[Paper Review] Hydrodynamics with triangular point group

Aaron J. Friedman, Caleb Q. Cook+1 more|arXiv (Cornell University)|Feb 16, 2022

Geomagnetism and Paleomagnetism Studies4 citations

TL;DR

This paper develops hydrodynamics for two-dimensional electron fluids with D₆ point group symmetry—breaking continuous rotational invariance to that of an equilateral triangle—identifying new symmetry-allowed dissipative transport coefficients that break spatial inversion and time-reversal symmetry while preserving their combination. It proposes two experimentally realizable setups, including nitrogen-vacancy center magnetometry in hexagonal devices, to detect these novel coefficients via local current imaging and Hall effect measurements in narrow channels.

ABSTRACT

When continuous rotational invariance of a two-dimensional fluid is broken to the discrete, dihedral subgroup $D_6$ - the point group of an equilateral triangle - the resulting anisotropic hydrodynamics breaks both spatial-inversion and time-reversal symmetries, while preserving their combination. In this work, we present the hydrodynamics of such $D_6$ fluids, identifying new symmetry-allowed dissipative terms in the hydrodynamic equations of motion. We propose two experiments - both involving high-purity solid-state materials with $D_6$-invariant Fermi surfaces - that are sensitive to these new coefficients in a $D_6$ fluid of electrons. In particular, we propose a local current imaging experiment (which is present-day realizable with nitrogen vacancy center magnetometry) in a hexagonal device, whose $D_6$-exploiting boundary conditions enable the unambiguous detection of these novel transport coefficients.

Motivation & Objective

• To develop a hydrodynamic theory for two-dimensional electron fluids with discrete D₆ rotational symmetry, which breaks continuous rotational invariance and spatial inversion symmetry.
• To identify new dissipative transport coefficients in the constitutive relations for current and stress tensor that arise due to the D₆ symmetry and broken inversion symmetry.
• To propose experimentally accessible platforms—specifically, hexagonal devices and narrow-channel Hall effect measurements—capable of detecting these previously unaccounted-for transport coefficients.
• To analyze the hydrodynamic response in the presence of broken reflection invariance using a Z₃ fluid model, extending the framework to anisotropic systems.
• To connect the emergent hydrodynamic coefficients to microscopic electronic structure, particularly in materials like ABA trilayer graphene with triangular Fermi surfaces.

Proposed method

• Derives the hydrodynamic equations of motion for D₆-symmetric fluids using representation theory of the dihedral group D₆, including continuity equations and constitutive relations for current and stress tensor.
• Applies Onsager reciprocal relations to constrain the transport coefficients under time-reversal and inversion symmetry, identifying new terms allowed by D₆ symmetry but not by O(2) isotropy.
• Introduces a stream function formalism to solve the steady-state hydrodynamic equations in polar and rotated coordinates, enabling analytical treatment of flow in symmetric geometries.
• Performs kinetic theory calculations using a Boltzmann-like formalism to estimate the magnitude of the new coefficient α on the Fermi surface of ABA trilayer graphene.
• Proposes a hexagonal device geometry with D₆-symmetric boundary conditions to enable local current imaging via nitrogen-vacancy center magnetometry for detecting the novel transport coefficients.
• Analyzes channel flow in narrow, D₆-symmetric geometries to probe the Hall voltage response, accounting for non-uniqueness due to symmetry-allowed anisotropic terms.

Experimental results

Research questions

• RQ1 What new dissipative transport coefficients emerge in hydrodynamic equations when continuous rotational invariance is broken to the D₆ point group?
• RQ2 How do the Onsager relations constrain the form of these new coefficients in the presence of broken spatial inversion and time-reversal symmetries?
• RQ3 Can the novel transport coefficients be probed experimentally in high-purity solid-state materials with triangular Fermi surfaces?
• RQ4 What is the role of the Z₃ fluid model in describing anisotropic hydrodynamics with broken reflection invariance?
• RQ5 How do the hydrodynamic responses in hexagonal and narrow-channel geometries differ from isotropic systems due to D₆ symmetry?

Key findings

• The hydrodynamic theory identifies new dissipative terms in the current and stress tensor constitutive relations that are symmetry-allowed by D₆ but forbidden under O(2) continuous rotation invariance.
• The coefficient α, which couples velocity gradients to charge density gradients with a 3φ dependence, is estimated to be on the order of 10⁻⁴–10⁻³ eV·nm² in ABA trilayer graphene based on Fermi surface geometry.
• In a hexagonal device with D₆-symmetric boundary conditions, the local current distribution becomes sensitive to the novel transport coefficients, enabling unambiguous detection via nitrogen-vacancy center magnetometry.
• In narrow-channel Hall effect measurements, the Hall voltage signal exhibits non-uniqueness due to symmetry-allowed anisotropic terms, providing a distinct signature of the new coefficients.
• The steady-state hydrodynamic equations in rotated coordinates reveal that the new terms break reflection invariance while preserving the combined CPT symmetry, leading to non-trivial flow patterns.
• The stream function formalism successfully solves the biharmonic equation in polar coordinates, yielding a general solution that captures the anisotropic flow response under D₆ symmetry.