Improved coarse-graining methods for two dimensional tensor networks including fermions

Abstract

We show how to apply renormalization group algorithms incorporating entanglement filtering methods and a loop optimization to a tensor network which includes Grassmann variables which represent fermions in an underlying lattice field theory. As a numerical test a variety of quantities are calculated for two dimensional Wilson-Majorana fermions and for the two flavor Gross-Neveu model. The improved algorithms show much better accuracy for quantities such as the free energy and the determination of Fisher’s zeros.

Tensor renormalization group for fermions

“…The TNR approaches mentioned at the beginning of this section attempt to remove the CDLs from the network. Some of the authors of this review article has checked the efficiency of the loop-TNR and the gilt-TNR for the Majorana-Wilson fermion system seen in section 3.1 and for the two-flavor Gross-Neveu model [120]. While the loop-TNR comes first in the chronological order, we show here how the gilt-TNR removes the CDL from a network for graphical simplicity.…”

“…and ´'s are over the auxiliary Grassmann variables, which is implicit in equation ( 8). This kind of Grassmann tensor network formulation has been widely applied: the Schwinger model [117][118][119] 5 , the Gross-Neveu model [102,120], free Wilson fermions [121,122], the N = 1 Wess-Zumino model [106], the NJL model [105], Wilson-Majorana fermions [120], infinitecoupling QCD [104], and SU(2) lattice gauge theory with reduced staggered fermions [107].…”

“…In [120], the performance of the Grassmann version of loop-TNR and gilt-TNR was inspected by computing the free energy and the determination of Fisher's zeros. In the following, we review the RG flow of the singular values that are believed to store the information of the system.…”

“…Note that these behaviors of the singular values are qualitatively the same as those observed for the two-dimensional Ising model; the equivalence between the Ising model and the Majorana-Wilson fermion system can be seen in a sense like this. For the growth of the entanglement entropy and relationship to the Calabrese-Cardy formula, we refer the reader to the discussion in [120].…”

“…Applications for relativistic models are presented in section 5 which covers the Gross-Neveu model [102,103], QCD at infinite coupling [104], the Nambu-Jona-Lasinio (NJL) model [105], the N = 1 Wess-Zumino model [106] and non-abelian gauge theories with fermion [107]. Finally, the Fermi-Hubbard model which has been extensively studied in condensed matter community [108][109][110][111][112][113][114] is considered in section 6 following the approach of [115] in 1+1 dimensions and [116] in 2+1 dimensions.…”