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High-and Low-temperature expansion for the spin statistical systems using the finite lattice method

Research Project

Project/Area Number 08640494
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 物性一般(含基礎論)
Research InstitutionOsaka Prefectural College of Technology

Principal Investigator

ARISUE Hiroaki  Osaka Prefectural College of Technology, Professor, 助教授 (10175987)

Project Period (FY) 1996 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1998: ¥200,000 (Direct Cost: ¥200,000)
Fiscal Year 1997: ¥200,000 (Direct Cost: ¥200,000)
Fiscal Year 1996: ¥1,600,000 (Direct Cost: ¥1,600,000)
KeywordsIsing model / Potts model / high-temperature expansion / finite lattice method / phase transition / critical exponent / magnetic susceptibility / correlation length / 低温展開 / Potts模型 / large-q展開 / Patts模型 / SOS模型 / 比熱
Research Abstract

Using the finite lattice method we calculated the high order terms of the various types of the parameter expansions for various kinds of spin systems.
We first calculated the low-temperature expansion series of the free energy, the magnetization and the magnetic susceptibility in the q-state Potts model in two dimensions for all of q=5-50, where the system exhibits first order phase transition to the 41st order in the expansion, parameter. We also applied the finite lattice method to the large-q expansion of the same system and obtained the series for the up to 6th order energy cumulants to the 23rd order in 1/√q. Analyzing the series, we found that the values of the free energy and the latent heat coincide in very high precision with those of the exact solutions at the critical point and the series for the specific heat gives converging results in the accuracy of a few percent even for q=5, where the correlation length of the system amount to several thousands of the lattice spacing and other methods such as the Monte Carlo simulations cannot give any meaningful result.
Secondly we calculated the low-temperature expansion series for the three kinds of quantities concerning the interface width for the Ising model in three dimensions and for the Absolute value SOS model and Discrete Gaussian model on the square lattice and on the triangular lattice, respectively, to 17th, 23rd, 22nd, 24th and 22nd order in the expansion parameter for the respective model. Assuming the critical exponent of the each quantity that is predicted by the renormalization group arguments, we obtained from the series the value of the roughening transition point for each model. They agree quite well with the results of the Monte Carlo simulation for the former three models and they give new prediction for the latter two models.

Report

(4 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • 1996 Annual Research Report
  • Research Products

    (18 results)

All Other

All Publications (18 results)

  • [Publications] H.Arisue, K.Tabata: "Low-Temperature Series for the Potts Model by Finite-Lattice Method"Nuclear Physics B[Proc.Suppl.]. 47. 739-742 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Arisue: "Low-temperature expansion of the surface width in the D=3 Ising model"Nuclear Physics B[Proc.Suppl.]. 47. 743-746 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Arisue, K.Tabata: "Low-temperature series for the square lattice Potts model by the Improved Finite-Lattice Method"Journal of Physics A. 30. 3313-3327 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Arisue: "Surface width of the Solid-On-Solid models"Nuclear Physics B[Proc.Suppl.]. 63. 649-651 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Arisue, K.Tabata: "Large-q expansion of the specific heat for the two-dimensional q-state Potts model"Physical Review E. 59. 186-188 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Arisue, K.Tabata: "Large-q expansion of the energy and magnetization cumulants for the two-dimensional q-state Potts model"Nuclear Physics B[Field Theory and Statistical Physics]. 546. 558-584 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Arisue, K.Tabata: "Low-Temperature Series for the Potts Model by Finite-Lattice Method"Nuclear Physics B [Proc. Suppl.]. Vol.47. 739-742 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Arisue: "Low-temperature expansion of the surface width in the D=3 Ising model"Nuclear Physics B [Proc. Suppl.]. Vol.47. 743-746 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Arisue, K.Tabata: "Low-temperature series for the square lattice Potts model by the Improved Finite-Lattice Method"Journal of Physics A. Vol.63. 3313-3327 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Arisue: "Surface width of the Solid-On-Solid models"Nuclear Physics B [Proc. Suppl.]. Vol.63. 649-651 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Arisue, K.Tabata: "Large-q expansion of the specific heat for the two-dimensional q-state Potts model"NuPhysical Review E. Vol.59. 186-188 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Arisue, K.Tabata: "Large-q expansion of the energy and magnetization cumulants for the two-dimensional q-state Potts model"Nuclear Physics B [Field Theory and Statistical Physics]. Vol.546. 558-584 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] H.Arisue: "Surface width of the Solid-On-Solid models" Nuclear Physics B(Proc.Suppl.). 63A-C. 649-651 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Arisue: "Large-q expansion of the specific heat for the two-dimensional g-state Potts model" Physical Review E. 59. 186-188 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Arisue: "Large-q expansion of the energy and magnetization cumulants for the two-dimensional g-state Potts model" Nuclear Physics B. (掲載予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] Hiroaki Arisue: "Low-temperature Series for the Square Lattice Potts model by the Inproved Finite-Lattice Method" Journal of Physics A:Mathematical and General. 30. 3313-3327 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Hiroaki Arisue: "Surface width of the Solid-On-Sodid models" Nuclear Physics B(Proc.Suppl.). 63A-C. 649-651 (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] Hiroaki Arisue: "Low-Temperature Series for the Square Lattice Potts Model by the Improved Finite-Lattice Method" Journal of Physics A : Mathematical and General. (掲載予定).

    • Related Report
      1996 Annual Research Report

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Published: 1996-04-01   Modified: 2016-04-21  

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