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arxiv: 2506.17386 · v2 · submitted 2025-06-20 · ❄️ cond-mat.mes-hall · cond-mat.mtrl-sci· cond-mat.str-el

Quantum Geometric Origin of the Intrinsic Nonlinear Hall Effect

Yannis Ulrich , Johannes Mitscherling , Laura Classen , Andreas P. Schnyder This is my paper

Pith reviewed 2026-05-19 08:02 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.mtrl-scicond-mat.str-el
keywords nonlinear Hall effectquantum metric dipoleBerry curvature dipoleintraband quantum geometrytopological antiferromagnetsmultiband systemstime-reversal symmetry breaking
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The pith

An intraband quantum metric dipole supplies a lifetime-independent term to the intrinsic nonlinear Hall effect in broken time-reversal systems.

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

The paper decomposes the intrinsic second-order nonlinear Hall effect of a generic multiband system into quantum-geometric contributions with a fully quantum-mechanical projector-based formalism. Expanding the nonlinear conductivity in powers of quasiparticle lifetime recovers the established Berry curvature dipole at order tau and addresses earlier discrepancies on the interband quantum metric dipole at order tau zero. The analysis uncovers an extra tau-zero contribution fixed by the intraband quantum metric dipole that originates from virtual interband transitions captured only in the complete quantum treatment. This intraband term is generically nonzero when time-reversal symmetry is broken and can be isolated by symmetry; explicit calculations on low-energy models show it becomes especially large in gapped Dirac cones inside antiferromagnets.

Core claim

A projector-based quantum treatment of the nonlinear conductivity decomposes the intrinsic NLHE into the Berry curvature dipole at order tau, the interband quantum metric dipole at order tau zero, and a new intraband quantum metric dipole also at order tau zero. The intraband term arises from additional virtual interband transitions and remains finite in systems that break time-reversal symmetry, as shown analytically for topological band crossings and through symmetry classification of magnetic space groups that flags candidate materials including Yb3Pt4, CuMnAs, CoNb3S6 and MnNb3S6.

What carries the argument

The intraband quantum metric dipole, which captures extra virtual interband transitions in the quantum-geometric decomposition of the nonlinear Hall conductivity.

Load-bearing premise

The expansion of the nonlinear conductivity in powers of quasiparticle lifetime tau remains valid and cleanly separates into distinct geometric contributions without higher-order mixing or scattering corrections.

What would settle it

A measurement of finite nonlinear Hall conductivity that stays constant as scattering rate approaches zero in a material where Berry curvature dipole and interband quantum metric dipole vanish by symmetry yet time-reversal symmetry is broken.

Figures

Figures reproduced from arXiv: 2506.17386 by Andreas P. Schnyder, Johannes Mitscherling, Laura Classen, Yannis Ulrich.

Figure 1
Figure 1. Figure 1: FIG. 1. The (a) inter- and (b) intraQMD contributions before [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

We decompose the intrinsic second-order nonlinear Hall effect (NLHE) of a generic multiband system into its quantum-geometric contributions within a fully quantum-mechanical, projector-based formalism. By expanding the nonlinear conductivity in powers of the quasiparticle lifetime $\tau$, we recover the established Berry curvature dipole at order $\tau$ and clarify discrepancies in previous literature concerning the (interband) quantum metric dipole (or Berry curvature polarizability) contribution at order $\tau^0\textrm{.}$ Crucially, our method reveals an additional contribution at order $\tau^0$, determined by the {\it intraband} quantum metric dipole (intraQMD), arising from additional virtual interband transitions captured within the fully quantum-mechanical treatment. The intraQMD contribution is generically nonzero in systems with broken time-reversal symmetry and can be distinguished from other geometric contributions by symmetry. Analytical results for low-energy models of topological band crossings, which are hotspots of quantum geometry, demonstrate how band topology influences each contribution. In particular, the intraQMD contribution is especially large in gapped Dirac cones in antiferromagnets. Through a comprehensive symmetry classification of all magnetic space groups, we identify several candidate materials that are expected to exhibit large intrinsic NLHE, including the topological antiferromagnets Yb$_3$Pt$_4$, CuMnAs, and CoNb$_3$S$_6$, as well as the nodal-plane material MnNb$_3$S$_6$.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a projector-based, fully quantum-mechanical formalism to decompose the intrinsic second-order nonlinear Hall conductivity of generic multiband systems into quantum-geometric contributions. Expanding the conductivity in powers of the quasiparticle lifetime τ recovers the Berry curvature dipole at order τ, clarifies the interband quantum metric dipole at order τ^0, and identifies a new intraband quantum metric dipole (intraQMD) contribution at τ^0 arising from virtual interband transitions. The intraQMD term is nonzero when time-reversal symmetry is broken, is analyzed analytically in low-energy models of topological band crossings, and is used to classify magnetic space groups, yielding candidate materials such as Yb₃Pt₄, CuMnAs, CoNb₃S₆, and MnNb₃S₆.

Significance. If the decomposition is robust, the work clarifies longstanding discrepancies in the literature on quantum-metric contributions to the nonlinear Hall effect and introduces a previously overlooked intraband geometric term that can be large near topological crossings in antiferromagnets. The analytic results on low-energy models and the exhaustive symmetry classification of magnetic space groups constitute concrete, falsifiable predictions that can guide experiments. The parameter-free character of the geometric decomposition in the projector formalism is a notable strength.

major comments (1)
  1. [Section deriving the nonlinear conductivity and the τ-expansion] The central separation of the nonlinear conductivity into cleanly isolated powers of τ (in particular the isolation of the intraQMD term at order τ^0) is performed under a constant-τ relaxation-time approximation. In multiband systems, momentum-dependent or interband scattering processes can generate self-energy or vertex corrections that mix different powers of τ. Without an explicit demonstration that such mixing is absent or higher-order, the load-bearing claim that the intraQMD contribution is unambiguously identified at τ^0 remains at risk. This issue should be addressed by either extending the derivation to include a more general scattering kernel or by providing numerical checks against known limits in the section deriving the conductivity.
minor comments (2)
  1. [Symmetry classification section] The symmetry classification of magnetic space groups is comprehensive, but a compact table listing the allowed NLHE contributions (Berry curvature dipole, interband QMD, intraQMD) for each relevant class would improve readability and allow readers to quickly identify the symmetry conditions under which the intraQMD term dominates.
  2. [Low-energy model analysis] In the low-energy model calculations, the explicit expressions for each geometric contribution (e.g., intraQMD) should be compared directly with the corresponding expressions obtained from the full projector formalism to make the connection between the analytic results and the general decomposition transparent.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comment on the relaxation-time approximation. We address this point directly below and will incorporate clarifications into the revised version.

read point-by-point responses

  1. Referee: [Section deriving the nonlinear conductivity and the τ-expansion] The central separation of the nonlinear conductivity into cleanly isolated powers of τ (in particular the isolation of the intraQMD term at order τ^0) is performed under a constant-τ relaxation-time approximation. In multiband systems, momentum-dependent or interband scattering processes can generate self-energy or vertex corrections that mix different powers of τ. Without an explicit demonstration that such mixing is absent or higher-order, the load-bearing claim that the intraQMD contribution is unambiguously identified at τ^0 remains at risk. This issue should be addressed by either extending the derivation to include a more general scattering kernel or by providing numerical checks against known limits in the section deriving the conductivity.

    Authors: We thank the referee for highlighting this subtlety. Our derivation is performed within the standard constant-τ relaxation-time approximation, which is widely employed in transport theory to isolate intrinsic geometric contributions and yields a well-defined power series in τ. Within this controlled approximation the intraQMD term appears unambiguously at order τ^0. While momentum-dependent or interband scattering can in principle introduce mixing, such corrections arise at higher order in the disorder strength and lie outside the leading intrinsic response we target. To address the concern we will add an explicit discussion of the approximation's scope in the revised manuscript and include a numerical verification on a minimal two-band model confirming that the τ^0 separation remains intact under constant τ. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained first-principles expansion

full rationale

The paper derives the decomposition of the intrinsic nonlinear Hall conductivity via a projector-based quantum-mechanical formalism followed by a direct perturbative expansion in powers of the quasiparticle lifetime τ. This recovers the known Berry curvature dipole at order τ and isolates both interband and intraband quantum metric dipole contributions at order τ^0 from the same conductivity expression without introducing fitted parameters, self-referential definitions, or load-bearing self-citations. The symmetry classification of magnetic space groups and the analysis of low-energy models (Dirac cones, antiferromagnets) follow from standard group theory and explicit model calculations. No step reduces the claimed intraQMD term to an input by construction; the separation into geometric contributions is presented as a consequence of the fully quantum treatment rather than an ansatz or renaming of prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on standard quantum mechanics and band theory assumptions plus the validity of the lifetime expansion; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Quasiparticle lifetime τ expansion separates geometric contributions without higher-order scattering mixing
    Invoked when expanding nonlinear conductivity in powers of τ to isolate τ^0 terms.
  • domain assumption Projector-based formalism captures all virtual interband transitions for the intraband quantum metric dipole
    Central to revealing the additional intraQMD contribution.

pith-pipeline@v0.9.0 · 5806 in / 1320 out tokens · 35947 ms · 2026-05-19T08:02:52.877994+00:00 · methodology

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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/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
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    Relation between the paper passage and the cited Recognition theorem.

    We perform a rigorous quantum geometrical decomposition of the second-order nonlinear Hall effect (NLHE). We find that the NLHE originates from three distinct quantum geometrical terms: the Berry curvature dipole (BCD) and the interband quantum-metric dipole (interQMD) ... and in addition an intraband term proportional to the momentum derivative of the quantum metric. This third term, which we refer to as the 'intraband quantum metric dipole' (intraQMD)

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supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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Forward citations

Cited by 2 Pith papers

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  1. Nonreciprocal current induced by dissipation in time-reversal symmetric systems

    cond-mat.mes-hall 2026-04 unverdicted novelty 7.0

    Dissipation induces nonreciprocal current in time-reversal symmetric noncentrosymmetric systems via interband processes, inversely proportional to lifetime and linked to the shift vector.

  2. Minimal Hamiltonian deformations as bulk probes of effective non-Hermiticity in Dirac materials

    cond-mat.mes-hall 2026-02 unverdicted novelty 6.0

    Minimal pseudo-Lorentz-symmetry-breaking Hamiltonian deformations act as bulk probes that separate renormalizable observables from those carrying irreducible non-Hermitian structure in two-dimensional Dirac semimetals...

Reference graph

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    To be concrete, the projectors take the form ˆPn(k) =P α |uα n(k)⟩⟨uα n(k)| with Bloch functions |uα n(k)⟩, where α spans the possibly degenerate eigenspace of eigenvalue En

    Insertion of band projectors We employ the fact that the Bloch Hamiltonian can be decomposed in terms of its band dispersionEn(k) and the correspond- ing projection ˆPn(k) to the eigenspace, ˆH(k) = X n En(k) ˆPn(k) , (S2) where we consider orthogonal projectors ˆPn(k) ˆPm(k) = δnm ˆPn(k). To be concrete, the projectors take the form ˆPn(k) =P α |uα n(k)⟩...

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    The single- and multi-band quantum geometric tensor are defined as Qab n = tr ˆPn(∂a ˆPn)(∂b ˆPn) , (S9) Qab mn = tr ˆPn(∂a ˆPm)(∂b ˆPn)

    Summary of quantum geometric invariants We briefly summarize and set the notation for the quantum geometric quantities in their projector form that we will use hereafter [6, 48]. The single- and multi-band quantum geometric tensor are defined as Qab n = tr ˆPn(∂a ˆPn)(∂b ˆPn) , (S9) Qab mn = tr ˆPn(∂a ˆPm)(∂b ˆPn) . (S10) The (two-band) quantum geometric ...

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    Quantum geometric decomposition We now turn to decomposing the two types of traces appearing in Eq. (S8). The first derivative of the Bloch Hamiltonian decomposes as [48] ˆPm ∂a ˆH ˆPn = δmn Ea n ˆPn − ϵmn ˆPm(∂a ˆPn) ˆPn (S13) with Ea n = ∂aEn and ϵmn = Em − En. Moreover, ˆPm(∂a ˆPn) ˆPn = − ˆPm(∂a ˆPm) ˆPn. Using this decomposition, the trace involving ...

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  60. [60]

    (S42) By integrating by parts in momentum, we may combine these contributions

    Contribution A: trivial geometry tr [ ˆPn] All three traces contribute a diagonal (in the band indices) term: A = Z k X n unnn 2 Ea nEb nEc ntr[ ˆPn] + X n unnn 2 Ea nEc nEb ntr[ ˆPn] + X n wnnEa nEbc n tr[ ˆPn] ! . (S42) By integrating by parts in momentum, we may combine these contributions. The main technique is to use momentum derivatives of the energ...

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  61. [61]

    Contribution B: quantum geometric tensors Qab nm and Qac nm Again all traces contribute a term proportional to Qab nm (or equivalently to Qac nm): B = Z k − X n̸=m wnmϵnmϵc nmQab nm − X n̸=m umnm 2 Ec m + umnn 2 Ec n ϵ2 nmQab nm ! + (b ↔ c) . (S48) It is convenient to separate Qab nm into its symmetric and antisymmetric part in n, m, which also correspond...

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  62. [62]

    Collecting all coefficients yields C = Z k X n̸=m 2wnnϵnmEa nQbc (nm) − X n̸=m ummn 2 Ea m + unnm 2 Ea n ϵ2 nmQbc nm !

    Contribution C: quantum geometric tensor Qbc nm The last quantum geometric tensor contribution is Qbc nm. Collecting all coefficients yields C = Z k X n̸=m 2wnnϵnmEa nQbc (nm) − X n̸=m ummn 2 Ea m + unnm 2 Ea n ϵ2 nmQbc nm ! . (S54) As previously, we separate into symmetric and antisymmetric components: C = Z k X n̸=m 2wnnϵnmEa nQbc (nm) − X n̸=m unnm 2 E...

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  63. [63]

    Contribution D: connections C b;ac nm , C c;ab nm The total contribution from connections is D = Z k − X m,n 1 2 wnmϵ2 nmC b;ac nm − X m,n 1 2 wnmϵ2 nmC c;ab nm ! . (S58)

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  64. [64]

    (S60) We indicate summing only over different band indices with the notation(l, m, n) (i.e

    Contribution E: three-band traces tr [ea mneb nlec lm], tr[ea mnec nleb lm] The contribution to the three-band traces tr[ea mneb nlec lm], tr[ea mnec nleb lm] is given by E = Z k X m,n i 2 wnmϵnm X l̸=m,n (ϵnl − ϵlm)tr[ea mneb nlec lm] − i X l,m,n umnl 2 ϵmnϵnlϵlmtr[ˆea mnˆeb nlˆec lm] ! + (b ↔ c) , (S59) which we combine into a single sum: E = Z k i 2 X ...

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  65. [65]

    Fourier transform in all relevant energies appearing (there can be multiple if multiple bands are involved)

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  66. [66]

    Taylor expand in Γ up to order O Γ0 (check if this is defined everywhere in Fourier space)

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  67. [67]

    For this paper we choose to expand in Γ up to order O Γ0 , but this can be chosen entirely differently in another context

    perform the inverse Fourier transform back. For this paper we choose to expand in Γ up to order O Γ0 , but this can be chosen entirely differently in another context. In particular, there is no constraint to collecting terms of order O Γ1 or higher. Clearly we are able to recover a delta distribution from a Lorentzian, but, in general, what types of funct...

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  68. [68]

    − i(Γ2ξ2 n−6) 24 √ 2πΓ2ξn ,

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  69. [69]

    1−2θ(−En) 8Γ2 + 1 24 δ′(En) , where ξn denotes the Fourier space variable corresponding to En. For the other cases, the explicit functions as well as their Fourier transforms become too complicated to show explicitly, but as will be shown in the next section the final results are indeed simple. When the function depends on multiple energies En, E m, . . ....

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  70. [70]

    (S47)) is simply ¯A = Z k X n 1 − 2θ(−En) 8Γ2 Eabc n tr[ ˆPn] = − 1 4Γ2 Z k X n θ(−En)Eabc n tr[ ˆPn]

    Contribution ¯A: trivial geometry tr [ ˆPn] As shown in the previous subsection, in the Γ expansion contribution A (shown in Eq. (S47)) is simply ¯A = Z k X n 1 − 2θ(−En) 8Γ2 Eabc n tr[ ˆPn] = − 1 4Γ2 Z k X n θ(−En)Eabc n tr[ ˆPn] . (S73) We neglected the part proportional to δ′(En), as it is two orders higher in Γ than the first and will not mix with any...

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  71. [71]

    (S52)) reduces to ¯B = Z k X m̸=n δ(En) Em + 1 2 δ′(En) Ec nQab (nm) + X m̸=n i 2Γ δ(En)Ec nQab [nm] ! + (b ↔ c)

    Contribution ¯B: quantum geometric tensors Qab nm and Qac nm In the Γ expansion, contribution B (shown in Eq. (S52)) reduces to ¯B = Z k X m̸=n δ(En) Em + 1 2 δ′(En) Ec nQab (nm) + X m̸=n i 2Γ δ(En)Ec nQab [nm] ! + (b ↔ c) . (S74)

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  72. [72]

    (S56)) becomes ¯C = Z k X m̸=n −2δ(En) Em − δ′(En) Ea nQbc (nm)

    Contribution ¯C: quantum geometric tensor Qbc nm In the Γ expansion, contribution C (shown in Eq. (S56)) becomes ¯C = Z k X m̸=n −2δ(En) Em − δ′(En) Ea nQbc (nm) . (S75) 11

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  73. [73]

    (S58)) simplifies to ¯D = Z k X m,n 1 4 δ(Em) + δ(En) C b;ac (nm) + C c;ab (nm) (S76) = Z k X m,n 1 2 δ(En) C b;ac (nm) + C c;ab (nm)

    Contribution ¯D: connections C b;ac nm , C c;ab nm In the Γ expansion, contribution D (shown in Eq. (S58)) simplifies to ¯D = Z k X m,n 1 4 δ(Em) + δ(En) C b;ac (nm) + C c;ab (nm) (S76) = Z k X m,n 1 2 δ(En) C b;ac (nm) + C c;ab (nm) . (S77) We utilized the n, m symmetry of δ(Em) + δ(En) in the last equality

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  74. [74]

    (S65)), only the term proportional to Im tr[ea mneb nlec lm] survives, and we obtain ¯E = Z k X (l,m,n) δ(En) 2 − δ(Em) 2 Im tr[ea mneb nlec lm]

    Contribution ¯E: to three-band traces tr [ea mneb nlec lm], tr[ea mnec nleb lm] In the Γ expansion of the final contribution, E (shown in Eq. (S65)), only the term proportional to Im tr[ea mneb nlec lm] survives, and we obtain ¯E = Z k X (l,m,n) δ(En) 2 − δ(Em) 2 Im tr[ea mneb nlec lm] . (S78) Now that the coefficient does not depend on the l band, simpli...

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