Energy-momentum tensor in the 2D Ising CFT in full modular space

TL;DR

This paper derives lattice operators for the energy-momentum tensor in the 2D Ising conformal field theory, validating their correctness through numerical Monte Carlo simulations and analyzing lattice transformations.

Contribution

It introduces a new set of lattice operators for the EM tensor in the Ising CFT applicable to various lattice geometries, with numerical validation and analysis of lattice transformation effects.

Findings

Operators satisfy conformal Ward identities

Validation through Monte Carlo confirms correctness

Analysis of lattice transformation effects on EM tensor

Abstract

A set of lattice operators for the energy-momentum (EM) tensor in the Ising CFT is derived in the spin variables. Our expression works under arbitrary affine transformation both on triangular and hexagonal lattices (where the former includes the rectangular lattices). The correctness of the operators is numerically confirmed in Monte Carlo calculations by comparing the results with the conformal Ward identity, including the operator normalization. In the derivation of the EM tensor, a staggered structure of the affine-transformed hexagonal lattice is analyzed, which shows a peculiar shift from the circumcenter dual lattice and appears as a mixing angle between the holomorphic part $T(z)$ and the antiholomorphic part $\tilde T(\bar z)$. The details of this contribution will appear in a subsequent paper.