Hiroki Yoshida and Takehito Yokoyama Tokyo have theoretically proposed a Hall effect driven by effective gravitational fields originating from quantum geometry. Their work identifies four distinct mechanisms: real-space gravity, momentum-space gravity, gravitational anomalous velocity and gravitational Lorentz force, all described by Christoffel symbols in real, momentum, or time spaces. The researchers constructed a unified theoretical framework based on semiclassical theory to systematically investigate how emergent gravity affects electron transport, demonstrating through model calculations that a measurable Hall response can arise. These findings open a new avenue for exploring gravitational effects within quantum systems.

Christoffel Symbols Define Emergent Gravity in Quantum Systems

The mathematical description of gravity within quantum systems extends beyond momentum-space considerations, as Christoffel symbols define four distinct mechanisms driving an “emergent-gravity Hall effect”. This theoretical work builds upon semiclassical wavepacket dynamics, extending them to include interband transitions beyond the standard adiabatic approximation, allowing for the natural emergence of Christoffel symbols within the equations governing wavepacket motion.

This approach enables a comprehensive analysis of transport phenomena influenced by quantum geometry in real, momentum, time and parameter spaces, suggesting that gravity-like effects are not limited to spatial dimensions within a material. Model calculations demonstrate the existence of conditions where a measurable Hall response, the transverse motion of electrons, can arise, indicating a pathway toward experimental verification of these emergent gravitational effects.

The weighted quantum metric and Christoffel symbols are defined for general two-band systems, with specific expressions provided for real space, including terms dependent on the weighted quantum metric and the Christoffel symbol. For example, the real-space Christoffel symbol is given by $Γ_{a, i j}^{r r r}$ = + + +, where summation over μ = x,y,z is taken. Explicit calculations of the Christoffel symbols for a magnetic skyrmion reveal their dependence on the spin structure, demonstrating how these symbols manifest in specific physical scenarios.

By considering the dependence on an external parameter λ, they showed that these symbols contribute to the current in insulating materials, where only specific terms in the Boltzmann equation are relevant. The equation ⟨ u|(∂ = + ℏ^2( $Γ_{a, i j} λ_{i} λ_{j}$ + G_(ai)λ_i) + O(ℏ^3) demonstrates the role of Christoffel symbols and the weighted quantum metric in determining the system’s response to external perturbations.

Four Mechanisms Drive the Emergent-Gravity Hall Effect

Calculations reveal that the emergent-gravity Hall conductivity, arising from these effects can reach 10^-9 to 10^-10 Amperes per Volt², a magnitude exceeding that of nonlinear Hall effects induced by the Berry curvature dipole, typically around 10^-12 Amperes per Volt². This difference in scale stems from the emergent-gravity Hall effect’s independence from scattering time (τ⁰ scaling), unlike the Berry curvature dipole effect which scales with τ¹, offering a potential pathway for experimental differentiation between the two phenomena.

The in-plane magnetic field induces a nonlinear planar Hall effect, but its intrinsic contribution also scales as τ⁰ and remains significantly smaller than the momentum-gravity Hall effect at approximately 10^-12 Amperes per Volt². The researchers demonstrate that the emergent gravitational field intensifies in regions where the spin texture changes rapidly, although spatial averaging of the relevant Christoffel symbol component, results in a net zero value.

A detailed examination of a two-dimensional single skyrmion, defined by a specific magnetic texture, provided a concrete system for calculating the emergent-gravity Hall conductivity. The equation conjugate to position, derived from the Lagrangian, incorporates corrections to the band energy, instantaneous Bloch states and terms involving Christoffel symbols and geometric quantities. The researchers’ work builds upon the concept of emergent electromagnetic fields, proposing an analogous “emergent gravitational field” as the driver of this novel Hall effect.

The equation ⟨ u|(∂ = + + + O(ℏ³) demonstrates the influence of Christoffel symbols on the system’s energy, where λ represents an external parameter. This equation highlights how these symbols contribute to the curvature of the energy landscape, influencing the motion of electrons within the material. The framework allows for a systematic investigation of how emergent gravity in these spaces affects transport phenomena, providing a comprehensive approach to understanding these complex interactions.

Real-Space Gravity Impacts Electron Motion via Potential Gradients

The equations governing electron motion reveal a direct link between real-space coordinates and the emergence of gravitational-like forces, according to new theoretical work. Calculations demonstrate that the Christoffel symbols, quantities traditionally associated with momentum-space gravity, also describe how electrons respond to gradients in a material’s potential energy landscape, effectively creating a “real-space gravity” effect. This extends the role of these symbols beyond their conventional interpretation and suggests a broader influence on electron behavior. This suggests that gravity-like effects aren’t limited to spatial dimensions, but can manifest in more abstract parameter spaces influencing electron dynamics.

The theoretical foundation rests on a Lagrangian formulation, where the equation of motion for an electron is derived using the Euler-Lagrange equation. A key result of this derivation is an equation detailing the electron’s velocity, incorporating terms related to potential gradients, the Berry curvature and the Christoffel symbols. These symbols appear in the equation alongside terms representing the curvature of the energy landscape, directly influencing how electrons navigate the potential.

The analysis extends to systems where the Bloch states, describing electron behavior, depend on momentum but not on real-space texture. In these scenarios, the equations of motion simplify, highlighting the influence of the Christoffel symbols on the electron’s momentum. This simplification allows for a clearer understanding of how these symbols contribute to nonlinear Hall responses, where electrons exhibit transverse motion in response to an applied electric field.

To illustrate the effect, the researchers considered a noncentrosymmetric magnet with a specific Hamiltonian, revealing how the Christoffel symbols manifest in the system. They then applied this framework to a magnetic skyrmion, a swirling magnetic texture, to explicitly calculate the real-space Christoffel symbols. The team investigated charge pumping in insulating systems, focusing on the contribution of terms in the Boltzmann equation.

They found that even in insulators, where charge transport is limited, the Christoffel symbols play a role, highlighting the fundamental nature of this emergent gravity effect. “We have investigated transport phenomena in systems with spatial, momentum, temporal, and external-parameter dependencies,” the researchers state, emphasizing the broad applicability of their framework.

Momentum-Space Gravity Arises from Weighted Quantum Metrics

The identification of these mechanisms stems from a unified theoretical framework constructed by Hiroki Yoshida and Takehito Yokoyama, both of Institute of Science Tokyo, Japan. The researchers’ approach centers on weighted quantum metrics, geometrical quantities defining the properties of Bloch states, and their derivatives which manifest as these emergent gravitational fields. They used the Boltzmann equation with a relaxation time approximation to derive the electron distribution function, allowing them to investigate these effects to arbitrary order.

They found that the weighted quantum metric, defined as $G_{i j}^{X Y}$ = 2Re[sum_(m ≠ n) $A_{n m, i}^{X} A_{m n, j}^{Y} / (ε_{m} - ε_{n})$ ], determines the energy landscape influencing electron behavior. This energy term, requiring consideration up to O(ℏ^2), combines with Berry-connection corrections to define the effective Lagrangian for a wavepacket initially prepared in band n: L = -ε_(n,c) + ℏq_c· $r_{c}$ + ℏsum_XX_iA^ $X_{n n, i}$ + ℏ^2/2sum_(X,Y) $X_{i} G_{i j}^{X Y} Y_{j}$ .

Nonadiabatic Corrections Yield Emergent Gravitational Fields

The team’s theoretical work builds on the discovery of the topological Hall effect induced by scalar spin chirality, demonstrating a growing interest in how spatial geometry and spin structure can generate novel transport responses. These symbols, defined from the weighted quantum metric as $Γ_{i, j k}^{a b c}$ , adhere to the symmetry relation $Γ_{i, k j}^{a c b}$ and are proposed as descriptors of emergent gravitational fields. This term directly describes the Hall response induced by the emergent gravitational field.

A key requirement for observing this effect is that the energy of the system is not an even function of momentum, a condition that distinguishes the emergent-gravity Hall effect from the more established topological Hall effect. To further differentiate the two, the researchers suggest investigating spin configurations lacking scalar spin chirality. Model calculations were then performed to demonstrate the effect, initially examining a one-dimensional domain-wall structure as an illustrative example.

While one-dimensional systems typically exhibit no Hall effect, analysis of the emergent gravitational field revealed it to be locally nonzero even in this simple spin configuration. Further calculations focused on a two-dimensional single skyrmion, described by a specific exchange field h(x,y) = h/(1 + |u|^2)(2Re(u),2Im(u),1-|u|^2), where u(x,y) represents the skyrmion’s spatial profile.

👉 More information
🗞 Emergent-Gravity Hall Effect from Quantum Geometry
✍️ Hiroki Yoshida and Takehito Yokoyama
🧠 DOI: http://link.aps.org/doi/10.1103/7cjk-63m2

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