Breakdown of a conservation law in incommensurate systems

Abstract

We show that invariance properties of the Lagrangian of an incommensurate system, as described by the Frenkel-Kontorova model, imply the existence of a generalized angular momentum that is an integral of motion if the system remains floating. The behavior of this quantity can therefore monitor the character of the system as floating (when it is conserved) or locked (when it is not). We find that, during the dynamics, the nonlinear couplings of our model cause parametric phonon excitations that lead to the appearance of Umklapp terms and to a sudden deviation of the generalized momentum from a constant value, signaling a dynamical transition from a floating to a pinned state. We point out that this transition is related but does not coincide with the onset of sliding friction, which can take place when the system is still floating.

Dynamical transitions in incommensurate systems

“…Qualitatively, one can understand this because the non-analiticity of the modulation function is caused by the excitation of so many phonon modes as to render the Fourier series representing it not absolutely convergent. The conservation of the GAM only breaks down when this excitation reaches the zone boundary, generating Umklapp terms [8]. This is shown more formally in the next section.…”

“…After describing the FK model, we show that the dynamical modulation function undergoes at a critical time t c1 the same breaking of analiticity that, in the static case, occurs at λ c . A second critical time t c2 > t c1 , at which the GAM conservation stops, was identified in [8]; here, we show explicitly the connection between these two quantities, by showing analytically that the analiticity of the modulation function implies conservation of the GAM. We also present some initial results about the order of this dynamical transition.…”

“…The static version of the model is characterized by the Aubry transition [7], from a floating to a pinned state, for a critical value λ c of the substrate modulation potential. Using the undamped dynamical version of this model, we addressed in previous papers the topic of "dissipation" (in the sense of transfer of energy from the center of mass to phonon modes) in incommensurate structures: we have studied the mechanism (parametric resonances) that governs the onset of sliding friction [6], and the conditions under which a new conserved quantity can be defined, which can be seen as a Generalized Angular Momentum (GAM) in the complex plane [8].In this work, we present new results, showing that, in the dynamics, a floating-to-pinned transition, analogous to the static Aubry transition, is found for all values of the potential λ < λ c . The transition is characterized by a region of critical times, with a remarkably complex behavior.…”

“…These excitations can further combine and give rise to the appearance of Umklapp terms. This phenomenon can be monitored by the breakdown of the conservation of the GAM, as derived in [8]:…”

“…For values of the coupling λ < λ c , the Fourier transform of the displacement related to q has thus the strongest amplitude, and amplitudes of higher harmonics nq scale with λ |n| . It is therefore natural to use |n| instead of k and relabel the modes (see [8] for the details of the relabeling procedure). In the left side of fig.…”