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Quantum Hall Effect and Quillen Metric

Abstract

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-Soulé curvature formula for the Quillen metric and the adiabatic curvature for the electromagnetic and geometric adiabatic transport of the integer Quantum Hall state. We then obtain the geometric part of the adiabatic phase in QHE, given by the Chern–Simons functional.

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References

  1. Abanov, A.G., Gromov, A.: Electromagnetic and gravitational responses of two-dimensional non-interacting electrons in background magnetic field. Phys. Rev. B. 90, 014435 (2014). arXiv:1401.3703 [cond-mat.str-el]

  2. Alvarez-Gaume L., Moore G., Vafa C.: Theta functions, modular invariance, and strings. Commun. Math. Phys. 106, 1–40 (1986)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  3. Atiyah M.F., Patodi V.K., Singer I.M.: Spectral asymmetry and Riemannian geometry. II. Bull Lond. Math. Soc. 5, 229–234 (1973)

    Article  MATH  Google Scholar 

  4. Atiyah M.F., Singer I.M.: The index of elliptic operators. IV. Ann. Math. (2) 93, 119–138 (1971)

    MathSciNet  Article  MATH  Google Scholar 

  5. Avron J.E., Seiler R.: Quantization of the Hall conductance for general, multiparticle Schrödinger hamiltonians. Phys. Rev. Lett. 54, 259 (1985)

    ADS  MathSciNet  Article  Google Scholar 

  6. Avron J.E., Seiler R., Zograf P.G.: Adiabatic quantum transport: quantization and fluctuations. Phys. Rev. Lett. 73(24), 3255–3257 (1994)

    ADS  Article  Google Scholar 

  7. Avron, J.E., Seiler, R., Zograf, P.G.: Viscosity of quantum Hall fluids. Phys. Rev. Lett. 75(4), 697–700 (1995). arXiv:cond-mat/9502011

  8. Belavin A., Knizhnik V.: Algebraic geometry and the geometry of quantum strings. Phys. Lett. B 168(3), 201–206 (1986)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  9. Belavin A., Knizhnik V.: Complex geometry and the theory of quantum strings. Sov. Phys. JETP 64(2), 215–228 (1986)

    MathSciNet  MATH  Google Scholar 

  10. Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften, Vol. 298, pp. viii+369. Springer-Verlag, Berlin (1992)

  11. Berman, R.: Kähler–Einstein metrics emerging from free fermions and statistical mechanics. JHEP. 10, 106 (2011). arXiv:1009.2942 [hep-th]

  12. Berman, R.: Determinantal point processes and fermions on complex manifolds: large deviations and bosonization. Commun. Math. Phys. 327, 1–47 (2014). arXiv:0812.4224 [math.CV]

  13. Berthomieu A.: Analytic torsion of all vector bundles over an elliptic curve. J. Math. Phys. 42(9), 4466–4487 (2001)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  14. Bismut J.-M.: The Atiyah-Singer Index Theorem for families of Dirac operators: two heat equation proofs. Invent. Math. 83, 91–151 (1986)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  15. Bismut J.-M., Bost J.-B.: Fibrés déterminants, métriques de Quillen et dégénérescence des courbes. Acta Math. 165(1-2), 1–103 (1990)

    MathSciNet  Article  Google Scholar 

  16. Bismut J.-M., Cheeger J.: \({\eta}\) -invariants and their adiabatic limits. J. Am. Math. Soc. 2(1), 33–70 (1989)

    MathSciNet  MATH  Google Scholar 

  17. Bismut J.-M., Freed D.: The analysis of elliptic families. I.. Commun. Math. Phys. 106(1), 159–176 (1986)

    ADS  Article  MATH  Google Scholar 

  18. Bismut J.-M., Freed D.: The analysis of elliptic families. II. Commun. Math. Phys. 107(1), 103–163 (1987)

    ADS  Article  MATH  Google Scholar 

  19. Bismut J.-M., Gillet H., Soulé C.: Analytic torsion and holomorphic determinant bundles. II. Direct images and Bott-Chern forms. Commun. Math. Phys. 115(1), 79–126 (1988)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  20. Bismut J.-M., Gillet H., Soulé C.: Analytic torsion and holomorphic determinant bundles. III. Quillen metrics on holomorphic determinants. Commun. Math. Phys. 115(2), 301–351 (1988)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  21. Bismut J.-M., Köhler K.: Higher analytic torsion forms for direct images and anomaly formulas. J. Algebraic Geom. 1(4), 647–684 (1992)

    MathSciNet  MATH  Google Scholar 

  22. Bismut J.-M., Vasserot E.: The asymptotics of the Ray–Singer analytic torsion associated with high powers of a positive line bundle. Commun. Math. Phys. 125, 355–367 (1989)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  23. Bost J.-B.: Intrinsic heights of stable varieties and abelian varieties. Duke Math. J. 82(1), 21–70 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  24. Bost J.-B., Jolicœur T.: A holomorphy property and the critical dimension in string theory from an index theorem. Nucl. Phys. B 286, 175–188 (1987)

    Article  Google Scholar 

  25. Bradlyn, B., Read, N.: Low-energy effective theory in the bulk for transport in a topological phase. Phys. Rev. B 91, 125303 (2015). arXiv:1407.2911 [cond-mat.mes-hall]

  26. Bradlyn, B., Read, N.: Topological central charge from Berry curvature: Gravitational anomalies in trial wave functions for topological phases. Phys. Rev. B 91, 165306 [cond-mat.mes-hall] (2015). arXiv:1502.04126

  27. Can, T., Laskin, M., Wiegmann, P.: Fractional quantum Hall effect in a curved space: gravitational anomaly and electromagnetic response. Phys. Rev. Lett. 113, 046803 (2014). arXiv:1402.1531 [cond-mat.str-el]

  28. Can, T., Laskin, M., Wiegmann, P.: Geometry of quantum Hall states: gravitational anomaly and transport coefficients. Ann. Phys. 362, 752–794 (2015). arXiv:1411.3105 [cond-mat.str-el]

  29. Catlin, D.: The Bergman kernel and a theorem of Tian, analysis and geometry in several complex variables (Katata, 1997), pp. 1–23. Trends Math., Birkhäuser Boston, Boston (1999)

  30. Dai X.: Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence. J. Am. Math. Soc. 4, 265–321 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  31. D’Hoker E., Phong D.H.: On determinants of Laplacians on Riemann surfaces. Commun. Math. Phys. 104, 537–545 (1986)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  32. D’Hoker E., Phong D.H.: The geometry of string perturbation theory. Rev. Mod. Phys. 60, 917 (1988)

    ADS  MathSciNet  Article  Google Scholar 

  33. Donaldson, S.K.: Scalar curvature and projective embeddings. II. Q. J. Math. 56(3), 345–356 (2005). arXiv:math/0407534 [math.DG]

  34. Douglas, M.R., Klevtsov, S.: Bergman kernel from path integral. Commun. Math. Phys. 293(1), 205–230 (2010). arXiv:0808.2451 [hep-th]

  35. Fay, J.: Kernel functions, analytic torsion and moduli spaces. Memoirs of AMS, Vol. 96 no. 464, Providence RI (1992)

  36. Ferrari, F., Klevtsov, S.: FQHE on curved backgrounds, free fields and large N. JHEP. 12, 086 (2014). arXiv:1410.6802 [hep-th]

  37. Ferrari, F., Klevtsov, S., Zelditch, S.: Gravitational actions in two dimensions and the Mabuchi functional, Nucl. Phys. B. 859(3), 341–369 (2012). arXiv:1112.1352 [hep-th]

  38. Forrester P.J.: Log-gases and random matrices. Princeton University Press, Princeton (2010)

    MATH  Google Scholar 

  39. Fröhlich J., Studer U.M.: \({U(1)\times SU(2)}\) -gauge invariance of non-relativistic quantum mechanics, and generalized Hall effects. Commun. Math. Phys. 148, 553–600 (1992)

    ADS  Article  MATH  Google Scholar 

  40. Griffiths P., Harris J.: Principles of Algebraic Geometry. Wiley, New York (1978)

    MATH  Google Scholar 

  41. Gromov, A., Abanov, A.G.: Density-curvature response and gravitational anomaly. Phys. Rev. Lett. 113, 266802 (2014). arXiv:1403.5809 [cond-mat.str-el]

  42. Gromov, A., Cho, G.Y., You, Y., Abanov, A.G., Fradkin, E.: Framing anomaly in the effective theory of fractional quantum Hall effect. Phys. Rev. Lett. 114, 016805 (2015). arXiv:1410.6812 [cond-mat.str-el]

  43. Kirby, R.: The topology of 4-manifolds. Lecture Notes in Mathematics, Vol. 1374, pp. 108. Springer-Verlag, Berlin (1989)

  44. Klevtsov, S.: Random normal matrices, Bergman kernel and projective embeddings. JHEP. 1401, 133 (2014). arXiv:1309.7333 [hep-th]

  45. Klevtsov, S., Wiegmann, P.: Geometric adiabatic transport in Quantum Hall states. Phys. Rev. Lett. 115, 086801 (2015). arXiv:1504.07198 [cond-mat.str-el]

  46. Knudsen F., Mumford D.: The projectivity of the moduli space of stable curves. I. Preliminaries on ‘det’ and ‘Div’. Math. Scand. 39, 19–55 (1976)

    Article  MATH  Google Scholar 

  47. Köhler K.: Holomorphic torsion on Hermitian symmetric spaces. J. Reine Angew. Math. 460, 93–116 (1995)

    MathSciNet  MATH  Google Scholar 

  48. Laskin, M., Can, T., Wiegmann, P.: Collective field theory for quantum Hall states. Phys. Rev. B, 92, 235141 (2015). arXiv:1412.8716 [cond-mat.str-el]

  49. Laughlin R.B.: Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50(18), 1395 (1983)

    ADS  Article  Google Scholar 

  50. Lévay P.: Berry phases for Landau Hamiltonians on deformed tori. J. Math. Phys. 36, 2792–2802 (1995)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  51. Lévay P.: Berry’s phase, chaos, and the deformations of Riemann surfaces. Phys. Rev. E 56(5), 6173–6176 (1997)

    ADS  MathSciNet  Article  Google Scholar 

  52. Lu Z.: On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch Amer. J. Math. 122(2), 235–273 (2000)

    Article  MATH  Google Scholar 

  53. Ma, X., Marinescu, G.: Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, 254, pp. xiv+422,. Birkhäuser Verlag, Basel (2007)

  54. Ma, X., Marinescu, G.: Berezin-Toeplitz quantization on Kähler manifolds. J. Reine Angew. Math. 662, 1–56 (2012). arXiv:1009.4405 [math.DG]

  55. Mumford D.: Tata lectures on theta I. Birkhäuser, Boston (1983)

    Book  MATH  Google Scholar 

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

    ADS  MathSciNet  Article  Google Scholar 

  57. Polyakov A.M.: Quantum gravity in two dimensions. Mod. Phys. Lett. A 2(11), 893–898 (1987)

    ADS  MathSciNet  Article  Google Scholar 

  58. Quillen D.: Determinants of Cauchy–Riemann operators over a Riemann surface. Funct. Anal. Appl. 19(1), 37–41 (1985)

    Article  MATH  Google Scholar 

  59. Ray D.B., Singer I.M.: Analytic torsion for complex manifolds. Ann. Math. (2) 98, 154–177 (1973)

    MathSciNet  Article  MATH  Google Scholar 

  60. Read, N.: Non-Abelian adiabatic statistics and Hall viscosity in quantum Hall states and \({p_x+ip_y}\) paired superfluids. Phys. Rev. B. 79(4), 045308 (2009). arXiv:0805.2507 [cond-mat.mes-hall]

  61. Read, N., Rezayi, E.H.: Hall viscosity, orbital spin, and geometry: paired superfluids and quantum Hall systems. Phys. Rev. B. 84(4), 085316 (2009). arXiv:1008.0210 [cond-mat.mes-hall]

  62. Simon B.: Holonomy, the quantum adiabatic theorem, and Berry’s phase. Phys. Rev. Lett. 51, 2167 (1983)

    ADS  MathSciNet  Article  Google Scholar 

  63. Son, D.T.: Newton-Cartan Geometry and the Quantum Hall Effect. arXiv:1306.0638 [cond-mat.mes-hall]

  64. Tao R., Wu Y.-S.: Gauge invariance and fractional quantum Hall effect. Phys. Rev. B 30, 1097 (1984)

    ADS  Article  Google Scholar 

  65. Tejero Prieto C.: Fourier-Mukai transform and adiabatic curvature of spectral bundles for Landau Hamiltonians on Riemann surfaces. Commun. Math. Phys. 265(2), 373–396 (2006)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  66. Thouless D.J., Kohmoto M., Nightingale M.P., den Nijs M.: Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405 (1982)

    ADS  Article  Google Scholar 

  67. Tokatly, I.V., Vignale, G.: Lorentz shear modulus of a two-dimensional electron gas at high magnetic field. Phys. Rev. B. 76, 161305 (2007). arXiv:0706.2454 [cond-mat.mes-hall]

  68. Tokatly, I., Vignale, G.: Lorentz shear modulus of fractional quantum Hall states. J. Phys. C. 21, 275603 (2009). arXiv:0812.4331 [cond-mat.mes-hall]

  69. Verlinde E.P., Verlinde H.L.: Chiral bosonization, determinants and the string partition function. Nucl. Phys. B 288, 357–396 (1987)

    ADS  MathSciNet  Article  Google Scholar 

  70. Wen X.G., Zee A.: Shift and spin vector: New topological quantum numbers for the Hall fluids. Phys. Rev. Lett. 69, 953 (1992)

    ADS  Article  Google Scholar 

  71. Weng L.: Regularized determinants of Laplacians for Hermitian line bundles over projective spaces. J. Math. Kyoto Univ. 35(3), 341–355 (1995)

    MathSciNet  MATH  Google Scholar 

  72. Witten E.: Global gravitational anomalies. Comm. Math. Phys. 100(2), 197–229 (1985)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  73. Witten, E.: \({SL(2,\mathbb{Z})}\) action on 3-dimensional conformal field theories with abelian symmetry. From fields to strings: circumnavigating theoretical physics, Vol. 2, pp. 1173–1200. World Sci. Publ., Singapore (2005)

  74. Witten, E.: Fermion path integrals and topological phases. Rev. Mod. Phys. 88, 35001 (2016). arXiv:1508.04715 [cond-mat.mes-hall]

  75. Zabrodin, A., Wiegmann, P.: Large N expansion for the 2D Dyson gas. J. Phys. A. 39, 8933–8963 (2006). arXiv:hep-th/0601009

  76. Zelditch, S.: Szegő kernels and a theorem of Tian. IMRN. 1998(6), 317–331 (1998). arXiv:math-ph/0002009

  77. Zograf, P.G., Takhtadzhyan, L.A.: A local index theorem for families of \({\bar\partial}\) -operators on Riemann surfaces, Uspekhi Mat. Nauk 42(6)(258), 133–150 (1987) (Russian); English translation in Russian Math. Surveys 42:169–190

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Correspondence to George Marinescu.

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Klevtsov, S., Ma, X., Marinescu, G. et al. Quantum Hall Effect and Quillen Metric. Commun. Math. Phys. 349, 819–855 (2017). https://doi.org/10.1007/s00220-016-2789-2

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