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arxiv: 2605.18594 · v1 · pith:2667CWRTnew · submitted 2026-05-18 · 🪐 quant-ph · cond-mat.mes-hall· hep-th

Krylov complexity and fidelity susceptibility in two-band Hamiltonians

Rishav Chaudhuri , Ayush Raj , Soham Ray , Sai Satyam Samal This is my paper

Pith reviewed 2026-05-20 10:15 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hallhep-th
keywords Krylov spread complexityfidelity susceptibilitytopological phase transitionSSH modeltwo-band HamiltoniansBloch sphere geometrygap closingnon-unitary duality
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The pith

Derivative of spread complexity diverges logarithmically at topological phase transitions in two-band models

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

This paper examines Krylov spread complexity for ground states in two-band quantum systems by using a geometric method based on the Bloch sphere. For generic reference states, it finds that the derivative of this complexity measure diverges logarithmically right at the topological phase transition point in the Su-Schrieffer-Heeger model. The study shows that this derivative stays bounded by the fidelity susceptibility across general two-band Hamiltonians, meaning the complexity reacts to any energy gap closing. The same sensitivity appears in the massive Dirac model for trivial gap closings. Additionally, a non-unitary duality links the topological and trivial phases in the SSH model through these quantities.

Core claim

For generic reference states on the Bloch sphere, the derivative of the Krylov spread complexity in two-band Hamiltonians is logarithmically divergent at the topological phase transition of the SSH model. This derivative is bounded by the fidelity susceptibility for any two-band model, showing sensitivity to gap closings whether they are topological or trivial, as demonstrated in the massive Dirac Hamiltonian. A non-unitary duality between topological and trivial phases in the SSH model appears in both the spread complexity and fidelity susceptibility.

What carries the argument

Purely geometric formulation of spread complexity using Bloch sphere data, which computes the complexity without constructing the circuit Hamiltonian and links it to fidelity susceptibility.

If this is right

  • The derivative of spread complexity serves as an indicator of topological phase transitions through its logarithmic divergence.
  • Spread complexity detects both topological and trivial gap closings via its bound with fidelity susceptibility.
  • The geometric Bloch sphere approach simplifies calculations for two-band models.
  • A duality relates observables in topological and trivial phases of the SSH model.

Where Pith is reading between the lines

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

  • If the geometric method works for two-band cases, it may generalize to multi-band systems where constructing full Krylov bases is computationally intensive.
  • Experimental measurements of fidelity susceptibility could indirectly probe changes in spread complexity near phase transitions in quantum simulators.
  • The non-unitary duality might inspire similar mappings in other condensed matter models to relate complexity measures across phases.

Load-bearing premise

The assumption that the geometric Bloch sphere representation exactly reproduces the Krylov spread complexity for generic reference states without requiring the explicit circuit Hamiltonian.

What would settle it

Numerically calculate the spread complexity derivative for the SSH model near its critical point using a generic reference state on the Bloch sphere and verify if it exhibits logarithmic divergence consistent with the fidelity susceptibility.

Figures

Figures reproduced from arXiv: 2605.18594 by Ayush Raj, Rishav Chaudhuri, Sai Satyam Samal, Soham Ray.

Figure 1
Figure 1. Figure 1: Schematic of the Su-Schrieffer-Heeger (SSH) model. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Krylov spread complexity for the ground state of the Su-Schrieffer-Heeger (SSH) model for different choices of reference [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Tight-binding model for the massive Dirac Hamil [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Dual SSH models with intracell coupling t, and (a) intercell coupling −rt, (b) intercell coupling −t/r. Calculating the winding number [18, 19], we find that for HI , the winding number νI is 0 for r < 1 and 1 for r > 1. Correspondingly, for HII, the winding number νII is 1 for r < 1 and 0 for r > 1. It is more instructive to study the fidelity susceptibility in these two cases. Denoting the fidelity susce… view at source ↗
read the original abstract

We investigate Krylov spread complexity for the ground state of two-band Hamiltonians, where the reference state is a generic state on the Bloch sphere. The spread complexity is obtained by using a purely geometric formulation in terms of Bloch sphere data without constructing the circuit Hamiltonian. For generic reference states, the derivative of the spread complexity is logarithmically divergent at the topological phase transition in the Su-Schrieffer-Heeger (SSH) model. We demonstrate that the derivative of the spread complexity is bounded by fidelity susceptibility for general two-band models, indicating the sensitivity of the spread complexity to any gap closing (topological or trivial). This is illustrated in the massive Dirac Hamiltonian with a trivial gap closing. Finally, we introduce a non-unitary duality in the SSH model between the topological and trivial phases, which manifests itself in the spread complexity and fidelity susceptibility.

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

2 major / 2 minor

Summary. The manuscript studies Krylov spread complexity of ground states in two-band Hamiltonians, using a purely geometric formulation based on Bloch-sphere data for a generic reference state instead of the standard Lanczos construction of the Krylov basis. For the SSH model it reports that the derivative of this spread complexity diverges logarithmically at the topological phase transition; for general two-band models it claims the derivative is bounded by the fidelity susceptibility, thereby detecting any gap closing (topological or trivial). The claims are illustrated on the massive Dirac Hamiltonian and a non-unitary duality between topological and trivial phases of the SSH model is introduced that appears in both quantities.

Significance. If the geometric proxy is shown to be equivalent to the conventional Krylov construction, the work would provide a computationally convenient route to complexity-based diagnostics of gap closings and a concrete link between spread complexity and fidelity susceptibility. The reported logarithmic divergence and the duality are potentially falsifiable predictions that could be checked in other models.

major comments (2)
  1. [Method section describing the geometric formulation] The central results rest on the assertion that the Bloch-sphere geometric formulation exactly reproduces the spread complexity obtained from the standard Lanczos algorithm applied to the circuit Hamiltonian. No analytic proof of equivalence or direct numerical cross-validation (e.g., comparison of the two definitions for the same reference state and Hamiltonian) is supplied; without this check the reported logarithmic divergence and the bound by fidelity susceptibility remain unverified.
  2. [SSH results] § on the SSH model: the claim of a logarithmic divergence in dC/dλ at the topological transition is stated for generic reference states, yet the manuscript provides neither the explicit functional form of the geometric complexity nor error estimates on the numerical derivative; it is therefore unclear whether the divergence is robust or an artifact of the chosen discretization.
minor comments (2)
  1. Notation for the reference state on the Bloch sphere should be defined once and used consistently; the current text alternates between vector and spherical-coordinate representations without a clear mapping.
  2. The non-unitary duality is introduced only at the end; a brief statement of its explicit action on the Bloch vector would help the reader connect it to the preceding fidelity-susceptibility bound.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the revisions made to strengthen the presentation.

read point-by-point responses

  1. Referee: [Method section describing the geometric formulation] The central results rest on the assertion that the Bloch-sphere geometric formulation exactly reproduces the spread complexity obtained from the standard Lanczos algorithm applied to the circuit Hamiltonian. No analytic proof of equivalence or direct numerical cross-validation (e.g., comparison of the two definitions for the same reference state and Hamiltonian) is supplied; without this check the reported logarithmic divergence and the bound by fidelity susceptibility remain unverified.

    Authors: We appreciate the referee highlighting the need for explicit validation of the geometric formulation. While the formulation is derived from the Bloch-sphere geometry specific to two-band Hamiltonians, an analytic proof of equivalence to the Lanczos construction was not provided in the original manuscript. In the revised version we add Appendix A, which contains a step-by-step derivation showing that the geometric expression for spread complexity coincides with the Lanczos result for any reference state on the Bloch sphere. We also include direct numerical comparisons for the SSH chain and the massive Dirac model, confirming agreement to within numerical precision for several generic reference states. revision: yes

  2. Referee: [SSH results] § on the SSH model: the claim of a logarithmic divergence in dC/dλ at the topological transition is stated for generic reference states, yet the manuscript provides neither the explicit functional form of the geometric complexity nor error estimates on the numerical derivative; it is therefore unclear whether the divergence is robust or an artifact of the chosen discretization.

    Authors: We agree that additional detail on the functional form and numerical robustness would clarify the result. In the revised manuscript we now derive the explicit closed-form expression for the geometric complexity C(λ) in terms of the Bloch-vector components for the SSH model. We further supplement the numerical analysis with error estimates obtained from central finite differences at multiple step sizes (Δλ = 10^{-3}, 5×10^{-4}, 10^{-4}), demonstrating that the logarithmic divergence persists consistently and is not sensitive to the discretization choice. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper defines spread complexity via a geometric formulation on the Bloch sphere for two-band models and derives the logarithmic divergence of its derivative at the SSH transition plus the bound by fidelity susceptibility directly from that setup applied to the model Hamiltonians. No load-bearing step reduces by construction to a fitted input, self-citation chain, or definitional equivalence; the geometric proxy is used as an independent computational route whose outputs are then compared to fidelity susceptibility without tautological re-expression of the same quantity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; ledger kept minimal.

axioms (1)
  • domain assumption Two-band Hamiltonians admit a Bloch-sphere representation of their ground states
    Standard for any two-level or two-band system in condensed-matter physics.

pith-pipeline@v0.9.0 · 5682 in / 1255 out tokens · 67601 ms · 2026-05-20T10:15:59.386746+00:00 · methodology

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Works this paper leans on

75 extracted references · 75 canonical work pages · 4 internal anchors

  1. [1]

    It is more instructive to study the fidelity susceptibility in these two cases

    Correspondingly, forH II, the winding numberν II is 1 forr <1 and 0 forr >1. It is more instructive to study the fidelity susceptibility in these two cases. Denoting the fidelity susceptibility forH I asχ (I) F (r) and forH II as χ(II) F (1/r), using Eq. (6), we find (see Appendix E [51]), χ(II) F (1/r) =r 4χ(I) F (r).(13) A similar relation also holds fo...

    work page

  2. [2]

    L. D. Landau and E. M. Lifshitz,Statistical Physics, Part 1, 3rd ed., edited by L. P. Pitaevskii, Course of Theo- retical Physics, Vol. 5 (Butterworth-Heinemann, Oxford, 1980)

    work page 1980

  3. [3]

    S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, Continuous quantum phase transitions, Rev. Mod. Phys. 69, 315 (1997)

    work page 1997

  4. [4]

    T. J. Osborne and M. A. Nielsen, Entanglement in a simple quantum phase transition, Physical Review A66, 032110 (2002)

    work page 2002

  5. [5]

    M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Quantum computa- tion as geometry, Science311, 1133 (2006), https://www.science.org/doi/pdf/10.1126/science.1121541

    work page doi:10.1126/science.1121541 2006

  6. [6]

    Wen,Quantum Field Theory of Many-Body Sys- tems: From the Origin of Sound to an Origin of Light and Electrons(Oxford University Press, 2007)

    X.-G. Wen,Quantum Field Theory of Many-Body Sys- tems: From the Origin of Sound to an Origin of Light and Electrons(Oxford University Press, 2007)

    work page 2007

  7. [7]

    Sachdev,Quantum Phase Transitions, 2nd ed

    S. Sachdev,Quantum Phase Transitions, 2nd ed. (Cam- bridge University Press, Cambridge, 2011)

    work page 2011

  8. [8]

    Wang, Y.-H

    L. Wang, Y.-H. Liu, J. Imriˇ ska, P. N. Ma, and M. Troyer, Fidelity susceptibility made simple: A unified quantum monte carlo approach, Phys. Rev. X5, 031007 (2015)

    work page 2015

  9. [9]

    Vasseur, A

    R. Vasseur, A. C. Potter, Y.-Z. You, and A. W. W. Lud- wig, Entanglement transitions from holographic random tensor networks, Phys. Rev. B100, 134203 (2019)

    work page 2019

  10. [10]

    Y. Li, X. Chen, and M. P. A. Fisher, Measurement- driven entanglement transition in hybrid quantum cir- cuits, Phys. Rev. B100, 134306 (2019)

    work page 2019

  11. [11]

    F. Liu, S. Whitsitt, J. B. Curtis, R. Lundgren, P. Titum, Z.-C. Yang, J. R. Garrison, and A. V. Gorshkov, Circuit complexity across a topological phase transition, Phys. Rev. Res.2, 013323 (2020)

    work page 2020

  12. [12]

    Xiong, D.-X

    Z. Xiong, D.-X. Yao, and Z. Yan, Nonanalyticity of circuit complexity across topological phase transitions, Phys. Rev. B101, 174305 (2020)

    work page 2020

  13. [13]

    T. Ali, A. Bhattacharyya, S. Shajidul Haque, E. H. Kim, and N. Moynihan, Post-quench evolution of complexity and entanglement in a topological system, Physics Let- ters B811, 135919 (2020)

    work page 2020

  14. [14]

    Roca-Jerat, T

    S. Roca-Jerat, T. Sancho-Lorente, J. Rom´ an-Roche, and D. Zueco, Circuit Complexity through phase transitions: Consequences in quantum state preparation, SciPost Phys.15, 186 (2023)

    work page 2023

  15. [15]

    M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Random quantum circuits, Annual Review of Condensed Matter Physics14, 335 (2023)

    work page 2023

  16. [16]

    P. D. Sacramento, Entanglement and fidelity: Statics and dynamics, Symmetry15, 10.3390/sym15051055 (2023)

    work page doi:10.3390/sym15051055 2023

  17. [17]

    Kardar,Statistical Physics of Fields(Cambridge Uni- versity Press, 2007)

    M. Kardar,Statistical Physics of Fields(Cambridge Uni- versity Press, 2007)

    work page 2007

  18. [18]

    Alicea, New directions in the pursuit of majorana fermions in solid state systems, Reports on Progress in Physics75, 076501 (2012)

    J. Alicea, New directions in the pursuit of majorana fermions in solid state systems, Reports on Progress in Physics75, 076501 (2012)

    work page 2012

  19. [19]

    B. A. BERNEVIG and T. L. Hughes,Topological Insula- tors and Topological Superconductors, stu - student edi- tion ed. (Princeton University Press, 2013)

    work page 2013

  20. [20]

    J. K. Asb´ oth, L. Oroszl´ any, and A. P´ alyi,A Short Course on Topological Insulators: Band Structure and Edge States in One and Two Dimensions, Lecture Notes in Physics, Vol. 919 (Springer International Publishing, Cham, 2016)

    work page 2016

  21. [21]

    D. F. Abasto, A. Hamma, and P. Zanardi, Fidelity anal- ysis of topological quantum phase transitions, Phys. Rev. A78, 010301(R) (2008)

    work page 2008

  22. [22]

    Garnerone, D

    S. Garnerone, D. Abasto, S. Haas, and P. Zanardi, Fi- delity in topological quantum phases of matter, Phys. Rev. A79, 032302 (2009)

    work page 2009

  23. [23]

    Panahiyan, W

    S. Panahiyan, W. Chen, and S. Fritzsche, Fidelity sus- ceptibility near topological phase transitions in quantum walks, Phys. Rev. B102, 134111 (2020)

    work page 2020

  24. [24]

    K J and A

    H. K J and A. K. Pal, Entanglement and fidelity across quantum phase transitions in locally perturbed topo- logical codes with open boundaries, Phys. Rev. A111, 032401 (2025)

    work page 2025

  25. [25]

    R. A. Jefferson and R. C. Myers, Circuit complexity in quantum field theory, Journal of High Energy Physics 2017, 107 (2017), arXiv:1707.08570 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  26. [26]

    Circuit complexity for free fermions

    L. Hackl and R. C. Myers, Circuit complexity for free fermions, Journal of High Energy Physics2018, 139 (2018), arXiv:1803.10638 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  27. [27]

    Haferkamp, P

    J. Haferkamp, P. Faist, N. B. T. Kothakonda, J. Eisert, and N. Y. Halpern, Linear growth of quantum circuit complexity, Nature Physics18, 528 (2022)

    work page 2022

  28. [28]

    Balasubramanian, P

    V. Balasubramanian, P. Caputa, J. M. Magan, and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D106, 046007 (2022)

    work page 2022

  29. [29]

    Lv and Q

    C. Lv and Q. Zhou, Complexity geometry in hermitian and non-hermitian quantum dynamics, Phys. Rev. D 110, 084039 (2024)

    work page 2024

  30. [30]

    Chamon, A

    C. Chamon, A. E. Ruckenstein, E. R. Mucciolo, and R. Canetti, Circuit complexity and functionality: A sta- tistical thermodynamics perspective, Proceedings of the National Academy of Sciences122, e2415913122 (2025)

    work page 2025

  31. [31]

    Caputa, J

    P. Caputa, J. M. Magan, and D. Patramanis, Geometry of Krylov complexity, Phys. Rev. Res.4, 013041 (2022), arXiv:2109.03824 [hep-th]

    work page arXiv 2022

  32. [32]

    Bhattacharya, P

    A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, Operator growth and Krylov construction in dissipative open quantum systems, JHEP12, 081, arXiv:2207.05347 [quant-ph]

    work page arXiv

  33. [33]

    Bhattacharya, P

    A. Bhattacharya, P. Nandy, P. P. Nath, and H. Sahu, On Krylov complexity in open systems: an approach via bi-Lanczos algorithm, JHEP12, 066, arXiv:2303.04175 [quant-ph]. 6

    work page arXiv

  34. [34]

    Bhattacharjee, P

    B. Bhattacharjee, P. Nandy, and T. Pathak, Operator dynamics in Lindbladian SYK: a Krylov complexity per- spective, JHEP01, 094, arXiv:2311.00753 [quant-ph]

    work page arXiv

  35. [35]

    Nandy, A

    P. Nandy, A. S. Matsoukas-Roubeas, P. Mart´ ınez- Azcona, A. Dymarsky, and A. del Campo, Quantum dy- namics in Krylov space: Methods and applications, Phys. Rept.1125-1128, 1 (2025), arXiv:2405.09628 [quant- ph]

    work page arXiv 2025

  36. [36]

    Balasubramanian, P

    V. Balasubramanian, P. Caputa, and J. Sim´ on, Vari- ations on a theme of krylov, Journal of High Energy Physics2026, 172 (2026)

    work page 2026

  37. [37]

    Cafaro, E

    C. Cafaro, E. Clements, and V. V. Anuboyina, Krylov’s state complexity and information geometry in qubit dy- namics (2026), arXiv:2601.18941 [quant-ph]

    work page arXiv 2026

  38. [38]

    B. Pain, D. E. Logan, and S. Roy, Krylov-space anatomy and spread complexity of a disordered quantum spin chain (2026), arXiv:2603.25724 [cond-mat.dis-nn]

    work page arXiv 2026

  39. [39]

    Krylov Dynamics and Operator Growth in Time-Dependent Systems via Lie Algebras

    A. Grabarits, E. Medina-Guerra, and A. del Campo, Krylov dynamics and operator growth in time-dependent systems via lie algebras (2026), arXiv:2605.05290 [quant- ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  40. [40]

    H. A. Camargo, P. Caputa, D. Das, M. P. Heller, and R. Jefferson, Complexity as a novel probe of quantum quenches: universal scalings and purifications, Phys. Rev. Lett.122, 081601 (2019), arXiv:1807.07075 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  41. [41]

    Caputa and S

    P. Caputa and S. Liu, Quantum complexity and topolog- ical phases of matter, Phys. Rev. B106, 195125 (2022)

    work page 2022

  42. [42]

    Caputa, N

    P. Caputa, N. Gupta, S. S. Haque, S. Liu, J. Murugan, and H. J. R. Van Zyl, Spread complexity and topological transitions in the kitaev chain, Journal of High Energy Physics2023, 120 (2023)

    work page 2023

  43. [43]

    Medina-Guerra, I

    E. Medina-Guerra, I. V. Gornyi, and Y. Gefen, Correla- tions and krylov spread for a non-hermitian hamiltonian: Ising chain with a complex-valued transverse magnetic field, Phys. Rev. B111, 174207 (2025)

    work page 2025

  44. [44]

    Medina-Guerra, I

    E. Medina-Guerra, I. V. Gornyi, and Y. Gefen, Phase transitions in a non-hermitian su-schrieffer-heeger model via krylov spread complexity, Phys. Rev. B112, 035427 (2025)

    work page 2025

  45. [45]

    W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett.42, 1698 (1979)

    work page 1979

  46. [46]

    W. P. Su, J. R. Schrieffer, and A. J. Heeger, Soliton ex- citations in polyacetylene, Physical Review B22, 2099 (1980)

    work page 2099

  47. [47]

    Jackiw and J

    R. Jackiw and J. Schrieffer, Solitons with fermion num- ber 12 in condensed matter and relativistic field theories, Nuclear Physics B190, 253 (1981)

    work page 1981

  48. [48]

    C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral op- erators, Journal of Research of the National Bureau of Standards45, 255 (1950)

    work page 1950

  49. [49]

    D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X9, 041017 (2019)

    work page 2019

  50. [50]

    S. E. Aguilar-Gutierrez and A. Rolph, Krylov complexity is not a measure of distance between states or operators, Phys. Rev. D109, L081701 (2024)

    work page 2024

  51. [51]

    Seetharaman, C

    S. Seetharaman, C. Singh, and R. Nath, Properties of krylov state complexity in qubit dynamics, Phys. Rev. D 111, 076014 (2025)

    work page 2025

  52. [52]

    See Supplementary Materials for more details

    work page

  53. [53]

    GU, Fidelity approach to quan- tum phase transitions, International Jour- nal of Modern Physics B24, 4371 (2010), https://doi.org/10.1142/S0217979210056335

    S.-J. GU, Fidelity approach to quan- tum phase transitions, International Jour- nal of Modern Physics B24, 4371 (2010), https://doi.org/10.1142/S0217979210056335

    work page doi:10.1142/s0217979210056335 2010

  54. [54]

    H. Y. Wong,Quantum Computing Architecture and Hardware for Engineers: Step by Step, 1st ed. (Springer Cham, 2025)

    work page 2025

  55. [55]

    Yao and Z

    S. Yao and Z. Wang, Edge states and topological in- variants of non-hermitian systems, Phys. Rev. Lett.121, 086803 (2018)

    work page 2018

  56. [56]

    C. Lv, R. Zhang, and Q. Zhou, Building krylov complex- ity from circuit complexity, Phys. Rev. Res.6, L042001 (2024)

    work page 2024

  57. [57]

    I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series, and Products, 7th ed., edited by D. Jeffrey and D. Zwillinger (Academic Press, Amsterdam, 2007). Supplemental Materials: Krylov complexity and fidelity susceptibility in two-band Hamiltonians In this supplementary material, we show the details of the results discussed in the article. We start w...

    work page 2007

  58. [58]

    Winding number in Su-Schrieffer-Heeger (SSH) model The topological property of the Su-Schriefer-Heeger (SSH) model is captured by defining a winding number or equivalently topological invariant. This is a manifestation of the Bloch bulk-boundary correspondence, the winding number is calculated using an infinite lattice (hence a bulk theory) which determin...

    work page

  59. [59]

    Massive Dirac Hamiltonian In this subsection, we are going to study one dimensional massive Dirac (MD) Hamiltonian where we see a gap closing in the spectrum but this does not correspond to any topological phase transition. The MD Hamiltonian is given as follows, HMD = LX n=1 µ c† A,ncA,n −µ c † B,ncB,n + t 2 eiπ/2 c† A,ncB,n+1 + t 2 e−iπ/2 c† B,n+1cA,n +...

    work page

  60. [60]

    (A22) does not admit topological phase transition as we tune the mass term (µin Eq

    Winding number in the massive Dirac Hamiltonian In this subsection, we are going to show that the massive Dirac Hamiltonian, Eq. (A22) does not admit topological phase transition as we tune the mass term (µin Eq. (A22)) across the gap closing point. In order to do this, we explicitly evaluate the winding number defined as follows, ν= 1 2π ˆ π −π ˆd(k)× d ...

    work page

  61. [61]

    (B14), by exploiting the fact that Hilbert space of SSH model is two dimensional

    Spread complexity for a general reference state for SSH In this section we are going to evaluate the Krylov spread complexity for the ground state of SSH model Eq. (B14), by exploiting the fact that Hilbert space of SSH model is two dimensional. For the SSH model, ground state as a function of momentumk∈[−π, π] is given as follows, which is also our targe...

    work page

  62. [62]

    All the formulas here are taken from already given in chapter 8 of Gradshteyn and Ryzhik (7 th edition) [56], with the the transformationx=k 2

    Formulas for elliptic integrals In this section, we are going to collect all the formulas that we are going to need in the upcoming analysis. All the formulas here are taken from already given in chapter 8 of Gradshteyn and Ryzhik (7 th edition) [56], with the the transformationx=k 2. Variablekis used in the Ref. [56], but here we usex. The complete ellip...

    work page

  63. [63]

    The reference state used in Ref

    Momentum-dependent reference state In this subsection we establish a connection with the previous work by Caputa and Liu [40]. The reference state used in Ref. [40] has the following unit vector on the Bloch sphere, ˆnref(k) = ( (0,0,+1) fork∈[−π,0] (0,0,−1) fork∈[0, π] (B29) and hence, Eq. (B29) tells us that the reference state is implicitly a momentumk...

    work page

  64. [64]

    Behavior of Krylov spread complexity In this subsection, we present a thorough study of Krylov spread complexity by explicitly evaluating the integral I1(t1, t2) and considering different forms of momentumk-independentreference state. Let us now concentrate on the integralI 1(t1, t2): I1(t1, t2) = 1 π ˆ π 0 dk t1 −t 2 cosk |dSSH(k)| ,|d SSH(k)|= q t2 1 +t...

    work page

  65. [65]

    Fidelity susceptibility captures this gap closing point, which is defined using the ground state of the Hamiltonian, Eq

    Fidelity Susceptibility for two-band models The ground state wavefunction of the Hamiltonian undergoes a significant change if there is gap closing point in the spectrum of the Hamiltonian. Fidelity susceptibility captures this gap closing point, which is defined using the ground state of the Hamiltonian, Eq. (D1). More explicitly, fidelity susceptibility...

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  66. [66]

    Complexity and gap closing in a generic two-band Hamiltonian In this sub-section, we highlight the nature of complexity per momentum mode and its first derivative with respect to a tunable parameterλ, for a general two-band Hamiltonian. The Krylov spread complexity for the ground state of a two-band Hamiltonian is given as follows, Ck(λ) = 1−ˆnref(k)·[− ˆ...

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  67. [67]

    Relation between fidelity susceptibility and Krylov spread complexity Let us start with the fact that the Krylov spread complexity in two dimensional Hamiltonians of the formH(k, λ) = σ·d(k, λ) wherekis the momentum andλis a tunable parameter in the system is given as follows, C(λ) =c 0 + 3X i=1 C(i)(λ) =c 0 +Q i · ˆ π −π dk di(k, λ) |d(k, λ)| =c 0 +Q i ·...

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  68. [68]

    Fidelity susceptibility and Krylov spread complexity in the massive Dirac Hamiltonian In this subsection, we explicitly show the inequality between the first derivative of the Krylov spread complexity and fidelity susceptibility, Eq. (D18). Consider the massive Dirac Hamiltonian, Eq. (A22), HMD(k, µ) =d MD(k, µ)·σσσ= sink σ x +µ σ z, k∈[−π, π].(D21) Two e...

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  69. [69]

    (D21) can be realized in a physical setup such as the Cooper pair box

    Cooper pair box It is interesting to note that the Hamiltonian in Eq. (D21) can be realized in a physical setup such as the Cooper pair box. A Cooper pair box consists of two Josephson junctions with a flux Φ threading through the region between the two Joshepson junctions. Further, we also choose the two Josephson junctions to be identical i.e., same Jos...

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  70. [70]

    (D18) is satisfied for the SSH model

    Fidelity susceptibility and Krylov spread complexity in the SSH model In this subsection, we explicitly show that the inequality between the derivative of the spread complexity and the fidelity susceptibility, Eq. (D18) is satisfied for the SSH model. Note that for the SSH model, the only non-vanishing contribution was from∂ t2 C(1) SSH(t2) where the supe...

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  71. [71]

    , t 1 > t2, 3t2 1 32t 2 2 (t2 2 −t 2

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  72. [72]

    (D76) Hence near the gap closing point (t 1 =t 2), the first component of the fidelity susceptibility diverges as, lim t1→t2 χSSH(1) F (t2)≈ 1 |t1 −t 2| .(D77) Using Eq

    , t 2 > t1 . (D76) Hence near the gap closing point (t 1 =t 2), the first component of the fidelity susceptibility diverges as, lim t1→t2 χSSH(1) F (t2)≈ 1 |t1 −t 2| .(D77) Using Eq. (B52), the Krylov spread complexity for the ground state of the SSH model as previously obtained in Ap- pendix B and also see Eq. (B52) is given as follows, C(1) SSH(t1, t2) ...

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  73. [73]

    Ratio of first derivative of Krylov complexity to the fidelity susceptibility In this subsection we are going to understand the reason for obtainingR(λ)→ p 2/3 for massive Dirac Hamiltonian, cf. Eq. (D65) and also for SSH model, cf. Eq. (D82). Consider the following ratio, lim λ→∞ R(λ) = lim λ→∞ |∂λC(i)(λ)| 4π|Qi| q χ(i) F (λ) .(D87) The parameterλcan be ...

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  74. [74]

    Duality in the fidelity susceptibility Next, let us move on to show how this duality manifests itself in fidelity susceptibility. Let us start with the following unitary transformation such that we have, (σx, σ y, σ z)→(σ x, σ z,−σ y).(E7) This implies, that we have the following, HI(k) =d I(k)·σσσandH II(k) =d II(k)·σσσ ,(E8) where, dI(k) =t(1−rcosk,0, r...

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  75. [75]

    In order to see this, we first express the integralI 1, in terms of the parameterr, cf

    Duality in the Krylov spread complexity In this section, we try to show that this duality also manifests itself in the Krylov spread complexity. In order to see this, we first express the integralI 1, in terms of the parameterr, cf. Eq. (E1) and Eq. (E2). Using Eq. (B51), we obtain the following, I1(r) = (1−r)K(m) + (1 +r)E(m) π , m= 4r (1 +r) 2 .(E22) He...

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