Elasticity bounds from effective field theory

Abstract

Phonons in solid materials can be understood as the Goldstone bosons of the spontaneously broken spacetime symmetries. As such their low energy dynamics are greatly constrained and can be captured by standard effective field theory (EFT) methods. In particular, knowledge of the nonlinear stress-strain curves completely fixes the full effective Lagrangian at leading order in derivatives. We attempt to illustrate the potential of effective methods focusing on the so-called hyperelastic materials, which allow large elastic deformations. We find that the self-consistency of the EFT imposes a number of bounds on physical quantities, mainly on the maximum strain and maximum stress that can be supported by the medium. In particular, for stress-strain relations that at large deformations are characterized by a power-law behaviour σ(ε) ∼ ε ν , the maximum strain exhibits a sharp correlation with the exponent ν.

Scale invariant solids

“…An obvious extension of this work, which is currently under investigation, amounts to generalize these results to the non-linear regime up to arbitrarily large deformations. An EFT description has been recently introduced in [7]. We plan to report in a forthcoming work [31] the analysis of nonlinear elasticity in scale invariant system, comparing also the spontaneously broken vs. manifest cases.…”

“…The relations are long known but we find illustrative to derive them using Effective Field Theory (EFT) methods, by treating the phonons as the Goldstone bosons associated to the spontaneous symmetry breaking pattern which takes place in solids. The resulting Solid EFT are discussed at lowest order in derivatives for the phonon fields in [3][4][5][6], see also [7]. Let us emphasize that the EFT description below corresponds to the spontaneous breaking of spacetime symmetries.…”

“…In order to ensure the consistency of this choice (4.28), and of the model (4.7) in general, one must impose a number of requirements. First, absence of ghosts, absence of gradient instabilities locally in the 4D theory leads to limit the parameters a, b in the range [7]:…”

“…Ref. [7], initiated a study in this direction, showing that nontrivial relations among several nonlinear observables can indeed be identified. This motivates us to continue the analysis to more sophisticated cases.…”

“…The complete study of the low energy dynamics (including the dissipative terms) can be obtained using Hydrodynamic techniques. See[33][34][35][36] 7. To make the distinction with the previous section clearer, lower-case latin indices a, b, .…”