Emergent-Gravity Hall Effect in Quantum Materials

• Emergent-gravity Hall effect is a phenomenon where effective gravitational fields from quantum geometry and spin textures induce transverse Hall-like currents in quantum materials.
• It leverages semiclassical corrections, gravitational anomalous velocity, and momentum-space effects to generate measurable Hall responses without the presence of external magnetic fields.
• Experimental realizations include quantized Hall steps in skyrmion systems and electron trajectory deflections analogous to gravitational lensing, offering new avenues in material transport studies.

The emergent-gravity Hall effect denotes a class of Hall responses in quantum and topological materials that originate not from conventional electromagnetic fields but from effective gravitational fields induced by quantum geometry, spin textures, or the emergent geometry of collective states. Substantial theoretical and experimental advances have revealed a wide spectrum of mechanisms where quantum geometric and gravitational analogies play a central role in the generation of transverse (Hall-like) currents, energy fluxes, or other nontrivial transport signatures, even in the absence of explicit magnetic fields.

1. Fundamental Concepts and Theoretical Foundations

The emergent-gravity Hall effect arises when collective degrees of freedom—such as spin textures, domain walls, or the geometric structure of Bloch wavefunctions—generate effective fields that mimic gravitational curvature or connections. In condensed matter systems, these analogs are not spacetime metrics but are geometric and gauge-theoretic structures entering wavepacket dynamics or the effective action, affecting the motion of low-energy excitations.

Several formalisms converge on this point:

• Semiclassical theory with nonadiabatic corrections: Hall currents can be generated by emergent gravitational fields described via Christoffel symbols derived from derivatives of the quantum (weighted) metric, as established in the unified framework for the emergent-gravity Hall effect (Yoshida et al., 24 Jul 2025). This structure extends the traditional Berry curvature/connection paradigm by including higher-order quantum geometric corrections.
• Chern–Simons functional and gravitational anomalies: In topological insulators and superconductors, boundary energy-momentum currents and related Hall effects are encoded by gravitational Chern–Simons terms, with anomaly inflow guaranteeing covariant anomaly cancellation (1201.4095, Golan et al., 2018).
• Analogy to emergent electromagnetism: The adiabatic motion of electrons in noncoplanar spin textures (e.g., in skyrmion lattices) leads to emergent gauge fields (Abelian or non-Abelian), with Hall responses that parallel gravitational or geometric Hall mechanisms (Kanazawa et al., 2015, Göbel et al., 1 Oct 2024).

2. Mechanisms of Emergent-Gravity Hall Effects

A comprehensive theoretical framework for the emergent-gravity Hall effect, as formulated in (Yoshida et al., 24 Jul 2025), identifies four principal mechanisms:

Mechanism	Defining Christoffel Symbol	Physical Origin
Real-space gravity	$\Gamma^{rrr}_{a,ij}$	Spatial derivatives of quantum metric; nonuniform textures
Momentum-space gravity	$\Gamma^{qqq}_{a,ij}$	Momentum-space quantum geometry; noncentrosymmetric bands
Gravitational anomalous velocity	$\Gamma^{qqt}_{a,i}$	Time dependence in Hamiltonian/parameters
Gravitational Lorentz force	$\Gamma^{rrt}_{a,i}$	Mixed real-space/time-variation, e.g., moving textures

2.1 Real-space Gravity Effects

Hall currents can be generated by spatial variations in quantum geometry, such as in skyrmions or domain walls, through corrections in the semiclassical equations of motion of the form

$$\dot{q}_a = -\frac{1}{\hbar}\partial_{r_a}V + \cdots - \Gamma^{rrr}_{a,ij}\partial_{k_i}\varepsilon\,\partial_{k_j}\varepsilon.$$

A finite Hall response exists when these corrections produce an antisymmetric contribution to the conductivity tensor and the underlying energy dispersion breaks even parity (Yoshida et al., 24 Jul 2025).

2.2 Momentum-space Gravity and Nonlinear Hall Responses

In crystals with broken inversion symmetry or in systems with complex band structures, the Berry curvature and quantum geometric tensor in momentum space also feature Christoffel symbols. Corrections of the type

$$\dot{r}_a = \frac{1}{\hbar}\partial_{q_a}\varepsilon + \cdots + \Gamma^{qqq}_{a,ij}\left(-\frac{1}{\hbar}\partial_{r_i}V\right)\left(-\frac{1}{\hbar}\partial_{r_j}V\right)$$

give rise to nonlinear Hall effects (NLHE), as confirmed in three-dimensional magnetic systems where the NLHE is proportional to the emergent toroidal moment (Hou et al., 6 Sep 2024).

2.3 Gravitational Anomalous Velocity and Lorentz Force

In time-dependent systems, additional mixed Christoffel symbols (e.g., $\Gamma^{qqt}_{a,i}$, $\Gamma^{rrt}_{a,i}$) in the equations of motion act as "gravitational anomalous velocity" and "gravitational Lorentz force" terms. These lead to Hall-like responses controlled by dynamics, such as moving skyrmions, domain wall propagation, or external parameter sweeps (Yoshida et al., 24 Jul 2025).

3. Emergent Gravity, Chern–Simons Functionals, and Thermal/Energy Hall Effects

In topological insulators (class DIII) and $p$-wave superconductors, the emergent-gravity Hall effect connects directly to gravitational anomalies and Chern–Simons effective actions. Key aspects include:

• The gravitational Chern–Simons functional in $(2+1)$D, upon variation, yields an energy–momentum flux proportional to spatial derivatives of curvature (specifically, tidal forces), not just uniform gravitational fields (1201.4095, Golan et al., 2018).
• Energy-momentum conservation on $(1+1)$D domain walls (edge modes) is maintained by anomaly inflow from the bulk, ensuring that Hall-like thermal or energy currents are precisely accounted for by the gravitational anomaly and the covariant form of the surface anomaly (1201.4095).
• In $2+1$D $p$-wave superconductors, both gravitational Chern–Simons (gCS) and gravitational pseudo Chern–Simons (gpCS) terms appear in the effective action, with the gCS term being topologically quantized and the gpCS term contributing non-topological, but still gravitational-like responses (Golan et al., 2018).

4. Manifestations in Quantum Materials: Experimental and Model Realizations

The emergent-gravity Hall effect is realized and detected across a range of materials and phenomena:

• Skyrmion systems and the quantized topological Hall effect: The Hall resistivity exhibits discrete steps corresponding to the creation/annihilation of individual skyrmions, each contributing a quantized emergent flux $\Phi_{\rm eff} = -Q (h/e)$ (Kanazawa et al., 2015, Jiang et al., 2016).
• Orbital Hall effects and bimerons: Skyrmion and antiferromagnetic skyrmion textures can induce not only charge Hall effects but also pronounced orbital Hall responses, which persist even when the net emergent field is compensated (e.g., in AFM bimerons), revealing the dominance of orbital over spin Hall conductivities in certain regimes (Göbel et al., 1 Oct 2024).
• Viscous electron fluids (graphene): Hall viscosity generates a Hall field that is opposite in sign to the classical Hall effect, interpreted as a transport signature of emergent gravitational (geometric) responses in the electron fluid (Berdyugin et al., 2018).
• Quantum Hall systems in gravitational fields: The Landau level spectrum and Hall resistivity are modified by (real or artificial) gravitational potentials or rotations, leading to energy splitting, plateau broadening, and a nonlinear dependence on gravitational parameters (Hammad et al., 2020, Landry et al., 11 Mar 2024).

5. Quantum Geometry, Emergent Metrics, and Electron Lensing

A key theoretical advance is the identification of an emergent metric encoding quantum geometry corrections to semiclassical electron dynamics:

• The effective Hamiltonian for electrons strongly coupled to a slowly varying spin texture is recast as

$$\mathcal{H} = -J + \frac{1}{2m} \pi_i\,g^{ij}(x)\,\pi_j + V(x),$$

where $g^{ij}(x) = \delta^{ij} - l_C^2 G_{ij}(x)$ with $$G_{ij} = (1/4) \partial_i \mathbf{S} \cdot \partial_j \mathbf{S}$$ defines the local quantum metric (Onishi et al., 4 Jun 2025).

• The resulting electron dynamics follows classical geodesics:

$$\ddot{x}^i + \Gamma^i_{jk} \dot{x}^j \dot{x}^k = 0,$$

where the Christoffel symbol is computed from the emergent metric. For inhomogeneous textures (e.g., radial spirals), electron trajectories exhibit lensing (analogous to gravitational lensing in general relativity), controlled by the integrated curvature of the emergent geometry.

• This lensing mechanism contributes to a transverse (Hall) deflection when electrons traverse interfaces with different emergent metrics, opening a path to engineer and experimentally probe emergent-gravity Hall effects separate from standard behaviors linked to Berry curvature or magnetization.

6. Experimental Consequences and Theoretical Implications

The emergent-gravity Hall effect underpins novel transport phenomena in a range of quantum materials and has diverse implications:

• Edge and energy transport: In topological insulators and superconductors, energy–momentum (thermal) Hall currents at domain walls or edges are signatures of gravitational anomaly inflow, and their quantization is protected by topological Chern–Simons terms (1201.4095, Golan et al., 2018).
• Hall viscosity and hydrodynamics: In hydrodynamic (viscous) electron systems such as high-purity graphene, Hall viscosity and geometric contributions to the current provide experimental access to emergent gravitational responses through specific transport geometries (Berdyugin et al., 2018).
• Control and detection of orbital currents: In skyrmion-based orbitronic devices, the ability to generate pure orbital Hall currents or large orbital torques in the absence of net charge currents yields practical avenues for data manipulation and storage (Göbel et al., 1 Oct 2024).
• Testing gravitational laws: In engineered quantum Hall systems subjected to artificial or real gravitational fields, precise measurements of resistivity plateaus and Landau level splitting could serve as sensitive probes for deviations from Newtonian gravity (Hammad et al., 2020).

7. Outlook and Open Directions

The systematic identification and classification of emergent-gravity Hall effects unveil possibilities for both applied and fundamental investigations:

• The unified framework based on quantum geometric corrections (via Christoffel symbols in real, momentum, and time spaces) offers a robust platform for future theoretical and experimental research, enabling exploration of gravitational analogues in quantum materials under engineered conditions, including time-dependent or non-centrosymmetric settings (Yoshida et al., 24 Jul 2025).
• Theoretical proposals suggest that careful engineering of quantum metrics, smooth spin textures, and control over electronic band structure can realize and tune emergent Hall responses governed by effective gravity-like fields across multiple platforms.
• The conceptual connection between emergent gravity, topological defects, and geometric phases continues to bridge condensed matter, quantum information, and high-energy theory, with the emergent-gravity Hall effect serving as a paradigmatic manifestation of this interplay.

In summary, the emergent-gravity Hall effect encompasses an array of phenomena where quantum geometry, topological invariants, and effective gravitational fields combine to generate Hall-like responses, energy-momentum flows, and nontrivial transport effects in quantum and topological materials. Such effects are deeply linked to the geometric and topological structure of quantum states and provide an expanding frontier for the paper of gravitational analogues in condensed matter physics.