# Anomalous tensorial properties of anisotropic 2D materials

## TL;DR

This paper classifies anomalous tensorial properties in anisotropic 2D materials, revealing how symmetry constraints influence odd transport phenomena and broadening understanding of emergent material behaviors.

## Contribution

It develops a representation framework for second and fourth order tensors in 2D materials with specific symmetries, identifying conditions for anomalous responses.

## Key findings

- Classifies anomalous tensor properties based on symmetry constraints.
- Provides representation theorems for tensors in 2D materials.
- Discusses implications for materials like twisted bilayer graphene.

## Abstract

Odd transport phenomena -- defined as a flux response orthogonal to an applied gradient -- have been recently observed in isotropic systems, with a multitude of proposed models and experiments to study these effects. Odd transport manifests in tensors that describe linear relations between fluxes and gradients that drive them, particularly when parity and time-reversal symmetries are broken. In this work, we identify such odd properties to be a subset of a broader class of major-symmetry-breaking behaviors, which we term ``anomalous." We develop a classification of anomalous properties described by $2^\mathrm{nd}$ and $4^\mathrm{th}$ order tensors in anisotropic 2D materials that maintain discrete rotational and reflection symmetries, characterized by the 17 wallpaper groups. To this end, we present representation theorems for these tensors, identifying which components are constrained for specific spatial symmetries and thereby allowing materials to be grouped into classes that exhibit anomalous responses or not. We focus our discussion on $2^\mathrm{nd}$ order tensors in the context of electrical resistivity and on $4^\mathrm{th}$ order tensors in the context of viscosity and elasticity. These findings are broadly applicable to the study of novel emergent material properties. To illustrate this, we discuss implications of our findings for two very different 2D materials that have recently garnered attention in condensed matter physics: knitted fabrics and twisted bilayer graphene.