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Nonlinear Odd Viscoelastic Effect

T0 review · 2 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper predicts that biaxial strain on a three-dimensional topological crystal produces a dissipationless momentum current in the third direction, controlled by quantum geometry and integer topological invariants.

desk verdict A genuinely new nonlinear odd viscosity tensor with topological fingerprints, but the second-order strain truncation needs to be fixed before the central formulas can be trusted. read the letter →

arxiv 2511.22706 v2 pith:UV3MYBDT submitted 2025-11-27 cond-mat.mes-hall quant-ph

Ashwat Jain , Wojciech J. Jankowski , M. Mehraeen , Robert-Jan Slager This is my paper
classification cond-mat.mes-hallquant-ph
keywords nonlinearoddviscosityviscoelasticityquantumgeometryHopfinvariantthree-stategeometrictensorstrainresponsetopologicalphasetransitionmomentumcurrent
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper predicts a class of nonlinear odd viscoelastic effects in three-dimensional crystalline electron fluids: when a crystal is strained in two orthogonal directions, a dissipationless momentum current flows in the third direction. The central claim is that this second-order stress-stress response is entirely governed by the Hilbert-space geometry of the occupied Bloch states, decomposing into a two-band term built from symplectic Berry-type connection forms and a three-band term built from three-state quantum geometric tensors. The effect scales with integer topological invariants — the Hopf invariant in two-band models, and a chiral invariant in three-band models — and changes sharply across topological phase transitions. A sympathetic reader would care because the effect offers a bulk experimental fingerprint of multiband quantum geometry and of topological invariants that are otherwise hard to access, and it extends odd-viscosity physics from linear to nonlinear order.

What carries the argument

The objects doing the work are strain-dressed geometric tensors: the non-Abelian Berry connection A^{ab}_{ij} = i⟨u_a|∂_{w_ij}u_b⟩, the symplectic form Ω^{ab}_{ij,kl} = -2 Im(A^{ab}_{ij}A^{ba}_{kl}), and the three-state quantum geometric tensor Q^{abc}_{ij,kl,mn} = A^{ab}_{ij}A^{bc}_{kl}A^{ca}_{mn}. The two-band response is essentially Σ (1/Δ_{ba}) Ω^{ba}_{ij,(kl}∂_{mn)}Δ_{ba}, and the three-band response is Σ (Δ_{cb}/Δ_{ba}Δ_{ac}) Re[Δ_{cb}Q^{abc} + Δ_{ba}Q^{cab} + Δ_{ac}Q^{bca}], with normalized symmetrization of index pairs. These tensors convert geometric phase structure of the Bloch states into a prediction for the nonlinear momentum current, and their symmetry properties determine when

What would settle it

Compute the second-order stress-stress correlator using the exact strain-dependent Hamiltonian including the O(w^2) terms: if the result differs from Eqs. (4)–(5), the predicted topological scaling is incomplete. Alternatively, apply biaxial strain to a candidate Hopf insulator and look for a transverse momentum current scaling linearly with the Hopf invariant.

Watch

Extended reading notes

Core claim

The paper's central claim is that the dc nonlinear viscoelastic tensor η_{ij;kl,mn} — the second-order response of a momentum current to two static strains — is nonzero in three-dimensional magnetic and geometrically nontrivial phases, and decomposes over the Brillouin zone into two gauge-invariant pieces. The two-band piece is built from the strain-dressed symplectic Berry form Ω^{ba}_{ij,kl} weighted by the inverse gap; the three-band piece is built from products of three non-Abelian Berry connections forming the three-state quantum geometric tensor Q^{abc}. Numerically, the two-band response grows in integer steps as the Hopf invariant is scaled in a two-band Hopf insulator, and the three

Load-bearing premise

The whole derivation rests on taking the stress operator as T_ij = ∂H/∂w_ij evaluated at zero strain, thereby dropping O(w^2) terms in the strain-dependent Hamiltonian; the paper's own Discussion states these higher-order derivative terms survive in the w→0 limit, which, if true, would make Eqs. (4)–(5) incomplete at second order.

Editorial extensions

If this is right

  • Magnetic topological insulators in three dimensions, including candidate axion insulators, should exhibit a nonlinear momentum current under biaxial strain, with the response strength tied to integer topological invariants.
  • The effect gives a bulk experimental fingerprint of multiband Hilbert-space geometry — specifically the skewness and chirality of Wannier functions encoded in three-state quantum geometric tensors.
  • Because the three-band term survives in flattened bands and after the protecting symmetry is broken, it separates geometric contributions from dissipative ones, allowing a clean readout of quantum geometry.
  • The tensor obeys the sum rule η_xx;yy,zz + η_yy;zz,xx + η_zz;xx,yy = 0, so only two independent components need to be measured to characterize the effect.
  • The formal mapping between nonlinear viscoelastic and nonlinear Hall responses means biaxial strain protocols can be adapted from existing Hall viscosity experiments, with the current response deviating from linear scaling in the presence of nonzero correlators.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial: the same two-band/three-band decomposition likely extends to higher-order viscoelastic responses, where the paper's linear-in-strain truncation of the Hamiltonian would break down; those higher orders are where the dropped O(w^2) terms might enter.
  • Editorial: a biaxial-strain experiment on a candidate Hopf insulator could serve as a bulk probe of the Hopf invariant itself, complementing surface-state probes, provided the O(w^2) terms do not spoil the scaling.
  • Editorial: because the three-state quantum geometric tensor also controls other second-order geometric responses (shift-current-like and photogalvanic), the NOVE may share a common geometric origin with those transport channels, suggesting correlated signatures.
  • Editorial: a clean internal check is to recompute η in the same toy models using the full strain-dependent Hamiltonian, including the O(w^2) terms the paper drops; if the result changes, the topological scaling law would need refinement.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proposes a class of nonlinear odd viscoelastic effects in three-dimensional electronic systems. It defines a second-order stress-stress response tensor η_{ij;kl,mn} via a nested Kubo-like commutator of stress operators, claims the response is dissipationless, and derives a momentum-resolved decomposition into a two-band term built from symplectic Berry-type forms and a three-band term built from three-state quantum geometric tensors. The effect is demonstrated numerically in a perturbed Moore-Ren-Wen Hopf insulator, where the response grows with the Hopf invariant under k_z rescaling, and in a flattened chiral three-band model, where it tracks the chiral invariant and the three-state geometry. A real-space Wannier-center calculation is presented as a complementary check. The central physical claim is that biaxial strain produces a transverse momentum current whose magnitude is controlled by quantum geometry and integer topological invariants.

Significance. If the computed tensor is the complete second-order odd viscoelastic response, the paper introduces a genuinely new class of dissipationless mechanical responses and connects them to multiband quantum geometry and topological invariants in a falsifiable way. The strengths of the manuscript are its detailed Kubo derivation in the Supplemental Material, the gauge-invariant two-band/three-band decomposition, the independent numerical computation of Hopf and chiral invariants, and the complementary Wannier-center calculation. The proposed effect is experimentally addressable in magnetic topological insulators. However, the completeness of Eqs. (4)-(5) is conditional on resolving the strain-Hamiltonian truncation issue, so the significance is currently prospective rather than established.

major comments (2)
  1. [Discussion (strain-implementation paragraph); SM Sec. II] The response is computed after truncating the strained Hamiltonian to H(w) ≈ H(0) - w_ab T_ab, with T_ab = ∂H/∂w_ab|0. For a second-order response this truncation is not automatically justified. The quadratic terms (1/2)w_ab w_cd U_ab,cd, where U_ab,cd = ∂²H/∂w_ab∂w_cd|0, contribute to ⟨T_ij⟩ at first order in perturbation theory, i.e., at O(w²), through kernels of the form ⟨[T_ij, U_ab,cd]⟩; additionally, the strain dependence T_ij(w) = T_ij + w_cd U_ij,cd produces mixed first-order contributions. SM Sec. II only proves that the first strain derivative at w=0 equals the stress tensor; it does not show that U is annihilated by the odd projector in SM Eq. (12). The Discussion's statement that 'the higher-order derivative terms survive in the w_ab → 0 limit' directly contradicts the preceding assertion that they can be neglected. Unless a proof of cancellation is supplied, or the U terms a
  2. [Main text Eq. (1); SM Eqs. (11)-(13)] Eq. (1) defines η_{ij;kl,mn} as a single nested commutator, (i/ω)⟨[T_mn,[T_kl,T_ij]]⟩. The derivation in the SM instead defines the odd response as the antisymmetrized combination σ^odd_{A;B1,B2} = (2i/ω)χ_{A;B1,B2} - (i/ω)χ_{B1;A,B2} - (i/ω)χ_{B2;B1,A} in SM Eq. (13). These two objects are not equal; the Jacobi identity does not reduce the antisymmetrized combination to the single commutator. Consequently, Eq. (1) is not the quantity actually computed in Eqs. (4)-(5), and the relation J_ij = η w_kl w_mn in Eq. (2) does not follow from Eq. (1) as written. Please replace Eq. (1) by the SM definition, or explicitly state that Eq. (1) is a schematic representative of the antisymmetrized correlator.
minor comments (5)
  1. [Main text Eq. (4); SM Eq. (32)] The two-band formula is written as (1/Δ_ba) Ω (∂Δ_ba) in the main text and as Ω ∂ ln Δ_ba in the SM. These are equivalent by the chain rule, but the notation should be harmonized to avoid apparent inconsistency.
  2. [Abstract] The abstract mentions 'nonmetricity,' but this term is never defined or used in the main text or SM. Please either define it and connect it to the formulas, or remove it.
  3. [Main text Eq. (5); SM Eq. (26)] The definition of Q^{abc} differs in band-index order: main text has A^{ac}A^{cb}A^{ba}, while the SM has A^{ab}A^{bc}A^{ca}. Since a real part is taken, the expressions may coincide up to conjugation and relabeling, but the notation should be made consistent.
  4. [Fig. 2 caption] The caption lists χ values (0, p, −2p, ...) but does not explain how the plotted curves correspond to those values. Please add explicit labels per curve or panel.
  5. [Nonlinear viscoelasticity in Wannier basis] The quantity d x_c/dw_kl is a static strain derivative of a Wannier center, not a time-dependent current. The text calls it a 'net current' without deriving the relation to the momentum current J_ij. Please clarify this connection.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the nonlinear viscoelastic tensor is derived from a standard Kubo expansion and computed from model wavefunctions; self-citations are contextual, not load-bearing.

full rationale

The derivation chain is self-contained. The nonlinear odd viscoelastic tensor is obtained by starting from a standard Kubo/Dyson expansion (SM Eqs. 1-18), substituting stress operators defined via the Peierls gauge substitution, and then algebraically splitting the band sums into two-band and three-band contributions. No parameter is fitted to reproduce the target response; the numerical values in Figs. 2-3 are computed from the model wavefunctions, and the topological invariants are evaluated independently (e.g., Hopf preimage counting and known formulas). Self-citations (Refs. [25,64,67]) supply definitions of multistate geometric tensors, model Hamiltonians, and background invariants, but the central reduction from the correlator to Eqs. (4)-(5) does not rely on them for its logical force. The apparent discrepancy between the main-text Eq. (4) and SM Eq. (32) (Delta_ba vs ln Delta_ba) is not real: since d(ln Delta_ba) = d(Delta_ba)/Delta_ba, the two expressions coincide exactly. The O(w^2) truncation of the strained Hamiltonian, discussed in the main text ('the higher-order derivative terms survive in the w_ab -> 0 limit'), is a physical/applicability caveat about the validity of the perturbative expansion, not a circular step: it does not assume the result being derived, and the paper's derivation does not use those higher-order terms as inputs. Overall, the predictions are genuine computations from stated model assumptions, not identities by construction.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central formulas use only standard Kubo response and known model Hamiltonians. The free parameters are model or protocol choices, none fitted to the target response. The only structurally new mathematical objects are the strain-dressed geometric tensors A^ab_{ij}, Omega^ab_{ij,kl}, Q^abc_{ij;kl,mn}, which are defined from the wavefunctions, not postulated as independent entities.

free parameters (4)
  • delta (symmetry-breaking perturbation strength) = delta = 1/2
    Added to MRW and chiral models (Eqs. 7-8) to break M_{ij}T and chiral symmetries; amplitude chosen by hand; results in Fig. 2/3 depend on it.
  • p (k_z -> p k_z momentum rescaling) = p = 1,2,3,4 in Fig. 2
    Integer rescaling scales the Hopf invariant chi; used to demonstrate 'scaling up' with topological invariant.
  • m (topological mass) = varied; phases at 1<|m|<3 (chi=1), |m|<1 (chi=-2)
    Controls TPTs in the MRW and chiral models; response changes sharply across transitions.
  • Flat-band energies (-1,0,1) = E=(-1,0,1)
    Band flattening in the three-band model to isolate eta^{3-band} by nullifying energy-derivative terms.
assumptions (7)
  • standard math Kubo/Dyson second-order response formalism
    SM Eqs. (1-7); foundational.
  • domain assumption Odd/dissipative decomposition via full symmetrization (Tsirkin-Souza)
    SM Eqs. (9-13); defines 'dissipationless' for nonlinear response.
  • domain assumption Peierls gauge substitution k_j -> k_j - w_ij sin(k_i a)/a
    SM Eq. (20), from Ref. [8]; maps strain to momentum shift.
  • domain assumption Stress operator T_ij = dH/dw_ij evaluated at w=0
    SM Sec. II; the exactness claim at w=0 is used to define T.
  • ad hoc to paper O(w^2) Hamiltonian terms can be neglected in second-order response
    Main-text Discussion and SM Sec. II; the paper itself states these terms 'survive in the w_ab -> 0 limit', making the neglect an unproven assumption.
  • domain assumption Hopf-invariant and chiral-invariant formulas remain valid under perturbations for 0<=delta<=1
    SM Sec. IV, verified numerically but not proven.
  • domain assumption Biaxial strain cannot be eliminated by coordinate choice
    Discussion, citing Ref. [75]; load-bearing for the effect being genuine.

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Cite this review

Pith. "Pith review of Nonlinear Odd Viscoelastic Effect." pith.science (2026). https://pith.science/paper/UV3MYBDT

@misc{pith2026251122706,
  author       = {Pith},
  title        = {Pith review of: Nonlinear Odd Viscoelastic Effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UV3MYBDT}},
  note         = {Machine review of arXiv:2511.22706}
}
read the original abstract

We uncover a class of nonlinear odd viscoelastic effects in three spatial dimensions. We show that these dissipationless effects arise upon combining geometric distortions in two orthogonal directions, yielding momentum flow in the third direction. We demonstrate that the effect arises from nontrivial geometric tensors in quantum states, and can be scaled up with integer topological invariants. We further show that the effect fingerprints the multiband Hilbert-space geometry of the underlying quantum states, as encoded in the nonmetricity and three-state quantum geometric tensors. Our findings unravel the role of multistate geometry in viscoelastic phenomena, paving a path for experimental observation of uncharted nonlinear odd viscoelastic responses in quantum systems.

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Reference graph

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.