Abstract
Quantum systems capable of chaotic motion in the classical limit can display linear, quadratic, or quartic level repulsion. For a given (time independent or time dependent) Hamiltonian the degree of level repulsion is determined by the full group of its unitary and antiunitary symmetries. We establish this connection for kicked systems, using Wigner's theory of corepresentations, and the appropriate generalization of Pechukas' phase space flow. We illustrate the theory in terms of two systems of kicked spins both of which are time reversal invariant, one showing linear and the other quadratic level repulsion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Robnik, M., Berry, M.V.: J. Phys. A19, 669 (1986)
Seligman, T.H., Verbaarschot, J.J.T.: Phys. Lett.108A, 183 (1985)
Izrailev, F.M.: Phys. Rev. Lett.56, 541 (1986)
Dyson, F.I.: J. Math. Phys. B3, 1199 (1962)
Haake, F., Kuś, M., Scharf, R.: Z. Phys. (in press)
Pechukas, P.: Phys. Rev. Lett.51, 943 (1983)
Yukawa, T.: Phys. Rev. Lett.54, 1883 (1985)
Yukawa, T.: Phys. Lett.116, 227 (1986)
This condition can, in fact, be relaxed. The dynamics to be discussed below leads to a flow periodic ink with period 16πj for halfintegerj. Ergodicity of the flow (4) can therefore not hold onk-scales that large. It may and does appear to hold at much smaller values ofk, however
Note that anyU(k) has a trivialk-dependentT invariance which is, however, physically irrelevant: the complex conjugation operatorKK} with respect to the eigenbasis ofU(k) trivially impliesKU K=U +
Wigner, E.: Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, New York: Academic Press 1959
We haveT −1 R v T=R v and thus from (23a)\(\bar d(I) = d(I),\bar d(R_x ) = - d(R_x ),\bar d(R_y ) = d(R_y ),\bar d(R_z ) = - d(R_z )\). This implies that the matrixZ commutes with σ y and anticommutes with σ x and σ z . We can therefore chooseZ=iσ y . Because ofT 2=−1 we haved(T 2)=−I 2=c z Z 2=−c z and thusc z=1>0
We haveT −1 R x T=R x, thus\(\bar d( \pm I) = \pm 1,\bar d( \pm R_x ) = \pm i;\) this is not equivalent tod(±I)=±1, d(±R x)=∓i
Indeed, fromU|φ〉=eiφ|φ〉 we get, withT 2=−1,UT|φ〉=eiφ T|φ〉 with the same eigenvalue ϕ. Moreover, |ϕ> andT|> are orthogonal since 〈φ|Tφ〉=〈Tφ|T 2φ〉*=−〈Tφ|Tφ〉=−〈φ|Tφ〉
For the first identity consider 〈φ m |R z VR −1 z |φ n+N 〉, for the second 〈φ|TUT −1|φ m 〉, and for the third 〈φ n |TR y VR −1 y |T −1|φ m 〉
Frahm, H., Mikeska, H.J.: Z. Phys. B—Condensed Matter60, 117 (1985)
Frahm, H., Mikeska, H.J.: (to be published)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kuś, M., Scharf, R. & Haake, F. Symmetry versus degree of level repulsion for kicked quantum systems. Z. Physik B - Condensed Matter 66, 129–134 (1987). https://doi.org/10.1007/BF01312770
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01312770