Spontaneous breaking of rotational symmetry in the presence of defects

We prove a strong form of spontaneous breaking of rotational symmetry for a simple model of two-dimensional crystals with random defects in thermal equilibrium at low temperature. The defects consist of isolated missing atoms.


Motivation
Solid state physics is about crystals. In spite of the tremendous achievements and numerous applications of solid state physics, existence of crystals is mathematically not rigorously understood. In particular, understanding the melting transition from crystals to liquids seems out of reach for mathematicians.
One gets a better understanding of crystallization phenomena (classical and liquid crystals) by studying spontaneous breaking or preservation of spatial symmetries like translations and rotations. Preservation of translational symmetry is well understood in two dimensions, see for example [Ric07]. Hardly any mathematical results in realistic models are known in three dimensions. Among the more recent results on translational symmetry breaking in crystalline systems, we mention Aizenman, Jansen, and Jung [AJJ10].
By the work of Theil [The06], crystallization at temperature zero in two dimensions is much better understood than crystals at positive temperature. On a macroscopic scale, geometric rigidity is well understood. This starts with a result of Liouville. Consider a continuously differentiable map such that the derivative at any point is a rotation. By Liouville's result it is indeed globally a rotation. Among the more recent advances, we highlight Friesecke, James, and Müller [FJM02] proving a geometric rigidity result.
In [MR09], spontaneous breaking of rotational symmetry was shown for a toy model of a crystal without defects. However, crystals at positive temperature exhibit defects. These can be all kinds of local defects (e.g. missing atoms) and various non-local defects. In this work, we consider a variant of the model from [MR09] which allows the simplest type of local defects, isolated missing single atoms. Our approach can be generalized in a straightforward way to isolated islands of missing atoms as long as the islands are of bounded size. The model forbids non-local defects like crystal boundaries and dislocation lines by definition. Furthermore, to make the presentation as simple as possible, we work in two dimensions although this is not essential. We see the current work as one step towards a better mathematical understanding of rotational symmetry breaking in crystals. The presence of defects makes a Fourier analysis technique inappropriate for our model. It is replaced by the geometric rigidity result from [FJM02], which therefore is an important ingredient in our analysis.

The model
Assumptions. Throughout, we fix (a) a real-valued potential function V defined in an open interval containing 1. We assume that V is twice continuously differentiable with V ′′ > 0 and V ′ (1) = 0.
(b) α > 0 sufficiently small, depending on V . (More specifically, α needs to be so small that V is defined on [1 − α, 1 + α] and Corollary 2.4 below holds.) (c) l ∈ (1−α/2, 1+α/2). This parameter equals the distance of neighboring particles in the standard configuration defined in (1.10) below. Thus, it is a control parameter for the "pressure" of the system.
Let N ∈ N. We define the set Ω * l,N of configurations ω with periodic boundary conditions to consist of all ω ∈ (C ∪ { }) A 2 such that and ω(x + Nz) = ω(x) for x, z ∈ A 2 with ω(x) = . For x ∈ A 2 , ω(x) ∈ C is interpreted as the location of the particle with index x. If ω(x) = , then there is a hole or a defect associated with x. Note that any ω ∈ Ω * l,N is uniquely determined by its restriction to the set of representatives (1.2) of A 2 /NA 2 . This allows us to identify Ω * l,N with (C ∪ { }) I N . Furthermore, two configurations ω, ω ′ ∈ Ω * l,N are identified if there exists z ∈ A 2 such that for all x ∈ A 2 one has ω(x) = ω ′ (x + Nz). Let Ω l,N be the quotient space with respect to the equivalence relation given by this identification. One may identify Ω l,N with a measurable set of representatives Ω l,N ⊂ Ω * l,N . We introduce the set of representatives for C/lNA 2 . Although the precise choice of the set of representatives for Ω l,N in Ω * l,N is irrelevant, a possible choice is ω(x) ∈ Λ lN for the lexicographically smallest x ∈ I N with ω(x) = if ω is not the constant configuration with value . Let denote the set of defects in the configuration ω. For x ∈ I N and z ∈ {1, τ }, let denote the open triangle with corner points x, x + z, and x + τ z. Let Note that the closures of the triangles in T N cover Λ 1N . Let denote the set of neighbors of 0 in A 2 . The space Ω l,N of allowed configurations consists of all ω ∈ Ω l,N satisfying the following properties (Ω1)-(Ω4): (Ω1) |ω(x) − ω(y)| ∈ (1 − α, 1 + α) for all x, y ∈ A 2 with x ∼ y, ω(x) = , and ω(y) = .
(Ω2) Defects are isolated in the following sense. For all x, y ∈ A 2 , one can have ω(x) = and ω(y) = only if x = y or |x−y| > 2. This means that nearest and next-nearest neighbors of defects are present. (1.8) Extendω piecewice affine linearly to a mapω : C → C requiring thatω is affine linear on the closure of every triangle in T . We require: (Ω3)ω : C → C is bijective.
(Ω4) For all x ∈ A 2 and all z ∈ N , one has In other words,ω preserves orientations.
We remark that we could drop condition (Ω4) because it follows from the other conditions (Ω1)-(Ω3). Since the proof of this fact is more analytic than stochastic and is not needed in the current paper, we skip it. Note that the standard configuration is an allowed configuration. Thus, Ω l,N = ∅. Let m ∈ R; m has the interpretation of a chemical potential. It parametrizes the energetic costs of a defect. Define the Hamiltonian for ω ∈ Ω l,N . Let λ denote the Lebesgue measure on C. We endow (C ∪ { }) I N with the reference measure (λ + δ ) I N . This yields a reference measure on Ω * l,N . Restricting this reference measure to Ω l,N and using the above identification, this defines in turn a reference measure µ N on Ω l,N . Note that µ N (Ω l,N ) < ∞ as a consequence of (Ω1).

Results
We remark that under the assumptions stated at the beginning of Section 1.2, for all β > 0, m ∈ R, N ∈ N with N ≥ 5, x ∈ A 2 , and z ∈ N , one has (1.14) This follows from (1.1) together with the translational invariance of P β,m,N . In particular, under P β,m,N , the distribution ofω(x + z) −ω(x) is not rotationally invariant. Note that |ω(x + z) −ω(x)| is bounded uniformly in N, and thus, equation (1.14) remains true when one takes subsequential weak limits as N → ∞. As a consequence, any infinite volume Gibbs measure obtained as such a subsequential limit is not rotationally invariant. We prove a much stronger form of breaking of rotational symmetry.
Theorem 1.1 Under the assumptions stated at the beginning of Section 1.2, there is a constant m 0 = m 0 (V ), such that the following holds: Corollary 1.2 Under the assumptions of Theorem 1.1, For every triangle ∆ ∈ T ,ω is affine linear on ∆. Hence, its Jacobian ∇ω is constant on ∆; we denote by ∇ω(∆) this constant value. (1.17) Finally, a remark on infinite volume limits. Since the above results are uniform in the size N of the underlying lattice, the finite-volume results carry over to infinite-volume Gibbs measures obtained as subsequential limits as N → ∞.
Organization. In our proof of these results, we proceed as follows. In Section 2, we compare the Hamiltonian of a configuration ω ∈ Ω l,N with the Hamiltonian of the standard configuration ω l . Subsequently, in Section 3, we use these estimates to bound the partition sum from below and the internal energy from above. Our proofs rely crucially on the following rigidity estimate. We use it both locally (in Lemma 2.6), and globally (in Lemma 3.2).
Theorem 1.4 (Friesecke, James, and Müller [FJM02, Theorem 3.1]) Let U be a bounded Lipschitz domain in R n , n ≥ 2. There exists a constant C(U) with the following property: (1.18) We are interested in bounded domains U ⊂ R 2 which are bounded by finitely many pieces of straight lines and in continuous functions v : U → R 2 that are piecewise affine linear with respect to a triangulation of U. Note that these functions belong to W 1,2 (U, R 2 ).

An estimate for the Hamiltonian
We identify C with R 2 . In this section, we prove the following.
Lemma 2.1 There exist constants c 2 (V ) > 0 and m 1 (V ) > 0 such that for all ω ∈ Ω l,N , one has Here and in the rest of the paper, the distance is taken with respect to an arbitrary norm · on 2 × 2-matrices.
First, we estimate the contribution of the Hamiltonian for single triangles. Then, we show that the defects are negligible.

Estimates for individual triangles
Let ∆ be a triangle in R 2 with corner points A 1 , A 2 , A 3 , i.e. the interior of the convex hull of {A 1 , A 2 , A 3 }. Let further ω : R 2 → R 2 be the affine linear map that maps 0, 1, τ to A 1 , A 2 , A 3 , respectively. We assume that (A 1 , A 2 , A 3 ) is positively oriented, i.e. det ∇ω > 0. We introduce the sides of the triangle: Recall that l∆ 0,1 is an equilateral triangle with side length l. Throughout, we write T ≍ S for terms T ≥ 0 and S ≥ 0 if there are uniform constants c, C > 0 such that cT ≤ S ≤ CT holds. If the constants depend on the fixed potential V , we write T ≍ V S.
Lemma 2.2 Let p(l) := 2 √ 3V ′ (l)/l. For sufficiently small α > 0 and side lengths a 1 , a 2 , a 3 ∈ (1 − α, 1 + α), one has Proof. Heron's formula gives the area of the triangle ∆ with side length a 1 , a 2 , and a 3 as The function A is twice continuously differentiable with All second derivatives of A(a 1 , a 2 , a 3 ) are bounded for a 1 , a 2 , a 3 ∈ (1 − α, 1 + α), with α > 0 small enough. Consequently, Since V is twice differentiable, we get using the last equation . The claim follows for α small enough.
We use now the following fact: Assume that S is a compact submanifold of R d , given as a set of zeros for some open set U ⊆ R d and some smooth function f : U → R m , m ≤ d. Assume further that ∇f has rank m on S. Then, there is a neighborhood U ′ ⊆ U of S such that for all (2.13) We apply this fact to S = SO(2), U = {Q ∈ R 2×2 : det Q > 0}, and f : U → R 2×2 sym , f (Q) = Q * Q − Id; its derivative has full rank on S. Forα > 0 sufficiently small and |a j − 1| <α, j = 1, 2, 3, M = ∇ω is close to SO(2); recall that det M > 0 by (Ω4). Consequently, (2)). (2.14) Together with (2.11), this implies the claim. Combining Lemmas 2.2 and 2.3 and scaling with l, which is close to 1, yields the following.

Proof of Lemma 2.1
Let ω ∈ Ω l,N . For x ∈ I N and y ∈ A 2 with x ∼ y, ω(x) = and ω(y) = , we call the undirected edge {x, y} • a boundary edge with respect to ω if there exists z ∈ A 2 with z ∼ x, z ∼ y, and ω(z) = ; • an inner edge with respect to ω otherwise.
We denote the set of boundary and inner edges with respect to ω by ∂E N (ω) and E • N (ω), respectively.
Proof of Lemma 2.1. For all x ∈ I N and y ∈ I N +1 with x ∼ y, one has |ω l (x)−ω l (y)| = l for the standard configuration ω l . Thus, any edge {x, y} contributes the amount V (l) to H m,N (ω l ).
Let ω ∈ Ω l,N . For ∆ ∈ T pres N (ω), let a j (∆), j = 1, 2, 3, denote the side lengths of the triangle ω(∆). For any x ∈ I N with ω(x) = , there are 6 edges incident to x which are neither boundary edges nor inner edges with respect to ω. Consequently, we obtain For the last equation, note that the first term counts only half of the contribution from boundary edges, although their contribution needs to be fully counted.
3 Uniform finite-volume estimates 3.1 Lower bound for the partition sum Let ε > 0. Since V is continuous, for all sufficiently small r > 0, for all N, for all ω ∈ S r,l,N and all x, y ∈ A 2 with x ∼ y, one has |V (|ω(x) − ω(y)|) − V (l)| < ε. Consequently, |H m,N (ω) − H m,N (ω l )| ≤ 3|I N |ε for all ω ∈ S r,l,N and we conclude for all β > 0 We now argue that S r,l,N ⊆ Ω l,N for r ∈ (0, α/4). Using |l − 1| < α/2, we get for all ω ∈ S r,l,N and x, y ∈ A 2 with x ∼ y, Hence, condition (Ω1) is satisfied. Condition (Ω2) is satisfied by absence of defects in S r,l,N . To see thatω is one-to-one, note that for sufficiently small r and ω ∈ S r,l,N , the Jacobi matrix ∇ω is close to the identity matrix and hence v, ∇ω(x)v > 0 for all v ∈ R 2 \ {0} and all x ∈ R 2 for whichω is differentiable at x. Further, the mapω is onto. This is a consequence of the following topological fact. Consider a lattice Γ ⊂ R 2 of rank 2. Then, every continuous map f : R 2 → R 2 with f (x + y) = f (x) + y for all x ∈ R 2 and y ∈ Γ is onto. This shows that condition (Ω3) is fulfilled.
Condition (Ω4) is satisfied for ω l and translations of it, and consequently also for ω ∈ S r,l,N for r sufficiently small. We conclude S r,l,N ⊆ Ω l,N . Thus, µ N (S r,l,N ) = (πr 2 ) |I N |−1 λ(Λ lN ) by the definition of µ N , since integration over ω(x) for all x = 0 given ω(0) yields the factor πr 2 and integration over ω(0) yields the volume λ(Λ lN ). Consequently, we get the assertion (3.1) of the lemma.
Lemma 3.3 There exists a uniform constant c 9 such that the following holds: For all δ > 0, there exist c 10 > 0 and c 11 ∈ R such that for any β ≥ c 9 , m ≥ m 0 := m 1 + 1 (with m 1 as in Lemma 2.1) and any N ≥ 5, one has (3.10) As a consequence, (3.14) For the remaining part, we first apply the inequality xe −x ≤ e −x/2 with x = βA m,l,N , then we use the exponential Chebyshev inequality. This yields We use again the notation σ ω (x) := l −1ω (x) − x for x ∈ C: (3.17) Take an equilateral triangle ∆ ∈ T N with corner points A, B, and C. We claim that with a constant c 12 > 0 not depending on the choice of ∆. Since σ ω is affine linear on ∆, the claim reduces to showing for any matrix M ∈ R 2×2  where the factor λ(Λ lN ) stems from integrating the root of S over the set of representatives Λ lN of C/lNA 2 ; a Gaussian integral arises for each of the |I N | − |D| − 1 edges of S. There exists a uniform constant c 14 > 0 such that and hence Take a uniform constant c 9 so large that for all β ≥ c 9 and m ≥ m 0 = m 1 + 1 one has For these β and m, we get (3.28) Next, we insert the lower bound for the partition sum from Lemma 3.1 with ε := δ/24 and r = r(ε). Using also |T N | ≥ 1, we obtain with constants c 10 > 0 and c 11 ∈ R depending on δ. This yields Claim (3.10). For any given δ > 0, −βδ/8 − log β + c 11 (δ) → −∞ as β → ∞. Consequently, Claim (3.11) follows. This can be seen as follows: For x ∈ A 2 , let θ x : Ω l,N → Ω l,N , θ x ω(y) = ω(y −x) for y ∈ A 2 , denote the shift operator. For any x ∈ A 2 , P β,m,N is invariant under θ x . Consequently, for any∆ ∈ T N and x ∈ I N , we get is a matrix norm and hence equivalent to any other matrix norm on R 2×2 . Thus Theorem 1.1 follows from Theorem 1.3.