Abstract
We give an upper bound on the modulus of the ground-state overlap of two non-interacting fermionic quantum systems with N particles in a large but finite volume L d of d-dimensional Euclidean space. The underlying one-particle Hamiltonians of the two systems are standard Schrödinger operators that differ by a non-negative compactly supported scalar potential. In the thermodynamic limit, the bound exhibits an asymptotic power-law decay in the system size L, showing that the ground-state overlap vanishes for macroscopic systems. The decay exponent can be interpreted in terms of the total scattering cross section averaged over all incident directions. The result confirms and generalises P. W. Anderson’s informal computation (Phys. Rev. Lett. 18:1049–1051, 1967).
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References
Anderson P.W.: Infrared catastrophe in Fermi gases with local scattering potentials. Phys. Rev. Lett. 18, 1049–1051 (1967)
Anderson P.W.: Ground state of a magnetic impurity in a metal. Phys. Rev. 164, 352–359 (1967)
Bauer H.: Measure and integration theory. de Gruyter, Berlin (2001)
Basor E.L., Chen Y.: The X-ray problem revisited. J. Phys. A 36, L175–L180 (2003)
Birman, M.Š., Èntina, S.B.: The stationary method in the abstract theory of scattering. Math. USSR Izv. 1, 391–420 (1967) [Russian original: Izv. Akad. Nauk SSSR Ser. Mat. 31, 401–430 (1967)]
Broderix K., Hundertmark D., Leschke H.: Continuity properties of Schrödinger semigroups with magnetic fields. Rev. Math. Phys. 12, 181–225 (2000)
Elstrodt J.: Maß- und Integrationstheorie. Springer, Berlin (2005)
Frank R.L., Lewin M., Lieb E.H., Seiringer R.: Energy cost to make a hole in the Fermi sea. Phys. Rev. Lett. 106, 150402 (2011)
Germinet F., Klein A.: Operator kernel estimates for functions of generalized Schrödinger operators. Proc. Amer. Math. Soc. 131, 911–920 (2003)
Hamann D.R.: Orthogonality catastrophe in metals. Phys. Rev. Lett. 26, 1030–1032 (1971)
Hentschel M., Guinea F.: Orthogonality catastrophe and Kondo effect in graphene. Phys. Rev. B 76, 115407 (2007)
Heyl M., Kehrein S.: Crooks relation in optical spectra: Universality in work distributions for weak local quenches. Phys. Rev. Lett. 108, 190601 (2012)
Heyl M., Kehrein S.: X-ray edge singularity in optical spectra of quantum dots. Phys. Rev. B 85, 155413 (2012)
Hislop P.D., Müller P.: The spectral shift function for compactly supported perturbations of Schrödinger operators on large bounded domains. Proc. Amer. Math. Soc. 138, 2141–2150 (2010)
Hunziker W., Sigal I.M.: The quantum N-body problem. J. Math. Phys. 41, 3448–3510 (2000)
Helmes R.W., Sindel M., Borda L., von Delft J.: Absorption and emission in quantum dots: Fermi surface effects of Anderson excitons. Phys. Rev. B 72, 125301 (2005)
Hentschel M., Ullmo D., Baranger H.U.: Fermi edge singularities in the mesoscopic regime: Anderson orthogonality catastrophe. Phys. Rev. B 72, 035310 (2005)
Kirsch W.: Small perturbations and the eigenvalues of the Laplacian on large bounded domains. Proc. Amer. Math. Soc. 101, 509–512 (1987)
Küttler, H., Otte, P., Spitzer, W.: Anderson’s orthogonality catastrophe for one-dimensional systems, e-print arXiv:1301.4923, Ann. Henri Poincaré. doi:10.1007/s00023-013-0287-z
Kaga H., Yosida K.: Orthogonality catastrophe due to local electron correlation. Prog. Theor. Phys. 59, 34–39 (1978)
Nozières P., de Dominicis C.T.: Singularities in the X-ray absorption and emission of metals. iii. one-body theory exact solution. Phys. Rev. 178, 1097–1107 (1969)
Ohtaka K., Tanabe Y.: Theory of the soft-X-ray edge problem in simple metals: historical survey and recent developments. Rev. Mod. Phys. 62, 929–991 (1990)
Otte P.: An adiabatic theorem for section determinants of spectral projections. Math. Nachr. 278, 470–484 (2005)
Röder G., Hentschel M.: Orthogonality catastrophe in ballistic quantum dots: Role of level degeneracies and confinement geometry. Phys. Rev. B 82, 125312 (2010)
Rivier N., Simanek E.: Exact calculation of the orthogonality catastrophe in metals. Phys. Rev. Lett. 26, 435–438 (1971)
Reed M., Simon B.: Methods of modern mathematical physics III. Academic Press, New York (1979)
Simon B.: Schrödinger semigroups. Bull. Amer. Math. Soc. (N.S.) 7 447–526 (1982)
Simon, B.: Trace ideals and their applications, Mathematical Surveys and Monographs, vol. 120, 2nd ed. American Mathematical Society, Providence (2005)
Stein E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton (1993)
Stollmann P.: Caught by disorder: bound states in random media, Progress in Mathematical Physics, vol. 20. Birkhäuser, Boston (2001)
Türeci, H.E., Hanl, M., Claassen, M., Weichselbaum, A., Hecht, T., Braunecker, B., Govorov, A., Glazman, L., Imamoglu, A., von Delft, J.: Many-body dynamics of exciton creation in a quantum dot by optical absorption: A quantum quench towards Kondo correlations. Phys. Rev. Lett. 106, 107402 (2011)
Weidmann, J.: Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Springer, New York (1980)
Yafaev D.: Scattering theory: some old and new problems, Lecture Notes in Mathematics, vol. 1735. Springer, Berlin (2000)
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Communicated by B. Simon
Work supported by Sfb/Tr 12 of the German Research Council (Dfg).
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Gebert, M., Küttler, H. & Müller, P. Anderson’s Orthogonality Catastrophe. Commun. Math. Phys. 329, 979–998 (2014). https://doi.org/10.1007/s00220-014-1914-3
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DOI: https://doi.org/10.1007/s00220-014-1914-3