gustavo Mana; Andrea Pelissetto; Alan D. Sokal

doi:10.1103/PhysRevD.54.R1252

Rapid Communication
Access by Xinjiang University

Asymptotic scaling in the two-dimensional SU(3) $σ$ model at correlation length 4 × $10^{5}$

Gustavo Mana^*

Andrea Pelissetto^†

Alan D. Sokal^‡

Department of Physics, New York University, 4 Washington Place, New York, New York 10003

Dipartimento di Fisica and INFN-Sezione di Pisa, Università degli Studi di Pisa, I-56100 Pisa, Italy

Department of Physics, New York University, 4 Washington Place, New York, New York 10003

^*Electronic address (internet): MANA@MAFALDA.PHYSICS.NYU.EDU
^†Electronic address (internet): PELISSET@SUNTHPI1.DIFI.UNIPI.IT
^‡Electronic address (internet): SOKAL@NYU.EDU

PDF

Phys. Rev. D 54, R1252(R) – Published 15 July, 1996

DOI: https://doi.org/10.1103/PhysRevD.54.R1252

Abstract

We carry out a high-precision simulation of the two-dimensional SU(3) principal chiral model at correlation lengths $ξ$ up to ≈4 × $10^{5}$ , using a multigrid Monte Carlo (MGMC) algorithm. We extrapolate the finite-volume Monte Carlo data to infinite volume using finite-size-scaling theory, and we discuss carefully the systematic and statistical errors in this extrapolation. We then compare the extrapolated data to the renormalization-group predictions. For $ξ ≳ 10^{3}$ we observe good asymptotic scaling in the bare coupling; at $ξ \approx 4 \times 10^{5}$ the nonperturbative constant is within 2-3% of its predicted limiting value.

References (22)

M. Creutz, Quarks, Gluons, and Lattices (Cambridge University Press, New York, 1983) H. J. Rothe, Lattice Gauge Theories: An Introduction (World Scientific, Singapore, 1992)
A. Patrasciouiu, Phys. Rev. Lett. 54, 1192 (1985) A. Patrasciouiu and E. Seiler, Max-Planck-Institut Report No. MPI-Ph/91-88, 1991 (unpublished) in Lattice '92, Proceedings of the International Symposium, Amsterdam, The Netherlands, edited by J. Smit and P. van Baal [Nucl. Phys. B (Proc. Suppl.) 30, 184 (1993)] Phys. Rev. Lett. 74, 1920 (1995) ibid.74, 1924 (1995)
UKQCD Collaboration, S. P. Booth et al., Nucl. Phys. B394, 509 (1993) A. Ukawa, in Lattice '92 [[2]], p. 3, Sec. 6.2 G. S. Bali and K. Schilling, Phys. Rev. D 47, 661 (1993) K. M. Bitar et al., hep-lat/9602010 [Phys. Rev. D (to be published)]
A. M. Polyakov, Phys. Lett. 59B, 79 (1975) E. Brézin and J. Zinn-Justin, Phys. Rev. B 14, 3110 (1976) W. A. Bardeen, B. W. Lee, and R. E. Shrock, Phys. Rev. D 14, 985 (1976) J. B. Kogut, Rev. Mod. Phys. 51, 659 (1979), Sec. VIII.C
S. Caracciolo, R. G. Edwards, A. Pelissetto, and A. D. Sokal, Phys. Rev. Lett. 75, 1891 (1995) A. Patrascioiu and E. Seiler, ibid. 76, 1178 (1996) S. Caracciolo, R. G. Edwards, A. Pelissetto, and A. D. Sokal, ibid. 76, 1179 (1996)
M. Lüscher, P. Weiz, and U. Wolff, Nucl. Phys. B359, 221 (1991)
J.-K. Kim, Phys. Rev. Lett. 70, 1735 (1993) in Lattice '93, Proceedings of the International Symposium, Dallas, Texas, edited by T. Draper [Nucl. Phys. B (Proc. Suppl.) 34, 702 (1994)] Phys. Rev. D 50, 4663 (1994) Europhys. Lett. 28, 211 (1994) Phys. Lett. B 345, 469 (1995)
S. Caracciolo, R. G. Edwards, S. J. Ferreira, A. Pelissetto, and A. D. Sokal, Phys. Rev. Lett. 74, 2969 (1995)
[[5]]
[[11]]
S. Caracciolo, R. G. Edwards, A. Pelissetto, and A. D. Sokal, Nucl. Phys. B403, 475 (1993)
P. Rossi and E. Vicari, Phys. Rev. D 49, 1621 (1994) ibid.49, 6072 (1994) ibid.50, 4718(E) (1994)
J. Balog, S. Naik, F. Niedermayer, and P. Weisz, Phys. Rev. Lett. 69, 873 (1992)
[[15]] [[12]]
H. Flyvbjerg, Nucl. Phys. B348, 714 (1991) P. Biscari, M. Campostrini, and P. Rossi, Phys. Lett. B 242, 225 (1990) S. Caracciolo and A. Pelissetto (unpublished)
E. Dagotto and J. B. Kogut, Nucl. Phys. B290[FS20], 451 (1987)
M. Hasenbusch and S. Meyer, Phys. Rev. Lett. 68, 435 (1992)
R. R. Horgan and I. T. Drummond, Phys. Lett. B 327, 107 (1994)
G. Mana, T. Mendes, A. Pelissetto, and A. D. Sokal, in Lattice '95, Proceedings of the International Symposium, Melbourne, Australia, edited by T. Kiev [Nucl. Phys. B (Proc. Suppl.) 47, 796 (1996)]
Omitted endnote
G. Mana, A. Pelissetto, and A. D. Sokal (unpublished)
G. Martinelli, G. Parisi, and R. Petronzio, Phys. Lett. 100B, 485 (1981) S. Samuel, O. Martin, and K. Moriarty, ibid. 153B, 87 (1985) G. P. Lepage and P. B. Mackenzie, Phys. Rev. D 48, 2250 (1993)

[c1] M. Creutz, Quarks, Gluons, and Lattices (Cambridge University Press, New York, 1983) H. J. Rothe, Lattice Gauge Theories: An Introduction (World Scientific, Singapore, 1992)

[c2] A. Patrasciouiu, Phys. Rev. Lett. 54, 1192 (1985) A. Patrasciouiu and E. Seiler, Max-Planck-Institut Report No. MPI-Ph/91-88, 1991 (unpublished) in Lattice '92, Proceedings of the International Symposium, Amsterdam, The Netherlands, edited by J. Smit and P. van Baal [Nucl. Phys. B (Proc. Suppl.) 30, 184 (1993)] Phys. Rev. Lett. 74, 1920 (1995) ibid.74, 1924 (1995)

[c3] UKQCD Collaboration, S. P. Booth et al., Nucl. Phys. B394, 509 (1993) A. Ukawa, in Lattice '92 [[2]], p. 3, Sec. 6.2 G. S. Bali and K. Schilling, Phys. Rev. D 47, 661 (1993) K. M. Bitar et al., hep-lat/9602010 [Phys. Rev. D (to be published)]

[c4] A. M. Polyakov, Phys. Lett. 59B, 79 (1975) E. Brézin and J. Zinn-Justin, Phys. Rev. B 14, 3110 (1976) W. A. Bardeen, B. W. Lee, and R. E. Shrock, Phys. Rev. D 14, 985 (1976) J. B. Kogut, Rev. Mod. Phys. 51, 659 (1979), Sec. VIII.C

[c5] S. Caracciolo, R. G. Edwards, A. Pelissetto, and A. D. Sokal, Phys. Rev. Lett. 75, 1891 (1995) A. Patrascioiu and E. Seiler, ibid. 76, 1178 (1996) S. Caracciolo, R. G. Edwards, A. Pelissetto, and A. D. Sokal, ibid. 76, 1179 (1996)

[c6] M. Lüscher, P. Weiz, and U. Wolff, Nucl. Phys. B359, 221 (1991)

[c7] J.-K. Kim, Phys. Rev. Lett. 70, 1735 (1993) in Lattice '93, Proceedings of the International Symposium, Dallas, Texas, edited by T. Draper [Nucl. Phys. B (Proc. Suppl.) 34, 702 (1994)] Phys. Rev. D 50, 4663 (1994) Europhys. Lett. 28, 211 (1994) Phys. Lett. B 345, 469 (1995)

[c8] S. Caracciolo, R. G. Edwards, S. J. Ferreira, A. Pelissetto, and A. D. Sokal, Phys. Rev. Lett. 74, 2969 (1995)

[c9] [[5]]

[c10] [[11]]

[c11] S. Caracciolo, R. G. Edwards, A. Pelissetto, and A. D. Sokal, Nucl. Phys. B403, 475 (1993)

[c12] P. Rossi and E. Vicari, Phys. Rev. D 49, 1621 (1994) ibid.49, 6072 (1994) ibid.50, 4718(E) (1994)

[c13] J. Balog, S. Naik, F. Niedermayer, and P. Weisz, Phys. Rev. Lett. 69, 873 (1992)

[c14] [[15]] [[12]]

[c15] H. Flyvbjerg, Nucl. Phys. B348, 714 (1991) P. Biscari, M. Campostrini, and P. Rossi, Phys. Lett. B 242, 225 (1990) S. Caracciolo and A. Pelissetto (unpublished)

[c16] E. Dagotto and J. B. Kogut, Nucl. Phys. B290[FS20], 451 (1987)

[c17] M. Hasenbusch and S. Meyer, Phys. Rev. Lett. 68, 435 (1992)

[c18] R. R. Horgan and I. T. Drummond, Phys. Lett. B 327, 107 (1994)

[c19] G. Mana, T. Mendes, A. Pelissetto, and A. D. Sokal, in Lattice '95, Proceedings of the International Symposium, Melbourne, Australia, edited by T. Kiev [Nucl. Phys. B (Proc. Suppl.) 47, 796 (1996)]

[c20] Omitted endnote

[c21] G. Mana, A. Pelissetto, and A. D. Sokal (unpublished)

[c22] G. Martinelli, G. Parisi, and R. Petronzio, Phys. Lett. 100B, 485 (1981) S. Samuel, O. Martin, and K. Moriarty, ibid. 153B, 87 (1985) G. P. Lepage and P. B. Mackenzie, Phys. Rev. D 48, 2250 (1993)

Physical Review D

Export citation

Export citation

Asymptotic scaling in the two-dimensional SU(3) $σ$ model at correlation length 4 × $10^{5}$

Abstract

References (22)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation

Asymptotic scaling in the two-dimensional SU(3) σ model at correlation length 4 × 105

Abstract

References (22)

Asymptotic scaling in the two-dimensional SU(3) $σ$ model at correlation length 4 × $10^{5}$