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Nuclear Physics B

Self-dual renormalization group analysis of the Potts models

D. Horn, Marek Karliner A, Shimon Yankielowicz B

Published Dec 1, 1980• 13 citations
Publisher
DOI
10.1016/0550-3213(80)90422-8

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The one-dimensional Hamiltonian version of the two-dimensional Potts model is studied by the application of a self-dual renormalization-group transformation. Blocks containing up to 11 sites are considered and mapped onto new, equivalent Potts spins by keeping the lowest-lying levels only. The higher-lying states are also taken into account in a perturbational way for smaller cells. Unfortunately, this renormalization-group treatment cannot reproduce the first-order transition for large values of the number of states, and the critical exponents for the three- and four-state Potts model do not seem to converge to the conjectured results as larger and larger blocks are taken.
The zero-temperature renormalization-group transformations for quantum spin systems are analyzed. The block transformation and the decimation-type transformation used in the study of the one-dimensional Ising model in transverse field are extended to the quantum version of the Potts model and Ashkin-Teller model. It is shown that for these self-dual models the two kinds of renormalization-group transformations are dual to each other and therefore give the same result for the critical behavior. The duality persists even if the higher-lying states are taken into account in a perturbational way.