Topological protection by local support symmetry and destructive interference

Domain: neuroscience
Published: 2026-02-13
Editorial outcome: major_revision

Abstract
Conventionally, symmetry-protected topological phases and band crossings are protected by global symmetries acting on the entire system. Here, we show that symmetries preserved only on a partial region of a system, termed local support symmetries, can protect topological features of the full system, even in the presence of symmetry-breaking couplings. We establish a unified framework by deriving explicit conditions for such protection in both insulating and metallic phases and show that destructive interference of Bloch wave functions plays a key role. Using representative tight-binding models, we demonstrate band crossings and topological bands protected by local support crystalline and time-reversal symmetries, and further present a realistic material realization in a fluorinated biphenylene network, where a band crossing is protected by a local support C2 symmetry.

Introduction
Triggered by the discovery of the topological insulator 1–8, researchers have almost exhaustively investigated a myriad of symmetry-protected topological (SPT) phases and topological semimetals8–23. Symmetry plays a critical role in the stabilization of those topological phases. For example, the nontrivial topology in Z2 topological insulators and topological crystalline insulators is protected by time-reversal and crystalline symmetries, respectively1,4,8,22. On the other hand, band-crossings, such as nodal points, nodal lines, and nodal planes, can circumvent gap opening with the aid of various crystalline symmetries, such as a mirror and various nonsymmorphic symmetries24–44. In each of these cases, the symmetry operations uniformly affect the entire system, as illustrated in Fig. 1.

In this paper, we demonstrate that topological protections can persist even when a system respects a symmetry in one part of it (S1) while the symmetry is broken in another (S2), as plotted schematically in Fig. 1. Such a symmetry is called a local support symmetry (LSS). For instance, we illustrate the existence of insulators lacking time-reversal symmetry (TRS) yet exhibiting occupied bands characterized by a well-defined Z2 invariant, protected by a local support time-reversal symmetry. We also exhibit semimetal examples hosting nodal points protected by local support C2 or nonsymmorphic symmetries. It is crucial to emphasize that the LSS alone cannot preserve these topological characteristics; rather, the hopping processes between two parts of the system (S1 and S2), denoted by h12, must satisfy a certain condition as follows. On the other hand, a band-crossing along an LSS-invariant momentum line can be protected if the column vectors of h12 are proportional to one of the eigenvectors of the unitary matrix representing the LSS. We note that, in many cases, these conditions for the inter-part coupling (h12) can be achieved through destructive interference of a Bloch wavefunction, a phenomenon commonly associated with stabilizing flat bands45–53. Consequently, the amplitude of the Bloch wavefunction relevant to the LSS protection becomes precisely zero at sites belonging to S2, allowing for the application of the conventional protection mechanism of SPT phases only to S1. This highlights the crucial role of destructive interference in this topological protection mechanism. Finally, we propose that Dirac points protected by the local support C2 symmetry can be observed in the biphenylene network54 fluorinated periodically. While the absorbed fluorine atoms break C2 symmetry in the entire system, a part of the system respects C2 symmetry. Moreover, the inter-part coupling satisfies the conditions for the destructive interference mentioned above, so that the Dirac nodes can be protected by the local support C2 symmetry even after the fluorination.

Results
In this section, we demonstrate that a topological phase protected by symmetry can remain robust even when coupled to a system that breaks this symmetry, provided that the coupling meets a specific condition. We highlight that destructive interference can be crucial in satisfying this condition. We consider a system with N orbitals per unit cell, which is partitioned into two distinct parts, labeled as S1 and S2, each comprising N1 and N2 orbitals, respectively. There is no overlap of orbitals between these two components, and N1 + N2 = N (generalizations are possible for different graphs). Then, the Bloch Hamiltonian is given by

H(k) = (h1(k) h12(k) / h21(k) h2(k)), (1)

where h1(k) is an N1 × N1 sub-Hamiltonian representing the S1 part, h2(k) is an N2 × N2 sub-Hamiltonian representing the S2 part, and h12(k) = h21(k)† is an N1 × N2 sub-matrix describing the inter-part coupling. However, the linear combination of v(u)y(k)’s is not always a finite sum of exponential phase factors, denoted as ein⋅k, where n is a vector consisting of integer components. Nevertheless, it has been shown that one can obtain an unnormalized eigenvector as a finite sum of Bloch phases in a flat band51,55. This can be understood briefly as follows. A general Bloch Hamiltonian H(k) corresponding to a tight-binding model with a finite hopping range consists of a finite sum of Bloch exponential factors in each element of it. Let us assume that this Hamiltonian hosts a flat band at E = Eflat. Then, the eigenvalue equation for the flat band is given by (H(k) − Eflat I)vk = 0, where the elements of H(k) − Eflat I are still a finite sum of the Bloch phases because Eflat is momentum-independent. Therefore, the components of the vectors in the kernel of H(k) − Eflat I can also be written as the finite sums of the Bloch factors if we allow unnormalized solutions51. This special form of the eigenvector explains the existence of compact localized eigenstates for a flat band, which can be obtained by a Fourier transformation of the unnormalized Bloch eigenvector. Consequently, if there are flat bands among the unoccupied bands, the condition (3) can be satisfied by designing every row of h21(k) to be a proper linear combination of the eigenvectors of the unoccupied flat bands of h1(k). Although we assume the flat bands are unoccupied for clarity, they may also be occupied if they are irrelevant to the system’s topology.

Methods
The density functional theory (DFT) calculations are performed using the Vienna Ab initio Simulation Package (VASP)68,69. The exchange-correlation energy functional was treated with the PBE functional70, and van der Waals (vdW) interactions between particles were accounted for using the DFT-D3 method71,72. For self-consistent calculations, we used a kinetic energy cutoff of 500 eV. We employed k-point sampling grids of 6 × 6 × 1 and 24 × 24 × 1 for structure relaxations and band calculations, respectively73,74. The geometry optimization is carried out until the energy convergence threshold of 10−6 eV is reached, and the Hellmann-Feynman forces acting on each atom are maintained below 0.001 eV/Å75.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript entitled \"Topological Protection by Local Support Symmetry and Destructive Interference\" proposes the concept of Local Support Symmetry (LSS) as a means of explaining how symmetry-protected topological features can persist even when the overall Hamiltonian does not possess the corresponding global symmetry. The authors show that if a certain subsystem of a lattice retains a local symmetry, and if the coupling to the symmetry-breaking part satisfies specific algebraic conditions or gives rise to compact localised states through destructive interference, then the topological properties of that subsystem can survive in the full system. They develop a general algebraic framework, support it with several tight-binding examples, and conclude with a density-functional theory study of fluorinated biphenylene, which is presented as a possible material realisation.\nIn my opinion, the work is clearly written and addresses a problem of wide current interest—the stability of topological features under partial symmetry breaking. The proposed framework bridges different areas of physics: flat-band theory, coupled-layer constructions, and symmetry-protected topology.\nHowever, the novelty of the manuscript lies more in its conceptual synthesis than in a genuinely new physical principle. The central mechanisms that underpin LSS—block decomposition of the Hamiltonian, destructive interference, and the existence of compact localised states—are well established in the literature on flat-band and compact-localisation phenomena. For these reasons, the authors should more clearly acknowledge this connection and present LSS as a unifying and generalisable reformulation rather than as a fundamentally new symmetry class. In my opinion, moderating claims of novelty will make the work stronger and more credible.\nIt would also help readers if the relationship between LSS and coupled-layer constructions (which are commonly used to build topological crystalline insulators) were clarified.\nIn fact, the algebraic structure employed here is similar, but the physical intention is reversed: rather than coupling layers to create topology, the LSS framework identifies conditions under which coupling does not destroy it. Framing the paper in this complementary light would situate it naturally among established approaches.\nA further weakness of the present version is the absence of a quantitative assessment of robustness. The protection mechanism depends on fine-tuned interference or on special coupling constraints, and its resilience to disorder, longer-range hopping, or structural perturbations remains uncertain. For Nature Communications, it would be important to demonstrate how the predicted topological features evolve under controlled symmetry-breaking perturbations, both in model Hamiltonians and in the DFT-based material. Such analyses would transform the work from a conceptual proposal into an experimentally testable claim.\nIn conclusion, in its current form, the manuscript offers an interesting and pedagogically useful perspective on a set of known physical ideas. Its main strength is to unify several mechanisms—symmetry protection, destructive interference, and localised-state formation—within a single algebraic and conceptual framework. Yet for publication in Nature Communications, the authors should more carefully situate the work within existing literature, add quantitative tests of robustness, and moderate the claims regarding generality and novelty.\nIn my opinion, with these improvements, the paper could appeal to a wide readership across condensed-matter theory, photonics, and materials science. I therefore recommend major revision.", "reviewer_2": "In this paper, the authors provide a framework for topological protection by local support symmetry (LSS), in which topological protections can persist even when a system respects a symmetry in one part of it while the symmetry is broken in another part. Here, LSS alone cannot preserve these topological characteristics; rather, through matrix analysis they find that the hopping processes between two parts of the system must satisfy a certain condition. In this work, they establish the explicit forms of these conditions for both the insulating and metallic cases, highlighting that a destructive interference of the Bloch wave function can play a crucial role in the fulfillment of these conditions. I think that these results are novel and they expand our understanding of symmetry-protected topological phases.\n\nA minor mistake:\n1)On page 8, the 6 × 6 matrix h1(k) should be the direct sum of two 3 × 3 matrices.", "reviewer_3": "Over the past decade, substantial progress has been made in the study of topological phases. Thanks to intensive efforts, our understanding of phases protected by exact global or crystalline symmetries has become quite mature. In this work, the authors investigate local subspace symmetries (LSS), where a designated subspace (or spatial subregion) respects a symmetry while the rest of the system does not.\nI find that the results presented in the manuscript are technically sound. However, in terms of novelty and potential impact on the field, I am afraid that the current manuscript is not sufficiently original or influential to warrant publication in Nature Communications. My assessment is based on the following two points:\n(i) Conceptual overlap with \"sub-symmetry\" (Nature Physics 19, 992-998, 2023).\nThe paper in Nature Physics introduced sub-symmetry-protected phases, where a symmetry is preserved only within a subspace of the full Hilbert space. For instance, the authors of that work considered a modified SSH chain in which either sublattice A or B preserves chiral symmetry, and they demonstrated that a zero mode remains protected. Much of the present manuscript appears closely related to this framework. While the authors certainly advance the idea, the fundamental concept seems very similar. Could the authors clarify the similarities and distinctions between topological phases protected by LSS and those protected by sub-symmetry?\n(ii) Topological features intrinsic to LSS.\nAs currently presented, LSS-protected phases seem to inherit topology that already exists within the symmetric subspace. If this is the case, it remains unclear whether LSS gives rise to genuinely new phases or topological invariants beyond those already known. Could the authors clarify whether LSS can protect new types of topology that are unique to LSS-protected topological phases?\nIn addition, I have a few minor comments:\n(iii) Scope of protected crossings.\nThe section \"General theory: semimetal case\" focuses on crossings enforced by symmetry representations. However, many gapless points are protected by higher-dimensional topological charges (such as a 1D winding number or a Chern number). Could the authors state explicitly whether such topological charges are outside the scope of this work, or explain how (if at all) the LSS framework constrains or incorporates them? A brief discussion of Weyl/Dirac nodes and nodal lines in the presence of LSS would be valuable.\n(iv) 3D example.\nFrom a materials perspective, it would be helpful if the authors could present an example involving a three-dimensional system."}, "author_response": "Comment 1.\nThe manuscript entitled ”Topological Protection by Local Support Symmetry and Destructive Interference” proposes the concept of Local Support Symmetry (LSS) as a means of explaining how symmetry-protected topological features can persist even when the overall Hamiltonian does not possess the corresponding global symmetry. The authors show that if a certain subsystem of a lattice retains a local symmetry, and if the coupling to the symmetry-breaking part satisfies specific algebraic conditions or gives rise to compact localised states through destructive interference, then the topological properties of that subsystem can survive in the full system. They develop a general algebraic framework, support it with several tight-binding examples, and conclude with a density-functional theory study of fluorinated biphenylene, which is presented as a possible material realisation.\nIn my opinion, the work is clearly written and addresses a problem of wide current interest—the stabilityoftopologicalfeaturesunderpartialsymmetrybreaking. Theproposedframeworkbridges different areas of physics: flat-band theory, coupled-layer constructions, and symmetry-protected topology.\nResponse.\nWesincerelythanktheReviewerforthispositiveandthoughtfulassessmentofourwork. Weare pleasedthattheReviewerfindsthemanuscriptclearlywrittenandthattheproposedconceptof Local Support Symmetry (LSS) and its implications for the robustness of topological features under partial symmetry breaking are viewed as timely and of broad interest. In particular, we appreciate the Reviewer’s recognition that our framework connects ideas from flat-band physics, coupled-layer constructions, and symmetry-protected topology, which was one of the central motivations of this study. We have carefully revised the manuscript to further improve clarity and presentation, while preserving the key physical insights highlighted by the Reviewer.\nComment 2.\nHowever, the novelty of the manuscript lies more in its conceptual synthesis than in a genuinely new physical principle. The central mechanisms that underpin LSS—block decomposition of the Hamiltonian, destructive interference, and the existence of compact localised states—are well established in the literature on flat-band and compact-localisation phenomena. For these reasons, the authors should more clearly acknowledge this connection and present LSS as a unifying and generalisable reformulation rather than as a fundamentally new symmetry class. In my opinion, moderating claims of novelty will make the work stronger and more credible.\nResponse.\nWe thank the referee for this thoughtful and constructive comment regarding the scope and framing of novelty in our manuscript. After carefully re-examining the manuscript in light of thiscomment,wefoundthatstrongwordingsuggestingafundamentallynewclassoftopological phases appeared in two sentences, namely in the Abstract and in the Discussion. Otherwise, throughout the most part of the main text, our intention has been to formulate a general and unifying framework that synthesizes and extends existing mechanisms, such as block decompo- sition, destructive interference, and compact localization, to a broader setting.\nWe agree that even these two instances of strong phrasing could give an unintended impression of claiming a new symmetry class or a fundamentally new physical principle. Following the referee’s suggestion, we have revised both sentences to moderate the claims and to emphasize more clearly the conceptual and unifying nature of the LSS framework. In the revised version, LSS is presented explicitly as a generalizable formulation for understanding topological protec- tion in systems with symmetry preserved only on a locally supported subspace, rather than as a new topological classification. We believe that these changes improve the clarity, balance, and credibility of the manuscript, and we sincerely thank the referee for prompting this refinement.\nThe corresponding revision can be found in Abstract and Discussion:\n[Abstract] “We formulate a unifying framework for solid-state phases in which symmetries pre- served only within a partial region of the system, termed local support symmetries (LSSs), can protect topological features of the full system, even in the presence of symmetry-breaking couplings.”\n[Discussion] “Our theory provides a unifying framework for interpreting protective mechanisms underlying a broad class of topological properties and band crossings in band structures, in- cluding situations where the relevant symmetry is preserved only on a restricted subspace.”\nComment 3.\nIt would also help readers if the relationship between LSS and coupled-layer constructions (which arecommonlyusedtobuildtopologicalcrystallineinsulators)wereclarified. Infact, thealgebraic structure employed here is similar, but the physical intention is reversed: rather than coupling layers to create topology, the LSS framework identifies conditions under which coupling does not destroy it. Framing the paper in this complementary light would situate it naturally among established approaches.\nResponse.\nWe thank the referee for this insightful comment and for suggesting a useful conceptual com- parison. Let us first clarify what is meant by coupled-layer constructions (CLC) in the context of topological crystalline insulators, and then explain how the present framework relates to, but also differs from, this established approach.\nCoupled-layerconstructionsareawidelyusedengineeringstrategyforgeneratinghigher-dimensional orcrystallineTopologicalphases. Inthisapproach,onestartsfromlower-dimensionaltopological building blocks (e.g., 1D SSH chains or 2D Chern/TCI layers) and introduces inter-layer cou- plings that are constrained by crystalline symmetries. The key idea is that individual layers can already host topologically nontrivial phases, while the specific pattern of symmetry-respecting interlayercouplingsorganizesandstabilizesthesefeaturesintoarobusthigher-dimensionalbulk topology with protected boundary states.\nThe referee is correct that the algebraic structure appearing in our work, namely, a block de- composition of the Hamiltonian and constraints on the inter-block hopping matrices, bears a formal resemblance to those used in CLC. However, the physical intention and logical direc- tion of the two approaches are fundamentally different, as the referee also mentioned. In CLC, inter-layer coupling is deliberately designed to create a new topological phase in the full system. By contrast, the LSS framework starts from a subsystem that already supports a symmetry- protected boundary or band-crossing feature, and asks under what conditions coupling to a symmetry-breaking sector fails to destroy this feature.\nIn this sense, LSS does not constitute a constructive recipe for generating topology, but rather a stability criterion: it identifies algebraic conditions, often realized through destructive interference or compact localization, under which hybridization with additional degrees of freedom becomesineffective. Importantly,theprotectedstatesintheLSSsettingneednotoriginatefrom stacking or layering, nor from crystalline symmetries acting globally; instead, their robustness follows from the fact that the relevant eigenstates have support only on a symmetry-preserving subspace that is annihilated by symmetry-breaking couplings.\nWe agree with the referee that framing the present work as complementary to coupled-layer constructions is helpful, and we will revise the discussion to emphasize this point more clearly. While CLC demonstrates how topology can be engineered by coupling, our work addresses the converse question: when and why topology can survive coupling. We believe that highlight- ing this distinction better situates the LSS framework within the broader landscape of modern approaches to symmetry-protected and crystalline topological phases.\nThe corresponding revision can be found in the first paragraph of the discussion section:\n“Understanding the role of coupling between block Hamiltonians in shaping topological phases has been a recurring theme in the study of symmetry-protected and crystalline topological matter. Aprominentexampleisthecoupled-layerconstruction, inwhichcouplingsbetweenstacked layers give rise to higher-dimensional topological phases. In these approaches, coupling plays a constructive role by generating a global topological obstruction in the full Bloch Hamiltonian. By contrast, the LSS framework focuses on identifying conditions under which coupling fails to destroy pre-existing symmetry-protected features.”\nComment 4.\nA further weakness of the present version is the absence of a quantitative assessment of ro- bustness. The protection mechanism depends on fine-tuned interference or on special coupling constraints, and its resilience to disorder, longer-range hopping, or structural perturbations re- mains uncertain. For Nature Communications, it would be important to demonstrate how the predicted topological features evolve under controlled symmetry-breaking perturbations, both in model Hamiltonians and in the DFT-based material. Such analyses would transform the work from a conceptual proposal into an experimentally testable claim.\nResponse.\nWe thank the referee for raising the important issue of quantitative robustness. We fully agree that, for a realistic assessment and experimental relevance, it is essential to characterize how LSS-protected band crossings evolve under controlled symmetry-breaking perturbations.\nTo address this point, we have introduced a quantitative measure of robustness, which we term thefragility ofanLSS-protectedbandcrossing. Thekeyobservationisthattheprotectionmech- anism relies on the stabilization of a compactly supported wave function through destructive interference. When the hopping parameters are modified such that this interference condition is violated, for example, by introducing longer-ranged hopping processes that break the com- patibility condition in Eq. (6) in the main text, the band-crossing is generally lifted and a finite gap is induced.\nWe quantify this effect by defining the fragility parameter\nF = ∆/t′ , (1)\nmax\nwhere ∆ denotes the induced gap and t′ is the maximum amplitude among the additional (symmetry-breaking) hopping terms. This dimensionless ratio directly measures the sensitivity of the band crossing to perturbations relative to their intrinsic energy scale. In generic situa- tions, one expects F ∼ O(1), indicating that the induced gap is comparable to the perturbative hopping strength. However, the precise value of F depends on how strongly the added hop- pings violate the destructive interference condition. In particular, F can vanish in the extreme case where longer-ranged hoppings are present but still satisfy the destructive interference con- straints, leaving the band crossing intact. Accordingly, the band crossing becomes increasingly robust as F → 0.\nIn the revised manuscript, we have explicitly evaluated the fragility F for all model Hamil- tonians considered in the paper, thereby providing a systematic and quantitative assessment of robustness across our examples. Importantly, this framework naturally extends to ab initio calculations: longer-ranged hopping amplitudes can be extracted from Wannierized DFT band structures, allowing the same fragility analysis to be applied to the DFT-based material ex- ample discussed in the manuscript. This establishes a direct and quantitative bridge between theconceptualLSSmechanism, realisticmicroscopicperturbations,andexperimentallytestable predictions.\nThedefinitionofthefragilitycanbefoundattheendofthesection“Generaltheory: semimetal case”:\n“The band crossing protected by the LSS can be lifted when the hopping parameters are tuned such that the compactly supported wave function is no longer stabilized. For instance, this occurs when longer-ranged hopping processes are introduced, thereby breaking the condition in Eq. (6). The resulting gap size depends on the detailed structure of the compactly supported wavefunction,includingitsspatialshapeandamplitudeconfigurationinrealspace. Wequantify the fragility of the band crossing against such perturbations by the ratio F = ∆/t′ , where max\n∆ denotes the induced gap and t′ represents the maximum amplitude among the additional hopping terms. Typically, one expects F ∼ 1, implying that the induced gap is comparable to theenergyscaleoftheperturbativehoppingprocesses. However, theprecisevalueofF depends on the extent to which these additional hoppings violate the destructive interference condition. In an extreme case, F can vanish if the added hoppings still satisfy the destructive interference condition. In general, the band crossing becomes more robust as F approaches zero.”\nIn the section “Model-I: topological insulator with local support time-reversal symmetry”, we use the fragility of the band-crossing at the time-reversal invariant momentum of the helical edge states:\n“By replacing ∆ in the definition of the fragility with the energy splitting of the edge states at k = 0, the fragility of the helical edge modes is estimated to be F ≈ 1.57×10−5.”\nIn the section “Model-II: Dirac fermions protected by local support C symmetry”, we include 2\nthe analysis of the fragility and the corresponding figure(Fig. 3) is also modified:\n“The condition for destructive interference can be violated by introducing additional hopping processes from the D site to its neighboring A and C sites, as illustrated in the inset of Fig. 3(f). When the hopping amplitudes for these four processes are identical (δt = 0), the destructive interference remains intact because the amplitudes arriving at the D site from the A and C sites cancel each other. In contrast, when δt ̸= 0, this cancellation no longer occurs and the band crossing becomes gapped. In Fig. 3(f), we quantify the fragility of the band crossing as a functionofδt. Onefindsthattheinducedbandgapremainsstronglysuppressedforsmallvalues of δt, indicating the robustness of the crossing against the longer-range hopping processes.”\nInthesection“Model-III:Diracfermionsenforcedbylocalsupportnonsymmorphicsymmetry”, we alsoincludetheanalysis ofthefragilityand thecorrespondingfigure(Fig.4)isalso modified:\n“One can ruin the destructive interference and open a gap by including the hopping processes from A and B sites to the neighboring C and D sites, as illustrated in the inset of Fig. 4(c). 2 2 1 1\nThe corresponding fragility is plotted as a function of t in Fig. 4(c). One can note that the 4\ngap is quite robust for small t .”\n“Thefragilitiesofthetwobandcrossings,DP andDP ,oftheunperturbedtight-bindingmodel 1 2\nshown in Fig. 5(d) against longer-ranged hopping processes t , t , and t are evaluated to be 6 7 8\nF = 0.00219 and 0.678, respectively. The exceptional robustness of DP originates from the 1\nfact that the destructive interference of Ψ is approximately preserved even after the inclusion 1\nof t , t , and t , since its amplitudes at A and A have opposite signs and the magnitudes 6 7 8 10 11\nof t and t are comparable. In contrast, for Ψ , the amplitudes at A and A have the same 7 8 2 10 11\nsign, and as a result, the destructive interference is strongly disrupted.”\nComment 5.\nIn conclusion, in its current form, the manuscript offers an interesting and pedagogically useful perspective on a set of known physical ideas. Its main strength is to unify several mecha- nisms—symmetry protection, destructive interference, and localised-state formation—within a single algebraic and conceptual framework. Yet for publication in Nature Communications, the authors should more carefully situate the work within existing literature, add quantitative tests of robustness, and moderate the claims regarding generality and novelty.\nIn my opinion, with these improvements, the paper could appeal to a wide readership across condensed-matter theory, photonics, and materials science. I therefore recommend major revision.\nResponse.\nWe sincerely thank the referee for the careful reading of our manuscript and for the thoughtful overallassessment. Weappreciatethereferee’srecognitionthatourmaincontributionistounify symmetry protection, destructive interference, and localized-state formation within a single algebraic and conceptual framework, which can serve a pedagogical role and help clarify the common structure behind these seemingly distinct mechanisms. We also take seriously all the referee’s concerns regarding positioning within the literature, quantitative robustness, and the toneofourclaims. Webelievethesechangessubstantiallystrengthenthemanuscriptandclarify its scope, while preserving the conceptual message highlighted by the referee. We are grateful for the recommendation for major revision, and we hope that the revised version now meets the standards for publication in Nature Communications and will indeed be of interest to readers across condensed-matter theory, photonics, and materials science.\nReviewer #2\nComment 1.\nIn this paper, the authors provide a framework for topological protection by local support symme- try (LSS), in which topological protections can persist even when a system respects a symmetry in one part of it while the symmetry is broken in another part. Here, LSS alone cannot pre- serve these topological characteristics; rather, through matrix analysis they find that the hopping processes between two parts of the system must satisfy a certain condition. In this work, they establish the explicit forms of these conditions for both the insulating and metallic cases, high- lighting that a destructive interference of the Bloch wave function can play a crucial role in the fulfillment of these conditions. I think that these results are novel and they expand our understanding of symmetry-protected topological phases.\nResponse.\nWe thank the referee for this positive and accurate summary of our work. We are pleased that the referee recognizes the novelty of the proposed framework and its contribution to broadening the understanding of symmetry-protected topological phases.\nComment 2.\nA minor mistake: (1)On page 8, the 6 × 6 matrix h1(k) should be the direct sum of two 3 × 3 matrices.\nResponse.\nWe thank the referee for pointing out this mistake. We have corrected the corresponding part\nThe corresponding revision can be found on Page X, Paragraph Y:\n“The S -part block of the Hamiltonian (??) is given by a 6×6 matrix h (k) = h (k)⊕h (k), 1 1 1,↑ 1,↓\nwhere...”\nReviewer #3\nComment 1.\nOver the past decade, substantial progress has been made in the study of topological phases. Thanks to intensive efforts, our understanding of phases protected by exact global or crystalline symmetries has become quite mature. In this work, the authors investigate local subspace sym- metries (LSS), where a designated subspace (or spatial subregion) respects a symmetry while the rest of the system does not.\nI find that the results presented in the manuscript are technically sound. However, in terms of novelty and potential impact on the field, I am afraid that the current manuscript is not suffi- ciently original or influential to warrant publication in Nature Communications. My assessment is based on the following two points:\nResponse.\nWe thank the referee for the careful reading of our manuscript and for the positive assessment of the technical correctness of our results. We also appreciate the referee’s comments regarding the novelty and potential impact of the work, which have helped us to clarify and strengthen the central contributions of the manuscript. In the revised manuscript, we have substantially improved the presentation to more clearly articulate the novelty and significance of our results. We now respond in detail to the two specific points raised by the referee below.\nComment 2.\nConceptual overlap with ”sub-symmetry” (Nature Physics 19, 992-998, 2023). The paper in Nature Physics introduced sub-symmetry-protected phases, where a symmetry is preserved only within a subspace of the full Hilbert space. For instance, the authors of that work considered a modified SSH chain in which either sublattice A or B preserves chiral symmetry, and they demonstrated that a zero mode remains protected. Much of the present manuscript appears closely related to this framework. While the authors certainly advance the idea, the fundamental concept seems very similar. Could the authors clarify the similarities and distinctions between topological phases protected by LSS and those protected by sub-symmetry?\nResponse.\nWe thank the referee for pointing out the conceptual relation to the recent work on sub- symmetry protection (Nature Physics 19, 992–998 (2023)). Below we briefly summarize that work and then clarify the essential differences from the present manuscript.\nSummary of the sub-symmetry approach. The Nature Physics paper introduced the concept of sub-symmetry (SubSy), in which a protecting symmetry is preserved only within a restricted subspace of the full Hilbert space, typically associated with a single sublattice. Using the SSH chain and the breathing kagome lattice as prototypical examples, the authors demonstrated that even when the global protecting symmetry and the corresponding bulk topological invariant are destroyed, certain boundary zero modes can remain pinned at zero energy if the perturbations respect the SubSy. In this framework, the protection mechanism relies on the fact that the boundary zero mode is entirely supported on a specific subspace (for example, a single sublattice). SubSy-preserving perturbations are defined such that they do not couple to thissubspace, oracttriviallyonit, whichimpliesthattheperturbationoperatorannihilatesthe boundary-state wave function. As a consequence, although the perturbations generally break the global protecting symmetry and destroy the bulk topological invariant, the boundary state remains an exact eigenstate with its eigenvalue pinned at zero energy.\nKey distinctions from the present work. While both approaches recognize that symme- try acting on a restricted subspace can have physical consequences, the scope, mechanisms, and implications are fundamentally different. The SubSy framework addresses the protection of specific boundary zero modes after the bulk topology has been destroyed. By contrast, the Local Support Symmetry (LSS) framework developed in the present manuscript provides a gen- eral algebraic criterion on the Hamiltonian that ensures the protection of topological features themselves, including band crossings and topological invariants, and applies to both insulating and metallic systems. In our work, the protection originates from destructive interference and the resulting local support of eigenstates, rather than from projector-based annihilation of per- turbations on a single sublattice.\nIn this sense, the conceptual overlap is limited to the general observation that a symmetry need not act on the full Hilbert space to be physically relevant, whereas the physical objects being protected, the underlying mechanisms, and the range of applicability are completely distinct.\nThe corresponding revision can be found at the final paragraph in the discussion section:\n“We note a conceptual relation to the recently introduced notion of sub-symmetry protection, where a symmetry preserved only within a restricted subspace can protect certain boundary zero modes even after the bulk topological invariant is destroyed. In that case, protection arises because the boundary state is fully supported on the subspace on which the sub-symmetry acts, so that symmetry-preserving perturbations do not shift its eigenvalue. By contrast, the LSS frameworkdevelopedhereprotectstopologicalfeaturesthemselvesthroughalgebraicconstraints on the Hamiltonian and the destructive interference plays a essential role.”\nComment 3.\nTopological features intrinsic to LSS. As currently presented, LSS-protected phases seem to inherit topology that already exists within the symmetric subspace. If this is the case, it remains unclear whether LSS gives rise to genuinely new phases or topological invariants beyond those already known. Could the authors clarify whether LSS can protect new types of topology that are unique to LSS-protected topological phases?\nResponse.\nWe thank the referee for raising this important conceptual question regarding whether LSS gives rise to genuinely new topological phases or invariants. As pointed out by the referee, we agree that the topological features protected by LSS, as presented in this work, are inherited fromthesymmetry-preservingsubspaceandcanbecharacterizedusingconventionaltopological invariants already established in the literature. In this sense, LSS does not introduce a new topological classification or fundamentally new invariants.\nAt the same time, following the another referee’s suggestion to emphasize the conceptual and unifying nature of the LSS framework, we have carefully revised the manuscript to clarify the intended scope of our contribution. The central aim of LSS is not to generate new topology, but to address a complementary and previously underexplored question: under what conditions do topological features defined within a restricted, symmetry-preserving subspace remain phys- ically robust when the corresponding symmetry is broken in the full Hamiltonian.\nFrom this perspective, LSS provides a general and physically transparent framework for under- standing the stability of known topological properties and band crossings under partial symme- try breaking. While the underlying invariants themselves are conventional, LSS identifies broad and model-independent conditions, such as block structure and destructive interference, under which these invariants continue to govern observable boundary or band-crossing phenomena in systems that would otherwise be classified as topologically trivial within standard symmetry- based approaches. We have clarified this point in the revised manuscript and believe that this framing better reflects the conceptual contribution and intended scope of the work.\nThe corresponding revision can be found in Abstract and Discussion:\n[Abstract] “We formulate a unifying framework for solid-state phases in which symmetries pre- served only within a partial region of the system, termed local support symmetries (LSSs), can protect topological features of the full system, even in the presence of symmetry-breaking couplings.”\n[Discussion] “Our theory provides a unifying framework for interpreting protective mechanisms underlying a broad class of topological properties and band crossings in band structures, in- cluding situations where the relevant symmetry is preserved only on a restricted subspace.”\nComment 4.\nScope of protected crossings. The section ”General theory: semimetal case” focuses on crossings enforced by symmetry representations. However, many gapless points are protected by higher-dimensional topological charges (such as a 1D winding number or a Chern number). Could the authors state explicitly whether such topological charges are outside the scope of this work, or explain how (if at all) the LSS framework constrains or incorporates them? A brief discussion of Weyl/Dirac nodes and nodal lines in the presence of LSS would be valuable.\nResponse.\nWe thank the referee for raising this important question regarding the scope of protected cross- ings and the role of higher-dimensional topological charges such as winding numbers, Chern numbers, and chiral charges.\nOur framework indeed focuses on band crossings enforced by local-support symmetries (LSSs) through representation constraints of the inter-part Hamiltonian. In this sense, the “General theory: semimetal case” addresses symmetry-enforced gapless points rather than topologically protected ones. We now clarify this explicitly in the revised manuscript.\n(i) Dirac and nodal-line semimetals: For Dirac points and nodal lines, crystalline symmetries, such as inversion, mirror, and rotational symmetries, play a central role in protecting the gap- less structures. Within our framework, one can indeed construct tight-binding models in which such crystalline symmetries act locally on one part of the system and are effectively realized through the LSS constraints, even when the global symmetry of the full system is broken. In other words, as long as the inter-part coupling satisfies the condition for the inter-part coupling derived in the section “General theory: semimetal case,” the Bloch wave functions localized in the symmetric part remain compactly supported and retain the Dirac or nodal-line structure. A nodal line semimetal example is discussed in the next question.\n(ii) Weyl semimetals: The situation for Weyl nodes is fundamentally different. Their stability does not rely on crystalline symmetries but rather on their topological monopole charge; thus, Weyl points do not require symmetry protection. For this reason, the analysis of Weyl nodes is not the principal target of the LSS framework, which is designed to capture symmetry-enforced, rather than topologically enforced, semimetallic structures.\nNonetheless, the LSS point of view can still offer insight. One can design models in which the wave functions forming the Weyl cones are compactly localized to a subsystem through de- structive interference, provided the inter-part hopping satisfies the LSS compatibility condition. This ensures that the Weyl-cone sector is preserved within one part of the lattice. One may also analyze, within the same framework, which forms of inter-part coupling prevent two Weyl nodes with opposite chiralities from merging and annihilating.\nWe have added a brief discussion in the discussion section in the revised text as follows.:\n“While we have examined this mechanism primarily in two-dimensional examples, it is equally applicable to three-dimensional systems, including Dirac and nodal-line semimetals. Dirac points and nodal lines, typically protected by crystalline symmetries such as inversion, mirror, or rotation, can still appear within our framework when these symmetries act locally on the S subsystem and the inter-part coupling satisfies the condition in Eq. (6). As a concrete 1\nillustration, Supplementary Sec. V demonstrates that a nodal line in a three-dimensional model can be protected by a local support mirror symmetry. In contrast, Weyl nodes fall outside the 10\nprimaryscopeofLSSprotection,astheirstabilityoriginatesfromatopologicalmonopolecharge rather than a crystalline symmetry. Nevertheless, if the inter-part hopping respects the LSS compatibility condition, destructive interference may still confine the Weyl-cone wave functions to S , thereby constraining how opposite-chirality Weyl nodes hybridize or annihilate through 1\ncouplings between the S and S parts.”\n1 2\nComment 5.\n3D example. From a materials perspective, it would be helpful if the authors could present an example involving a three-dimensional system.\nResponse.\nWe thank the referee for a fruitful suggestion. As a 3D example, we constructed a simple nodal line semimetal model, whose ring-shape band-crossing is protected by a local support mirror symmetry. We mentioned it in the discussion section of the revised manuscript as:\n“As a concrete illustration, Supplementary Sec. V demonstrates that a nodal line in a three- dimensional model can be protected by a local support mirror symmetry.\nDetails are included In Supplementary Information as:\n“We propose a three-dimensional toy model that hosts a ring-shaped nodal line protected by a local-support mirror symmetry along the z-axis. We start from a subsystem, denoted by S , 1\nwhich possesses a global mirror symmetry with respect to the z-axis, as illustrated in Fig. 5(a). This subsystem is constructed on a cubic lattice containing two sites per unit cell. We assume that s-orbitals reside on these sites. The sub-Hamiltonian for S , consisting of A and B sites, 1\nis given by\n(cid:18) (cid:19)\n0 f(k)\nh (k) = , (2)\n1 f(k)∗ 0\nwhere\nf(k) = −t (1+eikz)cosk −t (1+eikz)cosk −t (1+eikz)−it (1−eikz). (3)\nx x y y z 1\nWe have a nodal line at k = 0 plane for t = t = t = 1 and t = −0.2. This band-crossing is z x y 1 z\nprotected by the mirror symmetry with respect to the z-axis, whose mirror plane is indicated in Fig. 5(a). The mirror operation is represented by M = σ .\nz x\nWe now introduce two additional sites, labeled C and D, as shown on the right-hand side of Fig. 5(a). The subsystem composed of these two sites is denoted as S . When S and S are 2 1 2\ncoupled, the overall system no longer preserves the mirror symmetry. We assume that the C- site hosts an s-orbital, while the D-site is occupied by a p -orbital. Consequently, the hopping z\nbetween the neighboring B and C sites vanishes, and the hopping amplitudes from the D-site to the adjacent upper and lower A-sites acquire opposite signs due to the odd parity of the p z\norbital. The sub-Hamiltonian for the S part is given by\n2\nh (k) =\n(cid:18) ϵ\n0\nα+βe−ikz (cid:19)\n, (4)\n2 α+βeikz ϵ\n0\nwhere we set ϵ = −2, α = 1, and β = 0.5. On the other hand, the inter-part Hamiltonian is 0\ngiven by\n(cid:18) (cid:19)\nu 0\nh (k) = , (5)\n12 u w(1−eikz)\nwhere the second column reflects the properties of hopping processes between the p orbital at z\nthe D site and s orbitals at A and B sites, which are mentioned above. According to the general 11\ntheory for semimetal cases in the maintext, the columns of the inter-part Hamiltonian should be proportional to the eigenstate of the symmetry operation with an eigenvalue 1 or vanish at symmetry-invariant momenta for the nodal line to be protected although mirror symmetry is broken. In this nodal line semimetal model, the inter-part coupling indeed fullfills these conditions. The first column of h (k) is proportional to (1,1)T, which is the eigenvector of M 12 z\nwith an unity eigenvalue, and the second column vanishes at k = 0. Consequently, the nodal z\nline between the two bands of h (k) at k = 0 remains intact even when the parameters u and 1 z\nw are switched on, as shown in Fig. 5(b). The two crossing bands carry opposite local-support mirror eigenvalues, and the eigenstates with mirror eigenvalue −1 exhibit compact localization, having vanishing amplitudes on the C and D sites. Such eigenstates are stabilized through destructive interference: the opposite amplitudes on the A and B sites cancel at the C site, ensuring that the wave function remains confined within S . To quantify the robustness of the 1\nnodal line, we compute its fragility, defined as ∆/δt, against an additional hopping term that breaks the local-support condition by modifying the (2,2) element of h\n12\n(k) to w(1−eikz)+δt. Here, ∆ denotes the maximum gap opened along the nodal line. As shown in Fig. 5(c), the fragility decreases as δt increases, since the induced gap grows sublinearly as a function of δt.”"}, {"round": 2, "reviewer_comments": {"reviewer_1": "In the new version of their manuscript, the authors have addressed all the points I raised in my previous report. For this reason, I recommend the publication of their work on Nature Communications."}, "author_response": "We thank the reviewer for the positive assessment of our manuscript. We are pleased to hear that all points raised in the previous report have been satisfactorily addressed and appreciate the reviewer’s recommendation for publication in Nature Communications. No further changes were required in response to this comment."}]

Decision letter