Serge Aubry

Breathers in nonlinear lattices: existence, linear stability and quantization

Abstract It has been proved by the principle of anti-integrable limit (renamed here anti-continuous limit) and the implicit function theorem that, under rather weak hypotheses, there exist spatially localized and time periodic solutions (“breathers”) in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong. The models can be translationally invariant or not and be of any dimension. There are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions (“multibreathers”). The whole set of these solutions can be labeled by a discrete coding sequence. The condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called “phonobreather” which correspond to special codes (which can be also spatially chaotic) and where no oscillator is at rest. These results extend almost without change to models with coupled rotors. The existence of “rotobreathers” which are exact solutions consisting of a single rotor rotating while the other rotors oscillate, is proved. There are also multirotobreather solutions and “phono-rotobreathers”, etc. It is also shown that the phase of the single breathers constituting a multibreather (or multirotobreather) solution can be submitted to a “phase torsion”. These new dynamical structures carry a nonzero energy flow. In models with two dimensions and more, the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow. The dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti-continuous limit. The existence of two kinds of “polarobreathers”, which are spatially exponentially localized dynamical solutions, is proved for a small enough electronic transfer integral. The polarobreathers of the first kind include the standard static polaron. The lattice configuration is static (time independent) when the electron is in an excited state (i.e. the electronic oscillator oscillates periodically). They can be spatially chaotic. A polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation. The most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site. The standard Floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the Newton operator involved in the implicit function theorem. The Krein theory of bifurcations is reinterpreted from this point of view. The linear stability of the single breather (and rotobreather) solutions is then proved at least at small enough coupling. Finally, it is shown that in the quantum version of translationally invariant classical models (where there cannot exist in principle any strictly localized excitation) there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers. The bandwidth of these excitations goes exponentially to zero with the number of bound phonons. This theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist. These numerical applications are currently written elsewhere.

Chaotic trajectories in the standard map. The concept of anti-integrability

A rigorous proof is given in the standard map (associated with a Frenkel-Kontorowa model) for the existence of chaotic trajectories with unbounded momenta for large enough coupling constant k >k0. These chaotic trajectories (with finite entropy per site) are coded by integer sequences {mi} such that the sequence bi = |mi+1 + mi−1−2mi| be bounded by some integer b. The bound k0 in k depends on b and can be lowered for coding sequences {mi} fulfilling more restrictive conditions. The obtained chaotic trajectories correspond to stationary configurations of the Frenkel-Kontorowa model with a finite (non-zero) photon gap (called gap parameter in dimensionless units). This property implies that the trajectory (or the configuration {ui}) can be uniquely continued as a uniformly continuous function of the model parameter k in some neighborhood of the initial configuration. A non-zero gap parameter implies that the Lyapunov coefficient is strictly positive (when it is defined). In addition, the existence of dilating and contracting manifolds is proven for these chaotic trajectories. “Exotic” trajectories such as ballistic trajectories are also proven to exist as a consequence of these theorems. The concept of anti-integrability emerges from these theorems. In the anti-integrable limit which can be only defined for a discrete time dynamical system, the coordinates of the trajectory at time i do not depend on the coordinates at time i - 1. Thus, at this singular limit, the existence of chaotic trajectories is trivial and the dynamical system reduces to a Bernoulli shift. It is well known that the KAM tori of symplectic dynamical originates by continuity from the invariant tori which exists in the integrible limit (under certain conditions). In a similar way, it appears that the chaotic trajectories of dynamical systems originate by continuity from those which exists at the anti-integrable limits (also under certain conditions).

Mobility and reactivity of discrete breathers

Abstract Breathers may be mobile close to an instability threshold where the frequency of a pinning mode vanishes. The translation mode is a marginal mode that is a solution of the linearized (Hill) equation of the breather which grows linearly in time. In some cases, there are exact mobile breather solutions (found numerically), but these solutions have an infinitely extended tail which shows that the breather motion is nonradiative only when it moves (in equilibrium) with a particular phonon field. More generally, at any instability threshold, there is a marginal mode. There are situations where excitations by marginal modes produce new type of behaviors such as the fission of a breather. We may also have fusion. This approach suggests that breathers (which can be viewed as clusters of phonons) may react by themselves or with each other as well as in chemistry with atoms and molecules, or in nuclear physics with nuclei.

Anti-integrability in dynamic and variational problems

Abstract We show using a series of examples that the existence of a singular limit called “anti-integrable” can be useful for proving the existence of chaotic solutions and elucidating some of their properties leading to some new and unexpected results. In particular, the existence proof of chaotic trajectories in large classes of dynamical systems with discrete time, becomes straightforward. The method applies even for maps in infinite dimension, symplectic or not. Similar ideas can be used for proving the existence of breathers (or self-localised modes) and of chaotic distributions of breathers in any array of weakly coupled non linear oscillators (with continuous time). The existence of insulating bipolaronic structures in electron-phonon systems (electronic self-localisation) at large electron-phonon coupling can also be proven on the same basis.

Growth and decay of discrete nonlinear Schrödinger breathers interacting with internal modes or standing-wave phonons

We investigate the long-time evolution of weakly perturbed single-site breathers (localized stationary states) in the discrete nonlinear Schrodinger equation. The perturbations we consider correspond to time-periodic solutions of the linearized equations around the breather, and can be either (i) spatially localized or (ii) spatially extended. For case (i), which corresponds to the excitation of an internal mode of the breather, we find that the nonlinear interaction between the breather and its internal mode always leads to a slow growth of the breather amplitude and frequency. In case (ii), corresponding to interaction between the breather and a standing-wave phonon, the breather will grow provided that the wave vector of the phonon is such that the generation of radiating higher harmonics at the breather is possible. In other cases, breather decay is observed. This condition yields a limit value for the breather frequency above which no further growth is possible. We also discuss another mechanism for breather growth and destruction which becomes important when the amplitude of the perturbation is non-negligible, and which originates from the oscillatory instabilities of the nonlinear standing-wave phonons.

Standing wave instabilities in a chain of nonlinear coupled oscillators

We consider existence and stability properties of nonlinear spatially periodic or quasiperiodic standing waves (SWs) in one-dimensional lattices of coupled anharmonic oscillators. Specifically, we consider Klein-Gordon (KG) chains with either soft (e.g., Morse) or hard (e.g., quartic) on-site potentials, as well as discrete nonlinear Schrodinger (DNLS) chains approximating the small-amplitude dynamics of KG chains with weak inter-site coupling. The SWs are constructed as exact time-periodic multibreather solutions from the anticontinuous limit of uncoupled oscillators. In the validity regime of the DNLS approximation these solutions can be continued into the linear phonon band, where they merge into standard harmonic SWs. For SWs with incommensurate wave vectors, this continuation is associated with an inverse transition by breaking of analyticity. When the DNLS approximation is not valid, the continuation may be interrupted by bifurcations associated with resonances with higher harmonics of the SW. Concerning the stability, we identify one class of SWs which are always linearly stable close to the anticontinuous limit. However, approaching the linear limit all SWs with non-trivial wave vectors become unstable through oscillatory instabilities, persisting for arbitrarily small amplitudes in infinite lattices. Investigating the dynamics resulting from these instabilities, we find two qualitatively different regimes for wave vectors smaller than or larger than π/2, respectively. In one regime persisting breathers are found, while in the other regime the system rapidly thermalizes.

Equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for Frenkel-Kontorova models

There is a well-known correspondence between the dynamics of symplectic twist maps which represent an important class of Hamiltonian systems and the equilibrium states of a class of variational problems in solid-state physics, known as Frenkel-Kontorova models. In this paper it is shown that the key concepts of uniform hyperbolicity in the first context and phonon gap in the second context are equivalent. This allows one to transfer many ideas between the two and hence to deduce new results. For example, we prove the uniform hyperbolicity of certain invariant sets for symplectic twist maps constructed from so-called non-degenerate anti-integrable limits by Aubry and Abramovici, using the concept of phonon gap and deduce that they have measure zero; we prove existence of many multi-defect equilibrium states for a Frenkel-Kontorova system if it has a single non-degenerate defect equilibrium state, via standard results in the theory of uniform hyperbolicity.

Finite size effects on instabilities of discrete breathers

Abstract When the two arcs of the continuous phonon spectrum of the Floquet matrix of a discrete breather overlap on the unit circle, the breather solution in the infinite lattice might be stable while the corresponding solutions in finite systems appear to be unstable. More precisely, when the model parameters vary, the breather in the finite system exhibits a large number of collisions between the Floquet eigenvalues belonging to the phonon spectrum. These collisions correspond to complex cascades of instability thresholds followed near after by re-entrant stability thresholds. We interpret this complex structure on the basis of the band analysis of the matrix of the second variation of the action. Then we can predict that in the limit of an infinite system the number of instability and stability thresholds in the cascade diverges, but simultaneously the maximum amplitude of the instabilities vanishes, so that the breather in the infinite system recovers its linear stability (as long as all its other localized modes remain stable). This is the situation which is required in Cretegny et al. [Physica D 119 (1998) 73–87] for having inelastic phonon scattering with two channels. We also analyze the size effects when a Floquet eigenvalue associated with a localized mode collides with the Floquet continuous phonon spectrum with different Krein signature. In contrast to the previous case, the infinite system is unstable after the collision.

Chaotic polaronic and bipolaronic states in the adiabatic Holstein model

A rigorous proof for the existence of bipolaronic states is given for the adiabatic Holstein model for any lattice at any dimension, periodic or not, and for an arbitrary band filling, provided that the electron-phonon coupling (in dimensionless units) is large enough. The existence of mixed polaronic-bipolaronic states is also proven, but for larger electron-phonon coupling. These states consist of arbitrary distributions of bipolarons (or of bipolarons and polarons) localized in real space which can be simply labeled by pseudospin configurations as for a lattice gas model. The theory not only applies to periodic crystals, but also to quasicrystals, amorphous structures, polymer network, etc.When these bipolaronic and mixed polaronic-bipolaronic states exist, it is proven that: (1) These bipolaronic (and mixed polaronic-bipolaronic) states exhibit a nonzero phonon gap with a nonvanishing lower bound and an electronic gap at the Fermi energy. (2) These structures are insulating. The perturbation generated by any local change in the bipolaronic or polaronic distribution or by any charged impurity or defect decays exponentially at long distance. (3) These bipolaronic (and mixed polaronic-bipolaronic) states persist for any uniform magnetic field. (4) For large enough electron-phonon coupling, the ground state of the extended adiabatic Holstein model is a bipolaronic state when there is no uniform magnetic field or when it is small enough. It becomes a mixed polaronic-bipolaronic state for large enough magnetic field (note that the mixed polaronic-bipolaronic states are magnetic).In one-dimensional models, the ground state is an incommensurate (or commensurate) charge density wave (CDW) as predicted by Peierls (this result is not rigorous, but has been confirmed numerically). It is proven that the ground state becomes a “bipolaronic charge density wave” (BCDW) at large enough electron-phonon coupling. The existence of a transition by breaking of analyticity (TBA), which was numerically observed as a function of the electron-phonon coupling, is then confirmed. In that case, the shape of the effective bipolaron can be numerically calculated. It is observed that its size diverges at the TBA. The physical properties of BCDWs are rather different from those predicted by standard charge density wave theory. Bipolaronic charge density waves can also exist in models which are not only low-dimensional, but purely two- or three-dimensional.The technique for proving these theorems is an application of the concept of anti-integrability initially developed for Hamiltonian dynamical systems. It consists in proving that the eigenstates of the (trivial) Hamiltonian (called antiintegrable) obtained by canceling all electronic and lattice kinetic terms survive as a uniformly continuous function of the electronic kinetic energy terms in the Hamiltonian up to a certain threshold.

Analytic conditions for targeted energy transfer between nonlinear oscillators or discrete breathers

It is well known that any amount of energy injected in a harmonic oscillator which is resonant and weakly coupled with a second harmonic oscillator, tunnels back and forth between these two oscillators. When the two oscillators are anharmonic, the amplitude dependence of their frequencies breaks, in general, any eventual initial resonance so that no substantial energy transfer occurs unless, exceptionally, an almost perfect resonance persists. This paper considers this interesting situation more generally between two discrete breathers belonging to two weakly coupled nonlinear systems, finite or infinite. A specific amount of energy injected as a discrete breather in a nonlinear system (donor) which is weakly coupled to another nonlinear system (acceptor) sustaining another discrete breather, might be totally transferred and oscillate back and forth between these donor and acceptor breathers. The condition is that a certain well-defined detuning function is bounded from above and below by two coupling functions. This targeted energy transfer is selective, i.e., it only occurs for an initial energy close to a specific value. The explicit calculation of these functions in complex models with numerical techniques developed earlier for discrete breathers, allows one to detect the existence of possible targeted energy transfer, between which breathers, and at which energy. It should also help for designing models having desired targeted energy transfer properties at will. We also show how extra linear resonances could make the energy transfer incomplete and irreversible. Future developments of the theory will be able to describe more spectacular effects, such as targeted energy transfer cascades and avalanches, and energy funnels. Besides rather short-term applications for artificially built devices, this theory might provide an essential clue for understanding puzzling problems of energy kinetics in real materials, chemistry, and bioenergetics.