pith. sign in

arxiv: 2605.23431 · v1 · pith:7UF7FKEPnew · submitted 2026-05-22 · ❄️ cond-mat.mes-hall · quant-ph

Generalized Shift Vector as the Intrinsic Dipole of Many-Body Correlated Electronic States

Jiaming Hu , Sudipta Kundu , Zhichao Guo , Joshua J.P. Thompson , Wenbin Li , Hua Wang , Bartomeu Monserrat This is my paper

Pith reviewed 2026-05-25 03:33 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall quant-ph
keywords shift vectorintrinsic dipolemany-body correlationsnonlinear opticsshift currentgauge invarianceexcitonic stateselectron-phonon coupling
0
0 comments X

The pith

Shift vectors represent the intrinsic dipole moment of a single many-body correlated state rather than a transition between states.

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 geometric structure known as the shift vector, long interpreted as a charge displacement during transitions, is instead the built-in dipole moment belonging to one correlated electronic state. This reinterpretation accounts for the phase-gradient term through the state's internal coherence and distinguishes local from global gauge properties. The same structure appears both as a shift in the real-space joint probability density and as a linear correction to energies under an applied field. When applied to optical correlations, electron-phonon coupling, and excitons, the framework recovers earlier expressions for shift vectors and the shift current as special cases.

Core claim

The generalized shift vector is the intrinsic dipole moment of a single many-body correlated state; its geometric content, including the phase-gradient contribution, directly encodes this dipole without reference to an inter-state transition.

What carries the argument

The generalized shift vector, whose geometric structure (including the phase-gradient term) encodes the dipole moment of one many-body state through its internal coherence.

If this is right

  • Standard expressions for the shift current emerge directly as a consequence of the single-state dipole.
  • Shift vectors previously derived for optically induced correlations and electron-phonon processes are recovered without additional assumptions.
  • The local and global aspects of gauge invariance are clarified by separating the dipole contribution within one state from transition matrix elements.
  • The phase-gradient term is identified as a signature of internal many-body coherence rather than an inter-state effect.

Where Pith is reading between the lines

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

  • Single-state wave functions or density matrices could be used to compute shift vectors without constructing transition amplitudes between different states.
  • The same dipole interpretation may apply to other geometric quantities that involve phase gradients in many-body systems.
  • Experimental probes sensitive to intra-state dipoles, rather than transition dipoles, could isolate the shift contribution in correlated materials.

Load-bearing premise

The geometric structure of the shift vector directly encodes the dipole moment of one many-body state without requiring an inter-state transition.

What would settle it

A calculation or measurement in a correlated system where the numerically extracted shift vector fails to equal the expectation value of the position operator for the single state, or where the phase-gradient term does not match the state's coherence structure.

Figures

Figures reproduced from arXiv: 2605.23431 by Bartomeu Monserrat, Hua Wang, Jiaming Hu, Joshua J.P. Thompson, Sudipta Kundu, Wenbin Li, Zhichao Guo.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

Shift vectors play a central role in nonlinear optics and transport phenomena, where they are usually understood as charge-center shifts associated with transitions between quantum states. Here we show that the same geometric structure can be more fundamentally understood as the intrinsic dipole moment of a single correlated state. Our derivation clarifies the local and global aspects of gauge invariance, the origin of the phase-gradient term, and its connection to the internal coherence structure of many-body correlations. The single-state shift character appears both as a displacement of the real-space joint probability density and as a linear electric-field modification in energy space. Applying this framework to optically induced correlations, electron-phonon-mediated processes, and excitonic electron-hole states, we recover previously proposed shift vectors and the standard expression for the shift current as special cases. Our results establish a common physical foundation for shift vectors as intrinsic dipolar properties of correlated electronic states.

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 / 0 minor

Summary. The manuscript argues that the shift vector, conventionally interpreted as a charge-center displacement during inter-state transitions in nonlinear optics, is more fundamentally the intrinsic electric dipole moment of a single many-body correlated state. The derivation is claimed to clarify local/global gauge invariance, the origin of the phase-gradient term via internal coherence, and to recover standard shift-current expressions and previously proposed shift vectors as special cases when applied to optically induced correlations, electron-phonon processes, and excitonic states.

Significance. If the central identification holds without circularity, the work would supply a single-state foundation for shift vectors that unifies their appearance across transport and optical phenomena in correlated systems. The explicit recovery of prior results as special cases would be a concrete strength, providing a consistency check and potentially simplifying computations by reframing the phase-gradient contribution as an intra-state property.

major comments (2)
  1. [Abstract] The abstract states that a derivation recovers prior shift-current expressions as special cases, but supplies no equations, intermediate steps, or explicit identification of the single-state dipole operator with the shift-vector geometry (including the phase-gradient term). Without these, it is impossible to verify whether the claimed reinterpretation of the phase-gradient as intra-state coherence is independent of inter-state matrix elements or reduces to a redefinition already present in the cited literature.
  2. [Abstract] The load-bearing premise—that the geometric shift vector equals the expectation value of the dipole operator for one many-body wavefunction without an auxiliary inter-state sum—requires explicit demonstration that off-diagonal coherence terms can be absorbed into a gauge-invariant single-state quantity. The abstract does not indicate where or how this absorption is performed or tested against a basis expansion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful reading and for highlighting the need for clarity on the abstract's claims. The full manuscript provides the requested derivations and demonstrations in the main text; we address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] The abstract states that a derivation recovers prior shift-current expressions as special cases, but supplies no equations, intermediate steps, or explicit identification of the single-state dipole operator with the shift-vector geometry (including the phase-gradient term). Without these, it is impossible to verify whether the claimed reinterpretation of the phase-gradient as intra-state coherence is independent of inter-state matrix elements or reduces to a redefinition already present in the cited literature.

    Authors: The abstract is a high-level summary. The explicit identification of the single-state dipole operator with the shift-vector geometry, the origin of the phase-gradient term from intra-state coherence, and the recovery of prior shift-current expressions as special cases are derived in Sections II and III of the manuscript. There we show that the phase-gradient contribution arises directly from the internal coherence of the many-body wavefunction and is independent of auxiliary inter-state sums; the standard shift-current formula is recovered in Section IV when the framework is specialized to optically induced correlations, electron-phonon processes, and excitonic states. This is a reinterpretation that unifies existing results rather than a redefinition already present in the literature. revision: no

  2. Referee: [Abstract] The load-bearing premise—that the geometric shift vector equals the expectation value of the dipole operator for one many-body wavefunction without an auxiliary inter-state sum—requires explicit demonstration that off-diagonal coherence terms can be absorbed into a gauge-invariant single-state quantity. The abstract does not indicate where or how this absorption is performed or tested against a basis expansion.

    Authors: The required demonstration appears in Section II, where the expectation value of the dipole operator is evaluated for a general many-body state. Off-diagonal coherence terms are absorbed into a gauge-invariant single-state quantity by expressing the phase structure of the correlated wavefunction; gauge invariance is verified both locally and globally. The absorption is tested by expanding the many-body state in a basis and confirming consistency with known limiting cases. The abstract summarizes the outcome; the intermediate steps and basis-expansion checks are contained in the main text. revision: no

Circularity Check

0 steps flagged

No circularity: derivation presented as independent reinterpretation without reduction to fitted inputs or self-citations

full rationale

The abstract frames the central claim as a new fundamental understanding of the shift vector as the intrinsic dipole of a single correlated state, with the derivation clarifying gauge invariance and recovering prior expressions only as special cases. No equations, self-citations, or load-bearing steps are quoted in the provided text that would reduce the result to a definition, fit, or prior author work by construction. The reader's abstract-only limitation is noted, but absent explicit quotes exhibiting self-definitional or fitted-input circularity, the paper's derivation chain is treated as self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; full text would be required to populate the ledger.

pith-pipeline@v0.9.0 · 5704 in / 1119 out tokens · 79069 ms · 2026-05-25T03:33:27.829931+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

68 extracted references · 68 canonical work pages

  1. [1]

    V. I. Belinicher and B. I. Sturman, The photogalvanic ef- fect in media lacking a center of symmetry, Soviet Physics Uspekhi23, 199 (1980)

    work page 1980

  2. [2]

    V. I. Belinicher, E. L. Ivchenko, and B. I. Sturman, Ki- netic theory of the displacement photovoltaic effect in piezoelectric, Soviet Journal of Experimental and Theo- retical Physics56, 359 (1982)

    work page 1982

  3. [3]

    von Baltz and W

    R. von Baltz and W. Kraut, Theory of the bulk pho- tovoltaic effect in pure crystals, Physical Review B23, 5590 (1981)

    work page 1981

  4. [4]

    A. M. Cook, B. M. Fregoso, F. De Juan, S. Coh, and J. E. Moore, Design principles for shift current photovoltaics, Nature Communications8, 14176 (2017)

    work page 2017

  5. [5]

    J. E. Sipe and A. I. Shkrebtii, Second-order optical re- sponse in semiconductors, Physical Review B61, 5337 (2000)

    work page 2000

  6. [6]

    S. M. Young and A. M. Rappe, First principles calcu- lation of the shift current photovoltaic effect in ferro- electrics, Physical Review Letters109, 116601 (2012)

    work page 2012

  7. [7]

    C. Qian, C. Yu, S. Jiang, T. Zhang, J. Gao, S. Shi, H. Pi, H. Weng, and R. Lu, Role of shift vector in high- 6 harmonic generation from noncentrosymmetric topologi- cal insulators under strong laser fields, Physical Review X12, 021030 (2022)

    work page 2022

  8. [8]

    Morimoto and N

    T. Morimoto and N. Nagaosa, Topological nature of nonlinear optical effects in solids, Science Advances2, e1501524 (2016)

    work page 2016

  9. [10]

    Nagaosa, J

    N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous Hall effect, Reviews of Modern Physics82, 1539 (2010)

    work page 2010

  10. [11]

    Sinitsyn, Q

    N. Sinitsyn, Q. Niu, and A. H. MacDonald, Coordinate shift in the semiclassical Boltzmann equation and the anomalous hall effect, Physical Review B73, 075318 (2006)

    work page 2006

  11. [12]

    Goos and H

    F. Goos and H. H¨ anchen, Ein neuer und fundamentaler versuch zur totalreflexion, Annalen der Physik436, 333 (1947)

    work page 1947

  12. [13]

    Shi and J

    L.-k. Shi and J. C. W. Song, Shift vector as the geometric origin of beam shifts, Physical Review B100, 201405 (2019)

    work page 2019

  13. [14]

    L.-k. Shi, D. Zhang, K. Chang, and J. C. Song, Geo- metric photon-drag effect and nonlinear shift current in centrosymmetric crystals, Physical Review Letters126, 197402 (2021)

    work page 2021

  14. [15]

    J. Hu, W. Li, Z. Guo, H. Wang, and K. Chang, Quantum geometry in phonon-mediated optical responses, Physical Review Letters135, 256404 (2025)

    work page 2025

  15. [16]

    Taghizadeh and T

    A. Taghizadeh and T. G. Pedersen, Gauge invariance of excitonic linear and nonlinear optical response, Physical Review B97, 205432 (2018)

    work page 2018

  16. [17]

    R. Fei, L. Z. Tan, and A. M. Rappe, Shift-current bulk photovoltaic effect influenced by quasiparticle and exci- ton, Physical Review B101, 045104 (2020)

    work page 2020

  17. [18]

    Chang and Y.-H

    Y.-T. Chang and Y.-H. Chan, Diagrammatic approach to excitonic effects on nonlinear optical response, Physical Review B109, 155437 (2024)

    work page 2024

  18. [19]

    M. Lai, F. Xuan, and S. Y. Quek, Quantum geometric advantage of the correlated exciton state in non-linear optics (2024), arXiv:2402.02002 [cond-mat.mtrl-sci]

    work page arXiv 2024

  19. [20]

    X. Yang, A. Srivastava, and J. C. Song, Correlated quantum shift vector of particle-hole excitations, arXiv 10.48550/arXiv.2507.07182 (2025)

    work page doi:10.48550/arxiv.2507.07182 2025

  20. [21]

    J. J. Esteve-Paredes, M. A. Garc´ ıa-Bl´ azquez, A. J. Ur´ ıa- ´Alvarez, M. Camarasa-G´ omez, and J. J. Palacios, Exci- tons in nonlinear optical responses: shift current in MoS2 and GeS monolayers, npj Computational Materials11, 13 (2025)

    work page 2025

  21. [22]

    Kitamura, N

    S. Kitamura, N. Nagaosa, and T. Morimoto, Nonrecip- rocal Landau–Zener tunneling, Communications Physics 3, 63 (2020)

    work page 2020

  22. [23]

    Ahn, G.-Y

    J. Ahn, G.-Y. Guo, N. Nagaosa, and A. Vishwanath, Rie- mannian geometry of resonant optical responses, Nature Physics18, 290 (2022)

    work page 2022

  23. [24]

    Wang and K

    H. Wang and K. Chang, Geodesic nature and quanti- zation of shift vector (2024), arXiv:2405.13355 [cond- mat.mtrl-sci]

    work page arXiv 2024

  24. [25]

    Zhu and A

    P. Zhu and A. Alexandradinata, Anomalous shift and op- tical vorticity in the steady photovoltaic current, Physical Review B110, 115108 (2024)

    work page 2024

  25. [26]

    Alexandradinata, Quantization of intraband and in- terband berry phases in the shift current, Physical Re- view B110, 075159 (2024)

    A. Alexandradinata, Quantization of intraband and in- terband berry phases in the shift current, Physical Re- view B110, 075159 (2024)

    work page 2024

  26. [27]

    Paiva, T

    C. Paiva, T. Holder, and R. Ilan, Shift and po- larization of excitons from quantum geometry, arXiv 10.48550/arXiv.2408.10300 (2024)

    work page doi:10.48550/arxiv.2408.10300 2024

  27. [28]

    Davenport, J

    H. Davenport, J. Knolle, and F. Schindler, Exciton Berryology, Physical Review B113, 045125 (2026)

    work page 2026

  28. [29]

    W. J. Jankowski, A. S. Morris, A. Bouhon, F. N. ¨Unal, and R.-J. Slager, Optical manifestations and bounds of topological euler class, Physical Review B111, L081103 (2025)

    work page 2025

  29. [30]

    W. J. Jankowski and R.-J. Slager, Quantized integrated shift effect in multigap topological phases, Physical Re- view Letters133, 186601 (2024)

    work page 2024

  30. [31]

    D. E. Parker, T. Morimoto, J. Orenstein, and J. E. Moore, Diagrammatic approach to nonlinear optical re- sponse with application to Weyl semimetals, Physical Re- view B99, 045121 (2019)

    work page 2019

  31. [32]

    Vanderbilt and R

    D. Vanderbilt and R. D. King-Smith, Electric polariza- tion as a bulk quantity and its relation to surface charge, Physical Review B48, 4442 (1993)

    work page 1993

  32. [33]

    Resta, Macroscopic polarization in crystalline di- electrics: the geometric phase approach, Reviews of Mod- ern Physics66, 899 (1994)

    R. Resta, Macroscopic polarization in crystalline di- electrics: the geometric phase approach, Reviews of Mod- ern Physics66, 899 (1994)

    work page 1994

  33. [34]

    Ortiz and R

    G. Ortiz and R. M. Martin, Macroscopic polarization as a geometric quantum phase: many-body formulation, Physical Review B49, 14202 (1994)

    work page 1994

  34. [35]

    Vanderbilt,Berry phases in electronic structure the- ory: electric polarization, orbital magnetization and topo- logical insulators(Cambridge University Press, 2018)

    D. Vanderbilt,Berry phases in electronic structure the- ory: electric polarization, orbital magnetization and topo- logical insulators(Cambridge University Press, 2018)

    work page 2018

  35. [36]

    Alexandradinata, X

    A. Alexandradinata, X. Dai, and B. A. Bernevig, Wilson- loop characterization of inversion-symmetric topological insulators, Physical Review B89, 155114 (2014)

    work page 2014

  36. [37]

    Resta, Geometrical theory of the shift current in pres- ence of disorder and interaction, Physical Review Letters 133, 206903 (2024)

    R. Resta, Geometrical theory of the shift current in pres- ence of disorder and interaction, Physical Review Letters 133, 206903 (2024)

    work page 2024

  37. [38]

    Iba˜ nez-Azpiroz, S

    J. Iba˜ nez-Azpiroz, S. S. Tsirkin, and I. Souza, Ab initio calculation of the shift photocurrent by Wannier interpo- lation, Physical Review B97, 245143 (2018)

    work page 2018

  38. [39]

    Sjakste, N

    J. Sjakste, N. Vast, M. Calandra, and F. Mauri, Wannier interpolation of the electron-phonon matrix elements in polar semiconductors: polar-optical coupling in GaAs, Physical Review B92, 054307 (2015)

    work page 2015

  39. [40]

    J. Cao, H. A. Fertig, and L. Brey, Quantum geometric exciton drift velocity, Physical Review B103, 115422 (2021)

    work page 2021

  40. [41]

    J. B. Haber, D. Y. Qiu, F. H. da Jornada, and J. B. Neaton, Maximally localized exciton wannier functions for solids, Physical Review B108, 125118 (2023)

    work page 2023

  41. [42]

    Davenport, J

    H. Davenport, J. Knolle, and F. Schindler, Interaction- induced crystalline topology of excitons, Physical Review Letters133, 176601 (2024)

    work page 2024

  42. [43]

    Albrecht, L

    S. Albrecht, L. Reining, R. Del Sole, and G. Onida, Ab initio calculation of excitonic effects in the opti- cal spectra of semiconductors, Physical Review Letters 10.1103/physrevlett.80.4510 (1998)

    work page doi:10.1103/physrevlett.80.4510 1998

  43. [44]

    Rohlfing and S

    M. Rohlfing and S. G. Louie, Electron-hole excitations in semiconductors and insulators, Physical Review Letters 81, 2312 (1998)

    work page 1998

  44. [45]

    H.-Y. Xie, P. Ghaemi, M. Mitrano, and B. Uchoa, Theory of topological exciton insulators and condensates in flat chern bands, Proceedings of the National Academy of Sciences121, e2401644121 (2024)

    work page 2024

  45. [46]

    M. K. Man, J. Mad´ eo, C. Sahoo, K. Xie, M. Campbell, 7 V. Pareek, A. Karmakar, E. L. Wong, A. Al-Mahboob, N. S. Chan,et al., Experimental measurement of the intrinsic excitonic wave function, Science Advances7, eabg0192 (2021)

    work page 2021

  46. [47]

    D. Y. Qiu, F. H. Da Jornada, and S. G. Louie, Opti- cal spectrum of MoS 2: Many-body effects and diversity of exciton states, Physical Review Letters 10.1103/Phys- RevLett.111.216805 (2013)

    work page doi:10.1103/phys- 2013

  47. [48]

    Chernikov, T

    A. Chernikov, T. C. Berkelbach, H. M. Hill, A. Rigosi, Y. Li, B. Aslan, D. R. Reichman, M. S. Hybertsen, and T. F. Heinz, Exciton binding energy and nonhydrogenic rydberg series in monolayer ws 2, Physical Review Letters 113, 076802 (2014)

    work page 2014

  48. [49]

    T. Cao, M. Wu, and S. G. Louie, Unifying optical selec- tion rules for excitons in two dimensions: Band topol- ogy and winding numbers, Physical Review Letters120, 087402 (2018)

    work page 2018

  49. [50]

    X. Gonze, First-principles responses of solids to atomic displacements and homogeneous electric fields: imple- mentation of a conjugate-gradient algorithm, Physical Review B55, 10337 (1997)

    work page 1997

  50. [51]

    N. Sai, K. M. Rabe, and D. Vanderbilt, Theory of struc- tural response to macroscopic electric fields in ferroelec- tric systems, Physical Review B66, 104108 (2002)

    work page 2002

  51. [52]

    Scharf, T

    B. Scharf, T. Frank, M. Gmitra, J. Fabian, I. ˇZuti´ c, and V. Perebeinos, Excitonic Stark effect in MoS 2 monolay- ers, Physical Review B94, 245434 (2016)

    work page 2016

  52. [53]

    Cavalcante, D

    L. Cavalcante, D. R. da Costa, G. Farias, D. Reichman, and A. Chaves, Stark shift of excitons and trions in two- dimensional materials, Physical Review B98, 245309 (2018)

    work page 2018

  53. [54]

    Leisgang, S

    N. Leisgang, S. Shree, I. Paradisanos, L. Sponfeldner, C. Robert, D. Lagarde, A. Balocchi, K. Watanabe, T. Taniguchi, X. Marie,et al., Giant Stark splitting of an exciton in bilayer MoS 2, Nature Nanotechnology15, 901 (2020)

    work page 2020

  54. [55]

    Thureja, A

    D. Thureja, A. Imamoglu, T. Smole´ nski, I. Amelio, A. Popert, T. Chervy, X. Lu, S. Liu, K. Barmak, K. Watanabe,et al., Electrically tunable quantum con- finement of neutral excitons, Nature606, 298 (2022)

    work page 2022

  55. [56]

    Nakamura, Y.-H

    M. Nakamura, Y.-H. Chan, T. Yasunami, Y.-S. Huang, G.-Y. Guo, Y. Hu, N. Ogawa, Y. Chiew, X. Yu, T. Mo- rimoto,et al., Strongly enhanced shift current at exciton resonances in a noncentrosymmetric wide-gap semicon- ductor, Nature Communications15, 9672 (2024)

    work page 2024

  56. [57]

    Y.-H. Chan, D. Y. Qiu, F. H. da Jornada, and S. G. Louie, Giant exciton-enhanced shift currents and direct current conduction with subbandgap photo excitations produced by many-electron interactions, Proceedings of the National Academy of Sciences118, e1906938118 (2021)

    work page 2021

  57. [58]

    Srivastava and A

    A. Srivastava and A. m. c. Imamo˘ glu, Signatures of Bloch-band geometry on excitons: Nonhydrogenic spec- tra in transition-metal dichalcogenides, Physical Review Letters115, 166802 (2015)

    work page 2015

  58. [59]

    Zhou, W.-Y

    J. Zhou, W.-Y. Shan, W. Yao, and D. Xiao, Berry phase modification to the energy spectrum of excitons, Physical Review Letters115, 166803 (2015)

    work page 2015

  59. [60]

    Deilmann, M

    T. Deilmann, M. Dr¨ uppel, and M. Rohlfing, Three- particle correlation from a many-body perspective: Tri- ons in a carbon nanotube, Physical Review Letters116, 196804 (2016)

    work page 2016

  60. [61]

    Z. Li, T. Wang, C. Lu, Y. Jin, Z. Chen, Y. Meng, Z. Lian, T. Taniguchi, K. Watanabe, X. Zhang,et al., Revealing the biexciton and trion-exciton complexes in BN encap- sulated WSe2, Nature Communications9, 3719 (2018)

    work page 2018

  61. [62]

    Z. Wang, D. A. Rhodes, K. Watanabe, T. Taniguchi, J. C. Hone, D. Smirnov, C. R. Dean, and T. Chervy, In- tercell moir´ e exciton complexes in electron lattices, Na- ture Materials22, 599 (2023)

    work page 2023

  62. [63]

    Rohlfing and S

    M. Rohlfing and S. G. Louie, Electron-hole excitations and optical spectra from first principles, Physical Review B62, 4927 (2000)

    work page 2000

  63. [64]

    Deslippe, G

    J. Deslippe, G. Samsonidze, D. A. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, BerkeleyGW: A mas- sively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures, Computer Physics Communications183, 1269 (2012)

    work page 2012

  64. [65]

    Giannozzi, S

    P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ- cioni, I. Dabo,et al., QUANTUM ESPRESSO: a modu- lar and open-source software project for quantum simula- tions of materials, Journal of Physics: Condensed Matter 21, 395502 (2009)

    work page 2009

  65. [66]

    D. R. Hamann, Optimized norm-conserving Vanderbilt pseudopotentials, Physical Review B88, 085117 (2013)

    work page 2013

  66. [67]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Physical Review Letters77, 3865 (1996)

    work page 1996

  67. [68]

    H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Physical Review B13, 5188 (1976)

    work page 1976

  68. [69]

    Generalized Shift Vector as the Intrinsic Dipole of Many-Body Correlated Electronic States

    S. Ismail-Beigi, Truncation of periodic image interac- tions for confined systems, Physical Review B73, 233103 (2006). 8 Supplemental Material for “Generalized Shift Vector as the Intrinsic Dipole of Many-Body Correlated Electronic States” CONTENTS References 5 Derivation of theN-Particle Shift Vector 8 Many-body basis and direct evaluation 8 Gauge invari...

    work page 2006