{"id":"7a26ca5a-2562-481d-be0d-62a035a95668","arxiv_id":"2411.14831","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Zero-energy vortex modes coexist with Majorana corner modes in a 2D second-order topological superconductor only when the normal-state spectrum is gapless with Dirac cones.","lead":"This paper shows that in a two-dimensional second-order topological superconductor, zero-energy vortex modes can appear together with Majorana corner modes, but only when the normal-state material has gapless Dirac cone points. The result matters because it identifies when vortex Majorana-like modes survive in higher-order topological superconductors, which is relevant for topological quantum computing proposals.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof that the quadratic (k_x^2 - k_y^2) term leaves the vortex zero mode at zero energy rests on a single vanishing matrix element, not on operator annihilation; the analytic zero-mode condition (14) is therefore not rigorously established.","rationale":"The paper presents a coherent analytical derivation of vortex zero modes in a HOTSC, confirmed by numerical calculations at selected parameters. The derivation follows the standard Jackiw-Rossi/Fu-Kane approach, and the extension to two orbital bands and finite chemical potential with Bessel functions is plausible. The condition (14) for coexistence is derived from the low-energy expansion. However, the treatment of the quadratic (k_x^2 - k_y^2) term is insufficient: a vanishing diagonal matrix element does not imply the state is unaffected. This is the single most load-bearing weakness because the central claim is specifically about the 'gapless' nature of the vortex modes; if these modes acquire a small gap from higher-order lattice terms, the coexistence of gapless vortex modes and Majorana corner states is not realized. The numerical results provide strong evidence at the shown parameters, and the paper's own caveat that the vortex modes are not topologically protected is appropriate. The absence of code or data for exact reproduction is a secondary issue. Overall, the reader's CONDITIONAL verdict is appropriate; the concern is addressable by additional numerical or analytical checks, not fatal.","tokens_in":12266,"tokens_out":13098,"duration_ms":127390,"concrete_test":"Numerically diagonalize the full lattice Hamiltonian (1) with a vortex for a series of parameter sets satisfying Eq. (14) with nonzero delta_t (e.g., tx = 2, ty = -1.9, -1.5, -1.0, adjusting t1 and Delta_epsilon via (14)), while staying in the HOTSC phase, and compute the vortex-mode energy as a function of system size. If the energy is exactly zero to machine precision for all these cases, the concern is resolved. If it is nonzero and scales as delta_t^2 or delta_t^2 / Delta, the condition (14) is only approximate and the vortex modes are not truly gapless. Additionally, directly compute the matrix elements of the lattice version of (d_x^2 - d_y^2) between the vortex zero mode and low-lying excited states; nonzero matrix elements imply the zero mode is not an exact eigenstate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that zero-energy vortex modes coexist with Majorana corner modes depends on the assertion that the quadratic kinetic term proportional to (k_x^2 - k_y^2) in Eq. (13) does not affect the zero-energy solution (12). Appendix B shows only that Psi^dag(d_x^2 - d_y^2)Psi = 0 for state (12), i.e., the diagonal matrix element vanishes. It does not show that the differential operator annihilates Psi, nor that all off-diagonal matrix elements between Psi and other states vanish. Acting with (d_x^2 - d_y^2) on (12) yields components with cos(2phi) and sin(2phi) angular factors (see Eq. B1) that are not proportional to the original state, so the state is not an eigenstate when this term is retained. Generically, this produces a second-order energy shift from coupling to finite-energy states, so the vortex mode is only zero to leading order in the continuum low-energy expansion. Consequently, condition (14), which cancels only the ~epsilon and (k_x^2 + k_y^2) terms, may not be sufficient to guarantee truly gapless vortex modes in the full lattice model. The numerical demonstrations at specific parameter points are consistent with the claim, but they do not establish (14) as an exact condition, especially for delta_t != 0 where the second term is present. Since the paper's headline distinction between 'gapless' and 'gapped' vortex modes rests on this premise, the analytic derivation is the weakest load-bearing step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a two-orbital lattice model of a two-dimensional second-order topological superconductor with Rashba spin-orbit coupling and singlet pairing. It derives an approximate continuum solution for zero-energy vortex-localized modes when the normal-state bulk spectrum has Dirac cones, giving the explicit wavefunction in Eq. (12) and the parameter condition (14) for gapless vortex modes. It reports numerical spectra showing that these zero modes can coexist with Majorana corner modes when the superconducting bulk and Wannier gaps are open, and it analyzes how the modes hybridize when the vortex approaches an edge or a corner.","tokens_in":12590,"tokens_out":8135,"duration_ms":82651,"significance":"If the claims hold, the paper provides a useful and nontrivial extension of vortex zero-mode physics to higher-order topological superconductors: unlike first-order systems, the edge spectrum is gapped, yet zero-energy vortex modes can still appear and coexist with corner Majoranas, with the number of pairs set by the number of Dirac cones. The analytic construction is self-contained, the parameter condition (14) is explicit, and the finite-size numerics in Figs. 2 and 3 support the coexistence scenario. The main caveat is the treatment of the quadratic kinetic term, which is the subject of the major comments.","major_comments":[{"comment":"The statement that the quadratic term t cx sign tx in Eq. (13) 'does not influence' the zero-energy solution (12) is not established. Appendix B proves only that the diagonal matrix element Psi^dag (d_x^2 - d_y^2) Psi vanishes for state (12); it does not show that the operator annihilates Psi or that all off-diagonal matrix elements to other states vanish. Acting on (12) with the operator in Eq. (B1) generates components with cos 2phi and sin 2phi angular factors that are not proportional to the original state, so the state is not an eigenstate when this term is retained. Generically this produces a second-order energy shift, so the vortex mode is zero only to leading order in the continuum expansion. Consequently, condition (14), which cancels only the other terms in Eq. (13), may not be sufficient to guarantee truly gapless vortex modes in the full lattice model. The numerical demonstrations at specific parameter points are consistent with the claim, but they do not establish Eq. (14) as an exact condition. I request a proof of the vanishing of the off-diagonal couplings, an estimate of the residual gap, or a concrete finite-size scaling study showing that the lowest positive eigenvalue tends to zero at the parameters of Fig. 2.","section":"Sec. IV, Eq. (13) and Appendix B"}],"minor_comments":[{"comment":"The notation for the vortex phase uses arg[z((R_f+R_m)/2) - z(R_v)] for bond-centered pairings; please clarify the convention for the bond midpoint and the branch of arg.","section":"Eq. (4)"},{"comment":"The term 'gapless vortex modes' is potentially confusing because the modes are localized and do not form a band; 'zero-energy vortex modes' would be more precise throughout.","section":"Abstract and throughout"},{"comment":"The system size N is never stated; please give the lattice size and the numerical tolerance used to identify zero-energy states.","section":"Figs. 2 and 3"},{"comment":"The phase theta introduced in Eq. (A1) is not defined before it is used; please specify its relation to the sign choices in beta.","section":"Appendix A, Eq. (A1)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within scope for cond-mat.supr-con and the numerical evidence is credible. The central risk is overstatement of Eq. (14) as an exact condition; the requested revision should resolve this before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper gives a concrete analytical solution for zero-energy vortex modes in a two-band second-order topological superconductor, and shows numerically that they can coexist with Majorana corner modes. That is a real step forward, because the earlier literature left the vortex-mode question unsettled. The Jackiw-Rossi machinery is used cleanly, and the condition (14) for coexistence is explicit and testable.\n\nWhat I like: the derivation in Sec. III is coherent, the Bessel-function solution (12) is concrete, and the numerics in Fig. 2 confirm zero-energy states at the predicted parameters. The discussion of single-pair vs double-pair modes and the spin projection tied to vorticity is clear. The self-citations to [35,36] are not padding; they establish the HOTSC phase of the model.\n\nThe soft spot is real. Appendix B shows that the angular average of (∂_x^2 - ∂_y^2) in the zero-mode state vanishes, but that only means the first-order energy shift is zero. It does not show the operator annihilates the state, so second-order couplings to finite-energy states could shift the vortex-mode energy. The paper's central distinction between 'gapless' and 'gapped' vortex modes depends on this term being harmless, so condition (14) is not rigorously established for the lattice model. The numerics at the specific parameter points are consistent with the claim, but they don't cover the full parameter range. This should be fixable: either prove the operator annihilates the state under the same continuum assumptions, or present a perturbation analysis showing the second-order shift vanishes, or soften the claim to leading order.\n\nMinor point: no code or data included, so exact reproduction is not possible, but the numerical method is standard.\n\nOverall, the paper deserves a serious referee. I would send it to peer review with a request to address the Appendix B gap. It's a conditional accept from me.","headline":"A useful analytic step on vortex zero modes in 2D HOTSC, but the justification that the (k_x^2 - k_y^2) kinetic term leaves them gapless is weaker than the text suggests.","tokens_in":13100,"tokens_out":5171,"would_cite":true,"duration_ms":53771,"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 2D second-order topological superconductor can host zero-energy vortex-localized modes together with Majorana corner modes, provided the normal-state bulk spectrum has Dirac cones.","keywords":["second-order topological superconductor","Majorana corner modes","vortex zero modes","Dirac cones","spin-orbit coupling","topological insulator-superconductor interface","higher-order topology","superconducting vortex"],"falsifier":"Compute the exact lattice spectrum of the model with a vortex at the parameters satisfying Eq. (14) for a sequence of increasing system sizes and extract the lowest vortex-bound eigenvalue; if that energy approaches a nonzero saturation value rather than decaying to zero, the continuum zero mode is not exact. A complementary check is to include the next-order momentum terms omitted from Eq. (13) and test whether the vortex-mode energy becomes nonzero.","tokens_in":12087,"feed_emoji":"🌀","tokens_out":10246,"duration_ms":95055,"temperature":0.7,"pith_summary":"Second-order topological superconductors are expected to have gapped edge spectra, so vortex-bound states would be expected to acquire finite energy. This paper shows that in a two-dimensional second-order topological superconductor built from a normal layer with spin-orbit coupling and a superconducting layer, zero-energy vortex-localized modes can coexist with the Majorana corner modes that define the higher-order phase. The requirement is that the non-superconducting bulk spectrum be gapless and contain Dirac cones; the number of pairs of vortex zero modes equals the number of Dirac cones, and an explicit wavefunction is derived for them. If the normal-state spectrum is gapped, the vortex modes are gapped while corner Majoranas survive. The result matters because it gives concrete lattice-parameter conditions under which a single material could show both corner Majoranas and zero-bias vortex peaks in scanning tunneling experiments.","feed_headline":"Zero-energy vortex modes can coexist with Majorana corner modes","feed_subtitle":"They appear only when the normal-state bulk spectrum has Dirac cones; each cone yields one pair of vortex modes.","key_machinery":"The machinery is a continuum expansion around the Dirac points of the normal-state spectrum, combined with the standard zero-mode construction for vortices in topological systems. The key object is the zero-energy vortex wavefunction of Eq. (12), built from Bessel functions and the radial envelope $F(r)$, which solves the vortex equations when the chemical potential is finite. The argument hinges on the separation of the kinetic terms in Eq. (13): the combination $(\\partial_x^2 - \\partial_y^2)$ has zero overlap with the zero mode because of its angular dependence, whereas $(\\partial_x^2 + \\partial_y^2)$ would produce a finite gap. Vanishing of both the constant term and the $(\\partial_x^2 + \\partial_y^2)$ term in the expansion yields the coexistence condition Eq. (14). A second central mechanism is the absence of a boundary-localized zero-mode counterpart: in a first-order topological superconductor the increasing solution of the vortex equations corresponds to an edge Majorana, but in a second-order superconductor the edge spectrum is gapped, so only the vortex-localized solution survives.","core_discovery":"The central claim is that zero-energy vortex modes are not forbidden in a second-order topological superconductor; they appear precisely when the normal (non-superconducting) layer has Dirac cones in its bulk spectrum. Working in a two-orbital square-lattice model with spin-orbit coupling and spin-singlet pairing, the authors expand the Hamiltonian near the spin-orbit nodal points and solve the superconducting vortex equations. The resulting zero-energy solution, Eq. (12), is a combination of Bessel functions $J_0(r\\mu/2\\lambda)$ and $J_1(r\\mu/2\\lambda)$ times a radial envelope $F(r)=\\exp\\left(-\\int_0^r \\Delta(\\rho)/2|\\lambda|\\,d\\rho\\right)$, with only one spin projection selected by the vortex winding. The gapless modes survive when the kinetic parameters satisfy $t_1 = c_x \\,\\delta t\\, \\operatorname{sign} t_y / 2$ and $\\Delta\\varepsilon = -2c_x \\,\\delta t\\, \\operatorname{sign} t_x$, where $c_x = \\pm 1$ labels the Dirac point and $\\delta t = (|t_x| - |t_y|)/2$; then the number of zero-mode pairs equals the number of Dirac cones. When this condition is violated and the normal spectrum is gapped, the vortex modes acquire a gap while the Majorana corner modes remain, so the coexistence is controlled by lattice parameters rather than by topology alone.","pith_inferences":["Because the zero mode is not topologically protected, real materials will generically have small corrections that split it from zero; the practical prediction is a narrow but not exactly pinned zero-bias peak whose residual energy is set by the deviation from Eq. (14).","The Dirac-cone-to-vortex-mode counting may extend beyond this two-orbital lattice: any normal state with several symmetry-related Dirac points should contribute one vortex-mode pair per cone, provided the same kind of kinetic cancellation can be arranged.","The condition Eq. (14) can be read as a design rule: strain that changes $|t_x| - |t_y|$ or gates that shift $\\Delta\\varepsilon$ provide a knob to turn vortex zero modes on and off while leaving corner Majoranas intact, which could be used to manipulate the low-energy state content of a device."],"forward_implications":["In a sample in the second-order topological superconducting phase with $t_1 = c_x \\,\\delta t\\, \\operatorname{sign} t_y / 2$ and $\\Delta\\varepsilon = -2c_x \\,\\delta t\\, \\operatorname{sign} t_x$, scanning tunneling spectroscopy should simultaneously show zero-bias conductance peaks at vortex cores and at corners.","The number of zero-energy vortex pairs is fixed by the number of Dirac cones in the normal-state spectrum: one pair for a single cone and two pairs for two cones, so the vortex-mode degeneracy directly probes the normal-state band structure.","When the normal spectrum is gapped, the same model produces only gapped vortex modes while Majorana corner modes persist, so the absence of a zero-bias vortex peak does not exclude a second-order topological superconducting phase.","The vortex zero modes are not protected by a topological invariant, so approaching a boundary or corner hybridizes them with edge and corner states; near a corner one pair becomes corner-localized and may remain gapless or acquire a small gap depending on $\\Delta_0$ and other parameters.","The coexistence condition is a precise relation among hopping parameters and on-site energies, so it can be tuned by lattice deformations or gate potentials to switch between gapped and gapless vortex modes without destroying the corner Majoranas."],"supporting_citations":[{"why":"Supplies the two-orbital model of a normal layer with spin-orbit coupling whose superconducting version hosts Majorana corner states, the starting point of the paper.","marker":"[34]"},{"why":"Establishes the higher-order topological superconducting phase and the Dirac-mass mechanism for edge states in this model.","marker":"[35]"},{"why":"Provides the standard continuum derivation of zero-energy Majorana vortex states at a topological-insulator surface, which the authors adapt to the two-level case.","marker":"[19]"},{"why":"Gives the pinned-vortex Majorana solution and its boundary conditions, used to identify the vortex-localized zero mode.","marker":"[20]"},{"why":"Supplies the classic result that fermion-vortex systems support zero modes tied to Dirac cones, the conceptual basis for counting pairs by Dirac points.","marker":"[41]"},{"why":"Proposes a bulk-vortex correspondence for higher-order topological superconductors, motivating the search for vortex zero modes in this setting.","marker":"[27]"},{"why":"Shows that multiple Majorana vortex zero modes can appear in topological crystalline insulators, providing precedent for the double-pair modes found here.","marker":"[40]"},{"why":"Used for the quadrupole-moment calculation that verifies the higher-order topological phase at zero next-nearest-neighbor hopping and on-site splitting.","marker":"[8]"},{"why":"Supplies the Wannier-band polarization and quadrupole method that identifies the higher-order topological phase in Sec. II.","marker":"[37]"},{"why":"Demonstrates multiple Majorana modes at vortex ends in doped topological insulators, another precedent for several zero-energy vortex states.","marker":"[22]"}],"fun_headline_variants":["Vortex zero modes coexist with Majorana corners — via Dirac cones","Gapless vortex modes join Majorana corners when Dirac cones appear","Zero-energy vortex modes and Majorana corners: coexistence explained","Dirac cones enable vortex zero modes alongside Majorana corner states","Coexistence of gapless vortex and Majorana corner modes hinges on Dirac cones"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation assumes that lattice-scale corrections to the kinetic energy, beyond the terms kept in the continuum expansion, do not shift the zero-energy vortex mode; the paper verifies this for one combination of second derivatives, showing it has zero overlap with the wavefunction, but does not prove the full kinetic operator exactly kills the state, so neglected higher-order terms could open a small gap.","fun_headline_variants_meta":{"raw":{"variants":["Vortex zero modes coexist with Majorana corners — via Dirac cones","Gapless vortex modes join Majorana corners when Dirac cones appear","Zero-energy vortex modes and Majorana corners: coexistence explained","Dirac cones enable vortex zero modes alongside Majorana corner states","Coexistence of gapless vortex and Majorana corner modes hinges on Dirac cones"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000297,"raw_usage":{"total_tokens":1785,"prompt_tokens":1070,"completion_tokens":715,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":686,"completion_tokens_details":{"reasoning_tokens":639}},"tokens_in":686,"tokens_out":715,"duration_ms":6571,"temperature":1.0,"reasoning_tokens":639,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:50:33.858308+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the exact lattice spectrum of the model with a vortex at the parameters satisfying Eq. (14) for a sequence of increasing system sizes and extract the lowest vortex-bound eigenvalue; if that energy approaches a nonzero saturation value rather than decaying to zero, the continuum zero mode is not exact. A complementary check is to include the next-order momentum terms omitted from Eq. (13) and test whether the vortex-mode energy becomes nonzero.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the higher-order topological superconducting phase and the Dirac-mass mechanism for edge states in this model."},{"cited_title":"Langbehn, Y","cited_arxiv_id":null,"evidence_quote":"Provides the standard continuum derivation of zero-energy Majorana vortex states at a topological-insulator surface, which the authors adapt to the two-level case."},{"cited_title":"Zhu, Tunable majorana corner states in a two-dimensional second-order topological superconductor induced by magnetic fields, Phys","cited_arxiv_id":null,"evidence_quote":"Gives the pinned-vortex Majorana solution and its boundary conditions, used to identify the vortex-localized zero mode."},{"cited_title":"Zhang, Z","cited_arxiv_id":null,"evidence_quote":"Supplies the classic result that fermion-vortex systems support zero modes tied to Dirac cones, the conceptual basis for counting pairs by Dirac points."},{"cited_title":"Alicea, Y","cited_arxiv_id":null,"evidence_quote":"Proposes a bulk-vortex correspondence for higher-order topological superconductors, motivating the search for vortex zero modes in this setting."},{"cited_title":"Kheirkhah, Z","cited_arxiv_id":null,"evidence_quote":"Shows that multiple Majorana vortex zero modes can appear in topological crystalline insulators, providing precedent for the double-pair modes found here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Used for the quadrupole-moment calculation that verifies the higher-order topological phase at zero next-nearest-neighbor hopping and on-site splitting."},{"cited_title":"Hosur, P","cited_arxiv_id":null,"evidence_quote":"Supplies the Wannier-band polarization and quadrupole method that identifies the higher-order topological phase in Sec. II."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Demonstrates multiple Majorana modes at vortex ends in doped topological insulators, another precedent for several zero-energy vortex states."}],"review_version":1}