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arXiv:1211.3675 · Physical Review B

Accurate exponents from approximate tensor renormalizations

Yannick Meurice

Published Feb 27, 2013• 33 citations
arXivHTMLPDF

Abstract

We explain the recent numerical successes obtained by Tao Xiang's group, who developed and applied tensor renormalization group methods for the Ising model on square and cubic lattices, by the fact that their new truncation method sharply singles out a surprisingly small subspace of dimension two. We show that in the two-state approximation, their transformation can be handled analytically, yielding a value of 0.964 for the critical exponent $\ensuremath{\nu}$ much closer to the exact value 1 than the 1.338 value obtained in the Migdal-Kadanoff approximation. We propose two alternative blocking procedures that preserve the isotropy and improve the accuracy to $\ensuremath{\nu}=0.987$ and 0.993, respectively. We discuss applications to other classical lattice models, including models with fermions, and suggest that it could become a competitor for Monte Carlo methods suitable for accurate calculations of critical exponents, taking continuum limits, and the study of near-conformal systems in arbitrarily large volumes.

DOI
10.1103/physrevb.87.064422

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We explain the recent numerical successes obtained by Tao Xiang's group, who developed and applied tensor renormalization group methods for the Ising model on square and cubic lattices, by the fact that their new truncation method sharply singles out a surprisingly small subspace of dimension two. We show that in the two-state approximation, their transformation can be handled analytically, yielding a value of 0.964 for the critical exponent \ensuremathν\ensuremath{\nu}\ensuremathν much closer to the exact value 1 than the 1.338 value obtained in the Migdal-Kadanoff approximation. We propose two alternative blocking procedures that preserve the isotropy and improve the accuracy to \ensuremathν=0.987\ensuremath{\nu}=0.987\ensuremathν=0.987 and 0.993, respectively. We discuss applications to other classical lattice models, including models with fermions, and suggest that it could become a competitor for Monte Carlo methods suitable for accurate calculations of critical exponents, taking continuum limits, and the study of near-conformal systems in arbitrarily large volumes.
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