2007 Fiscal Year Final Research Report Summary
Research on Random Matrix Theory:Mathematical aspects and its application to quantum physics
| Project/Area Number |
16540342
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Mathematical physics/Fundamental condensed matter physics
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| Research Institution | Shimane University |
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Principal Investigator |
MOCHIZUKI Shinsuke (西垣 真祐) Shimane University, Department of Material Science, Associate Professor (00362913)
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| Co-Investigator(Kenkyū-buntansha) |
NAGAO Taro Nagoya University, Graduate School of Mathematics, Associate Professor (10263196)
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| Project Period (FY) |
2004 – 2007
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| Keywords | Random Matrix Theory / Energy Level Statistics / Gauge Field Theory / Quantum Chaos / Quantum Field Theory / Anderson Localization / Lattice Gaufre Theory |
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| Research Abstract |
Mathematical aspects of disordered systems
Aiming to construct an effective theory describing the energy level statistics of many-body quantum systems with arbitrary strength of both disorder and interactions, we have applied the following techniques to the problem.
1. Replica method for interacting disordered system. By developing a novel framework for the replica trick based on integrable systems including Pfaffian lattice hierarchy, we have partly reproduced numerically- generated transitional level statistics of ID disordered systems of global symmetry classes AIII and CI.
2. AdS/CFT-motivated method. As a complementary to 1., we are applying the gauge-gravity correspondence, the single major concept in string theory, to the derivation of spectral nonlinear sigma model from disordered systems. By choosing ad-hoc higher dimensional curved spacetimes whose boundary the disordered system resides on, we have tried to derive a scale-invariant effective theory describing the latter atop the
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second-order phase transition.
Application of RMT to quantum physics
We have numerically analyzed the chiral and confinement transition of QCD through the spectral statistics of the Dirac operator, namely for the quenched SU(2) gauge theory on nonisotropic lattices. Monte-Carlo-simulating at varied temperatures and boundary conditions, we have precisely measured the level spacing distribution and the two-level correlation function of Kogut-Susskind Dirac operator eigenvalues in the spectral bulk, and the fractality of the associated eigenfunctions, at and around the QCD phase transition. The measured level spacing distribution is fitted against the analytical result of the orthgonal deformed RMT that has been shown by the author to describe the critical level statisitics at the localization transition, and exhibited an excellent match with the latter. That the conductance parameter determined turns out to be very close to that of the 3D Anderson model provides a strong support for the Diakonov-Petkov conjecture (restoration of chiral symmetry by the localization of the quark wave functions). Less
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