When correlations exceed system size: ﬁnite-size scaling in free boundary conditions above the upper critical dimension

Yu. Honchar; B. Berche; Yu. Holovatch; R. Kenna

doi:10.5488/cmp.27.13603

Authors

Yu. Honchar Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine; Centre for Fluids and Complex Systems, Coventry University, Coventry CV1 5FB, UK; L4 Collaboration and Doctoral College for the Statistical Physics of Complex Systems, Lviv-Leipzig-Lorraine-Coventry, Europe https://orcid.org/0000-0003-2660-4593
B. Berche Laboratoire de Physique et Chimie Théoriques, Université de Lorraine - CNRS, Nancy Cedex, France; L4 Collaboration and Doctoral College for the Statistical Physics of Complex Systems, Lviv-Leipzig-Lorraine-Coventry, Europe; https://orcid.org/0000-0002-4254-807X
Yu. Holovatch Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine; Centre for Fluids and Complex Systems, Coventry University, Coventry CV1 5FB, UK; L4 Collaboration and Doctoral College for the Statistical Physics of Complex Systems, Lviv-Leipzig-Lorraine-Coventry, Europe; Complexity Science Hub Vienna, 1080 Vienna, Austria https://orcid.org/0000-0002-1125-2532
R. Kenna Centre for Fluids and Complex Systems, Coventry University, Coventry CV1 5FB, UK; L4 Collaboration and Doctoral College for the Statistical Physics of Complex Systems, Lviv-Leipzig-Lorraine-Coventry, Europe https://orcid.org/0000-0001-9990-4277

DOI:

https://doi.org/10.5488/cmp.27.13603

Keywords:

universality, ﬁnite-size scaling, upper critical dimension

Abstract

We progress finite-size scaling in systems with free boundary conditions above their upper critical dimension, where in the thermodynamic limit critical scaling is described by mean-field theory. Recent works show that the correlation length is not bound by the system's physical size, a belief that long held sway. Instead, two scaling regimes can be observed — at the critical and pseudo-critical temperatures. We demonstrate that both are manifest for free boundaries. We use numerical simulations of the d = 5 Ising model to analyse the magnetization, susceptibility, magnetization Fourier modes and the partition function zeros. While some of the response functions hide the dual finite-size scaling, the precision enabled by the analysis of Lee–Yang zeros allows this be brought to the fore. In particular, finite-size scaling of leading zeros at the pseudo-critical point confirms recent predictions coming from correlations exceeding the system size. This paper is dedicated to Jaroslav Ilnytskyi on the occasion of his 60th birthday.

References

Berche B., Kenna R., Walter J. C., Nucl. Phys. B, 2012, 865, 115, https://doi.org/10.1016/j.nuclphysb.2012.07.021. DOI: https://doi.org/10.1016/j.nuclphysb.2012.07.021

Lundow P. H., Markström K., Nucl. Phys. B, 2014, 889, 249, https://doi.org/10.1016/j.nuclphysb.2014.10.011. DOI: https://doi.org/10.1016/j.nuclphysb.2014.10.011

Flores-Sola E., Berche B., Kenna R., Weigel M., Phys. Rev. Lett., 2016, 116, 115701, https://doi.org/10.1103/PhysRevLett.116.115701. DOI: https://doi.org/10.1103/PhysRevLett.116.115701

Kenna R., Berche B., Condens. Matter Phys., 2013, 16, 23601, https://doi.org/10.5488/CMP.16.23601. DOI: https://doi.org/10.5488/CMP.16.23601

Kenna R., Berche B., J. Phys. A: Math. Theor., 2017, 50, 235001, https://doi.org/10.1088/1751-8121/aa6bd5. DOI: https://doi.org/10.1088/1751-8121/aa6bd5

Berche B., Chatelain C., Dhall C., Kenna R., Low R., Walter J. C., J. Stat. Mech.: Theory Exp., 2008, 2008, P11010, https://doi.org/10.1088/1742-5468/2008/11/P11010. DOI: https://doi.org/10.1088/1742-5468/2008/11/P11010

Ellis T., Kenna R., Berche B., Condens. Matter Phys., 2023, 26, No. 3, 33606 https://doi.org/10.5488/CMP.26.33606. DOI: https://doi.org/10.5488/CMP.26.33606

Berche B., Ellis T., Holovatch Yu., Kenna R., SciPost Phys. Lect. Notes, 2022, 60, https://doi.org/10.21468/SciPostPhysLectNotes.60. DOI: https://doi.org/10.21468/SciPostPhysLectNotes.60

Ivaneyko D., Berche B., Holovatch Yu., Ilnytskyi J., Physica A, 2008, 387, 4497–4512, https://doi.org/10.1016/j.physa.2008.03.034. DOI: https://doi.org/10.1016/j.physa.2008.03.034

Ilnytskyi J. M., Holovatch Yu., Condens. Matter Phys., 2007, 10, 539–552, https://doi.org/10.5488/CMP.10.4.539. DOI: https://doi.org/10.5488/CMP.10.4.539

Ivaneyko D., Ilnytskyi J., Berche B., Holovatch Yu., Physica A, 2006, 370, 163–178, https://doi.org/10.1016/j.physa.2006.03.010. DOI: https://doi.org/10.1016/j.physa.2006.03.010

Chaikin P. M., Lubensky T. C., Principles of Condensed Matter Physics, Cambridge University Press, 2000.

Berche B., Henkel M., Kenna R., J. Phys. Stud., 2009, 13, 3201, https://doi.org/10.30970/jps.13.3001. DOI: https://doi.org/10.30970/jps.13.3001

Janke W., Kenna R., Phys. Rev. B, 2002, 65, 064110, https://doi.org/10.1103/PhysRevB.65.064110. DOI: https://doi.org/10.1103/PhysRevB.65.064110

Privman V., Fisher M. E., J. Stat. Phys., 1983, 33, 385, https://doi.org/10.1007/BF01009803. DOI: https://doi.org/10.1007/BF01009803

Brézin E., Zinn-Justin J., Nucl. Phys. B, 1985, 257, 867, https://doi.org/10.1016/0550-3213(85)90379-7. DOI: https://doi.org/10.1016/0550-3213(85)90379-7

Jones J. L., Young A. P., Phys. Rev. B, 2005, 71, 174438, https://doi.org/10.1103/PhysRevB.71.174438. DOI: https://doi.org/10.1103/PhysRevB.71.174438

Flores-Sola E. J., Berche B., Kenna R., Weigel M., Eur. Phys. J. B, 2015, 88, 28, https://doi.org/10.1140/epjb/e2014-50683-1. DOI: https://doi.org/10.1140/epjb/e2014-50683-1

Kenna R., Berche B., Eur. Phys. Lett., 2014, 105, 26005, https://doi.org/10.1209/0295-5075/105/26005. DOI: https://doi.org/10.1209/0295-5075/105/26005

Binder K., Nauenberg M., Privman V., Young A. P., Phys. Rev. B, 1985, 31, 1498, https://doi.org/10.1103/PhysRevB.31.1498. DOI: https://doi.org/10.1103/PhysRevB.31.1498

Kenna R., Berche B., In: Order, Disorder, and Criticality: Advanced Problems of Phase Transition Theory, Holovatch Yu. (Ed.), Vol. 4, World Scientific, Singapore, 2015, 1–54. DOI: https://doi.org/10.1142/9789814632683_0001

Yang C. N., Lee T. D., Phys. Rev., 1952, 87, 404, https://doi.org/10.1103/PhysRev.87.404. DOI: https://doi.org/10.1103/PhysRev.87.404

Lee T. D., Yang C. N., Phys. Rev., 1952, 87, 410, https://doi.org/10.1103/PhysRev.87.410. DOI: https://doi.org/10.1103/PhysRev.87.410

Fisher M. E., In: Lectures in Theoretical Physics, Vol. 7C, Britten W. E. (Ed.), University of Colorado Press, Boulder, Colorado, USA, 1965, 1–159.

Wu F. Y., Int. J. Mod. Phys. B, 2008, 22, 1899, https://doi.org/10.1142/S0217979208039198. DOI: https://doi.org/10.1142/S0217979208039198

Lundow P. H., Markström K., Nucl. Phys. B, 2011, 845, 120–139, https://doi.org/10.1016/j.nuclphysb.2010.12.002. DOI: https://doi.org/10.1016/j.nuclphysb.2010.12.002

Lundow P. H., Nucl. Phys. B, 2021, 967, 115422, https://doi.org/10.1016/j.nuclphysb.2021.115422. DOI: https://doi.org/10.1016/j.nuclphysb.2021.115422

Wolff U., Phys. Rev. Lett, 1989, 62, 361, https://doi.org/10.1103/PhysRevLett.62.361. DOI: https://doi.org/10.1103/PhysRevLett.62.361

Kasteleyn P. W., Fortuin C. M., J. Phys. Soc. Jpn. Suppl., 1969, 26, 11.

Fortuin C. M., Kasteleyn P. W., Physica, 1972, 57, 536, https://doi.org/10.1016/0031-8914(72)90045-6. DOI: https://doi.org/10.1016/0031-8914(72)90045-6

Ferrenberg A. M., Swendsen R. H., Phys. Rev. Lett., 1988, 61, 2635, https://doi.org/10.1103/PhysRevLett.61.2635. DOI: https://doi.org/10.1103/PhysRevLett.61.2635

Ferrenberg A. M., Swendsen R. H., Phys. Rev. Lett., 1989, 63, 1195, https://doi.org/10.1103/PhysRevLett.63.1195. DOI: https://doi.org/10.1103/PhysRevLett.63.1195

Rudnick J., Gaspari G., Privman V., Phys. Rev. B, 1985, 32, 7594, https://doi.org/10.1103/PhysRevB.32.7594. DOI: https://doi.org/10.1103/PhysRevB.32.7594

Bena I., Droz M., Lipowski A., Int. J. Mod. Phys. B, 2005, 19, 4269, https://doi.org/10.1142/S0217979205032759. DOI: https://doi.org/10.1142/S0217979205032759

Krasnytska M., Berche B., Holovatch Yu., Kenna R., J. Phys. A: Math. Theor., 2016, 49, 135001, https://doi.org/10.1088/1751-8113/49/13/135001. DOI: https://doi.org/10.1088/1751-8113/49/13/135001

Krasnytska M., Berche B., Holovatch Yu., Kenna R., Europhys. Lett., 2015, 111, 60009, https://doi.org/10.1209/0295-5075/111/60009. DOI: https://doi.org/10.1209/0295-5075/111/60009

Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P., Numerical Recipes: The Art of Scientific Computing, 3rd Edition, Cambridge University Press, 2007.

Itzykson C., Pearson R. B., Zuber J. B., Nucl. Phys. B, 1983, 220, 415, https://doi.org/10.1016/0550-3213(83)90499-6. DOI: https://doi.org/10.1016/0550-3213(83)90499-6