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Study of topological condensed matter problems by random matrices

Research Project

Project/Area Number 21540380
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Mathematical physics/Fundamental condensed matter physics
Research InstitutionThe University of Tokyo

Principal Investigator

HIKAMI Shinobu  東京大学, 大学院・総合文化研究科, 教授 (30093298)

Project Period (FY) 2009 – 2011
Project Status Completed (Fiscal Year 2011)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2010: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2009: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Keywords数理物理 / メゾスコピック系 / リーマン面 / トポロジー / ランダム行列 / フトリックス模型 / 超対称理論 / 行列模系 / 統計力学 / 行列模型 / ランダムウオーク
Research Abstract

The topological invariance of the moduli space of Riemann surface is shown to be obtained from the Gaussian random matrix model with an external source by the tuning of external source. The invariance is classified by a parameter p, which characterizes the degeneracy of the external source. The invariance is related to the p-spin curves on the Riemann surface. This topological singularity is related the bifurcation of the growth of crystals. The parameter p is extended to the negative values(p=-1,-2,...) and it represents level k SL(2, R)/U(1) Wess-Zumino-Witten term. The case of p=-2 corresponds to the unitary matrix model, which has a phase transition. The strong and weak expansion are examined by Gaussian random matrix theory. This transition is studied in the relation of the phase transitions in the condensed matter problems. The time dependence of the random matrix model is studied by the reduction to 2-matrix model and the algebraic structure is investigated in details. The algebraic structure is found to be the associated W-algebra. The structure represents the deviation from N=2 supersymmetric minimal model and the change of the Ramond-Ramond term for the time dependence is studied.

Report

(4 results)
  • 2011 Annual Research Report   Final Research Report ( PDF )
  • 2010 Annual Research Report
  • 2009 Annual Research Report
  • Research Products

    (15 results)

All 2012 2011 2010 2009 Other

All Journal Article (6 results) (of which Peer Reviewed: 3 results) Presentation (4 results) Book (3 results) Remarks (2 results)

  • [Journal Article] On an Airy matrix model with a logarithmic potential2012

    • Author(s)
      E. Brezin and S. Hikami
    • Journal Title

      Journal of Physics A : Mathematical and Theoretical

      Volume: 45 Pages: 45203-45203

    • Related Report
      2011 Final Research Report
  • [Journal Article] On an Airy matrix model with a logarithmic potential2012

    • Author(s)
      E.Brezin, S.Hikami
    • Journal Title

      Journal of Physics A : Mathematical and Theoretical

      Volume: 45 Pages: 45203-45203

    • Related Report
      2011 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Duality and Replica for a Unitary Matrix Model2010

    • Author(s)
      E. Brezin and S. Hikami
    • Journal Title

      JHEP

      Volume: 07 Pages: 67-67

    • Related Report
      2011 Final Research Report
  • [Journal Article] Duality and replicas for a unitary matrix model2010

    • Author(s)
      E.Brezin, S.Hikami
    • Journal Title

      Journal of High Energy Physics

      Volume: 7 Pages: 67-67

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Duality and replicas for a unitary matrix model2010

    • Author(s)
      E.Brezin, S.Hikami
    • Journal Title

      Journal of high energy physics

      Volume: 7 Pages: 67-67

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Computing topological invariants with one and two-matrix models2009

    • Author(s)
      E. Brezin and S. Hikami
    • Journal Title

      JHEP

      Volume: 04 Pages: 110-110

    • Related Report
      2011 Final Research Report
  • [Presentation] On an Airy matrix model with a logarithmic potential2012

    • Author(s)
      S.Hikami
    • Organizer
      51th ENS Summer Conference
    • Place of Presentation
      Ecole Normale Superieure (Paris) FR(招待講演)
    • Year and Date
      2012-08-22
    • Related Report
      2011 Annual Research Report
  • [Presentation] On an Airy matrix model with a logarithmic potential2011

    • Author(s)
      S. Hikami
    • Place of Presentation
      Ecole Normale Superieure, Paris
    • Year and Date
      2011-08-22
    • Related Report
      2011 Final Research Report
  • [Presentation] Duality and replicas for a unitary matrix model2010

    • Author(s)
      S. Hikami
    • Organizer
      Conference Topological strings, modularity and non-perturbative physics
    • Place of Presentation
      The Erwin Schrodinger International Institute for Mathematical Physics, Wien
    • Year and Date
      2010-07-28
    • Related Report
      2011 Final Research Report
  • [Presentation] Duality and replicas for a unitary matrix model2010

    • Author(s)
      S.Hikami
    • Organizer
      Topological String Theory, Modularity and Nonperturbative Physic
    • Place of Presentation
      Schrodinger Ins.Wien
    • Year and Date
      2010-07-28
    • Related Report
      2009 Annual Research Report
  • [Book] The Handbook of Random Matrix Theory (Chap.19, Characteristic Polynomials)2012

    • Author(s)
      E.Brezin, S.Hikami
    • Publisher
      Oxford University Press
    • Related Report
      2011 Annual Research Report
  • [Book] The Oxford Handbook of Random Matrix Theory. Edited by G. Akemann et al, Chap 19, Characteristic polynomials2011

    • Author(s)
      E. Brezin and S. Hikami
    • Publisher
      Oxford University Press
    • Related Report
      2011 Final Research Report
  • [Book] The Oxford Hand book of random matrix theory, chapter 192011

    • Author(s)
      E.Brezin, S.Hikami
    • Total Pages
      17
    • Publisher
      Oxford University Publisher
    • Related Report
      2010 Annual Research Report
  • [Remarks]

    • Related Report
      2011 Final Research Report
  • [Remarks]

    • URL

      http://www.oist.jp/mathematical-and-theoretical-physics-unit

    • Related Report
      2011 Final Research Report

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Published: 2009-04-01   Modified: 2016-04-21  

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