Hydrodynamics with triangular point group
At a Glance
Section titled “At a Glance”| Metadata | Details |
|---|---|
| Publication Date | 2023-05-31 |
| Journal | SciPost Physics |
| Authors | Aaron J. Friedman, Caleb Q. Cook, Andrew Lucas |
| Institutions | University of Colorado Boulder, Stanford University |
| Citations | 10 |
| Analysis | Full AI Review Included |
Executive Summary
Section titled “Executive Summary”This research details the anisotropic hydrodynamics of two-dimensional (2D) electron fluids possessing discrete triangular (D6) point group symmetry, focusing on novel dissipative effects arising from broken spatial inversion (I) and time-reversal (Θ) symmetries.
- Novel Transport Coefficient: A new, symmetry-allowed dissipative coefficient (α or β, related by the coupling parameter ξ) is identified. This coefficient is only present when both I and Θ symmetries are individually broken, while their combination (IΘ) is preserved.
- Physical Manifestation: The new coefficient ξ induces a “piezoelectric-like” effect in the electron fluid, where a background electric field (E) generates shear stresses (τxy or τxx = -τyy) proportional to E. Crucially, this effect is dissipative (entropy-producing).
- Experimental Proposal: A hexagonal device geometry with symmetry-engineered boundary conditions is proposed for detection. This setup forces the current to vanish at the device center unless the fluid possesses D6 symmetry.
- Detection Method: Nitrogen-Vacancy (NV) center magnetometry is proposed to measure the resulting localized current anomaly at the device center (r → 0).
- Signal Strength: The hexagonal device yields a current signal proportional to ξ (O(ξ)), offering an unambiguous and strong detection mechanism. Conventional Hall effect measurements in narrow channels are shown to produce a much weaker, higher-order signal (O(ξ2)).
- Microscopic Feasibility: Kinetic theory modeling, using the Fermi surface of ABA trilayer graphene, confirms that the magnitude of the new coefficient (α) is comparable to standard viscosities and incoherent conductivities, suggesting the effect is experimentally discernible.
Technical Specifications
Section titled “Technical Specifications”The following parameters and scaling relations were derived from the hydrodynamic and kinetic theory models, particularly focusing on ABA trilayer graphene as a candidate material.
| Parameter | Value | Unit | Context |
|---|---|---|---|
| Point Group Symmetry | D6 | None | Dihedral group of order six (equilateral triangle) |
| Conserved Symmetries | IΘ (Inversion-Time Reversal) | None | I and Θ are individually broken |
| New Coupling Coefficient | ξ = α / ρ0 = β / c2 | Length | Relates the new dissipative terms (α, β) |
| Stability Constraint | D η ≥ ρ0 c2 ξ2 | None | Required for positive semidefinite dissipation matrix (σ) |
| Diffusive Length Scale (l0) | (D(η + ζ) / ρ0c2)1/2 | Length | Characteristic length scale for hydrodynamic modes |
| Hexagonal Current Signal | Proportional to ξ | Current | Unambiguous detection of D6 symmetry (O(ξ)) |
| Hall Voltage Signal | Proportional to ξ2 | Voltage | Minimal signal in narrow channel flow (O(ξ2)) |
| Model Fermi Energy (EF) | 0.008 | eV | ABA Trilayer Graphene (mid-range estimate) |
| Dimensionless D6 Coupling (α) | 0.1394 | None | Estimated value for EF = 0.008 eV |
| Dimensionless Shear Viscosity (η) | 0.35(2) | None | Estimated value for EF = 0.008 eV |
| Onsager Ratio (α2 / η σinc) | 0.13(6) | None | Ratio must be less than or equal to 1 (stability check) |
Key Methodologies
Section titled “Key Methodologies”The study employed a rigorous theoretical framework combining symmetry analysis, hydrodynamic expansion, and kinetic theory modeling to predict the behavior of D6-symmetric electron fluids.
- Symmetry Constraint Derivation:
- The continuous rotational group O(2) was restricted to the discrete D6 subgroup.
- Representation theory was used to identify the irreducible representations (irreps) and derive the unique rank-three invariant tensor (λijk) that is invariant under D6 but not O(2).
- Hydrodynamic Expansion:
- Linearized continuity equations for charge (ρ) and momentum (π) were constructed.
- Constitutive relations for current (j) and stress tensor (τ) were expanded to first order in derivatives, incorporating the new D6-invariant terms (coefficients α and β).
- Onsager reciprocal relations, enforced by the preserved IΘ symmetry, were used to relate the new dissipative coefficients: α = βχ (where χ is charge susceptibility).
- Steady-State Flow Analysis (Stream Function):
- The equations of motion were solved for steady-state flow using a stream function (ψ), resulting in a modified biharmonic equation perturbed by the D6 coupling ξ (Equation 3.52).
- The solutions showed that the D6 coupling causes angular harmonics to mix (m → m ± 3), a key signature used in the experimental proposal.
- Kinetic Theory Modeling and Estimation:
- The Boltzmann equation was applied to a microscopic model of a Fermi liquid with a triangular Fermi surface (specifically, the K-point Fermi surface of ABA trilayer graphene).
- The relaxation time approximation (τee) was used to calculate the dimensionless transport coefficients (η, ζ, ξ, α, σinc) for various Fermi energies (EF = 0.004 to 0.024 eV).
- Experimental Design (Hexagonal Device):
- Boundary conditions corresponding to the R2 irrep of D12 (equivalent to R1 of D6) were imposed on a circular/hexagonal device to ensure that a nonzero current at the center only appears if D6 symmetry is present (ξ ≠ 0).
Commercial Applications
Section titled “Commercial Applications”The findings and proposed experimental techniques are relevant to several high-tech fields focused on advanced materials and quantum transport phenomena.
- Anisotropic 2D Electronics: Designing and optimizing electronic devices (e.g., fluidic transistors, current rectifiers) that exploit the directional, viscous flow characteristics of D6-symmetric materials.
- Advanced Materials Characterization: The proposed hexagonal device and NV magnetometry technique provide a unique, unambiguous method for measuring fundamental symmetry-breaking transport coefficients (ξ) in candidate materials like ABA trilayer graphene, PdCoO2, PtSn4, and WTe2.
- Quantum Sensing and Metrology: Utilizing NV center magnetometry, a present-day realizable technology, to locally image and quantify electron current flow, moving beyond conventional bulk electrical measurements.
- Fundamental Condensed Matter Physics: Providing a robust theoretical framework for understanding electron hydrodynamics in anisotropic systems, which is critical for interpreting transport data in high-purity solid-state materials where viscous effects dominate.
- Soft and Active Matter Systems: The hydrodynamic framework derived for D6 symmetry may be adapted to model and predict the collective behavior of classical active matter systems (e.g., fluids composed of triangular or chiral microscopic components).
View Original Abstract
When continuous rotational invariance of a two-dimensional fluid is broken to the discrete, dihedral subgroup D_6 <mml:math xmlns:mml=“http://www.w3.org/1998/Math/MathML” display=“inline”> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:math> - 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 <mml:math xmlns:mml=“http://www.w3.org/1998/Math/MathML” display=“inline”> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:math> -symmetric 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 <mml:math xmlns:mml=“http://www.w3.org/1998/Math/MathML” display=“inline”> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:math> -invariant Fermi surfaces - that are sensitive to these new coefficients in a D_6 <mml:math xmlns:mml=“http://www.w3.org/1998/Math/MathML” display=“inline”> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:math> -invariant electron fluid. 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 <mml:math xmlns:mml=“http://www.w3.org/1998/Math/MathML” display=“inline”> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:math> -exploiting boundary conditions enable the unambiguous detection of these novel transport coefficients.