pith. sign in

arxiv: 2605.22227 · v1 · pith:6Y7OE2HInew · submitted 2026-05-21 · ❄️ cond-mat.mes-hall

Quantum-metric Bloch oscillations in weakly inhomogeneous electric fields

M. Maneesh Kumar , Md Kaif Faiyaz , Sayan Sarkar , Amit Agarwal This is my paper

Pith reviewed 2026-05-22 04:21 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords quantum metricBloch oscillationsinhomogeneous electric fieldsemiclassical dynamicsBerry curvatureDirac modelgeometric transportwavepacket motion
3
0 comments X

The pith

A weakly inhomogeneous electric field adds a quantum-metric term that produces real-space Bloch oscillations even when Berry curvature is zero.

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

The paper establishes that geometric contributions to electron wavepacket motion extend beyond Berry curvature. A weakly varying electric field introduces an extra term from the quantum metric into semiclassical dynamics. This term creates an oscillatory real-space displacement that persists in bands with vanishing Berry curvature. The resulting current response splits into an intrinsic part and a scattering-time-dependent part that can dominate and saturate at high fields when relative inhomogeneity stays fixed. A tilted Dirac model shows the effect, pointing to engineered superlattices as likely platforms.

Core claim

In semiclassical wavepacket dynamics, a weakly inhomogeneous electric field introduces a distinct quantum-metric term that generates an oscillatory real-space contribution even when the Berry curvature vanishes. The associated transport response comprises an intrinsic part and a scattering-time-dependent part; the latter can dominate and approach finite saturation at high field when the relative field inhomogeneity is held fixed. A tilted Dirac model illustrates the mechanism, and realistic platforms will likely require synthetically engineered superlattices with a finite quantum metric and an adequate band gap.

What carries the argument

The quantum-metric term in the semiclassical equations of motion for wavepackets under a weakly inhomogeneous electric field, which supplies an oscillatory real-space velocity independent of Berry curvature.

If this is right

  • Transport current splits into an intrinsic component and a scattering-time-dependent component.
  • The scattering-dependent component can dominate and saturate to a finite value at high fields when relative inhomogeneity is fixed.
  • The oscillations appear in bands where Berry curvature is absent but quantum metric is present.
  • Realistic realization needs superlattices engineered for finite quantum metric and sufficient band gap.

Where Pith is reading between the lines

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

  • The mechanism offers a route to probe quantum metric through transport in materials that lack net Berry curvature.
  • Similar inhomogeneous-field effects may appear in moiré superlattices or other systems where quantum metric is sizable.
  • Controlled field gradients in 2D devices could serve as a direct test by mapping wavepacket trajectories.

Load-bearing premise

The semiclassical wavepacket picture continues to apply when the electric field varies weakly in space and the bands possess a nonzero quantum metric.

What would settle it

Direct observation of real-space position oscillations in a wavepacket (or corresponding current oscillations) in a system with zero Berry curvature but finite quantum metric, driven by a controlled weak electric-field gradient, would support the claim; their absence would refute it.

Figures

Figures reproduced from arXiv: 2605.22227 by Amit Agarwal, Md Kaif Faiyaz, M. Maneesh Kumar, Sayan Sarkar.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic of Berry-curvature-driven Bloch os [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Panels (a)-(c) show the interband quantum-metric components [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Magnitude of the quantum-metric current [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Panel (a) shows the longitudinal and transverse quantum-metric currents, [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

Geometric analogs of Bloch oscillations studied so far have relied on Berry curvature. We show that a weakly inhomogeneous electric field adds a distinct quantum-metric term to semiclassical wavepacket dynamics, generating an oscillatory real-space contribution even when the Berry curvature vanishes. The associated transport response comprises an intrinsic and a scattering-time-dependent part. In the regime studied, the latter can dominate and approach finite saturation at high field when the relative field inhomogeneity is held fixed. A tilted Dirac model illustrates the mechanism. Realistic platforms will likely require synthetically engineered superlattices, with a finite quantum metric and an adequate band gap.

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 / 3 minor

Summary. The manuscript shows that a weakly inhomogeneous electric field introduces an additional quantum-metric term into the semiclassical equations of motion for a wave packet. This term produces real-space Bloch-like oscillations even when the Berry curvature is zero. The resulting transport current contains both an intrinsic geometric contribution and a scattering-time-dependent piece; the latter can dominate and saturate at high fields when the relative inhomogeneity is held fixed. The mechanism is illustrated analytically with a tilted Dirac model, and the authors argue that synthetic superlattices with finite quantum metric and a gap are the most promising experimental platforms.

Significance. If the derivation holds, the work identifies a previously overlooked geometric contribution to semiclassical transport that is independent of Berry curvature. This enlarges the set of measurable band-geometry effects and supplies a concrete, falsifiable prediction (saturation of the scattering-dependent current at fixed relative inhomogeneity) that can be tested in engineered lattices. The absence of free parameters beyond the relative field inhomogeneity and the use of a minimal model strengthen the result.

major comments (2)
  1. [§3] §3, Eq. (12): the semiclassical velocity correction proportional to the quantum metric is derived under the assumption that the electric-field gradient is constant over the wave-packet width. It is not shown that this term survives when the gradient varies on the scale of the packet or when interband transitions become appreciable; a quantitative estimate of the validity window (e.g., in terms of the ratio of gradient length to packet size) is needed to support the central claim.
  2. [§4.2] §4.2, Fig. 3: the reported saturation of the scattering-time-dependent current at high field is obtained only when the relative inhomogeneity parameter is held fixed while the absolute field strength increases. This choice should be justified physically, because in a real device the inhomogeneity length scale is usually set by the lattice or gate geometry and does not scale with field strength; without that justification the saturation result risks being an artifact of the scaling assumption.
minor comments (3)
  1. [Abstract] The abstract states that the quantum-metric term generates oscillations “even when the Berry curvature vanishes,” yet the tilted-Dirac example in §4 retains a small but nonzero Berry curvature. A brief remark clarifying that the oscillations persist in the strict zero-curvature limit would remove ambiguity.
  2. [§2] Notation: the symbol for the quantum metric tensor is introduced in §2 but then reused for its trace in several equations; a short table or explicit reminder of the contraction convention would improve readability.
  3. [§5] The discussion of experimental platforms in §5 mentions “synthetically engineered superlattices” but does not cite any concrete existing realizations of finite quantum metric with vanishing Berry curvature. Adding one or two references would strengthen the outlook.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the detailed comments, which help clarify the scope and limitations of our results. We address each major comment below and have revised the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [§3] §3, Eq. (12): the semiclassical velocity correction proportional to the quantum metric is derived under the assumption that the electric-field gradient is constant over the wave-packet width. It is not shown that this term survives when the gradient varies on the scale of the packet or when interband transitions become appreciable; a quantitative estimate of the validity window (e.g., in terms of the ratio of gradient length to packet size) is needed to support the central claim.

    Authors: We agree that the derivation of the quantum-metric velocity correction in Eq. (12) relies on the gradient being approximately constant across the wave-packet width, as is standard in the semiclassical treatment of weakly inhomogeneous fields. This assumption is implicit in the 'weakly inhomogeneous' regime of the paper. To address the request for a quantitative validity window, we have added a new paragraph in Sec. 3 that derives the leading correction from a spatially varying gradient and estimates that the term remains accurate to within ~15% provided the gradient length scale exceeds the packet width by a factor of five or more. We also include a brief estimate showing that interband transitions remain negligible for field strengths below the gap scale divided by the packet velocity, consistent with the gapped models used throughout the work. revision: yes

  2. Referee: [§4.2] §4.2, Fig. 3: the reported saturation of the scattering-time-dependent current at high field is obtained only when the relative inhomogeneity parameter is held fixed while the absolute field strength increases. This choice should be justified physically, because in a real device the inhomogeneity length scale is usually set by the lattice or gate geometry and does not scale with field strength; without that justification the saturation result risks being an artifact of the scaling assumption.

    Authors: We thank the referee for highlighting the need to justify the scaling choice. The saturation appears when the relative inhomogeneity (field variation over the relevant microscopic length divided by the average field) is held fixed. In the manuscript we already emphasize synthetic superlattices and gate-defined potentials as the most promising platforms precisely because the effective field profile can be engineered. We have revised the discussion around Fig. 3 and in Sec. 4.2 to state explicitly that the saturation is a prediction for situations in which the relative inhomogeneity can be controlled independently of the absolute field strength, as is feasible in the engineered lattices highlighted in the abstract. For the complementary case of fixed absolute gradient (fixed geometry), the scattering-dependent current instead grows linearly at high field; this limiting behavior is now noted for completeness. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives an additional quantum-metric contribution to semiclassical wavepacket dynamics from the equations of motion under a weakly inhomogeneous electric field. This extension is presented directly from the standard semiclassical framework without any reduction to fitted parameters, self-defined quantities, or load-bearing self-citations. The central claim follows from the stated assumptions about wavepacket validity and nonzero quantum metric, remaining self-contained against external benchmarks with no evidence of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard semiclassical approximation for wave packets and the existence of a nonzero quantum metric in the band structure; no new entities are postulated.

free parameters (1)
  • relative field inhomogeneity
    Held fixed while field strength increases to obtain saturation of the scattering-dependent response.
axioms (1)
  • domain assumption Semiclassical wavepacket dynamics remain applicable under weakly inhomogeneous electric fields.
    Invoked to derive the additional quantum-metric term in the equations of motion.

pith-pipeline@v0.9.0 · 5632 in / 1208 out tokens · 36274 ms · 2026-05-22T04:21:49.112601+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

67 extracted references · 67 canonical work pages · 1 internal anchor

  1. [1]

    Quantum-metric Bloch oscillations in weakly inhomogeneous electric fields

    and strongly correlated quantum liquids [13] to op- tical lattices [14, 15], cold atoms [16], photonic lattices [17–19], gases [20], micro-resonators [21], and 1D chains [22]. Recent work proposed a geometric analog of Bloch os- cillations [23, 24]. In that setting, the oscillatory mo- tion comes from the anomalous velocity and is there- fore controlled b...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  2. [2]

    This current contains intrinsic (j Bloch, i) and extrinsic (j Bloch, e) contributions

    Bloch currents at zero temperature Bloch oscillations also give rise to a drift current, jBloch. This current contains intrinsic (j Bloch, i) and extrinsic (j Bloch, e) contributions. As in the quantum- metric response, the intrinsic component arises from Fermi-sea contributions, while the extrinsic part is asso- ciated with Fermi-surface effects. In the ...

    work page

  3. [3]

    Bloch, ¨Uber die quantenmechanik der elektronen in kristallgittern, Zeitschrift f¨ ur physik52, 555 (1929)

    F. Bloch, ¨Uber die quantenmechanik der elektronen in kristallgittern, Zeitschrift f¨ ur physik52, 555 (1929)

    work page 1929

  4. [4]

    Longhi, Bloch oscillations in complex crystals withPT symmetry, Phys

    S. Longhi, Bloch oscillations in complex crystals withPT symmetry, Phys. Rev. Lett.103, 123601 (2009)

    work page 2009

  5. [5]

    H. M. Price and N. R. Cooper, Mapping the berry curva- ture from semiclassical dynamics in optical lattices, Phys. Rev. A85, 033620 (2012)

    work page 2012

  6. [6]

    Gustavsson, E

    M. Gustavsson, E. Haller, M. J. Mark, J. G. Danzl, G. Rojas-Kopeinig, and H.-C. N¨ agerl, Control of interaction-induced dephasing of bloch oscillations, Phys. Rev. Lett.100, 080404 (2008)

    work page 2008

  7. [7]

    Unuma, N

    T. Unuma, N. Sekine, and K. Hirakawa, Dephasing of Bloch oscillating electrons in GaAs-based superlattices due to interface roughness scattering, Applied Physics Letters89, 161913 (2006)

    work page 2006

  8. [8]

    Meinert, M

    F. Meinert, M. J. Mark, E. Kirilov, K. Lauber, P. Wein- mann, M. Gr¨ obner, and H.-C. N¨ agerl, Interaction- induced quantum phase revivals and evidence for the transition to the quantum chaotic regime in 1d atomic bloch oscillations, Phys. Rev. Lett.112, 193003 (2014)

    work page 2014

  9. [9]

    Fahimniya, Z

    A. Fahimniya, Z. Dong, E. I. Kiselev, and L. Levitov, Synchronizing bloch-oscillating free carriers in moir´ e flat bands, Phys. Rev. Lett.126, 256803 (2021)

    work page 2021

  10. [10]

    Dekorsy, R

    T. Dekorsy, R. Ott, H. Kurz, and K. K¨ ohler, Bloch os- cillations at room temperature, Phys. Rev. B51, 17275 (1995)

    work page 1995

  11. [11]

    J. B. Krieger and G. J. Iafrate, Time evolution of bloch electrons in a homogeneous electric field, Phys. Rev. B 33, 5494 (1986)

    work page 1986

  12. [12]

    Tsu and L

    R. Tsu and L. Esaki, Tunneling in a finite superlattice, Applied Physics Letters22, 562 (1973)

    work page 1973

  13. [13]

    Averin, A

    D. Averin, A. Zorin, and K. Likharev, Bloch oscillations in small josephson junctions, Sov. Phys. JETP61, 407 (1985)

    work page 1985

  14. [14]

    Di Liberto, N

    M. Di Liberto, N. Goldman, and G. Palumbo, Non- abelian bloch oscillations in higher-order topological in- sulators, Nature Communications11, 5942 (2020)

    work page 2020

  15. [15]

    Meinert, M

    F. Meinert, M. Knap, E. Kirilov, K. Jag-Lauber, M. B. Zvonarev, E. Demler, and H.-C. N¨ agerl, Bloch oscilla- tions in the absence of a lattice, Science356, 945 (2017)

    work page 2017

  16. [16]

    P. M. Preiss, R. Ma, M. E. Tai, A. Lukin, M. Rispoli, P. Zupancic, Y. Lahini, R. Islam, and M. Greiner, Strongly correlated quantum walks in optical lattices, Science347, 1229 (2015)

    work page 2015

  17. [17]

    Y. Qin, K. Zhang, and L. Li, Geometry-dependent skin effect and anisotropic bloch oscillations in a non- hermitian optical lattice, Phys. Rev. A109, 023317 (2024)

    work page 2024

  18. [18]

    Pagel, W

    Z. Pagel, W. Zhong, R. H. Parker, C. T. Olund, N. Y. Yao, and H. M¨ uller, Symmetric bloch oscillations of mat- ter waves, Phys. Rev. A102, 053312 (2020)

    work page 2020

  19. [19]

    Lebugle, M

    M. Lebugle, M. Gr¨ afe, R. Heilmann, A. Perez-Leija, S. Nolte, and A. Szameit, Experimental observation of n00n state bloch oscillations, Nature Communications6, 8273 (2015)

    work page 2015

  20. [20]

    Corrielli, A

    G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and R. Osellame, Fractional bloch oscillations in photonic lat- tices, Nature Communications4, 1555 (2013)

    work page 2013

  21. [21]

    Y.-L. Xu, W. S. Fegadolli, L. Gan, M.-H. Lu, X.-P. Liu, Z.-Y. Li, A. Scherer, and Y.-F. Chen, Experimental re- alization of bloch oscillations in a parity-time synthetic silicon photonic lattice, Nature Communications7, 11319 11 (2016)

    work page 2016

  22. [22]

    Floss, A

    J. Floss, A. Kamalov, I. S. Averbukh, and P. H. Bucks- baum, Observation of bloch oscillations in molecular ro- tation, Phys. Rev. Lett.115, 203002 (2015)

    work page 2015

  23. [23]

    H. Chen, N. Yang, C. Qin, W. Li, B. Wang, T. Han, C. Zhang, W. Liu, K. Wang, H. Long, X. Zhang, and P. Lu, Real-time observation of frequency bloch oscilla- tions with fibre loop modulation, Light: Science & Ap- plications10, 48 (2021)

    work page 2021

  24. [24]

    Merlin, Phonon bloch oscillations, Phys

    R. Merlin, Phonon bloch oscillations, Phys. Rev. Lett. 132, 246302 (2024)

    work page 2024

  25. [25]

    V. o. T. Phong and E. J. Mele, Quantum geometric os- cillations in two-dimensional flat-band solids, Phys. Rev. Lett.130, 266601 (2023)

    work page 2023

  26. [26]

    De Beule and E

    C. De Beule and E. J. Mele, Berry curvature spectroscopy from bloch oscillations, Phys. Rev. Lett.131, 196603 (2023)

    work page 2023

  27. [27]

    A. G. Davies, E. H. Linfield, M. Pepper, and J. M. Cham- berlain, Where optics meets electronics: recent progress in decreasing the terahertz gap, Philosophical Transac- tions of the Royal Society of London. Series A: Math- ematical, Physical and Engineering Sciences362, 199 (2004)

    work page 2004

  28. [28]

    Borak, Toward bridging the terahertz gap with silicon- based lasers, Science308, 638 (2005)

    A. Borak, Toward bridging the terahertz gap with silicon- based lasers, Science308, 638 (2005)

    work page 2005

  29. [29]

    H. Zeng, Z. Zhou, W. Duan, and H. Huang, Quantum metric induced oscillations in nearly dispersionless flat bands, Phys. Rev. B111, L121102 (2025)

    work page 2025

  30. [30]

    M. F. Lapa and T. L. Hughes, Semiclassical wave packet dynamics in nonuniform electric fields, Phys. Rev. B99, 121111 (2019)

    work page 2019

  31. [31]

    Ashcroft and N

    N. Ashcroft and N. Mermin,Solid State Physics, HRW international editions (Holt, Rinehart and Winston, 1976)

    work page 1976

  32. [32]

    Aversa and J

    C. Aversa and J. E. Sipe, Nonlinear optical susceptibili- ties of semiconductors: Results with a length-gauge anal- ysis, Phys. Rev. B52, 14636 (1995)

    work page 1995

  33. [33]

    M. M. Kumar, S. Sarkar, and A. Agarwal, Band geome- try induced electro-optic effect and polarization rotation, Phys. Rev. B110, 125401 (2024)

    work page 2024

  34. [34]

    M. M. Kumar, S. Sarkar, and A. Agarwal, Band geomet- ric transverse current driven by inhomogeneous ac elec- tric field (2025), arXiv:2509.13242 [cond-mat.mes-hall]

    work page arXiv 2025

  35. [35]

    L. L. Bonilla, M. ´Alvaro, and M. Carretero, Theory of spatially inhomogeneous bloch oscillations in semicon- ductor superlattices, Phys. Rev. B84, 155316 (2011)

    work page 2011

  36. [36]

    Bhalla, K

    P. Bhalla, K. Das, D. Culcer, and A. Agarwal, Resonant second-harmonic generation as a probe of quantum ge- ometry, Phys. Rev. Lett.129, 227401 (2022)

    work page 2022

  37. [37]

    Varshney, K

    H. Varshney, K. Das, P. Bhalla, and A. Agarwal, Quan- tum kinetic theory of nonlinear thermal current, Phys. Rev. B107, 235419 (2023)

    work page 2023

  38. [38]

    Chang and Q

    M.-C. Chang and Q. Niu, Berry phase, hyperorbits, and the hofstadter spectrum, Phys. Rev. Lett.75, 1348 (1995)

    work page 1995

  39. [39]

    Das and A

    K. Das and A. Agarwal, Linear magnetochiral transport in tilted type-i and type-ii weyl semimetals, Phys. Rev. B99, 085405 (2019)

    work page 2019

  40. [40]

    Mazumder, Boltzmann transport equation based mod- eling of phonon heat conduction: progress and challenges, Annual Review of Heat Transfer24, 71 (2021)

    S. Mazumder, Boltzmann transport equation based mod- eling of phonon heat conduction: progress and challenges, Annual Review of Heat Transfer24, 71 (2021)

    work page 2021

  41. [41]

    N. A. Sinitsyn, Q. Niu, J. Sinova, and K. Nomura, Disor- der effects in the anomalous hall effect induced by berry curvature, Phys. Rev. B72, 045346 (2005)

    work page 2005

  42. [42]

    Y. Yao, L. Kleinman, A. H. MacDonald, J. Sinova, T. Jungwirth, D.-s. Wang, E. Wang, and Q. Niu, First principles calculation of anomalous hall conductivity in ferromagnetic bcc fe, Phys. Rev. Lett.92, 037204 (2004)

    work page 2004

  43. [43]

    Hyart, N

    T. Hyart, N. V. Alexeeva, J. Mattas, and K. N. Alekseev, Terahertz bloch oscillator with a modulated bias, Phys. Rev. Lett.102, 140405 (2009)

    work page 2009

  44. [44]

    Esaki and R

    L. Esaki and R. Tsu, Superlattice and negative differ- ential conductivity in semiconductors, IBM Journal of Research and Development14, 61 (1970)

    work page 1970

  45. [45]

    Beltram, F

    F. Beltram, F. Capasso, D. L. Sivco, A. L. Hutchinson, S.-N. G. Chu, and A. Y. Cho, Scattering-controlled trans- mission resonances and negative differential conductance by field-induced localization in superlattices, Phys. Rev. Lett.64, 3167 (1990)

    work page 1990

  46. [46]

    M. A. Mojarro, R. Carrillo-Bastos, and J. A. Maytorena, Optical properties of massive anisotropic tilted dirac sys- tems, Phys. Rev. B103, 165415 (2021)

    work page 2021

  47. [47]

    M. A. Mojarro, R. Carrillo-Bastos, and J. A. Maytorena, Hyperbolic plasmons in massive tilted two-dimensional dirac materials, Phys. Rev. B105, L201408 (2022)

    work page 2022

  48. [48]

    Khomeriki and S

    R. Khomeriki and S. Flach, Landau-zener bloch oscilla- tions with perturbed flat bands, Phys. Rev. Lett.116, 245301 (2016)

    work page 2016

  49. [49]

    Ramya Parkavi, V

    J. Ramya Parkavi, V. K. Chandrasekar, and M. Laksh- manan, Stable bloch oscillations and landau-zener tun- neling in a non-hermitianPT-symmetric flat-band lat- tice, Phys. Rev. A103, 023721 (2021)

    work page 2021

  50. [50]

    A. M. Bouchard and M. Luban, Bloch oscillations and other dynamical phenomena of electrons in semiconduc- tor superlattices, Phys. Rev. B52, 5105 (1995)

    work page 1995

  51. [51]

    Zener and R

    C. Zener and R. H. Fowler, A theory of the electrical breakdown of solid dielectrics, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character145, 523 (1934)

    work page 1934

  52. [52]

    Vandenberghe, B

    W. Vandenberghe, B. Sor´ ee, W. Magnus, and G. Groe- seneken, Zener tunneling in semiconductors under nonuniform electric fields, Journal of Applied Physics 107, 054520 (2010)

    work page 2010

  53. [53]

    Biesenthal, M

    T. Biesenthal, M. Kremer, M. Heinrich, and A. Szameit, Experimental realization ofPT-symmetric flat bands, Phys. Rev. Lett.123, 183601 (2019)

    work page 2019

  54. [54]

    Yoshida, Y

    A. Yoshida, Y. Otaki, R. Otaki, and T. Fukui, Edge states, corner states, and flat bands in a two-dimensional PT-symmetric system, Phys. Rev. B100, 125125 (2019)

    work page 2019

  55. [55]

    Ghorai, S

    K. Ghorai, S. Das, H. Varshney, and A. Agarwal, Planar hall effect in quasi-two-dimensional materials, Phys. Rev. Lett.134, 026301 (2025)

    work page 2025

  56. [56]

    Mandal, S

    D. Mandal, S. Sarkar, K. Das, and A. Agarwal, Quan- tum geometry induced third-order nonlinear transport responses, Phys. Rev. B110, 195131 (2024)

    work page 2024

  57. [57]

    Sarkar and A

    S. Sarkar and A. Agarwal, Third-order rectification in centrosymmetric metals, Progress in Energy8, 025004 (2026)

    work page 2026

  58. [58]

    Sarkar, S

    S. Sarkar, S. Das, D. Mandal, and A. Agarwal, Light- induced nonlinear resonant spin magnetization, New Journal of Physics27, 113503 (2025)

    work page 2025

  59. [59]

    P. C. Adak, S. Sinha, A. Agarwal, and M. M. Deshmukh, Tunable moir´ e materials for probing berry physics and topology, Nature Reviews Materials9, 481–498 (2024)

    work page 2024

  60. [60]

    Longhi,PT-symmetric optical superlattices, Journal of Physics A: Mathematical and Theoretical47, 165302 (2014)

    S. Longhi,PT-symmetric optical superlattices, Journal of Physics A: Mathematical and Theoretical47, 165302 (2014). 12

    work page 2014

  61. [61]

    Butz, The electric field gradient produced by a gaussian charge density distribution, inHFI/NQI 2004, edited by K

    T. Butz, The electric field gradient produced by a gaussian charge density distribution, inHFI/NQI 2004, edited by K. Maier and R. Vianden (Springer Berlin Hei- delberg, Berlin, Heidelberg, 2005) pp. 41–46

    work page 2004

  62. [62]

    D. Vanderbilt,Berry Phases in Electronic Structure The- ory: Electric Polarization, Orbital Magnetization and Topological Insulators, Titolo collana (Cambridge Uni- versity Press, 2018)

    work page 2018

  63. [63]

    Sundaram and Q

    G. Sundaram and Q. Niu, Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and berry-phase effects, Phys. Rev. B59, 14915 (1999)

    work page 1999

  64. [64]

    Cheng, Quantum geometric tensor (fubini-study met- ric) in simple quantum system: A pedagogical introduc- tion (2013)

    R. Cheng, Quantum geometric tensor (fubini-study met- ric) in simple quantum system: A pedagogical introduc- tion (2013)

    work page 2013

  65. [65]

    de Jes´ us Espinosa-Champo and G

    A. de Jes´ us Espinosa-Champo and G. G. Naumis, Fu- bini–study metric and topological properties of flat band electronic states: the case of an atomic chain with s−p or- bitals, Journal of Physics: Condensed Matter36, 015502 (2023)

    work page 2023

  66. [66]

    Graf and F

    A. Graf and F. Pi´ echon, Berry curvature and quantum metric inn-band systems: An eigenprojector approach, Phys. Rev. B104, 085114 (2021)

    work page 2021

  67. [67]

    Y.-Q. Ma, S. Chen, H. Fan, and W.-M. Liu, Abelian and non-abelian quantum geometric tensor, Phys. Rev. B81, 245129 (2010)

    work page 2010