The first-order flexibility of a crystal framework
Four sets of necessary and sufficient conditions are obtained for the first-order rigidity of a crystallographic bond-node framework in . In particular, an extremal rank characterisation is obtained which incorporates a multi-variable matrix-valued transfer function defined on the product space . The first-order flex space of a crystal framework is shown to be finite-dimensional if and only if its geometric spectrum, associated with , is a finite set in . More generally the first-order flex space of a crystal framework is shown to be the closed linear span of a set of vector-valued polynomially weighted geometric multi-sequences whose geometric multi-factors in lie in a finite set. Paradoxically, first-order rigid crystal frameworks may possess nontrivial continuous motions which (necessarily) are non-differentiable. The examples given are associated with aperiodic displacive phase transitions between periodic states.
Key words and phrases: crystal, rigidity, flexibility, aperiodic phase transition
Supported by EPSRC grant EP/P01108X/1 Infinite bond-node frameworks
Let be a crystallographic bar-joint framework, or crystal framework, in , where . The vector space of real infinitesimal flexes, or first-order flexes, is the space of -valued velocity fields on the joints of which satisfy the first-order flex condition for every bar. This space contains the finite-dimensional vector space of rigid body motion flexes and, as in the theory of finite bar-joint frameworks (, ), a crystal framework is said to be infinitesimally rigid, or first-order rigid, if . See Owen and Power , for example. There have been a number of recent theoretical accounts of flexibility and rigidity in infinite periodic structures, such as , , , . Also in materials science, over a much longer period, there have been extensive studies of flexibility, stability and phonon modes, such as , , , , . However these accounts generally assume some form of periodic boundary conditions and so far there has been no characterisation given for first-order rigidity per se. In what follows we employ new methods, from a commutative algebra perspective, and obtain four sets of necessary and sufficient conditions for first-order rigidity.
By simple linearity, the rigidity condition is equivalent to the corresponding equality, , for complex scalars and so, as usual, we consider throughout complex velocity fields and complex infinitesimal flexes. An evident necessary condition is the triviality of a geometric flex spectrum associated with a periodic structure on . This spectrum is a subset of the -fold product of the punctured complex plane and may be defined as a natural extension of the rigid unit mode (RUM) spectrum in the -torus which underlies the analysis of low energy phonon modes (mechanical modes) and almost periodic flexes. See , ,  and  for example. A point in the geometric flex spectrum corresponds to nonzero, possibly unbounded, flexes which are -periodic modulo the nonzero multiplicative factors given by the components of . We refer to in this case as a geometrically periodic flex or factor-periodic flex with geometric index or multi-factor . It follows that infinitesimal rigidity implies that the geometric spectrum is trivial in the sense of reducing to the point . Additionally, the space of periodic flexes for the periodic structure, taken in the broader, flexible lattice, sense, must coincide with the -dimensional space of infinitesimal translations. We shall show, in particular, that these two conditions, stated in condition (iii) of Theorem 3.2, are sufficient as well as necessary.
Our main approach is to view the geometric flex spectrum in two other ways. Firstly, in difference equation terms, it is the set of solutions of the characteristic equations of a set of difference equations, for vector-valued multi-sequences, that arises from a choice of periodic structure for . These solutions are the points of rank degeneracy of a matrix-valued transfer function on which is the extension of the symbol function , with domain , which determines the rigid unit modes of . Secondly, in commutative algebra terms, the geometric flex spectrum is related to a natural -module associated with the periodic structure. This module is generated by the rows of the transfer function and the geometric spectrum is associated with maximal containing modules. Our proofs exploit these perspectives together with Noetherian module variants of fundamental arguments in algebraic spectral synthesis which are due to Marcel Lefranc . In particular we use the Hahn Banach separation theorem for topological vectors spaces of sequences, we appeal to Hilbert’s strong Nullstellensatz and Krull’s intersection theorem, and we make use of the Lasker-Noether primary decomposition of Noetherian modules.
For a general crystal framework the space of all first-order flexes is invariant under the natural translation operators and is closed with respect to the topology of coordinatewise convergence. It is of significance then to determine the structural properties of general closed shift-invariant subspaces of the space of vector-valued functions on . This topic is of independent interest and in Theorem 4.10 we generalise Lefranc’s spectral synthesis theorem for the case to this vector-valued setting.
As a further application of these methods we show in Theorem 3.4 that the flex space of a crystal framework is the closed linear span of flexes which are vector-valued polynomially weighted geometric multi-sequences. Moreover there is a dense linear span of this type where the associated geometric multi-factors of the velocity fields are finite in number, where this finiteness derive from the Lasker-Noether decomposition of a -module for . The theorem may thus be viewed as providing an answer, albeit an ambiguous one if the flex space is infinite-dimensional, to the informal question: What are the fundamental first-order modes of a crystal framework ? Another consequence is the following equivalence. A crystal framework has a finite-dimensional flex space if and only if the geometric spectrum is a finite set.
In the final section we show that, paradoxically, a first-order rigid crystal framework may possess a nontrivial continuous motion. Such a motion is necessarily non-smooth and our examples are associated with aperiodic displacive phase transitions between periodic states.
A crystal framework in is defined to be a bar-joint framework where is a countable simple graph and is an injective translationally periodic placement of the vertices as joints . It is assumed here, moreover, that the periodicity is determined by a basis of linearly independent vectors and that the corresponding translation classes for the joints and bars are finite in number. The assumption that is injective is not essential although with this relaxation one should assume that each bar has positive length .
The complex infinitesimal flex space is the vector space of -valued functions on the set of joints satisfying the first-order flex conditions
Coordinates for this vector space and the space of all velocity fields may be introduced, first, by making a (possibly different) choice of linearly independent periodicity vectors for , which we shall denote as
and, second, by choosing finite sets, and respectively, for the corresponding translation classes of the joints and the bars. We refer to the basis choice as a choice of periodic structure for (following terminology from Delgado-Freidrichs ) while the pair represents a choice of motif for this periodic structure , .
2.1. Transfer functions and -modules
Let be the ring of polynomials in the commuting variable over the field . Identify this with the algebra of multi-variable complex polynomials defined on and write for the containing ring of functions on generated by the coordinate functions and their inverses . This is the ring of multivariate complex trigonometric polynomials which we shall refer to as the Laurent polynomial ring.
Let and . Borrowing terminology from the theory of difference equations we now define the transfer function of which is an matrix of functions in determined by the pair . We label the vertices in , and hence the joints of , by pairs where and , for , is the joint .
Let be a crystal framework in with motif and let be the vector for the bar in associated with the edge .
(i) The transfer function is the matrix over the Laurent polynomial ring whose rows are labelled by the edges for the bars of and whose columns are labelled by the vertices for the joints of and coordinate indices in . The row for an edge with takes the form
while if it takes the form
(ii) The -module of , associated with the motif , is the submodule
of the -module , where are the vector-valued functions given by the rows of the transfer function.
For a given periodic structure one may rechoose the set , through an appropriate translation into the positive cone of , so that the multi-variable vector-valued polynomials are replaced by vector-valued polynomials , in for some fixed . Henceforth we assume that this choice has been made. We may therefore define the -module as the submodule of the left -module generated by the vector-valued polynomials . In particular we have
Different choices of for the same periodic structure give transfer functions that are equivalent in a natural way. Specifically, the replacement of a motif edge by an alternative representative results in the multiplication of the appropriate row by a monomial. Also any relabelling of the motif joints and bars corresponds to column and row permutations. It follows that any two transfer functions, and , for a given periodic structure satisfy the equation , where and are diagonal monomial matrices and are permutation matrices.
The values for which the rank of is less than lead to a finite-dimensional space of complex infinitesimal flexes which are periodic up to a multiplicative factor. Such flexes are referred to here as factor-periodic flexes since they are characterised by a set of equations of the form
which relate the (complex) velocity of a joint in to the velocity of the joint for .
Let be a crystal framework in with a choice of periodic structure and labelled motif, and associated transfer function .
(i) The geometric flex spectrum of is the set
(ii) The rigid unit mode spectrum or RUM spectrum of is the subset .
From our earlier remarks it follows that the sets depend only on the choice of periodic structure (up to coordinate relabelling).
2.2. Velocity fields and forms of rigidity
All variants of infinitesimal rigidity depend on a choice of vector space of preferred velocity fields. In this section we define such vector spaces and the resulting forms of periodic and aperiodic infinitesimal rigidity. We first describe a space of exponential velocity fields which plays a key role in our main results.
Let be a vector in which is in the nullspace of . Then the function
defines a factor-periodic velocity field which is an infinitesimal flex , . In this coordinate formalism a complex velocity field for the framework is given by a function (or vector-valued multi-sequence) in where is a combined velocity vector for the joints which are the translates of the motif joints by the vector . Explicitly, with , we have
where is the velocity vector at the joint , and where we have introduced notation for the vertices of the underlying graph .
We now introduce terminology for factor-periodic velocity fields and related velocity fields. Let and write for the geometric multi-sequence given by , for . More generally, a polynomially weighted geometric multi-sequence, or -sequence, is a multi-sequence in of the form , where is a polynomial in . Define , the space of exponential velocity fields, to be the subspace of formed by the linear span of the velocity fields , for all in , all polynomials in and all vectors in . This space does not depend on a choice of periodic structure.
An infinitesimal flex in is referred to as an exponential flex and these vectors determine a subspace, denoted . That is,
We say that is -rigid, or exponentially rigid if .
We next recall various forms of periodic rigidity, each of which is associated with a subspace of .
Given a choice of periodic structure for define to be the associated vector space of periodic velocity fields and write for the subspace of periodic first-order flexes. When there is cause for confusion these flexes are also referred to a strictly periodic flexes, with the periodic structure understood. The periodic flexes are the factor-periodic flexes for the multi-factor . The framework is said to be periodically rigid, or -rigid, if . The inclusion here is proper since infinitesimal rotations are not periodic infinitesimal flexes. The terms fixed lattice rigid, fixed torus rigid and strictly periodically rigid are also used for this notion of rigidity.
A weaker form of periodic rigidity, known as flexible lattice periodic rigidity (and also termed flexible torus rigidity or simply periodic rigidity) is associated with a larger space of velocity fields which have the form
These form an -dimensional space of velocity fields which are periodic modulo an affine correction in which the joints in the -cell each receive an additional velocity . We write this space as and note that we have a direct sum
where is the space of the axial velocity fields, .
Let be a crystal framework. Then
A translational infinitesimal flex , associated with the velocity , has the form with identically equal to . In particular it is strictly periodic. On the other hand let be the rotational infinitesimal flex associated with the orthogonal matrix in , let be the vector of joints from a motif for the periodic structure , and let . Then and
The right hand expression is linear in and so may be written in the form
where and is the linear polynomial with total degree . From these observations the inclusions follow. ∎
A crystal framework is said to be flexible lattice periodically rigid, or -rigid, if .
Let be the point in . A necessary and sufficient condition for periodic rigidity is that the scalar rigidity matrix has rank . Borcea and Streinu  have obtained an analogous necessary and sufficient condition for flexible lattice periodic rigidity. Another derivation of this characterisation is in Power . The rigidity condition is the maximality of the rank of a matrix, which we write here as , which is an augmentation of by columns associated with the entries of the variable matrix , as in the following definition. The maximal rank condition is then
Let be a crystal framework in with motif and let be the vectors associated with the bars in corresponding to edges . The flexible lattice periodic rigidity matrix is the matrix whose rows, labelled by the edges with , have the form
while the rows with take the form
3. The main results
In the next section we define a duality between -modules in and shift-invariant subspaces of the space of velocity fields. It is this duality that underlies the following definition.
The -rigidity module associated with a periodic structure for , and which is also denoted when the periodic structure is understood, is the annihilator of the space in .
We say that a transfer function is rank extremal if for all and the rank of is . Also we say that is rank extremal if its rank is .
The following statements are equivalent for a crystal framework in .
(i) is first-order rigid.
(ii) is exponentially rigid.
(iii) For a given periodic structure there are no nontrivial factor-periodic flexes or flexible lattice periodic flexes.
(iv) For a given periodic structure and motif the transfer function and the matrix are rank extremal.
(v) For a given periodic structure the -module agrees with the rigidity module .
Let be a crystal framework in with a periodic structure with translation classes of joints. A vectorial -sequence for , for this periodic structure, with geometric index , is a velocity field of the form
where is a vector-valued polynomial in .
The term root sequence in the following theorem is defined in Section 3.1 while the term closed refers to the topology for coordinate-wise convergence or, more precisely, the topology of pointwise convergence in the space of velocity fields.
Let be a crystal framework in with a given periodic structure and associated -module . Then is the closed linear span of -sequences in . Moreover, if is a root sequence for the Lasker-Noether decomposition of then is the closed linear span of the -sequences in with geometric indices .
It is possible, although unusual, for a crystal framework to have a finite-dimensional first-order flex space which is strictly larger than the finite-dimensional space of rigid motion flexes, and we give some examples below. The finiteness of the geometric spectrum is a simple necessary condition for this phenomenon and we shall show, from the primary decomposition structure of , that it is also a sufficient condition.
Let be a crystal framework in with a given periodic structure and associated geometric flex spectrum . Then the following statements are equivalent.
(i) is finite-dimensional.
(ii) is a finite set.
In the rest of this section we recall the Lasker-Noether theorem, we give some simple examples to illustrate the main results and some steps of the proofs, and we discuss primary ideals in formal power series rings.
3.1. The primary decomposition of modules for Noetherian rings
The Lasker-Noether theorem states that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules.
Let be a Noetherian ring, let be a submodule of an -module , and for , let be multiplication by . Then is a primary submodule of if is proper and for every the map is either injective or nilpotent. If then is a prime ideal and is said to be a -primary submodule of .
Let be a primary decomposition of the -module where is -primary for distinct primes . A root sequence for is a set of points in where for each the point is a root of in the sense that for all in .
For more details and discussion see Ash , as well as Atiyah and MacDonald , Krull  and Rotman . In particular (Chapter 1 of ) a strong form of the Lasker-Noether theorem asserts that every finitely generated submodule of a Noetherian ring has a decomposition as given in Definition 3.7, and this is called a primary decomposition. Moreover any such decomposition leads to a reduced primary decomposition with distinct primes ideals , and this set of prime ideals is uniquely determined by .
3.2. Examples and remarks
Consider first the simplest possible connected crystal framework in two dimensions, namely the 2D grid framework , whose joints lie on the integer lattice. We show that there is a set of vectorial -sequences with dense span in the flex space.
For a suitable choice of single joint motif, the transfer function has 2 row vector functions, . The corresponding -module in is
Consider and . Then . Moreover is a submodule of and is module-isomorphic to . Thus, for the map is injective if is not a factor of and is zero otherwise. Thus , and similarly , are primary submodules of . Also the ideals and ) are prime ideals in , and in fact they are associated prime ideals in for and , respectively, and is -primary.
We now see that a root sequence , for , can be any pair with in . Let us take . Theorem 3.4 predicts that there is a set of infinitesimal flexes of the form
whose closed linear span is . To see, independently, that this is true consider first the polynomials of the form , with a single variable polynomial. The velocity field
is a velocity field which gives a constant horizontal velocity to the joints on each horizontal line. These are infinitesimal flexes. Moreover it is straightforward to show by direct arguments that the closed span of these flexes give the space of all infinitesimal flexes with this horizontal constancy property, including, in particular, the localised translational flexes which are supported on a single horizontal line of joints. We remark that these localised translational flexes are evidently not in the (unclosed) linear span of vectorial -sequences.
Exchanging the roles of the variables it follows similarly that there are vectorial -flexes whose closed linear span contains the vertically localised flexes. The closed span of the vertically localised flexes and the horizontally localised flexes is the space of all flexes, since one can show that every infinitesimal flex is an infinite linear sum of the line-localised flexes. Thus the conclusion of the theorem is confirmed for .
Another favoured crystal framework example in is the kagome framework, , which is associated with the regular hexagonal tiling of the plane. There are joints and bars in a primitive motif and the -module in has decomposition length . A direct verification of the conclusion of Theorem 3.4 may be obtained as above by exploiting the fact that the line-localised flexes form a generalised basis for the flex space .
A simple 2-dimensional crystal framework which illustrates Theorem 3.5 may be obtained from by adding the diagonal bars , for even. This is Example 3 from the gallery of examples in Badri, Kitson and Power . An explicit primitive motif consists of 2 joints and 5 bars and the RUM spectrum and the geometric spectrum are equal to the set . One can verify directly that the first-order flex space is -dimensional and is spanned by a basis for the rigid motion flexes together with a single geometric flex, with , that restricts to alternating rotational flexes of each diagonalised square subframework.
For a simple -dimensional illustration, with , one may take an infinitesimally rigid crystal framework and attach a parallel copy (with the same period vectors) by means of parallel bars between corresponding joints. In this case the first-order flex space has dimension . More elaborate (connected) examples of the same flavour may be obtained from (disconnected) entangled frameworks  by the periodic addition of connecting bars.
While the pure geometric flexes alone need not have dense span in the flex space they may nevertheless be sufficient for restricted classes of first-order flexes with respect to other closure topologies. This has been shown to be the case for the space of uniformly almost periodic flexes . It would be of interest to develop further such analytic spectral synthesis and to find spectral integral representations for other classes of flex spaces.
The existence of generalised bases of localised geometric flexes for a crystal framework is considered in Badri, Kitson and Power . It seems, as in the case of the kagome framework for example, that such bases give the best way of understanding the first-order flex space and rigid unit modes in that every such flex is an infinite linear combination of basis elements. However such crystal flex bases need not exist and the considerations in  suggest that this is typical unless the geometric spectrum has sufficient linear structure.
One can also consider forms of rigidity, which one might call persistent rigidity, with respect to all periodic structures, both in the strict (fixed lattice) case and the flexible lattice case. The latter form is known as ultrarigidity (see Malestein and Theran ) while the former form we refer to as persistent periodic rigidity. Each may be defined in terms of the vector space of velocity fields which is the union over all periodic structures of the appropriate spaces of periodic velocity fields. These rigidity notions are weaker than strict periodic infinitesimal rigidity but stronger than first-order rigidity.
For a periodic structure for define the rational RUM spectrum to be the intersection of with the points in whose arguments are rational multiples of . Then it can be shown that a crystal framework is persistently periodically rigid if and only if the matrix values of the transfer function on the subset have extremal rank. The analogous characterisation for ultrarigidity, together with detailed algorithmic considerations, is given in .
In view of the unbounded nature of a -flex with nonunimodular geometric multi-factor it might appear that the results of Section 3 have little relevance to physical applications to material crystals. This is definitely not the case however since surface modes, associated with a hyperplane boundary wall for example, arise as bounded restrictions of unbounded flexes of the bulk crystal. See for example Lubensky et al , Power  and Rocklin et al . Thus the geometric spectrum in effect identifies free surfaces where one may find (genuine) surface modes while the crystal -module and its annihilator provides further information.
It seems to us, moreover, that the commutative algebra viewpoint can usefully extend the conceptual analysis of crystal frameworks which, for example, may now be described as primary or properly decomposable according to whether these properties hold for the -module associated with a primitive periodic structure.
3.3. Primary ideals in
In this section we show that a primary -module in with root may be recovered from the -module that it generates in , where is the ring of formal power series in . This connection plays a key role in our main proof, as we discuss in Section 4.1. Since we have not found a satisfactory reference we give the details of this connection in Proposition 3.12 and its proof.
Write for the ring of rational functions in that are continuous on some neighbourhood of . (The notation reflects the fact that if is the ideal in generated by then the set is multiplicative and is the localization .) Since is maximal and therefore prime, is a Noetherian local ring with unique maximal ideal . We also write for the formal power series ring which is also a Noetherian local ring, with unique maximal ideal . Thus we have the natural ring inclusions
That these rings are Noetherian is discussed in Atiyah and MacDonald , for example.
Let be a finitely generated submodule of , let be the corresponding -module in , and let be the corresponding -module in .
Let be a primary submodule in with associated root . Then .
Let be a Noetherian local ring with maximal ideal and let be a finitely generated -module. Then .
Let be a -module in . Then .
The inclusion of in the intersection is elementary. On the other hand the intersection is equal to the set
Write where is the partial sum of the series for for terms of total degree less than . Then the element belongs to . Also the element belongs to . Observe that also belongs to and so it belongs to . Thus lies in the intersection
By the Krull intersection theorem
and so the intersection of (1) is equal to , and the lemma follows. ∎
Proof of Lemma 3.12.
By the previous lemma it suffices to show is equal to , which is the set
Let belong to this set. Then is equal to the finite sum where for all . Thus .
On the other hand, since is a primary -module, the map
is either nilpotent or injective. Since does not vanish at the origin the map is not nilpotent and so it follows that . ∎
4. Shift-invariant subspaces of
Let and let be the topological vector space of vector-valued functions with the topology of coordinatewise convergence. Let be the generators of and let be the forward shift operators, so that , for all and each . A subspace of is said to be an invariant subspace if it is invariant for the shift operators and their inverses, or equivalently if for each . In this section we obtain a spectral synthesis property for closed shift-invariant subspaces of .
4.1. -modules and their reflexivity
There is a bilinear pairing such that, for in and in , Similarly, considering as the space , for and we have the corresponding pairing , where
It is elementary to show that with this pairing the vector space dual of can be identified with . Also, with the same pairing the dual space of the vector space is identified with . Thus both spaces are reflexive, that is, equal to their double dual, in the category of vector spaces. These dual space identifications also hold in the category of linear topological spaces when each is endowed with the topology of coordinatewise convergence, simply because all linear functionals are automatically continuous with these topologies.
For a subspace of we write for the annihilator in with respect to the pairing. Thus
Similarly for a subspace of we write for the annihilator in with respect to the same pairing.
Let be a closed subspace of and let be a closed subspace of . Then and .
This follows from the dual space identifications and from the Hahn-Banach theorem for topological vector spaces (, IV. 3.15). ∎
The following lemma provides a route for the analysis of shift-invariant subspaces in terms of the structure of their uniquely associated -modules . Note that it follows from the Noetherian property that -modules in are necessarily closed.
A closed subspace in is an invariant subspace if and only if is an -module of the module .
For all , and we have and the lemma follows. ∎
4.2. Primary decompositions of Noetherian modules
The Lasker-Noether theorem for a nonzero finitely generated module over a Noetherian ring ensures that is an intersection of a finite sequence of primary modules, , where is -primary for distinct prime ideals . In particular this decomposition applies to the crystal framework module over the Noetherian polynomial ring . The next lemma shows that it is also applicable to the -module .
The Laurent polynomial ring is a Noetherian ring.
The argument is elementary. (Alternatively, if is the multiplicative subset then the ring is isomorphic to the localization , and so is Noetherian by , Corollary 10.20.) ∎
For the rest of this section we let be a proper -module in with primary decomposition
as above, where the -modules are -primary.
Fix , with . Then there exists a point such that if is a polynomial in then .
To see this note that the complex variety is nonempty by Hilbert’s Nullstellensatz , since and hence is a proper ideal. Moreover, there is a point in this variety which is in . Indeed, if this were not the case then the monomial would be zero on the variety of . It then follows from the strong Nullstellensatz (, Theorem 5.99) that for some index the power is in . This implies which is a contradiction. ∎
Write for the -module and note that is recoverable from as the set of elements with in and . It follows from this that we have the decomposition
where the implied modules (the intersections ) are -primary -modules with distinct prime ideals . Moreover each prime ideal has a root in (rather than ).
4.3. Modules and dual spaces for power series rings
For each and associated root , as above, let