2003 Fiscal Year Final Research Report Summary
Study of Low Dimensional Quantum Systems with Infinite Degree of Freedom by Non-Commutative Functional Analysis
| Project/Area Number |
12440049
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Kyushu University |
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Principal Investigator |
MATSUI Taku Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (50199733)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAYASHIKI Atsushi Kyushu University, Faculty of Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (10237456)
WATATANI Yasuo Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (00175077)
KOSAKI Hideki Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (20186612)
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| Project Period (FY) |
2000 – 2003
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| Keywords | functional analysis / infinite dimensional / non-commutative / quantum spin systems / low dimensional |
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| Research Abstract |
(i)We proved the non-commutative central limit theorem of Goderis-Verbeure-Vets for states with uniform exponential decay of correlation. Our results can be applied to the KMS state of one-dimensional quantum spin chains for finite range translationally invariant interactions. Our limit theorem is valid for a class of quasilocal(non local) observables.
(ii)We obtained a fluctuation theorem for quantum systems with continuous spectrum. For a Gibbs state we impose an external field for a finite time and we have shown that the energy transition probability satisfies an inequality similar to Gallavotti-Cohen's fluctuation theorem. We use the modular theory of von Neumann algebras in crucial steps.
(iii)We consider characterization of certain non-equilibrium steady states of infinite quantum systems. If the time evolution satisfies the L1 asymptotic abelian condition, the non-equilibrium steady states are the KMS states for the Zubarev Hamiltonian. However, for the XY model in which the L1 asymptotic abelian condition is not valid, the non-equilibrium steady states fail to be KMS states for any continuous time evolution. Nevertheless, the states are characterized by the variational principle of free energy for a formal Zubarev Hamiltonian.
(iv)We obtained the complete set of ground states for the one-dimensional XXZ model with lsing-like anisotropy.
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