Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
Science, 2006 · paper · Physics
A theoretical proposal that HgTe/CdTe quantum wells can realize the quantum spin Hall effect. Above a critical well thickness, the electronic state inverts and a topological phase with one pair of helical edge states appears. Experimental detection methods are discussed.
Checked: PDF from arxiv.org, fetched 4 Oct 2026. Page numbers are those of that PDF.
7 findings
Substance Eq. 8 prints T† with k− in all four in-plane entries ("−(1/√2)Pk−", "−(1/√6)Pk−", "(1/√6)Pk−", "(1/√2)Pk−"), which cannot hold for couplings of opposite Δm_J and conflicts with Eq. 25's "Ak−" versus "−Ak+".
- p. 16 Eq. 8, matrix T†
1/√2 P k−
- p. 21 Eq. 25, lower block
−Ak+
Only the authors can say what is right here.
Judge's note: Viewed pages 16 and 21. Reworked using the stated basis (Γ6,±1/2; Γ8,+3/2,+1/2,−1/2,−3/2) and the axial symmetry the paper invokes. Row 1 (Γ8,+3/2 from Γ6,+1/2) and row 2 col 2 (Γ8,+1/2 from Γ6,−1/2) change m_J by +1. Row 3 col 1 (Γ8,−1/2 from Γ6,+1/2) and row 4 (Γ8,−3/2 from Γ6,−1/2) change m_J by −1. Since k± carry opposite angular momentum, the two groups must carry opposite k± factors; a basis phase choice cannot turn k− into k+. Eq. 25 does have Ak− for the E1+/H1+ pair and −Ak+ for E1−/H1−. Rows 3 and 4 of T† should carry k+. The formula is wrong as printed, though the derived effective Hamiltonian is unaffected.
Second check: another model agreed
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Reference or label The example "H^eff_23 = (1/√6) P k+ ∫dz⟨Γ6,+1/2(z)|Γ8,−1/2(z)⟩" is not the (2,3) element in the stated basis order (|E1,+>, |H1,+>, |E1,−>, |H1,−>); it couples E1+ and E1−, i.e. index pair 1,3.
- p. 6 text after Eq. 4
For example, H^eff_23 = (1/√6) P k+ ∫ dz⟨Γ6, +1/2(z)|Γ8, −1/2(z)⟩
- p. 5 basis definition, bottom of page 5 to top of page 6
ψ1,...,4 = (|E1, + >
- p. 6 text after Eq. 4
For example, Heff 23 = 1/√6 Pk+ ∫dz⟨Γ6, +1/2(z)|Γ8, −1/2(z)⟩
Judge's note: Viewed pages 4, 5, 6, 18 and 20. The basis order is stated on pages 5-6 and repeated on page 20. Index 2 is H1+, which page 4 and page 18 say is formed only from |Γ8,±3/2⟩ (f3), with no Γ6 component. The quoted overlap is between a Γ6,+1/2 component (E1+, index 1) and a Γ8,−1/2 component (E1−, index 3). Transposing indices under Eq. 4's convention only swaps 23 and 32. The label should be 13 (or 31).
Second check: another model agreed
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Reference or label SOM text says "For values of these parameters at d = 40 Å and d = 70 Å see Table 1", but Table 1 has rows for 58 Å and 70 Å, and the main text says "for values at d = 58 Å and d = 70 Å see SOM".
- p. 21 SOM, text below Eq. 25
For values of these parameters at d = 40 Å and d = 70 Å see Table 1
- p. 21 Table 1, first column
58
- p. 6 main text after Eq. 4
for values at d = 58 Å and d = 70 Å see SOM
- p. 6 main text, k·p section
for values at d = 58 ˚A and d = 70 ˚A see SOM
Judge's note: Viewed pages 6 and 21. Table 1 on page 21 has only two rows, d = 58 and d = 70. The sentence below Eq. 25 on the same page says 40 Å and 70 Å, while the main text on page 6 says 58 Å and 70 Å. 40 Å is used for the dispersion plots (Fig. 2B), but no 40 Å parameters are tabulated. The SOM sentence should read 58 Å.
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Reference or label Eq. 6 prints σxy = −(1/8π²)∫∫dkx dky d̂·∂x d̂×∂y d̂ "in units of e2/h" and says it is "quantized to take integer values"; with that prefactor the result is −n/(2π), not an integer, in units of e²/h.
- p. 7 Eq. 6 and following line
in units of e2/h, where d̂ denotes the unit di(k) vector
- p. 7 text after Eq. 6
σxy is quantized to take integer values which measures the Skyrmion number
Only the authors can say what is right here.
Judge's note: Viewed page 7. Reworked: d̂·(∂x d̂ × ∂y d̂) is the pulled-back area element of the unit sphere, so the Brillouin-zone integral is 4π times the winding number n. Then −(1/8π²)·4πn = −n/(2π). That is an integer multiple only of e²/ħ (n e²/(2πħ) = n e²/h). For integers, and ±1/2 for a meron, in units of e²/h as the text states, the prefactor would need to be 1/(4π). Either the unit label or the prefactor is off; the authors must say which. Treated as a unit/prefactor label slip, since the quantised results quoted afterwards are the standard ones.
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Reference or label Reference 13 reads "L. Sheng et. al., Phys. Rev. Lett. 65, 136602 (2005)."; volume 65 does not fit the year 2005, since the same list has PRL 94 and 95 for 2005.
- p. 11 Reference 13
L. Sheng et. al., Phys. Rev. Lett. 65, 136602 (2005).
- p. 11 Reference 7
Phys. Rev. Lett. 95, 226801 (2005)
- p. 11 ref 13
Phys. Rev. Lett. 65, 136602 (2005)
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the true volume of the cited paper (reviewers say 95) is not in the files. Viewed page 11: a bold 65 is printed. Internally, refs 5, 7 and 10 give PRL volumes 94 and 95 for 2005, refs 3 and 6 give 92-93 for 2004, and refs 8 and 11 give 96 for 2006, so volume 65 with year 2005 and a six-digit article number is inconsistent within the list itself. Whether the intended volume is 94 or 95 cannot be settled from the files.
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Reference or label The SOM text says the dispersions "are given in Fig. 1 of the supporting online material for several values of d", but the only SOM figure is captioned "Figure 4: Dispersion relations for the E1 and H1 subbands".
- p. 21 SOM, last sentence of page
are given in Fig. 1 of the supporting online material
- p. 22 Figure caption
Figure 4: Dispersion relations for the E1 and H1 subbands
- p. 21 last sentence of page
are given in Fig. 1 of the supporting online material for several values of d
- p. 21 last sentence
given in Fig. 1 of the supporting online material
Judge's note: Viewed pages 21 and 22. Page 21 ends with the pointer to "Fig. 1 of the supporting online material"; the dispersion figure on page 22 is labelled Figure 4, continuing the main-text numbering. No SOM figure is labelled Fig. 1, and the main-text Figure 1 is a different figure. The pointer and the label do not match.
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Wording Text says "gap parameter M, which is the energy difference between the E1 and H1 levels at the Γ point", but with H = ε(k) + d_iσ_i and d3(0) = M (Eqs. 2-3, 25) the levels are C+M and C−M, so the difference is 2M.
- p. 5 first paragraph
gap parameter M, which is the energy difference between the E1 and H1 levels at the Γ point
- p. 21 Eq. 25 diagonal
ǫk + M(k)
- p. 4 Eq. 3
d3 = M − B(k2x + k2y)
- p. 21 Table 1, column M(eV)
0.00922
Judge's note: Viewed pages 4, 5, 14 and 21. Eq. 25 has diagonal entries ε_k ± M(k), so at k=0 the E1/H1 splitting is 2M. Table 1 gives |M| = 0.00686 eV at 70 Å, i.e. a splitting of 0.0137 eV, which agrees with the roughly 0.015 eV E1–H1 separation in Fig. 2A at 70 Å. I found no definition elsewhere making M the full difference. Wording is off by a factor of 2; M is half the difference.
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10 minor (typos and the like; most checked by the judge only)
Reference or label Table 1 header gives "A(eV) B(eV) C(eV) D(eV) M(eV)", although in Eq. 25 A multiplies k and B, D multiply k^2, so only C and M can be plain energies for the continuum Hamiltonian.
- p. 21 Table 1 header
A(eV) B(eV) C(eV) D(eV) M(eV)
- p. 6 text after Eq. 4
for k ∈ [0, 0.01 Å−1] the dispersion relation is dominated by the Dirac linear terms
Only the authors can say what is right here.
Judge's note: Viewed page 21 (Table 1, Eq. 25) and page 6. Eq. 25 has A k± and ε_k = C − D(kx²+ky²), M(k) = M − B(kx²+ky²), with k quoted in Å⁻¹ on page 6 and nm⁻¹ in Fig. 4. Taking A as eV·Å, A·k at k = 0.04 Å⁻¹ is about 0.145 eV, which matches the scale of roughly 0.1–0.15 eV in Fig. 4 at 0.4 nm⁻¹; this supports Å-based units. The header is dimensionally loose unless a lattice constant is set to 1 as in the tight-binding form of Eq. 5, but no lattice constant is stated. The authors should confirm the intended units (eV·Å for A, eV·Å² for B and D).
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Reference or label Reference 27 reads "K. Ortner et. al., Applied Physics Letters 79, 3980 (2000)."; reviewer says volume 79 was published in 2001.
- p. 12 Reference 27
K. Ortner et. al., Applied Physics Letters 79, 3980 (2000).
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that Applied Physics Letters volume 79 appeared in 2001, not 2000. Viewed page 12: the entry is printed as quoted. No other Applied Physics Letters entry in the list allows an internal check of volume against year, so only the authors can confirm.
Second check: another model agreed
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Wording Text gives "At d = dc ∼ 64 Å these bands cross" (and caption 63.5 Å), while the E1/H1 crossing drawn in Fig. 2A sits at about 62 Å on the plotted axis.
- p. 5 second paragraph
At d = dc ∼ 64 Å these bands cross
- p. 9 last lines
d < dc ≈ 64 Å and d > dc ≈ 64 Å
- p. 14 Figure 2A, crossing point and 'dc' label
dc
Judge's note: Viewed pages 5, 9 and 14. On the page image the 50/60/70 ticks are about 103 px apart and the crossing (and the dc label) lies about 23 px past the 60 tick, i.e. about 62 Å. Linear interpolation of Table 1 M values (0.00922 at 58 Å, −0.00686 at 70 Å) gives a zero near 65 Å, which supports the text rather than the figure. The gap is about 2 Å on a low-resolution plot and the text uses ∼/≈, so this is only a minor consistency point.
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Wording In "T = Ec(z) + (−∂zA(z)∂z)" the symbol A(z) (and A^(Cd), A^(Hg) in Eqs. 15-17) is never defined, and the letter A is reused for the linear-in-k coefficient in Eqs. 3, 25 and Table 1.
- p. 17 definitions below Eq. 10
T = Ec(z) + (−∂zA(z)∂z)
- p. 21 Eq. 25 and text below
A, B, C, D, M depend on the specified quantum-well width
Judge's note: Viewed pages 16-19 and 21 and searched the text for A(z). No definition appears; Eq. 8 writes the conduction term as ħ²k²/2m*, so A(z) is presumably ħ²/2m*(z). T and U are also reused (interband matrix in Eq. 7, unitary matrix in Eq. 27). Meaning is recoverable from context, so this is a notation point only.
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Typo Eq. 1 prints the last two spinor components with no separating comma: "|Γ8, −1/2⟩|Γ8, −3/2⟩".
- p. 3 Eq. 1
|Γ8, −1/2⟩|Γ8, −3/2⟩
Judge's note: Viewed page 3. The first four components are comma-separated; the fifth and sixth are printed adjacent with no comma. Typeset that way, not an extraction artefact.
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Typo Reference 5 prints the article number as five digits: "Phys. Rev. Lett. 94, 47204 (2005)."; other entries use six digits including leading zeros.
- p. 11 Reference 5
Phys. Rev. Lett. 94, 47204 (2005).
Judge's note: Viewed page 11. Every other Phys. Rev. entry in the list has a six-digit identifier, and refs 9 and 12 keep their leading zero (085308, 045322). The entry is missing the leading zero (047204). Format-only; the reference remains identifiable.
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Typo Reference 25 prints the first author as "M. F. L. Goleterman"; reviewers say the surname is Golterman.
- p. 12 Reference 25
M. F. L. Goleterman, K. Jansen, D.B. Kaplan, Phys. Lett. B 301, 219 (1993).
- p. 12 Reference 25
M. F. L. Goleterman, K. Jansen, D.B. Kaplan
- p. 12 Ref. 25
M. F. L. Goleterman
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that the first author of Phys. Lett. B 301, 219 (1993) is spelled Golterman. Viewed page 12: the entry does read "Goleterman". Nothing in the files gives the name a second time, so the spelling cannot be cross-checked internally.
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Typo Eq. 18 lacks brackets: "Hf3(z) = Ev(z) − (γ1(z) − 2γ2(z))(−∂z²)f3(z) = Ef3(z).", so Ev(z) is not applied to f3(z).
- p. 19 Eq. 18
Hf3(z) = Ev(z) − (γ1(z) − 2γ2(z))(−∂z²)f3(z) = Ef3(z).
Judge's note: Viewed page 19. As printed, the middle expression is a bare potential minus an operator acting on f3. The β² and κ² expressions that follow correspond to [Ev(z) − (γ1 − 2γ2)(−∂z²)] f3(z). Typesetting omission.
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Typo Superfluous "and" in "state with only f6(z) non-zero and to get the E1− and H1− bands respectively".
- p. 20 SOM, first paragraph
state with only f6(z) non-zero and to get the E1− and H1− bands
Judge's note: Viewed page 20. The sentence reads "...on a state with only f6(z) non-zero and to get the E1− and H1− bands respectively"; the "and" before "to get" is a stray word.
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Typo Stray duplicated object in the SOM sentence: "and puts it H into the form".
- p. 21 SOM, text between Eq. 27 and Eq. 28
and puts it H into the form
Judge's note: Viewed page 21. Between Eq. 27 and Eq. 28 the page prints "...in the lower block and puts it ℋ into the form"; both the pronoun and the script H are on the page image, so this is not an extraction artefact. One of the two should be deleted.
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