🔬 mesoscale physics

Emergent curved space and gravitational lensing in quantum materials

This paper demonstrates that long-wavelength spin textures in quantum materials can induce an emergent curved space for itinerant electrons, leading to an analog of gravitational lensing caused by nonadiabatic quantum corrections.

Original authors: Yugo Onishi, Nisarga Paul, Liang Fu

Published 2026-02-11

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CC BY 4.0

Original authors: Yugo Onishi, Nisarga Paul, Liang Fu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are driving a car on a vast, flat desert. If you keep your steering wheel straight, you will travel in a perfectly straight line. This is how we usually think about electrons moving through a material: they move in straight lines unless they hit an obstacle or are pulled by a magnet.

But what if the desert itself wasn't flat? What if the ground was actually a series of hills, valleys, and funnels? Even if you kept your steering wheel perfectly straight, your car would curve, spiral, or even orbit around a central point because the shape of the ground is forcing you to turn.

This is the core discovery of the paper by Onishi, Paul, and Fu. They have found a way to make electrons "feel" like they are driving on a curved landscape, even when there are no magnets or hills present.

The Secret Ingredient: The "Spin" Texture

To understand how they do this, we have to look at the electrons' "spin." Think of every electron as a tiny, spinning compass needle. In most materials, these needles all point in the same direction.

However, in certain "quantum materials," these tiny compass needles can be arranged in beautiful, swirling patterns—like a whirlpool of needles or a spiral staircase. This is called a spin texture.

The researchers discovered that when an electron moves through these swirling patterns, it doesn't just see a magnetic field; it experiences a change in the very "fabric" of the space it is traveling through. Because the electron's own spin is trying to keep up with the swirling background, it creates a mathematical effect that is identical to curved space-time.

The Analogy: The "Stretchy" Map

Imagine you are walking on a map. Usually, one inch on the map equals one mile on the ground everywhere. But imagine a magical map where, in certain areas, the paper is stretched out like rubber.

If you try to walk "one inch" through a stretched area, you actually end up walking much further in real life. To an observer watching you, it would look like you slowed down or took a detour, even though you thought you were walking straight.

In these quantum materials, the swirling spin patterns act like that stretchy rubber. They "stretch" the distance the electron has to travel. This "stretching" is what the scientists call an emergent metric. To the electron, the space has literally become curved.

The Result: Gravitational Lensing

Because the space is curved, the electrons do something amazing: they perform gravitational lensing.

In outer space, massive objects like galaxies warp the space around them. When light from a distant star passes near a galaxy, the curved space bends the light, acting like a giant magnifying glass. This is "gravitational lensing."

The researchers showed that in these quantum materials, the swirling spin patterns act like those massive galaxies. As electrons fly past a swirl, their paths bend. They don't bend because a magnet is pulling them; they bend because the "straight" path has become a curve.

Why does this matter?

This isn't just a cool math trick. It opens up a new way to control electricity:

No Magnets Required: Usually, to bend an electron's path, you need a heavy, external magnetic field. This method uses the material's own internal "swirls," which could lead to much smaller, more efficient electronic components.
A Laboratory for Gravity: Gravity is incredibly hard to study because it is so weak. By using these materials, scientists can create a "miniature universe" on a tiny chip, where they can study the effects of curved space and "gravity" in a controlled, tabletop experiment.
New Types of Electronics: By designing specific "swirls" (spin textures), we could potentially create "lenses" for electrons, focusing them into specific areas to create ultra-fast or ultra-sensitive quantum computers.

In short: The researchers have found a way to turn a piece of solid matter into a playground of curved space, allowing us to bend the path of electricity using nothing but the geometry of quantum spins.

Technical Summary: Emergent Curved Space and Gravitational Lensing in Quantum Materials

Authors: Yugo Onishi, Nisarga Paul, and Liang Fu (MIT)
Date: October 7, 2025

1. Problem Statement

In condensed matter physics, it is well-established that itinerant electrons coupled to localized spin textures can give rise to emergent gauge fields (effective magnetic fields), leading to phenomena like the Topological Hall Effect. However, conventional treatments typically focus on the adiabatic limit (strong coupling), where the electron spin perfectly follows the local texture.

The authors address a higher-order physical consequence: what happens when the coupling $J$ is finite and non-adiabatic corrections are considered? They propose that these corrections do not merely manifest as scalar potentials or magnetic fields, but fundamentally alter the spatial metric experienced by the electrons, effectively creating an emergent curved space.

2. Methodology

The researchers employ a theoretical framework based on the Schrieffer-Wolff (SW) transformation to derive a low-energy effective Hamiltonian for itinerant electrons coupled to a localized spin texture $\mathbf{S}(\mathbf{x})$ .

Hamiltonian Setup: They start with the Hamiltonian $H = \frac{(\mathbf{p} - e\mathbf{A})^2}{2m} - J\mathbf{S}(\mathbf{x}) \cdot \boldsymbol{\sigma}$ .
Unitary Transformation: They apply a transformation $U(\mathbf{x})$ to rotate the electron's spin to align with the local spin axis, which introduces a $U(2)$ gauge field $\mathbf{a} = U^\dagger \nabla U$ .
Effective Theory Derivation: By treating the off-diagonal terms (which couple the spin-up and spin-down sectors) via the SW transformation, they derive the effective Hamiltonian up to $O(J^{-1})$ .
Geometric Mapping: They map the resulting kinetic energy term to a Riemannian manifold, where the quantum geometric tensor $\mathcal{Q}_{ij}$ (comprising the quantum metric $G_{ij}$ and Berry curvature $\Omega$ ) dictates the properties of the emergent space.

3. Key Contributions

Discovery of the Emergent Metric: The authors show that the effective metric $g_{ij}$ is modified by the quantum metric $G_{ij}$ of the spin texture: $g_{ij} = \delta_{ij} - \ell_C^2 G_{ij}$ , where $\ell_C = \hbar/\sqrt{mJ}$ acts as a "Compton wavelength."
Analogy to General Relativity: They establish a formal analogy where the coupling $J$ plays the role of $mc^2$ (rest mass energy) and the emergent metric $\tilde{g}_{\mu\nu}$ mimics the spacetime metric in Einsteinian gravity.
Identification of "Gravitational" Forces: They demonstrate that the electron experiences a force proportional to the square of its velocity due to the Christoffel symbols $\Gamma^i_{jk}$ , a hallmark of geodesic motion in curved space.
Geometric Interpretation: They prove that the emergent space can be visualized as a $d$ -dimensional surface embedded in a higher-dimensional manifold (e.g., a radial spiral texture corresponds to the geometry of a cone).

4. Results

Electron Lensing Effect: The most significant physical result is gravitational lensing. Unlike scalar potentials (which affect slow particles more) or magnetic fields, the emergent gravitational effect persists for—and is even more significant for—fast-moving electrons.
Specific Texture Behaviors:
- Ferromagnet-to-Helical Transition: Electrons passing through a domain wall experience a bending of their trajectory toward the normal, governed by a relation analogous to Snell's Law but without the possibility of total internal reflection.
- Radial Spiral Texture: This texture creates a conical geometry. Electrons traveling near the origin undergo "focusing," where trajectories are bent by the curvature, mimicking the deflection of light around a mass in (2+1)D gravity.
Deflection Angle: Using the Gauss-Bonnet theorem, they show that the deflection angle $\phi$ of an electron is directly proportional to the integrated Gaussian curvature $K$ enclosed by its path.

5. Significance

New Physics in Quantum Materials: This work introduces a "gravitational" paradigm to condensed matter, moving beyond the standard gauge-field (magnetic) descriptions.
Universality: The authors argue that because these effects arise from the fundamental non-adiabaticity of quantum coupling, emergent curved space should be a ubiquitous feature in various quantum systems (including pseudospin systems).
Experimental Potential: The predicted lensing effects are within observable ranges for typical Hund's coupling and spin texture wavelengths ( $\lambda_s \sim 10\text{--}100$ nm). This opens new avenues for "gravitational" spectroscopy and the study of analog gravity (e.g., black hole analogs or gravitational waves) within solid-state devices.
Symmetry Independence: Unlike the Hall effect, which requires time-reversal symmetry breaking, emergent gravitational lensing can occur in coplanar spin textures without an emergent magnetic field.