Two-dimensional nonlinear dynamical response of the magnetoelectrically driven dimerized spin-$1{/}2$ chain

Wolfram Brenig

REVIEW 3 major objections 6 minor 1 cited by

Two-dimensional nonlinear dynamical response of the magnetoelectrically driven dimerized spin-$1{/}2$ chain

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read In a dimerized spin-1/2 chain, an electric-field 2D nonlinear response reads off spinon scattering rates and reveals in-gap two-spinon bound states.

desk verdict Clean XY-limit 2D nonlinear response calculation with a promising spinon scattering-rate read-off; the XXZ bound-state claim rests on an RPA with an overcomplete basis and should be treated as suggestive until cleaned up. read the letter →

arxiv 2507.17823 v1 pith:SX4I2SY2 submitted 2025-07-23 cond-mat.str-el

Wolfram Brenig This is my paper · ORCID

classification cond-mat.str-el

keywords two-dimensionalnonlinearspectroscopydimerizedspin-1/2chainspinonmagnetoelectriccouplinggalvanoelectriclinesecond-orderresponseRPAvertexcorrectiontwo-spinonboundstate

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

This paper argues that two-dimensional nonlinear optical response to electric fields can expose the fractionalized excitations of a dimerized spin-1/2 chain. It derives the second-order response for the noninteracting XY limit and shows that a sharp antidiagonal line along $\omega_1 = -\omega_2$ — the galvanoelectric line — carries a Lorentzian profile whose perpendicular width is the spinon scattering rate, information that linear-response probes cannot provide. For the XXZ version, it includes interaction corrections to the light-matter coupling via RPA vertex renormalization, and finds that the $zz$-exchange can bind pairs of spinons into an in-gap bound state. For representative parameters this bound state sits at about $0.205\,J$ and concentrates the dominant response into a small low-frequency region of the two-dimensional frequency plane. If correct, the result gives a practical way to measure spinon lifetimes and to see collective bound states in magnetoelectric 2D spectroscopy.

What carries the argument

The central object is the fully symmetrized second-order nonlinear response function $\chi_2(\omega_1,\omega_2)$, built from three polarization operators and evaluated with Jordan-Wigner fermions. Its singular antidiagonal (galvanoelectric) line, where $\omega_1 + \omega_2 = 0$, is the feature that carries the spinon lifetime: the $\sim 1/\eta$ prefactor of the free-fermion expression is regularized by promoting the broadening $\eta$ to a physical scattering rate $\Gamma$, which then sets the Lorentzian width perpendicular to the line. For the XXZ case, the new machinery is the RPA vertex correction: the $zz$-exchange is rewritten as momentum-separated particle-hole channels, collected into a $10\times 10$ coupling matrix and a bubble matrix $D(i\omega_n)$, whose poles in the spin gap are the two-spinon bound states that reshape the dressed polarization vertex and hence the 2D response.

What would settle it

Compute the second-order response with an energy-dependent self-energy instead of a constant $\Gamma$; if the antidiagonal becomes non-Lorentzian or its width no longer matches the computed spinon decay rate, the central claim fails. Experimentally, compare the antidiagonal width with a spinon lifetime obtained from thermal transport or specific-heat measurements in a candidate magnetoelectric dimerized chain.

Watch

Extended reading notes

Core claim

In the XY limit, the paper obtains an exact analytic expression for the second-order nonlinear response function $\chi_2(\omega_1,\omega_2)$ of the dimerized chain coupled to an electric field through the spin-current magnetoelectric coupling. The response is nonzero only when a dc field breaks inversion symmetry, and it contains a singular galvanoelectric line along $\omega_1 = -\omega_2$ whose real part scales as $1/\Gamma$ while its perpendicular profile is a Lorentzian of width $\Gamma$. Identifying the causal broadening $\eta$ with a physical scattering rate, the paper concludes that the perpendicular width of this line equals the spinon scattering rate. In the XXZ case, treating the $zz$-exchange in a random-phase approximation renormalizes the polarization vertices, and the resulting RPA propagator shows a two-spinon bound state splitting off from the two-spinon continuum; with the paper's parameters this bound state lies at approximately $\epsilon_B = 0.205\,J$ inside the dimerization gap. The dressed response then concentrates its dominant weight near the lines $\omega_1 = \pm\epsilon_B$, $\omega_2 = \pm\epsilon_B$, and $\omega_1 + \omega_2 = \pm\epsilon_B$, transferring spectral weight from the two-spinon continuum to low frequencies. This is the paper's central claim: magnetoelectric 2D nonlinear spectroscopy of a dimerized chain can read off fractionalized quasiparticle lifetimes and can reveal interaction-induced spinon bound states.

Load-bearing premise

The entire read-off rests on the assumption that one fixed number describes how fast the fractional half-spin excitations lose energy; if real materials have several decay channels or energy-dependent lifetimes, the measured line width will not equal that number.

Editorial extensions

If this is right

In the XY limit, the antidiagonal line provides a direct spectroscopic measurement of the spinon scattering rate; no linear response quantity of this model carries that single-particle lifetime.
The response is a dc-field effect: it vanishes without the dc field and reverses sign when the dc field reverses, so a lock-in measurement can isolate the nonlinear signal.
For XXZ coupling, sufficiently strong $zz$-exchange produces a two-spinon bound state in the gap ($\epsilon_B \sim 0.205\,J$ for the paper's parameters), and the main nonlinear response contracts into a small low-frequency region of the 2D frequency plane.
The twelve crossing points of the resonance lines $\omega_1 = \pm\epsilon_B$, $\omega_2 = \pm\epsilon_B$, $\omega_1+\omega_2 = \pm\epsilon_B$ form a fingerprint of the bound state.
The paper suggests that analogous singular lines should appear at other locations in frequency space in higher-order response functions.

Reading between the lines

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

By extension, any magnetoelectric material whose low-energy excitations map onto free fermions should show a similar antidiagonal-width read-off, since the analytic form only uses the fermionic dispersion and dipole matrix elements.
The bound-state-induced low-energy concentration suggests an experimentally testable signature: tuning the dimerization or the dc field should move the twelve crossing points, which would distinguish bound-state physics from mere multi-spinon continua.
A natural next calculation is to include a frequency-dependent spinon self-energy; if the resulting antidiagonal is non-Lorentzian, the simple constant-$\Gamma$ read-off would need revision.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read this for the XY part: it is a clean, exact free-fermion evaluation of the second-order 2D nonlinear response of a dimerized spin-1/2 chain with KNB magnetoelectric coupling, and the galvanoelectric line's perpendicular width as a read-off of spinon scattering rates is a genuinely useful idea. The analytic expression (Eq. 13) is new for this model, the GEE-line analysis is clearly explained, and the energy scales check out (the 0.604 and 2.04 J bounds are the two-spinon edges).

The XXZ vertex-correction section is a different beast. It is openly a subset of all interaction diagrams, an RPA resummation of zz-exchange in the particle-hole channel. The qualitative conclusion that zz-coupling can pull spectral weight into the dimerization gap and possibly form a two-spinon bound state is plausible and consistent with earlier field-theory and ED work (refs 73-75). But the specific implementation has a real soft spot: the four exchange eigenfunctions h1...h4 in Eq. (22) are linearly dependent (h3 and h4 are complex linear combinations of h1 and h2), so the four 'eigenvalues' in Eq. (21) are not a faithful spectral decomposition of the rank-2 kernel -v_{k-k'}. Inverting the 10x10 matrix (1 - Δ c⟨BB†⟩) in this redundant basis can generate spurious poles; the claimed in-gap bound state at ε_B≈0.205 should be checked by a numerical calculation, for example ED on a small chain.

One note: the stress-test that ran alongside this letter claims a sign error in the decomposition, but that concern does not hold up. Substituting the stated h's and ν's reproduces -v_{k-k'} exactly; the reported 'wrong sign' comes from a sign slip in the stress-test itself. The overcompleteness concern is the substantive one.

The other caveat is the physical interpretation of Γ. The paper treats it as a single, frequency-independent scattering rate, acknowledges it is a free parameter, and that is enough for a proposal. If real spinons have strongly frequency-dependent or non-Lorentzian lineshapes, the width-to-rate mapping needs refinement.

Bottom line: the XY section deserves a serious referee and is likely correct; the XXZ section is suggestive and needs a mathematical cleanup. I would send it to referees, ask for a careful check of the RPA basis and the bound-state pole, and I would cite the XY result.

Referee Report

3 major / 6 minor

Summary. The paper studies the second-order nonlinear dynamical response of a dimerized spin-1/2 chain coupled to electric fields through the inverse Dzyaloshinskii-Moriya (KNB) mechanism. In the XY limit the model is noninteracting and the response is evaluated analytically as a three-point fermion Green's function, yielding a closed form for chi_2(omega_1, omega_2). The main feature is an antidiagonal 'galvanoelectric' line along omega_1 = -omega_2, whose perpendicular width is proposed as a direct measure of the spinon scattering rate. For the XXZ case, the paper adds RPA vertex corrections from the zz exchange and reports an in-gap two-spinon bound state, leading to a claimed dramatic transfer of spectral weight to low frequencies. The central quantitative claims are the XY-lifetime read-off and the XXZ bound-state-driven spectral contraction.

Significance. The topic is timely: electric-field-driven 2D coherent spectroscopy of frustrated and low-dimensional magnets is an emerging direction, and the paper offers one of the first concrete calculations for a spin chain. The XY-limit derivation is a transparent, closed-form evaluation that can be checked directly; the identification of energy scales with the two-spinon bounds (0.604 J and 2.04 J) and the explicit symmetry discussion are useful and credible. The proposed GEE-line read-off of spinon scattering rates, if established, would be a genuinely new spectroscopic tool for fractionalized systems, since linear response only shows continua. The XXZ RPA bound-state scenario, if correct, would also be a significant qualitative prediction. However, both central claims are conditional: the read-off relies on an assumed single-parameter scattering rate Gamma, and the bound-state result rests on an overcomplete RPA channel basis. The paper is honest in stating some of these limitations, but the abstract and conclusions present the claims more strongly than the current derivation supports.

major comments (3)

[IV.A, Eqs. (21)-(24); Figs. 4 and 5] The four 'eigenfunctions' h3 = delta^{1/2}(cos k + i sin k) and h4 = delta^{1/2}(cos k - i sin k) are linear combinations of h1 = cos k and h2 = sin k, so the Gram matrix of the set {h1,h2,h3,h4} is singular and the decomposition in Eq. (21) is not a faithful spectral decomposition but an overcomplete representation. I checked the algebraic sign issue explicitly: with nu = (-2,-2,-1,1) the imaginary part of the decomposition is -2i delta sin(k-k'), which correctly reproduces -v_{k-k'}; that particular objection does not land. The load-bearing problem is that the RPA denominator in Eq. (26), D = (1 - Delta c <BB^dag>)^-1, depends on the redundant channels. Different but equivalent decompositions of the same interaction will give different resolvents, and poles of D can be generated by the null space of the bubble matrix rather than by a physical two-spinon bound state. Since the bound-state energy epsilon_B ~ 0.205 and the twelve resonances in Fig. 5 are read off from D(omega), the XXZ central claim is not yet established. Please recompute the pole position using a minimal basis (for example the two independent functions e^{ik} and e^{-ik}) or explicitly project out the null space and show that the pole position and residue are unchanged; if the result survives, the paper should present that minimal-basis calculation.
[III.A, after Eq. (14)] The main XY-limit claim, that scattering rates of fractional spinons can be read off from the perpendicular width of the antidiagonal, rests on replacing the infinitesimal eta by a single frequency- and momentum-independent Gamma. The paper states immediately after Eq. (14) that Gamma 'remains a free parameter' and 'could, in principle, be also frequency and momentum dependent', and in Sec. IV it is absorbed without a microscopic calculation of the spinon self-energy. For the noninteracting XY chain there is no intrinsic scattering, so the finite width in Fig. 2 is purely a phenomenological regulator; for the interacting chain the actual spinon lineshape is not derived. As it stands, the statement that scattering rates can be read off is a proposal rather than a derived prediction. A quantitative version requires either a microscopic estimate of Gamma (e.g., from the imaginary part of a computed spinon self-energy) or a discussion of how a frequency- and momentum-dependent self-energy would distort the Lorentzian line shape and how the width should then be interpreted.
[IV.B, Figs. 4 and 5] The numerical content of the main XXZ figures is not reproducible from the text. The figures do not list the k-mesh or number of momentum points, the integration scheme, or the value of Gamma used in the main panels; the inset of Fig. 5 mentions Gamma = 0.01, but the main panels' broadening is not given, and Fig. 2 appears to use Gamma = 0.06. Since the claimed bound-state energy epsilon_B ~ 0.205 and the twelve crossing points are obtained from numerical data rather than from a closed-form expression, these parameters are needed to check the central claim and to assess how sensitive the pole position is to the overcomplete-basis issue described above.

minor comments (6)

[Eq. (15)] The definition of v_q is missing a closing parenthesis: it should read Delta v_q = Delta(2 cos q + 2i delta sin q).
[Figs. 2 and 4] The symbol rendered as a square (e.g., '□ = 0.06' in Fig. 2 and '□ = 0.002' in Fig. 4) appears to be Gamma; the typesetting should be corrected for clarity.
[III.A, after Eq. (14)] The statement that the real part of chi_2 has a Lorentzian line shape of width Gamma perpendicular to the antidiagonal is asserted without derivation; a short analytic illustration would make the central read-off claim easier to verify.
[Eq. (26) and surrounding text] The Matsubara quantity D(i omega_n) is written with a subscript q but q -> 0 is applied immediately afterward; it would be clearer to state explicitly that all later quantities are evaluated at q = 0.
[Sec. IV.B, Fig. 5] The inset in Fig. 5(a) is described as showing the twelve fingerprints of the bound state, but at the printed scale these features are difficult to distinguish; a larger separate panel would improve the presentation.
[References] References [34] and [47] are incomplete (missing article numbers/page ranges); they should be updated before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: XY-limit 2D NRF is an analytic free-fermion result; the XXZ RPA vertex corrections are computed from the zz-exchange, and Gamma is an openly free parameter, so no fitted input is relabeled as a prediction.

full rationale

The derivation is self-contained. The XY-limit response is obtained by explicitly evaluating the two bubble diagrams of Fig. 1 for noninteracting Jordan-Wigner fermions, giving the analytic expression Eq. (13); the galvanoelectric line Eq. (14) and the perpendicular Lorentzian width in Gamma follow from the poles of that three-point function, not from curve fitting. The broadening Gamma is explicitly declared to remain a free parameter (Sec. III, after Eq. 14), so the claim that the width reads off the spinon scattering rate is a propagation of an input parameter through an exact formula, not a back-fitted prediction. The XXZ RPA denominator D=(1-Delta c <BB^dagger>)^-1 is built self-consistently from the zz-exchange vertex, with the bound-state position epsilon_B read off from the computed spectrum Im[D11(omega)]; it is not imposed to match a target. Self-citations [39] and [52] are used for consistency checks and context (e.g., 'See also a similar conclusion for the Kitaev model'), but the central Lorentzian-width statement is justified in the text as 'straightforward to show' from the derived Eq. (14), so those citations are not load-bearing. Concerns about the algebraic correctness of the separable decomposition of v_{k-k'} would be correctness risks, not circularity, because the RPA pole is an output of the stated equations rather than an equivalent restatement of their inputs.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The XY-limit claim rests on standard free-fermion response theory and is self-contained. The XXZ claim relies on the RPA approximation and the phenomenological Gamma, both of which are uncontrolled. The possible sign inconsistency in the interaction decomposition is a mechanical point that should be corrected.

free parameters (5)

Gamma (scattering rate) = 0.06 (Fig. 2), 0.002 (Fig. 4), 0.01 (Fig. 5 inset)
Phenomenological linewidth replacing the causal broadening eta in the GEE line (Eq. 14). The central claim that the perpendicular width measures the spinon scattering rate depends on the physical identity of this parameter.
E_dc (dc electric field amplitude) = 0.2 (units of J)
The paper absorbs the dipole coupling constant into the definition of E; E_dc breaks inversion symmetry and is required for a nonzero second-order response. Its value is chosen for the numerical examples, not independently measured.
Exchange dimerization delta = 0.3
Model parameter chosen for the numerical plots; not fitted to data.
KNB dimerization gamma = 0.2
Model parameter chosen for the numerical plots; not fitted to data.
Temperature T = 0.01
Chosen to be below the dimerization gap; temperature dependence is not analyzed.

assumptions (7)

standard math Jordan-Wigner transformation maps the spin chain to free spinless fermions in the XY limit
Used in Sec. II to diagonalize the xy part. Exact for one-dimensional spin-1/2.
domain assumption The KNB inverse Dzyaloshinskii-Moriya interaction is the only magnetoelectric coupling considered; exchange-striction and spin-dependent hybridization are discarded
Sec. II explicitly rules out the other mechanisms; this fixes the form of the light-matter coupling.
domain assumption Electric field is confined to the y direction, so the Jordan-Wigner string terms vanish
Sec. II: only the y-component keeps the polarization string-free.
standard math The 2D response is computed from the fully symmetrized retarded response function via Matsubara analytic continuation
Sec. III, standard nonlinear response formalism.
ad hoc to paper The GEE singularity is regularized by replacing the causal broadening eta with a physical scattering rate Gamma
Sec. III after Eq. (14). Gamma is a free parameter, not derived; central to the spectroscopic read-off claim.
domain assumption The RPA vertex corrections capture the relevant interaction effects in the two-particle limit; one-spinon self-energies are neglected
Sec. IV: the paper states a complete dressing is beyond the analytic treatment; the RPA subset is assumed sufficient for qualitative effects.
ad hoc to paper The interaction -v_{k-k'} admits the 10-channel decomposition with the given eigenvalues
Eqs. (21)-(22). This is a specific technical device introduced to make the RPA tractable. The sign of the imaginary part appears inconsistent with the stated v_q, so the decomposition needs verification.

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Cite this review

Pith. "Pith review of Two-dimensional nonlinear dynamical response of the magnetoelectrically driven dimerized spin-$1{/}2$ chain." pith.science (2026). https://pith.science/paper/SX4I2SY2

@misc{pith2026250717823,
  author       = {Pith},
  title        = {Pith review of: Two-dimensional nonlinear dynamical response of the magnetoelectrically driven dimerized spin-$1/2$ chain},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SX4I2SY2}},
  note         = {Machine review of arXiv:2507.17823}
}

abstract

A study of the dynamical two-dimensional (2D) nonlinear response of the dimerized spin-$1{/}2$ chain to external electric fields is presented. The coupling of the spin system to those fields is set to arise from the inverse Dzyaloshinskii-Moriya interaction. In the XY-limit, we provide analytical expressions for the second-order nonlinear dynamical response function. Apart from multi spinon continua, this response displays a strong antidiagonal, i.e. galvanoelectric, feature in the 2D frequency plane. This allows to read off scattering rates of the fractional spinon excitations. For the XXZ-case, we focus on the interaction-driven renormalization of the light-matter coupling by considering vertex corrections which are induced by the zz-exchange. We show this renormalization to modify the spinon joint density of states significantly, potentially allowing for the formation of in-gap bound states. As a result, the vertex corrected light-matter coupling can induce a dramatic spectral weight transfer to lower energies for the dynamical response function within the 2D frequency plane.

Figures

Figures reproduced from arXiv: 2507.17823 by the authors.

**Figure 1.** Diagrams for the 2D NRF χ2(ω1, ω2) at order O(2) in Eac(t). The solid lines carry a band index µ = 1, 2, the dots refer to the 2 × 2 polarization operator matrices pµν. I.e., the diagrams correspond to 2×8 expressions. We note that for Edc = δ = γ = 0, the quasiparticle dispersion simplifies to ϵk = cos(k) for k ∈ BZ and P turns diagonal with p11k = sin(k). III. 2D RESPONSE FUNCTION IN THE XY-LIMIT In this section w… view at source ↗

**Figure 2.** Contours of (a) the real and (b) imaginary part [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Panel (a) Vertex corrected bare diagrams of Fig. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Spectrum of the 11 matrix-element of the RPA 10 × 10 series D(ω) for a representative set of parameters, varying ∆, as listed in legend. y-axis values are scaled by arbitrary factors listed. sum of operators. In the diagonal particle representation it reads B † i,q=0 =…

**Figure 5.** Figure 5: Contours of (a) the real and (b) imaginary part [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

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Forward citations

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Reference graph

Works this paper leans on

76 extracted references · 67 canonical work pages · cited by 1 Pith paper

[1]

Panel (c) The interacting spinon particle- hole pair is generated by RPA scattering from the bare four- spinon vertex

Panel (b) The dressed polarization vertex (solid triangle) is a sum of the bare vertex (solid circle) and the bare vertex coupled to an interacting spinon particle-hole pair (solid rect- angle with circle). Panel (c) The interacting spinon particle- hole pair is generated by RPA scattering from the bare four- spinon vertex. This vertex is momentum-separat...

work page
[2]

Dzyaloshinskii, Sov

I.E. Dzyaloshinskii, Sov. Phys. JETP10, 628 (1960)

work page 1960
[3]

Fiebig, J

M. Fiebig, J. Phys. D: Appl. Phys.38, R123 (2005)

work page 2005
[4]

Tokura, S

Y. Tokura, S. Seki, and N. Nagaosa, Rep. Prog. Phys. 77, 076501 (2014)

2014
[5]

Dong, J.-M

S. Dong, J.-M. Liu, S.-W. Cheong, and Z. Ren, Advances in Physics64, 519 (2015)

work page 2015
[6]

Fiebig, T

M. Fiebig, T. Lottermoser, D. Meier, and M. Trassin, Nat Rev Mater1, 16046 (2016)

work page 2016
[7]

S. Dong, H. Xiang, and E. Dagotto, National Science Review6, 629 (2019)

work page 2019
[8]

Liang, H

X. Liang, H. Chen, and N. X. Sun, APL Materials9, 041114 (2021)

work page 2021

Show all 76 references

[9]

Liang, A

X. Liang, A. Matyushov, P. Hayes, V. Schell, C. Dong, H. Chen, Y. He, A. Will-Cole, E. Quandt, P. Martins, J. McCord, M. Medarde, S. Lanceros-Mendez, S. Van Dijken, N. X. Sun, and J. Sort, IEEE Trans. Magn.57, 1 (2021)

2021
[10]

V. G. Baryakhtar and I. E. Chupis , Sov. Phys. Solid State11, 2628 (1970)

1970
[11]

G. A. Smolenskii and I. E. Chupis, (1982). G.A. Smolen- ski and I.E. Chupis, Usp. Fiziol. Nauk137, 415 (1982); [Sov. Phys. Usp.25, 475 (1982)]

1982
[12]

A.Pimenov, A.A.Mukhin, V.Yu.Ivanov, V.D.Travkin, A.M.Balbashov, andA.Loidl, NaturePhys2, 97(2006)

2006
[13]

A. B. Sushkov, R. V. Aguilar, S. Park, S.-W. Cheong, and H. D. Drew, Phys. Rev. Lett.98, 027202 (2007)

2007
[14]

A. M. Shuvaev, V. D. Travkin, V. Yu. Ivanov, A. A. Mukhin, and A. Pimenov, Phys. Rev. Lett.104, 097202 (2010)

2010
[15]

Mochizuki, N

M. Mochizuki, N. Furukawa, and N. Nagaosa, Phys. Rev. Lett.104, 177206 (2010)

2010
[16]

Takahashi, R

Y. Takahashi, R. Shimano, Y. Kaneko, H. Murakawa, and Y. Tokura, Nature Phys8, 121 (2012)

2012
[17]

S. Tóth, B. Wehinger, K. Rolfs, T. Birol, U. Stuhr, H. Takatsu, K. Kimura, T. Kimura, H. M. Rønnow, and C. Rüegg, Nat Commun7, 13547 (2016)

2016
[18]

R. S. Fishman, S. Bordács, V. Kocsis, I. Kézsmárki, J. Viirok, U. Nagel, T. Rõõm, A. Puri, U. Zeitler, Y. Toku- naga, Y. Taguchi, and Y. Tokura, Phys. Rev. B95, 024423 (2017)

2017
[19]

Kocsis, S

V. Kocsis, S. Bordács, Y. Tokunaga, J. Viirok, L. Peedu, T. Rõõm, U. Nagel, Y. Taguchi, Y. Tokura, and I. Kézs- márki, Phys. Rev. B100, 155124 (2019)

2019
[20]

Fang and J

C. Fang and J. Hu, Europhys. Lett.82, 57005 (2008)

2008
[21]

A. B. Sushkov, M. Mostovoy, R. Valdés Aguilar, S.-W. Cheong, and H. D. Drew, J. Phys.: Condens. Matter20, 434210 (2008)

2008
[22]

H. Ueda, H. Jang, S. H. Chun, H.-D. Kim, M. Kim, S.- Y. Park, S. Finizio, N. Ortiz Hernandez, V. Ovuka, M. Savoini, T. Kimura, Y. Tanaka, A. Doll, and U. Staub, Phys. Rev. Research4, 023007 (2022)

2022
[23]

Azimi, M

M. Azimi, M. Sekania, S. K. Mishra, L. Chotorlishvili, Z. Toklikishvili, and J. Berakdar, Phys. Rev. B94, 064423 (2016)

2016
[24]

Zhang, J

Y. Zhang, J. Dai, X. Zhong, D. Zhang, G. Zhong, and J. Li, Advanced Science8, 2102488 (2021)

2021
[25]

Mukamel, Principles of Nonlinear Optical Spec- troscopy, OxfordUniversityPress, RevisedEdition, 1999

S. Mukamel, Principles of Nonlinear Optical Spec- troscopy, OxfordUniversityPress, RevisedEdition, 1999

1999
[26]

P. N. Butcher, D. Cotter, The Elements of Nonlinear Optics, Cambridge University Press, Cambridge, 1990

1990
[27]

Mukamel, Annu

S. Mukamel, Annu. Rev. Phys. Chem.51, 691 (2000)

2000
[28]

Brixner, T

T. Brixner, T. Mančal, I. V. Stiopkin, and G. R. Fleming, J. Chem. Phys.121, 4221 (2004)

2004
[29]

Cho, Chem

M. Cho, Chem. Rev.108, 1331 (2008)

2008
[30]

N. Huse, B. D. Bruner, M. L. Cowan, J. Dreyer, E. T. J. Nibbering, R. J. D. Miller, and T. Elsaesser, Phys. Rev. Lett.95, 147402 (2005)

2005
[31]

M. L. Cowan, B. D. Bruner, N. Huse, J. R. Dwyer, B. Chugh, E. T. J. Nibbering, T. Elsaesser, and R. J. D. Miller, Nature434, 199 (2005)

2005
[32]

Kuehn, K

W. Kuehn, K. Reimann, M. Woerner, and T. Elsaesser, The Journal of Chemical Physics130, 164503 (2009)

2009
[33]

Kuehn, K

W. Kuehn, K. Reimann, M. Woerner, T. Elsaesser, and R. Hey, J. Phys. Chem. B115, 5448 (2011)

2011
[34]

Suemoto, and K

J.Lu, X.Li, H.Y.Hwang, B.K.Ofori-Okai, T.Kurihara, T. Suemoto, and K. A. Nelson, Phys. Rev. Lett.118, 207204 (2017)

2017
[35]

D. A. S. Kaib, M. Möller, and R. Valentí, arXiv:2502.01746

arXiv
[36]

Wan and N

Y. Wan and N. P. Armitage, Phys. Rev. Lett.122, 257401 (2019)

2019
[37]

W. Choi, K. H. Lee, and Y. B. Kim, Phys. Rev. Lett. 124, 117205 (2020). 9

2020
[38]

M. K. Negahdari and A. Langari, Phys. Rev. B107, 134404 (2023)

2023
[39]

Qiang, V

Y. Qiang, V. L. Quito, T. V. Trevisan, and P. P. Orth, Phys. Rev. Lett.133, 126505 (2024)

2024
[40]

Brenig and O

W. Brenig and O. Krupnitska, J. Phys.: Condens. Matter 36, 505806 (2024)

2024
[41]

Z.-L. Li, M. Oshikawa, and Y. Wan, Phys. Rev. X11, 031035 (2021)

2021
[42]

G. Sim, J. Knolle, and F. Pollmann, Phys. Rev. B107, L100404 (2023)

2023
[43]

Q. Gao, Y. Liu, H. Liao, and Y. Wan, Phys. Rev. B107, 165121 (2023)

2023
[44]

Hart and R

O. Hart and R. Nandkishore, Phys. Rev. B107, (2023)

2023
[45]

G. Sim, F. Pollmann, and J. Knolle, Phys. Rev. B108, 134423 (2023)

2023
[46]

Li and Y

Z.-L. Li and Y. Wan, Phys. Rev. B108, 165151 (2023)

2023
[47]

Potts, R

M. Potts, R. Moessner, and O. Benton, Phys. Rev. B 109, 104435 (2024)

2024
[48]

Watanabe, S

Y. Watanabe, S. Trebst, and C. Hickey, Phys. Rev. B 110, (2024)

2024
[49]

R. M. Nandkishore, W. Choi, and Y. B. Kim, Phys. Rev. Research3, 013254 (2021)

2021
[50]

Mahmood, D

F. Mahmood, D. Chaudhuri, S. Gopalakrishnan, R. Nandkishore, and N. P. Armitage, Nat. Phys.17, 627 (2021)

2021
[51]

Kappl, F

P. Kappl, F. Krien, C. Watzenböck, and K. Held, Phys. Rev. B107, 205108 (2023)

2023
[52]

Srivastava, S

A. Srivastava, S. Birnkammer, G. Sim, M. Knap, and J. Knolle, arXiv:2502.17554

arXiv
[53]

Wolfram Brenig, arXiv:2504.07177

arXiv
[54]

C. Jia, S. Onoda, N. Nagaosa, and J. H. Han, Phys. Rev. B76, 144424 (2007)

2007
[55]

Katsura, M

H. Katsura, M. Sato, T. Furuta, and N. Nagaosa, Phys. Rev. Lett.103, 177402 (2009)

2009
[56]

Katsura, N

H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev. Lett.95, 057205 (2005)

2005
[57]

Brockmann, A

M. Brockmann, A. Klümper, and V. Ohanyan, Phys. Rev. B87, 054407 (2013)

2013
[58]

You, G.-H

W.-L. You, G.-H. Liu, P. Horsch, and A. M. Oleś, Phys. Rev. B90, 094413 (2014)

2014
[59]

Menchyshyn, V

O. Menchyshyn, V. Ohanyan, T. Verkholyak, T. Krokhmalskii, and O. Derzhko, Phys. Rev. B92, 184427 (2015)

2015
[60]

Thakur and P

P. Thakur and P. Durganandini, Phys. Rev. B97, 064413 (2018)

2018
[61]

Baran, V

O. Baran, V. Ohanyan, and T. Verkholyak, Phys. Rev. B98, 064415 (2018)

2018
[62]

Yi, W.-L

T.-C. Yi, W.-L. You, N. Wu, and A. M. Oleś, Phys. Rev. B100, 024423 (2019)

2019
[63]

Strečka, L

J. Strečka, L. Gálisová, and T. Verkholyak, Phys. Rev. E 101, 012103 (2020)

2020
[64]

Richter, V

J. Richter, V. Ohanyan, J. Schulenburg, and J. Schnack, Phys. Rev. B105, 054420 (2022)

2022
[65]

W. A. B. Evans, Proc. Phys. Soc.88, 723 (1966)

1966
[66]

Rostami, M

H. Rostami, M. I. Katsnelson, G. Vignale, and M. Polini, Annals of Physics431, 168523 (2021)

2021
[67]

O. A. Aktsipetrov, A. A. Fedyanin, E. D. Mishina, A. N. Rubtsov, C. W. van Hasselt, M. A. C. Devillers, and Th. Rasing, Phys. Rev. B54, 1825 (1996)

1996
[68]

A. Y. Bykov, T. V. Murzina, M. G. Rybin, and E. D. Obraztsova, Phys. Rev. B85, 121413 (2012)

2012
[69]

D. E. Parker, T. Morimoto, J. Orenstein, and J. E. Moore, Phys. Rev. B99, 045121 (2019)

2019
[70]

J. E. Sipe and A. I. Shkrebtii, Phys. Rev. B61, 5337 (2000)

2000
[71]

R. Fei, W. Song, and L. Yang, Phys. Rev. B102, 035440 (2020)

2020
[72]

Ishizuka and M

H. Ishizuka and M. Sato, Phys. Rev. Lett.129, 107201 (2022)

2022
[73]

A. Raj, S. Chaudhary, and G. A. Fiete, Phys. Rev. Re- search6, 013048 (2024)

2024
[74]

G. S. Uhrig and H. J. Schulz, Phys. Rev. B54, R9624 (1996); [Erratum: Phys. Rev. B58, 2900 (1998)]

1996
[75]

Fledderjohann and C

A. Fledderjohann and C. Gros, Europhys. Lett.37, 189 (1997)

1997
[76]

Augier, D

D. Augier, D. Poilblanc, S. Haas, A. Delia, and E. Dagotto, Phys. Rev. B56, R5732 (1997)

1997

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Panel (c) The interacting spinon particle- hole pair is generated by RPA scattering from the bare four- spinon vertex

Panel (b) The dressed polarization vertex (solid triangle) is a sum of the bare vertex (solid circle) and the bare vertex coupled to an interacting spinon particle-hole pair (solid rect- angle with circle). Panel (c) The interacting spinon particle- hole pair is generated by RPA scattering from the bare four- spinon vertex. This vertex is momentum-separat...

work page

[2] [2]

Dzyaloshinskii, Sov

I.E. Dzyaloshinskii, Sov. Phys. JETP10, 628 (1960)

work page 1960

[3] [3]

Fiebig, J

M. Fiebig, J. Phys. D: Appl. Phys.38, R123 (2005)

work page 2005

[4] [4]

Tokura, S

Y. Tokura, S. Seki, and N. Nagaosa, Rep. Prog. Phys. 77, 076501 (2014)

2014

[5] [5]

Dong, J.-M

S. Dong, J.-M. Liu, S.-W. Cheong, and Z. Ren, Advances in Physics64, 519 (2015)

work page 2015

[6] [6]

Fiebig, T

M. Fiebig, T. Lottermoser, D. Meier, and M. Trassin, Nat Rev Mater1, 16046 (2016)

work page 2016

[7] [7]

S. Dong, H. Xiang, and E. Dagotto, National Science Review6, 629 (2019)

work page 2019

[8] [8]

Liang, H

X. Liang, H. Chen, and N. X. Sun, APL Materials9, 041114 (2021)

work page 2021