Latent Haldane Models

Anouar Moustaj; Cristiane Morais Smith; Lumen Eek; Malte Rontgen

REVIEW 3 major objections 4 minor 1 cited by

Latent Haldane Models

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Decorated two-dimensional lattices can hide a Haldane model: after an isospectral reduction to two sites, their gap-closing energies and Chern phase boundaries become analytically accessible.

desk verdict A useful 2D extension of isospectral reduction with analytic phase diagrams, but the load-bearing step—transferring Chern numbers from the full lattice to the latent Haldane model—is asserted rather than proved. read the letter →

arxiv 2411.08202 v2 pith:GBTFY7PK submitted 2024-11-12 cond-mat.mes-hall cond-mat.str-elquant-ph

Anouar Moustaj , Lumen Eek , Malte Rontgen , Cristiane Morais Smith This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.str-elquant-ph

keywords isospectralreductionHaldanemodellatentsymmetriesSemenoffmassChernnumbertopologicalphasediagramdecoratedlatticesgraphyne

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 works out a way to turn certain complicated two-dimensional decorated lattices into much simpler effective models: after an isospectral reduction (ISR), a family of such lattices is described by a $2\times2$ Bloch Hamiltonian with the same form as the Haldane model, but with energy-dependent coefficients. The reduced Hamiltonian can carry a latent Semenoff mass, an inversion-breaking term that appears without any staggered on-site potential, and, when the decoration uses complex nearest-neighbor hoppings with the right phase differences, a latent Haldane mass. Because the effective model is the Haldane model, the paper can use that model's known gap-closing condition and Chern-number phase diagram to predict, analytically, the gap-closing energies and topological phase boundaries of the original multi-band lattices. If the prediction is right, then for fillings such as $\nu=1$, $4$, $7$, and $9$ in the worked examples the decorated lattices host the Chern phases assigned by the reduced model. The construction generalizes: replacing the decorating substructures with arbitrary graphs still produces the same latent masses, so the family of lattices with a hidden Haldane description is broad.

What carries the argument

The central object is the isospectral reduction (ISR), which replaces the full Hamiltonian $H$ by the energy-dependent effective Hamiltonian $\tilde H_S(E)=H_{SS}-H_{S\bar S}(H_{\bar S\bar S}-EI)^{-1}H_{\bar S S}$ on a chosen set of sites $S$ (with $\bar S$ the complement of $S$); it preserves the spectrum exactly. In this paper the reduction is performed onto the two sites $A$ and $B$ of each unit cell, collapsing many bands into a $2\times2$ Bloch Hamiltonian of Haldane form with energy-dependent coefficients $A(E)$, $T(E)$, $M(E)$, and $\Lambda(E)$. The reduction is what makes the latent masses visible: an asymmetry between the hopping neighborhoods of $A$ and $B$ shows up as $M(E)$ on the diagonal, and complex nearest-neighbor hoppings with phases $2\pi/3$ show up as $\Lambda(E)$ multiplying the Haldane phase function $f_\phi(k)$ of Eq. (2). The paper then uses the low-energy Dirac expansion of the Haldane model, where the Chern number is controlled by the sign of $M-3\sqrt3\lambda\sin\phi$, to convert gap-closing equations into topological phase boundaries.

What would settle it

Compute the full Bloch Hamiltonian of the modified $\alpha$-graphyne model at filling $\nu=1$, integrate the Berry curvature over the Brillouin zone for parameter pairs on both sides of the predicted critical value $\delta_c\approx 0.271558$, and compare the resulting Chern number with the $C=1$ or $C=0$ label from the latent Haldane model; a mismatch would show that the topology does not transfer from the reduced model to the original lattice.

Watch

Extended reading notes

Core claim

The central claim is that the physics of the Haldane model is latent in a family of decorated two-dimensional lattices and can be made explicit by an isospectral reduction onto two sites per unit cell. The reduction produces an energy-dependent Haldane-Bloch Hamiltonian whose diagonal terms contain a latent Semenoff mass $M(E)$ when the hopping neighborhoods of the two reduced sites differ, and whose coupling terms contain a latent Haldane mass $\Lambda(E)$ when complex nearest-neighbor hoppings with relative phases $2\pi/3$ are attached to those sites. The paper derives explicit formulas for these quantities in a modified $\alpha$-graphyne lattice and in a decorated hexagonal plaquette, and it shows that the energies at which the original gaps close are the solutions of simple energy-dependent equations, such as $E-A(E)=0$ for the gapless case and Eq. (22) for the full model. Phase diagrams for fillings $\nu=1$ and $\nu=4$ (and topological edge states for $\nu=1$, $7$, and $9$) then follow from the reduced model's Chern numbers, so the topological transitions of the complicated lattice are predicted without diagonalizing the full multi-band problem.

Load-bearing premise

The load-bearing assumption is that the Chern number of a filled band cluster of the original multi-band lattice equals the Chern number of the energy-dependent $2\times2$ reduced Haldane model evaluated at the corresponding gap-closing energy, a transfer the paper checks only through ribbon edge states at selected parameters.

Editorial extensions

If this is right

Gap-closing energies of the decorated lattices can be obtained from energy-dependent equations such as $E-A(E)=0$, so topological phase-transition points can be located without diagonalizing the full multi-band problem.
A latent Semenoff mass appears whenever the hopping neighborhoods of the two reduced sites differ, so a gap can be opened and closed by tuning a hopping asymmetry $\delta$ rather than by adding an on-site staggered potential.
Complex nearest-neighbor hoppings with a $2\pi/3$ phase difference generate a latent Haldane mass, so the decorated lattices realize Chern insulators without the next-nearest-neighbor complex hoppings of the original Haldane model.
Replacing the decorating substructure by an arbitrary graph $G$ attached at a single site still produces the latent Haldane mass, and a reflection-symmetric graph $G$ produces the latent Semenoff mass, so the family of lattices with a hidden Haldane description is broad.
For fillings $\nu=1$, $4$, $7$, and $9$ in the worked examples, the phase diagrams assign definite Chern numbers and the ribbon-geometry edge states match those assignments, including the always-topological $\nu=4$ case.

Reading between the lines

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

If the Chern-number transfer holds in general, the ISR becomes a general dimension-reduction diagnostic: any lattice that reduces to a known two-band model would inherit that model's full phase diagram, not just its spectrum.
Because the latent masses are set by hopping amplitudes and decoration geometry rather than by site energies, engineered lattices such as photonic, phononic, or electric-circuit arrays could realize the predicted Chern phases by tuning bond strengths, which is often experimentally easier than controlling on-site potentials.
A direct test of the paper's implicit assumption would be a systematic calculation of the original lattice's many-band Berry curvature for every gap in the parameter plane, comparing its Chern numbers with the reduced model's labels instead of relying on selected ribbon spectra.
The same reduction recipe applied with spin degrees of freedom should produce latent Kane-Mele type models, and the paper's own generalization principle suggests that constructing such spinful decorated lattices is a concrete next step.

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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, here is my take on Moustaj et al. The paper moves the isospectral reduction (ISR) program from 1D to 2D in a genuinely useful way: it shows that a family of decorated lattices reduce to energy-dependent Haldane-like Hamiltonians, with latent Semenoff and Haldane masses emerging from hopping geometry rather than staggered potentials or next-nearest-neighbor terms. The explicit ISR algebra in Sec. III and Appendix A is checkable, and the gap-closing energies predicted from Eqs. (19) and (20) match the band structures and ribbon spectra shown. That part is solid and is the real contribution. The construction principles in Appendix A are a nice generalization that should let others build similar models.

The soft spot is exactly where the reader puts it: the phase diagrams in Figs. 6 and 10 assign C=0/C=1 labels from the 2x2 reduced model, and the paper never proves that the full multi-band lattice carries the same Chern number at those fillings. ISR preserves the spectrum, but the reduced eigenstate is related to the full one by a k-dependent operator that enters the Berry connection; nothing in the manuscript shows those corrections integrate to zero over the Brillouin zone. The ribbon edge-state checks are good evidence, but only at selected points. I also note that Eq. (23) appears without derivation and Eq. (22) is stated rather than derived—minor, but annoying in a paper whose selling point is analytic control. No code is shipped, which is not fatal but would have helped verify the Chern numbers numerically.

I do not see circularity: the latent masses are derived from the original Hamiltonians, not fitted to the target results, so the reader's low circularity burden is correct. The self-citation is heavy, but the cited ISR work is genuinely theirs and relevant.

Who should read this: people designing topological lattice models, and anyone wanting a concrete 2D example of ISR beyond 1D. It deserves a serious referee: the construction is new and checkable, and the missing Chern-transfer theorem is a fixable gap rather than a fatal flaw. I would send it to an expert in topological band theory, asking for either a proof or a numerical sweep of Chern numbers over the full Brillouin zone, plus derivations of Eqs. (22) and (23).

Referee Report

3 major / 4 minor

Summary. The manuscript develops a construction principle by which decorated two-dimensional tight-binding lattices reduce, under an isospectral reduction (ISR), to energy-dependent 2x2 Bloch Hamiltonians of Haldane form. It identifies an energy-dependent latent Semenoff mass arising from asymmetric hopping neighborhoods and an energy-dependent latent Haldane mass arising from complex nearest-neighbor hoppings with 2pi/3 phase differences. Using the standard Haldane gap-closing condition M = +/-3sqrt(3) lambda sin(phi), the authors predict gap-closing energies (Eqs. (19), (20), (22)) and construct phase diagrams in the parameters delta and phi (Figs. 6 and 10) for fillings nu=1, 4, 7, and 9, with selected points checked against ribbon edge spectra (Figs. 7 and 11). A generalization to arbitrary substructures is presented in Section IV and Appendix A.

Significance. If the topological transfer is supplied, this is a useful framework: it converts the spectral problem of a family of multi-band lattices into a low-dimensional energy-dependent Haldane problem, gives analytic predictions for gap-closing positions, and provides explicit, checkable construction principles for the latent Semenoff and Haldane masses. The predictions for gap-closing energies follow rigorously from isospectrality of the ISR and are confirmed by the band structures shown in the figures. The paper also offers a concrete route to lattices whose topological response is controlled by hopping parameters rather than by an on-site staggered potential. The main open point is whether the Chern labels of the full lattice coincide with those of the reduced energy-dependent model; this is a correctness risk that the manuscript currently does not resolve.

major comments (3)

[Sections III C and III D; Figs. 6 and 10] The phase diagrams label regions C=0 and C=1 using the reduced 2x2 Haldane model, but the manuscript does not prove that the Chern number of a filled band cluster of the original decorated lattice equals the Chern number of the energy-dependent reduced model evaluated at the gap-closing energy. The ISR preserves the spectrum, and the eigenvectors of the reduced model correspond pointwise in k to the S-component of the full eigenvectors, but the full eigenvector contains additional components on the eliminated sites; its Berry connection therefore differs by k-dependent terms involving W(k) and grad_k W(k), and no argument is given that these integrate to zero over the Brillouin zone. Since the reduced model is itself energy-dependent, its Chern number is only defined after fixing E at a gap-closing position, and it is not automatic that this invariant equals the full-model Chern number. The ribbon edge-state checks in Figs. 7 and 11 cover selected parameters but not the entire phase boundaries. Please provide a proof (for example an adiabatic or homotopy argument relating the full and reduced eigenvectors, or a direct calculation of the full-model Chern numbers along the phase boundaries), or explicitly qualify the phase diagrams as predictions obtained from the latent model that still require verification in the full lattice.
[Section III D, Eq. (23)] The analytic phase boundary for nu=1 shown in Fig. 10(a) is stated through Eq. (23), with only the remark that the energies corresponding to the fillings are plugged into Eq. (9). The intermediate step giving the explicit solutions E* of Eq. (22) and their substitution into the mass condition is not shown, so Eq. (23) is not checkable as printed. Please include the derivation or at least a clearly defined expression for the plotted critical curve, so that the analytic phase boundary can be reproduced.
[Section III C, Eqs. (19)-(20) and Fig. 6(b)] The phase diagrams use fillings nu=1, 4, 7, and 9, but the text does not spell out which band gap of the full lattice corresponds to each filling, nor why the reduced-model gap-closing condition evaluated at that filling is the relevant one. For example, the statement after Eq. (20) that the nu=4 filling is always topological regardless of delta is inferred from the vertical line phi=0 in Fig. 6(b), but the argument identifying the nu=4 gap with the relevant solution of the latent-model condition is not given. Please state the filling-to-gap correspondence explicitly for every phase diagram.

minor comments (4)

[Fig. 5 caption] The caption says the band structures are for unequal values of the latent Haldane mass, but the model in Section III C has no latent Haldane mass; the caption should refer to different values of delta or of the hopping asymmetry (t1, t2).
[Fig. 11 caption] The caption describes the model as the latent Semenoff mass model, but the plotted system is the full latent Haldane mass model of Section III D; please correct the terminology.
[Eq. (23) and Fig. 10] Equation (23) is typeset ambiguously: it is unclear whether the final term delta(2t3^2 - 9 t_tilde^2) is inside or outside the square root, and the figure axes label the hopping parameter as g while the text uses t_tilde; please clarify both points.
[Section III B, paragraph after Eq. (15)] The proof-of-principle model adds the Semenoff and Haldane terms by hand to the reduction sites, which is correctly described as a warm-up, but it should be stated even more explicitly that this is not a latent-mass example; the latent masses appear only in Sections III C and III D.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the latent masses and gap-closing conditions are derived algebraically from the explicit decorated-lattice Hamiltonians, with no fitted parameter or self-citation carrying the central predictions.

full rationale

The central derivation is self-contained. The reduced 2x2 Hamiltonians in Eqs. (14), (17)-(18), and (21) are obtained by evaluating the isospectral reduction formula (10) on the explicit tight-binding Hamiltonians, so the latent Semenoff mass M(E) and the latent Haldane mass Lambda(E) are explicit functions of the original hopping parameters (t1, t2, t3, delta, t-tilde) rather than quantities fitted to the target phase boundaries. The gap-closing energies are solved from algebraic conditions such as E - A(E) = 0 and M(E*) = +/-3 sqrt(3) lambda sin(phi), which follow from isospectrality together with the textbook Haldane mass relation; the critical values like delta_c in Eqs. (20) and (23) are then obtained by solving those equations, not by reading them off the phase diagram. The paper does not invoke any uniqueness theorem or ansatz from the authors' prior work to force its conclusions; the self-citations to earlier ISR applications are contextual and do not carry the derivation. The only load-bearing step beyond the spectrum is the transfer of Chern numbers from the full multi-band lattice to the energy-dependent 2x2 reduced model, which is assumed rather than proved and checked only through ribbon edge states at selected parameters; however, that is an unproven topological-transfer assumption, not a circular reduction of the prediction to its inputs, because the phase labels are not used to define M(E) or Lambda(E) and the gap-closing positions follow already from isospectrality alone.

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

The paper introduces no new physical entities; the latent Semenoff and Haldane masses are derived effective parameters, not postulated objects. The central claims rest on standard tight-binding assumptions, the isospectral reduction theorem, and an unproven transfer of Chern numbers between the full lattice and the reduced model. The model parameters are construction inputs, not fits to external data.

free parameters (5)

t1 and t2 (set to t±δ)
Asymmetric hopping strengths in the modified graphyne model; their difference generates the latent Semenoff mass. Values are model inputs, not fitted.
t3
Intermediate hopping in the decorated graphyne chain; enters A(E) and M(E).
t~
Complex hopping amplitude to auxiliary sites in the decorated hexagonal plaquette; sets the latent Haldane mass scale.
λ and φ
Haldane term added by hand in the proof-of-principle and latent Semenoff models; the paper studies phase diagrams in δ and φ.
δ
Inversion-symmetry-breaking parameter swept to find phase boundaries; critical δc is derived, not fitted.

assumptions (5)

standard math ISR is isospectral: the eigenvalues of eH_S(E) coincide with those of the original Hamiltonian (Ref. [21]).
Used throughout Section III to replace the full lattice by a 2x2 energy-dependent model.
standard math The Haldane model gap-closing conditions M=±3√3 λ sinφ and E=-3λ cosφ at K and K' (Eqs. 4-6).
Basis for translating gap-closing conditions into phase boundaries in the latent models.
standard math Bulk-boundary correspondence: edge states in a ribbon imply a nonzero Chern number.
Used to validate phase diagrams in Figs. 7 and 11.
domain assumption The Chern number of the full multi-band lattice equals the Chern number of the 2x2 latent Haldane model for the fillings considered.
Not proven in the paper; the phase diagrams in Figs. 6 and 10 label C from the reduced model and edge states are checked only for selected parameters.
domain assumption Independent decorations contribute additively to the reduced Hamiltonian when their auxiliary sites are disjoint.
Used in Section III D when combining the latent Semenoff and Haldane decorations into a single onsite term and mass; no explicit proof of additivity is given.

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Cite this review

Pith. "Pith review of Latent Haldane Models." pith.science (2026). https://pith.science/paper/GBTFY7PK

@misc{pith2026241108202,
  author       = {Pith},
  title        = {Pith review of: Latent Haldane Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GBTFY7PK}},
  note         = {Machine review of arXiv:2411.08202}
}

abstract

Latent symmetries, which materialize after performing isospectral reductions, have recently been shown to be instrumental in revealing novel topological phases in one-dimensional systems, among many other applications. In this work, we explore how to construct a family of seemingly complicated two-dimensional models that result in energy-dependent Haldane models upon performing an isospectral reduction. In these models, we find energy-dependent latent Semenoff masses without introducing a staggered on-site potential. In addition, energy-dependent latent Haldane masses also emerge in decorated lattices with nearest-neighbor complex hoppings. Using the Haldane model's properties, we then predict the location of the topological gaps in the aforementioned family of models and construct phase diagrams to determine where the topological phases lie in parameter space. This idea yielded, for instance, useful insights in the case of a modified version of $\alpha$-graphyne and hexagonal plaquettes with additional decorations, where the gap-closing energies can be calculated using the ISR to predict topological phase transitions.

Figures

Figures reproduced from arXiv: 2411.08202 by the authors.

**Figure 2.** FIG. 2. Band structure of the Haldane model for a ribbon [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Unit cell of the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: FIG. 4. (a) Modified graphyne model and (b) ISR to graphene [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. (a)-(c) Band structures of [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. Phase diagram of the modified graphyne model (a) [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 8.** Figure 8: FIG. 8. Cell of a decorated honeycomb lattice. a) Side view [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

**Figure 7.** Figure 7: FIG. 7. Bandstructure of the latent Semenoff mass model [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 9.** Figure 9: FIG. 9. Band structures of the full latent Haldane model. The [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

**Figure 11.** Figure 11: FIG. 11. Bandstructure of the latent Semenoff mass model, [PITH_FULL_IMAGE:figures/full_fig_p008_11.png]

**Figure 8.** Figure 8: There, we coupled all the A sites via complexvalued hoppings to a single site and all the B sites, also via complex-valued hoppings, to another single site; cf [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 10.** Figure 10: FIG. 10. Phase diagram of the full latent Haldane model for [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]

**Figure 12.** Figure 12: FIG. 12. Cell of a decorated hexagonal plaquette. (a), (c) [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]

**Figure 13.** Figure 13: FIG. 13. As in Fig. 12(a), though with enumerated sites. [PITH_FULL_IMAGE:figures/full_fig_p011_13.png]

**Figure 14.** Figure 14: FIG. 14. Half of the plaquette depicted in Fig. 12(c), and with enumerated sites. [PITH_FULL_IMAGE:figures/full_fig_p012_14.png]

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Investigation of non-Hermitian and Hermitian models of Altermagnets
cond-mat.mes-hall 2025-09 reject novelty 4.0 of 10

A model of an insulating altermagnet with g-wave order and non-Hermitian terms is claimed to have Chern number +1, implying a quantum anomalous Hall insulator.

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Namely, HSS ΨS + HSSΨS = EΨS and HSS ΨS + HSS ΨS = EΨS
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(12), the eigenvalues of eHS(E) are the values of E for which Det( eHS(E) − EI ) = 0

As can be deduced from Eq. (12), the eigenvalues of eHS(E) are the values of E for which Det( eHS(E) − EI ) = 0
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With k being equal to either K or K′, for which g(k) vanishes
[58]

This equation can be derived in a similar manner as in the latent Haldane model discussed in Section III B
[59]

M permutes the two sites in G that are connected to S

To see this fact, one has to write the permutation ma- trix M corresponding to the reflection symmetry of G. M permutes the two sites in G that are connected to S. From the commutation of M and HG (the Hamiltonian describing G), it follows that the corresponding diago- nal ent...
[60]

Latent Haldane mass Let us start by the principle for generating a latent Haldane mass, as depicted in Fig. 12(a). This figure is again depicted in Fig. 13(a), though with additional numbers imprinted on each site to ease the discussion. The central six sites, that is, the red...
[61]

Latent Semenoff mass 1 2 3 4 6 7 8 9 5 FIG. 14. Half of the plaquette depicted in Fig. 12(c), and with enumerated sites. Finally, we come to the construction principle for a latent Semenoff mass. A simple setup for understanding the details of the underlying principle is given...

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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This results in the reduced Bloch- Hamiltonian HE(k) + λfϕ(k) 0 0 λf−ϕ(k)

(18) Once again, we can add the TR-symmetry breaking term (Haldane mass) by hand, connecting the sites where we perform the ISR. This results in the reduced Bloch- Hamiltonian HE(k) + λfϕ(k) 0 0 λf−ϕ(k) . Note that in this setup, inversion symmetry has been broken by the presence of the third hopping, which is why it induces the mass term in the ISR pictu...

work page

[3] [3]

(17) For simplicity, we now set t1 = t + δ and t2 = t − δ; the above two equations then become A(E) ≡ 3E(t2 + δ2) (E2 − t2 3) M (E) ≡ − 6Etδ (E2 − t2

work page

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(18), we obtain three equations that tell us for which parameter values the spectrum becomes gapless

(19) Inserting these values into M (E∗) = ±3 √ 3λ sin ϕ and using Eq. (18), we obtain three equations that tell us for which parameter values the spectrum becomes gapless. Note that, by analogy to the original Haldane model, these points also correspond to phase transitions. As an example, we take t3 = 1, t = 1, λ = 0 .2 and ϕ = π/2, with varying δ. For t...

work page

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Materi- als for the Quantum Age

− 9˜t2 2E = 0. (22) The solutions of this equation are represented in Fig. 9(b) by red lines. When the energies corresponding to the fillings ν = 1 and ν = 9 are plugged in Eq. (9), the 8 K' M K -2.7 0.0 2.7 E(k) (a) K' M K (b) K' M K (c) FIG. 9. Band structures of the full latent Haldane model. The bands cross at the energies predicted from Eq. (22). Par...

work page

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