Hydrostatic equilibrium in multi-Weyl semimetals

TL;DR

This work develops a covariant framework to study hydrostatic equilibrium in multi-Weyl semimetals, whose anisotropic dispersion $\omega^2 \sim k_{\parallel}^2 + k_{\perp}^{2n}$ breaks Lorentz boost invariance and reduces spacetime symmetry to $(SO(1,1)\times SO(2))\ltimes \mathbb{R}^4$. By gauging this spacetime group and coupling to a $U(1)$ gauge field, the authors construct an Aristotelian (stringy Newton–Cartan–like) background and derive a covariant partition function to obtain non-dissipative hydrostatic data, including zeroth-order constitutive relations and Ward identities. The approach identifies intrinsic versus extrinsic torsion as essential geometric data and provides explicit formulas for the spin and affine connections, enabling a consistent description of currents and stress in curved backgrounds. The framework generalizes to other systems with the same symmetry, such as Bjorken flow, and opens avenues for anomaly analyses and holographic connections in non-Lorentzian hydrodynamics with anisotropic topological quasiparticles.

Abstract

We study the hydrostatic equilibrium of multi-Weyl semimetals, a class of systems with Weyl-like quasi-particles but anisotropic dispersion relation $ω^2 \sim k_\parallel^2 + k_\perp^{2n}$, with $n$ a possitive integer. A characteristic feature of multi-Weyl systems is the lack of Lorentz invariance, instead, they possess the reduced spacetime symmetry $(SO(1,1)\times SO(2))\ltimes \mathbb R^4$. In this work we propose a covariant formulation for the low energy theory, allowing for a minimal coupling of the fermion field to external geometric background and $U(1)$ gauge field. The non-Lorentzian structure of the field theory demands introducing an Aristotelian spacetime analogous to the so-called stringy Newton-Cartan geometry \cite{Andringa:2012uz}. Our proposal allows for a systematic study of the hydrostatic properties via the derivation of the partition function of the system. In addition to multi-Weyl models, our formulation can be applied to systems with similar spacetime symmetry groups, such as Bjorken flow.