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.
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For any time-independent scalar field configurations, the stress-energy tensor components are
Ttt ≡ ρ ¼ V; ð6Þ
x Tx ≡ p ¼ −V þ XVX þ 2ZVZ; ð7Þ
x I I Ty ¼ 2∂xϕ ∂yϕ VX; ð8Þ
where VX ≡∂V=∂X, etc. Henceforth, we shall work with FIG. 1. The nonlinear shear stress-strain curve σðεÞ for the the deformed field configuration (3), which introduces benchmark model (15) for B ¼ 1.6 and A ¼ 0.05, 0.2, 0.35, 0.5, both the shear and bulk deformation. In particular, when “ 0.61 (from bottom to top). The black stars represent the break- setting α ¼ 1, it describes a pure shear strain (i.e., volume ing” points of the material arising due to the onset of gradient preserving) in the ðx; yÞ directions induced by ε ≠ 0.For instability; the red dot indicates the onset of superluminality. ε ¼ 0 and α ≠ 1, the same setup encodes a pure bulk strain. In the considered scalar field background configuration, Therefore, the effective low energy description (2) must be j ¼ α2ð2 þ ε2Þ j ¼ α4 X and Z take the values X str , Z str . physically invalid at least for such a large deformation. In particular, the full nonlinear stress-strain curve for In any case, one can translate the constraints i and ii as pure shear deformations as a function of ε reads upper bounds on the maximum allowed strain that is ð Þ rffiffiffiffiffiffiffiffiffiffiffiffiffi compatible with the given choice of V Z; X . We remark ε2 that these bounds arise even for smooth choices of the σðεÞ ≡ ¼ 2ε 1 þ ð2 þ ε2 1Þ ð Þ Txy 4 VX ; : 9 effective Lagrangian VðZ; XÞ, and yet they naturally lead to ε ¼ ε stress-strain curves that terminate at some point max; The analogous stress-strain curve for pure bulk defor- see Fig. 1 for some illustrative examples. mations can also be found by expressing ΔTx ¼ Tx − Txj Additionally, demanding that none of these pathologies x x x eq as a function of the bulk strain, α − 1. It is thus clear that occurs for materials that we know admit large deformations from the knowledge (measurement) of both shear and bulk (elastomers) significantly constrains the stress-strain curves diagrams one can extract the shape of VðX; ZÞ—the full and therefore the possible nonlinear response of materials effective Lagrangian. For instance, under the assumption on quite general grounds. We illustrate the point by that the Z-dependence is negligible, then from a given σðεÞ focusing on materials/EFTs which allow for large defor- shear stress-strain curve, one can extract mations and which realize stress-strain curves with a power-law scaling, Z pffiffiffiffiffiffiffiffiffiffiffi X σð − 2Þ ð Þ ≃ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix ν V X dx : σ ∼ ε for ε ≫ 1: ð4Þ 2 x2=4 − 1
Henceforth, we shall refer to ν as the strain exponent.As To make the connection to the linear elasticity theory we show below, both the maximum strain and the exponent explicit, one considers small shear and bulk deformations, ν are bounded from above, and there is a general relation i.e., small values of ε and α − 1. Then, as usual, elastic between the two. It is unclear to us to what extent these deformations at the linear level are described in terms of the results were already known before. Nonetheless, our main displacement tensor goal is to show how the EFT perspective presented here brings some additional layer of understanding to these 1 ε ¼ ð∂ δϕ þ ∂ δϕ Þ ð Þ phenomena. ij 2 i j j i ; 10 First, let us obtain the corresponding stress-energy tensor δϕI ≡ ϕI − ϕI by varying the action with respect to the curved spacetime where eq is the displacement away from the metric gμν and evaluating it on the Minkowski background, equilibrium state, ϕI ¼ xI. A deformation of the body that ¼ η eq gμν μν: changes its volume is given by the compression or bulk i strain as εii ¼ ∂iδϕ . In turn, a deformation that only affects 2 δS 1 ¼ − pffiffiffiffiffiffi its shape—pure shear—is given by ε − δ ε . Tμν − δ μν ik 2 ik jj g g g¼η Expanding both the stress-energy tensor components (7), I (8) and the displacement tensor (10) up to linear order in ε ¼ −ημνV þ 2∂μϕ ∂νϕIVX and α − 1, one recovers the usual expression in 2 þ 1 þ 2ð∂ ϕI∂ ϕ − ∂ ϕI∂ ϕJI Þ ð Þ μ ν IX μ ν IJ VZ: 5 dimensions,
065015-3 ALBERTE, BAGGIOLI, CASTILLO, and PUJOLAS` PHYS. REV. D 100, 065015 (2019) 1 different EFT. The specific nature of this transition remains lin ¼ð þ ε Þδ þ 2 ε − δ ε ð Þ Tij p K kk ij G ij 2 ij kk ; 11 hidden in the leading order low energy EFT presented in this work. For instance, whether the gradient instability where p is the equilibrium pressure and G and K are the develops as a soft (slow) or hard (fast) process depends on shear and bulk elastic moduli. In the case of a pure shear the nature of the next-to-leading order corrections to ð Þ deformation, this gives Txy ¼ 2Gεxy þ , and we can read V X; Z . One may speculate that the hard case corresponds off the shear modulus G as to a breaking of the material and the soft case corresponds to the necking phenomenon—a decrease in the cross sec- G ¼ 2VXð2; 1Þ: ð12Þ tionalarea ofa materialsamplethat is often seen undertensile ε ¼ 0 stress. This would resemble the so-called soft phonon Similarly for the case of the pure bulk strain ( ), we instability observed in some materials; see Refs. [19–26]. first note that Eq. (7) holds at nonlinear level, i.e., for α Concerning superluminality, let us emphasize that in arbitrarily large values of . In order to find the linear bulk contrast to ghost and gradient instabilities the issue of modulus, we expand both the bulk strain and the bulk stress superluminal propagation relates to the possibility of a Δ x α ¼ 1 Tx around the equilibrium value . For the stress, this Lorentz invariant UV completion, not to the stability of Δ ¼ 2 ε þ gives Tii K ii with the equilibrium pressure propagation [27]. In order to apprehend the physical ε ¼ 2ðα − 1Þ given in (7) and ii . The bulk modulus is then picture, it is instructive to recall a classic in field theory: K ¼ 2ZV þ 4Z2V þ 4XZV þ X2V ; ð13Þ the example given by high spin fields where the problem of Z ZZ XZ XX superluminality is known to arise [28]. As discussed in where all the quantities are evaluated at X ¼ 2, Z ¼ 1. Refs. [29,30], there are two ways to resolve the problem, All the details concerning the consistency and stability of which require augmenting the EFT either by higher order perturbations around the strained background configuration operators or with additional light d.o.f. Any of the two are given in the Appendix. There, we find that the spectrum resolutions makes it manifest that the naive EFT truncation of perturbations contains two gapless phonon modes, [akin to the one that we are doing in Eq. (2)] breaks down. Moreover, it also gives an idea of how—what that trunca- ω ¼ c ðα; εÞk; ð14Þ tion might be missing. In our case, this means that in the vicinity of violating the no-superluminality condition with the sound speeds bearing a nonlinear dependence on corrections to the particular shape of VðX; ZÞ that we the strain parameters α, ε. For the consistency and stability consider must become important either by the presence of of a given VðX; ZÞ around the background (3), we require additional operators or light fields. The possibility that the absence of i) modes with negative kinetic energy, i.e., higher order operators (with more derivatives) can fix the ghosts; ii) negative sound speeds squared, i.e., gradient superluminality problem while keeping the rest of the instability; and iii) superluminal propagation. In each case, elastic response properties is nontrivial, and we leave it this leads to a certain value of maximal strain, ε , beyond max for future research. On the other hand, the possibility that which one of these consistency conditions is violated. one needs to supplement the benchmark model with other A typical stress-strain curve exhibiting this behavior, light d.o.f. seems quite reasonable—after all, in real-world obtained for a given choice of VðX; ZÞ, is shown in Fig. 1. materials, phonons do couple to many other modes. If this It is important to remark that our expressions for ε max is the resolution, then the physical interpretation of the derived in Appendix should be interpreted as giving an bound given by superluminality is that ε can be under- upper bound on the maximum strain that the material can max stood as an upper limit on when these light d.o.f. have to be support, since other effects not included here can enter taken into account. before, thus lowering the actual maximum ε. For instance, one expects plastic/dissipative effects to enter at some point III. RESULTS IN A SCALING MODEL in real materials. However, this alters our analysis only for ε ε plastic < max; thus, we still obtain an upper bound on the For concreteness, we shall focus on the simple potential maximum reversible deformability. ð Þ¼ρ A ðB−AÞ=2 ð Þ It is interesting to consider the possibility that it really is V X; Z eqX Z ; 15 ε the max found here (or a value very close to it) that ρ corresponds to the physical limitation to the material where eq is the dimensionful energy density set by the deformation. In this case, the EFT gives partial information equilibrium configuration. The reason for choosing this on how the material might “break.” As was mentioned form is that it realizes a power-law scaling like (4) at large earlier, there are two main options: that the breakdown is due deformations, ε ≫ 1. This behavior is observed in hypere- to gradient instability or due to reaching superluminality. lastic rubberlike materials, and there are many phenom- In the case of gradient instability, one expects that, like enological models [4,12,31–36] that reduce to (15) at large any instability, this is physically resolved by a transition strains with various strain exponents ν. Here, we are to another ground state, most likely described by a interested in characterizing how the stress-strain curves
065015-4 ELASTICITY BOUNDS FROM EFFECTIVE FIELD THEORY PHYS. REV. D 100, 065015 (2019)
(and mainly the maximum stress and strain) depend on the This is shown in Fig. 1 for various values of A and B. parameters A and B. Let us also note that there are two Notably, the stress-strain curves obtained from the special “corners” in parameter space: for A ¼ 0, the bench- benchmark models mimic a large variety of materials mark potential describes a perfect fluid [7,9]; for A ¼ 1, including fibers, glasses, and elastomers [12]. More pre- B ¼ 1, the model reduces to two free scalar fields. cisely, Eq. (18) describes Neo-Hookean systems which We first find that the linear elastic moduli for the follow Hooke’s law at small strain but exhibit nonlinear potential (15) take the simple form power-law scalings at large deformations [5]. Similarly, the nonlinear response to a pure compression, that we define as ¼ ρ 2A ¼ ρ 2A ð − 1Þ ð Þ κ ≡ α − 1 G eq A; K eq B B : 16 , reads
Δ ðκÞ¼ρ 2Aþ1ð − 1Þ½ðκ þ 1Þ2B − 1 ð Þ They are both positive for A>0, B>1. Moreover, the Tii eq B : 19 Poisson’s ratio—the negative ratio of transverse to axial strain—for our models is readily obtained as (see Ref. [37]) We show the full nonlinear response to pure bulk defor- mation for various values of B in Fig. 3. As per con- ε κ ≫ 1 K − G BðB − 1Þ − A struction, at large strains, , , the nonlinear stresses r ≡ ¼ : ð17Þ display power-law scalings of the form K þ G BðB − 1ÞþA σðεÞ ∼ Aε2A; ΔT ðκÞ ∼ ðB − 1Þκ2B; ð20Þ The result is shown in Fig. 2. At large B, the ratio is close to ii its upper bound, meaning that the models are close to from where we read off the shear and bulk strain exponents perfect incompressible elastic materials. At small values of as ν ¼ 2A and ν ¼ 2B. Note that, as can be seen B and large A, the ratio tends to its lower negative bound. shear bulk from Eq. (16), A and B also control the linear shear and A negative Poisson’s ratio is typical of more exotic (the so- bulk moduli. called auxetic) materials like some foams and metamate- Combining the requirements of the absence of ghosts, rials. Interestingly, the limit of free canonical scalars is in gradient instabilities, and superluminal propagation with that regime. Finally, for most of the models described, the positivity of the elastic moduli, K and G, constrains the −0.5 < r < 0.5, as is common for steels and rigid allowed range of parameters. In the simple case of linear polymers. deformations, we obtain the following allowed region for For the full nonlinear response to pure shear, Eq. (9) the exponents A and B: gives pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ≤ A ≤ 1 and 1 ≤ B ≤ 1 − A þ 1: ð21Þ σðεÞ¼ρ ε ε2 þ 4ðε2 þ 2ÞA−1 ð Þ eqA : 18 The analysis can be extended to a finite strain and as mentioned above leads us to another important result: the ε existence of a maximum strain max that can be supported by the system before the onset of one of the aforementioned ε pathologies. How max depends on the strain exponents is shown in Fig. 4; the exact analytic expressions can be found 2 ε in the Appendix. We must emphasize that the max obtained in this way is not meant to be the actual maximum deformation that a material with the aforementioned scaling properties can withstand but rather an upper bound on it. Still, this already provides quite a lot of information. For ε instance, in the large (yellowish) area of Fig. 4 where max only reaches values of approximately 1, one can already discard the existence of very elastic materials that exhibit scaling as in (4) with those scaling exponents. We note that the regions in the A − B parameter space where large strains can be supported are near the special points A ¼ 1, B ¼ 1 (free scalars) or A ¼ 0 (fluid limit).
2Let us remark that, as can be inferred from Eq. (18), the power-law scaling can really be reached only for ε ≳ 2. There- FIG. 2. Poisson’s ratio r in the allowed parameter region given fore, the limits shown in Fig. 4 can only be extended to a material in Eq. (21). following (20) at large strains in the bluish part of the diagram.
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Intriguingly, for A ≪ 1, a number of “universal” corre- lations appear. First, we find a universal scaling of the maximum strain pffiffiffi B − 1 1=4 ε ≃ 2 : ð22Þ max A
Inserting this in the expression (9) for the nonlinear shear stress, we further obtain
σ ≡ σðε Þ¼ρ ð Þ max max eqA: 23
This shows a linear dependence of the maximal stress supported by a material on the strain exponent A, which in our simple model controls also the linear elastic modulus. Similar linear correlations are observed experimentally in various materials [38–42]. Additionally, we also find a clear relation between the hardness and the maximum strain, σ ∼ ε−4 FIG. 3. The nonlinear bulk stress-strain curve for the bench- max max. Let us emphasize, however, that whether the mark model and the parameter values A ¼ 0.5 and B ¼ 1.1, 1.3, correlations that we find can be extrapolated to real-world 2B materials strongly depends on i) whether their stress-energy 1.5, 1.7, 1.9. The large strain scaling is set by ΔTii ∝ κ . function V behaves as a power law at large strain and ii) whether they can support large deformations. Finally, let us note that within the benchmark model (15) Therefore, for the model (15), we expect the real-world κ (nonrelativistic) solids to lie near the A ¼ 0 axis. In this there are no constraints on the bulk strain arising from the limit, the maximum strain is set from the absence of consistency and stability requirements. This is a conse- gradient instability for almost all values of B. quence of (15) being a monomial. For more general choices, additional bounds can arise. Let us also mention that for B ∈ ð0; 1Þ it is possible to achieve a negative bulk modulus, K<0, in a way that is perfectly consistent from the EFT perspective. In particular, as long as K>−G, the 2 stability constraint cþ > 0 is still satisfied. This has also been studied in four dimensions [43,44] and observed experimentally [45].
IV. NONRELATIVISTIC SOLIDS The benchmark model, Eq. (15), considered above has been useful for exhibiting the constraining power of the EFT methods; however, it has one disadvantage: the region in parameter space giving small sound speeds as in real- world elastic materials is very small. To be more specific, the typical sound speeds are at most of order approximately 10−4 in the units of the speed of light. In the parameter space A, B, this corresponds to the corner where both A and B − 1 are of order 10−8, or less. The problem with this is that in the benchmark model (15) A and B also control the exponents in the stress-strain relation at large strain, σ ∼ εν FIG. 4. The allowed parameter region (21) for the benchmark ν ¼ 2 ν ¼ 2 with shear A for pure shear and bulk B for pure bulk model (15). The left, bottom, and right edges are respectively deformations respectively. It follows that the benchmark given by the gradient instability, positivity of the bulk modulus, models can only cover realistic materials with very specific and superluminality. The red line separates the region where the ν ∼ 10−8 ν ≃ 2 maximum strain is due to the gradient instability (left) and the exponents, basically shear and bulk . Clearly, region where it is due to superluminality (right). Large strains there has to be a way around this limitation because elastic ν [and therefore the power-law behavior (4)] are realized in the materials with more generic values for bulk=shear do exist bluish area. and one expects that a similar EFT construction should
065015-6 ELASTICITY BOUNDS FROM EFFECTIVE FIELD THEORY PHYS. REV. D 100, 065015 (2019) describe them. The obvious guess is that the benchmark choice, Eq. (15), is too restrictive. In this section, we show how to deform the model in order to have small speeds of sound while keeping large deformability and generic exponents. Fortunately, there is a well-motivated and unique way to ensure that the sound speeds become as small as needed while keeping the stress-strain relations untouched. Thisp isffiffiffiffi achieved by adding an extra term to the potential δV ∝ Z with a large coefficient in front. This term is special for many reasons. Physically, it is proportional to the mass density of the material [6]. This immediately explains why the coefficient in front of it must be large in the non- relativistic materials. The mass density contributes to the Lagrangian (an energy density) weighted by c2 [6] and is much larger than the typical stresses in solids. Related to this, in the fluctuations around any background, this term only produces temporal kinetic terms, as can be easily seen in Eqs. (A4)–(A9) in Appendix, noting that this term 2 ¼ 0 2 δ þ 2 δ ¼ 0 FIG. 5. Expanded parameter space for v . . The red line satisfies VZ Z VZZ . Therefore, this new term splits the regions where the limit on the maximal strain comes only contributes to the denominators in the formulas for from superluminality (on the right) and from gradient instability the speeds, and so enhancing it decreases the speeds. (on the left). The green dashed line is A ¼ B. In the region A ≥ B, Moreover, an important feature of this term is that it does the maximum strain is only dictated by subluminality. not affect the bulk stress Tii nor the shear stress Tij,soit does not alter the stress-strain relations [this is clear from A ¼ B at small A, B. Importantly, both regions contain Eqs. (7) and (9)]. This term only appears in the energy sizeable values for A, B. The first conclusion, then,pffiffiffiffi is that, density T00, as it must be, since it only accounts for the indeed, adding a large mass-density term Z to the inertial mass and thus it contributes like “dust” (pressure- Lagrangian opens up the possibility to model nonrelativ- less fluid). This is crucial to retain the predictive/ istic materials with sizeable shear and bulk exponents, ν ¼ 2 ν ¼ 2 constraining power of the EFT framework because in order shear A and bulk B. to go to the nonrelativistic regime it suffices to add one single parameter in the full nonlinear Lagrangian. For these reasons, it suffices to switch to the following model, pffiffiffiffi A 2 X B−A ð Þ¼ρ þ 2 ð Þ V X; Z 0 Z v 2 Z ; 24 with v a small parameter (which is a measure of the typical speeds in the units of the speed of light). This guarantees that the material is nonrelativistic while keeping the non- linear static elastic response the same as in the benchmark model (15). The discussion about the stability and consistency of this model is also mentioned in the Appendix. In summary, we find that for v ≪ 1 there are two new regions in the A − B parameter space that allow i) small velocities and ii) ε ≫ 1 (i.e., a very elastic material), as can be seen in Figs. 5 and 6. The first region is close to the line A ¼ B but with (A 065015-7 ALBERTE, BAGGIOLI, CASTILLO, and PUJOLAS` PHYS. REV. D 100, 065015 (2019) In the new region with A Interestingly enough, even though this differs from Eq. (22) [valid for the benchmark model (15) at A ≪ 1], we still ε ð Þ have some relation max A; B . The second new region (for A>B) instead is only constrained by the subluminality condition, and so the bounds are less powerful. Specifically, for v ≪ 1, we find 2 1 FIG. 7. Speed ratio constraints for r ¼ 10, 10 . Dashed lines 2 −1=A ε ¼ 3 ε ∼ 2 ðAðA þ 1ÞþBðB − 3ÞÞ : ð26Þ show where max , 5, 8 (blue, orange, green). max v2=A ν S 2 ε Notice that the maximum strain scales as εmax ∼ 1=v , corresponding to large deformation ( max significantly where νS ≡ 2A. This scaling can be understood because for greater than 1), moderate r, and large exponents. The size large shear deformation the phonon speed cþ grows as of this region in parameter space depends on the criteria for 2 2 2A ε cþ ∼ v ε . Since the constraint obtained within the EFT is r and max, but one can say that it extends to next to the really only an upper limit on the strain, one obtains only a A ¼ B line within a few percent. In this region, nontrivial very large upper bound—a very loose bound. relations such as (27) or (25) should hold. As is clear from Figs. 5 and 6, the two regions actually As a final remark, let us emphasize the most basic touch each other; therefore, at some point, one of the speeds property of this new region: it is close to A ¼ B. In other must increase also in the A This new constraint is shown in Fig. 7. This figure shows 3For recent EFT-like efforts to include dissipation in fluids and that there is indeed an overlap between the regions viscoelastic materials; see Refs. [46–59]. 065015-8 ELASTICITY BOUNDS FROM EFFECTIVE FIELD THEORY PHYS. REV. D 100, 065015 (2019) intrinsically nonlinear response parameters, such as the Kostya Trachenko, and Alessio Zaccone for useful dis- maximum stress and the strain exponent. cussions and comments about this work and the topics An interesting case is represented by the conformal considered. M. B. is supported in part by the Advanced solids limit, realized by potentials of the form VðX; ZÞ¼ ERC grant SM-grav, Grant No. 669288. V. C. C. and O. P. X3=2FðX=Z1=2Þ, which preserve scale invariance and imply acknowledge support by the Spanish Ministry of Education μ Tμ ¼ 0 [60] (see also Refs. [61,62]). In this case, the bulk & Science under Grants No. FPA2014-55613-P and modulus is directly proportional to the energy density No. FPA2017-88915-P and the Severo Ochoa excellence K ¼ 3=4ρ, as observed in earlier holographic models program of MINECO (Grants No. SO-2012- 0234 and [62]. Concerning the strain exponents, scale invariance No. SEV-2016- 0588), as well as by the Generalitat de ν ¼ 3 ν ≤ 3 2 fixes bulk and bounds shear = . Let us emphasize, Catalunya under Grant No. 2014-SGR-1450. M. B. would however, that the notion of a conformal solid, understood as like to thank Iceland University and Queen Mary an EFT with a Lagrangian of the above form, should be University for the warm hospitality during the completion distinguished from a system of which the low energy of this work. dynamics is controlled by a strongly coupled infrared fixed point. In that case, the standard EFT methods are not APPENDIX: FLUCTUATIONS AND granted to apply. A study of the nonlinear elasticity for that CONSISTENCY case using holographic techniques is deferred to a separate work [63]. In order to study the stability of perturbations around the We have also shown how to extend the analysis to strained background configuration, we expand the scalar ϕI ¼ ϕI þ πI nonrelativistic materials, with realistically small sound fields as str . To identify the propagating d.o.f., speeds. Our main conclusion—that the EFT method we perform the decomposition into longitudinal and trans- πI ¼ πI þ πI π provides nontrivial relations between nonlinear response verse fluctuations by splitting L T, with L=T parameters—remains true also in this regime. Moreover, let satisfying us make a remark about the region close to A ¼ B of these OI ∂ πK ¼ 0; εIJOK∂ πT ¼ 0: ðA1Þ nonrelativistic solids. In this region, the EFT method is the K I L I K J most informative, so it is worth trying to compare its This gives two dynamical scalar modes that can be defined predictions to data. A proper analysis of the experimental through data on real-world elastomers is well beyond the scope of πI ¼ IK∂ π πI ¼ εIJ K∂ πT ð Þ this work, but we would like to make one comment. It is L O K L; T OJ K : A2 known [3] that a very successful way to fit the nonlinear π ¼ π ð Þ response of some rubbers consists of writing VðX; ZÞ as a Constraining the spatial dependencepffiffiffiffiffiffiffiffi to L=T L=T t; x IJ I μ J 2 sum of a few powers of the matrix X ¼ ∂μϕ ∂ ϕ as and redefining πL=T → πL=T= −∂x, we obtain the follow- IJ pn V ¼ ΣnμnTr½ðX Þ with some constants μn, pn. At large ing quadratic action for the fluctuations, deformations, these models are dominated by the term with Z the highest power; call it p. It is easy to see that taking δ ¼ 3 ½ π_ 2 þ π_ 2 þ 2 π_ π_ − 2 ð∂ π Þ2 S2 d x NT T NL L NTL T L cT x T V ¼ Tr½ðXIJÞp does not strictly coincide with our bench- mark models for any A, B; however, it does lead to a − 2 ð∂ π Þ2 − 2 2 ∂ π ∂ π ð Þ cL x L cTL x T x L ; A3 response at large strains very similar to that in our bench- mark model with A ¼ B ¼ p (for instance in the stress- 2 2 2 where the parameters NT, NL, NTL and cT, cL, cTL depend strain relations). This is encouraging because it would on both the shear and bulk strains; i.e., they are functions of suggest that the models in the region near A ¼ B could ε and α. The explicit expressions in terms of the derivatives correspond to these rubbers. It would be interesting to see of the function VðX; ZÞ, defined in Eq. (15), are found to be whether (27) or (25) holds for them. We leave these questions for the future. 1 N ¼ ððX2 − 2ZÞV þ XV Þ; ð Þ Furthermore, it would be desirable to introduce dis- T 2 Z X A4 sipative and thermal effects within the EFT picture of condensed matter systems [54,64]. In this regard, the holo- X NL ¼ ZVZ þ VX; ðA5Þ graphic description could provide valuable supplementary 2 – – qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi insight [61 63,65 69]. We hope to return to some of these 1 points eventually. ¼ ð 2 − 4 Þ ð Þ NTL 2 Z X Z VZ; A6 1 ACKNOWLEDGMENTS 2 ¼ ð þ 2 Þþ ð þ 4 þ Þ cL Z VZ ZVZZ 2 X VX ZVXZ XVXX ; We thank Alex Buchel, Carlos Hoyos, Karl Landsteiner, ð Þ Mikael Normann, Giuliano Panico, Napat Poovuttikul, A7 065015-9 ALBERTE, BAGGIOLI, CASTILLO, and PUJOLAS` PHYS. REV. D 100, 065015 (2019) 1 Let us point out that evaluating the sound speeds c at 2 ¼ ðð 2 − 4 Þð þ 2 Þþ2 ÞðÞ α ¼ 1 ε ¼ 0 cT 4 X Z VZ ZVZZ XVX A8 and we find that the result coincides with the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi standard relationships obeyed by the transverse and longi- 1 ϕI ¼ I 2 ¼ ð 2 − 4 Þð þ 2 þ Þ ð Þ tudinal phonons of the equilibrium state eq x , cTL Z X Z VZ ZVZZ XVXZ ; A9 2 sffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi þ with all the quantities evaluated on the scalar field back- G K G cT ¼ ;cL ¼ ; ðA18Þ ϕI ρ þ p ρ þ p ground solution str. I Let us emphasize that for a nondiagonal matrix OJ the transverse and longitudinal modes remain mixed both with where ρ and p are the equilibrium energy density and respect to time and spatial derivatives. In order to study the pressure, as in Eqs. (6) and (7). The K and G refer to the stability of fluctuations, we therefore first introduce the linearized bulk and shear moduli, defined in Eqs. (13) kinetic matrix as and (12). The conditions necessary to ensure the positivity of λ N N N ¼ T TL : ðA10Þ then read NTL NL 4d c>0;d≥ 0; 1 − ≥ 0: ðA19Þ The absence of ghostlike excitations then requires that the c2 eigenvalues of the kinetic matrix, λ , are positive. This gives the first condition for stable propagation of the The first two constraints above can be expressed as λ 0 2 modes: > . inequalities for quadratic polynomials in ε . For the bench- It is straightforward to determine the true dynamical mark model, we find that upon setting modes described by the action (A3) by working at the level π of the equations of motion of the mixed fields L=T. After A − B<0;A>0 ðA20Þ iωt−ikx Fourier transforming as πL=T ¼ aL=Te , we can solve for the spectrum of perturbations to obtain these are satisfied for any choice of ε, while the last condition is fulfilled automatically for arbitrary choice ω2 ¼ 2 ðα εÞ 2 ð Þ c ; k : A11 of A, B, ε. The conditions necessary for avoiding the gradient The other conditions for consistency that we are going to instability are in turn impose are thus: 2 (i) c ≥ 0, i.e., the absence of gradient instabilities; 4bd 2 ≤ 1 a>0;b≥ 0; 1 − ≥ 0 ðA21Þ (ii) c , i.e., the absence of superluminal modes. a2 The exact expressions of the kinetic eigenvalues can be put in the form and are slightly harder to satisfy. It is easy to see that by rffiffiffiffiffiffiffiffiffiffiffiffiffi setting c 4d λ ¼ 1 1 − ; ðA12Þ 2 c2 A þ B>1 ðA22Þ with and assuming that (A20) holds the condition a>0 can be satisfied for arbitrary values of ε. However, for these values ¼ þ ð Þ c NL NT; A13 of A, B, the equation b ¼ 0 defines an inverse parabola in 2 2 the ε space with two real roots ε only when ¼ − 2 ¼ N ð Þ d NTNL NTL det : A14 B − 1 > 0: ðA23Þ Similarly, the sound speeds can be expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hence, the condition b ≥ 0 is only satisfied for a 4bd 2 2 2 2 ¼ 1 1 − ð Þ ε ∈ ½ε−; εþ . Since we are only interested in positive c 2 A15 2d a values of ε2, then we conclude that the condition b ≥ 0 imposes a constraint on the maximal allowed strain applied with to our system given by ¼ 2 þ 2 − 2 2 ð Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a cTNL cLNT cTLNTL; A16 AðB − AÞþA þ BðB − 1Þ ε2 ¼ 2 − 2: ðA24Þ ¼ 2 2 − 4 ð Þ max ð − Þ b cTcL cTL: A17 A B A 065015-10 ELASTICITY BOUNDS FROM EFFECTIVE FIELD THEORY PHYS. REV. D 100, 065015 (2019) Analyzing the last condition in (A21) analytically becomes decouple at the level of the quadratic action (A3). Indeed, ε ¼ 0 ¼ 2 ¼ 0 more involved. We find, however, that in the parameter for , we find that NTL cTL . From the positivity region 2 2 of the remaining quantities NT, NL, cT, cL, we arrive at the 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi following set of conditions on the parameters of our model: B ≤ ð2 − A þ 4 − 3A2ÞðA25Þ 2 2 A>0;Aþ BðB − 1Þ > 0; 1 þ Bv > 0: ðA28Þ the maximal strain is determined by the onset of the gradient instability and is thus given by (A24). Only in The propagation speeds of the canonically normalized the region complementary to (A25) is the maximal strain modes are then given as fixed by requiring the absence of superluminal propagation, 2 2 2 2ð þ ð − 1ÞÞ cT ¼ v A cL ¼ v A B B ð Þ finding 2 ; 2 : A29 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NT 1 þ Bv NL 1 þ Bv ð þ − 2Þ ε2 ¼ 2 A A B − 2 ð Þ max 2 : A26 We thus see that with this choice of potential both A þ AðB − 1ÞþðB − 2ÞB propagation speeds in the infinitesimal strain limit scale with v. Hence, in order to go to nonrelativistic speeds, we We present the full constraints on the parameter space just need to set v ≪ 1. Let us also point out that the two obtained numerically in the main text. speeds are related as c2 ¼ c2 þ v2BðB − 1Þ. The second Finally, let us quote our results for the simple case of L T term in this relation comes from the linear bulk modulus, linear deformations, i.e., of zero background shear strain, ε ¼ 0 defined in Eq. (13). For the new choice of potential, it i.e., . We obtain the following allowed region for the ¼ ρ 2ð − 1Þ − 1 exponents A and B: equals K 0v B B, and thus for a negative B , pffiffiffiffiffiffiffiffiffiffiffi the bulk modulus becomes negative. Henceforth, we shall 0 ≤ A ≤ 1 and 1 ≤ B ≤ 1 − A þ 1: ðA27Þ only consider B ≥ 1. The additional term in the potential also enables us to More specifically, the two kinetic eigenvalues in this case expand the allowed parameter space for A, B. In particular, −1þA−2B are equal and given by λ ¼ 2 B imposing the by analyzing the stability conditions (A21), we find that the constraint B>0. The sound speeds are in turn given by maximal strain is only set by the requirement of the absence 2 ¼ A 2 ¼ − 1 þ A c− B and cþ B B. 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