REVIEW 4 major objections 4 minor 56 references
Higher-order continuum models for twisted bilayer graphene
T0 review · 4 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash
Pith's one-line read This paper proves that a second-order correction to the Bistritzer–MacDonald model approximates twisted bilayer graphene electron dynamics to accuracy $O(\varepsilon^{1+\eta_{-}})$, one power of $\varepsilon$ better than the first-order…
desk verdict Solid multiscale derivation with a significant gap in the single-PDE ellipticity claim. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the corrected Hamiltonian $H^{\mathrm{eff}} := H^{(1)} + \zeta(\varepsilon)H^{(\mathrm{NNN})} + \xi(\varepsilon)H^{(\nabla,\mathrm{NN})} + \varepsilon H^{(2)}$, whose four pieces are all independent of $\varepsilon$; each is a partial differential operator with smooth, moiré-periodic coefficients. The argument is carried by a multiple-scales expansion: the continuum wave function is decomposed as $f = f^{(1)} + \zeta(\varepsilon)f^{(\mathrm{NNN})} + \xi(\varepsilon)f^{(\nabla,\mathrm{NN})} + \varepsilon f^{(2)}$, with each piece solving one of four $\varepsilon$-independent Schrödinger equations driven by the previous order. The error analysis then reduces to comparing the exact interlayer coupling with filtered first-order Taylor expansions $\hat{\mathcal{h}}_{12}(k,q;\varepsilon)$ of the hopping function's Fourier transform about the reciprocal-lattice shells at distances $|K|$, $2|K|$, and $\sqrt{7}|K|$ from the Dirac point, where the decay bounds of Assumption 2.2 are tuned so that every residual has the claimed order.
What would settle it
Numerically evaluate the Fourier transform of a physical interlayer hopping function (for instance the Slater–Koster form) with the layer separation scaling like $|\log\varepsilon|$, and check whether $\hat{h}_{12,\mathrm{rad}}(|K|;\varepsilon) = \varepsilon$, $|\hat{h}'_{12,\mathrm{rad}}(|K|;\varepsilon)| \leq C\varepsilon^{(1+\eta)/2}$, and $|\hat{h}_{12,\mathrm{rad}}(2|K|;\varepsilon)| \leq C\varepsilon^{(3+\eta)/2}$ can all hold with a single $\eta > 0$; a family that fails one of these bounds would lie outside the theorem. A second test is to simulate the tight-binding versus continuum error below $\varepsilon \approx 9 \times 10^{-4}$, where the paper's coefficient estimates predict the observed $O(\varepsilon^3)$ error should cross over to $O(\varepsilon^{\sqrt{7}})$.
Extended reading notes
Core claim
The paper's central claim is that the first-order Bistritzer–MacDonald (BM) model admits a rigorously justified second-order refinement. The corrected Hamiltonian is $H^{\mathrm{eff}} := H^{(1)} + \zeta(\varepsilon)H^{(\mathrm{NNN})} + \xi(\varepsilon)H^{(\nabla,\mathrm{NN})} + \varepsilon H^{(2)}$, where $H^{(1)}$ is the BM operator, $H^{(\mathrm{NNN})}$ couples next-nearest-neighbor reciprocal-lattice shells, $H^{(\nabla,\mathrm{NN})}$ is a momentum-dependent interlayer term, and $H^{(2)}$ contains the quadratic Dirac-cone and twist corrections; the prefactors $\zeta, \xi$ are $o(1)$ powers of $\varepsilon$ fixed by the decay of the interlayer hopping function. Given spectrally localized, sufficiently regular initial data, Theorem 3.1 constructs from the solution an approximate tight-binding wave function satisfying $\|\phi(T) - \psi(T)\|_H \leq C\|f_0\|_{H^{6+\eta}}\varepsilon^{1+\eta_{-}}(\varepsilon T + \varepsilon^{\eta-\eta_{-}}(\varepsilon T)^2)$ uniformly in $\varepsilon$ and $T$, i.e. $O(\varepsilon^{1+\eta_{-}})$ accuracy on time scales of order $\varepsilon^{-1}$. Theorem 3.2 obtains the same order for a single effective Schrödinger equation with $H^{\mathrm{eff}}$ under an ellipticity condition ($|\alpha_d| \neq |\alpha_o|$) and slightly more regularity. Together these improve the first-order BM error rate of $O(\varepsilon^{1+\eta_*}T)$ by one power of $\varepsilon$, extend the Dirac-cone theorem to general symmetric hopping functions, and identify which symmetries of the BM model (notably particle–hole symmetry) are broken at second order.
Load-bearing premise
The proof rests on Assumption 2.2: the Fourier transform of the interlayer hopping function must satisfy sharp decay and normalization bounds, including $\hat{h}_{12,\mathrm{rad}}(|K|;\varepsilon) = \varepsilon$ and controlled derivatives at the neighboring shells $|K|$, $2|K|$, and $\sqrt{7}|K|$, and the paper itself notes that the Slater–Koster hopping function used in the numerics may not satisfy these bounds, so the theorems may not cover the numerical examples.
Editorial extensions
If this is right
- Over time scales of order $\varepsilon^{-1}$, the second-order continuum solution is $O(\varepsilon^{1+\eta_{-}})$-accurate and converges uniformly on the longer interval $t_0 \varepsilon^{-(3+\eta_{-})/2}$, upgrading the first-order BM bound $\varepsilon^{1+\eta_*}T$ to $\varepsilon^{2+\eta_{-}}T$.
- The second-order model reproduces qualitative features the first-order model misses: flat-band wave packets develop the tight-binding spiral pattern, and the accidental particle–hole symmetry of the BM Hamiltonian is broken — a property expected to matter for many-body models built on the continuum.
- The validity class widens: any $2\pi/3$-rotation-symmetric, super-algebraically decaying intralayer hopping function with nonzero Fermi velocity yields Dirac cones, and interlayer hopping may carry angular dependence beyond the radial case.
- The multiscale construction extends to arbitrary order: keeping further terms in the Taylor expansions of the hopping functions gives $O(\varepsilon^s)$ accuracy on $O(\varepsilon^{-1})$ time scales for any $s$.
- For analytic interlayer hopping with layer separation scaling like $|\log\varepsilon|$, the geometric decay rate is $O(\varepsilon^{\sqrt{7}})$ in the limit, though small coefficients can make $O(\varepsilon^3)$ dominate at the numerically accessible values, with a predicted crossover near $\varepsilon \approx 9 \times 10^{-4}$.
Reading between the lines
- Because the four Schrödinger equations in the multiscale expansion are all $\varepsilon$-independent, the components $f^{(1)}, f^{(\mathrm{NNN})}, f^{(\nabla,\mathrm{NN})}, f^{(2)}$ could be precomputed once per geometry and reassembled for any twist angle by rescaling — a practical shortcut for parameter sweeps that the paper does not exploit.
- The exponent $\sqrt{7}-2$ is dictated by the shell structure of the hexagonal reciprocal lattice (nearest neighbors at $|K|$, $2|K|$, $\sqrt{7}|K|$); the same shell-counting argument should transfer to other moiré materials, giving a geometric rule of thumb for how many expansion terms any effective model needs at a given accuracy.
- The spiral pattern traced to the momentum-dependent interlayer term suggests a diagnostic: selectively switching off $H^{(\nabla,\mathrm{NN})}$ in simulations of other flat-band systems could reveal whether similar spiral dynamics are generic in moiré materials or particular to the twisted bilayer graphene band geometry.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper derives a second-order correction to the Bistritzer-MacDonald continuum model for twisted bilayer graphene, starting from a tight-binding model with rapidly decaying, partially symmetric hopping functions. The main analytical results are Theorem 3.1, which bounds the difference between the tight-binding solution and a multiple-scales continuum approximation by an O(epsilon^{1+eta_-}) error on time scales of order epsilon^{-1}, and Theorem 3.2, which claims the same accuracy for a single second-order PDE with Hamiltonian H_eff. The paper also proves symmetry correspondences between the discrete and continuum models and reports numerical simulations using Slater-Koster hopping functions, where the second-order model is more accurate and reproduces a spiral pattern absent from the first-order model.
Significance. If Theorem 3.1 stands, the paper is a genuine and nontrivial advance: it rigorously improves the first-order BM validity theorem of Watson-Kong-MacDonald-Luskin by one power of epsilon, covers more general intralayer and interlayer hopping functions, and gives a parameter-free derivation of the next-order terms rather than fitting them to the target dynamics. The symmetry analysis is also a useful contribution. However, the advertised single-PDE version, Theorem 3.2, currently rests on an unproved ellipticity assertion, so the paper's central claim is only partially established. The numerical section is suggestive but uses hopping functions that the paper itself says may lie outside Assumption 2.2, so it cannot serve as a substitute for the missing proof.
major comments (4)
- [Section 3.2, Eq. (3.12)] The ellipticity lower bound in (3.12) is asserted to follow from |alpha_d| != |alpha_o| because the symbol of S grows quadratically. This is not correct for H_eff, whose constant-coefficient high-frequency part is L + epsilon S, not epsilon S. For real alpha, alpha_d, alpha_o and p=(p,0), the eigenvalues of the upper 2x2 constant-coefficient block are alpha p + (epsilon/2)(alpha_d+alpha_o)p^2 and -alpha p + (epsilon/2)(alpha_d-alpha_o)p^2. If either coefficient alpha_d +/- alpha_o has the opposite sign to alpha, one of these eigenvalues vanishes at |p| ~ epsilon^{-1}. Since |alpha_d| != |alpha_o| excludes both alpha_d = alpha_o and alpha_d = -alpha_o, such a crossing always exists for this branch. The bounded periodic term T(r) is O(1) and cannot repair the loss of ellipticity at such frequencies. Thus the proof of Theorem 3.2 in Appendix A.2 is invalid as written; (3.12) must either be added as an explicit hypothesis of Theorem 3.2 or proved under genuinely stronger spectral assumptions.
- [Appendix A.2, proof of Theorem 3.2] The proof begins by reducing to the case zeta(epsilon)=xi(epsilon)=epsilon and then states that the general case follows from the same arguments. This is a load-bearing reduction, not a cosmetic one: the definition of f^(2,all), the identity (A.69) for the residual of f + epsilon^2 f^(3), and the resulting powers of epsilon all depend on the three correction terms carrying exactly the same prefactor epsilon. Under Assumption 2.2 the general prefactors zeta and xi are only known to be O(epsilon^{(1+eta)/2}), and the 'same arguments' must be carried out with fractional powers and possibly distinct orderings of zeta and xi. The authors should provide the general-case calculation or restrict Theorem 3.2 to zeta=xi=epsilon and state the restricted result.
- [Section 5 and Section 6] The numerical experiments use the Slater-Koster and Fang-Kaxiras hopping functions, while Section 6 explicitly concedes that Slater-Koster may fail the bounds in Assumption 2.2. The convergence rates in Figure 5.4 are therefore not a test of Theorems 3.1-3.2, and the text's statement that the simulations are 'consistent with' the theorems is too strong. The authors should either verify (2.20)-(2.21) for the specific functions used in the numerics, simulate with a function known to satisfy Assumption 2.2, e.g. Example 2.1, or clearly label the Slater-Koster results as outside the assumptions and explain what evidence they provide for the theorems.
- [Section 3.1, Lemma A.3] The residual estimate in Lemma A.3 is central to Theorem 3.1, and the proof appears to rely on an inequality whose typesetting is ambiguous: the displayed estimate '1+eta/2 < (4+2eta)/sqrt(7)-1' should read '(1+eta)/2 < (4+2eta)/sqrt(7)-1'. The inequality in the intended sense is true for 0<eta<=1, but it is not needed: the bounds |zeta|, |xi| <= C epsilon^{(1+eta)/2} from (3.10) already give |zeta|^2, |zeta xi|, |xi|^2 <= C epsilon^{1+eta} and epsilon|zeta| <= C epsilon^{1+eta}. The authors should clarify this step and correct the typesetting.
minor comments (4)
- [Section 1.5] In the notation list, 'deonted' should be 'denoted'.
- [Section 3.2, paragraph after (3.12)] The sentence explaining why the lower bound of (3.12) holds should be replaced by a precise statement of the additional hypothesis needed, given Major Comment 1.
- [Section 5, paragraph after Figure 5.4] The discussion of the error model error(epsilon)=F1 epsilon^{sqrt(7)}+F2 epsilon^3 is helpful, but the claim that the numerical rate O(epsilon^{2.98}) is 'consistent with Theorem 3.2' should be softened: under the paper's own Slater-Koster estimate eta <= sqrt(7)-2, the theorem predicts a rate no better than epsilon^{2+eta_-}, which is strictly weaker than 2.98.
- [Appendix A.1, Lemma A.2] The proof invokes an ellipticity estimate (A.2) for the first-order operator H^(1) without proof or reference; since this estimate is reused in the proof of Theorem 3.1, a short proof or an explicit citation to [35] would improve readability.
Circularity Check
No significant circularity: the higher-order continuum model is derived from the same microscopic tight-binding Hamiltonian by Taylor and multiple-scales expansion, and it is validated against independent tight-binding numerics.
full rationale
The derivation chain is non-circular. The continuum operators H(1), H(NNN), H(∇,NN), and H(2) are defined by Taylor-expanding the Bloch transform of the tight-binding hopping functions about the monolayer Dirac point (Section 3.1, equations (3.5), (A.15), (A.21)), with no parameter fitted to the target wave-function dynamics. The normalization h12,rad(|K|; ε)=ε in Assumption 2.2 is an assumption on the microscopic interlayer hopping function, not a fit to the error being predicted. Theorem 3.1 is proved from the microscopic model by constructing a residual and bounding it through Lemmas A.1–A.10, so the claimed accuracy bound is not an input. Theorem 3.2 adds H^eff and invokes the ellipticity bound (3.12); the paper's assertion that (3.12) follows from |α_d| ≠ |α_o| is not proved and is an internal correctness gap, but it is not a circular reduction. The numerical section benchmarks against tight-binding dynamics with physical Slater-Koster parameters rather than against fitted outputs, and the paper explicitly acknowledges in Sections 5 and 6 that Slater-Koster may not satisfy Assumption 2.2, so the numerics are honestly presented as outside the theorem's hypotheses. Self-citations to [35] appear, including [35, equation (44)] in Lemma A.8, but this is an elementary Bloch-transform identity, and the first-order result of [35] is the extension target rather than an input to the new theorem. Thus there is no load-bearing circularity.
Assumptions & free parameters
assumptions (7)
- domain assumption Assumption 2.1: the intralayer hopping function h is super-algebraically decaying and invariant under 2pi/3 rotation.
- domain assumption Assumption 2.2: the interlayer hopping function h12 has a separable Fourier transform with prescribed decay and normalization bounds.
- domain assumption Self-adjointness condition h(-r) = h(r) for the intralayer hopping function.
- domain assumption The exact twist-angle relation theta = 2 sin^{-1}(beta epsilon/2).
- domain assumption Separability of the interlayer hopping function, hat h12(k) = hat h12,rad(|k|) hat h12,ang(k/|k|).
- domain assumption Initial data f0 lies in H^{6+eta} for Theorem 3.1 and H^{8+eta} for Theorem 3.2, and is spectrally localized near monolayer Dirac points.
- domain assumption Ellipticity condition (3.12), including |alpha_d| != |alpha_o| for Theorem 3.2.
Cite this review
Pith. "Pith review of Higher-order continuum models for twisted bilayer graphene." pith.science (2026). https://pith.science/paper/CPAKNM2U
@misc{pith2026250208120,
author = {Pith},
title = {Pith review of: Higher-order continuum models for twisted bilayer graphene},
year = {2026},
howpublished = {\url{https://pith.science/paper/CPAKNM2U}},
note = {Machine review of arXiv:2502.08120}
}
read the original abstract
The first-order continuum PDE model proposed by Bistritzer and MacDonald in \cite{bistritzer2011moire} accurately describes the single-particle electronic properties of twisted bilayer graphene (TBG) at small twist angles. In this paper, we obtain higher-order corrections to the Bistritzer-MacDonald model via a systematic multiple-scales expansion. We prove that the solution of the resulting higher-order PDE model accurately approximates the corresponding tight-binding wave function under a natural choice of parameters and given initial conditions that are spectrally localized to the monolayer Dirac points. Numerical simulations of tight-binding and continuum dynamics demonstrate the validity of the higher-order continuum model. Symmetries of the higher-order models are also discussed. This work extends the analysis from \cite{watson2023bistritzer}, which rigorously established the validity of the (first-order) BM model.
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[38]
For k, q∈ R2 and 1{·} the indicator function, let ˆ𝒽12(k, q; ε) := ˆh12(q; ε) 1{|q| ≤2|K|} + (k − q) · ∇ˆh12(q; ε) 1{|q| ≤ |K|} (A.21) denote a filtered first-order Taylor expansion of ˆh12(k; ε) about k = q. Then, as shown in Lemma A.9, (^F ′ twist)σ(k1, T) = 1 ε|Γ|2 e−iET X ...
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[39]
Therefore, for every G2 ∈ R∗ 2, there exists a unique G1 ∈ R∗ 1 such that the delta-function does not identically vanish. We can 36 therefore write ( ^H12ϕ2)σ(k1, T) = 1 |Γ| X G2∈R∗ 2 ˆ Γ∗ 2 X σ′ ei(G1(G2)·τ σ 1 −G2·τ σ′ 2 )δ(k1 + G1(G2) − k2 − G2) × ˆh12(k2 + G2; ε) ˜ϕσ′ 2 (k...
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[40]
It is a well-known property of the honeycomb lattice that Λ1 1 ∪Λ1 2 ∪Λ1 3 = R∗ 2, and thus Λ1 ∪Λ2 ∪Λ3 = R∗ 2 ⊕R∗ 2
∈ R∗ 2 ⊕ R∗ 2 : G2 − G′ 2 ∈ Λ1 j }, where the sets Λ 1 1, Λ1 2, Λ1 3 ⊂ R∗ 2 are given by Λ1 1 := {G2 ∈ R∗ 2 : |G2 + K2| = |K2|}, Λ1 2 := {G2 ∈ R∗ 2 : |G2 + K2| = 2|K2|}, Λ1 3 := {G2 ∈ R∗ 2 : |G2 + K2| ≥ √ 7|K2|}. It is a well-known property of the honeycomb lattice that Λ1 1 ∪...
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[41]
∈ Λ1. Similalry, the terms ∆ 2 and ∆3 can be written explicitly as ∆σ 2 (k1, T) = 1 ε|Γ|2 e−iET X (G2,G′ 2)∈Λ2 X σ′∈{A,B} ei(G1(G2)·τ σ 1 +(G′ 2−G2)·τ σ′ 2 ) × (ˆh12(K2 + G2 − G′ 2; ε) − ˆh12(k1 + G1(G2); ε)) ˆf σ′ 2 k1 + G1(G2) − G2 − K2 + G′ 2 ε , εT ∆σ 3 (k1, T) = − 1 ε|Γ|2...
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[42]
For j ∈ {1, 2}, let Uj : ℓ2(Rj; C2) → ℓ2(Rj; C2) such that U = diag(U1, U2)
Observe that Π σ j is a bijection with (Π σ j )−1(Rj) = R−2π/3(Rj + τ σ j ) − τ σ j . For j ∈ {1, 2}, let Uj : ℓ2(Rj; C2) → ℓ2(Rj; C2) such that U = diag(U1, U2). The 2 π/3-rotation invariance of h in Assumption 2.1 implies that [ Hjj , Uj] = 0 for j = 1, 2. Moreover, a direct...
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[43]
We conclude that [ H, U ] = 0 and the proof is complete. 48
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[44]
We will prove that [ H, Mx] = 0; the argument for My is similar. Writing Mx = 0 Mx,12 M† x,12 0 with Mx,12 : ℓ2(R2; C2) → ℓ2(R1; C2) defined by ( Mx,12ψ)σ R1 = ψσ Πσ x,1(R1), it follows that HMx = H12M† x,12 H11Mx,12 H22M† x,12 H † 12Mx,12 ! , MxH = Mx,12H † 12 Mx,12H22 M† x,1...
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[45]
+ τ σ 1 − τ σ′ 2 ; ε)ψσ′ R′ 1 , (A.74) where the last equality follows from the fact that Π σ x,2 : R2 → R1 is a bijection with (Π σ x,2)−1 = Πσ x,1. Next, we write (Mx,12H † 12ψ)σ R1 = X R1∈R′ 1 X σ′∈{A,B} h12(Πσ x,1(R1) − R′ 1 + τ σ 2 − τ σ′ 1 ; ε)ψσ′ R′ 1 , (A.75) and verif...
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[46]
Therefore, our assumption that h12(mx(r)) = h12(r) implies that H12M† x,12 = Mx,12H † 12 as desired
+ τ σ 1 − τ σ′ 2 = R1 + τ σ 1 − mx(R′ 1 + τ σ′ 1 ), Πσ x,1(R1) − R′ 1 + τ σ 2 − τ σ′ 1 = mx(R1 + τ σ 1 ) − R′ 1 − τ σ′ 1 . Therefore, our assumption that h12(mx(r)) = h12(r) implies that H12M† x,12 = Mx,12H † 12 as desired. It remains to prove that H11Mx,12 = Mx,12H22. We find...
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[47]
We observe that the diagonal blocks of H eff are differential operators with constant coefficients, and thus commute with Tv
Let v ∈ Rm. We observe that the diagonal blocks of H eff are differential operators with constant coefficients, and thus commute with Tv. It remains to show that the off-diagonal blocks also commute with Tv. We have v = n1am,1 + n2am,2 for some ( n1, n2) ∈ Z2, which implies th...
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[48]
We write D = diag(PC σ1, PC σ1) and observe that LPC σ1f (r) = L f B(−r) f A(−r) = −α(Dr1 − iDr2 ) ¯f A(−r) − ¯α(Dr1 + iDr2 ) ¯f B(−r) = PC ¯α(Dr1 + iDr2 )f A(r) α(Dr1 − iDr2 )f B(r) = PC σ1Lf (r) (A.80) while the functions defined in (A.76) satisfy σ1t† j(−r)σ1 = tj(r). Using...
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[49]
Define the operator ℛ1f (r) := diag(1, ei2π/3)f (R⊤ 2π/3r) so that ℛ = diag(ℛ1, ℛ1). We then have Lℛ1f (r) = αei2π/3(1, −i) · R2π/3∇f B(R⊤ 2π/3r) ¯α(1, i) · R2π/3∇f A(R⊤ 2π/3r) ! = α(1, −i) · ∇f B(R⊤ 2π/3r) ¯αei2π/3(1, i) · ∇f A(R⊤ 2π/3r) ! = ℛ1Lf (r) and diag(1, ei2π/3)tj(R⊤ ...
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[50]
The same argument establishes that T† ∇,NN = λ0t† 0(r)e1 · D + λ2t† 1(r)(R2π/3e1) · D + λ4t† 2(r)(R⊤ 2π/3e1) · D also commutes with ℛ1, and thus [ H (∇,NN), ℛ] = 0
Writing D = (D1, D2), e1 = (1, 0) and T∇,NN = λ0t0(r)e1 · D + λ2t1(r)(R2π/3e1) · D + λ4t2(r)(R⊤ 2π/3e1) · D, it follows that T∇,NNℛ1f (r) = λ0t0(r) diag(1, ei2π/3)(e1 · R2π/3D) + λ2t1(r) diag(1, ei2π/3)(e1 · D) + λ4t2(r) diag(1, ei2π/3)(e1 · R⊤ 2π/3D) f (R⊤ 2π/3r) and ℛ1T∇,NNf...
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[51]
We begin with the diagonal blocks of H eff
We write Mx = 0 C𝓂x C𝓂x 0 , 𝓂xf (r1, r2) := f (−r1, r2), with C the complex conjugation operator defined in (4.8). We begin with the diagonal blocks of H eff . Observe that [diag(L, L), Mx] = 0 [ L, C𝓂x] [L, C𝓂x] 0 = 0 0 0 0 , as the assumption α ∈ R implies that LC𝓂xf (r) = α...
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[52]
We write My = 0 σ1𝓂y σ1𝓂y 0 , 𝓂yf (r1, r2) := f (r1, −r2) and, using the assumption that α ∈ R to justify the last equality below, Lσ1𝓂yf (r) = L f B(r1, −r2) f A(r1, −r2) = α(Dr1 + iDr2 )f A(r1, −r2) ¯α(Dr1 − iDr2 )f B(r1, −r2) = σ1𝓂yLf (r); hence [diag(L, L), My]. Moreover, ...
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[53]
The assumed rotation invariance of h12 implies that ˆh12(R2π/3k; ε) = ˆh12(k; ε) for all (k; ε) ∈ R2×(0, 1), thus the result follows from (2.19) and (3.2)
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[54]
Since h is assumed to be even in its first argument, the real part of the sum vanishes and thus α = ∂k1 ˜hA,B(K) ∈ R
A direct calculation reveals that ∂k1 ˜hA,B(K) = −i X R∈R R1e−iR·Kh(R + τ A,B) = −i X R∈R R1e−i4πR1/3ah(R1, R2 − a/ √ 3), where R = (R1, R2) above. Since h is assumed to be even in its first argument, the real part of the sum vanishes and thus α = ∂k1 ˜hA,B(K) ∈ R. Similarly, ...
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[55]
It follows that ˆh12,ang(k1, −k2) = ˆh12,ang(k1, k2) and thus ˆh′ 12,ang(k1, −k2) = −ˆh′ 12,ang(k1, k2) for all (k1, k2) ∈ S
The assumption on h12 implies that ˆh12(k1, −k2; ε) = ˆh12(k1, k2; ε) for all ( k1, k2; ε) ∈ R2 × (0, 1). It follows that ˆh12,ang(k1, −k2) = ˆh12,ang(k1, k2) and thus ˆh′ 12,ang(k1, −k2) = −ˆh′ 12,ang(k1, k2) for all (k1, k2) ∈ S. The result then follows from the definitions ...
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[56]
Therefore, ˆh12,ang(k1, −k2) = ˆh12,ang(k1, k2), ˆh′ 12,ang(k1, −k2) = −ˆh′ 12,ang(k1, k2) for all ( k1, k2) ∈ S, and the result follows from (3.2)
We write ˆh12(k1, −k2; ε) = ˆ R2 e−i(−k1r1+k2r2)h12(r; ε)dr = ˆ R2 e−i(−k1r1+k2r2)h12(−r1, r2; ε)dr = ˆh12(k; ε), with the second equality above justified by our assumption on h12. Therefore, ˆh12,ang(k1, −k2) = ˆh12,ang(k1, k2), ˆh′ 12,ang(k1, −k2) = −ˆh′ 12,ang(k1, k2) for a...
Reviewed August 8, 2026 · model on record in the stance chip above.
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