🔬 mesoscale physics

Nonlinear Odd Viscoelastic Effect

This paper uncovers a class of dissipationless nonlinear odd viscoelastic effects in three-dimensional quantum systems, where geometric distortions in orthogonal directions generate momentum flow in a third direction, a phenomenon rooted in multiband Hilbert-space geometry and scalable by topological invariants.

Original authors: Ashwat Jain, Wojciech J. Jankowski, M. Mehraeen, Robert-Jan Slager

Published 2026-07-16

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CC BY 4.0

Original authors: Ashwat Jain, Wojciech J. Jankowski, M. Mehraeen, Robert-Jan Slager

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 the world around you as a giant, invisible dance floor. When you push on a piece of jelly, it wobbles and flows; when you push on a brick, it resists and snaps back. Scientists call these behaviors "viscosity" (how sticky or fluid something is) and "elasticity" (how springy it is). Usually, if you push a material in one direction, it moves or squishes in that same direction. But in the strange, quantum world of electrons inside crystals, things get weird. Sometimes, if you push an electron fluid in one direction, it doesn't just move forward—it might slide sideways, like a car drifting around a corner without turning the steering wheel. This is called "odd viscosity," a mysterious, friction-free flow that happens because of the hidden, geometric shapes of the quantum states the electrons live in.

Now, imagine pushing that material not just once, but with two different pushes at the same time, or pushing it in a way that twists its shape. Scientists have long known about the simple, one-push version of this sideways drift. But what happens if you get fancy and apply complex, twisting forces? Does the material still dance sideways, or does it do something even stranger? This is the big question that has been waiting in the wings of physics. Understanding this isn't just about playing with math; it's about finding new ways to see the invisible geometry of the universe. If we can figure out how these materials react to complex pushes, we might be able to "fingerprint" the hidden shapes of quantum matter, revealing secrets about how electrons are arranged that we've never been able to see before.

In this paper, the authors uncover a brand-new class of these weird, friction-free effects, which they call "nonlinear odd viscoelastic effects." They predict that if you squeeze a quantum material in two different directions at once—say, squishing it from the left and stretching it from the top—it will suddenly start flowing with momentum in a third, completely perpendicular direction (like a stream shooting out of a pipe that wasn't even connected to the water source). This isn't just a small wobble; it's a robust, dissipationless flow that arises from the deep, geometric structure of the quantum states themselves.

The researchers show that this effect is like a secret code written in the material's geometry. They found that the strength of this sideways flow is directly linked to specific mathematical shapes called "quantum geometric tensors." Think of these tensors as the DNA of the electron's shape. The authors discovered that this new effect is a combination of two types of geometric interactions: one involving two electron states dancing together, and another involving a trio of states interacting in a complex loop. They demonstrated that in certain "topological" materials—crystals with special, knotted quantum structures—this effect is not only present but can be scaled up. By changing the material's properties, they showed the effect can grow stronger, acting like a volume knob for this strange quantum flow.

Crucially, the paper suggests that this effect is a powerful tool for measurement. Because the flow depends so heavily on the specific "shape" of the quantum states, measuring this nonlinear flow could allow scientists to detect and map out the internal geometry of materials in ways that were previously impossible. They simulated this behavior in specific models, like a "Hopf insulator" (a type of magnetic topological material), and found that the effect persists even when the material is in a "flat-band" state, where the usual energy differences between electron levels vanish. This proves that the effect is purely geometric, not just a side effect of energy changes.

The authors also explain that this isn't just a theoretical curiosity; it's a real phenomenon that could be measured in the lab. They propose that by applying specific combinations of normal and shear stresses (pushing and twisting) to magnetic topological insulators, researchers could observe this unique current. If successful, this would open a new window into the quantum world, allowing us to "see" the hidden, multi-dimensional shapes of electron states by watching how they flow when pushed in just the right way. It's a bit like figuring out the shape of a hidden object by watching how the wind swirls around it when you blow on it from two different angles at once.

Technical Summary: Nonlinear Odd Viscoelastic Effect

Problem Statement
Viscoelasticity, traditionally understood as a bulk property of matter responding to physical deformations, includes dissipationless momentum flows orthogonal to perturbation directions, known as odd viscosity and odd elasticity. While linear odd viscoelastic effects (e.g., Hall viscosity) are well-established in quantum fluids and linked to topological invariants and Berry curvature, the existence and nature of nonlinear odd viscoelastic effects (NOVEs) remain unexplored. Specifically, it is unknown whether nonlinear responses to geometric distortions can arise in quantum fluids, how they relate to the multiband Hilbert-space geometry of quantum states, and whether they can exist independently of electronic Hall currents.

Methodology
The authors employ quadratic Kubo response theory generalized to higher-order tensor distortion field perturbations to derive the nonlinear elastic response of crystalline solids.

Theoretical Framework: The system is subjected to elastic distortion fields $w_{ij} = \partial u_i / \partial x_j$ . The nonlinear elastic current density $J_{ij}$ is expressed as a second-order response to these distortions, characterized by the tensor $\eta_{ij;kl,mn}$ .
Operator Expansion: The derivation utilizes the stress tensor operator ( $T_{ij}$ ), the elastic modulus tensor operator ( $U_{ij,kl}$ ), and the nonlinear elastic modulus tensor operator ( $V_{ij,kl,mn}$ ). These are expressed in terms of band-geometric quantities.
Geometric Decomposition: The dc quadratic response tensor is decomposed into gauge-invariant contributions from two-band and three-band virtual transitions. These contributions are formulated using:
- Two-state terms: Involving the adiabatic curvature (Berry curvature), the quantum metric, and the quantum nonmetricity tensor ( $N$ ).
- Three-state terms: Involving the three-state quantum geometric tensor (QGT) ( $Q$ ), which captures amplitudes of interband transitions between three states.
Model Systems: The theory is applied to specific topological models:
1. A two-band Moore-Ran-Wen (MRW) Hopf insulator model to demonstrate the role of the Hopf invariant.
2. A flattened three-dimensional chiral model (Altland-Zirnbauer class AIII) to isolate and compare two-band and three-band contributions, specifically in the flat-band limit where energy dispersion effects are removed.
Real-Space Analysis: The momentum-space results are mapped to the real-space Wannier function representation to visualize the transport of Wannier centers under stress.

Key Contributions and Results

Prediction of NOVEs: The paper uncovers a class of dissipationless nonlinear odd viscoelastic effects in three spatial dimensions. These effects arise when geometric distortions in two orthogonal directions generate momentum flow in a third direction (e.g., normal distortions $w_{xx}, w_{yy}$ inducing current $J_{zz}$ ).
Geometric Origin: The authors demonstrate that NOVEs are uniquely constituted by two-state and three-state quantum geometric tensors.
- The quantum nonmetricity tensor ( $N$ ) captures the failure of the Hermitian connection to preserve the two-state quantum metric under parallel transport.
- The three-state QGT ( $Q$ ) captures the nonlinear character of the effect through triplets of virtual transitions.
Topological Scaling: In the MRW Hopf insulator model, the nonlinear viscoelastic response scales linearly with the integer topological Hopf invariant ( $\chi$ ). By scaling the momentum $k_z \to p k_z$ , the invariant and the response scale by an integer factor $p$ .
Multiband Interplay: In multiband systems, the paper shows that NOVEs naturally arise from the interplay of two-band and three-band contributions. In the flat-band limit, the response becomes purely geometric, reflecting the internal geometry of nonmetricities and three-state QGTs, independent of band dispersion.
Symmetry and Chirality: The vanishing of the three-band term under specific symmetries (e.g., $M_{xz}T$ ) reflects the chirality of the electronic ground states. Breaking these symmetries allows the retrieval of geometric responses even when topological invariants are trivialized, proving the effect is intrinsically geometric.
Wannier Representation: The nonlinear odd viscoelastic transport is shown to correspond to the velocity of Wannier centers ( $d x_c / dt$ ) induced by stress rates, providing a real-space visualization of the momentum-space currents.

Significance and Claims
The paper claims to establish a new set of unique viscoelastic effects that serve as experimental fingerprints for the multistate geometry of underlying quantum states.

Beyond Topology: The findings suggest that viscoelastic responses can probe nontrivial multistate geometries (encoded in nonmetricity and three-state QGTs) even in the absence of nontrivial topological invariants or electronic Hall currents.
Experimental Feasibility: The authors propose that magnetic topological insulators, such as axion insulators, are strong candidates for observing these effects due to their accessible three-dimensional multiband quantum geometry. They suggest that the scaling of current responses under orthogonal distortions should deviate from linearity, reflecting the identified odd effects.
Theoretical Foundation: This work paves the way for studying analogous time-dependent responses and establishes viscoelasticity as a probe for higher-order geometric tensors in quantum systems, extending the understanding of quantum geometry beyond the Berry curvature and quantum metric.

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