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REVIEW 4 major objections 5 minor 49 references

Transmission through rectangular potentials in semimetals featuring quadratic dispersion

T0 review · 4 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read For quadratic-band-crossing semimetals, the previous tunneling calculation missed evanescent waves and is now corrected.

desk verdict Self-correction that fixes a real 2d QBCP tunneling error; the 3d section is undone by singular, unnormalized spinor basis. read the letter →

arxiv 2502.06265 v2 pith:OUT75DMO submitted 2025-02-10 cond-mat.mes-hall

Ipsita Mandal This is my paper · ORCID
classification cond-mat.mes-hall PACS 73.23.-b73.40.Gk72.10.-d
keywords quadraticbandcrossingevanescentwavesrectangularpotentialbarriertransmissioncoefficientFanofactorconductanceLuttingersemimetalKleintunneling
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 corrects the theory of tunneling through a rectangular potential barrier in semimetals whose bands touch at a quadratic band-crossing point (QBCP), in both two and three dimensions. The earlier treatment of this problem left out the evanescent-wave solutions, so the wavefunction could not satisfy all the boundary conditions at the barrier edges. With those solutions included, the paper recomputes the transmission and reflection coefficients, the ballistic conductance, and the Fano factor for representative parameters. If the correction is right, the published curves for these quantities are outdated and the corrected results, including a new small-energy formula, are the valid ones. The conclusion that QBCPs do not show Klein tunneling survives the correction.

What carries the argument

The load-bearing object is the piecewise scattering wavefunction that includes evanescent modes with imaginary wavevectors, $k_x=\pm i\sqrt{2mE+q_n^2}$ outside the barrier and similarly inside, in addition to the propagating modes $k_x=\pm\sqrt{2mE-q_n^2}$. The paper defines a QBCP as a nodal point where the low-energy bands disperse quadratically in momentum, and the Hamiltonian is quadratic in derivatives. The matching procedure integrates the Schrödinger equation across the barrier edges twice, yielding continuity of the two spinor components and their $x$-derivatives at $x=0$ and $x=L$; in 3d the same conditions are imposed on the four spinor components at $z=0$ and $z=L$. This gives exactly eight equations for the eight coefficients in 2d and sixteen in 3d, closing the system that the earlier work left open.

What would settle it

Solve the same rectangular-barrier scattering problem on a lattice QBCP model by direct numerical integration of the time-independent Schrödinger equation and compare the transmission at normal incidence with the small-$E$ formula $T=4E\,\sec^2\phi\,\csc^2(L\sqrt{2mV_0})/V_0$; a mismatch would show that the assumed derivative-matching conditions are incomplete or incorrect.

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Extended reading notes

Core claim

The central claim is that a correct scattering solution for a rectangular potential in a QBCP semimetal must include evanescent plane-wave parts alongside the propagating ones, for the incoming, reflected, and transmitted waves and inside the barrier. For the two-dimensional model with Hamiltonian $H^{\mathrm{kin}}_{2d}(k_x,k_y)=\frac{1}{2m}[2k_x k_y\,\sigma_x+(k_y^2-k_x^2)\sigma_z]$, the dispersion $\varepsilon_{2d}(k_x,k_y)=(k_x^2+k_y^2)/(2m)$ gives four wavevector solutions for fixed transverse momentum, two real and two imaginary; dropping the imaginary pair leaves the derivative-matching equations underdetermined. The paper keeps all eight amplitude coefficients in 2d (sixteen in 3d), fixes them by continuity of both spinor components and their $x$- (or $z$-) derivatives at the two barrier edges, and obtains the transmission amplitude $t_n$. It then computes $T=|t_n|^2$, the conductance, and the Fano factor via the Landauer formula, and derives the small-$E$ limit $T(E,V_0,\phi)=4E\,\sec^2\phi\,\csc^2(L\sqrt{2mV_0})/V_0+O(E^2)$ in 2d. The corrected results are shown in polar and conductivity plots, and compared with normal electron gases and graphene.

Load-bearing premise

The calculation assumes that requiring both the wavefunction and its derivative to be continuous at the two barrier edges is the complete and correct set of interface conditions for the quadratic Hamiltonian; if the true conditions differ, the solved coefficients and all resulting transmission curves would change.

Editorial extensions

If this is right

  • The published $T(E,V_0,\phi)$, $\sigma(E,V_0)$, and $F(E,V_0)$ curves for QBCP barriers are superseded by the corrected results in Figs. 2–9.
  • In two dimensions, the conductivity vanishes and the Fano factor approaches unity as $E\to 0$, so a QBCP junction becomes sub-Poissonian at low energy while still suppressing transmission below the Sharvin limit.
  • QBCPs continue to show no Klein tunneling: transmission is not unity at normal incidence, in contrast to Dirac, Weyl, and triple-point fermions.
  • The appearance of evanescent waves is tied to the quadratic-in-momentum dispersion along the transport direction; the same feature occurs in bilayer graphene, semi-Dirac, and multi-Weyl semimetals, so the method carries over to those problems.
  • For 3d QBCPs, the calculation shows how to handle the two degenerate conduction bands by assigning separate amplitudes to each, with only the $\Psi_{+1}$ incident component needed because the cross-amplitude vanishes by symmetry.

Reading between the lines

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

  • Editorial inference: the same evanescent-wave omission could affect other QBCP transport calculations, such as tunneling through delta-function potentials or Josephson-junction Andreev spectra; the paper explicitly lists those problems as next steps, so a consistent treatment will need the full 8- or 16-dimensional matching.
  • Editorial inference: the low-energy scaling $\sigma\propto E$ and $F\to 1$ in 2d is sharp enough to test experimentally in a ballistic QBCP device; a measurement of conductance versus gate voltage at low temperature would distinguish the corrected curves from the old ones and from graphene's $E$-linear conductivity.
  • Editorial inference: the boundary-condition counting suggests a general recipe for any band structure whose group velocity is not linear: keep all wavevector solutions of the characteristic equation, including complex ones, and match enough derivatives to close the system.
  • Editorial inference: for tilted or anisotropic quadratic dispersions, the number and form of evanescent modes will change, and the same matching procedure could be run again to produce corrected transmission formulas for those variants.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper re-examines transmission through a rectangular electrostatic barrier in semimetals with quadratic band-crossing points (QBCPs). The author argues that Ref. [11] discarded evanescent solutions of the Schrödinger equation, so the boundary conditions were only partially satisfied; keeping the evanescent modes gives a square system of eight matching equations in two dimensions and sixteen in three dimensions. The transport quantities T(E,V0,phi), conductivity sigma, and Fano factor F are recomputed and shown in polar and line plots, with a closed low-energy expression for the 2d case in Eq. (14). The paper is framed as a correction of previously published results.

Significance. If the 2d calculation is correct, it provides a nontrivial correction to a published transport calculation and yields an explicit falsifiable prediction at small E in Eq. (14), including the absence of Klein tunneling. The 2d formalism is standard and the inclusion of evanescent modes is physically reasonable; I agree with the skeptical assessment that the boundary-condition matching itself is not the central weakness. The 3d section, however, is not in a usable state as written: the outgoing evanescent term in Eq. (30) grows exponentially, the eigenvectors in Eq. (24) are singular at normal incidence with unspecified normalizations, and the conductance formula relies on an unproved statement that interband transmission vanishes. The paper also does not provide code, data, or full transmission amplitudes, which would be needed to replace previously published curves. No circularity concern attaches to the central calculation: the transmittance follows from solving a boundary-value problem with no tunable parameters.

major comments (4)
  1. [III.A, Eq. (30)] The transmitted evanescent term in the right-side wavefunction is written with the factor e^{kappa(z-L)}, which grows without bound for z>L because kappa = sqrt(2mE + k_perp^2) > 0. This makes the 3d scattering state non-normalizable and does not correspond to an outgoing evanescent mode; the analogous 2d term in Eq. (11) correctly uses e^{-kappa(x-L)}. Since this term enters the matching equations Eq. (31), the 3d results in Figs. 7-9 are not well-defined as written.
  2. [III.A, Eq. (24)] The four 3d eigenvectors contain explicit factors 1/(k_x - i k_y), 1/(k_x + i k_y), and their squares, so they are singular at normal incidence k_perp = 0; the normalization factors N_{+-s} are not given. Since theta = 0 is included in the polar plots and is an endpoint of the conductance integral Eq. (33), the transmission amplitudes are not well-defined at normal incidence. The author should specify the orthonormalization convention, the branch of the square root for imaginary k_z, and a k_perp -> 0 limiting procedure, and show that the limit is independent of the azimuthal direction in the transverse plane.
  3. [III.B, footnote 2 and Eq. (33)] The assertion that |t_{n,2}| = 0 for an incident Psi_{+,1} state is stated without derivation. The boundary-condition system Eq. (31) couples all four spinor components, and no symmetry is displayed that would obviously decouple the two positive-energy bands at fixed k_perp; a scalar barrier can in general scatter between degenerate bands. Because Eq. (33) uses only |t_{n,1}|^2 for the conductance, this assumption is load-bearing. Please provide the symmetry argument or a numerical check that includes |t_{n,2}|^2 before the 3d conductance can be accepted.
  4. [II.A and III.A, Eqs. (11)-(13) and (30)-(32)] The paper's central quantitative claim is that the previous transmission coefficients from Ref. [11] are incorrect, yet the new amplitudes are not shown ('extremely long') and no code or data are supplied. The low-energy formula Eq. (14) provides one check, but it does not validate the intermediate-energy curves in Figs. 2-9. I request the full amplitudes, or at least machine-readable data or scripts that solve Eqs. (12) and (31), so the corrected curves can be independently verified.
minor comments (5)
  1. [II.A, Eq. (12)] The last boundary condition should read d_x phi_M(L) = d_x phi_R(L), not d_x phi_L(x)|_{x=L}; Eq. (31) has the analogous typo (and an x versus z slip in the derivative argument).
  2. [I, Eq. (2)] The denominator in the Fano factor should be sum_n T_n, not sum_{n~} T_{n~}; the tilde over n appears to be a typographical artifact.
  3. [Fig. 3, Fig. 5, Fig. 8] The lower-panel captions of Figs. 3, 5, and 8 label the reflection coefficient as T(E,V0,phi) or T(E,V0,theta); these should be R(E,V0,phi) and R(E,V0,theta), respectively.
  4. [II and III headings] The section headings 'F ormalism' and 'T ransmission coefficients' contain TeX-spacing artifacts and should be corrected.
  5. [II.A and III.A, barrier-region modes] The special case E = V0 is not discussed; inside the barrier the four k_x solutions change their character and the independent modes can coalesce, so a brief remark about this non-generic point would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the transmission calculation is a self-contained boundary-value problem solved without fitted parameters or imported uniqueness claims.

full rationale

The paper's central derivation is self-contained. It starts from the stated low-energy Hamiltonian (Eq. 7 for 2d, Eq. 26 for 3d), the piecewise-constant potential (Eq. 6 and Eq. 25), and the matching conditions (Eq. 12 and Eq. 31). The evanescent-wave solutions are derived directly from the dispersion relations (Eq. 9 and Eq. 28), not taken from any prior work. The transmission and reflection amplitudes are obtained by solving the resulting linear system for the unknown coefficients, with no free parameters fitted to data and no use of the target transmission coefficients as inputs. The self-citations, including Ref. [11] which is the paper being corrected, are used for context, comparison, and as prior work to be improved upon; they do not supply any load-bearing premise. The claim that Ref. [11] missed evanescent waves is justified by the independent solution count and the explicit appearance of imaginary wavevectors in Eq. (9). The technical gap concerning the 3d eigenvectors (Eq. 24) — their unspecified normalization and singular behavior at normal incidence — is a correctness or rigor concern, not a circularity, because the derivation does not rely on the result being true to define its inputs. Therefore, the paper does not reduce its predictions to its assumptions by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The calculation rests on standard low-energy k.p models for QBCPs and standard scattering-matrix matching. No parameters are fitted to data, and no new physical entities are introduced; the evanescent waves are mathematical solutions of the stated Hamiltonian. The main unproved steps are the completeness of the four-wavevector ansatz and the boundary-condition count.

assumptions (4)
  • domain assumption The low-energy physics of 2d and 3d QBCPs is captured by the Hamiltonians in Eqs. (3) and (20).
    All transport results depend on these effective k.p models; if real materials deviate from these, the predictions change.
  • standard math The squared dispersion relation kx^2 + qn^2 = +/- 2mE yields a complete set of four independent scattering solutions for each transverse mode, including evanescent modes.
    The ansatz in Eq. (11) uses all four kx roots; if this set is not complete for the original linear operator, the boundary matching would be wrong.
  • domain assumption Continuity of the two spinor components and their x-derivatives at x=0 and x=L (Eq. 12) is the correct set of boundary conditions for the step potential.
    The central matching calculation and all extracted coefficients depend on this count and form of boundary conditions.
  • domain assumption For E < V0, the Fermi level inside the barrier lies in the valence band, so the barrier region must use the negative-energy eigenstates as selected by the Heaviside theta functions in Eq. (11).
    This band-index choice determines the form of the barrier wavefunction and therefore the transmission amplitudes.

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Cite this review

Pith. "Pith review of Transmission through rectangular potentials in semimetals featuring quadratic dispersion." pith.science (2026). https://pith.science/paper/OUT75DMO

@misc{pith2026250206265,
  author       = {Pith},
  title        = {Pith review of: Transmission through rectangular potentials in semimetals featuring quadratic dispersion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OUT75DMO}},
  note         = {Machine review of arXiv:2502.06265}
}
read the original abstract

We revisit the problem of transmission of quasiparticles through a rectangular potential barrier, for semimetals featuring quadratic-in-momentum band-crossings at a nodal point. Although this was considered in Annals of Physics 419 (2020) 168235, the solutions corresponding to evanescent waves were missed, leading to a partial fulfillment of the boundary conditions required to determine the piecewise-continuous wavefunctions. In this paper, our aim is to correct those shortcomings, recompute the transmission coefficients, and show the resulting behaviour of the conductivity and the Fano factor for some representative parameter values.

Figures

Figures reproduced from arXiv: 2502.06265 by the authors.

Figure 1
Figure 1. FIG. 1. Schematics of transmission through a potential barrier in a system hosting a QBCP. The upper panel shows the schematic [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. 2d QBCP: The polar plots show the transmission coefficient, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. 2d QBCP: The polar plots show the transmission coefficient, [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4. 2d electron-gas: The polar plots show the transmission coefficient, [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. 2d electron-gas: The polar plots show the transmission coefficient, [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Plots of the conductivity ( [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. 3d QBCP: The polar plots show the transmission coefficient, [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. 3d QBCP: The polar plots show the transmission coefficient, [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Plots of the conductivity ( [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

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Reference graph

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.