Quantum Hall effect and Quillen metric
| Autor: | Semyon Klevtsov, Xiaonan Ma, George Marinescu, Paul Wiegmann |
|---|---|
| Přispěvatelé: | Université Paris Diderot - Paris 7 (UPD7), Université Paris Diderot - Paris 7 ( UPD7 ) |
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
High Energy Physics - Theory
Mathematics - Differential Geometry FOS: Physical sciences anomaly Quantum Hall effect holomorphic Curvature determinant 01 natural sciences Hall effect: quantum [ PHYS.HTHE ] Physics [physics]/High Energy Physics - Theory [hep-th] Condensed Matter - Strongly Correlated Electrons Line bundle 0103 physical sciences Riemann surface: compact FOS: Mathematics [ PHYS.PHYS.PHYS-GEN-PH ] Physics [physics]/Physics [physics]/General Physics [physics.gen-ph] Compact Riemann surface Complex Variables (math.CV) 0101 mathematics 010306 general physics Adiabatic process magnetic field: flux Mathematical Physics Mathematical physics Physics asymptotic expansion Strongly Correlated Electrons (cond-mat.str-el) Mathematics - Complex Variables Chern-Simons term [PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th] 010102 general mathematics Statistical and Nonlinear Physics [PHYS.PHYS.PHYS-GEN-PH]Physics [physics]/Physics [physics]/General Physics [physics.gen-ph] regularization electromagnetic High Energy Physics - Theory (hep-th) Differential Geometry (math.DG) gravitation Regularization (physics) adiabatic curvature spectral quantization Asymptotic expansion Laplace operator |
| Zdroj: | Commun.Math.Phys. Commun.Math.Phys., 2017, 349 (3), pp.819-855. ⟨10.1007/s00220-016-2789-2⟩ Commun.Math.Phys., 2017, 349 (3), pp.819-855. 〈10.1007/s00220-016-2789-2〉 |
| Popis: | We study the generating functional, the adiabatic curvature and the adiabatic phase for the integer quantum Hall effect (QHE) on a compact Riemann surface. For the generating functional we derive its asymptotic expansion for the large flux of the magnetic field, i.e., for the large degree $k$ of the positive Hermitian line bundle $L^k$. The expansion consists of the anomalous and exact terms. The anomalous terms are the leading terms of the expansion. This part is responsible for the quantization of the adiabatic transport coefficients in QHE. We then identify the non-local (anomalous) part of the expansion with the Quillen metric on the determinant line bundle, and the subleading exact part with the asymptotics of the regularized spectral determinant of the Laplacian for the line bundle $L^k$, at large $k$. Finally, we show how the generating functional of the integer QHE is related to the gauge and gravitational (2+1)d Chern-Simons functionals. We observe the relation between the Bismut-Gillet-Soul\'e curvature formula for the Quillen metric and the adiabatic curvature for the electromagnetic and geometric adiabatic transport of the integer Quantum Hall state. Then we relate the adiabatic phase in QHE to the eta invariant and show that the geometric part of the adiabatic phase is given by the Chern-Simons functional. Comment: 36 pages, v4: greatly expanded version, added: references, Sec. 1.1 and Appendix A with background material, examples in Sec. 2.3 and Sec. 4, Thm. 3 and Sec. 5 expanded with more details on the relation between the adiabatic phase, eta invariant and Chern-Simons functional. To appear in Commun. Math. Phys |
| Databáze: | OpenAIRE |
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