Many models for structures in condensed matter can be associated with Hamiltonian dynamical systems with discrete time. This connection is due to the fact that both models are defined by the minimisation or the extremalisation of the same variational form called "free energy" in the first case and "action" in the second case. Thus, the results obtained for the first class of problems turn out to have applications for the second class and vice-versa, but however, with a physical interpretation which is totally different. For example, the breaking of the KAM tori and the occurrence of chaos in the standard map turns out to correspond to a pinning transition and the occurrence of chaotic metastable states in the associated Frenkel-Kontorowa models. The anti-integrable limit for structural problems is a very natural limit where the "atoms" of the structure become disconnected. It corresponds to a highly singular limit for the associated dynamical system which up to now did not focus much attention. The associated dynamical system becomes undeterministic and just reduces to a Bernoulli shift. Nevertheless, a perturbation theorem can be established at this limit which proves the persistence of chaotic trajectories when the dynamical system returns to be deterministic. This result is extended to a large class of dynamical systems with discrete time including non-Hamiltonian systems. An anti-integrable limit can also be found in the adiabatic Holstein model describing electrons coupled to phonons at several dimensions and some extensions which are not apparently connected to any dynamical system. Then the anti-integrable limit is obtained when the electronic kinetic energy vanishes. Treating this kinetic energy in an exact perturbation theory, allows one to prove new results concerning the existence of bipolaronic, polaronic and mixed polaronic-bipolaronic insulators. The possible extension of the KAM theory to the small electron-phonon coupling regime and especially in the one-dimensional adiabatic Holstein model is still an open problem. A similar anti-integrable limit is also found in the Holstein-Hubbard model involving in addition an on-site electron-electron repulsion. Some partial but exact results were recently obtained in that model proving the persistance of bipolaronic, polaronic magnetic and mixed structures with a non-vanishing electronic kinetic energy while this Hubbard term is rigorously taken into account. Another family of open problems concerns dynamical systems with a continuous time such as arrays of coupled nonlinear oscillators. Such models can be useful for understanding the transport of the vibrational energy in nonlinear solids such as quasi-crystals, in polymers or in DNA chains (etc.). The anti-integrable limit corresponds to the uncoupled limit when the oscillators oscillate independently with random amplitudes, periods and phase. Can these solutions or some of these solutions be continued at non-zero coupling? R. MacKay recently proved that the anti-integrable solution corresponding to a single oscillator moving can be continued as a localised breather solution when the coupling constant is not too large. © 1994.
Aubry, S. (1994). The concept of anti-integrability applied to dynamical systems and to structural and electronic models in condensed matter physics. Physica D: Nonlinear Phenomena, 71(1–2), 196–221. https://doi.org/10.1016/0167-2789(94)90190-2