Dislocation Lines in Three-Dimensional Solids at Low Temperature

We propose a model for three-dimensional solids on a mesoscopic scale with a statistical mechanical description of dislocation lines in thermal equilibrium. The model has a linearized rotational symmetry, which is broken by boundary conditions. We show that this symmetry is spontaneously broken in the thermodynamic limit at small positive temperatures.


Motivation and background
The perhaps most fundamental mathematical problem of solid state physics is that of crystallization, which in a classical version could be formulated as follows. Let v : R >0 → R be a two-body potential of Lennard-Jones type, and consider the N -particle energy The Grand Canonical Gibbs measure at inverse temperature β > 0 and fugacity z > 0 in finite volume Ω ⊂ R 3 is the point process on Ω given by modulo permutations of particles with the appropriate normalizing constant Z Ω,β,z . The crystallization problem is to prove that at low temperature and high density, i.e., large but finite inverse temperature β and fugacity z, the corresponding infinite volume Gibbs measures have configurations that are non-trivially periodic with the symmetry of a crystal lattice. This problem remains far out of reach. Unfortunately, even the zero temperature case, i.e., studying the limiting minimizers of the finite volume energy, is open and appears to be very difficult. For the zero temperature case in two dimensions, see Theil [21] and references therein; for important progress on the problem in three dimensions, see Flatley and Theil [9] and references therein. Detailed understanding of the zero temperature case is a prerequisite for understanding the low temperature regime, but in addition, any description at finite temperature must explain spontaneous breaking of the rotational symmetry and take into account the possibility of crystal dislocations. This significantly complicates the problem because proving spontaneous breaking of continuous symmetries is already notoriously difficult in models where the ground states are obvious, such as in the O(3) spin model, for which the only robust method is the very difficult work of Balaban, see [3] and references therein.
With a realistic microscopic model out of reach, we start from a mesoscopic rather than microscopic perspective to understand the effect of the dislocations. We expect the latter to be a fundamental aspect also of the original problem. Our model does not account for the full rotational symmetry, but is only invariant under linearized rotations. More precisely, our model for deformed solids in three dimensions consists of a gas of closed vector-valued defect lines which describe crystal dislocations on a mesoscopic scale. For this model, we show that the breaking of the linearized rotational symmetry persists in the thermodynamic limit.
Our model is strongly motivated by the one introduced and studied by Kosterlitz and Thouless [17], and refined by Nelson and Halperin [19], and Young [23]. The Kosterlitz-Thouless model has an energy that consists of an elastic contribution and a contribution due to crystal dislocations. These two contributions are assumed independent. This KTHNY theory explains crystallization and the melting transition in two dimensions, as a transition mediated by vector-valued dislocations effectively interacting through a Coulomb interaction. For a textbook treatment of this phenomenology, see Chaikin and Lubensky [6]. The Kosterlitz-Thouless model for two-dimensional melting is closely related to the two-dimensional rotator model, studied by Kosterlitz and Thouless in the same paper as the melting problem, following previous insight by Berezinskiȋ [4]. For their model of a two-dimensional solid, the assumption that the energy consists of elastic and dislocation contributions, which can be assumed to be essentially independent, is not derived from a realistic microscopic model. On the other hand, the rotator model admits an exact description in terms of spin waves (corresponding to the elastic energy) and vortices described by a scalar Coulomb gas. In this description, the spin wave and vortex contributions are not far from independent, and, in fact, in the Villain version of the rotator model [22], they become exactly independent. Based on a formal renormalization group analysis, Kosterlitz and Thouless proposed a novel phase transition mediated by unbinding of the topological defects, the Berezinskiȋ-Kosterlitz-Thouless transition. In the two-dimensional rotator model the existence of this transition was proved by Fröhlich and Spencer [12]. For recent results on the two-dimensional Coulomb gas, see Falco [8]. In higher dimensions, the description of the rotator model in terms of spin waves and vortex defects remains valid, except that the vortex defects, which are point defects in two dimensions, now become closed vortex lines [13] as in our solid model. Using this description and the methods they had introduced for the two-dimensional case, Fröhlich and Spencer [12,13] proved long-range order for the rotator model at low temperature in dimensions d 3, without relying on reflection positivity. The latter is a very special feature used in [11] to establish long-range order for the O(n) model exactly with nearest neighbor interaction on Z d , d 3. In general, proving spontaneous symmetry breaking of continuous symmetries remains a difficult problem. However, aside from the most general approach of Balaban and reflection positivity, for abelian spin models, several other techniques exist [13,16].
As discussed above, our model, defined precisely in (25), is closely related to the Kosterlitz-Thouless model, see for example [6, (9.5.1)]. Our analysis is based on the Fröhlich-Spencer approach for the rotator model [12,13].
In a parallel study of Giuliani and Theil [14] following Ariza and Ortiz [1], a model very similar to ours is examined, but with a microscopic interpretation, describing locations of individual atoms. In particular, it also has a linearized rotational symmetry.
In [15,18] (see also [2]), some of us studied other simplified models for crystallization. These models have full rotational symmetry, but do not permit dislocations. In [18] defects were excluded, while in the model in [15] isolated missing single atoms were allowed.

A linear model for dislocation lines on a mesoscopic scale 1.2.1 Linearized elastic deformation energy
An elastically deformed solid in continuum approximation can be described by a deformation map f : R 3 → R 3 with the interpretation that for any point x in the undeformed solid f (x) is the location of x after deformation. The Jacobi matrix ∇f : R 3 → R 3×3 describes the deformation map locally in linear approximation. Only orientation preserving maps, det ∇f > 0, make sense physically. The elastic deformation energy E el (f ) is modeled to be an integral over a smooth elastic energy density where the second representation holds under the assumption of rotation invariance; see Appendix A.1.1. From now on, we consider only small perturbations f = id + εu : R 3 → R 3 of the identity map as deformation maps. The parameter ε corresponds to the ratio between the microscopic and the mesoscopic scale. We Taylor-expandρ el around the identity matrix Id using thatρ el is smooth near Id, obtainingρ with a positive definite quadratic form F on symmetric matrices. Under the assumption of isotropy (see Appendix A.1), writing |·| for the Euclidean norm, the general form for F is In elasticity theory, the constants λ and µ are the so-called Lamé coefficients. Even for cubic monocrystals, the isotropy assumption is restrictive for realistic models. While it is not important for our analysis, we nonetheless assume isotropy to keep the notation somewhat simpler. We refer to [6, Chapters 6.4.2 and 6.4.3] for a discussion on the number of elastic constants necessary in order to describe various crystal systems. Summarizing, we have the following model for the linearized elastic deformation energy: for measurable w : R 3 → R 3×3 and for F as in (5).

Burgers vector densities
The following model is intended to describe dislocation lines on a mesoscopic scale as they appear in solids at positive temperature. We describe the solid by a smooth map w : R 3 → R 3×3 replacing the map ∇u : R 3 → R 3×3 from Section 1.2.1. If dislocation lines are absent, the model described now boils down to the setup of Section 1.2.1 with w = ∇u being a gradient field. The field b : is intended to describe the Burgers vector density. It vanishes if and only if w = ∇u is a gradient field. One can interpret b ijk as the k-th component of the resulting vector per area if one goes through the image in the deformed solid of a rectangle which is parallel to the i-th and j-th coordinate axis. The antisymmetry b ijk = −b jik can be interpreted as the change of sign if the orientation of the rectangle is changed. Any smooth field b : R 3 → R 3×3×3 which is antisymmetric in its first two indices is of the where denotes the exterior derivative with respect to the first two indices. Being antisymmetric in its first two indices, it is convenient to write the Burgers vector density b in the form whereb : R 3 → R 3×3 and ε ijk = det(e i , e j , e k ) with the standard unit vectors e i ∈ R 3 , i ∈ [3] := {1, 2, 3}. The integrability condition (9) can be written in the form In view of this equation, one may visualizeb to be a sourceless vector-valued current.

Model assumptions
In linear approximation, the leading order total energy of a deformed solid described by w : R 3 → R 3×3 is modeled to consist of an "elastic" part and a local "dislocation" part: where H el (w) was introduced in (7). The field w consists of an exact contribution (modeling purely elastic fluctuations) and a coexact contribution representing the elastic part of the energy induced by dislocations. Both these contributions are contained in H el (w), while H disl (d 1 w) is intended to model only the local energy of dislocations. The dislocation part H disl (b) ∈ [0, ∞] is defined for measurable b : R 3 → R 3×3×3 being antisymmetric in its first two indices.
We describe now a coarse-grained model for dislocation lines: Dislocation lines are only allowed in the set Λ of undirected edges of a mesoscopic lattice in R 3 . Let V Λ denote its vertex set. As a lattice, the graph (V Λ , Λ) is of bounded degree. To model boundary conditions, we only allow dislocation lines on a finite subgraph G = (V, E) of (V Λ , Λ), ultimately taking the thermodynamic limit E ↑ Λ. We write E Λ if E is a finite subset of Λ. We denote the edge between adjacent vertices x, y ∈ V Λ by {x, y}. The graph (V Λ , Λ) is not intended to describe the atomic structure of the solid, as it lives on a mesoscopic scale. Rather, it is just a tool to introduce a coarse-grained structure which eventually makes the model discrete.
To every edge e = {x, y}, we associate a counting direction, which has no physical meaning but serves only for bookkeeping purposes. Let σ ∈ {1, −1, 0} V Λ ×Λ be the signed incidence matrix of the graph (V Λ , Λ), being defined by its entries if e is an ingoing edge into v, −1 if e is an outgoing edge from v, 0 otherwise.
The Burgers vectors on the finite subgraph G = (V, E) are encoded by a family I = (I e ) e∈E ∈ (R 3 ) E of vector-valued currents flowing through the edges in counting direction. They should fulfill Kirchhoff's node law We now impose an additional discreteness condition on I, which encodes the restriction that Burgers vectors should take values in a microscopic lattice reflecting the atomic structure of the solid. Let Γ ⊂ R 3 be a lattice, interpreted as the microscopic lattice (scaled to length scale 1). We set Note that the current I is indexed by the edges in the mesoscopic graph (V, E) but takes values in the microscopic lattice Γ. One should not confuse the mesoscopic graph (V, E) nor the mesoscopic lattice Λ with the microscopic lattice Γ; they have nothing to do with each other. For every edge e ∈ E, let n e ∈ R 3 denote the unit vector pointing in its counting direction. Before smearing out, we model the vector-valued current associated to the current configuration I to be described by the following matrix-valued measure J(I) : Borel(R 3 ) → R 3×3 on R 3 , supported on the union of all edges: The Burgers vector densityb(I) associated to I is then modeled by the convolution of J(I) with the form function ϕ:b It is shown in Appendix A.2 thatb(I) is sourceless, i.e., (12) holds for it. Altogether, this yields the Burgers vector density as a function of I: For a graphical illustration of I and b(I) see Figure 1. From now on, we abbreviate H disl (I) := H disl (b(I)) and supp I := {e ∈ E : I e = 0}. We require the following general assumptions: Assumption 1.
• Locality: For I = I 1 + I 2 with I 1 , I 2 ∈ I such that no edge in supp I 1 has a common vertex with another edge in supp I 2 we have H disl (I) = H disl (I 1 ) + H disl (I 2 ). Moreover, H disl (0) = 0.
• Lower bound: For some constant c > 0 and all I ∈ I, Condition (20) reflects the local energetic costs of dislocations in addition to the elastic energy costs reflected by H el (w).
One obtains typical examples for H disl (I) by requiring a number of assumptions on the form function ϕ and Roughly speaking, the last condition means that different edges e ∈ Λ do not overlap too much after broadening with supp ϕ. Finally, for all b, for some constant c 1 > 0, and One particular example is obtained by taking equality in (22).

Model and main result
One summand in the Hamiltonian of our model is defined by for all I ∈ (R 3 ) E satisfying (15). The condition that w is compactly supported reflects the boundary condition: Close to infinity the solid must not be moved away from its reference location. The symmetry with respect to linearized global rotations is reflected by the fact H el (w) = H el (w + w const ) for every constant antisymmetric matrix w const ∈ R 3×3 . Only the boundary condition, i.e., only the restriction that w should have compact support, breaks this global symmetry. This paper is about the question whether this symmetry breaking persists in the thermodynamic limit E ↑ Λ.
Because H el is positive semidefinite, H * el is positive semidefinite as well, cf. (63) below. This gives us the following linearized model for the dislocation lines at inverse temperature β < ∞: here δ I denotes the Dirac measure in I ∈ I and we use the convention e −∞ = 0 throughout the paper.
Whenever the E-dependence is kept fixed, we use the abbreviations Z β = Z β,E and P β = P β,E .
Remark. If we compare our model to the rotator model, purely elastic deformations correspond to "spin wave" contributions, while deformations induced by the Burgers vectors correspond to "vortex" contributions. In our model, the purely elastic deformations are orthogonal to the deformations induced by the Burgers vectors in a suitable inner product · , · F ; this is made precise in Eq. (67) below. Therefore, we do not model the purely elastic part stochastically. It would not be relevant for our purposes, because in a linearized model, it is expected to be independent of the Burgers vectors anyway.
Our first result shows that any sequence of smooth configurations satisfying the boundary conditions (i.e., being compactly supported) with prescribed Burgers vectors has a limit w * in L 2 provided that the energy is approaching the infimum of all energies within the class. We show later that this limit is a unique minimizer of H el in a suitable Sobolev space. An explicit description of w * is provided in Lemma 5 below.
Lemma 2 (Compactly supported approximations of the minimizer). For any I ∈ I, there is a bounded smooth function w * (·, In the whole paper, constants are denoted by c 1 , c 2 , etc. They may depend on the fixed model ingredients: the microscopic lattice Γ, the mesoscopic lattice Λ, the constant c from formula (20), and the form function ϕ. All constants keep their meaning throughout the paper. Similarly, the expression "β large enough" means "β > β 0 with some constant β 0 depending also only on Γ, Λ, c, and ϕ". The following theorem shows that the breaking of linearized rotational symmetry w ; w + w const induced by the boundary conditions persists in the thermodynamic limit E ↑ Λ, provided that β is large enough.

Theorem 3 (Spontaneous breaking of linearized rotational symmetry).
There is a constant c 2 > 0 such that for all β large enough and for all t ∈ R, and consequently We remark that the symmetry I ↔ −I implies that w * (x, I) is a centered random matrix. Since w * (x, I) encodes in particular the orientation of the crystal at location x, this result may be interpreted as the presence of long range orientational order in the thermodynamic limit.
Organization of the paper. In Section 2, we identify the minimal energy configuration w * in the sense of Lemma 2 in the appropriate Sobolev space. Section 3 deals with the statistical mechanics of Burgers vector configurations by means of a sine-Gordon transformation and a cluster expansion in the spirit of the Fröhlich-Spencer treatment of the Villain model [12,13]. Section 4 provides the bounds for the observable, which manifests the spontaneous breaking of linearized rotational symmetry. It uses variants of a dipole expansion, which we provide in the appendix.

Minimizing the elastic energy
In this section, we collect various properties of H * el (I) defined in (23). In particular, we prove Lemma 2.

Sobolev spaces
Let V be a finite-dimensional C-vector space with a norm | · | coming from a scalar product · , · V . For integrable f : denote its Fourier transform, normalized such that the transformation becomes unitary. For any We set The Fourier transform f →f gives rise to a natural isometric isomorphism For any α ∈ R, the sesquilinear form extends to a continuous sesquilinear form endowed with the norm For any α ∈ R, we introduce the exterior derivative d j : for j ∈ N 0 and d −1 := 0, d * −1 := 0. They are adjoint to each other in the sense that for any α ∈ R, The Laplace operator ∆ : To see this, we calculate and

Elastic Hamiltonian
Given I ∈ I and b = b(I), we calculate now H * el (I). To begin with, we observe that H el , introduced in (7), is a quadratic form, and therefore it can be written as with a sesquilinear form · , · F depending on F defined in (5). More precisely, using A , B = Tr(AB t ) for the Euclidean scalar product for matrices A, B, we introduce · , · F : Because of the stability condition (5), the inner product · , · F is positive semidefinite. For any α ∈ R, we consider the restriction of · , We claim that a convenient choice is To see this, we observe that ∆ −1 commutes with d 1 and d * 1 because ∆ −1 corresponds to multiplication with the scalar |k| −2 in Fourier space, and d 1 , d * 1 correspond to (multi-component) multiplication operators in Fourier space, as well. Therefore, Using that ker(d 1 :
At this moment, we are most interested in the case α = −1; the case of general values for α is needed for regularity considerations in the proof of Lemma 2 later on.
This is best seen using a Fourier transform and the Cauchy-Schwarz inequality: where k · ψ(k) denotes the Euclidean scalar product in C 3 . Using fact (59) and the stability condition for µ and λ given in (5), which implies µ + λ > 0, we obtain also claim (56).
Definition of the minimizer. In the next lemma, it is shown that the minimizer of the elastic energy has the following form: with w b defined in (47), Lemma 5 (Minimizer of the elastic energy). The infimum in (49) is a minimum: It is unique in the following sense: For all w ∈ L 2∨ 0 (V 1 ) with d 1 w = b(I) and w , w F = w * , w * F , we have w = w * . The summands of the minimizer w * given in (61) have the following components: Proof. The calculation shows that the function ψ * solves the system of equations or equivalently, using (51) and (61), By the above, the following calculation shows that w * is a minimizer in (49) as claimed: For all f ∈ L 2∨ 1 (V 0 ): Furthermore, using (56) we obtain : In particular, d 0 f = 0 implies d 0 f , d 0 f F > 0, which yields the claimed uniqueness of the minimizer. Let i, j ∈ [3]. The identity (64) follows from the definition (47) of w b . Using it, we express v b ∈ L 2∨ −1 (V 0 ) as follows: One term dropped out in the last step because of the antisymmetry Using l,m ∂ m ∂ l b lmk = 0 from the antisymmetry b lmk = −b mlk , this equals This shows that d 0 ψ * has the form given in (65).

Regularity of the minimizer.
Proof of Lemma 2. We set L 2∨ >α (V) : . By Sobolev's embedding theorem, w * is a bounded smooth function with all derivatives being bounded. In particular, pointwise evaluation w * (x) of w * makes sense for every x ∈ R 3 .

Cluster expansion
We now develop a cluster expansion (polymer expansion) of the measures P β,E defined in (25), using the strategy of Fröhlich and Spencer [13]. In the following, E Λ is a given finite set of edges in the mesoscopic lattice. We take the thermodynamic limit E ↑ Λ only in the end.

Sine-Gordon transformation
Because the quadratic form H * el defined in (23) is positive semidefinite the function exp{−βH * el } is the Fourier transform of a centered Gaussian random vector φ = (φ e ) e∈E on some auxiliary probability space with corresponding expectation operator denoted by E: For any observable σ : I → R and β > 0, we define Here, in order to exchange expectation and summation, we used that e −βH disl (I) is summable over the set I by (20). Note that Z β = Z β (0) implies

Preliminaries on cluster expansions
In this section, we collect some background on cluster expansions (polymer expansions). For recent treatments of cluster expansions, see in particular Poghosyan and Ueltschi [20] or Bovier and Zahradník [5] and references. To make our presentation most accessible, we use the textbook version given in [10]. Let B denote the set of all non-empty connected subsets of E. We call X, Y ∈ B compatible, X ∼ Y , if no edge in X has a common vertex with an edge in Y . Otherwise X, Y are called incompatible, X ∼ Y . In particular, X ∼ X. Recall supp I = {e ∈ E : I e = 0} for I ∈ I. Let J = {I ∈ I : supp I ∈ B}.
The incompatibility relation ∼ on B is inherited to an incompatibility relation, also denoted by ∼, on J via Every subset of E can be uniquely decomposed in a set of pairwise compatible connected components, which is a subset of B. For n ∈ N, let J n ∼ ={(I 1 , . . . , I n ) ∈ J n : I i ∼ I j for all i = j}.
Consider I ∈ I and the connected components X 1 , . . . , X n (n ∈ N 0 ) of supp I. We set I j := I1 X j ∈ J .
Here it is crucial that the Kirchhoff rule (15) holds for I if and only if it holds for all I j . Then, using the locality of H disl given in Assumption 1, we obtain For I ∈ I and some β > 0, we abbreviate K(I, φ) := e i φ , I e −βH disl (I) .
The function K has the following important factorization property: For I ∈ I with connected components I 1 , . . . , I n as above, one has In view of the definition (76) of Z β,φ , this yields The summand 1 comes from the contribution of I = 0, using H disl (0) = 0. Recall that by (20) |K(I, φ)| = e −βH disl (I) ≤ e −βc I 1 is summable over I ∈ I, which shows that all expressions in (85) are absolutely summable. To control Z β,φ , we use a cluster expansion. Next we cite the relevant theorems. Let LetJ be any finite set endowed with a reflexive and symmetric incompatibility relation ∼. We defineJ n ∼ by (81) with J replaced byJ .
Moreover, in this case, the series (88) is absolutely convergent.

Partial partition sums
We take a sequence ( By monotone convergence for series, For I ∈ I we define its size size I := I 1 + diam supp I. Here diam denotes the diameter in the graph distance in the mesoscopic lattice G = (V, E). The size has the following properties. For I 1 , I 2 ∈ I with I 1 ∼ I 2 , one has Recall that I takes values in the microscopic lattice Γ. We set and observe for all I ∈ I: If in addition supp I is connected, we have diam supp I ≤ | supp I| and hence The following lemma serves to verify the hypothesis (89) of the cluster expansion.
Dropping the condition that J should fulfill the Kirchhoff rules, we obtain the following bound for any given X ∈ B: with the abbreviation Because Γ is a three-dimensional lattice, for any k ∈ N there are at most c 9 k 2 lattice points within distance [ηk, η(k + 1)) from 0, where c 9 > 0 is a constant only depending on Γ. Thus for all large β, uniformly in I ∈ J m . Here we have used that e −βc 7 M ≤ βc 5 η for large β.
For (I 1 , . . . , I n ) ∈ J n m with U (I 1 , . . . , I n ) = 0 and I = I 1 + · · · + I n as in the above summation, we have We choose a "reference edge" o ∈ supp I. Substituting (117) The inner sum on the right-hand side can be extended to run over all n-tuples (I 1 , . . . , I n ) in J n m with o ∈ supp I 1 ∪ . . . ∪ supp I n , since by definition, for any I which cannot be written as a sum of such I 1 , . . . , I n , for some n ∈ N, z + m (β, I) = 0 or o / ∈ supp I. It follows As we observed above, it suffices to consider only finite m in the claim (113). It remains to show that for β large enough it is true that yielding claim (120). Since |z m (β, I)| ≤ z + m (β, I), the claim (114) is an immediate consequence of (113).

Gaussian lower bound for Fourier transforms
Next, we apply a cluster expansion with K(I, φ) defined in (83) to obtain a representation of Z β,φ and finally a bound for the Fourier transform of the observable.
The last series converges absolutely and its absolute value is bounded by I∈I z + (β, I) < ∞.
We set Since size I is bounded away from 0, there is a constant c 12 > 0 such that R(I, o) ≤ c 12 size I. By the definition (19) of b(I), one has the bound b lij (I) 1 ≤ c 13 I 1 ≤ c 13 size I for all its components, with some constant c 13 > 0. Hence, we obtain Because b is compactly supported and divergence-free in the sense of equation (12), by the fundamental theorem of calculus. Recall the representation w * = w b + d 0 ψ * with w b , d 0 ψ * as in (64) with the constant c 14 = 72c 12 c 13 /π. The stability condition (5) implies |λ|/|2µ + λ| ≤ 1. In the same way as in (148) with the constant c 15 = 1944c 12 c 13 /π. It follows still in the case v(o) = 0 The next step involves translation-invariance: Shifting both x and I by a mesoscopic lattice vector v ∈ V Λ does not change w * ij (x, I) because (x, I) → b(I)(x) has the same translation-invariance. Because the inequality (150) is written in a translation-invariant form, it holds also if we drop the assumption v(o) = 0. This yields for all β ≥ β 1 for sufficiently large β 1 , neither depending on o, x, nor I. Case 2: Next we consider the case |v(o) − x| < 2R (I, o). We recall the definition of J jk (I) from (17). We now use the symbol · 1 in two different ways. On the one hand, I 1 = e∈E |I e | for I. On the other hand, J jk (I) 1 denotes the total unsigned mass of the signed measure J jk (I) given by the following definition: For any signed measureJ on R 3 with Hahn decompositionJ =J + −J − , we define J 1 :=J + (R 3 ) +J − (R 3 ). With this interpretation, we have Combining this with (210) and (211)

Identifying long-range order
We finally prove now our main result.
Proof of Theorem 3. Applying Lemma 11 yields for t ∈ R and E Λ We may drop the summand indexed by I = 0 because σ ij (x, y) , 0 = 0. Inserting (143) and employing Lemma 9 in the last line in (159) with a constant c 2 = c 2 (c 11 , c 7 ) > 0, where c 11 was defined in (142). Mind that c 2 does not depend on x, y, i, j, E, β. This proves the first claim. By Theorem 3.3.9 in [7], for β large enough, the variance of σ ij (x, y) , I exists and fulfills var P β,E ( σ ij (x, y) , I ) ≤ E P β,E [ σ ij (x, y) , where the last inequality is a consequence of the lower bound (26).
We remark that the reflection symmetries H el (−w) = H el (w) and H disl (−I) = H disl (I) imply that w * (x, −I) = −w * (x, I) and w * (x, I) are equal in distribution with respect to P β,E , jointly in x ∈ R 3 . In particular, the first inequality in (160) is actually an equality.
Note that reflections R ∈ O(3) \ SO(3) or singular or orientation reversing linearized deformations M ∈ R 3×3 with det M ≤ 0 do not make sense physically in this context.
For M, N ∈ GL + (3) it is equivalent that M t M = N t N and that there exists R ∈ SO(3) such that N = RM . As a consequence, ρ el (M ) is a function of M t M . We setρ el (M t M ) := ρ el (M ).
Note that the assumption (rot inv) of rotational invariance does not imply isotropy of the solid, which is defined by ρ el (M R) = ρ el (M ), (R ∈ SO(3), M ∈ GL + (3)). (isotropy) Thus, anisotropy means that first rotating the solid and then deforming it with a given deformation might cost a different elastic energy than deforming it with the same deformation without rotating it first. Although the isotropy assumption is an oversimplification for any real monocrystal, we assume it to keep the presentation simple.
If the assumption (isotropy) holds, theñ ρ el (A) =ρ el (R t AR) holds for all positive definite matrices A = A t and R ∈ SO(3). Taylor-expanded this means for all symmetric matrices U = U t and R ∈ SO(3). Thus, F (U ) depends only on the list a, b, c of eigenvalues of U (with multiplicities). For diagonal matrices U = diag(a, b, c) the only quadratic forms which are symmetric in a, b, c are linear combinations of (Tr U ) 2 = (a + b + c) 2 and |U | 2 = a 2 + b 2 + c 2 .
Thus, under an isotropy assumption, we have with real constants λ and µ and e t = (1, 1, 1). The matrix µId + λ 2 ee t has the double eigenvalue µ with eigenspace e ⊥ and a single eigenvalue µ + 3λ/2 with eigenspace Re. Hence the quadratic form F is positive definite if and only if µ and λ satisfy the conditions given in (5). Summarizing, we have the model for the linearized elastic deformation energy given in (6)-(7).

A.2 From Kirchhoff 's node rule to continuum sourceless currents
We show that Kirchhoff's node rule for the discrete current I implies absence of sources for its smoothed variantb(I). For e ∈ E from x ∈ V to y ∈ V in its counting direction, we write x = v − (e) and y = v + (e). We rewrite (18) (j, k ∈ [3], I ∈ (R 3 ) E with (15), x ∈ R 3 ).
As a consequence of Kirchhoff's rule (15),b(I) is indeed divergence-free: (divb(I)) k (x) = In this appendix, we derive a simplified version of the dipole expansion for the Coulomb potential, which we need as ingredient to identify long-distance bounds for the observable w * in the proof of Lemma 12.