Emergent-gravity Hall effect from quantum geometry

Problem Statement
Transport phenomena in solids are fundamentally governed by the geometry and topology of electronic wavefunctions. While the anomalous Hall effect is well-understood through the lens of Berry curvature (momentum-space geometry) and the topological Hall effect through emergent electromagnetic fields from spin textures, the role of quantum geometry in generating effective gravitational fields remains less explored. Specifically, the derivative of the quantum metric constitutes the Christoffel symbol, representing a form of "momentum-space gravity." The central question addressed in this work is whether emergent gravitational fields, arising from quantum geometry in real space, momentum space, time, and parameter space, can drive novel Hall effects in electronic transport.

Methodology
The authors develop a unified theoretical framework based on semiclassical wavepacket dynamics, extending the standard adiabatic approximation to include nonadiabatic interband transitions.

1. Lagrangian Formulation: Starting from a general Lagrangian for an electron with dependencies on real space ($r$), momentum space ($q$), time ($t$), and an arbitrary time-dependent parameter ($\lambda$), the authors derive the effective Lagrangian by expanding the quantum state beyond the adiabatic limit. This expansion incorporates first-order corrections to the adiabatic approximation.
2. Equations of Motion: Using the Euler-Lagrange equations, they derive the equations of motion for the wavepacket centers. Crucially, the inclusion of nonadiabatic processes introduces correction terms proportional to $\hbar^2$ that take the form of Christoffel symbols ($\Gamma$), defined via the weighted quantum metric ($G$). These symbols are identified as "emergent gravitational fields."
3. Transport Calculation: The authors employ the Boltzmann equation within the relaxation time approximation to derive the electron distribution function to arbitrary order in the relaxation time ($\tau$). By combining the derived equations of motion with the distribution function, they calculate the electric current, focusing on terms that contribute to transverse (Hall) responses.
4. Model Systems: To demonstrate the theory, the authors perform model calculations for:
   • Real-space textures: One-dimensional magnetic domain walls and two-dimensional magnetic skyrmions.
   • Momentum-space geometry: Two-dimensional systems with Rashba-type spin-orbit interaction and Zeeman coupling.
   • Time-domain dynamics: Systems with moving textures or time-varying external fields.

Key Contributions and Results
The paper identifies four distinct mechanisms for an "emergent-gravity Hall effect," all described by Christoffel symbols in different spaces:

1. Real-Space Gravity:

   • In systems with spatially varying spin textures (e.g., skyrmions), the real-space Christoffel symbol ($\Gamma^r_{rr}$) induces a Hall response.
   • The authors demonstrate that for a single magnetic skyrmion in a noncentrosymmetric magnet, a finite emergent-gravity Hall conductivity arises. The magnitude is estimated to be approximately $10^{-4} e^2/\hbar$ per skyrmion.
   • A key condition for a finite Hall response in this regime is that the energy dispersion must not be an even function of momentum ($h_0(q) \neq h_0(-q)$), distinguishing it from the topological Hall effect which relies on scalar spin chirality.

2. Momentum-Space Gravity:

   • In spatially uniform systems, the momentum-space Christoffel symbol ($\Gamma^q_{qq}$) leads to a nonlinear Hall effect.
   • The authors calculate the nonlinear Hall conductivity for a system with Rashba spin-orbit coupling and a magnetic field. They find that the momentum-gravity-induced effect scales as $\tau^0$ (independent of relaxation time), distinguishing it from the Berry curvature dipole mechanism (scaling as $\tau^1$).
   • The estimated magnitude ($\sim 10^{-9}$ to $10^{-10}$ Am/V$^2$) is predicted to be larger than the nonlinear Hall effect induced by the Berry curvature dipole.

3. Gravitational Anomalous Velocity and Lorentz Force (Time Domain):

   • Christoffel symbols involving time derivatives ($\Gamma^t$) give rise to new transport terms.
   • The antisymmetric part of $\Gamma^q_{qt}$ induces a "gravitational anomalous velocity" perpendicular to the momentum velocity, analogous to the Berry curvature effect.
   • The antisymmetric part of $\Gamma^r_{rt}$ acts as a "gravitational Lorentz force," perpendicular to the real-space velocity, mimicking a magnetic field.
   • These effects are shown to generate Hall responses in systems with moving textures (e.g., a drifting skyrmion) or time-varying magnetic fields.

Significance and Claims
The authors claim to have uncovered a new class of Hall effects driven by effective gravitational fields arising from quantum geometry. By treating real space, momentum space, time, and parameter space on an equal footing, the work provides a unified framework for investigating transport phenomena governed by Christoffel symbols.

• The paper posits that these effects open a "new avenue for exploring gravitational effects in quantum systems."
• It clarifies the specific conditions required for a finite Hall response, such as the breaking of inversion symmetry and the specific symmetry properties of the energy dispersion.
• The authors note that while related works have discussed emergent gravity from spin textures or nonadiabatic momentum-space effects, this work is distinct in its systematic derivation of Hall effects across all four spaces (real, momentum, time, and parameter) and its explicit identification of the four mechanisms listed above.

The findings suggest that emergent gravity is not merely a theoretical curiosity but a tangible contributor to transport properties, potentially observable in systems with noncentrosymmetric magnetic textures or under specific dynamic conditions.