Nonlinear Odd Viscoelastic Effects

• Nonlinear odd viscoelasticity is defined by antisymmetric, dissipationless stress responses that produce transverse momentum under applied strain.
• It utilizes high-rank tensors that couple quantum geometry and topology to capture multiband and nonlinear behaviors in both crystalline solids and active fluids.
• These effects enable the design of advanced metamaterials and active matter systems with programmable flow, spontaneous pattern formation, and robust energy transport.

Nonlinear odd viscoelastic effects encompass a class of dissipationless, nonreciprocal responses to strain beyond linear order, arising in systems with broken time-reversal and parity symmetry, and displaying fundamentally quantum-geometric and topological characteristics in solids as well as unique dynamical instabilities and nonlinearities in complex fluids. These effects are encoded in higher-rank viscoelastic tensors, manifest as momentum or stress responses transverse to the directions of applied strain, and reveal new connections between multiband quantum geometry, active matter, and the rheology of nonequilibrium materials (Jain et al., 27 Nov 2025, Matus et al., 2023, Floyd et al., 2022).

1. Formal Definition and General Setting

Odd viscoelasticity refers to the set of material responses where the constitutive relations between stress and strain (or strain-rate) include components that are antisymmetric under exchange of certain tensor indices, corresponding to nonconservative, dissipationless, and time-reversal-odd/mechanically nonreciprocal effects. In the nonlinear regime, the stress response is second (or higher) order in the applied strains, resulting in momentum or current generation in directions orthogonal to those of the applied deformations.

In crystalline solids, the unsymmetrized strain tensor $w_{ij} = \partial_i u_j$ acts as a gauge field coupling to the lattice momentum via the Peierls substitution. The generalized current operators $T_{ij} = \partial H / \partial w_{ij}$ capture the system's viscoelastic response (Jain et al., 27 Nov 2025).

The second-order, time-independent (dc) nonlinear odd viscoelastic tensor is defined as:

$$\eta^\text{odd}_{ij;kl,mn} \equiv \sigma^\text{odd}_{\tilde{T}_{ij};\tilde{T}_{kl},\tilde{T}_{mn}}(0;0,0)$$

where $\sigma^\text{odd}$ is the fully antisymmetrized, second-order nonlinear viscosity/conductivity extracted in a Kubo–Dyson formalism and incorporating only the dissipationless part under all permutations of operator slots.

2. Quantum-Geometric and Topological Foundations

Nonlinear odd viscoelastic effects in quantum solids fundamentally originate from nontrivial quantum geometry and topological characteristics of the many-band Hilbert space. The response is controlled by geometric tensors that generalize Berry connections and curvatures to multi-state and multi-parameter regimes:

• The dressed Berry connection $$A_{ij}^{ab}(k) = i\langle u^a | \partial_{w_{ij}} u^b \rangle$$
• The two-band dressed curvature $$\Omega_{ij,kl}^{ab}(k) = -2 \text{Im}[A_{ij}^{ab} A_{kl}^{ba}]$$
• The three-band geometric tensor $$Q_{ij;kl,mn}^{abc}(k) = A_{ij}^{ab}A_{kl}^{bc}A_{mn}^{ca}$$

In second-order nonlinear response, both two-band and three-band (genuinely multi-state) contributions are present. The full tensor can be written (at zero frequency) as (Jain et al., 27 Nov 2025):

$$\eta^\text{odd}_{ij;kl,mn} = 3i \sum_{a \text{ occ.}, b, c \text{ unocc.}} \frac{\Delta^{ba} S^{acb} - \Delta^{ca} S^{bac}}{(\Delta^{ca})^2 (\Delta^{ba})^2}$$

where $S^{abc}$ is a partially antisymmetrized triple-matrix element combination and $\Delta^{ba}=E^b-E^a$.

The tensor satisfies the cyclic sum rule:

$$\eta^\text{odd}_{ij;kl,mn} + \eta^\text{odd}_{kl;mn,ij} + \eta^\text{odd}_{mn;ij,kl} = 0$$

ensuring only the fully antisymmetric ("odd") part survives and confirming the dissipationless, nonreciprocal character (Jain et al., 27 Nov 2025).

Topological quantization plays an essential role: under suitable conditions, integrals of the geometric tensors over the Brillouin zone give integer invariants such as Chern numbers or Hopf invariants, linking $\eta^\text{odd}$ to quantized, robust transport coefficients in analogy with Hall and Hall-viscosity responses. While explicit formulas for 3D integer quantization in terms of Chern or Pontryagin invariants remain to be fully detailed, numerical calculations in perturbed Hopf models confirm their correlation with nonzero $\eta^\text{odd}$ (Jain et al., 27 Nov 2025).

3. Continuum and Lattice Models: Constitutive Laws

In both quantum solids and active matter systems, the nonlinear odd viscoelastic response is embedded in high-rank constitutive laws. For crystalline lattices, the second-order Kubo–Dyson expansion forms the backbone of the theoretical analysis:

$$\langle A(t)\rangle = \sum_{ij} \int d\tau_1 d\tau_2 \chi_{A;B_i,B_j}(t;\tau_1,\tau_2)\,\phi_i(t-\tau_1)\,\phi_j(t-\tau_2)$$

whereupon extracting the frequency dependences, antisymmetrizing, and taking the DC limit, yields the nonlinear odd tensor.

In active viscoelastic fluids, the microscopic origin may lie in nonreciprocal interactions, e.g., chiral active dumbbells connected by an odd spring (Matus et al., 2023). Coarse-graining these microscopic equations (via Fokker–Planck and virial calculus) produces modified constitutive equations such as the odd Jeffreys model:

$$\begin{aligned} \partial_t T_{\langle ij \rangle} &+ \chi_s T_{\langle ij \rangle} + \chi_o T^o_{\langle ij \rangle} = -\gamma_s \partial_t D_{\langle ij \rangle} - \gamma_o \partial_t D^o_{\langle ij \rangle} \ T^A &= \frac{\kappa_o}{\kappa_e}(T + \eta_b D) \end{aligned}$$

where both even and odd relaxation rates and moduli $\chi_{s,o},\,\gamma_{s,o},\,\zeta_{s,o}$ appear. Nonlinearities beyond the linear Jeffreys-Maxwell regime are detected as algebraic growth of trace and antisymmetric stress components under finite strain (Matus et al., 2023).

In continuum hydrodynamics, the Cauchy stress receives both even and odd contributions, with an elastic modulus tensor $C_{ijkl}$ incorporating odd (nonreciprocal) parts such as $K^o E_{ijkl}$ and $A \epsilon_{ij} \delta_{kl}$ (Floyd et al., 2022). Corotational MAXWELL-type evolution equations then generalize to include nonlinear odd responses.

4. Dynamical Phenomena and Pattern Formation

Nonlinear odd viscoelasticity leads to unique dynamical instabilities and pattern formation not present in reciprocal media. In chiral active viscoelastic fluids, the odd elastic modulus $K^o$ can drive an oscillatory instability, resulting in an array of vortices with selected wavelength $k^* \approx K^o / 4\eta_s$ and growth rate $\omega_{\max}\approx (K^o)^2 / 8\eta_s\rho_s$ (Floyd et al., 2022).

Nonlinear saturation is governed by mechanisms such as shear-thickening in the polymeric viscosity:

$$\eta_p(\Psi_{ij}) = \eta_p^0\left(1 + 2\beta^2 \Psi_{ij}\Psi_{ij}\right)^{(n-1)/2}$$

with $n > 1$, which leads to a Landau-type amplitude equation for the vortex envelope:

$$\partial_t A = \omega_{\max} A - \gamma |A|^2 A + \xi \nabla^2 A + \cdots$$

These dynamics produce steady-state vortex lattices and oscillatory patterns, directly linked to the magnitude and sign of $K^o$. The conditions for instability and pattern formation are captured by a dimensionless "oddness" parameter $\mathrm{Od} = K^o/\mu$; for $\mathrm{Od}^2 \gtrsim \mathcal{O}(1)$, pattern-forming instabilities arise (Floyd et al., 2022).

5. Experimental Manifestations and Measurement

Experimental signatures of nonlinear odd viscoelastic effects are multifaceted. In solid-state and cold-atom systems, the theory suggests applying two orthogonal static strains and measuring the emergent momentum flow in the third direction, with resonant torsion or shear-wave transducers as possible probes (Jain et al., 27 Nov 2025). For engineered quantum materials or metamaterials, tracking shifts of Wannier centers under strain via finite-difference or maximally localized Wannier function techniques may provide an indirect fingerprint of nonlinear odd viscoelasticity.

In active fluid systems, oscillatory cross-correlations in stress relaxation, frequency-dependent odd viscoelastic moduli $G^{(o)}(\omega)$ (as measured via rheometry), and observation of nonlinear growth in antisymmetric stresses under finite shear constitute key experimental observables (Matus et al., 2023).

The realization of macroscopic odd viscoelastic effects requires broken time-reversal and parity individually (but not their product), hence magnetic, noncentrosymmetric crystals, or chiral active matter are favored platforms. In fluids, mechanical energy injection via, e.g., ATP hydrolysis or external driving sustains the nonreciprocal, entropy-reducing work characteristic of odd viscosity and elasticity (Floyd et al., 2022).

6. Applications and Theoretical Implications

Nonlinear odd viscoelastic effects open avenues in the design of metamaterials with programmable mechanical response, active gels supporting topological waveguides, and microfluidic chiral transport. The quantum-geometric underpinning reveals a direct connection between higher-order transport coefficients and Hilbert-space topology, providing tools for probing the multiband quantum geometry of solids (Jain et al., 27 Nov 2025). In active matter, odd viscoelasticity suggests new mechanisms for spontaneous pattern formation, flow control, and robust energy transfer in nonreciprocal media (Floyd et al., 2022).

The persistence of odd viscosity and elasticity in the zero-temperature, non-diffusive limit, as well as the spectral signatures in stress--stress correlators, indicate that genuinely nonlinear and topologically protected transport phenomena may be engineered in both quantum and classical systems (Matus et al., 2023).

7. Outlook and Open Questions

Open theoretical questions include quantitative relations between nonlinear odd viscoelastic tensors and higher Chern or Pontryagin invariants in three dimensions, explicit construction and classification of lattice models realizing quantized nonlinear odd responses, and the amplification or control of these effects in synthetic and biological settings.

Measurement protocols for isolating nonlinear odd viscoelastic effects, especially in multiband solids and chiral active matter, are under development. The interplay between dissipationless transport, energy injection, and entropy production in nonequilibrium environments remains a critical area for further theoretical and experimental exploration (Jain et al., 27 Nov 2025, Matus et al., 2023, Floyd et al., 2022).