Abstract
We discuss several quantum lattice systems and show that there may arise chaotic structures in the form of incommensurate or irregular quantum states. As a first example we consider a tight-binding model in which a single electron is strongly coupled with phonons on a one-dimensional (1D) chain of atoms. In the adiabatic approximation the system is described by a discrete nonlinear Schrödinger equation. We have reformulated this equation to the form of a two-dimensional (2D) mapping. By doing this we may investigate a quantum problem in terms usually applied to classical nonlinear dynamic problems. We find three types of solutions: periodic, quasiperiodic and chaotic. The first one are periodic solutions associated with localized deformations of the lattice, like Peierls’ charge-density waves or lattice solitons. In the two latter cases the periodicity of solitons breaks down, i.e., the distance between two nearest solitons deviates slightly from the period, that changes randomly, i.e. is unpredictable and therefore chaotic. Thus, we show that the wave function of an electron on a deformable lattice may exhibit incommensurate and irregular structures, analogous to structures arising in classical chaos. We also discuss some other many-body systems including a Kagomé antiferromagnet, where the ground state may have incommensurate structure.
Preview
Unable to display preview. Download preview PDF.
References
Ashcroft N.W. and Mermin N.D., Solid State Physics, (Cornell University, 1976).
Bernu B., Lhuillier C., Lecheminant P., and Zindzingre P., Phys. Rev. Lett. 1997, submitted.
Callaway J., Quantum Theory of the Solid State, (Academic Press, NY, 1976).
Christov C.I. and Nicolis G., Physica A 228, 326 (1996).
Duffing G., Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung (Vieweg & Sohn, Braunschweig, 1918).
Emin D, Phys. Today, 35, 111 (1985); Phys. Rev. Lett. 62, 1544 (1989).
Gernoth, K.A., Clark, J.W., Senger, G., and Ristig, M.L.: Phys. Rev. B 49, 15836 (1994).
Holstein H., Ann. Phys. (N. Y.) 8, 343 (1959).
Kürten K.E. and Nicolis G., Physica A (1997).
Kusmartsev F.V. and Rashba E.I., Sov. Phys.-JETP 59, 668 (1984).
Luther A., Timonen J. and Pokrovsky V., in Phase Transitions in Surface Films, eds. J.G. Dash and J. Ruvalds, (Plenum, New York, 1980).
Pekar S.I., Untersuchungen über die Elektronentheorie der Kristalle, (Akademie Verlag, Berlin, 1954).
Rashba E.I., Opt. Spectr. 2, 88 (1957).
Rashba E.I., in Excitons, ed. by E.I. Rashba and M.D. Sturge, (North-Holland, Amsterdam, 1982) p. 543.
Ristig, M.L. and Kim, J.W.: Phys. Rev. B 40, 6665 (1996).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag
About this paper
Cite this paper
Kusmartsev, F.V., Kürten, K.E. (1997). Effects of chaos in quantum lattice systems. In: Clark, J.W., Ristig, M.L. (eds) Theory of Spin Lattices and Lattice Gauge Models. Lecture Notes in Physics, vol 494. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0104309
Download citation
DOI: https://doi.org/10.1007/BFb0104309
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63207-8
Online ISBN: 978-3-540-69211-9
eBook Packages: Springer Book Archive