Accurate exponents from approximate tensor renormalizations

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 0.964 for the critical exponent nu much closer to the exact value 1 than 1.338 obtained in the Migdal-Kadanoff approximation. We propose two alternative blocking procedures that preserve the isotropy and improve the accuracy to 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 to calculate accurately critical exponents, take continuum limits and study near-conformal systems in arbitrarily large volumes.

5 pages, 3 figures, intro and conclusions expanded + two new paragraphs on work in progress

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• Phase Transitions of Ferromagnetic Potts Models on the Simple Cubic Lattice
• Understanding the entanglement entropy and spectra of 2D quantum systems through arrays of coupled 1D chains
• The fine structure of the entanglement entropy in the classical XY model
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• Examples of symmetry-preserving truncations in tensor field theory
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• RG inspired Machine Learning for lattice field theory
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• Comparing Tensor Renormalization Group and Monte Carlo calculations for spin and gauge models
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