- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Finite-size scaling for quantum criticality using the finite-element method

Edwin Antillon1, Birgit Wehefritz-Kaufmann1,2, and Sabre Kais1,3,*

  • 1Department of Physics, Purdue University, West Lafayette, Indiana 47907, USA
  • 2Departments of Mathematics and Physics, Purdue University, West Lafayette, Indiana 47907, USA
  • 3Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, USA

  • *kais@purdue.edu
PDF

Phys. Rev. E 85, 036706 – Published 15 March, 2012

DOI: https://doi.org/10.1103/PhysRevE.85.036706

Abstract

Finite size scaling for the Schrödinger equation is a systematic approach to calculate the quantum critical parameters for a given Hamiltonian. This approach has been shown to give very accurate results for critical parameters by using a systematic expansion with global basis-type functions. Recently, the finite-element method was shown to be a powerful numerical method for ab initio electronic-structure calculations with a variable real-space resolution. In this work, we demonstrate how to obtain quantum critical parameters by combining the finite-element method (FEM) with finite size scaling (FSS) using different ab initio approximations and exact formulations. The critical parameters could be atomic nuclear charges, internuclear distances, electron density, disorder, lattice structure, and external fields for stability of atomic, molecular systems and quantum phase transitions of extended systems. To illustrate the effectiveness of this approach we provide detailed calculations of applying FEM to approximate solutions for the two-electron atom with varying nuclear charge; these include Hartree-Fock, local density approximation, and an “exact” formulation using FEM. We then use the FSS approach to determine its critical nuclear charge for stability; here, the size of the system is related to the number of elements used in the calculations. Results prove to be in good agreement with previous Slater-basis set calculations and demonstrate that it is possible to combine finite size scaling with the finite-element method by using ab initio calculations to obtain quantum critical parameters. The combined approach provides a promising first-principles approach to describe quantum phase transitions for materials and extended systems.

View figure in article

Article Text

References (62)

  1. M. P. Nightingale, Physica A 83, 561 (1976).
  2. J. L. Cardy, Finite-Size Scaling (Elsevier Science, New York, 1988).
  3. M. E. Fisher, in Critical Phenomena, Proceedings of the 51st Enrico Fermi Summer School, Verenna, Italy, edited by M. S. Green (Academic Press, New York, 1971); M. E. Fisher and M. N. Barber, Phys. Rev. Lett. 28, 1516 (1972).
  4. B. Widom, in Critical Phenomena in Fundamental Problems in Statistical Mechanics, edited by E. G. D. Cohen (Elsevier, New York, 1975).
  5. M. N. Barber, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Labors (Academic Press, London, 1983), Vol. 8.
  6. V. Privman, Finite Size Scaling and Numerical Simulations of Statistical Systems (World Scientific, Singapore, 1990).
  7. P. J. Reynolds, H. E. Stanley, and W. Klein, J. Phys. A 11, L199 (1978).
  8. P. J. Reynolds, H. E. Stanley, and W. Klein, Phys. Rev. B 21, 1223 (1980).
  9. W. Moy, S. Kais, and P. Serra, Mol. Phys. 106, 203 (2008).
  10. J. P. Neirotti, P. Serra, and S. Kais, Phys. Rev. Lett. 79, 3142 (1997).
  11. E. Antillon, W. Moy, Q. Wei, and S. Kais, J. Chem. Phys. 131, 104105 (2009).
  12. E. Tsuchida and M. Tsukada, Phys. Rev. B 54, 7602 (1996).
  13. D. Stiles (private communication).
  14. J. E. Pask and P. A. Sterne, Modell. Simul. Mater. Sci. Eng. 13, R71 (2005).
  15. L. R. Ram-Mohan, Finite Element and Boundary Element Applications in Quantum Mechanics (Oxford University Press, London, 2002).
  16. D. W. Pepper and J. C. Heinrich, The Finite Element Method (Taylor & Francis, New York, 2006).
  17. O. Taisuke and T. Masayuki, Comput. Phys. Commun. 182, 1245 (2011).
  18. J. E. Bylaska, M. Holst, and J. H. Weare, J. Chem. Theor. Comp. 5, 937 (2009).
  19. R. Alizadegan, K. J. Hit, and T. J. Martinez, J. Chem. Phys. 132, 034101 (2010).
  20. J. E. Pask, B. M. Klein, C. Y. Fong, and P. A. Sterne, Phys. Rev. B 59, 12352 (1999).
  21. C. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (1952).
  22. T. D. Lee and C. N. Yang, Phys. Rev. 87, 410 (1952).
  23. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 1999).
  24. J. P. Neirotti, P. Serra, and S. Kais, Phys. Rev. Lett. 79, 3142 (1997).
  25. P. Serra, J. P. Neirotti, and S. Kais, Phys. Rev. Lett. 80, 5293 (1998).
  26. S. Kais, J. P. Neirotti, and P. Serra, Int. J. Mass Spectrom. 182, 23 (1999).
  27. P. Serra, J. P. Neirotti, and S. Kais, Phys. Rev. A 57, R1481 (1998).
  28. P. Serra, J. P. Neirotti, and S. Kais, J. Phys. Chem. A 102, 9518 (1998).
  29. J. P. Neirotti, P. Serra, and S. Kais, J. Chem. Phys. 108, 2765 (1998).
  30. Q. Shi and S. Kais, Mol. Phys. 98, 1485 (2000).
  31. S. Kais and Q. Shi, Phys. Rev. A 62, 060502 (2000).
  32. S. Kais and P. Serra, Int. Rev. Phys. Chem. 19, 97 (2000).
  33. S. Kais and P. Serra, Adv. Chem. Phys. 125, 1 (2003).
  34. P. Serra and S. Kais, Chem. Phys. Lett. 372, 205 (2003).
  35. A. Ferron, P. Serra, and S. Kais, J. Chem. Phys. 120, 8412 (2004).
  36. Q. Shi and S. Kais, Mol. Phys. 98, 1485 (2000).
  37. Q. Shi and S. Kais, Int. J. Quantum Chem. 85, 307 (2001).
  38. A. Ferron, P. Serra, and S. Kais, J. Chem. Phys. 128, 044307 (2008).
  39. B. Simon, J. Funct. Anal. 25, 338 (1977).
  40. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, Vol. 3, 3rd ed. (Butterworth-Heinemann, London, 1981).
  41. S. Chandrasekhar, Astrophys. J. 100, 176 (1944).
  42. F. H. Stillinger and D. K. Stillinger, Phys. Rev. A 10, 1109 (1974).
  43. J. D. Baker, D. E. Freund, R. N. Hill, and J. D. Morgan III, Phys. Rev. A 41, 1247 (1990).
  44. J. M. Thijssen, Computational Physics (Cambridge University Press, Cambridge, 1999).
  45. W. Schweizer. Numerical Quantum Dynamics (Kluwer Academic Publishers, Dordrecht, 2001).
  46. G. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, Cambridge, 2005).
  47. H. Mitler, Phys. Rev. 99, 1835 (1955).
  48. C. W. Ufford and J. G. Thomas, Phys. Rev. 133, A121 (1964).
  49. W. Kohn and L. Sham, Phys. Rev. A 140, 1133 (1965).
  50. C. J. Umrigar and X. Gonze, Phys. Rev. A 50, 3827 (1994).
  51. J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992).
  52. D. Porezag, [http://beam.acclab.helsinki.fi/∼akrashen/escalc/mathsub.f].
  53. G. Breit, Phys. Rev. 35, 569 (1930).
  54. F. S. Levin and J. Shertzer, Phys. Rev. A 32, 3285 (1985).
  55. F. M. Gomes, [http://www.ime.unicamp.br/∼chico/arpack++].
  56. C. Geuzaine and J. Remacle, [http://geuz.org/gmsh/].
  57. R. Bulirsch and J. Stoer, Num. Math 6, 413 (1964).
  58. P. A. Sterne, J. E. Pask, and B. M. Klein, Appl. Surf. Sci. 149, 238 (1999).
  59. E. Tsuchida, J. Chem. Phys. 121, 4740 (2004).
  60. D. A. Mazziotti, Phys. Rev. Lett. 106, 083001 (2011).
  61. D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004).
  62. B. Tanatar and D. M. Ceperley, Phys. Rev. B 39, 5005 (1989).

Outline

Information

Phys. Rev. E 85, 036706– Published 15 March, 2012
  • Received 12 September 2011
Article part of CHORUS

©2012 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation