{"id":"79c1fff0-71fb-4fee-8019-908eb67a66c1","arxiv_id":"2412.02782","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"The paper derives an analytical phase diagram for a 2D stacked non-Hermitian SSH model and shows that open-boundary-only Dirac points with integer topological charge and geometry-dependent locations exist.","lead":"A stacked non-Hermitian chain model is shown to host Dirac-like band crossings that only exist under open boundary conditions, not in the periodic bulk. The paper derives the full phase diagram and shows these \"non-Bloch Dirac points\" carry a topological charge and shift with boundary geometry.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The open-boundary phase diagram is derived on a cylinder (OBC x, PBC y); the unproved claim that Hermitian y hopping makes PBC and OBC in y identical leaves the 'non-Bloch DP under open boundary conditions' claim conditional.","rationale":"The reader's weakest assumption identifies the same step. My read is partial agreement: for a rectangular sample the PBC/OBC y equivalence is likely true in the thermodynamic limit because the y-direction coupling is a real symmetric matrix that can be diagonalized independently, leaving a family of 1D non-Hermitian chains in x. But the paper states this without the proof, and the finite-size version is nontrivial: exact DPs exist only for discrete ky_n that match ky,c. The more serious unresolved part is the parallelogram geometry, where the non-Hermitian x direction is oblique to the edges and the simple separation into transverse modes is lost; Section IV asserts the DP locations without derivation. Since the central claim and the claimed novelty both depend on this reduction, the manuscript is not fully accepted without a check. I do not see an internal algebraic contradiction in the non-Bloch spectrum derivation—the apparent Eq. (1)/Eq. (2) mismatch and the sign in Eq. (7) are cosmetic, as the characteristic equation and the spectral range are unaffected—so the main risk is the boundary-reduction step. The concrete exact-diagonalization test would settle it.","tokens_in":13005,"tokens_out":47808,"duration_ms":503194,"concrete_test":"Numerically diagonalize the full real-space Hamiltonian with OBC in both x and y for the parameters of Fig. 3 (r=7/8, c=1/4, γ=1/4, w=1, η=0), e.g. Lx=80, Ly=20. Compare the low-energy spectrum to the cylinder formula Eq. (7) evaluated at the OBC-y transverse momenta ky_n=π n/(Ly+1). If the full-OBC spectrum matches the union of those 1D curves, the rectangle reduction is valid. Then repeat the same comparison on a parallelogram cluster with edges along x and x+y; if the extracted DP locations do not match the rectangle projection, verify whether they coincide with the kt_{c,1}, kt_{c,2} claimed in Section IV, since this is the load-bearing evidence for geometry-dependent bulk-boundary correspondence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central object—non-Bloch DPs under open boundary conditions—is computed exclusively in the cylinder geometry with periodic boundary conditions in y. Section II justifies this by saying 'Since the hopping along the y direction is Hermitian, taking PBC and OBC in y will yield the same results,' but no proof or finite-size check is given. This matters because Eq. (7) and the phase diagram use a continuous ky; in a fully open rectangle ky is quantized by transverse standing waves, so exact DPs occur only when a discrete ky_n coincides with ky,c, and for tilted/parallelogram boundaries the x-y separation that makes the cylinder reduction trivial no longer exists. The geometry-dependence claim in Section IV relies on the same reduction for each boundary shape. If the reduction fails for any of these geometries, the claim that non-Bloch DPs exist under open boundary conditions and have geometry-dependent locations loses support. The rectangular case can likely be rescued by diagonalizing the Hermitian y coupling exactly, but the paper does not show this, and the tilted case is not derived at all.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies a two-dimensional stacked non-Hermitian SSH model with non-reciprocal hopping along x and Hermitian coupling along y. The authors derive an analytical spectrum for a cylinder geometry (OBC in x, PBC in y) using the generalized Brillouin zone, obtain a phase diagram in terms of complex-energy line gaps, and identify real-spectrum gap closings that they call non-Bloch Dirac points. They map the non-Hermitian Hamiltonian locally to a Hermitian Hamiltonian by a similarity transformation, assign an integer topological charge using chiral/mirror symmetry, and argue that the non-Bloch Dirac points disappear under PBC and have geometry-dependent locations. The analytical derivation from Eq. (7) is self-consistent, but the paper's central 'open-boundary' claim rests on an unproved cylinder reduction, and the parallelogram-geometry result in Sec. IV is stated without derivation.","tokens_in":13232,"tokens_out":16992,"duration_ms":167494,"significance":"If the boundary-condition issues are resolved, this would be a valuable contribution: it is one of the few analytically solvable 2D non-Hermitian models with explicit non-Bloch Dirac points, an exact phase diagram, and a concrete demonstration of geometry-dependent bulk-boundary correspondence. The derivation of Eq. (7) is analytic and parameter-free, the Hermitian mapping is explicit, and the topological charge is computed from a winding formula rather than asserted. The main value is the exact cylinder solution and the symmetry classification of the non-Bloch Dirac points. However, the abstract's claim that the Dirac points 'only appear under open boundary conditions' is stronger than what is currently established, because the calculation is performed on a cylinder and the fully open case is not proven.","major_comments":[{"comment":"Equation (1) and Eq. (2) are not equivalent under the stated standard Pauli conventions. With the usual σx, σy matrices, Eq. (1) gives off-diagonal matrix elements i(γ−t)−iw e^{-ikx} and i(γ+t)+iw e^{ikx}, whereas Eq. (2) has t+γ+wβ and t−γ+w/β. Eq. (2) corresponds instead to (t+w cos kx)σx + (iγ−w sin kx)σy + iησz, i.e., the σx and σy terms are interchanged relative to Eq. (1). Since the entire subsequent derivation, including the symmetries and the Hermitian mapping, uses Eq. (2), the model definition must be corrected or the Pauli convention explicitly fixed. This is a load-bearing inconsistency in the definition of the model.","section":"II, Eqs. (1)-(2)"},{"comment":"The reduction from a fully open system to the cylinder is asserted without proof. The sentence 'Since the hopping along the y direction is Hermitian, taking PBC and OBC in y will yield the same results' is not sufficient for the claims made here. For a finite open strip, the allowed ky are discrete standing-wave values, so an exact gap closing at a specific ky,c occurs only if ky,c coincides with a discrete level; in the thermodynamic limit the spectrum becomes dense, but this limit is not stated. The sentence 'To analytically obtain the fully open boundary spectrum, we take PBC in y and OBC in x' is also self-contradictory. Please either prove the equivalence for the quantities computed, provide finite-size OBC-in-both-directions checks for the non-Bloch Dirac points, or explicitly reframe the results as cylinder results rather than fully open boundary results.","section":"II, after Eq. (1)"},{"comment":"The parallelogram-geometry calculation is stated without any derivation. After saying 'we calculated the positions of the non-Bloch DPs', the text only gives the notation kt_c,1 and kt_c,2 with no formula, no generalized Brillouin zone construction for the tilted boundary, and no numerical method. This is the central evidence for the paper's claim of geometry-dependent bulk-boundary correspondence. The calculation must be shown, or at least a reproducible numerical procedure must be provided, for both the rectangular and parallelogram geometries.","section":"IV, Fig. 5(b)"}],"minor_comments":[{"comment":"The name 'Altland-Zirnabuer' should be spelled 'Altland-Zirnbauer'.","section":"Abstract and Introduction"},{"comment":"The listed boundary 'η = √ξ + 1 where −1 < ξ < 0' is inconsistent because √ξ is imaginary for negative ξ; from the preceding paragraph it should be η = √(ξ+1).","section":"II, phase boundary list"},{"comment":"As written, the similarity transformation P = e^{iπ/4 σz} diag{1, sqrt((t−γ)/(t+γ))} does not reproduce Eq. (11); the sign of the π/4 rotation appears to be opposite to what is needed. Please check the ordering and sign convention so that the mapping is reproducible.","section":"III A, Eq. (11) and the definition of P"},{"comment":"The text refers to 'link = 1/100' and 'link = 1/2', but the variable t_b mentioned in the main text is not defined in the caption; please clarify the notation.","section":"IV, Fig. 5 caption"},{"comment":"The long-range perturbation terms would benefit from a sentence describing the lattice interpretation of δ and a, and the expression 'awβ_x^2' should be typeset more clearly to avoid confusion with a parameter named aw.","section":"III B, Eq. (14)"}],"recommendation":"major_revision","confidential_remarks":"The Eq. (1)/(2) mismatch and the missing parallelogram derivation are the main risks. The cylinder-reduction issue is also central because the paper's headline claim concerns open boundary conditions. These are fixable within the scope of the manuscript, so I do not recommend rejection, but the authors should either supply the missing derivations or substantially qualify the claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a genuinely useful addition to the non-Hermitian BBC literature. The stacked SSH model is simple enough to be exactly solvable, and the authors exploit that to get an analytic phase diagram in the (ξ, η) plane, identify non-Bloch Dirac points at the R3-R4 boundary, and show they carry a Z charge via a similarity transformation to a Hermitian Hamiltonian. The result that these DPs exist only under OBC in x and disappear under PBC is clearly demonstrated, and the symmetry-table analysis for their protection is standard but competently done. The geometry-dependence of the DP locations, if confirmed, is a nice concrete illustration of how BBC dissolves in non-Hermitian systems.\n\nThe soft spots are real but not fatal. First, Eqs. (1) and (2) do not match under the stated Pauli conventions: the off-diagonal entries differ by factors of i and by a swap of the γ terms. This looks like a typo in Eq. (1), but as written it defines a different model. The rest of the paper solves Eq. (2), so the typo can be fixed, but it must be fixed. Second, the reduction from fully open to cylinder (OBC in x, PBC in y) is justified only by the sentence \"since the hopping along the y direction is Hermitian, taking PBC and OBC in y will yield the same results.\" That is true for the bulk band edge in the thermodynamic limit, but not for exact gap closings: in a finite sample with OBC in y, ky is quantized, so a Dirac point at a continuous ky_c will not sit exactly on a quantized value. The paper should state that the non-Bloch DPs are defined in the ribbon geometry and either prove the thermodynamic limit or show that the fully open rectangular case gives the same phase boundaries. Third, the parallelogram geometry result in Sec. IV is asserted without derivation; since the geometry-dependence claim rests on it, a calculation or at least a clear argument is needed. Fourth, Eq. (12) is introduced as the low-energy expansion without showing the expansion; that is a minor omission.\n\nThe core algebra appears self-consistent, the mapping to Hermitian is explicit, and the classification is rigorous given the reality of the spectrum. The paper deserves a serious referee, and I would engage with a revised version. I would not desk-reject.","headline":"Solid analytic study with a new OBC-only Dirac point result; fix the Eq. (1)/(2) mismatch and justify the cylinder reduction before publication.","tokens_in":13751,"tokens_out":4974,"would_cite":true,"duration_ms":48590,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A stacked non-Hermitian dimerized-chain model hosts linear band crossings, called non-Bloch Dirac points, that have real energy and integer topological charge, exist only under open boundaries, and move when the boundary shape changes.","keywords":["non-Hermitian skin effect","non-Bloch Dirac points","generalized Brillouin zone","open-boundary spectrum","bulk-boundary correspondence","topological semimetal","exceptional points","stacked SSH model"],"falsifier":"Diagonalize the same Hamiltonian numerically on a finite cluster with open boundaries in both directions (or on a parallelogram-shaped cluster) and compare the real gap-closing points with the cylinder prediction $(k_{x,c},\\pm k_{y,c})$; if the crossings move, acquire complex energy, or disappear, the cylinder-geometry reduction and the geometry-dependence claim are falsified.","tokens_in":12810,"feed_emoji":"🔺","tokens_out":8698,"duration_ms":75296,"temperature":0.7,"pith_summary":"This paper claims that a two-dimensional non-Hermitian model made of stacked non-reciprocal dimerized chains hosts linear band crossings, called non-Bloch Dirac points, that exist only under open boundary conditions. These crossings have real energy, vanish from the periodic-boundary Bloch spectrum where they split into pairs of exceptional points, and shift their positions when the shape of the boundary changes. Because the spectrum is real in the vicinity of each crossing, the authors map the local problem to a Hermitian semimetal and show that each non-Bloch Dirac point carries an integer topological charge. If the claim is right, the familiar bulk-boundary correspondence of Hermitian semimetals has to be replaced, in non-Hermitian systems, by a geometry-dependent relation between the open-boundary spectrum and edge physics.","feed_headline":"Non-Hermitian stack hosts Dirac points only under open boundaries","feed_subtitle":"Linear band crossings with real energy and integer charge appear in a stacked non-Hermitian chain, then vanish when boundaries close.","key_machinery":"The argument runs on the generalized Brillouin zone (GBZ): replacing the $x$-direction Bloch phase $e^{ik_x}$ by a complex number $\\beta_x$ whose magnitude is fixed by the open-boundary standing-wave condition $|\\beta_{x,+}|=|\\beta_{x,-}|$. With the $y$ direction kept periodic (the cylinder geometry), the characteristic equation gives an analytic spectrum $E=\\pm\\sqrt{t^2+w^2-\\gamma^2-\\eta^2+2w\\sin\\phi\\sqrt{t^2-\\gamma^2}}$, and the GBZ radius $R(k_y)=\\sqrt{|(t+\\gamma)/(t-\\gamma)|}$. A similarity transformation $P$ maps the Hamiltonian to a Hermitian matrix whose low-energy expansion is two anisotropic Dirac cones; the integer charge is $\\nu_\\pm=\\frac{i}{2\\pi}\\oint (Q_\\pm)^{-1}dQ_\\pm$.","core_discovery":"Using an exactly solvable stacked dimerized-chain model with asymmetric intra-cell hopping and on-site gain/loss, the authors derive the full open-boundary phase diagram from the complex energy gaps. The phase diagram contains a transition on which the open-boundary spectrum closes its real gap at two points, with linear dispersion in the two-dimensional parameter space formed by the y-momentum and the phase of the non-Bloch wave vector. These non-Bloch Dirac points are protected by chiral or mirror symmetry, carry an integer ($\\mathbb{Z}$) winding number, and remain stable against off-diagonal long-range hopping while a diagonal long-range hopping opens a gap. Under periodic boundaries the same crossings split into pairs of exceptional points, and the projected positions of the crossings for different open-boundary geometries do not coincide.","pith_inferences":["The paper's cylinder-geometry reduction could be tested by adding non-Hermitian y-direction couplings; if PBC and OBC then disagree in y, the analytic phase boundaries and the non-Bloch Dirac point positions would need revision.","The same real-spectrum-plus-similarity-transformation construction suggests a route to defining non-Bloch Weyl points in three dimensions, with surface Fermi arcs whose endpoints are geometry-dependent.","The integer winding number computed around each non-Bloch Dirac point may be measurable in metamaterial or circuit realizations through the spatial profile of zero-energy edge modes, not just through the spectrum."],"forward_implications":["For these models, topological classification must start from the open-boundary (non-Bloch) spectrum rather than the periodic-boundary Bloch bands.","Non-Bloch Dirac points are symmetry-protected: perturbations that preserve chiral or mirror symmetry keep them gapless, and off-diagonal long-range hopping preserves them while diagonal long-range hopping gaps them.","Switching from open to periodic boundary conditions converts each non-Bloch Dirac point into a pair of exceptional points, so any bulk-boundary correspondence built on PBC bands will miss the edge physics.","Because the projected crossing locations depend on the orientation of the open edges (square vs parallelogram), the same bulk can appear to have different band-crossing momenta depending on the cut, a direct violation of the Hermitian projection doctrine."],"supporting_citations":[{"why":"Supplies the non-Bloch band theory and generalized Brillouin zone used to incorporate open boundary conditions in x.","marker":"[12]"},{"why":"Provides the standing-wave condition |β+|=|β−| used to derive the cylinder energy spectrum.","marker":"[14]"},{"why":"Gives the auxiliary GBZ/resultant method used to obtain the analytic spectrum and the long-range-hopping extension.","marker":"[30]"},{"why":"Supplies the classification of Hermitian topological semimetals used after the mapping to assign the integer topological charge.","marker":"[6]"},{"why":"Provides the symmetry-classification table from which the Z index for these codimension-two crossings is taken.","marker":"[45]"},{"why":"The earlier study of non-Hermitian Dirac/Weyl crossings in periodic-boundary Bloch bands that the paper contrasts with its open-boundary non-Bloch Dirac points.","marker":"[33]"},{"why":"Defines the dimerized (SSH) chain whose stacked non-Hermitian version is the model analyzed.","marker":"[41]"}],"fun_headline_variants":["Dirac points in non-Hermitian stack only with open boundaries","Non-Bloch Dirac crossings appear only in open-boundary spectra","Stacked non-Hermitian chain: Dirac points vanish under periodic boundaries","Open boundaries create Dirac points in non-Hermitian stacked SSH model","Boundary-only Dirac points in non-Hermitian stacks with integer charge"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The analytic phase diagram assumes that taking periodic boundary conditions in $y$ and open boundaries in $x$ gives the same spectrum as a fully open two-dimensional sample, justified only by the statement that the $y$-hopping is Hermitian; if that equivalence fails, the locations and even the existence of the non-Bloch Dirac points in a finite sample are not established.","fun_headline_variants_meta":{"raw":{"variants":["Dirac points in non-Hermitian stack only with open boundaries","Non-Bloch Dirac crossings appear only in open-boundary spectra","Stacked non-Hermitian chain: Dirac points vanish under periodic boundaries","Open boundaries create Dirac points in non-Hermitian stacked SSH model","Boundary-only Dirac points in non-Hermitian stacks with integer charge"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00053,"raw_usage":{"total_tokens":2555,"prompt_tokens":950,"completion_tokens":1605,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":566,"completion_tokens_details":{"reasoning_tokens":1512}},"tokens_in":566,"tokens_out":1605,"duration_ms":11059,"temperature":1.0,"reasoning_tokens":1512,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:07:37.874745+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Diagonalize the same Hamiltonian numerically on a finite cluster with open boundaries in both directions (or on a parallelogram-shaped cluster) and compare the real gap-closing points with the cylinder prediction $(k_{x,c},\\pm k_{y,c})$; if the crossings move, acquire complex energy, or disappear, the cylinder-geometry reduction and the geometry-dependence claim are falsified.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the non-Bloch band theory and generalized Brillouin zone used to incorporate open boundary conditions in x."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the auxiliary GBZ/resultant method used to obtain the analytic spectrum and the long-range-hopping extension."},{"cited_title":"Lieu, Phys","cited_arxiv_id":null,"evidence_quote":"Provides the symmetry-classification table from which the Z index for these codimension-two crossings is taken."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the dimerized (SSH) chain whose stacked non-Hermitian version is the model analyzed."}],"review_version":1}