2007 Fiscal Year Final Research Report Summary
Studies on quantum impurities and quantum Brownian motion by the exact WKB method
| Project/Area Number |
17540354
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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 | Shizuoka University |
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Principal Investigator |
SUZUKI Junji Shizuoka University, Department of Physics, Associate Professor (40222062)
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| Project Period (FY) |
2005 – 2007
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| Keywords | exact WKB method / quantum impurties / ODE / IM correspodence / Bethe ansatz |
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| Research Abstract |
Studies on Quantum Impurities:
The quantum impurities on the one dimensional lattice are often simplified by incorporating the effect by relevant boundary conditions or effective fields. The spin 1/2 anisotropic magnetic chain is studied in this prospect.
In the presence of magnetic field, I found that the corresponding transfer matrix is completely factorized in terms of Q operators. This enables simpler evaluation of the finite size property of the model by the exact WKB method than the existing method.
The study on the effective impurity in terms of the boundary condition is performed for a discrete version of the super-symmetric sine-Gordon model. As a result of the application of the idea of functional equations and analyticity argument, we derive a set of nonlinear integral equations.
It enables the quantitative study of any excited states in the model at arbitrary length scale.
We are thus able to check the consistency of the model in the ultra-violet and infra-red regions.
These resu
… More
lts deserves the basis for the future studies.
The ODE/IM correspondence
The ODE/IM correspondence is realized as a curious interplay between the spectral problem of 2nd order ordinary differential equations (Schroedinger equation) and the zeros of Q operator in the conformal field theory.
This yields us to explore several fundamental problems in ID quantum mechanics though well developed technology in the many body physics. As example, we refer to the study of the phase boundary of the PT symmetry breaking.
The generalization to higher ODE, however, has defied simple attempts.
We instead consider integro-differential equations.
By utilizing the knowledge of algebraic structure associated to the quantum sine-Gordan model at special coupling constant value, we successfully determine the explicit forms of integro-differential equations.
The numerical studies confirm that the spectral problem of the integro-differential equations is solved by the Bethe ansatz equations which encode the zeros of (eigenvalues of) Q operators. Less
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