{"id":"cb52cc9a-e485-453d-ba99-1e42e43e5359","arxiv_id":"2412.17084","paper_version":3,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A projection method with fewer trial orbitals than bands gives exponentially localized Wannier functions for a trivial subspace of Chern or Z2 topological bands.","lead":"This paper shows how to build exponentially localized Wannier functions that cover most, but not all, of a topologically obstructed band manifold, using supercells and a reduced set of trial orbitals. The approach splits the band manifold into a localized trivial part and a small topological remainder, which may help in modeling correlated flat-band systems.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim depends on the rectangular overlap matrix A_k having a uniform positive singular-value gap over the entire BZ, but full-BZ checks are missing for the 5x5 Haldane and 3x3 Kane-Mele supercells; only high-symmetry-path data are shown for the 2x2 cells.","rationale":"The reader's identified weakest assumption is the same one I find load-bearing. The method's logic is otherwise sound: if A_k is uniformly full rank, the projected states of Eq. (12) constitute a smooth frame for a J-dimensional subbundle of the occupied bundle, whose Chern number is zero; the complement then necessarily carries the original topology. The numerical demonstration of exponential decay and finite spreads is consistent with this, and the decomposition in Eq. (25) is internally coherent. No mathematical contradiction with the topology is present, because a Chern bundle can contain a trivial subbundle. The gap is between what is claimed (a substantial Wannierizable subspace at large supercell sizes) and what is verified (full rank only along paths for small cells). Since the missing check is straightforward but currently absent, the CONDITIONAL verdict stands unchanged.","tokens_in":815,"tokens_out":2397,"duration_ms":158351,"concrete_test":"For the trial-function subsets used in Figs. 8 and 14, compute the minimum singular value of A_k on a dense uniform BZ grid (e.g., 200x200) for the 5x5 Haldane and 3x3 Kane-Mele supercells, and check that it remains above a positive threshold (say 1e-3) as the grid is refined. Also repeat the 2x2 Haldane and Kane-Mele singular-value scans over the full BZ rather than only the high-symmetry path, and locate the global minimum. As an analytic cross-check for the 2x2 Haldane case, derive det(A_k^T A_k) as a trigonometric polynomial in (k_x,k_y) and verify positivity. If any singular value vanishes, test whether a different choice of omitted site restores a uniform gap; if none does, the construction is not generic.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The construction's exponential localization rests on the J-dimensional subspace defined by Eq. (12) being a smooth subbundle of the occupied manifold. This requires the M by J inner-product matrix A_k of Eq. (7) to have rank J with singular values bounded below by a positive constant for every k in the supercell BZ. The paper verifies this only along a high-symmetry path for the 2x2 Haldane (Fig. 2(b)) and 2x2 Kane-Mele (Fig. 11(b)) cases. For the largest Wannier fractions - 5x5 Haldane f_W=24/25 in Sec. III D and 3x3 Kane-Mele f_W=16/18 in Sec. IV D - no singular-value data or analytic bound are reported. Rank deficiency of a rectangular A_k is a codimension-two condition in k, so it can occur at generic points missed by a path; if it does, the reduced 'trivial' subspace is singular and the Wannier functions are not exponentially localized. The local check at the band-inversion point is therefore not sufficient to support the quantitative Wannier-fraction claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a construction of exponentially localized Wannier functions that span a subspace of a topologically obstructed band manifold. The key idea is to use fewer trial orbitals than occupied bands in the projection method, so that the rectangular overlap matrix A_k can have full column rank everywhere in the Brillouin zone, yielding a smooth gauge on the selected subspace. The complement of this 'reduced Wannier subspace' inherits the topological index. The method is demonstrated on the Haldane model in its Chern-insulating phase and on the Kane-Mele model in its Z2-odd phase, using supercells to increase the fraction of the occupied manifold that can be Wannierized. A Wannier fraction fW is defined, with a bound derived from the maximal null-space dimension of A_k.","tokens_in":16658,"tokens_out":8652,"duration_ms":91934,"significance":"If the construction works as claimed, it offers a practical way to separate a large 'trivial' part of a topological band manifold from the small 'itinerant' part that carries the Chern or Z2 obstruction. This could be useful in flat-band and moiré contexts where one wants a localized Wannier basis for correlation calculations while keeping the topological subspace accessible. The paper is clearly written, the numerical spreads and Berry-curvature results are internally consistent, and the authors provide an explicit rank-nullity bound on the achievable Wannier fraction. The implementation in an extension of PythTB is a further strength, as it makes the procedure reproducible in principle.","major_comments":[{"comment":"The central smoothness assumption is verified only for the initial projection in the small supercells, not for the final subspaces at the claimed maximum Wannier fractions. The bound fW <= 1 - Nmax/Nb is necessary but not sufficient: a fixed set of omitted columns can still fail to have full rank at some k, because the null space at rank-deficient points need not be contained in the omitted columns. The paper reports singular values only along a high-symmetry path for the 2x2 Haldane and Kane-Mele cases, and these are for the initial projection, before the subspace-selection and maximal-localization steps. No full-BZ singular-value or minimum-gap data are given for the 5x5 Haldane (fW=24/25) or 3x3 Kane-Mele (fW=16/18) supercells, nor is an analytic argument supplied. Since rank deficiency is a codimension-two condition that can occur at generic points missed by a path, this missing check is load-bearing for the claim that the reduced Wannier functions are exponentially localized.","section":"Secs. III.D and IV.D, Eq. (32), Figs. 2(b) and 11(b)"},{"comment":"The evidence for exponential localization is indirect. The figures plot bin-averaged density weights on a finite discrete-k supercell; they demonstrate decay of the density, not exponential localization of the Wannier amplitudes in the thermodynamic limit. Please either prove the existence of a smooth periodic gauge (for example, by establishing a uniform lower bound on the relevant singular values over the full BZ for the final optimized subspace), or provide a convergence study of the Wannier tails as the supercell and k-mesh sizes are increased. Without this, the central claim remains an assertion.","section":"Secs. III.B and IV.B, Figs. 3 and 12"},{"comment":"The statement that 'The dimensionality of the null space at the singularities remains invariant through band folding' is asserted without proof. In a supercell, multiple primitive-cell k-points may fold to the same supercell k-point, and it is not obvious that the null-space dimension at those points is simply the primitive-cell value. Please provide a justification or a direct numerical check for the 5x5 and 3x3 supercells.","section":"Sec. III.D, paragraph after Eq. (35)"}],"minor_comments":[{"comment":"The object (A_K)_{4,n} with n varying is a row of the inner-product matrix, but the text calls it 'an entire column of zeros.' Please correct the row/column terminology.","section":"Eq. (23)"},{"comment":"The sentence 'If A is singular, then V W† becomes semi-unitary' is incorrect; for a square matrix A, V and W from the SVD are unitary even when A is singular, and the breakdown is the divergence of S^{-1/2}. The semi-unitary case is the rectangular one of Eq. (12). Please rephrase to avoid confusion about the mechanism.","section":"Sec. II.B, discussion after Eq. (11)"},{"comment":"The notation 1_{M x J} should be defined explicitly, for example as the first J columns of the M x M identity matrix or as the rectangular diagonal matrix with unit entries.","section":"Eq. (12)"},{"comment":"The text says that trial wave functions are placed on 'randomly chosen subsets of low-energy sites' for the 5x5 supercell. Please specify the number of realizations, the random seed or distribution, and whether the plotted spreads are averaged over realizations, so that the results are reproducible.","section":"Sec. III.D, Fig. 8"},{"comment":"The phrase 'the supercell conjugate to the discrete k-mesh' is unclear. Please specify the real-space supercell size and the k-mesh used for the plots, and clarify whether the plotted quantity is the Wannier function density or the site-projected weight.","section":"Figs. 3 and 12, insets"},{"comment":"Reference [31] is cited as the implementation but is not described. Please indicate whether this is a publicly available code repository and provide a URL or DOI if applicable.","section":"References, [31]"}],"recommendation":"major_revision","confidential_remarks":"The paper presents an interesting and potentially useful decomposition, and I do not see a fundamental inconsistency in the approach. The main obstacle to acceptance is the missing verification of the uniform singular-value gap for the maximum-Wannier-fraction supercells, which is the load-bearing condition for exponential localization. This is fixable within the scope of the manuscript by adding full-BZ numerical checks or an analytic argument. I would not reject on the basis of disagreement with current consensus; the claim is a new construction rather than a challenge to known no-go theorems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper proposes a simple and sensible trick: to Wannierize a topological band, deliberately use fewer trial functions than there are bands, so the overlap matrix becomes rectangular and full rank even at the band-inversion point. The resulting Wannier functions span a trivial subspace that can be exponentially localized, and the complement carries the Chern number or Z2 invariant. That is a genuinely new combination of existing tools (projection, disentanglement, maximal localization), and it is demonstrated on Haldane and Kane-Mele models with supercells. The rank-nullity bound on the Wannier fraction is clean and correct, and the numerical spreads, Berry curvature, and this decomposition are internally consistent. I find the central idea credible.\n\nThe real soft spot is the proof of exponential localization. The construction requires the rectangular overlap matrix A_k to have a uniform positive singular-value gap over the whole BZ. The paper shows singular values only along a high-symmetry path for the 2x2 supercells. For the 5x5 Haldane (fraction 24/25) and the 3x3 Kane-Mele (16/18), no such data are shown. Since rank deficiency is a codimension-two condition in 2D, it could occur at a generic point missed by a path. If that happened, the reduced subspace would be singular and the Wannier functions would not be exponentially localized. The exponential decay plots for the 2x2 cases are convincing, but they don't cover the largest fractions. I'd want to see full-BZ singular-value spectra or an argument that the gap is generically nonzero and bounded below. This is fixable and not obviously wrong, but it is the difference between a conditional and a solid claim.\n\nMinor points: no code is released, which makes reproduction harder, though the method is simple enough to reimplement. The paper is honest about the trade-offs: broken translational symmetry, larger spreads near the omitted site, and the fact that the Wannier states are not energy eigenstates.\n\nThis is a methods paper for the Wannier and flat-band communities. If the full-BZ checks come through, it will be useful for partially Wannierizing obstructed bands in moiré models. I would send it to a serious referee; it deserves a round of revision rather than a desk reject.","headline":"A credible and potentially useful method for partially Wannierizing Chern and Z2 bands, but the exponential-localization claim would be stronger with full-BZ singular-value checks for the larger supercells.","tokens_in":17148,"tokens_out":4626,"would_cite":true,"duration_ms":43293,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"By using fewer trial orbitals than bands, exponentially localized Wannier functions can be built for most of a topologically obstructed manifold, leaving a small complement that carries the Chern or Z2 index.","keywords":["Wannier functions","topological obstruction","Chern insulator","Haldane model","Kane-Mele model","Z2 index","supercell","projection method"],"falsifier":"Compute the smallest singular value of $A_k$ on a dense full Brillouin-zone mesh for the $5\\times5$ Haldane supercell at $f_W=24/25$ and the $3\\times3$ Kane-Mele supercell at $f_W=16/18$; if it reaches zero at any point, the claimed exponential localization of the reduced Wannier functions fails.","tokens_in":16188,"feed_emoji":"🔬","tokens_out":6963,"duration_ms":54643,"temperature":0.7,"pith_summary":"The paper claims that a topologically obstructed band manifold can be split into a large trivial subspace that admits exponentially localized Wannier functions and a small complementary subspace that carries the topological index. The construction uses fewer trial orbitals than there are bands, so the projection overlap matrix stays full rank everywhere in the Brillouin zone instead of becoming singular at the band-inversion momentum. The authors demonstrate this on Haldane and Kane-Mele models in supercells, reaching Wannier fractions of 3/4 and 6/8 in 2x2 cells and up to 24/25 and 16/18 in larger cells. If correct, the method gives a practical route to localized tight-binding-like descriptions of most of a Chern or Z2 band, with the obstructive sector cleanly separated.","feed_headline":"Topological bands yield localized orbitals if you drop one state","feed_subtitle":"New construction splits a Chern or Z2 band into a localizable part and a tiny topological remnant.","key_machinery":"The key object is the rectangular inner-product matrix $A_{mn}(k)=\\langle\\psi_{mk}|g_n\\rangle$ between Bloch eigenstates of the target manifold and a set of $J$ trial orbitals. When $J=M$ and the band is topological, $A_k$ drops rank at the inversion momentum and the $S_k^{-1/2}$ normalization diverges; when $J<M$, $A_k$ is $M\\times J$ and full rank, so its singular value decomposition yields a semi-unitary projector $V\\,1_{M\\times J}\\,W^\\dagger$ defining a smooth $J$-dimensional subspace. That subspace is then refined by subspace selection, which minimizes the gauge-invariant spread $\\Omega_I$, and maximal localization, which minimizes the gauge-dependent spread $\\tilde\\Omega$, producing the exponentially localized reduced Wannier functions whose complement carries the topology.","core_discovery":"The central discovery is that the topological obstruction to exponential Wannier localization can be confined to a lower-dimensional subspace rather than poisoning the whole band manifold. In the projection method, the obstruction appears as a rank deficiency of the $M \\times J$ inner-product matrix $A_k$ at momenta where the band character inverts. By omitting trial functions so that $A_k$ becomes rectangular with full rank at every $k$, the projected Bloch-like states remain smooth and periodic, and the resulting $J$ Wannier functions decay exponentially. The complement, obtained by subtracting the trivial projector from the occupied projector, inherits the Chern number or the Z2 oddness; for a $C=1$ Haldane band it is one band, and for a $Z_2=1$ Kane-Mele insulator it is one Kramers pair. The number of Wannier functions that can be constructed is bounded by $1 - N_{\\max}/N_b$, where $N_{\\max}$ is the maximal null-space dimension of $A_k$, and supercell folding raises the fraction toward one while breaking primitive translational symmetry.","pith_inferences":["The maximal null-space dimension $N_{\\max}$ of $A_k$ is effectively a real-space measure of the obstruction; it might be possible to prove a general identity tying $N_{\\max}$ to the Chern number or $Z_2$ index, making the Wannier-fraction formula exact rather than an upper bound.","The omitted trial site acts like a localized defect whose position is a free choice; treating that position as a variational parameter could reduce spreads further, and in magic-angle twisted bilayer graphene the 'fidget spinner' charge pattern suggests which three sites to keep.","The decomposition suggests a concrete strategy for interacting flat-band models: if a fractional filling occupies the topological sector, the trivial Wannier sector still supplies a local basis for the remaining degrees of freedom, an application the paper motivates but does not test."],"forward_implications":["For a Chern insulator with $C=1$, all but one band of the occupied manifold can be Wannierized, with Wannier fraction bounded by $f_W \\le 1 - 1/N_{sc}^2$ on an $N_{sc}\\times N_{sc}$ supercell.","For a $Z_2$-odd insulator, all but one Kramers pair admit exponentially localized Wannier functions respecting time-reversal symmetry, with fraction bounded by $f_W \\le 1 - 2/n_{occ}$.","The trivial subspace yields Wannier-interpolated bands that track the low-energy occupied bands, while the topological subspace reproduces the inverted character at the band-inversion point and carries the net Chern number.","The procedure works without borrowing opposite-topology bands, so it remains applicable when no nearby band of opposite Chern number exists, and it can be incorporated into first-principles Wannier workflows.","In flat-band or moiré systems, a supercell commensurate with a charge-density wave could isolate a Wannierizable trivial sector, potentially simplifying strongly correlated calculations."],"supporting_citations":[{"why":"Supplies the projection, subspace selection, and maximal-localization formalism that the reduced construction builds on.","marker":"[29]"},{"why":"Provides the disentanglement and subspace-selection procedure used to minimize the gauge-invariant spread of the projected subspace.","marker":"[28]"},{"why":"Gives the spread functional and gradient formulas used to carry out maximal localization.","marker":"[32]"},{"why":"Shows how the Z2 obstruction appears in Wannier constructions and how breaking time-reversal symmetry can resolve it, the starting point the paper generalizes.","marker":"[19]"},{"why":"Establishes the behavior of the gauge-invariant spread across the Chern-insulator transition, which motivates the finite-metric observations.","marker":"[18]"},{"why":"Provides the alternative of optimally localized algebraic Wannier functions with power-law tails, the comparison case the paper deliberately avoids.","marker":"[23]"},{"why":"Defines Wannier obstructions and fragile topology, the broader context of why such decompositions matter.","marker":"[17]"}],"fun_headline_variants":["Localized Wannier states for topological bands: drop one state","Split a Chern band: localize everything but a single remnant","Reduced Wannier trick: keep topology in a tiny subspace","Exponential localization in topological bands by omitting a state","Topological band decomposition: localizable part plus a remnant"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method requires that after omitting enough trial functions, the rectangular overlap matrix $A_k$ has full rank with all singular values bounded away from zero at every $k$ in the Brillouin zone; the paper checks singular values along high-symmetry paths but does not report a full-zone verification for every supercell or an analytic proof.","fun_headline_variants_meta":{"raw":{"variants":["Localized Wannier states for topological bands: drop one state","Split a Chern band: localize everything but a single remnant","Reduced Wannier trick: keep topology in a tiny subspace","Exponential localization in topological bands by omitting a state","Topological band decomposition: localizable part plus a remnant"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000257,"raw_usage":{"total_tokens":1563,"prompt_tokens":913,"completion_tokens":650,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":529,"completion_tokens_details":{"reasoning_tokens":564}},"tokens_in":529,"tokens_out":650,"duration_ms":6231,"temperature":1.0,"reasoning_tokens":564,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:49:03.560712+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the smallest singular value of $A_k$ on a dense full Brillouin-zone mesh for the $5\\times5$ Haldane supercell at $f_W=24/25$ and the $3\\times3$ Kane-Mele supercell at $f_W=16/18$; if it reaches zero at any point, the claimed exponential localization of the reduced Wannier functions fails.","supporting_citations":[],"review_version":1}