Non-perturbative Approaches in Nanoscience and Corrections to Finite-Size Scaling

  • J. KaupužsEmail author
  • R. V. N. Melnik
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 232)


Non-perturbative approaches in nanoscience are discussed. Traditional applications of these approaches cover description of charge transport and optical phenomena in nano-scale systems. We focus on finite-size effects in spin systems near the critical point, based on Monte Carlo (MC) method and some analytical arguments. We have performed MC simulations of the 3D Ising model for small, as well as large linear lattice sizes up to \(L=2560\), providing a numerical evidence for a recent challenging prediction, according to which the asymptotic decay of corrections to finite-size scaling is remarkably slower than it was expected before. Our approach along with several other non-perturbative approaches, like, e.g., the non-perturbative nonequilibrium Greens functions (NEGF) method, reveals a potential application of non-perturbative methods to nanoscience and nanotechnology through condensed matter physics, including semiconductor physics and physics of disordered systems like spin glasses.


Ising model Non-perturbative methods Finite-size effects Corrections to scaling Critical exponents Monte Carlo simulation 



This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET: The authors acknowledge the use of resources provided by the Latvian Grid Infrastructure and High Performance Computing centre of Riga Technical University. R. M. acknowledges the support from the NSERC and CRC program.


  1. 1.
    M.E.J. Newman, G.T. Barkema, Monte Carlo Methods in Statistical Physics (Clarendon Press, Oxford, 1999)zbMATHGoogle Scholar
  2. 2.
    C. Riddet, A. Asenov, Proceedings Simulation of Semiconductor Processes and Devices (2008), pp. 261–264Google Scholar
  3. 3.
    A. Martiner, M. Bescond, J.R. Barker, A. Svizhenko, M.P. Anantram, C. Millar, A. Asenov, IEEE Trans. Electron Devices 54, 2213–2222 (2007)CrossRefGoogle Scholar
  4. 4.
    D.A. Areshkin, B.K. Nikolic, Phys. Rev. B 81, 155450 (2010)CrossRefGoogle Scholar
  5. 5.
    Z. Papić, E.M. Stoudenmire, D.A. Abanin, Ann. Phys. 362, 714–725 (2015)CrossRefGoogle Scholar
  6. 6.
    H.L. Calve, P.M. Perez-Piskunov, H.M. Pastawski, S. Roche, L.E.F.F. Torres, J. Phys. Condens. Matter 25, 144202 (2013)Google Scholar
  7. 7.
    M. Hasenbusch, Phys. Rev. B 82, 174433 (2010)CrossRefGoogle Scholar
  8. 8.
    M. Castellana, G. Parisi, Sci. Rep. 5, 12367 (2015)CrossRefGoogle Scholar
  9. 9.
    L. Lin, A.H. Larsen, N.A. Romero, V.A. Morozov, C. Glinsvad, F. Abild-Pedersen, J. Greeley, K.W. Jacobsen, J.K. Nørskov, J. Phys. Chem. Lett. 4, 222–226 (2013)CrossRefGoogle Scholar
  10. 10.
    J. Kaupužs, R.V.N. Melnik, J. Rimšāns, Int. J. Mod. Phys. C 27, 1650108 (2016)CrossRefGoogle Scholar
  11. 11.
    S.K. Ma, Modern Theory of Critical Phenomena (W. A. Benjamin Inc, New York, 1976)Google Scholar
  12. 12.
    J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 1996)zbMATHGoogle Scholar
  13. 13.
    J. Kaupužs, Ann. Phys. (Berlin) 10, 299–331 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    J. Kaupužs, Canadian. J. Phys. 9, 373 (2012)Google Scholar
  15. 15.
    A.A. Pogorelov, I.M. Suslov, J. Exp. Theor. Phys. 106, 1118 (2008)CrossRefGoogle Scholar
  16. 16.
    J. Kaupužs, J. Rimšāns, R.V.N. Melnik, Ukr. J. Phys. 56, 845 (2011)Google Scholar
  17. 17.
    J. Kaupužs, R.V.N. Melnik, J. Rimšāns, Int. J. Mod. Phys. C 28, 1750044 (2017)CrossRefGoogle Scholar
  18. 18.
    U. Wolff, Phys. Rev. Lett. 62, 361 (1989)CrossRefGoogle Scholar
  19. 19.
    J. Kaupužs, J. Rimšāns, R.V.N. Melnik, Phys. Rev. E 81, 026701 (2010)CrossRefGoogle Scholar
  20. 20.
    S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, A. Vichi, J. Stat. Phys. 157, 869 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    M. Hasenbusch, Int. J. Mod. Phys. C 12, 911 (2001)CrossRefGoogle Scholar
  22. 22.
    J. Kaupužs, Int. J. Mod. Phys. A 27, 1250114 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    F. Kos, D. Poland, D. Simmons-Duffin, JHEP 1406, 091 (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Materials Science and Applied ChemistryInstitute of Technical Physics, Riga Technical UniversityRigaLatvia
  2. 2.Institute of Mathematical Sciences and Information TechnologiesUniversity of LiepajaLiepajaLatvia
  3. 3.The MS2 Discovery Interdisciplinary Research InstituteWilfrid Laurier UniversityWaterlooCanada
  4. 4.BCAM - Basque Center for Applied MathematicsBilbaoSpain

Personalised recommendations