More about the Grassmann tensor renormalization group

Shinichiro Akiyama,Daisuke Kadoh

doi:10.1007/jhep10(2021)188

Abstract

We derive a general formula of the tensor network representation for d-dimensional lattice fermions with ultra-local interactions, including Wilson fermions, staggered fermions, and domain-wall fermions. The Grassmann tensor is concretely defined with auxiliary Grassmann variables that play a role in bond degrees of freedom. Compared to previous works, our formula does not refer to the details of lattice fermions and is derived by using the singular value decomposition for the given Dirac matrix without any ad-hoc treatment for each fermion. We numerically test our formula for free Wilson and staggered fermions and find that it properly works for them. We also find that Wilson fermions show better performance than staggered fermions in the tensor renormalization group approach, unlike the Monte Carlo method.

Highlights

JHEP10(2021)188 tensors requires separate handling of tensors for the initial and the renormalized steps
We find that Wilson fermions show better performance than staggered fermions in the tensor renormalization group approach, unlike the Monte Carlo method
We derived a general formula of the tensor network representation for lattice fermions

Summary

Formalism of the Grassmann tensors

The variables ηi (i = 1, · · · , N ) are single-component Grassmann numbers which satisfy the anti-commutation relation {ηi, ηj} = 0. We begin with defining a Grassmann tensor and its contraction rule with single-component index ηi.. We define a Grassmann contraction from η1 to ζ1 as dξdξ e−ξξAξη2...ηN Bξζ2...ζM. To define the Grassmann contraction with multi-dimensional indices, we consider the case of N = mK, M = mL in eq (2.2) for simplicity. The Grassmann contraction is given for the multi-component case: dΞdΞ AΞΨ2...ΨK BΞΦ2...ΦL (2.4). The tensor network is defined by a product of T1T2 · · · where all indices are contracted as eq (2.4)

General formula of tensor network for arbitrary lattice fermions

Numerical applications

Summary and outlook

A Truncation technique

B Tensor networks for 2D lattice fermions

Staggered fermions