[Paper Review] Mixed axial-gravitational anomaly from emergent curved spacetime in nonlinear charge transport

Tobias Holder, Daniel T. Kaplan+2 more|arXiv (Cornell University)|Nov 15, 2021

Quantum, superfluid, helium dynamics4 citations

TL;DR

This paper demonstrates that nonlinear charge transport in time-reversal- and inversion-symmetry-broken materials generates a mixed axial-gravitational anomaly (AGA) via emergent curved spacetime in quasiparticle dynamics, leading to a measurable dc current at second order in electric field. The AGA effect is isolated in a purely electrical, multi-contact geometry, with a predicted anomalous Hall conductivity of ~50 Ω⁻¹μm⁻¹ and a Hall angle of ~0.5%, enabling direct detection without magnetic fields or thermal gradients.

ABSTRACT

In 3+1 dimensional spacetime, two vector gauge anomalies are known: The chiral anomaly and the mixed axial-gravitational anomaly. While the former is well documented and tied to the presence of a magnetic field, the latter instead requires a nonzero spacetime curvature, which has made it rather difficult to study. In this work, we show that a quantum anomaly arises in the second-order electrical response for zero magnetic field, which creates a dc-current that can be either longitudinal or transverse to electric field. Consequently, the continuity equation for the chiral current is not conserved at order $τ^{-1}$, where $τ$ is the quasiparticle relaxation time. We can identify the anomaly as a mixed axial-gravitational one, and predict a material in which the anomaly-induced current can be isolated in a purely electrical measurement. Our findings indicate that charge transport generically derives from quasiparticle motion in an emergent curved spacetime, with potentially far-reaching consequences for all types of response functions.

Motivation & Objective

• To establish that nonlinear electrical response in Weyl semimetals generates a mixed axial-gravitational anomaly (AGA) without magnetic fields or thermal gradients.
• To demonstrate that quasiparticles in a periodic lattice move in an emergent curved spacetime, even at zero temperature and in static lattices.
• To propose a multi-contact electrical measurement setup that isolates the AGA-induced current, enabling direct observation in a bulk, all-electrical configuration.
• To show that the AGA effect persists at T=0 and is robust against relaxation time variations, distinguishing it from thermal or Berry-curvature-driven anomalies.
• To link changes in quasiparticle occupation numbers to emergent spacetime curvature, providing a new framework for understanding quantum anomalies in condensed matter.

Proposed method

• Derive the second-order electrical conductivity using perturbation theory and the Kubo formalism, identifying virtual transitions as the source of emergent spacetime curvature.
• Use a semiclassical picture to show that Fermi surface deformation due to electric fields mimics local curvature, breaking chiral charge conservation at order τ⁻¹.
• Identify the AGA contribution via the difference between cyclically permuted second-order conductivities, defined as σ^(gr) = σ^(2)_{xx;y} - σ^(2)_{yx;x}, which is independent of relaxation time τ.
• Distinguish the AGA term σ^(gr) from the τ²-dependent Drude-like term σ^(dr) by analyzing their different scaling with relaxation time τ.
• Propose a multi-contact electrical geometry to measure the transverse current induced by σ^(gr), enabling isolation of the AGA effect from other nonlinear responses.
• Use symmetry analysis to show that σ^(gr) vanishes upon restoration of inversion symmetry above the Néel temperature, confirming its topological origin.

Experimental results

Research questions

• RQ1 Can the mixed axial-gravitational anomaly be observed in a purely electrical, nonlinear transport measurement without external magnetic fields or thermal gradients?
• RQ2 How does emergent spacetime curvature in a periodic lattice influence the conservation of chiral charge in quasiparticle transport?
• RQ3 What is the scaling behavior of the AGA-induced current with quasiparticle relaxation time τ, and how does it differ from conventional nonlinear responses?
• RQ4 Can the AGA be isolated experimentally using a multi-contact electrical setup, and what is its magnitude in realistic materials?
• RQ5 To what extent do changes in quasiparticle occupation numbers—via virtual transitions—simulate gravitational effects in the effective spacetime geometry?

Key findings

• The mixed axial-gravitational anomaly (AGA) manifests in second-order nonlinear charge transport as a transverse dc current, even in zero magnetic field and at T=0.
• The AGA contribution σ^(gr) is independent of relaxation time τ, while the dominant Drude-like term σ^(dr) scales as τ², enabling clean separation of the anomaly.
• A measurable anomalous Hall conductivity of ~50 Ω⁻¹μm⁻¹ is predicted for an electric field of 1 V cm⁻¹, comparable to that of Fe thin films.
• The effective Hall angle is estimated at ~0.5%, with σ ≈ 10⁴ S cm⁻¹, indicating a detectable signal in realistic experimental conditions.
• The AGA signal is strongest in the xy and xz planes and vanishes when inversion symmetry is restored above the Néel temperature, confirming its topological origin.
• The AGA effect is robust against temperature variations and is insensitive to longitudinal conductivity, distinguishing it from the Berry-curvature-induced nonlinear anomalous Hall effect.