Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T00:45:58.879Z Has data issue: false hasContentIssue false

III - Semiclassical Approach to Spectral Correlation Functions

Published online by Cambridge University Press:  05 January 2012

By
Martin Sieber
Edited by
Jens Bolte  and
Frank Steiner
Martin Sieber
Affiliation:
University of Bristol
Jens Bolte
Affiliation:
Royal Holloway, University of London
Frank Steiner
Affiliation:
Universität Ulm, Germany
Get access

Summary

Introduction

One motivation for deriving semiclassical methods for chaotic systems like the Gutzwiller trace formula was to find a substitute for the EBK quantisation rules which provide semiclassical approximations for energy levels and wave functions in integrable systems [1]. Although the Gutzwiller trace formula allows to determine energy levels semiclassically from a knowledge of classical periodic orbits, in practise only a relatively small number of the lowest energies can be determined in this way due to the exponential proliferation of the number of long periodic orbits in chaotic systems.

However, the trace formula is very powerful in applications in which one is interested not in individual levels, but in statistical properties of the spectrum in the semiclassical regime. These spectral statistics are in the centre of interest in quantum chaos, because they are found to be universal and to provide a clear signature for chaos in the underlying classical system. All numerical evidence points to an agreement of spectral statistics in generic chaotic systems with those of random matrix theory (RMT) [2]. The aim of the semiclassical method is to find a theoretical explanation and, possibly, a proof for this connection between quantum chaos and random matrix theory. Similarly as on the quantum side, what is needed for this task is not a knowledge of individual periodic orbits, but of statistical properties of long periodic orbits.

The first steps in this direction were made in the seminal works of Hannay and Ozorio de Almeida, and Berry.

Type
Chapter
Information
Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology , pp. 121 - 142
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. M. C., Gutzwiller: Chaos in Classical and Quantum Mechanics, Springer, New York, (1990).Google Scholar
2. O., Bohigas, M. J., Giannoni and C., Schmit: Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws, Phys. Rev. Lett. 52 (1984) 1–4.Google Scholar
3. J. H., Hannay and A. M., Ozorio de Almeida: Periodic Orbits and a Correlation Function for the Semiclassical Density of States, J. Phys. A 17 (1984) 3429–3440.Google Scholar
4. M. V., Berry: Semiclassical Theory of Spectral Rigidity, Proc. R. Soc. Lond. A 400 (1985) 229–251.Google Scholar
5. N., Argaman, F., Dittes, E., Doron, J., Keating, A., Kitaev, M., Sieber and U., Smilansky: Correlations in the Actions of Periodic Orbits Derived from Quantum Chaos, Phys. Rev. Lett. 71 (1993) 4326–4329.Google Scholar
6. U., Smilansky and B., Verdene: Action Correlations and Random Matrix Theory, J. Phys. A 36 (2003) 3525–3549.Google Scholar
7. F., Haake: Quantum Signatures of Chaos, Springer, Berlin, (2000), 2nd edn.Google Scholar
8. E., Bogomolny, B., Georgeot, M. J., Giannoni and C., Schmit: Chaotic Billiards Generated by Arithmetic Groups, Phys. Rev. Lett. 69 (1992) 1477–1480.Google Scholar
9. J., Bolte, G., Steil and F., Steiner: Arithmetic Chaos and Violations of Universality in Energy Level Statistics, Phys. Rev. Lett. 69 (1992) 2188–2191.Google Scholar
10. E., Bogomolny, B., Georgeot, M. J., Giannoni and C., Schmit: Arithmetic Chaos, Phys. Rep. 291 (1997) 219–324.Google Scholar
11. B., Randol: The Length Spectrum of a Riemann Surface is Always of Unbounded Multiplicity, Proc. of the AMS 78 (19080) 455–456.
12. J., Marklof: Selberg's Trace Formula: An Introduction, this volume.
13. N., Argaman, Y., Imry and U., Smilansky: Semiclassical Analysis of Spectral Correlations in Mesoscopic Systems, Phys. Rev. B 47 (1993) 4440–4457.Google Scholar
14. R., Aurich and M., Sieber: An Exponentially Increasing Spectral Form Factor K(τ) for a Class of Strongly Chaotic Systems, J. Phys. A 27 (1994) 1967–1979.Google Scholar
15. M., Sieber and K., Richter: Correlations between Periodic Orbits and their Rôle in Spectral Statistics, Physica Scripta T90 (2001) 128–133.Google Scholar
16. M., Sieber: Leading Off-Diagonal Approximation for the Spectral Form Factor for Uniformly Hyperbolic Systems, J. Phys. A 35 (2002) L613–L619.Google Scholar
17. P. A., Braun, S., Heusler, S., Müller and F., Haake: Statistics of Self-Crossings and Avoided Crossings of Periodic Orbits in the Hadamard-Gutzwiller Model, Eur. Phys. J. B 30 (2002) 189–206.Google Scholar
18. M., Turek and K., Richter: Leading Off-Diagonal Contribution to the Spectral Form Factor of Chaotic Quantum Systems, J. Phys. A 36 (2003) L455–L462.Google Scholar
19. D., Spehner: Spectral Form Factor of Hyperbolic Systems: Leading Off-Diagonal Approximation, J. Phys. A 36 (2003) 7269–7290.Google Scholar
20. S., Müller: Classical Basis for Quantum Spectral Fluctuations in Hyperbolic Systems, Eur. Phys. J. B 34 (2003) 305–319.Google Scholar
21. S., Heusler, S., Müller, P., Braun and F., Haake: Universal Spectral Form Factor for Chaotic Dynamics, J. Phys. A 37 (2004) L31–L37.Google Scholar
22. P., Buser: Hyperbolic Geometry, Fuchsian Groups and Riemann Surfaces, this volume.
23. S., Heusler: Universal Spectral Fluctuations in the Hadamard-Gutzwiller Model and beyond, Ph.D. thesis, Universität Essen, (2003).
24. G., Berkolaiko, H., Schanz and R. S., Whitney: Leading Off-Diagonal Correction to the Form Factor of Large Graphs, Phys. Rev. Lett. 88 (2002) 104101–1–104101–4.Google Scholar
25. G., Berkolaiko, H., Schanz and R. S., Whitney: Form Factor for a Family of Quantum Graphs: An Expansion to Third Order, J. Phys. A 36 (2003) 8373–8392.Google Scholar
26. G., Berkolaiko: Form Factor for Large Quantum Graphs: Evaluating Orbits with Time-Reversal, Waves Random Media 14 (2004) S7–S27.Google Scholar
27. S., Gnutzmann and A., Altland: Universal Spectral Statistics in Quantum Graphs, Phys. Rev. Lett. 93 (2004) 194101–1 – 194101–4.Google Scholar
28. S., Gnutzmann and A., Altland: Spectral Correlations of Individual Quantum Graphs, Phys. Rev. E 72 (2005) 056215–1 – 056215–14.Google Scholar
29. K., Richter and M., Sieber: Semiclassical Theory of Chaotic Quantum Transport, Phys. Rev. Lett. 89 (2002) 206801–1 – 206801–4.Google Scholar
30. S., Heusler, S., Müller, P., Braun, and F., Haake: Semiclassical Theory of Chaotic Conductors, Phys. Rev. Lett. 96 (2006) 066804–1 – 066804-4.Google Scholar
31. P., Braun, S., Heusler, S., Müller, and F., Haake: Semiclassical Prediction for Shot Noise in Chaotic Cavities, J. Phys. A 39 (2006) L159-L165.Google Scholar
32. S., Heusler: The Semiclassical Origin of the Logarithmic Singularity in the Symplectic Form Factor, J. Phys. A 34 (2001) L483–L490.Google Scholar
33. J., Bolte and J., Harrison: The Spin Contribution to the Form Factor of Quantum Graphs, J. Phys. A 36 (2003) L433–L440.Google Scholar
34. J., Bolte and J., Harrison: The Spectral Form Factor for Quantum Graphs, in: G., Berkolaiko, R., Carlson, S., Fulling, P., Kuchment (eds.): Quantum Graphs and Their Applications, Contemporary Mathematics, vol. 415 (AMS, Providence, 2006).Google Scholar
35. T., Nagao and K., Saito: Form Factor of a Quantum Graph in a Weak Magnetic Field, Phys. Lett. A 311 (2003) 353–358.Google Scholar
36. K., Saito and T., Nagao: Spectral Form Factor for Chaotic Dynamics in a Weak Magnetic Field, Phys. Lett. A 352 (2006) 380-385.Google Scholar
37. M., Turek, D., Spehner, S., Müller, and K., Richter: Semiclassical Form Factor for Spectral and Matrix Element Fluctuations of Multidimensional Chaotic Systems, Phys. Rev. E 71 (2005) 016210–1 – 016210–15.Google Scholar
38. S., Müller, S., Heusler, P., Braun, F., Haake and A., Altland: Semiclassical Foundation of Universality in Quantum Chaos, Phys. Rev. Lett. 93 (2004) 014103–1 – 014103–4.Google Scholar
39. S., Müller, S., Heusler, P., Braun, F., Haake and A., Altland: Periodic-Orbit Theory of Universality in Quantum Chaos, Phys. Rev. E 72 (2005) 046207–1 – 046207–30.Google Scholar
40. E. B., Bogomolny and J. P., Keating: Gutzwiller's Trace Formula and Spectral Statistics: Beyond the Diagonal Approximation, Phys. Rev. Lett. 77 (1996) 1472–1475.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×