2002 Fiscal Year Final Research Report Summary
Foundation of the Numerical Renormalization Group and Its Higher-Dimensional Generalization
| Project/Area Number |
12640393
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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 |
物理学一般
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| Research Institution | Osaka University |
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Principal Investigator |
AKUTSU Yasuhiro Osaka University, Graduate School of Science, Professor (10191850)
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| Project Period (FY) |
2000 – 2002
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| Keywords | numerical renormalization / DMRG / density matrix / polymer / eigenvalue distribution / phase transition / higher dimension |
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| Research Abstract |
2000 : Higher-dimensional generalization of DMRG and its application to simple systems
Generalizing the matrix-product structure of the fixed-point wave function in the DMRG (1D quantum, 2D classical), we introduced higher-rank tensors to
express the "target state" (largest-eigenvalue state of the transfer matrix) for 3D classical statistical systems. The sum-of-the-products-of-tensors structure of
the wave function allows us to calculate the density matrix and other physical quantities by using the lower-dimensional algorithm ; our higher-dimensional
algorithm is the one with "nested" structure.
2001 : Application of the higher-dimensional DMRG to specific models and foundation of the density-matrix algorithm (esp. eigenvalue distribution)
Taking a class of polymer models, expressed as 3D vertex models, we applied the 3D DMRG and studied their thermal behavior
including the phase transitions. For another class of vertex models, we calculated the zero-point entropy and compared the result
to t
… More
he one derived via the Pauling approximation, seeing a good agreement. As for the universal asymptotics of the density-matrix
eigenvalues, we analyzed the exactly solvable models and extracted two exponents (main and sub) to characterize the asymptotics.
Physically, the main exponent is found to be related to the entropy of a semi-infinite 1D spin system where "weight" of sites grows
linearly with the distance from the origin ; thus the value of the exponent is expected to be fairly robust. As for the sub exponent, our
studies made on the solvable models shows a remarkable coincidence of the value among them, though a simple physical interpretation
has not been attained yet.
2002 : Relation between the accuracy of the physical quantities and the universal asymptotics of the density matrix.
Expecting that the universal asymptotics leads to the "universal truncation error" with respect to the retained bases, we performed
both analytical and numerical calculations, but with no decisive answer, hitherto Less
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Research Products
(12 results)
