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arxiv: 1907.11791 · v1 · pith:SO6ASBOHnew · submitted 2019-07-26 · 🧮 math-ph · cond-mat.mes-hall· math.MP· math.OA

A Guide to the Bott Index and Localizer Index

Terry A. Loring This is my paper

Pith reviewed 2026-05-24 14:53 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.mes-hallmath.MPmath.OA
keywords Bott indexlocalizer indexChern insulatortopological invariantspseudospectral indexmathematical physics
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The pith

The localizer index can be tuned to behave like the global Bott index on a Chern insulator model.

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

This paper guides computation of two topological indices on a standard Chern insulator. The Bott index draws on global system information while the localizer index uses only local data. The authors detail effective programming approaches for both on the model. They show a tuning method that lets the localizer index reproduce the global behavior of the Bott index. A reader cares because this bridges local calculations with global topological properties in physical systems.

Core claim

The Bott index is inherently global while the localizer index, formerly the pseudospectral index, is local. On a standard Chern insulator model the localizer index can be tuned so that it behaves like a global index, and the paper emphasizes practical programming steps for both quantities.

What carries the argument

The localizer index tuned via adjustable parameters to match global topological invariants on the Chern insulator lattice.

If this is right

  • Programming routines for the indices become practical on finite lattices with standard Chern models.
  • Local data suffice for global index values once the tuning parameters are chosen.
  • The localizer index serves as a drop-in replacement for the Bott index after tuning.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The tuning approach may allow index calculations on systems with disorder where global phases are hard to define.
  • Similar local-to-global matching could apply to other topological invariants beyond the Chern case.

Load-bearing premise

The standard Chern insulator model and the described tuning procedure generalize beyond the specific examples shown.

What would settle it

Compute the tuned localizer index and the Bott index on a lattice model without Chern number or on a different insulator Hamiltonian and check whether the values continue to agree after tuning.

Figures

Figures reproduced from arXiv: 1907.11791 by Terry A. Loring.

Figure 2.1
Figure 2.1. Figure 2.1: Some of the pseuodospectrum of a round Chern insulator. Along the x-y plane the gap kLλ(X0, Y0, H) −1k −1 is indicated by color. Also shown are portions of some contour surfaces for a few gap sizes very close to zero. where C is the norm of the infinite-area Hamiltonian. We have here a fuzzy version of S(C, c) = ([−1, 1] × [−1, 1] × [−C, C]) \ B(c) where B(c) is the open ball of radius c at the origin. T… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Some of the pseuodospectrum of a round Chern insulator with a hole in it. Where this is zero is the Clifford spectrum. We can more easily compute where this is small, and get an estimate on where lives the Clifford spectrum. Now that we are working with just three observables, we can specify a choice for the Γj , specifically Γ1 = σx, Γ2 = σy, Γ3 = σz. Then (2.3) Lλ(M1, M2, M3) = (M1 − λ1) ⊗ σx + (M2 − λ… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Bott index averaged over disorder as a function of the Fermi level: L = 20 (12,664 samples); L = 40 (7,681 samples); L = 60 (3,013 samples); L = 80 (1,121 samples). scalar logarithm of each, and add those. The heart of the algorithm is as follows, when implemented in Matlab. U = W'*exp_x*W; V = W'*exp_y*W; T = eig(commutator); % Just a list of eigenvalues index = -sum(imag(log(T)))/(2*pi); index = round(… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: For L = 10 to L = 100 the average time. Left panel is the pre￾ferred method using the matrices as in equation 3.2, right panel the method using the larger matrices described in Equation 3.5. The solid line with squares indicates the time in seconds needed for a full diagonalization of the Hamil￾tonian. The dashed line indicates the time in seconds for the rest of the calculation, with the energy cutoff a… view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: For L = 10 to to L = 100 the average error, defined as the distance to the closest integer. The solid line is the average error while, for reference, the dotted line plots the curve y = (0.4 × 10−17) L 2 . This plot is based on 32 samples for each value of L. feasable to work up to L = 80 instead of only L = 60, as in [26, [PITH_FULL_IMAGE:figures/full_fig_p010_4_3.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: For L = 40 to L = 400 the average time for the localizer method, time to compute the gap and to compute the index plotted separately. The testing was performed on a 8-core computer with each core rated at 2.67GHz. For reference, the dotted line plots the curve y = (5 × 10−9 ) L 4 . These plots are based one Fermi level of of 10 disordered Hamiltonians for each value of L. this is not a thorough analysis … view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: A slice of the pseudospectrum at λ3 = EF = 0. No disorder, various κ [PITH_FULL_IMAGE:figures/full_fig_p013_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: A slice of the pseudospectrum at λ3 = EF = −2.4. No disorder, various κ [PITH_FULL_IMAGE:figures/full_fig_p014_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: A slice of the pseudospectrum at λ3 = EF = 0. Strong disorder, various κ [PITH_FULL_IMAGE:figures/full_fig_p015_5_4.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: A slice of the pseudospectrum at λ3 = EF = −2.4. Strong disorder, various κ [PITH_FULL_IMAGE:figures/full_fig_p016_5_5.png] view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Pseudospectrum. Computed index. Extended by using image editing software. Index overlayed (darken only mode) on pseudospectrum. Blue is index 0. Red is index 1. Total black is where the Clifford spectrum and points where localizer gap is too small for the index to make even a little sense. and the matrix on the right has spectrum that is symmetric across 0. Also the localizer index is zero if κ is large … view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: Index as color overlay on pseudospectum, no disorder, at λ3 = EF = 0 and λ3 = EF = −2.4. κ = 1, E = 0 disorder 8 κ = 1, E = −2.4 disorder 8 [PITH_FULL_IMAGE:figures/full_fig_p018_5_7.png] view at source ↗
Figure 5.8
Figure 5.8. Figure 5.8: Index as color overlay on pseudospectum, with disorder, at λ3 = EF = 0 and λ3 = EF = −2.4. We also want a stability against disorder. With the rather strong disorder as used in Section 4, and just one sample, the same slices are shown in Figures 5.4 and 5.5. At EF = 0, as shown in [PITH_FULL_IMAGE:figures/full_fig_p018_5_8.png] view at source ↗
Figure 5
Figure 5. Figure 5: is harder to interpret without the [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the same but now with disorder, showing “in the bulk” mainly index [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows what happens for various values of [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Disorder averaged localizer index, with κ = C/L [PITH_FULL_IMAGE:figures/full_fig_p020_6_1.png] view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: Various κ values, searching for appropriate value for L. There is some theory when one has a bulk gap of a known size, as in [21]. However, we have just a mobility gap. Presumably, one could work in an appropriate C ∗ -algebra, find a gapped Hamiltonian that lives in that C ∗ -algebra and can extend the methods of [21] to that setting. For now, we are happy to gather numerical evidence. In [PITH_FULL_IM… view at source ↗
Figure 6.3
Figure 6.3. Figure 6.3: Localizer index averaged over disorder as a function of the Fermi level: κ = 0.01 (16,955 samples with L=80); κ = 0.001 (14,878 samples with L=100); κ = 0.0001 (5,598 samples with L=120); κ = 0.00001 (8,031 samples with L=200); κ = 0.000001 (870 samples with L=400). 7. Local nature of the localizer Better algorithms are needed to make the spectral localizer effective for three-dimensional systems, and in… view at source ↗
read the original abstract

The Bott index is inherently global. The pseudospectal index is inherently local, and so now the preferred name is the localizer index. We look at these on a rather standard model for a Chern insulator, with an emphasis how to program these effectively. We also discuss how to tune the localizer index so it behaves like a global index.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript is an expository guide to the Bott index (global topological invariant) and the localizer index (local version, formerly called the pseudospectral index). It demonstrates both on a standard Chern insulator model, with emphasis on practical programming implementation details, and discusses tuning the localizer index so that it behaves like a global index.

Significance. The paper supplies concrete implementation guidance and an explicit construction on the Chern insulator model showing that the localizer index can be tuned to align with the Bott index. This practical focus, together with the absence of any asserted general theorem beyond the demonstrated examples, provides useful reference material for numerical work in topological insulators.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'with an emphasis how to program' is missing the preposition 'on' and should read 'with an emphasis on how to program'.
  2. The manuscript would benefit from a short table or pseudocode listing the key tuning parameters and their effect on the localizer index for the Chern model, to make the tuning discussion more immediately usable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript as a practical guide to the Bott and localizer indices on Chern insulator models. The recommendation for minor revision is noted. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Expository guide; no derivation chain or fitted predictions present

full rationale

The paper presents an expository guide to the Bott index and localizer index on a standard Chern insulator model, focusing on programming and tuning guidance. No new first-principles derivations, predictions, or uniqueness theorems are asserted that could reduce to self-definition, fitted inputs, or self-citation chains. The central discussion of tuning is explicit construction on the given model rather than a general claim requiring external justification. No load-bearing steps match any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No new free parameters, axioms, or invented entities are introduced in the abstract; the work relies on standard definitions of Bott and localizer indices from prior literature.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The Bott Metric: A Real-Space Bridge Between Topology and Quantum Metric

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    The Bott metric, derived from the plaquette operator, unifies topology and quantum geometry by converging to the trace of the integrated quantum metric in the thermodynamic limit for disordered and amorphous systems.

  2. Adiabatic charge transport through non-Bloch bands

    cond-mat.mes-hall 2025-11 unverdicted novelty 5.0

    Non-Bloch bands in a non-Hermitian extended SSH model support adiabatic charge transport that preserves quantized flow when the bands remain gapped during time evolution.

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Works this paper leans on

33 extracted references · 33 canonical work pages · cited by 2 Pith papers · 2 internal anchors

  1. [1]

    mathworks.com/help/matlab/ref/ldl.html

    Matlab function ldl documentation. mathworks.com/help/matlab/ref/ldl.html. Accessed: 2019-07-24

    work page 2019

  2. [2]

    mumps.enseeiht.fr/doc/userguide_5.2.1.pdf

    MUltifrontal Massively Parallel Solver Users guide. mumps.enseeiht.fr/doc/userguide_5.2.1.pdf. Ac- cessed: 2019-07-24

    work page 2019

  3. [3]

    math.unm.edu/%7Eloring/BottLocalizerGuide/

    Supplementary files. math.unm.edu/%7Eloring/BottLocalizerGuide/

    work page

  4. [4]

    Topological photonic quasicrystals: Fractal topological spectrum and protected transport.Physical Review X, 6(1):011016, 2016

    Miguel A Bandres, Mikael C Rechtsman, and Mordechai Segev. Topological photonic quasicrystals: Fractal topological spectrum and protected transport.Physical Review X, 6(1):011016, 2016

    work page 2016

  5. [5]

    Bellissard, A

    J. Bellissard, A. van Elst, and H. Schulz-Baldes. The noncommutative geometry of the quantum Hall effect. J. Math. Phys. , 35(10):5373–5451, 1994. Topology and physics

    work page 1994

  6. [6]

    Decay properties of spectral projectors with applications to electronic structure.SIAM review, 55(1):3–64, 2013

    Michele Benzi, Paola Boito, and Nader Razouk. Decay properties of spectral projectors with applications to electronic structure.SIAM review, 55(1):3–64, 2013

    work page 2013

  7. [7]

    Matrix embeddings on flatR3 and the geometry of membranes

    David Berenstein and Eric Dzienkowski. Matrix embeddings on flatR3 and the geometry of membranes. Physical Review D, 86(8):086001, 2012

    work page 2012

  8. [8]

    Mapping topological order in coordinate space.Physical Review B, 84(24):241106, 2011

    Raffaello Bianco and Raffaele Resta. Mapping topological order in coordinate space.Physical Review B, 84(24):241106, 2011

    work page 2011

  9. [9]

    Shape theory forC ∗-algebras

    Bruce Blackadar. Shape theory forC ∗-algebras. Math. Scand., 56(2):249–275, 1985

    work page 1985

  10. [10]

    Déformations, morphismes asymptotiques etK-théorie bivariante

    Alain Connes and Nigel Higson. Déformations, morphismes asymptotiques etK-théorie bivariante. C. R. Acad. Sci. Paris Sér. I Math. , 311(2):101–106, 1990

    work page 1990

  11. [11]

    Dynamical preparation of Floquet Chern insulators.Nature commu- nications, 6:8336, 2015

    Luca D’Alessio and Marcos Rigol. Dynamical preparation of Floquet Chern insulators.Nature commu- nications, 6:8336, 2015

    work page 2015

  12. [12]

    Søren Eilers and Terry A. Loring. Computing contingencies for stable relations.Internat. J. Math. , 10(3):301–326, 1999

    work page 1999

  13. [13]

    Loring, and Gert K

    Søren Eilers, Terry A. Loring, and Gert K. Pedersen. Morphisms of extensions ofC ∗-algebras: pushing forward the Busby invariant.Adv. Math., 147(1):74–109, 1999

    work page 1999

  14. [14]

    Ruy Exel and Terry A. Loring. Invariants of almost commuting unitaries.J. Funct. Anal., 95(2):364–376, 1991

    work page 1991

  15. [15]

    Aperiodic weak topological superconductors.Phys- ical review letters, 116(25):257002, 2016

    Ion C Fulga, Dmitry I Pikulin, and Terry A Loring. Aperiodic weak topological superconductors.Phys- ical review letters, 116(25):257002, 2016

    work page 2016

  16. [16]

    Hastings and Terry A

    Matthew B. Hastings and Terry A. Loring. Topological insulators andC ∗-algebras: Theory and numer- ical practice. Ann. Physics, 326(7):1699–1759, 2011. A GUIDE TO THE BOTT INDEX AND LOCALIZER INDEX 25

    work page 2011

  17. [17]

    Numerical study of the topological anderson insulator in HgTe/CdTe quantum wells.Physical Review B, 80(16):165316, 2009

    Hua Jiang, Lei Wang, Qing-feng Sun, and XC Xie. Numerical study of the topological anderson insulator in HgTe/CdTe quantum wells.Physical Review B, 80(16):165316, 2009

    work page 2009

  18. [18]

    Möbius transformations and monogenic functional calculus.Electron

    Vladimir Kisil. Möbius transformations and monogenic functional calculus.Electron. Res. Announc. Math. Sci, 2(1):26–33, 1996

    work page 1996

  19. [19]

    Anyons in an exactly solved model and beyond.Annals of Physics , 321(1):2–111, 2006

    Alexei Kitaev. Anyons in an exactly solved model and beyond.Annals of Physics , 321(1):2–111, 2006

    work page 2006

  20. [20]

    Approximating spectral densities of large matrices.SIAM review, 58(1):34–65, 2016

    Lin Lin, Yousef Saad, and Chao Yang. Approximating spectral densities of large matrices.SIAM review, 58(1):34–65, 2016

    work page 2016

  21. [21]

    The spectral localizer for even index pairings

    Terry Loring and Hermann Schulz-Baldes. The spectral localizer for even index pairings.J. Noncommut. Geom., to appear. arXiv preprint arXiv:1802.04517

    work page internal anchor Pith review Pith/arXiv arXiv

  22. [22]

    Loring.K-theory and asymptotically commuting matrices.Canad

    Terry A. Loring.K-theory and asymptotically commuting matrices.Canad. J. Math. , 40(1):197–216, 1988

    work page 1988

  23. [23]

    Loring.C ∗-algebra relations

    Terry A. Loring.C ∗-algebra relations. Math. Scand., 107(1):43–72, 2010

    work page 2010

  24. [24]

    Loring.K-theory and pseudospectra for topological insulators

    Terry A. Loring.K-theory and pseudospectra for topological insulators. Ann. Physics, 356:383–416, 2015

    work page 2015

  25. [25]

    Bulk spectrum and k-theory for infinite-area topological quascrystal.arXiv preprint arXiv:1811.07494, 2018

    Terry A Loring. Bulk spectrum and k-theory for infinite-area topological quascrystal.arXiv preprint arXiv:1811.07494, 2018

    work page arXiv 2018

  26. [26]

    Loring and Matthew B

    Terry A. Loring and Matthew B. Hastings. Disordered topological insulators viaC ∗-algebras. Europhys. Lett. EPL, 92:67004, 2010

    work page 2010

  27. [27]

    Loring and Hermann Schulz-Baldes

    Terry A. Loring and Hermann Schulz-Baldes. Finite volume calculation ofK-theory invariants. New York J. Math., 23:1111–1140, 2017

    work page 2017

  28. [28]

    Spectral flow argument localizing an odd index pairing

    Terry A Loring and Hermann Schulz-Baldes. Spectral flow argument localizing an odd index pairing. Canadian Mathematical Bulletin , 62(2):373–381, 2019

    work page 2019

  29. [29]

    Amorphous topological insulators constructed from random point sets.Nature Physics, 14(4):380, 2018

    Noah P Mitchell, Lisa M Nash, Daniel Hexner, Ari M Turner, and William TM Irvine. Amorphous topological insulators constructed from random point sets.Nature Physics, 14(4):380, 2018

    work page 2018

  30. [30]

    Disordered topological insulators: a non-commutative geometry perspective.J

    Emil Prodan. Disordered topological insulators: a non-commutative geometry perspective.J. Phys. A , 44(11):113001, 50, 2011

    work page 2011

  31. [31]

    The fuzzy space construction kit

    Andreas Sykora. The fuzzy space construction kit.arXiv preprint arXiv:1610.01504 , 2016

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  32. [32]

    Time-dependent topological systems: A study of the bott index.Physical Review B , 98(23):235425, 2018

    Daniele Toniolo. Time-dependent topological systems: A study of the bott index.Physical Review B , 98(23):235425, 2018

    work page 2018

  33. [33]

    Chern numbers as half-signature of the spectral localizer.J

    Jonas; Viesca, Edgar Lozano; Schober and Hermann Schulz-Baldes. Chern numbers as half-signature of the spectral localizer.J. Math. Phys. , to appear. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131, USA E-mail address: loring@math.unm.edu

    work page