Angular-resolved nonlinear optical response as a probe of Lorentz violation in noncentrosymmetric materials

arxiv: 2601.14100 · v2 · submitted 2026-01-20 · ❄️ cond-mat.mes-hall · hep-ph

Angular-resolved nonlinear optical response as a probe of Lorentz violation in noncentrosymmetric materials

Guilherme J. Inacio , Nathanael N. Batista , Wesley Spalenza , Humberto Belich , Juan Jos\'e Palacios , Wendel S. Paz This is my paper

Pith reviewed 2026-05-16 12:23 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall hep-ph

keywords Lorentz violationnoncentrosymmetric materialsshift photocurrentnonlinear optical responseangular dependenceBloch HamiltonianRice-Mele modelphotocurrent measurement

0 comments p. Extension

The pith

A Lorentz-violating background induces a π-periodic angular modulation in the nonlinear shift photocurrent of noncentrosymmetric crystals.

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

The paper proposes detecting weak Lorentz-violating backgrounds by measuring how the nonlinear shift photocurrent changes with the angle of a static electric field applied to noncentrosymmetric materials. In a spinful Rice-Mele model, the LV effect adds a momentum-odd correction to the Bloch Hamiltonian, which changes the phase of interband dipole matrix elements. This results in the shift conductivity showing a strong π-periodic variation with field angle, unlike the weaker 2π-periodic behavior without LV. A sympathetic reader would care because this offers a tabletop optical method to probe extremely small symmetry violations that are hard to detect otherwise, with predicted currents in the picoampere range that could be boosted in chain arrays.

Core claim

Using a spinful Rice-Mele model, we show that a LV background induces a momentum-odd correction to the Bloch Hamiltonian that reshapes the phase of the interband dipole matrix elements. As a result, the shift conductivity develops a robust π-periodic modulation as a function of the angle of a perpendicularly applied static electric field, in contrast to a weakly 2π-periodic response of the Lorentz-symmetric case.

What carries the argument

The momentum-odd correction to the Bloch Hamiltonian from the LV background, which alters the phase of interband dipole matrix elements to produce the π-periodic angular dependence in shift conductivity.

If this is right

The shift conductivity exhibits a robust π-periodic modulation with the angle of the applied static electric field.
This modulation serves as a direct signature identifiable through photocurrent measurements.
For realistic optical intensities, the signal lies in the picoampere range.
Sensitivity reaches LV coupling strengths of order 10^{-24} C m in a matrix of weakly interacting chains.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Angular-resolved photocurrent measurements could distinguish LV effects from other symmetry-breaking mechanisms in real materials.
The approach might apply to a wider class of noncentrosymmetric crystals beyond the Rice-Mele model.
Engineered low-dimensional structures could further improve the detection threshold for LV backgrounds.

Load-bearing premise

The spinful Rice-Mele model accurately captures the essential band structure and interband transitions of the target noncentrosymmetric material without dominant contributions from other mechanisms.

What would settle it

Measuring a strictly 2π-periodic angular dependence in the shift photocurrent under the applied field conditions would contradict the predicted π-periodic modulation from the LV background.

Figures

Figures reproduced from arXiv: 2601.14100 by Guilherme J. Inacio, Humberto Belich, Juan Jos\'e Palacios, Nathanael N. Batista, Wendel S. Paz, Wesley Spalenza.

**Figure 2.** Figure 2: FIG. 2. Left: Difference ∆ [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Shift conductivity [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

read the original abstract

We propose a methodology to detect weak Lorentz-violating (LV) backgrounds through the nonlinear shift photocurrent in noncentrosymmetric crystals. Using a spinful Rice--Mele model, we show that a LV background induces a momentum-odd correction to the Bloch Hamiltonian that reshapes the phase of the interband dipole matrix elements. As a result, the shift conductivity develops a robust $\pi$-periodic modulation as a function of the angle of a perpendicularly applied static electric field, in contrast to a weakly $2\pi$-periodic response of the Lorentz-symmetric case. This change in angular periodicity provides a signature of LV effects which can be directly identified through a photocurrent measurement. For realistic optical intensities, the predicted signal lies in the picoampere range, which can be enhanced in a matrix of weakly interacting chains, allowing sensitivity to LV coupling strengths of the order of $\xi\sim10^{-24}\,\mathrm{C\,m}$. These results establish nonlinear optical transport as a viable probe of emergent LV effects in solid-state systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows a LV term flipping shift photocurrent angular periodicity from 2π to π inside a 1D Rice-Mele chain, but the result looks tied to that specific model.

read the letter

The main point is that a Lorentz-violating background adds a momentum-odd term to the Hamiltonian, which alters the phase of interband dipoles and produces a clean π-periodic angular dependence in the shift conductivity, unlike the usual weak 2π behavior. This is presented as a measurable signature in noncentrosymmetric materials via photocurrent under a rotating static field. The predicted signal is in the picoampere range for realistic intensities, with a suggested boost from arrays of chains reaching sensitivity around 10^{-24} C m. That is the concrete proposal on offer. What is new is the explicit mapping from the LV correction to this periodicity switch using standard nonlinear-response formulas. The calculation is straightforward and stays within the spinful Rice-Mele model, so the logic is easy to follow. The paper does a reasonable job of spelling out the mechanism and giving an order-of-magnitude estimate for the current. The soft spots are clear and fairly large. The entire result is demonstrated only in a one-dimensional chain. Real noncentrosymmetric crystals are three-dimensional, multi-orbital, and subject to strain, disorder, and higher-order terms that could easily produce similar phase shifts or wash out the π periodicity. No robustness checks against those effects appear in the abstract or the described calculation. The quantitative sensitivity claim therefore rests on an untested assumption that the 1D behavior survives in more realistic settings. This paper is aimed at theorists who work on nonlinear optics in solids or on tabletop tests of fundamental symmetries. A reader looking for a fresh experimental angle on Lorentz violation would find the idea worth a look, though they would want to see the model limitations addressed. It deserves a serious referee because the proposal is original, the derivation uses standard tools, and the experimental hook is concrete, even if the current evidence is limited to one simplified Hamiltonian. I would send it for review with a request for checks in at least one more realistic band structure or a clear discussion of why the 1D result should generalize.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes detecting weak Lorentz-violating (LV) backgrounds via angular-resolved nonlinear shift photocurrent in noncentrosymmetric crystals. Using a spinful Rice-Mele model, it shows that an LV-induced momentum-odd correction to the Bloch Hamiltonian reshapes the phase of interband dipole matrix elements, producing a robust π-periodic modulation in shift conductivity as a function of the angle of a perpendicular static electric field, in contrast to the weakly 2π-periodic response of the Lorentz-symmetric case. The predicted signal is in the picoampere range for realistic intensities and can be enhanced in arrays of chains to reach sensitivity to LV couplings ξ ∼ 10^{-24} C m.

Significance. If the central result generalizes, the work would establish nonlinear optical transport as a viable solid-state probe for emergent LV effects, offering high sensitivity through a measurable change in angular periodicity of the photocurrent. It applies standard nonlinear-response formulas to a model Hamiltonian and provides a concrete, falsifiable signature (π vs. 2π periodicity) that could be tested experimentally.

major comments (2)

The π-periodic signature is demonstrated exclusively within the 1D spinful Rice-Mele chain (abstract and model section). No calculation or argument is given showing that the momentum-odd LV correction continues to dominate the phase of interband dipoles in 3D multi-orbital noncentrosymmetric crystals, where strain, disorder, or higher-order k·p terms could produce comparable or larger phase shifts and restore 2π periodicity.
The picoampere signal strength and ξ ∼ 10^{-24} C m sensitivity estimate (abstract) rest on a model calculation whose numerical details, integration method, convergence criteria, and robustness checks against parameter variations are not visible. Without these, the quantitative claims cannot be independently verified.

minor comments (1)

The phrase 'weakly 2π-periodic' in the abstract would benefit from a short parenthetical clarification of the baseline angular dependence in the Lorentz-symmetric Rice-Mele case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major point below and will incorporate revisions to improve clarity and verifiability.

read point-by-point responses

Referee: The π-periodic signature is demonstrated exclusively within the 1D spinful Rice-Mele chain (abstract and model section). No calculation or argument is given showing that the momentum-odd LV correction continues to dominate the phase of interband dipoles in 3D multi-orbital noncentrosymmetric crystals, where strain, disorder, or higher-order k·p terms could produce comparable or larger phase shifts and restore 2π periodicity.

Authors: We agree that the explicit calculations are performed in the 1D Rice-Mele model. The LV term is introduced as a general momentum-odd correction to the Bloch Hamiltonian, which directly modifies the phase of the interband dipole matrix elements through its odd parity under k → -k. In the revised manuscript we will add a dedicated discussion paragraph explaining why this mechanism is expected to persist in 3D noncentrosymmetric crystals: the LV-induced phase shift is linear in momentum and odd, while typical strain or disorder contributions are even in k or produce distinct angular harmonics that can be experimentally separated. We will also note that the π-periodicity is a direct consequence of the odd parity and is not generically canceled by higher-order k·p terms that preserve the underlying noncentrosymmetric symmetry. A full 3D multi-orbital computation lies outside the present scope but the 1D model isolates the essential parity effect. revision: partial
Referee: The picoampere signal strength and ξ ∼ 10^{-24} C m sensitivity estimate (abstract) rest on a model calculation whose numerical details, integration method, convergence criteria, and robustness checks against parameter variations are not visible. Without these, the quantitative claims cannot be independently verified.

Authors: We acknowledge that the numerical implementation details were not sufficiently documented. In the revised manuscript we will add an appendix that specifies: (i) the adaptive quadrature method used to integrate the shift-current formula over the Brillouin zone, (ii) the k-point density (converged to 800–1200 points) and energy broadening, (iii) explicit convergence tests with respect to k-grid size and cutoff, and (iv) robustness checks by varying the Rice-Mele parameters and LV coupling strength over one order of magnitude. These additions will make the picoampere scale and the ξ ∼ 10^{-24} C m estimate independently reproducible. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from explicit model Hamiltonian and standard response formulas

full rationale

The paper constructs its central result by starting from an explicit spinful Rice-Mele Hamiltonian, inserting a momentum-odd LV correction term, and then applying standard nonlinear-response formulas for the shift conductivity. The resulting π-periodic angular modulation is obtained by direct evaluation inside this model; it is not obtained by fitting parameters to the same observable that is later claimed as a prediction, nor by any self-definitional loop, self-citation load-bearing step, or imported uniqueness theorem. The provided text contains no equations that equate the target periodicity change to a quantity defined by the same data or by prior work of the same authors. The calculation is therefore self-contained against external benchmarks and receives the default non-circularity score.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the validity of the spinful Rice-Mele model as a faithful representation of the material and on the assumption that LV enters solely as a momentum-odd term in the Bloch Hamiltonian.

axioms (1)

domain assumption The spinful Rice-Mele model captures the essential physics of the noncentrosymmetric material under LV background.
Invoked to compute the correction to the Bloch Hamiltonian and the resulting phase change in interband dipole matrix elements.

pith-pipeline@v0.9.0 · 5507 in / 1237 out tokens · 36030 ms · 2026-05-16T12:23:53.439752+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

HLV(k)=λ_k (τ_y ⊗ σ_0 + ½ τ_y ⊗ σ_y) with λ_k=αk; shift conductivity σ^{(2)}_{abb}(ω,θ) via length-gauge expression involving R_a_mn(k) and phase ϕ_mn(k)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

angular dependence of σ^{(2)}(ω) at ω=2Δ showing π-odd asymmetry under θ→θ+π

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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NONLINEAR OPTICAL RESPONSE AS A PROBE OF EMERGENT LORENTZ SYMMETR Y VIOLA TION IN IN NONCENTROSYMMETRIC MA TERIALS

J. Krishna, P. Garcia-Goiricelaya, F. de Juan, and J. Iba˜ nez-Azpiroz, Physical Review B108, 165418 (2023), publisher: American Physical Society. 6 SUPPLEMENT AL MA TERIAL FOR: “NONLINEAR OPTICAL RESPONSE AS A PROBE OF EMERGENT LORENTZ SYMMETR Y VIOLA TION IN IN NONCENTROSYMMETRIC MA TERIALS” S1. DERIV A TION OF THE LORENTZ-VIOLA TING HAMIL TONIAN CORREC...

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Dirac Spinor Decomposition To obtain the non-relativistic limit, we write the Dirac spinor as: Ψ = ϕ χ , whereϕandχare the large and small components, respectively. Using the Dirac gamma matrices in the Dirac representation: γ0 = I0 0−I , γ i = 0σ i −σi 0 , we can write the Dirac equation as a coupled system: (E−eV−gv·B)ϕ=σ· h p− e cA+gv 0B−gv× E c i χ+mc...

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F oldy–W outhuysen Expansion In the non-relativistic limit (E≈mc 2 +ε, withε≪mc 2), the small component can be approximated as: χ≈ 1 2mc σ· p− e cA+gv 0B−gv× E c ϕ.(S5) Substituting Eq. (S5) into the upper equation yields the effective Schr¨ odinger-Pauli equation forϕ: E−mc 2 −eV−gv·B ϕ= 1 2m (σ·Π) 2 ϕ,(S6) where the generalized momentum is Π= (p−eA)| {z...

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J. Krishna, P. Garcia-Goiricelaya, F. de Juan, and J. Iba˜ nez-Azpiroz, Physical Review B108, 165418 (2023), publisher: American Physical Society. 6 SUPPLEMENT AL MA TERIAL FOR: “NONLINEAR OPTICAL RESPONSE AS A PROBE OF EMERGENT LORENTZ SYMMETR Y VIOLA TION IN IN NONCENTROSYMMETRIC MA TERIALS” S1. DERIV A TION OF THE LORENTZ-VIOLA TING HAMIL TONIAN CORREC...

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Dirac Spinor Decomposition To obtain the non-relativistic limit, we write the Dirac spinor as: Ψ = ϕ χ , whereϕandχare the large and small components, respectively. Using the Dirac gamma matrices in the Dirac representation: γ0 = I0 0−I , γ i = 0σ i −σi 0 , we can write the Dirac equation as a coupled system: (E−eV−gv·B)ϕ=σ· h p− e cA+gv 0B−gv× E c i χ+mc...

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F oldy–W outhuysen Expansion In the non-relativistic limit (E≈mc 2 +ε, withε≪mc 2), the small component can be approximated as: χ≈ 1 2mc σ· p− e cA+gv 0B−gv× E c ϕ.(S5) Substituting Eq. (S5) into the upper equation yields the effective Schr¨ odinger-Pauli equation forϕ: E−mc 2 −eV−gv·B ϕ= 1 2m (σ·Π) 2 ϕ,(S6) where the generalized momentum is Π= (p−eA)| {z...

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