Pith. sign in

REVIEW 4 major objections 5 minor 63 references

Discrete Shift and Polarization from Response to Symmetry Defects in Interacting Topological Phases

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Crystalline topological invariants survive interactions: discrete shift 1/2 and polarization 1/2 in an interacting Hofstadter insulator, with an interaction-driven shift 1 in the CDW phase.

desk verdict Credible DMRG demonstration that defect-bound charges stay quantized in interacting IQH and CDW phases; the CDW S_o=1 claim is plausible but rests on a background subtraction that needs more scrutiny. read the letter →

arxiv 2510.19483 v3 pith:W3XZHLWZ submitted 2025-10-22 cond-mat.str-el

Lu Zhang , Min Long , Yuxuan Zhang , Zi Yang Meng , Xue-Yang Song This is my paper
classification cond-mat.str-el
keywords discreteshiftpolarizationdisclinationdislocationHofstadtermodelDMRGchargedensitywavecrystallinetopologicalinvariants
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 tries to establish that two crystalline symmetry-protected topological invariants—the discrete shift S_o and the polarization vector P_o—can be read off from the charge bound to rotation and translation defects even when interactions are strong. It studies spinless fermions on a square lattice in a magnetic field (the interacting Hofstadter model) with nearest-neighbor repulsion, using DMRG to compute the local density around a disclination and a dislocation. In the Chern-insulator phase, the excess charge δQ=1/8 gives S_o=1/2 and the dislocation charge gives P_o,y=1/2; deep inside the charge-density-wave phase, δQ=1/4 gives S_o=1. The central claim is that these invariants remain quantized beyond single-particle band theory and can be extracted from local density alone, which is measurable in cold-atom or photonic systems.

What carries the argument

The cut-and-glue construction: remove a wedge or half-row and reconnect dangling bonds with gauge fields chosen so every new plaquette has the same magnetic flux as the clean lattice. This creates a disclination (angle π/2) or dislocation (Burgers vector (0,1)) without altering the local Hamiltonian away from the defect. The excess charge δQ = Q_W − ν n_W (mod 1), with partial-site weights for edge sites, converts the measured density into the invariants via Ω_W/(2π) S_o = δQ and δQ = b·P_o. DMRG on a matrix-product-state chain provides the ground-state densities, and flux-threading checks give the Chern number.

What would settle it

An independent determination of ν from a much larger clean DMRG run, or a different edge-weighting convention, that changes δQ away from 1/8 (or 1/4) in the same clusters would falsify the quantization claim; likewise, a DMRG run at V=1.3 showing S_o drifting outside 0.5±0.02 would contradict the claimed robustness.

Watch

Extended reading notes

Core claim

The paper claims that the charge bound to a π/2 disclination and to a dislocation in the interacting Hofstadter model is quantized: δQ=1/8 in the Chern-insulator phase (so S_o=1/2) and δQ=1/4 deep in the CDW phase (so S_o=1), while the dislocation gives P_o,y=1/2. These values match the topological field theory prediction S_o = C/2 mod 1. The CDW quantization is described as a purely interaction-driven effect that cannot be obtained from single-particle bands. The extraction uses only the local density around the defect, measured with DMRG, and converges for region sizes R≥2.

Load-bearing premise

The extraction assumes the clean bulk charge per unit cell ν is known exactly and uniform through the finite defective cluster, and that the partial-site weights unambiguously define the enclosed charge; any error in ν shifts all extracted invariants by a constant.

Editorial extensions

If this is right

  • The discrete shift and polarization stay quantized for interaction strengths up to V≈1.2 in the Chern-insulator phase, so the crystalline response survives correlations.
  • The CDW phase, which has no band topology, still shows a quantized discrete shift S_o=1, extending the invariant beyond single-particle band analysis.
  • Only local-density measurements are needed, avoiding nonlocal operators; this makes matrix-product-state and similar tensor-network methods practical for defect responses.
  • The dislocation construction gives a gauge-invariant polarization even in a Chern insulator, resolving the usual polarization ambiguity.
  • The same response can be probed in cold-atom or photonic implementations of the Hofstadter model with symmetry defects.

Reading between the lines

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

  • One could test whether fractional excess charges appear in fractional Chern insulators using the same local-density recipe; if they do, the method would give a many-body diagnostic for fractional crystalline invariants.
  • The sublattice dependence of S_o in the CDW phase (S_o=1 at one sublattice, 0 at the other) is a sharp prediction that could be checked by placing the disclination center on either sublattice in the same DMRG setup.
  • The random-interaction robustness shown in the supplement suggests the defect-bound charge is stabilized by topology rather than by fine-tuned symmetry; this could be turned into a quantitative study of disorder strength versus quantization.
  • Near the IQH-CDW transition the quantization is expected to fail or require much larger systems; tracking δQ as a function of V might provide a finite-size signature of the transition.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper uses DMRG to compute the local excess charge bound to a disclination and a dislocation in the interacting Hofstadter model, extracting the discrete shift S_o and polarization P_o,y. In the IQH phase (C = -1) the authors report δQ ≈ 1/8 at a π/2 disclination and δQ ≈ 1/2 at a dislocation, yielding S_o = 1/2 and P_o,y = 1/2, respectively, for V up to ~1.2. In the strong-interaction CDW phase, they report δQ ≈ 1/4 at the disclination and hence S_o = 1, which they interpret as a purely interaction-driven crystalline response beyond the single-particle band picture. The core claim is that local density measurements around symmetry defects provide a robust way to extract crystalline topological invariants in interacting systems.

Significance. If fully supported, this would be a useful numerical demonstration that crystalline symmetry-protected invariants remain well defined and measurable in strongly correlated phases, beyond the single-particle Hofstadter analysis of Refs. [33,34]. The method uses only local densities, avoiding nonlocal operators that are awkward in MPS, and the noninteracting results reproduce earlier single-particle calculations. The main new claim is the CDW-phase discrete shift S_o = 1, which is presented as an interaction-driven effect. However, this claim rests on a background-subtraction prescription that is not independently validated, and the manuscript contains an internal inconsistency in the magnetic flux used for the CDW calculation. The IQH results are plausible but lack quantitative error estimates.

major comments (4)
  1. [Strong interacting regime, Eq. (4) and Fig. 4(b)] The central new result, S_o = 1 in the CDW phase, is extracted from δQ = 1/4 using Eq. (4), δQ = Q_W − ν n_W (mod 1). In the CDW phase the bulk density is not uniform but alternates between sublattices, so a single scalar ν is only meaningful if the integration region W has exactly equal weight on the two sublattices; the disclination itself breaks that balance. The paper does not report the R-dependence of δQ for the CDW phase, does not give ν measured independently in the defective bulk, and provides no raw Q_W, n_W, or error bars. The admission near V ~ 1.5 that 'the computation of δQ requires very large size to reach quantization' further underscores the delicacy. If a different legitimate choice of ν or w(i) shifts δQ away from 1/4, the S_o = 1 claim is a subtraction artifact rather than a quantized response. This concern must be addressed before the central claim can be accepted.
  2. [Model Hamiltonian and Strong interacting regime (flux inconsistency)] The text states in the Model section that 'we set t = 1 and φ = 13π/12 throughout unless specified otherwise', and Table I and Figs. 2–3 are presumably computed at φ = 13π/12. The Strong interacting regime section, however, says 'with φ = π to form a commensurate filling'. This is not a trivial typo: the model, the filling relation Eq. (8), the Chern number, and the existence of the CDW phase all depend on φ. The reader cannot reproduce the CDW calculation from the inconsistent text. Moreover, the Chern number in the CDW phase is never computed; Eq. (3) is invoked to relate S_o = 1 to an even Chern number, but the actual value of C in the CDW phase is not established. The authors should state the flux for each result and either measure C in the CDW phase or explain why the relation is still valid.
  3. [Table I and Fig. 2(a)] The quantitative support for the 'quantized' claim is weakened by the absence of error bars or truncation extrapolation. Table I shows S_o drifting monotonically from 0.505 at V = 0 to 0.493 at V = 1.2, and P_o,y from 0.498 to 0.506; these are deviations of order 1%, comparable to the claimed precision. The text attributes the drift to finite-size effects, but no systematic large-size or bond-dimension extrapolation is shown. To claim quantization 'to great precision', the authors should provide a convergence analysis in bond dimension and system size (as was partially done for the dislocation in Fig. 3) and assign uncertainties to the extracted values.
  4. [Eq. (3) and interpretation of S_o = 1] Equation (3) defines S_o modulo 1. Therefore S_o = 1 is equivalent to S_o = 0 in that convention. If the CDW value is meant as an integer invariant distinct from 0, the manuscript should specify the convention and explain what physically distinguishes the representative 1 from 0. As written, the statement that the CDW discrete shift is 'a purely interaction-driven effect beyond the single-particle band analysis' is not justified: an integer S_o with C = 0 (or any even C) would be the trivial value in the mod-1 sense. The nontrivial content is the fractional charge δQ = 1/4 at a π/2 disclination, but the manuscript should carefully separate this from the mod-1 invariant.
minor comments (5)
  1. [Fig. 1 caption vs. main text] Fig. 1(b)/(d) states the Burgers vector as b⃗ = (0,−1), while the text in the Excess charge section says 'which is (0,1) in our study'. The sign convention should be made consistent.
  2. [SI, Details of DMRG calculation] The SI says 'we keep up to m = 1800 states in the DMRG simulation for disclination' but the main text says m = 3000 for the disclination and m = 1800 for the dislocation. This appears to be a typo: the SI sentence should refer to the dislocation.
  3. [Table I caption] The caption reads 'DMRG calculation of bond dimension m = 3000 and m = 1800' but does not state which value applies to S_o and which to P_o,y. Please clarify.
  4. [Introduction, paragraph 1] Typo: 'U(1) conversation' should be 'U(1) conservation'.
  5. [DMRG calculation, paragraph 1] 'we choose a multiple (around 20) initial ansatz' should be 'we choose a number (around 20) of initial ansatze' or similar.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: DMRG excess-charge measurements are compared with, not fitted to, prior theoretical predictions.

full rationale

The paper is a numerical measurement paper. The central quantities (δQ, S_o, P_o,y, and C) are obtained directly from DMRG ground-state densities and adiabatic flux insertion, not by fitting the predicted values. In Eq. (4), the background subtraction is defined by ν as the bulk charge per unit cell far from defects, and the same ν is used across interaction strengths and for both defect types; the 1/8, 1/4, and 1/2 results are not produced by tuning ν. The relations Eqs. (3), (5), and (6) are theoretical inputs from field theory and from Refs. [33,34]. Although some of those references share an author with the present paper, they are published, independently derived results that the present DMRG calculation verifies rather than assumes. The CDW S_o=1 result uses Eq. (15) from Ref. [34] to relate disclination centers, but the primary measured input is the DMRG excess charge δQ = 0.25 in Fig. 4(b), which is an independent measurement of the same response. The caveat that δQ requires very large sizes near V∼1.5 is a convergence limitation, not a circular step. No fitted parameter is relabeled as a prediction, and no non-interacting ansatz is smuggled in through a self-citation. Thus I find no significant circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

No new particles, forces, or conserved quantities are introduced; the symmetry gauge fields A_ω and A_T are borrowed from existing field-theory literature. The central claim rests on standard field-theory identifications and on numerical convergence assumptions.

free parameters (5)
  • magnetic flux φ = 13π/12 (IQH scan), π (CDW scan)
    Chosen to fix Chern number and commensurate filling; a model parameter, not fitted to the defect response.
  • background charge per unit cell ν = determined by filling/band structure (not explicitly quoted)
    Used as subtractive reference in Eq. (4); if set from the expected topological filling, it could bias δQ toward quantization.
  • DMRG bond dimension m = 3000 (disclination), 1800 (dislocation)
    Convergence control; truncation errors 1e-4 to 1e-6 are quoted but not propagated to δQ and no systematic m-extrapolation is shown for Table I.
  • region radius R and weights w(i) = R=2 for disclination; R≈5 for dislocation; w=1/4,2/4,3/4 on edges
    Analysis choices; R-scaling is checked but the final R is chosen by hand, and the weighting scheme must be exact for the mod-1 result.
  • chemical potential μ = -1
    Chosen to place the filling in the single-particle gap; determines electron number and thus ν in the interacting runs.
assumptions (6)
  • domain assumption The topological response is captured by the mutual Chern-Simons field theory Eq. (2) with coefficients C, S_o, and P_o.
    Standard framework from refs [33,34,41,42]; not re-derived in this paper.
  • domain assumption Discrete shift satisfies S_o = C/2 mod 1 (Eq. 3) for C4-symmetric Hofstadter lattices.
    Cited to [34]; used to interpret the DMRG result as a topological invariant.
  • domain assumption Excess charge around a defect equals the symmetry-flux response, δQ = (Ω_W/2π) S_o and δQ = b·P_o (Eqs. 5-6).
    The key identification that turns a local density measurement into a topological invariant; assumes the finite region W encloses all defect-bound charge.
  • domain assumption The cut-and-glue construction with gauge matching (Eqs. 10-14) preserves flux per plaquette and does not introduce spurious charge at the seam.
    The defect Hamiltonian is constructed by hand; a gauge ambiguity exists and is fixed using λ_i; incorrect gauge wiring would shift the excess charge.
  • domain assumption DMRG ground states on 13×13 and 8×30 clusters are converged enough to resolve O(10^-2) deviations around 1/8 and 1/4 fractional charges.
    No exhaustive bond-dimension extrapolation or error bars are reported for the headline values.
  • domain assumption In the CDW phase, C4 rotation symmetry is preserved and polarization is defined on an enlarged unit cell with Burgers vector (0,2).
    Used to interpret the strong-interaction results in the SI and main text.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Discrete Shift and Polarization from Response to Symmetry Defects in Interacting Topological Phases." pith.science (2026). https://pith.science/paper/W3XZHLWZ

@misc{pith2026251019483,
  author       = {Pith},
  title        = {Pith review of: Discrete Shift and Polarization from Response to Symmetry Defects in Interacting Topological Phases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W3XZHLWZ}},
  note         = {Machine review of arXiv:2510.19483}
}
read the original abstract

We extend the previous study of extracting crystalline symmetry-protected topological invariants to the correlated regime. We construct the interacting Hofstadter model defined on square lattice with the rotation and translation symmetry defects: disclination and dislocation. The model realizes Chern insulator and the charge density wave state as one tunes interactions. Employing the density matrix renormalization group (DMRG) method, we calculate the excess charge around the defects and find that the topological invariants remain quantized in both phases, with the topological quantity extracted to great precision. This study paves the way for utilizing matrix product state, and potentially other quantum many-body computation methods, to efficiently study crystalline symmetry defects on 2D interacting lattice systems.

Figures

Figures reproduced from arXiv: 2510.19483 by the authors.

Figure 1
Figure 1. FIG. 1. The disclination (a,c) with Ω = [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. With periodic boundary condition in [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Transition from IQH to CDW as the interaction [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (7 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Cut-and-glue procedure for constructing the disclination with the center [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. cut-and-glue procedure for constructing the dislocation. The procedure is same as the disclination with sites in the [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Single particle level of the cluster with disclination(a) and dislocation(b). The red dashed line shows the chemical [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The excess charge around the disclination(a) and dislocation(b) by adding the random nearest interaction. The [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The excess charge obtained from single particle calculation. The figure shows the excess charge versus the region width [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. CDW phase of Hofstadter square lattice model cylinder with dislocation [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The fully polarized CDW phase with disclination. The disclination center is unoccupied [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

63 extracted references · 7 linked inside Pith

  1. [1]

    Kitaev, Periodic table for topological insulators and superconductors, inAIP conference proceedings, Vol

    A. Kitaev, Periodic table for topological insulators and superconductors, inAIP conference proceedings, Vol. 1134 (American Institute of Physics, 2009) pp. 22–30

    2009

  2. [2]

    S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. Ludwig, Topological insulators and superconductors: tenfold way and dimensional hierarchy, New Journal of Physics12, 065010 (2010)

    2010

  3. [3]

    Hohenadler, Z

    M. Hohenadler, Z. Y. Meng, T. C. Lang, S. Wessel, A. Muramatsu, and F. F. Assaad, Quantum phase transi- tions in the kane-mele-hubbard model, Phys. Rev. B85, 115132 (2012)

    2012

  4. [4]

    C.-K. Chiu, J. C. Teo, A. P. Schnyder, and S. Ryu, Classi- fication of topological quantum matter with symmetries, Reviews of Modern Physics88, 035005 (2016)

    2016

  5. [5]

    Chen, Z.-C

    X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Physical Review B—Condensed Matter and Materials Physics87, 155114 (2013)

    2013

  6. [6]

    Kapustin, Symmetry protected topological phases, anomalies, and cobordisms: beyond group cohomology, arXiv preprint arXiv:1403.1467 (2014)

    A. Kapustin, Symmetry protected topological phases, anomalies, and cobordisms: beyond group cohomology, arXiv preprint arXiv:1403.1467 (2014)

    arXiv 2014

  7. [7]

    Kapustin, R

    A. Kapustin, R. Thorngren, A. Turzillo, and Z. Wang, Fermionic symmetry protected topological phases and cobordisms, Journal of High Energy Physics2015, 1 (2015)

    2015

  8. [8]

    Gu and X.-G

    Z.-C. Gu and X.-G. Wen, Symmetry-protected topologi- cal orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory, Physical Review B90, 115141 (2014)

    2014

Show all 63 references
  1. [9]

    D. V. Else and C. Nayak, Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge, Phys. Rev. B90, 235137 (2014)

    2014

  2. [10]

    Wang and T

    C. Wang and T. Senthil, Interacting fermionic topological insulators/superconductors in three dimensions, Phys. Rev. B89, 195124 (2014)

    2014

  3. [11]

    Senthil, Symmetry-protected topological phases of quantum matter, Annu

    T. Senthil, Symmetry-protected topological phases of quantum matter, Annu. Rev. Condens. Matter Phys.6, 299 (2015)

    2015

  4. [12]

    He, H.-Q

    Y.-Y. He, H.-Q. Wu, Y.-Z. You, C. Xu, Z. Y. Meng, and Z.-Y. Lu, Bona fide interaction-driven topological phase transition in correlated symmetry-protected topological states, Phys. Rev. B93, 115150 (2016). 6

    2016

  5. [13]

    He, H.-Q

    Y.-Y. He, H.-Q. Wu, Z. Y. Meng, and Z.-Y. Lu, Topo- logical invariants for interacting topological insulators. i. efficient numerical evaluation scheme and implementa- tions, Phys. Rev. B93, 195163 (2016)

    2016

  6. [14]

    He, H.-Q

    Y.-Y. He, H.-Q. Wu, Z. Y. Meng, and Z.-Y. Lu, Topo- logical invariants for interacting topological insulators. ii. breakdown of single-particle green’s function formalism, Phys. Rev. B93, 195164 (2016)

    2016

  7. [15]

    Song and A

    X.-Y. Song and A. P. Schnyder, Interaction effects on the classification of crystalline topological insulators and superconductors, Phys. Rev. B95, 195108 (2017)

    2017

  8. [16]

    Thorngren and D

    R. Thorngren and D. V. Else, Gauging spatial sym- metries and the classification of topological crystalline phases, Physical Review X8, 011040 (2018)

    2018

  9. [17]

    Barkeshli, Y.-A

    M. Barkeshli, Y.-A. Chen, P.-S. Hsin, and N. Manjunath, Classification of (2+ 1) d invertible fermionic topological phases with symmetry, Physical Review B105, 235143 (2022)

    2022

  10. [18]

    Aasen, P

    D. Aasen, P. Bonderson, and C. Knapp, Characterization and classification of fermionic symmetry enriched topo- logical phases, arXiv preprint arXiv:2109.10911 (2021)

    2021 arXiv

  11. [19]

    Zhang, S.-Q

    J.-H. Zhang, S.-Q. Ning, Y. Qi, and Z.-C. Gu, Con- struction and classification of crystalline topological su- perconductor and insulators in three-dimensional inter- acting fermion systems, arXiv preprint arXiv:2204.13558 (2022)

    2022 arXiv

  12. [20]

    charge bound to magnetic flux

    initial ansatz and determine the ground state with the lowest energy. Details about the cut-and-glue pro- cedure and DMRG simulation are presented in SI [39]. U(1) symmetry is utilized in DMRG simulation. To computeS o and ⃗Po, we construct the lattice with corresponding symme...

  13. [21]

    T. Ando, Y. Matsumoto, and Y. Uemura, Theory of hall effect in a two-dimensional electron system, Journal of the Physical Society of Japan39, 279 (1975)

    1975

  14. [22]

    Avron, R

    J. Avron, R. Seiler, and P. G. Zograf, Viscosity of quan- tum hall fluids, Physical review letters75, 697 (1995)

    1995

  15. [23]

    Read and E

    N. Read and E. Rezayi, Hall viscosity, orbital spin, and geometry: Paired superfluids and quantum hall systems, Physical Review B—Condensed Matter and Materials Physics84, 085316 (2011)

    2011

  16. [24]

    Haldane, ” hall viscosity” and intrinsic metric of incompressible fractional hall fluids, arXiv preprint arXiv:0906.1854 (2009)

    F. Haldane, ” hall viscosity” and intrinsic metric of incompressible fractional hall fluids, arXiv preprint arXiv:0906.1854 (2009)

    2009 arXiv

  17. [25]

    Haldane, Geometrical description of the fractional quantum hall effect, Physical review letters107, 116801 (2011)

    F. Haldane, Geometrical description of the fractional quantum hall effect, Physical review letters107, 116801 (2011)

    2011

  18. [26]

    A. G. Abanov and A. Gromov, Electromagnetic and gravitational responses of two-dimensional noninteract- ing electrons in a background magnetic field, Physical Review B90, 014435 (2014)

    2014

  19. [27]

    Gromov, G

    A. Gromov, G. Y. Cho, Y. You, A. G. Abanov, and E. Fradkin, Framing anomaly in the effective theory of the fractional quantum hall effect, Physical review letters 114, 016805 (2015)

    2015

  20. [28]

    S. Liu, A. Vishwanath, and E. Khalaf, Shift insu- lators: Rotation-protected two-dimensional topological crystalline insulators, Physical Review X9, 031003 (2019)

    2019

  21. [29]

    W. A. Benalcazar, J. C. Teo, and T. L. Hughes, Classi- fication of two-dimensional topological crystalline super- conductors and majorana bound states at disclinations, Physical Review B89, 224503 (2014)

    2014

  22. [30]

    T. Li, P. Zhu, W. A. Benalcazar, and T. L. Hughes, Fractional disclination charge in two-dimensional c n- symmetric topological crystalline insulators, Physical Re- view B101, 115115 (2020)

    2020

  23. [31]

    Y. You, J. Bibo, and F. Pollmann, Higher-order entangle- ment and many-body invariants for higher-order topolog- ical phases, Physical Review Research2, 033192 (2020)

    2020

  24. [32]

    May-Mann and T

    J. May-Mann and T. L. Hughes, Crystalline responses for rotation-invariant higher-order topological insulators, Phys. Rev. B106, L241113 (2022)

    2022

  25. [33]

    C. W. Peterson, T. Li, W. Jiang, T. L. Hughes, and G. Bahl, Trapped fractional charges at bulk defects in topological insulators, Nature589, 376 (2021)

    2021

  26. [34]

    Zhang, N

    Y. Zhang, N. Manjunath, G. Nambiar, and M. Barkeshli, Fractional disclination charge and discrete shift in the hofstadter butterfly, Physical Review Letters129, 275301 (2022)

    2022

  27. [35]

    Zhang, N

    Y. Zhang, N. Manjunath, G. Nambiar, and M. Barkeshli, Quantized charge polarization as a many-body invariant in (2+ 1) d crystalline topological states and hofstadter butterflies, Physical Review X13, 031005 (2023)

    2023

  28. [36]

    Zhang and M

    Y. Zhang and M. Barkeshli, Fractionally quantized elec- tric polarization and discrete shift of crystalline frac- tional chern insulators, arXiv preprint arXiv:2411.04171 (2024)

    2024 arXiv

  29. [37]

    Kobayashi, Y

    R. Kobayashi, Y. Zhang, N. Manjunath, and M. Barkeshli, Crystalline invariants of fractional chern insulators, arXiv preprint arXiv:2405.17431 (2024)

    2024 arXiv

  30. [38]

    S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett.69, 2863 (1992)

    1992

  31. [39]

    Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96–192 (2011)

    U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96–192 (2011)

    2011

  32. [40]

    In this supplemental information, we provide the details about the construction of the Hamiltonian in the presence of symmetry defects, details of DMRG calculation, and Disclination and dislocation with random interaction

  33. [41]

    For example, the translation symmetry generatorsT x andT y satisfies the following relation:T xTyT −1 x T −1 y =e iϕ

    It means that the crystalline symmetry is realized projec- tively under the background magnetic field. For example, the translation symmetry generatorsT x andT y satisfies the following relation:T xTyT −1 x T −1 y =e iϕ

  34. [42]

    Song, Y.-C

    X.-Y. Song, Y.-C. He, A. Vishwanath, and C. Wang, Electric polarization as a nonquantized topological re- sponse and boundary luttinger theorem, Physical Review Research3, 023011 (2021)

    2021

  35. [43]

    Manjunath and M

    N. Manjunath and M. Barkeshli, Crystalline gauge fields and quantized discrete geometric response for abelian topological phases with lattice symmetry, Physical Re- view Research3, 013040 (2021)

    2021

  36. [44]

    The details can be found in paper [34]

    The discrete shift and polarization vector in lattice de- pends on the choice of the origino, which is not the focus of the current paper. The details can be found in paper [34]

  37. [45]

    R. B. Laughlin, Quantized hall conductivity in two di- mensions, Phys. Rev. B23, 5632 (1981)

    1981

  38. [46]

    Pollmann and A

    F. Pollmann and A. M. Turner, Detection of symmetry- protected topological phases in one dimension, Physical Review B—Condensed Matter and Materials Physics86, 125441 (2012)

    2012

  39. [47]

    M. P. Zaletel, Z. Zhu, Y.-M. Lu, A. Vishwanath, and S. R. White, Space group symmetry fractionalization in a chiral kagome heisenberg antiferromagnet, Physical re- view letters116, 197203 (2016)

    2016

  40. [48]

    M. P. Zaletel, Y.-M. Lu, and A. Vishwanath, Measur- ing space-group symmetry fractionalization in z 2 spin liquids, Physical Review B96, 195164 (2017)

    2017

  41. [49]

    Cincio and Y

    L. Cincio and Y. Qi, Classification and detection of symmetry fractionalization in chiral spin liquids, arXiv preprint arXiv:1511.02226 (2015)

    2015 arXiv

  42. [50]

    Sun, Y.-C

    G.-Y. Sun, Y.-C. Wang, C. Fang, Y. Qi, M. Cheng, and Z. Y. Meng, Dynamical signature of symmetry fraction- 7 alization in frustrated magnets, Phys. Rev. Lett.121, 077201 (2018)

    2018

  43. [51]

    Huang, X

    C.-Y. Huang, X. Chen, and F. Pollmann, Detection of symmetry-enriched topological phases, Physical Review B90, 045142 (2014)

    2014

  44. [52]

    Shiozaki, H

    K. Shiozaki, H. Shapourian, and S. Ryu, Many-body topological invariants in fermionic symmetry-protected topological phases: Cases of point group symmetries, Physical Review B95, 205139 (2017)

    2017

  45. [53]

    T. M. Gunawardana, F. Schindler, A. M. Turner, and R. Barnett, Microscopic theory of chern polarization, arXiv preprint arXiv:2502.17735 (2025)

    2025

  46. [54]

    Zhang and M

    Y. Zhang and M. Barkeshli, Electric polarization in chern insulators: Unifying many-body and single-particle ap- proaches, Physical Review B112, 115124 (2025)

    2025

  47. [55]

    Aidelsburger, M

    M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Realization of the hofstadter hamiltonian with ultracold atoms in optical lattices, Physical review letters111, 185301 (2013)

    2013

  48. [56]

    Miyake, G

    H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Bur- ton, and W. Ketterle, Realizing the harper hamiltonian with laser-assisted tunneling in optical lattices, Physical review letters111, 185302 (2013)

    2013

  49. [57]

    C. J. Kennedy, W. C. Burton, W. C. Chung, and W. Ket- terle, Observation of bose–einstein condensation in a strong synthetic magnetic field, Nature Physics11, 859 (2015)

    2015

  50. [58]

    Hafezi, S

    M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. Taylor, Imaging topological edge states in silicon photonics, Na- ture Photonics7, 1001 (2013)

    2013

  51. [59]

    Ozawa, H

    T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zil- berberg,et al., Topological photonics, Reviews of Modern Physics91, 015006 (2019)

    2019

  52. [60]

    Beijing PARATERA Tech CO.,Ltd

  53. [61]

    Dawning Information Industry Co.,Ltd

  54. [62]

    Fishman, S

    M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor Software Library for Tensor Network Calcula- tions, SciPost Phys. Codebases , 4 (2022)

    2022

  55. [63]

    Fishman, S

    M. Fishman, S. R. White, and E. M. Stoudenmire, Code- base release 0.3 for ITensor, SciPost Phys. Codebases , 4 (2022). SUPPLEMENT AL INFORMA TION Construction of the Hamiltonian in the presence with symmetry defects In this section, we briefly explain how to construct the Hof...

    2022

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.