PHYSICAL REVIEW D 100, 065015 (2019) Elasticity bounds from effective field theory Lasma Alberte* Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151, Trieste, Italy † Matteo Baggioli Crete Center for Theoretical Physics, Institute for Theoretical and Computational Physics Department of Physics, University of Crete, 71003 Heraklion, Greece ‡ Víctor Cáncer Castillo and Oriol Pujol`as § Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology (BIST) Campus UAB, 08193 Bellaterra, Barcelona, Spain (Received 12 September 2018; published 23 September 2019) 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 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 effective field theory 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 behavior σðεÞ ∼ εν, the maximum strain exhibits a sharp correlation with the exponent ν. DOI: 10.1103/PhysRevD.100.065015 I. INTRODUCTION well-defined material properties that go deep into the nonlinear response regime. Typically, these parameters A prominent and early example of an effective field are difficult to compute from the microscopic constituents, theory (EFT) is the theory of elasticity: the continuum-limit so there is a chance that EFT methods may help in the description of a solid’s mechanical response, including its understanding of some nonlinear elasticity phenomena. sound wave excitations—the phonons [1,2]. As in hydro- From the viewpoint of quantum field theory (QFT), it is dynamics, elasticity theory can be phrased as a derivative clear that elasticity theory can be treated as a nontrivial (i.e., expansion for an effective degree of freedom (d.o.f.)—the interacting) EFT. The way this theory works as an EFT, displacement vector of the solid elements with respect to however, is quite different from other well-known exam- equilibrium. Importantly, the classic elasticity theory can be ples, mostly because the underlying symmetry breaking promoted to the nonlinear regime, addressing the response pattern involves spacetime symmetries. The purpose of this to finite deformations [3–5]. Operationally, this is done by work is to revisit finite elasticity theory from the viewpoint finding the stress-strain relations for both the finite shear or of QFT. We aim at clarifying how the EFT methodology bulk strain applied to the material. These diagrams encode works for broken spacetime symmetries and find novel proportional limit several response parameters (such as the relations between (and bounds on) various nonlinear elasti- failure point or the ; see Ref. [5] for definitions), which are city parameters. *[email protected] II. FROM GOLDSTONES TO † [email protected]; www.thegrumpyscientist.com. STRESS-STRAIN CURVES ‡ [email protected] §[email protected] We start by stating the precise QFT sense in which elasticity theory can be treated as an EFT. The first Published by the American Physical Society under the terms of requirement is that the material must have a separation the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to of scales; we shall consider only low frequency (acoustic) the author(s) and the published article’s title, journal citation, phonons; any other mode is considered as much heavier and DOI. Funded by SCOAP3. and integrated-out. (Materials displaying scale invariance 2470-0010=2019=100(6)=065015(12) 065015-1 Published by the American Physical Society ALBERTE, BAGGIOLI, CASTILLO, and PUJOLAS` PHYS. REV. D 100, 065015 (2019) violate this assumption and deserve a separate treatment.) characterizing the materials state of deformation. It encodes Under this condition, we can exploit the fact that the the full nonlinear response for the so-called Cauchy phonons can be viewed as the Goldstone bosons of trans- hyperelastic solids, for which plastic and dissipative effects lational symmetry breaking [6–8]. As such, we obtain their can be ignored [4]. fully nonlinear effective action by the means of the standard The form of V can then be found from the stress-strain coset construction [9]. For simplicity, we shall work in relations measured in both the shear and the bulk channels 2 þ 1 spacetime dimensions, where the dynamical d.o.f. are of real solids (see, e.g., Refs. [3,4,12]). More specifically, contained in two scalar fields ϕIðxÞ. The internal symmetry the response of the material to constant and homogeneous group is assumed to be the two-dimensional Euclidean deformations can be deduced from configurations of the group, ISOð2Þ, acting like translations and rotations in the form. These can be reduced to configurations of the form pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! scalar fields space. The theory then must be shift invariant 2 I 1 þ ε =4 ε=2 in the ϕ ’s, implying that any field configuration that is ϕI ¼ I J I ¼ α str OJx ;OJ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; linear in the spacetime coordinates will satisfy the equa- ε=2 1 þ ε2=4 tions of motion. The equilibrium configuration of an ð3Þ isotropic material is given by where ε and α − 1 are the shear and the bulk strains ϕI ¼ δI J ð Þ eq Jx : 1 respectively, and they induce constant but nontrivial values j ¼ α2ð2 þ ε2Þ j ¼ α4 of X str and Z str . The amount of stress This vacuum expectation value spontaneously breaks the in the material generated by (or needed to support) such a ð2Þ ð2 1Þ symmetry group ISO × ISO ; down to the diagonal configuration depends only on the strains ε and α and on subgroup. the shape of VðX; ZÞ; see, e.g., Eq. (9). The upshot is that it Following the coset construction method, one concludes is possible to reconstruct the full form of the effective that the effective action at lowest order in derivatives takes Lagrangian (up to an irrelevant overall constant) by just the form measuring the stress-strain relations, that is, from the Z response to time-independent and homogeneous deforma- pffiffiffiffiffiffi S ¼ − d3x −gVðX; ZÞ; ð2Þ tions. This already illustrates how the solid EFTs retain predictive power. 1 The next apparent challenge from the QFT viewpoint is with X and Z defined in terms of the scalar fields matrix that the real-world stress-strain curves typically exhibit a IIJ ¼ μν∂ ϕI∂ ϕJ ¼ ðIIJÞ ¼ ðI IJÞ g μ ν as X tr , Z det . The dramatic feature: they terminate at some point, correspond- function VðX; ZÞ is “free,” and its form depends on the ing to the breaking (or elastic failure) of the material. It is solid. In this language, the phonons πI are identified as then natural to ask how exactly is the breaking seen in the the small excitations around the equilibrium configuration EFT. Must the function VðX; ZÞ be singular? Or does the ϕI ¼ ϕI þ πI defined through eq . Plugging this decomposi- breaking correspond to a dynamical process (e.g., an tion into (2), one can find the phonon kinetic terms and instability) that can be captured within the EFT with their self-interactions ð∂πÞn. The leading phonon effective a regular VðX; ZÞ? We argue below that the latter possibil- operators are determined by a few Wilson coefficients ity can certainly arise, allowing one to extract relations that are related to the lowest derivatives of V evaluated between the parameters that control the large deformations. on the equilibrium configuration; see Ref. [10] for details. The main task then is to analyze the stability properties (Analogous results can be found in Ref. [11] for super- of the strained configuration (3). This can be done by ϕI ¼ ϕI þ πI “ ” πI conductors.) The effective action (2) also encodes the setting str in (2) and expanding for small . response to finite (large) deformations, and for that, the In doing so, one easily finds that the phonon sound I global form of VðX; ZÞ is needed. speeds depend on the applied strain OJ. This is a long- By symmetry considerations, one cannot restrict the known phenomenon, the acoustoelastic effect; see, e.g., action (2) any further. To identify what is the function Refs. [13–18]. Still, we argue here that this can have a great VðX; ZÞ for a given material, one needs more information, impact on the stress-strain relations, eventually limiting the some kind of constitutive relation. According to the finite maximal stress that a material can withstand. The reason is elasticity literature (see, e.g., Ref. [5]), the function VðX; ZÞ that generically, increasing the strain results in increasing/ is naturally identified with the so-called strain-energy decreasing the various sound speeds—typically in an function. This is a function of the principal invariants unbounded fashion. In particular, in most cases, past some ε large enough strain value, max, one of the sound speeds 1 becomes either i) imaginary or ii) superluminal. Case i We retain the curved spacetime metric gμν only to make it clear how the energy-momentum tensor arises from this action. In implies that the material develops an instability and it must practice, we shall always work on the Minkowski background evolve to a different ground state. Case ii prevents the ημν ¼ diagð−1; þ1; þ1Þ. existence of a Lorentz invariant ultraviolet completion. 065015-2 ELASTICITY BOUNDS FROM EFFECTIVE FIELD THEORY PHYS.