Crystalline-equivalent topological phases of many-body fermionic systems in one dimension

Chang-Tse Hsieh; Chen-Shen Lee; Ken Shiozaki

REVIEW 4 major objections 5 minor 46 references

Crystalline-equivalent topological phases of many-body fermionic systems in one dimension

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Interacting reflection-symmetric fermion chains get a Z4 many-body classification

desk verdict Useful real-space invariants for 1D fermionic SPTs, but the reflection classification rests on an assumed AHSS differential. read the letter →

arxiv 2411.19268 v1 pith:6EJDTDNI submitted 2024-11-28 cond-mat.str-el cond-mat.mes-hallhep-th

Chen-Shen Lee , Ken Shiozaki , Chang-Tse Hsieh This is my paper · ORCID

classification cond-mat.str-elcond-mat.mes-hallhep-th

keywords symmetry-protectedtopologicalphasesone-dimensionalfermionscrystallineequivalencereflectionsymmetrychiralmany-bodyinvariantsAtiyah-Hirzebruchspectralsequencebulk-centercorrespondence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies one-dimensional charge-conserving fermionic systems whose gapped phases are protected by either chiral symmetry or reflection symmetry. Because of the crystalline equivalence principle, both families are classified by the same Z4 group, but the paper shows that the concrete physics differs: the chiral invariant counts unbound fermions at the edges, while the reflection invariant is carried by charges and reflection parities localized at the reflection center. Working with extensions of the SSH model, with and without interactions, the authors compute these many-body invariants and verify that gapped deformations between phases occur exactly when all invariants stay fixed. They further show that adding translation symmetry changes the reflection classification to Z×Z2×Z4, while leaving the chiral classification unchanged. A sympathetic reader would take away that real-space, ground-state quantum numbers are enough to fix the topology of these interacting phases.

What carries the argument

The load-bearing object is the Atiyah-Hirzebruch spectral sequence (AHSS) for generalized homology, applied to a Z2-symmetric cell decomposition of the line. Its first differential, set by an adiabatic-pump picture to Im(d1_{1,0})=Z(2,1) (or Z(2,1,−2,1) with translation), imposes the equivalence relation (N_C,R_C)+n(2,1)∼(N_C,R_C) that quotients the center quantum numbers down to Z4. The quantum numbers themselves are the charge number N_C and reflection eigenvalue R_C of the state localized at the reflection center, which for decomposable systems are directly read from the product ground state. The partial-reflection invariant Z^R then acts as the bridge: the paper argues that 2 Arg[Z^R]/π and (N_C,R_C) carry the same information, making the abstract cobordism invariant computable from real-space data.

What would settle it

Construct a decomposable reflection-symmetric chain whose center quantum numbers differ by the equivalence (2,1) but whose partial-reflection invariant Z^R evaluated at α=−π/2 takes different values; that would disprove Eq. (16). Equivalently, derive Im(d1_{1,0}) directly from local unitary pumps on the lattice models and check whether it equals Z(2,1); any other value invalidates the claimed Z4 quotient.

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Extended reading notes

Core claim

For a decomposable one-dimensional system with U(1)×Z2^R symmetry, the partial-reflection partition function Z^R(H_d) evaluates, at α=−π/2, to the pair of symmetry quantum numbers at the reflection center: (N_C,R_C) is equivalent to (2 Arg[Z^R(H_d)]/π, 0), under the equivalence (N_C,R_C)+n(2,1)∼(N_C,R_C). This quotient produces the Z4 classification. With translation symmetry the classification is the triple (Z_f, 2Arg[Z^R_C]/π, 2Arg[Z^R_C']/π), valued in Z×Z2×Z4, where Z_f is the filling per subsystem and the two reflection factors are evaluated at the two inequivalent reflection centers. The paper demonstrates this bulk-center correspondence numerically for the SSH extensions and their interacting versions, and shows that the resulting phase boundaries coincide with the gap-closing points of the many-body spectra.

Load-bearing premise

The entire Z4 and Z×Z2×Z4 classification hangs on the assigned value of the first spectral-sequence differential, Im(d1_{1,0}) = Z(2,1) (and Z(2,1,−2,1) with translation), which the paper takes from an adiabatic-pump picture rather than deriving from the lattice Hamiltonians; a different assignment would change the classification and the correspondence.

Editorial extensions

If this is right

Gapped deformations between models exist exactly when all topological invariants remain unchanged; a change in any chiral or reflection invariant forces a phase transition.
H2 and H−2, which carry opposite free-fermion winding numbers, can be adiabatically connected through an interacting chiral-symmetric path because they share the same many-body invariant Arg[Z^S]=π.
For reflection symmetry, H0 and H2 belong to the same reflection phase, so a reflection-preserving deformation connecting them exists even without interactions once chiral symmetry is broken.
Adding translation symmetry upgrades the reflection classification from Z4 to Z×Z2×Z4, while leaving the chiral classification untouched.
For decomposable systems, topological invariants can be replaced by simple symmetry quantum numbers, removing the need to evaluate partition-function traces.

Reading between the lines

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

One untested consequence is that the same center-quantum-number prescription should work for any decomposable system that respects the symmetry, not just the SSH-like models considered here; a reader could check this on other lattice families.
The paper's reliance on an assumed first differential suggests a direct lattice derivation of Im(d1_{1,0}) would be the natural next step; without it, the classification rests on a phenomenological input.
Because the equivalence (N_C,R_C)+n(2,1) is central, computing the partial-reflection invariant for systems whose center quantum numbers differ by (2,1) would provide a sharp test of the correspondence.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a useful paper that packages known classifications into concrete real-space formulas and checks them numerically. The soft spot is that the reflection-sector classification hangs on an unproved assignment of an AHSS differential, so Eq. (16) is only as secure as that input.

What's new: Eqs. (9), (16), (26) — real-space bulk-boundary and bulk-center correspondences for decomposable 1D fermionic SPT phases, and the Z×Z2×Z4 invariant triple when translation symmetry is present. The classifications themselves are from Shiozaki–Xiong–Gomi and cobordism, so the novelty is in the physical packaging, not the group-theoretic result. That packaging is worthwhile: counting unbound fermions or reading charges/parities at the reflection center gives people a direct way to identify phases in interacting models, without computing a partition-function invariant.

What the paper does well: the numerical checks are credible and extensive. Tables 1–5 systematically compare invariants to gap-closing in deformations, including the H2 → Hint → H−2 path where the many-body Z^S stays fixed while the free-fermion winding changes — a nice demonstration. Appendix A's analytical evaluation of Z^S and Z^R for H1 is clean and reproducible. The paper is also honest about the speculative nature of the translation construction.

The soft spots: the load-bearing input is the value of the first AHSS differential. The paper sets Im(d1) = Z(2,1) using the adiabatic-pump picture in Fig. 5, but it never computes that differential from the definition (45) or from the generalized homology functor. A different convention for the pin^c structure or a different identification of the 0-cell anomaly could change the quotient, and with it the claimed Z4 and the correspondence (16). The stress-test note is right about this. The same unproved input propagates to Section 6's Z×Z2×Z4. Second, Appendix D verifies the translation-invariant generators using Eq. (16) itself to convert quantum numbers into invariant values, so that part of the verification is circular. Third, there's a sign typo in Table 1's caption (the reader flagged it); minor.

Are these fatal? Not quite. The paper's core phase-structure verification is independent — Z^S and Z^R are evaluated from ground states and compared to spectra, and the results are consistent with the assumed d1. The d1 assignment is physically motivated and clearly flagged as a picture rather than a theorem. If a referee insists that the classification claim be proven from the homology functor, the authors should either supply that computation or explicitly re-cast the reflection correspondence as conditional on the d1 value. As it stands, I'd send this to peer review with a request to justify or de-emphasize the AHSS step, and to state the translation section as a proposal rather than a settled result.

Who it's for: anyone computing invariants of 1D interacting fermion models, especially those using decomposable or nearly decomposable systems. The paper is clearly written and the numerics are transparent. It deserves a serious referee.

Referee Report

4 major / 5 minor

Summary. The paper studies one-dimensional fermionic SPT phases with U(1)×Z2^S (chiral) and U(1)×Z2^R (reflection) symmetries, which are related by the crystalline equivalence principle and both classified by Z4. For extensions of the SSH model (H_α and interacting Hint_α), the authors compute the many-body invariants Z^S and Z^R from ground states, verify a chiral bulk-boundary correspondence 2Arg[Z^S]/π = N_b−N_a mod 4, and propose a reflection bulk-center correspondence relating Z^R to quantum numbers (N_C,R_C) at the reflection center via an Atiyah-Hirzebruch spectral sequence argument. They also treat the role of translation symmetry, claiming the classification becomes Z×Z2×Z4 and is described by the triple (Z_f, 2Arg[Z^R_C]/π, 2Arg[Z^R_C']/π).

Significance. If correct, the bulk-center correspondence provides a physically transparent real-space characterization of these interacting phases and a concrete illustration of how crystalline-equivalent SPT phases can differ microscopically. The paper's main strengths are its systematic evidence: Tables 1–5 show consistency between the invariants and the gap-closing behavior of the deformations; Appendix A gives clean analytic computations of Z^S and Z^R for H_1; and the deformations H_2→Hint±2→H_−2 and −H_0→−Hint0 give explicit interacting examples where the many-body picture differs from the free-fermion one. However, the mathematical classification claim rests on an unproved assignment of the first AHSS differential, and the translation-symmetry verification in Appendix D uses Eq. (16) in a circular way. These points need to be resolved before the central classification claim is fully established, though the numerical core of the paper appears sound.

major comments (4)

[§5.1, Eq. (15), Fig. 5] The value Im(d1_1,0)=Z(2,1) is assigned by the adiabatic-pump wave-packet argument in Fig. 5 rather than computed from the boundary map in Eq. (45) or from the generalized homology h^{U(1)×Z2^R}_0. Since this image determines the equivalence relation (N_C,R_C)+n(2,1)∼(N_C,R_C) in Eq. (15), and hence the Z4 quotient and the bulk-center correspondence Eq. (16), the central reflection classification is load-bearing on an unproved input. Please derive this differential from the cell decomposition and the underlying generalized homology theory, or clearly state it as an assumption and discuss whether alternative conventions (for the pin^c structure, the choice of generator on the 1-cell, or the anomaly contribution on the 0-cell) could change the quotient.
[§6, Eq. (23), Fig. 11] The translation-symmetry classification Z×Z2×Z4 in Eq. (23) relies on the same kind of unproved assignment, Im(d1_1,0)=Z(2,1,−2,1), justified by the adiabatic pump in Fig. 11. Because the entire change from the Z4 classification of §5.1 to the Z×Z2×Z4 classification of §6 comes from this image, this assignment needs the same level of mathematical justification as the nontranslation case. In particular, the paper should show that the differential follows from the Z⋊Z2-equivariant cell decomposition rather than from a heuristic wave-packet motion.
[Appendix D, §6.1] The verification of the proposed invariants for the generators in Eq. (29) is circular with respect to the bulk-center correspondence. Appendix D states: 'By considering the restriction (24), eq. (16), and the fact that ... we can readily verify that the generators (28) satisfy the statement (27).' Since Eq. (16) is precisely the bulk-center correspondence under investigation, the generator check does not independently establish the classification by the triple (Z_f, 2Arg[Z^R_C]/π, 2Arg[Z^R_C']/π). Please compute Z^R_C and Z^R_C' directly for the representing Hamiltonians in Eqs. (49), (52), and (55), or explicitly state that Eq. (16) is being assumed as a hypothesis rather than verified.
[§5.2, §6.3] The assignment of the quantum numbers (N_C,R_C) to a decomposable system is not unique, and the paper does not specify a canonical rule. In §6.3, the naive assignment for −H0 with translation symmetry C1/4 and C3/4 gives (0,0,0,0), and the authors replace it by (2,1,0,0) or (0,0,2,1) to satisfy restriction (24). Without a canonical prescription for localizing charges on the reflection center(s), the map from models to equivalence classes is not well-defined, and the agreement in Tables 2–5 could depend on this choice. Please specify a canonical assignment (e.g., from the ground-state product structure and the decomposition into subsystems) and prove that the resulting equivalence class is independent of the allowed charge moves.

minor comments (5)

[Abstract and §1] The phrase 'transition symmetry' in the abstract and in the introduction should read 'translation symmetry'.
[Table 3 caption] The caption says 'The way to calculate Z^R_C1/2 and Z^R_C1/2' but the second entry should be Z^R_C2/2.
[§6.3] The labels for the reflection centers in the translation-symmetry discussion are inconsistent: the triples for translation j→j+4 with centers C1/4 and C3/4 (and C2/4 and C4/4) are written with C1/2 and C2/2, and 'Table 5' for odd α should be 'Table 4'.
[§6.3] Two sentences are incomplete because the intended figures are missing: 'the picture of its charges in a subsystem is Here we make ...' and 'such as Because of ...' should either include the diagram or be rewritten as a self-contained verbal description.
[Appendix D, Eqs. (51)–(53)] The notation in Eq. (51) has unbalanced parentheses, and '∈ Z' should be '∈ Z × Z4 × Z4' or should specify the image of the map. In Eq. (53), the symbol '∼' is used inside a numerical expression; it should be replaced by an explicit statement about equality of equivalence classes in the quotient.

Circularity Check

1 steps flagged · score 4.0 of 10

Appendix D's generator check for the translation-symmetric triple is a consistency check: it computes the invariants from quantum numbers via Eq. (16), so the proof of Eq. (26) is partially circular.

self definitional [Sec. 6.1 / Appendix D, Eqs. (26)-(29)]
"By considering the restriction (24), eq. (16), and the fact that (NC, RC, NC′, RC′)+n(2, 1, −2, 1) ∼ (NC, RC, NC′, RC′) where n is an integer, we can readily verify that the generators (28) satisfy the statement (27). ... Using the quantum number, the relation (16), and the definition of Zf (25), we have Zf (H(ngZ)) = n, 2 Arg[Z R_C H(ngZ)]/π = n mod 4, 2 Arg[Z R_C′ H(ngZ)]/π = 0."

The generator verification in Appendix D is not an independent computation: it substitutes the chosen quantum numbers into Eq. (16), the bulk-center correspondence that Sec. 6 is trying to exploit, to obtain the invariant values reported in Eq. (29). The second and third entries of the triple (Zf, 2Arg[ZR_C]/π, 2Arg[ZR_C′]/π) are therefore restatements of Eq. (16) rather than values evaluated from the partial-reflection partition functions of the constructed Hamiltonians. Since Eq. (16) is itself established only conditionally, through the asserted AHSS differential Im(d1_{1,0}) = Z(2,1), this generator check is a consistency check with the assumed correspondence, not a proof that the triple classifies the Z×Z2×Z4 phases.

full rationale

The paper's core phase-structure verification is largely self-contained and non-circular. The chiral bulk-boundary correspondence (9) is proposed and then checked against independently computed partial-transpose partition functions (Table 1), while the reflection invariants ZR and the assigned quantum numbers (NC, RC) are separately evaluated for the lattice models and compared in Tables 2-5; those tables constitute real evidence for Eq. (16) for the models studied. The AHSS computation of the Z4 quotient uses an asserted first differential Im(d1_{1,0}) = Z(2,1), motivated by the adiabatic-pump picture in Fig. 5 rather than derived from the lattice Hamiltonians; this is a conditional physical input and a correctness risk, but it is not itself circular, since the pump argument is an independent heuristic and the resulting classification matches the external cobordism result. The genuine circular step is in Sec. 6.1/Appendix D: the 'verification' of the proposed triple invariant for the translation-symmetric generators computes 2Arg[ZR]/π by inserting quantum numbers into Eq. (16), so the outputs (29) reduce to the very correspondence whose validity the generator check is meant to support. Because the AHSS quotient (23) and the direct model tables provide independent content, the circularity is partial rather than total; the central classification claim does not collapse to a single fitted parameter. Score 4.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claims depend on the external many-body invariants, the AHSS framework and its d1 assignment, and the product-state decomposability of the models. No parameters were fitted to data; the only hand-set convention is the partial-reflection U(1) phase alpha = -pi/2. No new physical entities are postulated.

free parameters (1)

Partial-reflection U(1) phase alpha = -pi/2
Chosen in Eq. (8) so that Z^R realizes the Z4 invariant; Eq. (16) is stated only for this value. It is a convention from the literature, not fitted to data.

assumptions (6)

domain assumption Many-body invariants Z^S and Z^R from refs. [23,24,26] are valid complete invariants for 1D U(1) x Z2^S and U(1) x Z2^R fermionic SPT phases.
The paper's phase-structure claims assume these invariants classify SPTs; they are taken from prior work by, among others, the current authors.
domain assumption Cobordism classification Hom(Omega_2^{pin^c}(pt), U(1)) = Z4 and crystalline equivalence between reflection and chiral symmetries.
Used in Sec. 3 to identify Z^S and Z^R as Z4 invariants.
domain assumption AHSS machinery and G-symmetric cell decomposition for generalized homology from ref. [16].
Used in Sec. 5.1 and Sec. 6 to derive quotient classifications.
ad hoc to paper First differential image Im(d1_1,0) = Z(2,1), and its translation analogue Z(2,1,-2,1).
Assigned via the adiabatic pump picture in Fig. 5 rather than derived from the lattice Hamiltonians; it is load-bearing for Eqs. (15), (16), and (23).
domain assumption Decomposable models H_alpha and Hint_alpha have unique ground states at half-filling that are product states.
Needed to define Na, Nb, and the center quantum numbers; stated in Sec. 2 and used throughout.
domain assumption Translation-symmetry classification group h^{Z semidirect Z2}_0(R, dR) and the restriction Zf = N_C + N_C'.
Used in Sec. 6 to set the Z x Z2 x Z4 classification and the invariant triple.

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Pith. "Pith review of Crystalline-equivalent topological phases of many-body fermionic systems in one dimension." pith.science (2026). https://pith.science/paper/6EJDTDNI

@misc{pith2026241119268,
  author       = {Pith},
  title        = {Pith review of: Crystalline-equivalent topological phases of many-body fermionic systems in one dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6EJDTDNI}},
  note         = {Machine review of arXiv:2411.19268}
}

read the original abstract

We explore one-dimensional fermionic symmetry-protected topological (SPT) phases related by the crystalline equivalence principle. In particular, we study charge-conserving many-body topological phases of fermions protected respectively by chiral and reflection symmetries. While the classifications of the two crystalline-equivalent SPT phases are identical, their topological properties and phase structures can be very different, depending on the microscopic details. Specifically, we consider certain extensions of the Su-Schrieffer-Heeger model, with and without interactions, that preserve both chiral and reflection symmetries, and explicitly compute the many-body topological invariants based on the systems' ground states. The phase structures determined by these topological invariants align perfectly with the many-body spectra of deformations among the models. As expected, gapped deformations exist only when all the topological invariants remain unchanged. Moreover, we show that decomposable systems -- those that can be decomposed into local and decoupled subsystems -- can be topologically characterized by real-space quantum numbers directly associated with the symmetries. For reflection-symmetric systems, these quantum numbers are related to the many-body topological invariants via a bulk-center correspondence, which can be justified using the Atiyah-Hirzebruch spectral sequence in generalized homology theory. Finally, we discuss the role of transition symmetry in the many-body topologies of these SPT phases.

Figures

Figures reproduced from arXiv: 2411.19268 by the authors.

**Figure 1.** Schematic diagrams of Hα with α = −2, −1, 0, 1, 2 in the right semi-infinite limits. The shadow part represents the unit cell. (a) (b) (c) (d) (e) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Schematic diagrams of Hintα with α = −2, −1, 0, 1, 2 in the right semi-infinite limits. The blue parallelograms represent the four-site fermions interacting through quartic interactions. Hintα =    X j=1 (c † 4j−3 c † 4j−1 c4j+2|α|−2c4j+2|α| + h.c.), for α ≤ −1 X j=1 (c † 4j−3 c4j−2c † 4j−1 c4j + h.c.), for α = 0 X j=1 (c † 4j−2 c4j−1c † 4j c4j+1 + h.c.), for α = 1 X j=1 (c † 4j−2 c † 4j c… view at source ↗

**Figure 3.** (a)(b)(c) Arg[Z S] and (d)(e)(f) spectra of the deformations H1 → H0, H2 → H−2, and Hint2 → Hint0 with PBCs. As a concrete example, let us consider the following deformed Hamiltonian: (1 − t)Ha + tHb, t ∈ [0, 1] . (10) Tuning the parameter t from 0 to 1 in the above equation represents the adiabatic deformation from Ha to Hb, which we will denote as Ha → Hb in the subsequent discussion. The energy spectra of such sy… view at source ↗

Figures from the paper (14 more)

**Figure 4.** Figure 4: (a) Arg[Z S] of H2 → Hint2 (the same for H2 → Hint−2 , H−2 → Hint2 , and H−2 → Hint−2 ). (b)(c)(d)(e) Spectra of H±2 → Hint±2 with PBCs. and interesting case is deforming H2 to H−2. Since H2 and H−2 are characterized by different single-particle topological invariants …

**Figure 5.** Figure 5: Schematic of the first differential d 1 1,0 . The red dashed line denotes the reflection center, while the green and red wave packets represent complex fermions with charges +e and −e, respectively. To see the relation between Z R and (NC , RC ), we need to delve into…

**Figure 6.** Figure 6: For ±H0, the green wave packets represent the charges ψj = (c † 2j−1 ∓ c † 2j )/ √ 2. There are two types of reflection symmetry that ±H0 (or generally ±Hα) can respect, where the reflection centers are C 1 2 and C 2 2 . We highlight them with blue and red dashed lines…

**Figure 7.** Figure 7: For ±Hintα with even α, two possible types of reflection centers are labeled by C 1 4 and C 3 4 and highlighted with dashed blue and yellow lines. For odd α, we employ red and green lines to characterize the possible types of reflection centers with the labels C 2 4 an…

**Figure 8.** Figure 8: We consider the systems with Ntot = 12. (a) Arg[Z R] of H1 → H0 with the reflection center C 1 2 . The interval I = {c2, ..., c7} with the reflection center located between c4 and c5 is assigned here. (b)(c) Arg[Z R] of H2 → Hint0 and Hint2 → Hint0 with the reflection …

**Figure 9.** Figure 9: We consider the system with Ntot = 12. (a) Arg[Z R] of H0 → Hint0 with the reflection center C 1 4 . We pick up the interval I = {c1, ..., c4} with the reflection center situated between c2 and c3. (b) The energy spectrum of H0 → Hint0 with PBC. (a) [PITH_FULL_IMAGE:f…

**Figure 10.** Figure 10: The energy spectrum of −H0 → −Hint0 with PBC. 6 Role of translation symmetry In this section, we mainly discuss, when the translation symmetry gets involved, how the SPT phases of the 1d charge-conserved systems with chiral symmetry and with reflection symmetry will 1…

**Figure 11.** Figure 11: The cell decomposition and the schematic of the first differential [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

**Figure 12.** Figure 12: After rearranging, H2 is equivalent to stacking two dimerized SSH chains H1. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]

**Figure 13.** Figure 13: The purple arrows represent the directions of reflection symmetry. (a) The reflection [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]

**Figure 14.** Figure 14: The energy spectra of H1 → H0 with different values of Ntot. A level crossing occurs for the lowest two states (at t = 0.5) for Ntot = 8 and Ntot = 12, indicating a phase transition for this deformation. In topological band theory, a topological phase transition is ty…

**Figure 15.** Figure 15: The energy spectra of H2 → H−2, H2 → Hint0 , and Hint2 → Hint0 with different values of Ntot. A level crossing occurs in each deformation for either Ntot = 12 or Ntot = 16, indicating a phase transition. (a) (b) (c) (d) (e) (f) (g) (h) [PITH_FULL_IMAGE:figures/full_f…

**Figure 16.** Figure 16: Energy spectra of H±2 → Hint±2 with Ntot = 12 and Ntot = 16 . Since there are no level crossings in any of the spectra, we can reasonably conclude no phase transition occurs for these deformations. Appendix C Atiyah-Hirzebruch spectral sequence For the interacting sys…

**Figure 17.** Figure 17: The energy spectrum of H0 → H2 and H0 + 0.02Hcb → H2 + 0.02Hcb within k ∈ [0, 2π). References [1] Xiao-Liang Qi and Shou-Cheng Zhang. Topological insulators and superconductors. Rev. Mod. Phys., 83:1057–1110, Oct 2011. [2] Zheng-Cheng Gu and Xiao-Gang Wen. Tensor-enta…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

46 extracted references · 28 canonical work pages

[1]

Topological insulators and superconductors

Xiao-Liang Qi and Shou-Cheng Zhang. Topological insulators and superconductors. Rev. Mod. Phys., 83:1057–1110, Oct 2011

work page 2011
[2]

Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order

Zheng-Cheng Gu and Xiao-Gang Wen. Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order. Phys. Rev. B , 80:155131, Oct 2009

2009
[3]

Periodic table for topological insulators and superconductors

Alexei Kitaev. Periodic table for topological insulators and superconductors. AIP Conference Proceedings, 1134(1):22–30, 05 2009

work page 2009
[4]

Ching-Kai Chiu, Jeffrey C. Y. Teo, Andreas P. Schnyder, and Shinsei Ryu. Classification of topological quantum matter with symmetries. Reviews of Modern Physics , 88(3):035005–, 08 2016

work page 2016
[5]

Freed and Gregory W

Daniel S. Freed and Gregory W. Moore. Twisted Equivariant Matter. Annales Henri Poincar´ e, 12 2013

2013
[6]

Topology of crystalline insulators and superconductors

Ken Shiozaki and Masatoshi Sato. Topology of crystalline insulators and superconductors. Physical Review B , 90(16):165114–, 10 2014

work page 2014
[7]

On the K-Theoretic Classification of Topological Phases of Matter

Guo Chuan Thiang. On the K-Theoretic Classification of Topological Phases of Matter. Annales Henri Poincar´ e, 04 2016

work page 2016
[8]

Symmetry protected topo- logical orders and the group cohomology of their symmetry group

Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, and Xiao-Gang Wen. Symmetry protected topo- logical orders and the group cohomology of their symmetry group. Phys. Rev. B , 87:155114, Apr 2013. 22

work page 2013

Show all 46 references

[9]

Symmetry-protected topological orders for interact- ing fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory

Zheng-Cheng Gu and Xiao-Gang Wen. Symmetry-protected topological orders for interact- ing fermions: Fermionic topological nonlinear σ models and a special group supercohomology theory. Phys. Rev. B , 90:115141, Sep 2014

2014
[10]

Topological phases protected by point group symmetry

Hao Song, Sheng-Jie Huang, Liang Fu, and Michael Hermele. Topological phases protected by point group symmetry. Physical Review X , 7(1):011020–, 02 2017

2017
[11]

Building crystalline topo- logical phases from lower-dimensional states

Sheng-Jie Huang, Hao Song, Yi-Ping Huang, and Michael Hermele. Building crystalline topo- logical phases from lower-dimensional states. Physical Review B , 96(20):205106–, 11 2017

2017
[12]

Real-space recipes for general topological crystalline states

Zhida Song, Chen Fang, and Yang Qi. Real-space recipes for general topological crystalline states. Nature Communications, 11(1), August 2020

2020
[13]

Rotation symmetry-protected topological phases of fermions

Meng Cheng and Chenjie Wang. Rotation symmetry-protected topological phases of fermions. Phys. Rev. B , 105:195154, May 2022

2022
[14]

Real-space construction of crys- talline topological superconductors and insulators in 2d interacting fermionic systems

Jian-Hao Zhang, Shuo Yang, Yang Qi, and Zheng-Cheng Gu. Real-space construction of crys- talline topological superconductors and insulators in 2d interacting fermionic systems. Phys. Rev. Res., 4:033081, Jul 2022

2022
[15]

Construction and classifi- cation of crystalline topological superconductor and insulators in three-dimensional interacting fermion systems

Jian-Hao Zhang, Shang-Qiang Ning, Yang Qi, and Zheng-Cheng Gu. Construction and classifi- cation of crystalline topological superconductor and insulators in three-dimensional interacting fermion systems. arXiv e-prints , page arXiv:2204.13558, April 2022

2022 arXiv
[16]

Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena

Ken Shiozaki, Charles Zhaoxi Xiong, and Kiyonori Gomi. Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena. Progress of Theoretical and Experimental Physics , page ptad086, 07 2023

2023
[17]

Reflection positivity and invertible topological phases

Daniel S Freed and Michael J Hopkins. Reflection positivity and invertible topological phases. Geometry & Topology, 25:1165–1330, may 2021

2021
[18]

Symmetry protected topological phases, anomalies, and cobordisms: Beyond group cohomology, 2014

Anton Kapustin. Symmetry protected topological phases, anomalies, and cobordisms: Beyond group cohomology, 2014

2014
[19]

Fermionic symmetry pro- tected topological phases and cobordisms

Anton Kapustin, Ryan Thorngren, Alex Turzillo, and Zitao Wang. Fermionic symmetry pro- tected topological phases and cobordisms. Journal of High Energy Physics , 12 2015

2015
[20]

On the Cobordism Classification of Symmetry Protected Topological Phases

Kazuya Yonekura. On the Cobordism Classification of Symmetry Protected Topological Phases. Communications in Mathematical Physics , 06 2019

2019
[21]

Ryan Thorngren and Dominic V. Else. Gauging spatial symmetries and the classification of topological crystalline phases. Physical Review X , 8(1):011040–, 03 2018

2018
[22]

Cpt theorem and classification of topological insulators and superconductors

Chang-Tse Hsieh, Takahiro Morimoto, and Shinsei Ryu. Cpt theorem and classification of topological insulators and superconductors. Physical Review B , 90(24):245111–, 12 2014

2014
[23]

Many-body topological invariants in fermionic symmetry-protected topological phases: Cases of point group symmetries

Ken Shiozaki, Hassan Shapourian, and Shinsei Ryu. Many-body topological invariants in fermionic symmetry-protected topological phases: Cases of point group symmetries. Phys. Rev. B, 95:205139, May 2017

2017
[24]

Many-body topologi- cal invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries

Ken Shiozaki, Hassan Shapourian, Kiyonori Gomi, and Shinsei Ryu. Many-body topologi- cal invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries. Phys. Rev. B , 98:035151, Jul 2018

2018
[25]

Frank Pollmann and Ari M. Turner. Detection of symmetry-protected topological phases in one dimension. Phys. Rev. B , 86:125441, Sep 2012

2012
[26]

Many-body topological invariants for fermionic symmetry-protected topological phases

Hassan Shapourian, Ken Shiozaki, and Shinsei Ryu. Many-body topological invariants for fermionic symmetry-protected topological phases. Phys. Rev. Lett. , 118:216402, May 2017

2017
[27]

One-dimensional symmetry pro- tected topological phases and their transitions

Ruben Verresen, Roderich Moessner, and Frank Pollmann. One-dimensional symmetry pro- tected topological phases and their transitions. Physical Review B , 96(16):165124–, 10 2017

2017
[28]

Effects of interactions on the topological classification of free fermion systems

Lukasz Fidkowski and Alexei Kitaev. Effects of interactions on the topological classification of free fermion systems. Phys. Rev. B , 81:134509, Apr 2010

2010
[29]

Interacting one-dimensional fermionic symmetry-protected topological phases

Evelyn Tang and Xiao-Gang Wen. Interacting one-dimensional fermionic symmetry-protected topological phases. Phys. Rev. Lett. , 109:096403, Aug 2012. 23

2012
[30]

C. L. Kane and E. J. Mele. Z2 topological order and the quantum spin hall effect. Phys. Rev. Lett., 95:146802, Sep 2005

2005
[31]

Topological origin of zero-energy edge states in particle-hole symmetric systems

Shinsei Ryu and Yasuhiro Hatsugai. Topological origin of zero-energy edge states in particle-hole symmetric systems. Phys. Rev. Lett. , 89:077002, Jul 2002

2002
[32]

Hui Li and F. D. M. Haldane. Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum hall effect states. Phys. Rev. Lett. , 101:010504, Jul 2008

2008
[33]

Hatsugai

Y. Hatsugai. Bulk-edge correspondence in graphene with/without magnetic field: Chiral sym- metry, dirac fermions and edge states. Solid State Communications , 149(27):1061–1067, 2009

2009
[34]

Delplace, D

P. Delplace, D. Ullmo, and G. Montambaux. Zak phase and the existence of edge states in graphene. Phys. Rev. B , 84:195452, Nov 2011

2011
[35]

Roger S. K. Mong and Vasudha Shivamoggi. Edge states and the bulk-boundary correspondence in dirac hamiltonians. Phys. Rev. B , 83:125109, Mar 2011

2011
[36]

Bulk-edge correspondence for two-dimensional topo- logical insulators

Gian Michele Graf and Marcello Porta. Bulk-edge correspondence for two-dimensional topo- logical insulators. Communications in Mathematical Physics , 324(3):851–895, October 2013

2013
[37]

Asb´ oth, L´ aszl´ o Oroszl´ any, and Andr´ as P´ alyi.A Short Course on Topological Insulators

J´ anos K. Asb´ oth, L´ aszl´ o Oroszl´ any, and Andr´ as P´ alyi.A Short Course on Topological Insulators. Springer International Publishing, 2016

2016
[38]

Boundary green functions of topological insulators and superconductors

Yang Peng, Yimu Bao, and Felix von Oppen. Boundary green functions of topological insulators and superconductors. Phys. Rev. B , 95:235143, Jun 2017

2017
[39]

Winding number and zak phase in multi-band ssh models

Chen-Shen Lee, Iao-Fai Io, and Hsien-chung Kao. Winding number and zak phase in multi-band ssh models. Chinese Journal of Physics , 78:96–110, August 2022

2022
[40]

Bulk and Boundary Invariants for Complex Topo- logical Insulators

Emil Prodan and Hermann Schulz-Baldes. Bulk and Boundary Invariants for Complex Topo- logical Insulators. Springer International Publishing, 2016

2016
[41]

Essin and Victor Gurarie

Andrew M. Essin and Victor Gurarie. Bulk-boundary correspondence of topological insulators from their respective green’s functions. Phys. Rev. B , 84:125132, Sep 2011

2011
[42]

A linear algebra-based approach to understanding the relation between the winding number and zero-energy edge states

Chen-Shen Lee. A linear algebra-based approach to understanding the relation between the winding number and zero-energy edge states. SciPost Phys. Core , 7:003, 2024

2024
[43]

Turner, Yi Zhang, and Ashvin Vishwanath

Ari M. Turner, Yi Zhang, and Ashvin Vishwanath. Entanglement and inversion symmetry in topological insulators. Physical Review B , 82(24):241102–, 12 2010

2010
[44]

Hughes, Emil Prodan, and B

Taylor L. Hughes, Emil Prodan, and B. Andrei Bernevig. Inversion-symmetric topological insulators. Physical Review B , 83(24):245132–, 06 2011

2011
[45]

Topological phases protected by point group symmetry

Hao Song, Sheng-Jie Huang, Liang Fu, and Michael Hermele. Topological phases protected by point group symmetry. Phys. Rev. X , 7:011020, Feb 2017

2017
[46]

Xie Chen, F. J. Burnell, Ashvin Vishwanath, and Lukasz Fidkowski. Anomalous symmetry fractionalization and surface topological order. Phys. Rev. X , 5:041013, Oct 2015. 24

2015

Pith tools

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

[1] [1]

Topological insulators and superconductors

Xiao-Liang Qi and Shou-Cheng Zhang. Topological insulators and superconductors. Rev. Mod. Phys., 83:1057–1110, Oct 2011

work page 2011

[2] [2]

Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order

Zheng-Cheng Gu and Xiao-Gang Wen. Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order. Phys. Rev. B , 80:155131, Oct 2009

2009

[3] [3]

Periodic table for topological insulators and superconductors

Alexei Kitaev. Periodic table for topological insulators and superconductors. AIP Conference Proceedings, 1134(1):22–30, 05 2009

work page 2009

[4] [4]

Ching-Kai Chiu, Jeffrey C. Y. Teo, Andreas P. Schnyder, and Shinsei Ryu. Classification of topological quantum matter with symmetries. Reviews of Modern Physics , 88(3):035005–, 08 2016

work page 2016

[5] [5]

Freed and Gregory W

Daniel S. Freed and Gregory W. Moore. Twisted Equivariant Matter. Annales Henri Poincar´ e, 12 2013

2013

[6] [6]

Topology of crystalline insulators and superconductors

Ken Shiozaki and Masatoshi Sato. Topology of crystalline insulators and superconductors. Physical Review B , 90(16):165114–, 10 2014

work page 2014

[7] [7]

On the K-Theoretic Classification of Topological Phases of Matter

Guo Chuan Thiang. On the K-Theoretic Classification of Topological Phases of Matter. Annales Henri Poincar´ e, 04 2016

work page 2016

[8] [8]

Symmetry protected topo- logical orders and the group cohomology of their symmetry group

Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, and Xiao-Gang Wen. Symmetry protected topo- logical orders and the group cohomology of their symmetry group. Phys. Rev. B , 87:155114, Apr 2013. 22

work page 2013