[Une démonstration du lien entre la conductivité de Hall et le nombre de Chern pour le problème à N corps, basée sur la règle d’or de Fermi et les relations de Kramers-Kronig]
Ce travail présente une démonstration simple et originale du lien qui relie la conductivité de Hall quantifiée des isolants corrélés au nombre de Chern du problème à N corps, un invariant topologique défini dans l’espace des conditions aux bords généralisées. Contrairement aux démonstrations conventionnelles, qui sont généralement basées sur la formule de Kubo, cette approche s’appuie entièrement sur les relations de Kramers-Kronig et la règle d’or de Fermi, dans le cadre du dichroisme circulaire. Cette dérivation pédagogique illustre un fait remarquable, à savoir que la conductivité de Hall des isolants corrélés peut être déterminée en mesurant des excitations à un corps sous l’effet d’une force circulaire. Cette observation est particulièrement pertinente pour les systèmes quantiques topologiques pour lesquels les taux d’excitation peuvent être directement mesurés au laboratoire.
This work provides a surprisingly simple demonstration that the quantized Hall conductivity of correlated insulators is given by the many-body Chern number, a topological invariant defined in the space of twisted boundary conditions. In contrast to conventional proofs, generally based on the Kubo formula, our approach entirely relies on combining Kramers–Kronig relations and Fermi’s golden rule within a circular-dichroism framework. This pedagogical derivation illustrates how the Hall conductivity of correlated insulators can be determined by monitoring single-particle excitations upon a circular drive, a conceptually simple picture with direct implications for quantum-engineered systems, where excitation rates can be directly monitored.
Révisé le :
Accepté le :
Publié le :
Mots-clés : effets Hall quantiques, états topologiques de la matière, matière quantique fortement corrélée, dichroisme circulaire, règle d’or de Fermi, relations de Kramers-Kronig
Nathan Goldman 1, 2 ; Tomoki Ozawa 3
CC-BY 4.0
@article{CRPHYS_2024__25_G1_289_0,
author = {Nathan Goldman and Tomoki Ozawa},
title = {Relating the {Hall} conductivity to the many-body {Chern} number using {Fermi{\textquoteright}s} {Golden} rule and {Kramers{\textendash}Kronig} relations},
journal = {Comptes Rendus. Physique},
pages = {289--302},
publisher = {Acad\'emie des sciences, Paris},
volume = {25},
year = {2024},
doi = {10.5802/crphys.191},
language = {en},
}
TY - JOUR AU - Nathan Goldman AU - Tomoki Ozawa TI - Relating the Hall conductivity to the many-body Chern number using Fermi’s Golden rule and Kramers–Kronig relations JO - Comptes Rendus. Physique PY - 2024 SP - 289 EP - 302 VL - 25 PB - Académie des sciences, Paris DO - 10.5802/crphys.191 LA - en ID - CRPHYS_2024__25_G1_289_0 ER -
%0 Journal Article %A Nathan Goldman %A Tomoki Ozawa %T Relating the Hall conductivity to the many-body Chern number using Fermi’s Golden rule and Kramers–Kronig relations %J Comptes Rendus. Physique %D 2024 %P 289-302 %V 25 %I Académie des sciences, Paris %R 10.5802/crphys.191 %G en %F CRPHYS_2024__25_G1_289_0
Nathan Goldman; Tomoki Ozawa. Relating the Hall conductivity to the many-body Chern number using Fermi’s Golden rule and Kramers–Kronig relations. Comptes Rendus. Physique, Volume 25 (2024), pp. 289-302. doi : 10.5802/crphys.191. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.191/
[1] New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance, Phys. Rev. Lett., Volume 45 (1980) no. 6, pp. 494-497 | DOI
[2] Two-Dimensional Magnetotransport in the Extreme Quantum Limit, Phys. Rev. Lett., Volume 48 (1982) no. 22, pp. 1559-1562 | DOI
[3] Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett., Volume 49 (1982) no. 6, pp. 405-408 | DOI
[4] Holonomy, the quantum adiabatic theorem, and Berry’s phase, Phys. Rev. Lett., Volume 51 (1983) no. 24, L012054, pp. 2167-2170 | DOI
[5] Homotopy and quantization in condensed matter physics, Phys. Rev. Lett., Volume 51 (1983) no. 1, pp. 51-53 | DOI
[6] Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett., Volume 95 (2005) no. 14, 146802 | DOI
[7] Quantum Spin Hall Effect in Graphene, Phys. Rev. Lett., Volume 95 (2005) no. 22, 226801, 106802 | DOI
[8] Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science, Volume 314 (2006) no. 5806, 015005, pp. 1757-1761 | DOI
[9] Quantum Spin Hall Effect, Phys. Rev. Lett., Volume 96 (2006) no. 10, 106802 | DOI
[10] et al. Quantum spin Hall insulator state in HgTe quantum wells, Science, Volume 318 (2007) no. 5851, 125132, pp. 766-770 | DOI
[11] et al. Topological photonics, Rev. Mod. Phys., Volume 91 (2019) no. 1, 015006 | DOI
[12] Topological bands for ultracold atoms, Rev. Mod. Phys., Volume 91 (2019) no. 1, 015005 | DOI
[13] Quantized Hall conductance as a topological invariant, Phys. Rev. B, Volume 31 (1985) no. 6, 236601, pp. 3372-3377 | DOI
[14] Impurity effect, degeneracy, and topological invariant in the quantum Hall effect, Phys. Rev. B, Volume 33 (1986) no. 6, 085426, pp. 3844-3850 | DOI
[15] Quantization of the Hall conductance for general, multiparticle Schrödinger Hamiltonians, Phys. Rev. Lett., Volume 54 (1985) no. 4, pp. 259-262 | DOI
[16] Zero modes and the quantized Hall conductance of the two-dimensional lattice in a magnetic field, Phys. Rev. B, Volume 39 (1989) no. 16, pp. 11943-11949 | DOI
[17] Magnetic field induced multi-component QED 3 and quantum Hall effect, Z. Phys. C - Particles and Fields, Volume 33 (1986), 10610, pp. 41-45 | DOI
[18] A microscopic theory of the quantum Hall effect, Nucl. Phys., B, Volume 280 (1987), pp. 523-548 | DOI
[19] Field theory in a strong magnetic field and the quantum Hall effect: Integer Hall effect, Phys. Rev. B, Volume 42 (1990) no. 16, 10610 | DOI
[20] Topological order parameters for interacting topological insulators, Phys. Rev. Lett., Volume 105 (2010) no. 25, 256803 | DOI
[21] Single-particle Green’s functions and interacting topological insulators, Phys. Rev. B, Volume 83 (2011) no. 8, 085426 | DOI
[22] Bulk-boundary correspondence of topological insulators from their respective Green’s functions, Phys. Rev. B, Volume 84 (2011) no. 12, 125132, 035114 | DOI
[23] Simplified topological invariants for interacting insulators, Phys. Rev. X, Volume 2 (2012) no. 3, 031008, 146802 | DOI
[24] Connecting the Many-Body Chern Number to Luttinger’s Theorem through Středa’s Formula, Phys. Rev. Lett., Volume 131 (2023) no. 23, 236601, 226801 | DOI
[25] Probing topology by “heating”: Quantized circular dichroism in ultracold atoms, Sci. adv., Volume 3 (2017) no. 8, e1701207 | DOI
[26] Tracing the nonequilibrium topological state of Chern insulators, Phys. Rev. B, Volume 96 (2017) no. 15, 155122 | DOI
[27] Quantized Rabi oscillations and circular dichroism in quantum Hall systems, Phys. Rev. A, Volume 97 (2018) no. 6, 061602 | DOI
[28] et al. Measuring quantized circular dichroism in ultracold topological matter, Nat. Phys., Volume 15 (2019) no. 5, pp. 449-454 | DOI
[29] Probing localization and quantum geometry by spectroscopy, Phys. Rev. Res., Volume 1 (2019) no. 3, 032019, 146601 | DOI
[30] Detecting fractional Chern insulators through circular dichroism, Phys. Rev. Lett., Volume 122 (2019) no. 16, 166801, 125106 | DOI
[31] Quantization in chiral higher order topological insulators: Circular dichroism and local chern marker, Phys. Rev. Lett., Volume 123 (2019) no. 24, 247401 | DOI
[32] et al. Experimental measurement of the quantum metric tensor and related topological phase transition with a superconducting qubit, Phys. Rev. Lett., Volume 122 (2019) no. 21, 210401, 033385 | DOI
[33] et al. Experimental measurement of the quantum geometric tensor using coupled qubits in diamond, Natl. Sci. Rev., Volume 7 (2020) no. 2, pp. 254-260 | DOI
[34] Detecting chiral pairing and topological superfluidity using circular dichroism, Phys. Rev. Res., Volume 2 (2020) no. 3, 033385 | DOI
[35] How circular dichroism in time-and angle-resolved photoemission can be used to spectroscopically detect transient topological states in graphene, Phys. Rev. X, Volume 10 (2020) no. 4, 041013, 032019 | DOI
[36] Local Berry curvature signatures in dichroic angle-resolved photoelectron spectroscopy from two-dimensional materials, Sci. adv., Volume 6 (2020) no. 9, eaay2730, 015006 | DOI
[37] Interacting stochastic topology and Mott transition from light response, Phys. Rev. B, Volume 103 (2021) no. 3, 035114, 247401 | DOI
[38] Global and local topological quantized responses from geometry, light, and time, Phys. Rev. B, Volume 105 (2022) no. 12, 125106, 166801 | DOI
[39] A synthetic monopole source of Kalb–Ramond field in diamond, Science, Volume 375 (2022) no. 6584, 063316, pp. 1017-1020 | DOI
[40] et al. Experimental demonstration of topological bounds in quantum metrology (2022), eaay2730 | DOI
[41] Characterization of topological insulators: Chern numbers for ground state multiplet, J. Phys. Soc. Jpn., Volume 74 (2005) no. 5, 041013, pp. 1374-1377 | DOI
[42] Polarization and localization in insulators: Generating function approach, Phys. Rev. B, Volume 62 (2000) no. 3, pp. 1666-1683 | DOI
[43] Field theories of condensed matter physics, Cambridge University Press, 2013, 054438 | DOI
[44] Introduction to quantum mechanics, Cambridge University Press, 2018, 155122 | DOI
[45] Many-body chern number without integration, Phys. Rev. Lett., Volume 122 (2019) no. 14, 146601 | DOI
[46] Spectroscopy of edge and bulk collective modes in fractional Chern insulators, Phys. Rev. Res., Volume 6 (2024) no. 1, L012054, 061602 | DOI
[47] Quantized circular dichroism on the edge of quantum Hall systems: The many-body Chern number as seen from the edge (2024), e1701207 (preprint, arXiv:2407.04639)
[48] Faraday effect in solids, Phys. Rev., Volume 137 (1965) no. 2A, p. A448-A461 | DOI
[49] On the theory of dispersion of x-rays, Josa, Volume 12 (1926) no. 6, pp. 547-557 | DOI
[50] La diffusion de la lumiere par les atomes, Atti Cong. Intern. Fisica (Transactions of Volta Centenary Congress) Como, Volume 2 (1927), pp. 545-557
[51] Kramers–Kronig in two lines, Am. J. Phys., Volume 57 (1989) no. 9, 210401, p. 821-821 | DOI
[52] Classical electrodynamics, John Wiley & Sons, Inc. Hoboken, 1999, 256803
[53] Dichroic -sum rule and the orbital magnetization of crystals, Phys. Rev. B, Volume 77 (2008) no. 5, 054438, 031008 | DOI
[54] Berry phase effects on electronic properties, Rev. Mod. Phys., Volume 82 (2010) no. 3, pp. 1959-2007 | DOI
[55] Fractional Chern insulators of few bosons in a box: Hall plateaus from center-of-mass drifts and density profiles, Phys. Rev. A, Volume 102 (2020) no. 6, 063316 | DOI
[56] et al. Realization of a fractional quantum Hall state with ultracold atoms, Nature, Volume 619 (2023), p. 495–499 | DOI
Cité par Sources :
Commentaires - Politique