[Paper Review] Hall viscosity from elastic gauge fields in Dirac crystals

Alberto Cortijo, Yago Ferreirós+2 more|arXiv (Cornell University)|Jun 16, 2015

Topological Materials and Phenomena37 references26 citations

TL;DR

This paper proposes that elastic gauge fields in Dirac crystals—particularly in Weyl semimetals—generate a topological Hall viscosity via chiral anomaly mechanisms, analogous to electromagnetic responses. The effect arises from strain-induced gauge fields coupling to electrons, yielding a Hall viscosity coefficient orders of magnitude larger than previously known contributions, significantly enhancing experimental detectability in 2D and 3D topological materials.

ABSTRACT

The combination of Dirac physics and elasticity has been explored at length in graphene where the so--called "elastic gauge fields" have given rise to an entire new field of research and applications: Straintronics. The fact that these elastic fields couple to fermions as the electromagnetic field, implies that many electromagnetic responses will have elastic counterparts not explored before. In this work we will first show that the presence of elastic gauge fields will be the rule rather than the exception in most of the topologically non--trivial materials in two and three dimensions. In particular we will extract the elastic gauge fields associated to the recently observed Weyl semimetals, the "three dimensional graphene". As it is known, quantum electrodynamics suffers from the chiral anomaly whose consequences have been recently explored in matter systems. We will show that, associated to the physics of the anomalies, and as a counterpart of the Hall conductivity, elastic materials will have a Hall viscosity in two and three dimensions with a coefficient orders of magnitude bigger than the previously studied response. The magnitude and generality of the new effect will greatly improve the chances for the experimental observation of this topological, non dissipative response.

Motivation & Objective

• To establish the existence and structure of elastic gauge fields in three-dimensional Weyl semimetals, extending the known framework from 2D Dirac systems like graphene.
• To demonstrate that the presence of elastic gauge fields in topological materials naturally leads to a Hall viscosity response, even in the absence of external magnetic fields.
• To identify the origin of this Hall viscosity as a gravitational anomaly effect linked to the chiral anomaly in elastic degrees of freedom.
• To quantify the magnitude of the Hall viscosity and show it is significantly larger than phonon- or metric-induced contributions, improving experimental feasibility.

Proposed method

• Derive the effective action for elastic deformations in Dirac crystals using the low-energy effective Hamiltonian with strain-coupled gauge fields.
• Apply the Kubo formula to compute the stress-stress correlation function and extract the antisymmetric part of the viscosity tensor, identifying the Hall viscosity component.
• Use the chiral anomaly framework to relate the Hall viscosity to the topological response of the elastic gauge field, analogous to the electromagnetic Hall conductivity.
• Compute the Hall viscosity coefficient explicitly in a 3D Weyl semimetal model, obtaining η_H ∝ β² / a³ × ((b₃² − m²)/t²)^{3/2}, where β is the coupling strength.
• Analyze the symmetry breaking conditions required for Hall viscosity in 3D, showing that time reversal and rotational symmetry breaking are essential.
• Compare the magnitude of the elastic gauge field-induced Hall viscosity with conventional phonon- or metric-induced contributions, showing it is orders of magnitude larger.

Experimental results

Research questions

• RQ1 Can elastic gauge fields in 3D Dirac crystals give rise to a non-dissipative Hall viscosity response?
• RQ2 How does the Hall viscosity from elastic gauge fields compare in magnitude to conventional phonon- or metric-induced contributions in topological materials?
• RQ3 What is the role of the chiral anomaly in generating the Hall viscosity in elastic degrees of freedom, and how is it analogous to electromagnetic responses?
• RQ4 Under what symmetry conditions does Hall viscosity emerge in three-dimensional topological crystals?
• RQ5 Can the Hall viscosity be detected through phonon dispersion measurements, and what experimental signatures might it produce?

Key findings

• The Hall viscosity in 3D Dirac crystals arises from elastic gauge fields coupling to electrons, with a coefficient η_H = β² / (24π²a³) × ((b₃² − m²)/t²)^{3/2}, which is significantly larger than conventional contributions.
• The Hall viscosity is generated via the chiral anomaly mechanism in the elastic gauge field, making it a topological, non-dissipative response analogous to Hall conductivity.
• In Weyl semimetals, the Hall viscosity coefficient is enhanced by the presence of nodal points and broken time-reversal symmetry, with a distinct tensor component η_{3132} along the vector λ.
• For graphene, the characteristic frequency scale of the Hall viscosity is ω_H ≈ 95 eV, indicating strong effects on phonon dispersions, though challenging to resolve with current experimental techniques.
• The Hall viscosity can be probed via in-plane phonon dispersion measurements using X-ray scattering, Brillouin scattering, or electron energy loss spectroscopy.
• The inclusion of elastic gauge fields in the effective action for phonons introduces a new term ∫ d²r η_H^{ijlr} ∂_i u_j ∂_l ̇u_r, which modifies the elastic response and enables detection of topological effects in lattice dynamics.