pith. sign in

arxiv: 2311.12280 · v4 · pith:CREEDJRUnew · submitted 2023-11-21 · ❄️ cond-mat.dis-nn · cond-mat.quant-gas· cond-mat.stat-mech· quant-ph

Characterizing the Many Body Localization Crossover as a Metal-Insulator Transition: Localization length from Polarization and Quantum Metric

W. N. Faugno , Tomoki Ozawa This is my paper

Pith reviewed 2026-05-24 06:00 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.quant-gascond-mat.stat-mechquant-ph
keywords many-body localizationquantum metriclocalization lengthBose-Hubbard modelAnderson insulatorpolarizationtwist boundary conditions
0
0 comments X

The pith

The many-body quantum metric supplies a localization length that marks the MBL regime in interacting disordered chains.

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

The paper establishes that the many-body quantum metric, computed in twist-boundary parameter space, together with the localization parameter from polarization theory, can identify insulating character in disordered many-body systems. It first verifies the relation between these two quantities for excited states of the one-dimensional Anderson insulator in finite systems. It then applies the same comparison to the disordered Bose-Hubbard model and shows that the two quantities track each other across the ergodic-MBL crossover, allowing extraction of a localization length that quantifies the real-space spread of many-body wave functions.

Core claim

In the MBL regime of the disordered one-dimensional Bose-Hubbard model the many-body quantum metric and the localization parameter remain related, so that a localization length characterizing the real-space spread of the wave function can be read directly from the quantum metric.

What carries the argument

Many-body quantum metric (MBQM) defined on the space of twist boundary conditions, compared with the localization parameter of the modern theory of polarization.

If this is right

  • The MBL phase can be treated as an insulator whose localization length is defined consistently with other insulating phases.
  • The quantum metric offers an experimentally accessible route to that length without direct wave-function imaging.
  • The ergodic-MBL crossover acquires a geometric signature based on the mismatch or agreement of the two quantities.

Where Pith is reading between the lines

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

  • The same geometric diagnostic could be tested in other interacting localized phases, such as many-body mobility edges or Floquet MBL.
  • Cold-atom experiments that already measure polarization or Berry curvature could extract the proposed localization length.
  • If the relation survives in higher dimensions, it would supply a dimension-independent way to quantify MBL insulation.

Load-bearing premise

The relation between many-body quantum metric and localization parameter that holds for non-interacting Anderson insulators continues to diagnose localization once interactions are added.

What would settle it

In the MBL regime of an interacting disordered chain, the localization length inferred from the quantum metric differs systematically from the actual real-space spread of the many-body eigenstates.

Figures

Figures reproduced from arXiv: 2311.12280 by Tomoki Ozawa, W. N. Faugno.

Figure 1
Figure 1. Figure 1: , where all three quantities agree and are inde￾pendent of system size. For intermediate disorder, the three quantities still agree, but there is some noticeable dependence on system size suggesting that we have not achieved a large enough system to saturate the localiza￾tion length. For low disorder, discrepancies between the MBQM, variance and localization parameter arise due to finite size effects since… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The parameter ∆ for the half-filled non interacting [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Many body quantum metric [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: We observe that the region where ∆ is approx [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The parameter ∆ as a function of disorder strength [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Phase diagram in the [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Localization length for system sizes [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
read the original abstract

Many body localization (MBL) represents a unique physical phenomenon, providing a testing ground for exploring thermalization, or more precisely its failure. Here we characterize the MBL regime geometrically by the many-body quantum metric (MBQM), defined in the parameter space of twist boundary, and the localization parameter as defined in the modern theory of polarization and insulators. First, we demonstrate that the quantum metric can be used to characterize disordered insulating states by applying this theoretical framework to excited states of the 1D Anderson insulator. There we observe that the MBQM and localization parameter are related in finite realizations despite the states being gapless in the thermodynamic limit. Then, we consider a disordered 1D Bose-Hubbard model and find that we can characterize the ergodic-MBL crossover by comparing the MBQM and localization parameter. We find that we can extract a natural localization length in the MBL regime that characterizes the real space spread of the wave function and can be measured by extracting the quantum metric. Our analysis provides complementary insight into the MBL regime focusing on its insulating properties and providing a localization length whose definition is consistent across a range of insulating phases.

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

1 major / 2 minor

Summary. The manuscript claims that the many-body quantum metric (MBQM), defined via twist-boundary conditions, is related to the localization parameter from modern polarization theory in finite realizations of the 1D Anderson insulator, and that comparing these quantities in the disordered 1D Bose-Hubbard model characterizes the ergodic-MBL crossover while allowing extraction of a natural localization length from the MBQM alone in the MBL regime. This provides a geometric characterization of MBL as an insulator with a localization length consistent across phases.

Significance. If the reported numerical relation between MBQM and the localization parameter extends to interacting systems, the work supplies a complementary geometric diagnostic for the MBL crossover and a definition of localization length that is uniform across insulating phases and potentially extractable from quantum metric measurements. The approach is grounded in established polarization theory and offers insight into the insulating character of MBL states.

major comments (1)
  1. [abstract and Bose-Hubbard section] The abstract and the paragraph on the disordered 1D Bose-Hubbard model: the functional relation between MBQM and localization parameter is established numerically only for non-interacting Anderson-insulator excited states; no analytic argument or additional numerical test is supplied demonstrating that the same quantitative relation remains valid once on-site interactions are added. This extrapolation is load-bearing for the claim that the comparison marks the ergodic-MBL crossover and yields a localization length in the interacting model.
minor comments (2)
  1. [abstract] The abstract states that the quantities are related in finite Anderson realizations and that the comparison marks the crossover in the Bose-Hubbard model, but provides no error bars, system sizes, or details on how the crossover point is identified.
  2. [MBL regime discussion] The localization length is extracted from the same quantities used to define the crossover; explicit fitting procedures or cross-validation against independent measures (e.g., participation ratio) would clarify the diagnostic power.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting this important point regarding the scope of our numerical evidence. We address the comment below.

read point-by-point responses

  1. Referee: [abstract and Bose-Hubbard section] The abstract and the paragraph on the disordered 1D Bose-Hubbard model: the functional relation between MBQM and localization parameter is established numerically only for non-interacting Anderson-insulator excited states; no analytic argument or additional numerical test is supplied demonstrating that the same quantitative relation remains valid once on-site interactions are added. This extrapolation is load-bearing for the claim that the comparison marks the ergodic-MBL crossover and yields a localization length in the interacting model.

    Authors: We acknowledge that the quantitative functional relation between the many-body quantum metric and the localization parameter is established numerically only for excited states of the non-interacting 1D Anderson insulator. Both quantities are defined in a manner that does not presuppose the absence of interactions: the many-body quantum metric is obtained from the fidelity susceptibility under twist boundary conditions, and the localization parameter follows from the modern theory of polarization. In the disordered Bose-Hubbard model we therefore employ the comparison as a diagnostic that becomes meaningful once the system enters the MBL regime, where both quantities are expected to reflect strong real-space localization. We agree that an explicit numerical check of the same quantitative relation inside the interacting model, or an analytic derivation, would strengthen the presentation. We will revise the abstract and the Bose-Hubbard section to state explicitly that the relation is demonstrated in the non-interacting case and is used as a benchmark diagnostic when applied to the interacting model. revision: partial

Circularity Check

0 steps flagged

No significant circularity; numerical relation observed independently in each model without reduction to definition or fit.

full rationale

The paper first numerically observes a relation between MBQM and the localization parameter on finite Anderson insulator realizations, then compares the same two quantities in the interacting Bose-Hubbard model to characterize the ergodic-MBL crossover. The claim that MBQM alone yields a localization length in the MBL regime follows directly from this cross-model numerical correspondence rather than from any self-definitional equation, fitted parameter renamed as prediction, or self-citation chain. No load-bearing step reduces by construction to its own inputs; the derivation is an empirical characterization that remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the central claim rests on the numerical observation that MBQM and the polarization localization parameter remain related in finite gapless realizations and that this relation diagnoses the crossover when interactions are added.

axioms (1)
  • domain assumption The many-body quantum metric defined in twist-boundary parameter space and the localization parameter from polarization theory remain meaningfully related for excited states of the 1D Anderson model even though the spectrum is gapless in the thermodynamic limit.
    Invoked in the first demonstration paragraph of the abstract.

pith-pipeline@v0.9.0 · 5756 in / 1221 out tokens · 29245 ms · 2026-05-24T06:00:33.248690+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

83 extracted references · 83 canonical work pages

  1. [1]

    Bloch, J

    I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008)

    work page 2008

  2. [2]

    Bilokon, E

    V. Bilokon, E. Bilokon, M. C. Ba˜ nuls, A. Cichy, and A. Sotnikov, Many-body correlations in one-dimensional optical lattices with alkaline-earth(-like) atoms, Scientific Reports 13, 10.1038/s41598-023-37077-1 (2023)

    work page doi:10.1038/s41598-023-37077-1 2023

  3. [3]

    Chomaz, I

    L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe- Tolra, B. L. Lev, and T. Pfau, Dipolar physics: a review of experiments with magnetic quantum gases, Reports on Progress in Physics 86, 026401 (2022)

    work page 2022

  4. [4]

    Malz and J

    D. Malz and J. I. Cirac, Few-body analog quantum sim- ulation with rydberg-dressed atoms in optical lattices, PRX Quantum 4, 020301 (2023)

    work page 2023

  5. [5]

    Blatt and C

    R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nature Physics 8, 10.1038/nphys2252 (2012)

    work page doi:10.1038/nphys2252 2012

  6. [6]

    C. S. Adams, J. D. Pritchard, and J. P. Shaffer, Ry- dberg atom quantum technologies, Journal of Physics B: Atomic, Molecular and Optical Physics 53, 012002 (2019)

    work page 2019

  7. [7]

    X. Wu, X. Liang, Y. Tian, F. Yang, C. Chen, Y.-C. Liu, M. K. Tey, and L. You, A concise review of rydberg atom based quantum computation and quantum simulation*, Chinese Physics B 30, 020305 (2021)

    work page 2021

  8. [8]

    Kinoshita, T

    T. Kinoshita, T. Wenger, and D. S. Weiss, A quan- tum newton’s cradle, Nature 440, 10.1038/nature04693 (2006)

    work page doi:10.1038/nature04693 2006

  9. [9]

    Vasseur and J

    R. Vasseur and J. E. Moore, Nonequilibrium quantum dynamics and transport: from integrability to many- body localization, Journal of Statistical Mechanics: The- ory and Experiment 2016, 064010 (2016)

    work page 2016

  10. [10]

    Advances in Measuring the Environmental and Social Impacts of Environmental Programs

    R. Nandkishore and D. A. Huse, Many-body lo- calization and thermalization in quantum statisti- cal mechanics, Annual Review of Condensed Matter Physics 6, 15 (2015), https://doi.org/10.1146/annurev- conmatphys-031214-014726

    work page doi:10.1146/annurev- 2015

  11. [11]

    A. D. Luca and A. Scardicchio, Ergodicity breaking in a model showing many-body localization, Europhysics Let- ters 101, 37003 (2013)

    work page 2013

  12. [12]

    E. Levi, M. Heyl, I. Lesanovsky, and J. P. Garrahan, Robustness of many-body localization in the presence of dissipation, Phys. Rev. Lett. 116, 237203 (2016)

    work page 2016

  13. [13]

    A. P. Luca D’Alessio, Yariv Kafri and M. Rigol, From quantum chaos and eigenstate thermal- ization to statistical mechanics and thermody- namics, Advances in Physics 65, 239 (2016), https://doi.org/10.1080/00018732.2016.1198134. 9

    work page doi:10.1080/00018732.2016.1198134 2016

  14. [14]

    C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papi´ c, Quantum scarred eigenstates in a rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations, Phys. Rev. B 98, 155134 (2018)

    work page 2018

  15. [15]

    P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958)

    work page 1958

  16. [16]

    Fleishman and P

    L. Fleishman and P. W. Anderson, Interactions and the anderson transition, Phys. Rev. B 21, 2366 (1980)

    work page 1980

  17. [17]

    A. M. Finkel’shtein, Influence of Coulomb interaction on the properties of disordered metals, Soviet Journal of Ex- perimental and Theoretical Physics 57, 97 (1983)

    work page 1983

  18. [18]

    Giamarchi and H

    T. Giamarchi and H. J. Schulz, Anderson localization and interactions in one-dimensional metals, Phys. Rev. B 37, 325 (1988)

    work page 1988

  19. [19]

    B. L. Altshuler, Y. Gefen, A. Kamenev, and L. S. Levitov, Quasiparticle lifetime in a finite system: A nonperturba- tive approach, Phys. Rev. Lett. 78, 2803 (1997)

    work page 1997

  20. [20]

    I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Interact- ing electrons in disordered wires: Anderson localization and low-t transport, Phys. Rev. Lett. 95, 206603 (2005)

    work page 2005

  21. [21]

    Basko, I

    D. Basko, I. Aleiner, and B. Altshuler, Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states, Annals of Physics 321, 1126 (2006)

    work page 2006

  22. [22]

    Bauer and C

    B. Bauer and C. Nayak, Area laws in a many-body lo- calized state and its implications for topological order, Journal of Statistical Mechanics: Theory and Experi- ment 2013, P09005 (2013)

    work page 2013

  23. [23]

    Serbyn, Z

    M. Serbyn, Z. Papi´ c, and D. A. Abanin, Universal slow growth of entanglement in interacting strongly disordered systems, Phys. Rev. Lett. 110, 260601 (2013)

    work page 2013

  24. [24]

    Huang, Extensive entropy from unitary evolution, Preprints 10.20944/preprints202104.0254.v1 (2021)

    Y. Huang, Extensive entropy from unitary evolution, Preprints 10.20944/preprints202104.0254.v1 (2021)

    work page doi:10.20944/preprints202104.0254.v1 2021

  25. [25]

    D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Col- loquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019)

    work page 2019

  26. [26]

    Sierant and J

    P. Sierant and J. Zakrzewski, Many-body localization of bosons in optical lattices, New Journal of Physics 20, 043032 (2018)

    work page 2018

  27. [27]

    Lukin, M

    A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kaufman, S. Choi, V. Khemani, J. L´ eonard, and M. Greiner, Probing entanglement in a many- body–localized system, Science 364, 256 (2019), https://www.science.org/doi/pdf/10.1126/science.aau0818

    work page doi:10.1126/science.aau0818 2019

  28. [28]

    Bar Lev, G

    Y. Bar Lev, G. Cohen, and D. R. Reichman, Absence of diffusion in an interacting system of spinless fermions on a one-dimensional disordered lattice, Phys. Rev. Lett. 114, 100601 (2015)

    work page 2015

  29. [29]

    Schreiber, S

    M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨ uschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many- body localization of interacting fermions in a quasir- andom optical lattice, Science 349, 842 (2015), https://www.science.org/doi/pdf/10.1126/science.aaa7432

    work page doi:10.1126/science.aaa7432 2015

  30. [30]

    Sels and A

    D. Sels and A. Polkovnikov, Dynamical obstruction to localization in a disordered spin chain, Phys. Rev. E 104, 054105 (2021)

    work page 2021

  31. [31]

    Kiefer-Emmanouilidis, R

    M. Kiefer-Emmanouilidis, R. Unanyan, M. Fleischhauer, and J. Sirker, Slow delocalization of particles in many- body localized phases, Phys. Rev. B 103, 024203 (2021)

    work page 2021

  32. [32]

    Sels and A

    D. Sels and A. Polkovnikov, Thermalization of dilute im- purities in one-dimensional spin chains, Phys. Rev. X 13, 011041 (2023)

    work page 2023

  33. [33]

    Sierant, M

    P. Sierant, M. Lewenstein, A. Scardicchio, L. Vidmar, and J. Zakrzewski, Many-body localization in the age of classical computing*, Reports on Progress in Physics 88, 026502 (2025)

    work page 2025

  34. [34]

    Sierant, M

    P. Sierant, M. Lewenstein, and J. Zakrzewski, Polyno- mially filtered exact diagonalization approach to many- body localization, Phys. Rev. Lett. 125, 156601 (2020)

    work page 2020

  35. [35]

    H. P. L¨ uschen, P. Bordia, S. Scherg, F. Alet, E. Alt- man, U. Schneider, and I. Bloch, Observation of slow dynamics near the many-body localization transition in one-dimensional quasiperiodic systems, Phys. Rev. Lett. 119, 260401 (2017)

    work page 2017

  36. [36]

    Rispoli, A

    M. Rispoli, A. Lukin, R. Schittko, S.-S. Kim, M. Tai, J. L´ eonard, and M. Greiner, Quantum critical behaviour at the many-body localization transition, Nature 573, 385 (2019)

    work page 2019

  37. [37]

    L´ eonard, S.-S

    J. L´ eonard, S.-S. Kim, M. Rispoli, A. Lukin, R. Schittko, J. Kwan, E. Demler, D. Sels, and M. Greiner, Probing the onset of quantum avalanches in a many-body localized system, Nature Physics 19, 481 (2023)

    work page 2023

  38. [38]

    S. Bera, H. Schomerus, F. Heidrich-Meisner, and J. H. Bardarson, Many-body localization characterized from a one-particle perspective, Phys. Rev. Lett. 115, 046603 (2015)

    work page 2015

  39. [39]

    Pal and D

    A. Pal and D. A. Huse, Many-body localization phase transition, Phys. Rev. B 82, 174411 (2010)

    work page 2010

  40. [40]

    Weisse, R

    A. Weisse, R. Gerstner, and J. Sirker, Operator growth in disordered spin chains: Indications for the absence of many-body localization, Phys. Rev. Res. 7, 033018 (2025)

    work page 2025

  41. [41]

    Serbyn, Z

    M. Serbyn, Z. Papi´ c, and D. A. Abanin, Local conserva- tion laws and the structure of the many-body localized states, Phys. Rev. Lett. 111, 127201 (2013)

    work page 2013

  42. [42]

    D. A. Huse, R. Nandkishore, and V. Oganesyan, Phe- nomenology of fully many-body-localized systems, Phys. Rev. B 90, 174202 (2014)

    work page 2014

  43. [43]

    Chandran, I

    A. Chandran, I. H. Kim, G. Vidal, and D. A. Abanin, Constructing local integrals of motion in the many-body localized phase, Phys. Rev. B 91, 085425 (2015)

    work page 2015

  44. [44]

    Rademaker and M

    L. Rademaker and M. Ortu˜ no, Explicit local integrals of motion for the many-body localized state, Phys. Rev. Lett. 116, 010404 (2016)

    work page 2016

  45. [45]

    S. J. Thomson and M. Schir´ o, Time evolution of many- body localized systems with the flow equation approach, Phys. Rev. B 97, 060201 (2018)

    work page 2018

  46. [46]

    Pekker, B

    D. Pekker, B. K. Clark, V. Oganesyan, and G. Refael, Fixed points of wegner-wilson flows and many-body lo- calization, Phys. Rev. Lett. 119, 075701 (2017)

    work page 2017

  47. [47]

    Resta, Quantum-mechanical position operator in ex- tended systems, Phys

    R. Resta, Quantum-mechanical position operator in ex- tended systems, Phys. Rev. Lett. 80, 1800 (1998)

    work page 1998

  48. [48]

    Resta and S

    R. Resta and S. Sorella, Electron localization in the in- sulating state, Phys. Rev. Lett. 82, 370 (1999)

    work page 1999

  49. [49]

    Souza, T

    I. Souza, T. Wilkens, and R. M. Martin, Polarization and localization in insulators: Generating function approach, Phys. Rev. B 62, 1666 (2000)

    work page 2000

  50. [50]

    Resta, Why are insulators insulating and metals con- ducting?, Journal of Physics: Condensed Matter 14, R625 (2002)

    R. Resta, Why are insulators insulating and metals con- ducting?, Journal of Physics: Condensed Matter 14, R625 (2002)

    work page 2002

  51. [51]

    Lu and X

    X.-M. Lu and X. Wang, Operator quantum geometric tensor and quantum phase transitions, Europhysics Let- ters 91, 30003 (2010)

    work page 2010

  52. [52]

    Valen¸ ca Ferreira de Arag˜ ao, D

    E. Valen¸ ca Ferreira de Arag˜ ao, D. Moreno, S. Battaglia, G. L. Bendazzoli, S. Evangelisti, T. Leininger, N. Suaud, and J. A. Berger, A simple position operator for periodic 10 systems, Phys. Rev. B 99, 205144 (2019)

    work page 2019

  53. [53]

    Het´ enyi and B

    B. Het´ enyi and B. D´ ora, Quantum phase transitions from analysis of the polarization amplitude, Phys. Rev. B 99, 085126 (2019)

    work page 2019

  54. [54]

    Het´ enyi and S

    B. Het´ enyi and S. m. c. Cengiz, Geometric cumulants as- sociated with adiabatic cycles crossing degeneracy points: Application to finite size scaling of metal-insulator transi- tions in crystalline electronic systems, Phys. Rev. B 106, 195151 (2022)

    work page 2022

  55. [55]

    Resta, Electron localization in the quantum hall regime, Phys

    R. Resta, Electron localization in the quantum hall regime, Phys. Rev. Lett. 95, 196805 (2005)

    work page 2005

  56. [56]

    Wilkens and R

    T. Wilkens and R. M. Martin, Quantum monte carlo study of the one-dimensional ionic hubbard model, Phys. Rev. B 63, 235108 (2001)

    work page 2001

  57. [57]

    Thonhauser and D

    T. Thonhauser and D. Vanderbilt, Insulator/chern- insulator transition in the haldane model, Phys. Rev. B 74, 235111 (2006)

    work page 2006

  58. [58]

    Marsal, P

    Q. Marsal, P. Holmvall, and A. M. Black-Schaffer, Quan- tum metric and localization in a quasicrystal (2025), arXiv:2506.15575 [cond-mat.mes-hall]

    work page arXiv 2025

  59. [59]

    Filippone, P

    M. Filippone, P. W. Brouwer, J. Eisert, and F. von Op- pen, Drude weight fluctuations in many-body localized systems, Phys. Rev. B 94, 201112 (2016)

    work page 2016

  60. [60]

    Pouranvari and S.-F

    M. Pouranvari and S.-F. Liou, Characterizing many-body localization via state sensitivity to boundary conditions, Phys. Rev. B 103, 035136 (2021)

    work page 2021

  61. [61]

    Hamazaki, K

    R. Hamazaki, K. Kawabata, and M. Ueda, Non- hermitian many-body localization, Phys. Rev. Lett. 123, 090603 (2019)

    work page 2019

  62. [62]

    Heußen, C

    S. Heußen, C. D. White, and G. Refael, Extracting many- body localization lengths with an imaginary vector po- tential, Phys. Rev. B 103, 064201 (2021)

    work page 2021

  63. [63]

    O’Brien and G

    L. O’Brien and G. Refael, Probing localization proper- ties of many-body hamiltonians via an imaginary vector potential, Phys. Rev. B 108, 184207 (2023)

    work page 2023

  64. [64]

    Sierant, A

    P. Sierant, A. Maksymov, M. Ku´ s, and J. Zakrzewski, Fi- delity susceptibility in gaussian random ensembles, Phys. Rev. E 99, 050102 (2019)

    work page 2019

  65. [65]

    Maksymov, P

    A. Maksymov, P. Sierant, and J. Zakrzewski, Energy level dynamics across the many-body localization transi- tion, Phys. Rev. B 99, 224202 (2019)

    work page 2019

  66. [66]

    Resta, The insulating state of matter: a geometrical theory, The European Physical Journal B79, 121 (2011)

    R. Resta, The insulating state of matter: a geometrical theory, The European Physical Journal B79, 121 (2011)

    work page 2011

  67. [67]

    Resta, Geometry and topology in many-body physics (2020), arXiv:2006.15567 [cond-mat.str-el]

    R. Resta, Geometry and topology in many-body physics (2020), arXiv:2006.15567 [cond-mat.str-el]

    work page arXiv 2020

  68. [68]

    Q. Niu, D. J. Thouless, and Y.-S. Wu, Quantized hall conductance as a topological invariant, Phys. Rev. B 31, 3372 (1985)

    work page 1985

  69. [69]

    Banerjee, Topological aspects of the berry phase, Fortschritte der Physik/Progress of Physics 44, 323 (1996), https://onlinelibrary.wiley.com/doi/pdf/10.1002/prop.2190440403

    D. Banerjee, Topological aspects of the berry phase, Fortschritte der Physik/Progress of Physics 44, 323 (1996), https://onlinelibrary.wiley.com/doi/pdf/10.1002/prop.2190440403

    work page doi:10.1002/prop.2190440403 1996

  70. [70]

    Campos Venuti and P

    L. Campos Venuti and P. Zanardi, Quantum critical scal- ing of the geometric tensors, Phys. Rev. Lett. 99, 095701 (2007)

    work page 2007

  71. [71]

    Watanabe, Insensitivity of bulk properties to the twisted boundary condition, Phys

    H. Watanabe, Insensitivity of bulk properties to the twisted boundary condition, Phys. Rev. B 98, 155137 (2018)

    work page 2018

  72. [72]

    Ozawa and B

    T. Ozawa and B. Mera, Relations between topology and the quantum metric for chern insulators, Phys. Rev. B 104, 045103 (2021)

    work page 2021

  73. [73]

    S. S. Kondov, W. R. McGehee, W. Xu, and B. De- Marco, Disorder-induced localization in a strongly corre- lated atomic hubbard gas, Phys. Rev. Lett. 114, 083002 (2015)

    work page 2015

  74. [74]

    Serbyn, Z

    M. Serbyn, Z. Papi´ c, and D. A. Abanin, Criterion for many-body localization-delocalization phase transition, Phys. Rev. X 5, 041047 (2015)

    work page 2015

  75. [75]

    Mondaini and M

    R. Mondaini and M. Rigol, Many-body localization and thermalization in disordered hubbard chains, Phys. Rev. A 92, 041601 (2015)

    work page 2015

  76. [76]

    J. Chen, C. Chen, and X. Wang, Many-body localization transition in the disordered bose-hubbard chain (2023), arXiv:2104.08582 [cond-mat.dis-nn]

    work page arXiv 2023

  77. [77]

    Ozawa and N

    T. Ozawa and N. Goldman, Probing localization and quantum geometry by spectroscopy, Phys. Rev. Res. 1, 032019 (2019)

    work page 2019

  78. [78]

    Gianfrate, O

    A. Gianfrate, O. Bleu, L. Dominici, V. Ardizzone, M. De Giorgi, D. Ballarini, G. Lerario, K. W. West, L. N. Pfeiffer, D. D. Solnyshkov, D. Sanvitto, and G. Malpuech, Measurement of the quantum geometric tensor and of the anomalous hall drift, Nature 578, 381 (2020)

    work page 2020

  79. [79]

    E. J. Wilkens, G. Lawrence, A. Walsh, K. Morita, S. Simpson, C. Ritter, G. B. G. Stenning, A. M. Arevalo- Lopez, and A. C. Mclaughlin, Observation of an exotic insulator to insulator transition upon electron doping the mott insulator cemnaso, Nat. Comm. 14, 7037 (2023)

    work page 2023

  80. [80]

    J. M. Romeral, A. W. Cummings, and S. Roche, Scaling of the quantum geometry metrics in disordered topolog- ical phases (2024), arXiv:2406.12677 [cond-mat.dis-nn]

    work page arXiv 2024

Showing first 80 references.