Orbital transmutation and the electronic spectrum of FeSe in the nematic phase

Andrey V. Chubukov; Morten H. Christensen; Rafael M. Fernandes

REVIEW 4 major objections 4 minor 45 references

Orbital transmutation and the electronic spectrum of FeSe in the nematic phase

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Spin-orbit coupling or surface hybridization can change an excitation's dominant orbital character in FeSe between the tetragonal and nematic phases, explaining why the xz and yz modes at the M point do not merge at T_nem.

desk verdict Orbital transmutation is a genuinely new mechanism for the FeSe M-point puzzle, but the resolution is conditional on unmeasured couplings and fitted level crossings; still worth a serious referee. read the letter →

arxiv 1908.04889 v2 pith:PQV4C4JP submitted 2019-08-13 cond-mat.supr-con cond-mat.str-el

Morten H. Christensen , Rafael M. Fernandes , Andrey V. Chubukov This is my paper · ORCID

classification cond-mat.supr-concond-mat.str-el

keywords FeSenematicorderorbitaltransmutationspin-orbitcouplingsurface-inducedhybridizationangle-resolvedphotoemissionspectroscopyiron-basedsuperconductorselectronpockets

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to explain a specific ARPES anomaly in FeSe: deep in the nematic phase two sharp excitations at $M=(\pi,\pi)$ are dominated by the $xz$ and $yz$ orbitals, yet as the temperature rises through $T_{\rm nem}$ they approach each other and remain split, even though $xz$ and $yz$ are degenerate in the tetragonal phase. The paper argues that the standard nematic model fails here because it ignores hybridization already present above $T_{\rm nem}$. Including spin-orbit coupling, or the surface-allowed $xz/yz$ hybridization probed by ARPES and STM, produces an orbital transmutation: the excitation that looks $xz$ deep in the nematic phase becomes mostly $xy$ near $T_{\rm nem}$, so it merges with a different doublet and the two observed modes never meet. A sympathetic reader would care because the mechanism turns a puzzling experimental observation into a sharp, testable prediction about how orbital character changes with temperature, and it also explains why one of FeSe's electron pockets is hard to see in photoemission.

What carries the argument

The load-bearing object is the effective $2\times2$ Hamiltonian at the $M$ point in the $xz$--$xy$ and $yz$--$xy$ orbital sectors, built from a $k\cdot p$ expansion that respects the glide-plane symmetry of FeSe. The central identity is the mixing angle $\tan 2\phi = -\lambda/(\epsilon_1-\epsilon_3+\phi_1+\phi_3)$; when the nematic combination $\phi_1+\phi_3$ grows large enough to overcome the small orbital splitting $\epsilon_1-\epsilon_3$, the angle moves from near zero to near $-\pi/2$ and the dominant orbital weight of the two eigenstates is exchanged. That sign change, named orbital transmutation, is what lets one excitation from each doublet swap its orbital identity between high and low temperature. In the surface scenario the same work is done by the hybridization $\eta\,\hat d^\dagger_{xz,\sigma} d_{yz,\sigma}+\mathrm{H.c.}$, which is forbidden in the bulk by glide-plane symmetry but allowed at the surface probed by ARPES and STM.

What would settle it

A decisive test is polarization-resolved ARPES at the $M$ point just above $T_{\rm nem}$ with resolution better than the expected splittings: the spin-orbit scenario predicts one doubly degenerate $xz/yz$ doublet and one doubly degenerate $xy$ doublet, whereas the surface-hybridization scenario predicts two split $xz/yz$ modes with equal $xz$ and $yz$ orbital weight. Observing neither the predicted doublet structure nor the two split singlets would falsify the orbital-transmutation mechanism; seeing the two modes actually merge above $T_{\rm nem}$ would indicate the puzzle was a thermal-broadening artifact.

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Extended reading notes

Core claim

The central claim is that an excitation from each doublet at the $M$ point undergoes an orbital transmutation below $T_{\rm nem}$: its dominant orbital contribution changes relative to the tetragonal phase. In the spin-orbit-coupling scenario, the doublet closer to the Fermi level splits into a state that remains predominantly $yz$ and a state that becomes predominantly $xy$ at $T\ll T_{\rm nem}$; the other doublet splits into a state that remains predominantly $xy$ and a state that becomes predominantly $xz$. Because the $xz$-dominated low-temperature mode has become $xy$ as $T$ approaches $T_{\rm nem}$, it does not merge with the $yz$ mode; instead it joins the lower $xy$ doublet. The paper argues this reproduces the ARPES spectra, including the two sharp peaks at low temperature, the two weaker intermediate peaks identified with $xy$ orbitals, and the missing outer electron pocket if $xy$ fermions are incoherent. In the complementary surface-hybridization scenario, the same non-merging follows from a tetragonal-phase splitting of the $xz/yz$ doublet into equal $xz\pm yz$ mixtures, with the low-temperature eigenstates becoming almost pure $yz$ and $xz$.

Load-bearing premise

The argument needs the spin-orbit coupling $\lambda$ (or the surface hybridization $\eta$) to be large enough—the paper uses 10 meV for each—and the orbital splitting $\epsilon_1-\epsilon_3$ to be small enough that the nematic combination $\phi_1+\phi_3$ overturns the sign of the mixing-angle denominator; neither $\lambda$ nor $\eta$ has been measured directly on the electron pockets.

Editorial extensions

If this is right

The measured splitting between the $xz$ and $yz$ modes deep in the nematic phase is not simply $2|\phi_1|$; it is the modified expression in Eq. (21) that involves $\lambda$, $\phi_1+\phi_3$, and the orbital splitting, so extracting the nematic order parameter from ARPES requires accounting for spin-orbit coupling.
The absence of the outer electron pocket in ARPES finds a natural explanation: when $xy$ fermions are incoherent, a pocket that is predominantly $xy$ contributes only weak, broad spectral weight.
The $xz$ spectral weight measured along the diagonal of the peanut-shaped inner electron pocket is a direct fingerprint of spin-orbit coupling, and its magnitude is sensitive to the values of $\lambda$ and the nematic order parameters.
The two scenarios can be separated by measurements in the tetragonal phase: SOC gives two doubly degenerate bands, while surface-induced hybridization splits the $xz/yz$ doublet into distinct singlet excitations even above $T_{\rm nem}$.

Reading between the lines

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

If the mechanism is generic, orbital transmutation should be strongest in materials where the relevant orbital splitting is comparable to the nematic order parameter, as in FeSe; in materials with larger orbital splittings the same hybridization would produce only small weight transfers rather than an identity swap.
A testable extension would be to look for the predicted temperature dependence of spectral weight in detwinned crystals: the low-temperature $xz$-dominated peak should continuously lose $xz$ weight and gain $xy$ weight as $T$ approaches $T_{\rm nem}$, which could be seen by tracking peak intensities under two polarizations.
The same level-crossing logic may apply to other multiorbital systems with a nematic-like order parameter that can flip the sign of a mixing-angle denominator, suggesting orbital transmutation as a general spectroscopic signature of orbital order.

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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

Colleague,

The thing to know about this paper: it proposes a specific mechanism — orbital transmutation — to explain why the xz/yz doublet at the M point of FeSe does not merge above T_nem, and it does so with a clean 2x2 model. That is actually new. The ingredients (SOC and surface hybridization) were known, but nobody had identified that the dominant orbital character of a band can flip between tetragonal and nematic phases, so that the low-T xz excitation actually connects to the xy-dominated doublet above T_nem. The paper works through both SOC and SIH scenarios, computes spectral functions and Fermi surfaces, and gives a concrete way to distinguish them with polarized ARPES. Credit is due for that.

What it does well: the symmetry analysis is careful, the diagonalization is transparent, and the qualitative match to ARPES data — two sharp peaks deep in the nematic phase, two weaker xy features in between, and the persistence of splitting above T_nem — is plausible. It also explains the missing outer electron pocket via xy incoherence, which is consistent with earlier work. The paper is honest about what is not known; Sec. V explicitly says neither lambda nor eta has been measured directly.

Where it is soft: the whole transmutation mechanism only operates if the bare xz and xy levels cross as nematic order grows, i.e. |phi1+phi3| > epsilon1-epsilon3. That condition is satisfied only for the chosen parameters: phi1,0 = -24 meV and epsilon1-epsilon3 = 7.4 meV. These are not independently pinned down. phi1 is treated as known, but in the presence of SOC the splitting at low T is not simply -2 phi1 (Eq. 21), so there is no clean experimental handle on phi1 separately. Similarly, lambda and eta are set to 10 meV. If the actual values are smaller, or the orbital splitting larger, the transmutation is suppressed and the puzzle re-emerges. This makes the explanation conditional, not wrong. The paper does not pretend to have a parameter-free prediction; it offers a credible scenario and a test. The two scenarios (SOC vs SIH) are not distinguished by current data, but the paper gives a clear experiment to do.

In short: this is a solid theory paper that resolves a known puzzle in a very specific parameter regime. That regime may be right — FeSe has unusually small Fermi energies and the level crossing is visible in the standard model itself — but it is asserted, not yet established. The paper deserves a serious referee and would be a good reading-group piece for exactly that reason: it's a clear example of a mechanism that can look like a prediction but is actually fitted to the data. I'd cite it if I write about FeSe nematicity, and I'd accept it for peer review with the expectation that the authors sharpen the parameter-dependence discussion.

Referee Report

4 major / 4 minor

Summary. This manuscript proposes an explanation for the persistent splitting of the xz/yz-derived excitations at the M point in FeSe above the nematic transition. Working in the 2-Fe Brillouin zone, the authors add to the standard nematic k·p model either the spin-orbit coupling allowed at M or a surface-induced xz/yz hybridization. They show that in both cases the dominant orbital character of one low-energy branch changes between the tetragonal and deeply nematic regimes ('orbital transmutation'), so the state identified as xz at low T does not return to the upper xz/yz doublet as T approaches T_nem, but instead connects to the lower xy-dominated doublet. The model is used to compute M-point energies, spectral functions, Fermi-surface shapes, and orbital weights, and is compared with ARPES and STM data. The two scenarios are distinguished by predictions for polarized ARPES above T_nem.

Significance. If established, the orbital-transmutation mechanism would resolve a long-standing ARPES puzzle in FeSe and would be a useful concept for other small-Fermi-energy iron-based materials. The paper's strengths are its analytical transparency (the 2x2 diagonalizations leading to Eqs. (14)-(19) are clean), its symmetry-based derivation of the allowed SOC and surface terms, and the fact that it makes concrete falsifiable predictions: two degenerate doublets above T_nem in the SOC scenario versus split singlets in the SIH scenario, with distinct orbital contents measurable by polarized ARPES. The main limitation is that the effect is demonstrated in a parameter regime whose key inputs are not independently measured; the paper therefore establishes an internally consistent scenario rather than a closed explanation.

major comments (4)

[Sec. III, Eqs. (13)-(19); Sec. V] The central claim that the low-temperature xz branch does not merge at T_nem depends on the sign change of D(T) = eps1 - eps3 + phi1(T) + phi3(T) (Eq. (16)): only if |phi1+phi3| exceeds |eps1-eps3| does the mixing angle phi reach the transmuted branch. In the manuscript this inequality is guaranteed by choices that are fitted to the very data under study: phi1,0 = -24 meV, eps1,0 - eps3,0 is set to 7.4 meV in Sec. II and to 10.2 meV in the SOC scenario (Fig. 4), and phi3,0 is varied as 0 or +/-10 meV although its sign is stated to be experimentally unverified (Sec. I). Moreover, Sec. V explicitly states that neither lambda nor eta has been measured directly; the value 10 meV is assumed. If the actual lambda were much smaller, or if eps1 - eps3 were larger, the sign change would not occur and the standard-model contradiction would remain. Please provide an independent constraint on this parameter regime, for example a full fit to the M-point dispersions and orbital weights with lambda and eta as free parameters, or a threshold analysis showing how large lambda and eta must be for the transmutation to survive.
[Sec. III, Eq. (21); Sec. II] The quantitative comparison with ARPES is partly constructed. Eq. (21) shows that the low-temperature splitting Delta E is a combination of phi1, phi3, lambda, and eps1 - eps3, and the text (Sec. II) states that eps1,0 and eps3,0 are adjusted in each scenario to maintain the peanut-shaped pocket; the values differ between the SOC scenario (-24.8, -35.0 meV) and the SIH scenario (-26.3, -32.0 meV). Thus the measured M-point splitting constrains phi1 only through an assumed lambda (or eta), while the onsite energies are themselves re-fit to reproduce the pocket shapes. The agreement therefore demonstrates consistency, not prediction. I recommend an explicit statement of this circularity and, if possible, a two-parameter scan of (lambda, eta) showing the range over which the data are reproduced.
[Sec. III, Eq. (20) and Fig. 6] The observability of the non-merging in the SOC scenario relies on the ad hoc assumption that Gamma_xy = 10 meV is much larger than Gamma_xz = Gamma_yz = 3 meV. If the xy orbital were coherent, the transmuted lower branch would remain a sharp peak and the spectral function would show four peaks; if Gamma_xy is made very large, the apparent merging is produced by broadening rather than by the physics of transmutation. The paper motivates this input by orbital-selective physics (Refs. [38,39]) but does not calibrate or test it. This premise should be stated as a separate assumption and its consequences checked by varying Gamma_xy over a wide range.
[Sec. IV, Eq. (12); Sec. V] The SIH scenario has the same status as a consistency check rather than an explanation, because the splitting above T_nem is exactly the assumed surface hybridization 2 eta and eta is not measured. The scenario is only distinguished from SOC by the orbital content above T_nem (Sec. V), which is not currently available. The authors should either soften the wording that these scenarios 'describe the data' to 'are consistent with the data' or provide additional evidence, for example a surface-sensitive measurement or a comparison between ARPES and bulk probes, that an xz/yz hybridization of order 10 meV exists.

minor comments (4)

[Eq. (16) and surrounding text] The mixing angle phi in Eq. (16) is defined through tan 2phi, which is invariant under 2phi -> 2phi + pi; the text's statement that phi varies from approximately 0 to approximately -pi/2 is only correct for one branch of the inverse tangent. Please state the chosen branch or define phi directly from the eigenvector components to avoid ambiguity.
[Throughout] There are several typographical errors: 'ro' should be 'to' in Sec. IV, 'intentisty' should be 'intensity' and 'variation with' should probably be 'contrast with' in Sec. V, and 'fartherst' should be 'farthest' in the Fig. 9 caption.
[Sec. II, Table I] The temperature-dependent onsite energies eps1(T) = eps1,0 + 0.083T and eps3(T) = eps3,0 + 0.083T are given without stating the units of the slope; please specify meV/K and note that this slope is also an input fitted to the ARPES data.
[Fig. 7 and Fig. 11] The caption color code references the same orbital colors as Fig. 2, but the printed figures use overlapping blue and green shades; for accessibility, consider using distinct line styles or explicitly labeling the pockets in the lower panels.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the non-merging follows from an exact diagonalization and band-identity swap, not from a fitted parameter renamed as a prediction.

full rationale

The central mechanism, orbital transmutation, is a direct algebraic consequence of diagonalizing the 2x2 Hamiltonians in Eqs. (13)-(19): when |phi1 + phi3| exceeds epsilon1 - epsilon3, the mixing angle phi in Eq. (16) varies from 0 to -pi/2 and the dominant orbital character of the two coupled branches is exchanged. The paper then shows that the low-temperature xz-dominated excitation connects to the lower xy doublet above Tnem, rather than to the upper xz/yz doublet, so the two low-temperature xz/yz peaks do not merge; this is a derived consequence, not an imposed output. Parameters such as epsilon1,0, epsilon3,0, and phi1,0 are indeed fitted to ARPES/STM pocket shapes, tetragonal dispersions, and the low-temperature splitting, and the level-crossing condition is already visible in the standard-model fit of Fig. 2. However, the non-merging above Tnem is not one of the fitting targets; it emerges from the identity swap induced by SOC or SIH. The form of the SOC is taken from the external symmetry analysis of Ref. [10], and the SIH term is justified both by a symmetry argument in this paper and by a self-citation [20]; that self-citation is not load-bearing because the SOC scenario independently explains the non-merging. The paper explicitly states that lambda and eta have not been measured directly, making the scenarios conditional on the chosen parameter regime, but conditionality is a limitation, not circularity. No step in the derivation reduces by construction to its own input, so no circularity is found.

Assumptions & free parameters 9 free parameters · 5 assumptions · 0 invented entities

The central scenario rests on a specific parameter regime: small orbital splitting, sizeable nematic order, and finite, unmeasured couplings lambda and eta. Many parameters are fitted to the very data the paper explains. The symmetry-based Hamiltonians are standard, and no new physical entities are introduced.

free parameters (9)

epsilon1,0 (xz/yz onsite energy at M) = -24.6 meV (SOC: -24.8; SIH: -26.3)
Fitted to reproduce the ARPES spectra at M in the tetragonal phase.
epsilon3,0 (xy onsite energy at M) = -32.0 meV (SOC: -35.0)
Fitted to ARPES data; adjusted within 3 meV to maintain pocket shape.
Temperature slope of onsite energies = 0.083 meV/K
Chosen to match the temperature dependence of the M-point energies in ARPES.
Dispersion parameters v, p1, p2, (2m1)^-1, (2m3)^-1, a1, a3 = See Table I: -122.90, -137.22, -11.67, 1.41, 186.11, 136.12, -403.84 (meV)
Fitted to reproduce the shape of the electron pockets seen in ARPES and STM.
phi1,0 (nematic order for xz/yz at M) = -24 meV
Chosen to match the observed low-temperature splitting of the xz and yz states; also sets the transmutation condition.
phi3,0 (nematic order for xy at M) = 0, +/-10 meV
Not measured; three values tested to show results are robust.
lambda (electron-pocket spin-orbit coupling) = 10 meV
Not measured on electron pockets; value assumed by comparison with hole-pocket SOC of about 20 meV.
eta (surface-induced xz/yz hybridization) = 10 meV
Not measured; chosen to illustrate the surface scenario, variation does not alter qualitative results.
Damping rates Gamma_xy and Gamma_xz/yz = Gamma_xy = 10 meV; Gamma_xz = Gamma_yz = 3 meV
Phenomenological damping chosen by hand to mimic the presumed incoherence of xy fermions.

assumptions (5)

domain assumption The space group of FeSe is P4/nmm, whose glide-plane symmetry restricts the allowed hybridizations and SOC at the M point.
Used in Sec. II to fix the form of HSOC in Eq. (8) and to forbid or introduce SIH in Eq. (12).
domain assumption Nematic order parameters follow the mean-field form phi_i(T) = phi_i,0 sqrt(1 - T/T_nem).
Used in Sec. II and throughout to compute the temperature evolution of the M-point energies.
domain assumption xy orbital fermions are largely incoherent (Z_xy << 1), represented by Gamma_xy > Gamma_xz,yz.
Invoked in Secs. III and IV to explain why the outer electron pocket and xy peaks are weak in ARPES; relies on Refs. 38-40.
standard math The k.p expansion around the M point from Ref. 10 captures the low-energy electronic structure.
Underlies the Hamiltonian in Eqs. (4)-(11).
domain assumption The orbital energy difference epsilon1 - epsilon3 is small (about 7 meV) so that |phi1 + phi3| can exceed it.
Stated in Sec. III as the key to the phenomenon of orbital transmutation; this parameter regime is essential for the mechanism.

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Cite this review

Pith. "Pith review of Orbital transmutation and the electronic spectrum of FeSe in the nematic phase." pith.science (2026). https://pith.science/paper/PQV4C4JP

@misc{pith2026190804889,
  author       = {Pith},
  title        = {Pith review of: Orbital transmutation and the electronic spectrum of FeSe in the nematic phase},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PQV4C4JP}},
  note         = {Machine review of arXiv:1908.04889}
}

abstract

We consider the electronic spectrum near $M=(\pi,\pi)$ in the nematic phase of FeSe ($T<T_{{\rm nem}}$) and make a detailed comparison with recent ARPES and STM experiments. Our main focus is the unexpected temperature dependence of the excitations at the $M$ point. These have been identified as having $xz$ and $yz$ orbital character well below $T_{{\rm nem}}$, but remain split at $T>T_{{\rm nem}}$, in apparent contradiction to the fact that in the tetragonal phase the $xz$ and $yz$ orbitals are degenerate. Here we present two scenarios which can describe the data. In both scenarios, hybridization terms present in the tetragonal phase leads to an orbital transmutation, a change in the dominant orbital character of some of the bands, between $T > T_{\rm nem}$ and $T \ll T_{\rm nem}$. The first scenario relies on the spin-orbit coupling at the $M$ point. We show that a finite spin-orbit coupling gives rise to orbital transmutation, in which one of the modes, identified as $xz$ ($yz)$ at $T \ll T_{{\rm nem}}$, becomes predominantly $xy$ at $T > T_{{\rm nem}}$ and hence does not merge with the predominantly $yz$ ($xz$) mode. The second scenario, complementary to the first, takes into consideration the fact that both ARPES and STM are surface probes. In the bulk, a direct hybridization between the $xz$ and $yz$ orbitals is not allowed at the $M$ point, however, it is permitted on the surface. In the presence of a direct $xz/yz$ hybridization, the orbital character of the $xz/yz$ modes changes from pure $xz$ and pure $yz$ at $T \ll T_{{\rm nem}}$ to $xz \pm yz$ at $T > T_{{\rm nem}}$, i.e., the two modes again have mono-orbital character at low $T$, but do not merge at $T_{{\rm nem}}$. We discuss how these scenarios can be distinguished in polarized ARPES experiments.

Figures

Figures reproduced from arXiv: 1908.04889 by the authors.

**Figure 3.** FIG. 3. Fermi surfaces and orbital weights at [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 6.** The orbital transmutation can be gleaned by looking at the excitations in the standard model of Eq. (1). From [PITH_FULL_IMAGE:figures/full_fig_p003_6.png] view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: FIG. 4. Evolution of the energies of the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. Evolution of cos [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. Spectral function as a function of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. Fermi surfaces for the SOC scenario, with [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 8.** Figure 8: FIG. 8. Dispersion along [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: FIG. 9. Temperature evolution of the energies at the [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

**Figure 10.** Figure 10: FIG. 10. Spectral function as function of [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]

**Figure 11.** Figure 11: FIG. 11. Fermi surfaces in the presence of a surface-induced [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]

**Figure 12.** Figure 12: FIG. 12. Temperature evolution of the energies at the [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]

**Figure 13.** Figure 13: FIG. 13. Spectral function as a function of [PITH_FULL_IMAGE:figures/full_fig_p010_13.png]

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In this case, the yz dominated excitation undergoes an orbital transmutation

In the other domain, the situation is reversed, as φΓ < 0 13 and φ1 > 0. In this case, the yz dominated excitation undergoes an orbital transmutation

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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To simplify the presentation we will use this notation without specifying the value of kz

Strictly speaking, the notations Γ, X and Y hold only for kz = 0. To simplify the presentation we will use this notation without specifying the value of kz

work page