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arXiv:2208.03227 · Journal of High Energy Physics

Bond-weighting method for the Grassmann tensor renormalization group

Shinichiro Akiyama

Published Nov 7, 2022• 15 citations
arXivHTMLPDF

Abstract

Abstract Recently, the tensor network description with bond weights on its edges has been proposed as a novel improvement for the tensor renormalization group algorithm. The bond weight is controlled by a single hyperparameter, whose optimal value is estimated in the original work via the numerical computation of the two-dimensional critical Ising model. We develop this bond-weighted tensor renormalization group algorithm to make it applicable to the fermionic system, benchmarking with the two-dimensional massless Wilson fermion. We show that the accuracy with the fixed bond dimension is improved also in the fermionic system and provide numerical evidence that the optimal choice of the hyperparameter is not affected by whether the system is bosonic or fermionic. In addition, by monitoring the singular value spectrum, we find that the scale-invariant structure of the renormalized Grassmann tensor is successfully kept by the bond-weighting technique.

DOI
10.1007/jhep11(2022)030

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We propose an improved tensor renormalization-group (TRG) algorithm, the bond-weighted TRG (BTRG). In BTRG, we generalize the conventional TRG by introducing bond weights on the edges of the tensor network. We show that BTRG outperforms the conventional TRG and the higher-order tensor renormalization group with the same bond dimension, whereas its computation time is almost the same as that of TRG. Furthermore, BTRG can have nontrivial fixed-point tensors at an optimal hyperparameter. We demonstrate that the singular value spectrum obtained by BTRG is invariant under the renormalization procedure in the case of the two-dimensional Ising model at the critical point. This property indicates that BTRG performs the tensor contraction with high accuracy whereas keeping the scale-invariant structure of tensors.
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We review the basic ideas of the tensor renormalization group method and show how they can be applied for lattice field theory models involving relativistic fermions and Grassmann variables in arbitrary dimensions. We discuss recent progress for entanglement filtering, loop optimization, bond-weighting techniques and matrix product decompositions for Grassmann tensor networks. The new methods are tested with two-dimensional Wilson-Majorana fermions and multi-flavor Gross-Neveu models. We show that the methods can also be applied to the fermionic Hubbard model in 1+1 and 2+1 dimensions.
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