 1. Introduction
 2. Elementary bicrystallography
 3. Space groups of homophase bilayers with coincidence lattices
 4. Conclusion
 A1. Graphene bilayers with coincidence lattices
 A2. Hexagonal and rectangular coordinates
 A3. Coincidence and union lattices
 A4. The coincidence pattern [{\cal P}]
 A5. Symmetry groups of twisted graphene bilayers
 B1. Homogeneous dilatation–rotation coincidence lattices for heterophase bilayers
 C1. Pure shear deformation
 C2. 1D dilatation
 References
 1. Introduction
 2. Elementary bicrystallography
 3. Space groups of homophase bilayers with coincidence lattices
 4. Conclusion
 A1. Graphene bilayers with coincidence lattices
 A2. Hexagonal and rectangular coordinates
 A3. Coincidence and union lattices
 A4. The coincidence pattern [{\cal P}]
 A5. Symmetry groups of twisted graphene bilayers
 B1. Homogeneous dilatation–rotation coincidence lattices for heterophase bilayers
 C1. Pure shear deformation
 C2. 1D dilatation
 References
research papers
Crystallography of homophase twisted bilayers: coincidence, union lattices and space groups
^{a}CNRS UMR 8247, Institut de Recherche de Chimie ParisTech, 11 rue Pierre et Marie Curie, 75005 Paris, France
^{*}Correspondence email: denis.gratias@chimieparistech.psl.eu
This paper presents the basic tools used to describe the global symmetry of socalled bilayer structures obtained when two differently oriented crystalline monoatomic layers of the same structure are superimposed and displaced with respect to each other. The 2D nature of the layers leads to the use of complex numbers that allows for simple explicit analytical expressions of the symmetry properties involved in standard bicrystallography [Gratias & Portier (1982). J. Phys. Colloq. 43, C615–C624; Pond & Vlachavas (1983). Proc. R. Soc. Lond. Ser. A, 386, 95–143]. The focus here is on the twist rotations such that the superimposition of the two layers generates a coincidence lattice. The set of such coincidence rotations plotted as a function of the lengths of their coincidence lattice unitcell nodes exhibits remarkable arithmetic properties. The second part of the paper is devoted to determination of the space groups of the bilayers as a function of the rigidbody translation associated with the coincidence rotation. These general results are exemplified with a detailed study of graphene bilayers, showing that the possible symmetries of graphene bilayers with a coincidence lattice, whatever the rotation and the rigidbody translation, are distributed in only six distinct types of space groups. The appendix discusses some generalized cases of heterophase bilayers with coincidence lattices due to specific lattice constant ratios, and mechanical deformation by elongation and shear of a layer on top of an undeformed one.
1. Introduction
The discovery of strong electronic correlations and superconductivity in twisted bilayer graphene (Trambly de Laissardière et al., 2010, 2012), with a socalled magic rotation angle close to 1.05° where the Fermi velocity vanishes, has significantly increased the interest in detailed study (Cao, Fatemi, Demir et al., 2018; Cao, Fatemi, Fang et al., 2018) of these kinds of lowdimension structures [see, for transition metal dichalcogenides, Naik & Jain (2018), Wu et al. (2019), Soriano & Lado (2020), Venkateswarlu et al. (2020)]. The eventual aim is to determine which symmetry property may explain the existence of flat bands in the electronic structure (Suarez Morell et al., 2010): what, in the symmetry properties (if any) of twisted bilayers, is at the origin of this electronic localization?
A robust answer to this question requires a practical and simple crystallographic description of bilayer structures. This is the focus of the present work. The fundamental mathematical aspects of coincidence lattices at any dimensions are to be found in the very elaborated studies of Pleasants et al. (1996), Baake & Grimm (2006), Baake & Zeiner (2017). We focus here on the very elementary practical aspect of investigating the unique case of 2D bilayer structures.
Investigation of the symmetry properties of the superimposition of two 3D crystals, called bicrystals, was carried out in the 1980s (Gratias & Portier, 1982; Pond & Vlachavas, 1983) in the study of the properties of grain boundaries in metals and alloys. Although, at that time, these bicrystals were only theoretical concepts, their 2D versions of superimposing two monoatomic layers make sense in the present context as the idealization of a twisted bilayer considered as the superimposition of two infinitely thin monoatomic layers differently oriented by a twist rotation of angle α perpendicular to the layer plane and displaced with respect to each other by a translation τ in the plane.
The paper is organized as follows. Our first task is to enumerate which specific rotation angles α lead to a situation where two homophase layers share a common say , of index Σ in Λ, and to explicitly give the expressions of these sublattices and those generated by the union of the lattices of the two layers. Our second task is to understand how these specific coincidence angles are distributed with respect to the values of the square length σ (identical to Σ for the square and hexagonal systems) of the coincidence unitcell vectors. Our third task is to determine which is generated for bilayers with coincidence lattices when the rigidbody translation τ varies at constant rotation α. Three appendices give the explicit illustration of the whole process in the case of twisted graphene bilayers and the conditions for coincidence and union lattices to exist in the case of heterophase bilayers obtained by dilatation and/or rotation or mechanical deformation.
We use the following notation: point groups are noted in capital letters like G or W; space groups and translation groups are noted in calligraphic letters like or ; space symmetry operators (or functions in the complex plane as discussed next) are noted as or whereas operators are simply written as α or g.
2. Elementary bicrystallography
As already mentioned, homophase bilayers are ideally defined here as the superimposition of two identical monolayers on top of each other, forming an infinitely thin layer of matter. The twist operation, that transforms the monolayer I into II, is either a rotation–translation acting as = = , or a mirror translation (in all 2D enantiomorphic structures, these two descriptions are equivalent as they describe the same twist operation) oriented along a direction of angle θ with the x axis, acting as = = .
The original monolayer I has [we use the notations of Hahn (2005)] with Γ and lattice Λ showing the holohedral symmetry class of with according to:
(i) Oblique system : ;
(ii) Rectangular system : ;
(iii) Square system : ;
(iv) Hexagonal system : .
The corresponding group and lattice of the second monolayer II are given by
Since any point in the orbit of r under can be equivalently chosen, we characterize the transformation from layer I to II by the set . The inverse transformation from II to I is given by as shown in Fig. 1.
2.1. Using complex numbers for 2D crystallography
2D crystallography is particularly simple to handle using complex numbers. In fact, any 2D vector V = (x,y) in an orthonormal reference frame of the plane is equivalently described by a complex number . Concerning the nodes of a 2D lattice defined by its of vectors a and b, we choose the unitcell vector a along the real axis and its length as the length unit with no loss of generality. The unit vector b is the complex number where ρ is the length of vector b in units and φ the angle of b with the real axis as shown in Fig. 2. A general primitive lattice of unit vectors a = 1 and is then the set of complex numbers
[In addition, ctype lattices encountered in the rectangular symmetry class () are defined as = .] The complex notations of the 2D lattices are given in Table 1.

The symmetry operations act as functions of complex variable f(z) as elementary transformations of complex numbers:
(i) A translation acts on a point z as ;
(ii) A rotation ϕ around the origin transforms z into ;
(iii) A mirror passing through the origin and oriented in the direction θ transforms z into .
Space operators are the usual combinations of point symmetries and translations as shown in Table 2.

2.2. Coincidence angles for homophase bilayers
(This includes bilayers with different monolayers but sharing identical lattices.) General twisted bilayers are quasiperiodic structures built on a module of rank 4. Specific cases arise for particular values of the rotation angle α, called coincidence angles, where the two initial lattices Λ and share a 2D , called the coincidence lattice characterized by the index Σ [defined by equation (9)], the ratio of the unitcell sizes of and . This makes the nodes of the general module of rank 4 condense on a 2D lattice called the union lattice, discussed later, in a similar way to generating periodic approximants from quasicrystals. In fact, as will be shown next, coincidence angles occur an infinite countable number of times and form a uniformly dense set of values on the real axis: any generic twisted bilayer is infinitely close to a coincidence situation which is the only case leading to exact space symmetries of the bilayer.
Finding the proper coincidence angles has been the subject of a very large number of publications for 2D and 3D crystals (see, for instance, Ranganathan, 1966; Grimmer, 1973, 1974, 1984). The most complete and recent analysis of coincidence lattices in 2D crystals has been given by Romeu et al. (2012), a work that we reconsider here briefly using complex notations and that leads to a derivation which is simple and gives explicit expressions for the coincidence and union (homophase) lattices, as discussed next.
Let α be the rotation angle from the first monolayer to the second, both of G. The coincidence lattice, if any, is the common subset of the lattice translations of the monolayers:
A first necessary condition for a coincidence lattice to possibly exist is that a lattice row defined by the node (n,m) with superimposes on another one of the same orbit under by the rotation α around the origin:
The possible generic solutions are listed below according to the crystalline system of the structure. {There are a few specific cases, in particular for the square system, with n^{2}+m^{2} = where the nodes (n,m) and do not belong to the same orbit under [for instance the nodes (3, 4) and (5, 0)]. These cases are not explicitly considered here.}
(i) Oblique system of
2: the generic orbit contains only two terms , so that there are no solutions but the trivial rotation .(ii) Rectangular system {this includes the special case of those specific oblique lattices where which should be considered as ctype rectangular lattices [ cm(m)]} : the generic orbit contains four terms ; the two nontrivial solutions are those where is deduced from (n,m) by the mirrors and .
(iii) Square system of m_{x} and the mirror rotated by up to additional rotations.
: in addition to the rectangle case, new solutions are . All these solutions can be generated by using the mirror(iv) Hexagonal system : here too, all possible solutions are obtained by using the mirror m_{x} and the mirror rotated from the x axis by up to additional rotations of .
Hence, with the exception of the oblique system which presents no generic solutions, the rotation of a lattice node on top of one of its equivalents can be achieved using the two mirror generators of these point groups (see, for instance, Coxeter, 1963) according to: rectangle () mirrors in the directions and ; square () mirrors in the directions and ; hexagonal () mirrors in the directions and .
Let be the rotation around the origin that superimposes the node , on top of z = related to by the mirror oriented along the direction θ as shown in Fig. 3. Putting , we note that = and therefore = so that
and thus
These three relations apply for the rectangle, square and hexagonal systems with the following specific forms:
Rectangle :
Square :
Hexagonal :
These relations are necessary conditions for ensuring two equivalent lattice rows superimpose on each other by the rotation and . Because these two solutions differ only by the constant rotation θ, we consider from now on the unique solution defined by the basic relations
remembering that with each solution δ is associated the solution .
2.2.1. Coincidence lattices in the rectangle system
Ensuring one row in coincidence is of course not sufficient to generate a 2D coincidence lattice: this requires another noncollinear row of lattice nodes to be in coincidence for the same rotation angle.
We discard the oblique system that we know has no generic rows of coincidence whatever the rotation angle and thus no possible coincidence lattice. We focus now on the unique rectangle system since the square and hexagonal systems are specific highsymmetry cases of the rectangle one.
Let {T_{1},T_{2}} be the of the coincidence lattice we seek with in the rectangle system. Because , if it exists, shares at least the same symmetry class as the lattice of the monolayer (see, for instance, Gratias & Portier, 1982) – here the rectangular symmetry 2mm or higher – another coincidence vector exists that is aligned along up to a certain ratio r:
This requires , thus and which is achieved if and only if , i.e. , where p and q are coprime positive integers. Thus, σ is a rational number:
and is a multiple of .
These results confirm in a few calculation steps those obtained by Romeu et al. (2012) following a seminal paper by Ranganathan (1966) in the context of classical 3D crystallography. Here, coincidence lattices in homophase bilayers in the rectangular system exist if and only if the ratio is the square root of a rational number:
[Oblique lattices with as the hexagonal lattice can be considered as rectangular ctype lattices of parameters and therefore show a 2D coincidence lattice when , with .] We conclude therefore that the coincidence angles for the rectangle system are distributed as a uniformly dense countable set of points on the real axis as with .
2.2.2. Explicit expression of the coincidence lattice in the rectangle system
The unit vector T_{2} of is the smallest vector along with integer coordinates
It is obtained by multiplying by q and then dividing the result by :
We first note that putting and with , we obtain , which explicitly shows that, indeed, T_{2} belongs to Λ. We then observe that, as required, T_{2} is orthogonal to T_{1}, but the length of T_{2} is in the ratio ρ with the length of T_{1} only when and therefore, although with at least the same symmetry class as Λ, the coincidence lattice is not necessarily homothetic to Λ in the general case as illustrated in Fig. 4.
Because of relations (7), we have
so that the coincidence lattice is explicitly given by
showing that the coincidence lattice is generated by a lattice characterized by and , rotated by δ with respect to Λ and linearly dilated by .
Since Σ is the index of the translation group in Λ, i.e. the ratio of the surfaces of the of the coincidence lattice with respect to the one of the lattice Λ, we find
which is, indeed, an integer since is a divisor of n^{2}q+m^{2}p.
2.2.3. The union lattice
The other fundamental translation group is the group generated by the union of the lattice translation groups of the two crystals:
or
where , , , , .
Therefore
This shows that, for any coincidence angle and any symmetry class larger than or equal to the rectangular one, is homothetic to in the linear ratio (this ratio applies on each unit vector leading thus to a A) systems with the coincidence lattices given by
of nodes ). It is easily demonstrated that this relation holds for the square and hexagonal (see Appendix2.3. Coincidence patterns
A classical scheme in metallurgy consists of collecting all the possible coincidence angles α, each associated with its Σ index, in a general pattern of points which is the superimposition of all the coincidence angles equivalent to α with respect to the intrinsic symmetries of the layer, each associated with its Σ. In the case of a rectangle system, this pattern can exhibit quite a complicated fine structure due to the arithmetic irregularities introduced by the term in the definition of Σ seen in equation (9). Moreover, this kind of pattern is heavily redundant because of the superimposition of several rotations that are equivalent with respect to the inner symmetry of the layer. In fact, as shown in Fig. 5, a simpler and equally informative pattern is obtained by plotting only one rotation representative in the elementary sector of the of the monolayer, as a function of the square length of the superposition node (n,m) instead of Σ:
A very basic fact is that since the coincidence angles are defined by lattice vectors (n,m), where n and m are coprime integers, these vectors point to those nodes of a 2D lattice known as the set of points visible from the origin, noted here , as shown in Fig. 6. All points of the coincidence pattern are in a onetoone correspondence with those (n,m), of .
In particular, rational rows in the set faithfully mirror the branches in that are asymptotically converging to specific angles δ characterized by their coincidence nodes (n,m) with as exemplified by the rows and corresponding branches drawn in cyan and purple in Figs. 6 and 7.
The simplest way to classify and order these branches is to label them according to Farey sequences f(N) (see, for instance, Hardy & Wright, 1979). The Farey sequence of order N, noted f(N), is the set of fractions m/n where m and n are coprime integers, associated with the nodes (n,m) of the set [see, for instance, in a different context Philippon (2008)], and such that , ordered by size.
We note the following properties:
(i) For any two elements of a sequence, corresponding in the set to the nodes (n_{0},m_{0}) pointing in the direction tan(m_{0}/n_{0}) and (n_{1},m_{1}) pointing in the direction tan(m_{1}/n_{1}), with , the vector (n_{0}+n_{1},m_{0}+m_{1}) pointing along their diagonal is such that
with = = .
(ii) If two elements i and j are consecutive ( j = i+1) in a sequence with then . Because of Bezout's identity, we deduce that beyond (n_{i},m_{i}) and (n_{j},m_{j}) being coprimes, the pairs (n_{i},n_{j}) and (m_{i},m_{j}) are also coprimes.
In fact, because the coincidence angles α run between 0 and π for the rectangle system, the sequences we are interested in here are extended Farey sequences (Halphn, 1877), noted , made of the standard Farey sequence f(N) between (1, 0) and (1, 1) completed by the sequence from (1, 1) to (0, 1), obtained in adding to the original sequence the inverse fractions n/m in opposite order. Such typical extended sequences for the rectangular system, where , are
etc.
For the square system, the possible twist angles run from 0 to with the basic sequences ()
etc.
For the hexagonal system, with twist angles extending from 0 to , the sequences () are
etc.
2.4. Invariance property of the branches
Defining branches of points in the coincidence pattern is pertinent when the points of the same branch, described by a running index k, share the same property independent of this index. To determine which invariance property a branch corresponds to, we note that, because of relations (1), (2) and (3), any two points , associated with the coincidence node (n,m), and , associated with , of the same coincidence pattern are related by
where .
Relation (13) is the key for characterizing the invariance rule for each branch of the pattern.
We consider the case of the rectangle system () and choose two Farey neighbor terms (n_{0},m_{0}) and (n_{1},m_{1}) such that and . We put , , and .
We consider the set of nodes
under their irreducible form [], defining the points in :
with .
As shown in Fig. 6, at constant and running k, these nodes describe rows in that are parallel to the direction (n_{0},m_{0}). At constant k and running , they describe rows in the direction (n_{1},m_{1}). These two rows intersect at the node (n_{0}+n_{1},m_{0}+m_{1}).
Observing that
we note that at constant and running k, the points describe a set of branches in , one for each value of , asymptotic (by upper values for and by lower values for ) to for where all points share the invariance property:
Similarly, from relation (17), at constant k and running , corresponds a set of branches asymptotic (by upper values for and by lower values for ) to for sharing the invariance property
Concerning the irreducibility property, we note that and are both multiples of and therefore k and must be coprime for the node to belong to . Thus, any row in the set generated by a running at constant exhibits only the points that are not multiples of the prime factors of the constant . For example, in the Farey sequence F(0) = [(1,0),(0,1)] where = , the rows parallel to the x corresponding to running k at constant show, in increasing order: all k values for , only odd values of k for , k not a multiple of 3 for , k not a multiple of 2 and 3 for etc. The densest rows correspond to being a prime number. The same behavior is to be found for the rows parallel to the y direction and, extraordinarily enough, for any row parallel to a rational direction.
The branches associated with the smallest values of , designated here as optimal branches because they generate the smallest coincidence ) and (17) is the unity. These are the branches and associated rows colored, respectively, in cyan and purple in Figs. 6 and 7.
are those where the constant in relations (16The two optimal branches in k and defined by the neighbor nodes (n_{0},m_{0}), (n_{1},m_{1}) in the Farey sequence intersect at the node defined by , i.e. at the node (n_{0}+n_{1},m_{0}+m_{1}) which is precisely the term inserted between the two original nodes in the Farey sequence next to the original one.
2.5. Analytical expression of the optimal branches
Although the coincidence angles form a dense enumerable set of points on the trigonometric circle, the proximity of two alpha values does not ensure that of their corresponding σ values. This happens only when the two angles are on the same branch. Two branches are particularly important which are asymptotic to the angles of the generating mirrors of the of the lattice of the monolayer, i.e. for all systems, with additional for a rectangle, for a square and for a hexagonal system. They have a particular importance for bilayers with very small rotations as they allow us to choose the smallestsized coincidence lattices closest to the angle we seek generating the smallest atomic model to be used in electronic calculations.
In the rectangle system, the two extreme asymptotic angles are and associated with the two extreme branches defined by the Farey sequence [ (n_{0},m_{0}) = (1, 0), (n_{1},m_{1}) = (0, 1)]. The relation (4) leads to
We assume (). Using with and , we find
We first observe, as shown in Fig. 8, that each time k and p share the same divisor, changes its value so that the initially unique branch splits into ν subbranches , where ν is the number of divisors of p. [Let be the positive integer of prime factors a, b, c,…; the number of its divisors is .] Similarly, splits into μ subbranches , where μ is the number of divisors of q.
We then note that the same angle δ is shared by the two branches at steps, respectively, k and when , i.e. for for one branch and , for the other, . At that stage and therefore = = . The two branches superimpose every p steps for one branch and q steps for the other with the same Σ values. Hence, the optimal branch for small angles in the rectangle system () is found to be
with a primitive lattice of parameters A = , B = that condenses to a ctype lattice (A+B)/2, when p, q and l are all three simultaneously odd.
For the square system, the situation is much simpler since here . The Farey sequence to be used here is [(1,0),(1,1)] out of which we obtain
The same angle δ is shared by the two branches each time with, then, . This is easily understood by noting that implies and both even, which lead directly to = , = with a σ value twice smaller. The branch asymptotic to is therefore the optimal solution with the smallest defined by
with a primitive lattice defined by A = (1+l+il), .
The case of the hexagonal system is treated in Appendix A and leads to
These results are easily understood by noting that the smallest coincidence angles are obtained when the two superimposed nodes are as close as possible to each other.
Indeed, applying relation (4) to the rectangle system with and n = 1, m = ql leads to
For the square system with and n = l+1, m = l, relation (4) gives
and for the hexagonal system, with and n = 2l+1, m = l
3. Space groups of homophase bilayers with coincidence lattices
of a homophase bilayer with a coincidence lattice is a simple work in principle that follows the same general scheme: the symmetry group of a set of two identical objects taken as a whole is the union of the symmetry elements that are common to both objects and are intrinsic symmetries of these objects plus extra elements, if any, that exchange the two objects as illustrated in Fig. 9It is easily demonstrated that the union of these two sets forms a group (see, for instance, Gratias & Quiquandon, 2020). Here, both the rotation α and the rigidbody translations τ are to be considered in the computations of these two basic sets:
(i) The intersection group contains those symmetry elements of the original layers that are of the same nature and superimpose in space,
where ; this group is never empty since it contains at least the identity and the translation group of the coincidence lattice.
(ii) The additional set of symmetry elements correspond to those extra new elements that exchange the two layers, transforming layer I into II and simultaneously II into I, defined by the intersection of the cosets with designated, if not empty, the exchange set :
The symmetry group of the homophase bilayer, say of translation group , is thus the union
In addition to the group , another fundamental symmetry group of interest is the α and a given rigidbody translation τ, all translations that generate equivalent bilayers, i.e. bilayers that can superimpose on top of each other by an Any two such translations τ and for the same rotation α are said to be equivalent. They form the orbit of τ in .
which generates, for a given rotationBecause the rigidbody translation acts as a global translation of the layers, it is sufficient to consider only the orientational symmetry, i.e. the Γ, instead of the whole . The is obtained as in the preceding case, by considering the intersection of the point groups I and II and the exchange set but with the major change that, since the elements of the exchange set transform τ into its opposite, we must multiply the exchange set by the inversion operation:
where stands for the inversion operator. [A very unfortunate mistake is to be corrected in the work of Gratias & Quiquandon (2020) where the inversion operation has been forgotten in the expression of and improperly added in the one of .] Here again, it is easily shown that is a group.
The translation τ any translation of the lattices of either crystal transforms the bilayer into one of its equivalents. This translation group is the union group introduced in Section 2.2.3. The set of equivalent translations to τ is therefore the orbit of τ in the generated by the product of with the translation group :
of is found owing to the fact that adding toHence, the number of different symmetry groups of the bilayers induced by varying the rigidbody translation τ for a given rotation α is the number of strata of the group (). For example, as shown in Appendix A and Fig. 13, there are only six different space groups for graphene bilayers whatever the rigidbody translation (and whatever the α rotation). It also shows that the natural reference frame to be used for labeling τ is the union lattice. Moreover, the domain of definition of the rigidbody translation τ becomes very narrow and decreases linearly as for coincidence angles α tending towards zero. As a consequence, at very small angles of rotation where the coincidence lattice increases dramatically, the of the union lattice becomes small enough for the rigidbody translation to become physically meaningless. Therefore, in the case of large moiré patterns due to small disorientations, it is not necessary to consider the rigidbody translation in the description (it can be chosen to be the null vector).
3.1. Finding the bilayer groups: point symmetry
The
elements to consider are rotations and mirrors with {0, , , , }.The rotation α commutes with all the rotations of the lattice crystal since
The intersection α.
is thus the set of all the rotations of the of the crystal whatever the value of the coincidence angleOn the other hand, the exchange set contains the mirrors generated by the product of the rotation α by the original mirrors, i.e. mirrors rotated by from the original ones. Indeed, the elements of act on z as = whereas those of act as = .
Therefore the exchange sets contain all the mirrors obtained by a rotation of of the original mirrors of the structure whatever the value of the coincidence angle α. This explains why the coincidence lattice has the same as the original symmetry class of the lattice. For the group , the exchange set that contains all the mirrors is multiplied by the inversion, generating thus an equivalent set of mirrors but rotated by with respect to the initial ones.
3.2. Finding the bilayer groups: space symmetry
The
is easily determined since it is the of with .Concerning the group , the calculation requires a few steps.
Elements of are the elements such that = and elements of are the elements such that = .
From = and = , with being either a rotation = , or a mirror = , we have the general explicit expressions
For an element being possibly included in either set or , the arguments of the variable z must be identical for the equalities to hold for any value of z.
Concerning , the comparison between the lines (28) and (29) shows that the only possible solution for the elements and to be in is two rotations of the same angle ϕ such that
Concerning , the comparison of (28) with (30) shows that the pertinent elements are obtained with and being parallel mirrors such that
with or possibly for the ctype space groups cm and c2mm and the nonsymmorphic ones pg, p2mg, p2gg and p4gm.
Since is a vector of or , we find that:
(i) The rotation is in if τ is such that is a vector of (or ) which is achieved for τ pointing to special positions of the group .
(ii) The mirror is in if τ is such that is a vector of or requiring thus τ to point along the perpendicular bisector of a mirror of and thus τ to align along a mirror of .
These two conditions lead to nontrivial solutions for τ being located at special positions of .
This shows that the τ according to the different symmetry strata of : the number of different possible space groups of the bilayer is equal to the number of symmetry strata of the group .
depends on the value ofMoreover, the type of α: whatever the value of α in that set is, the groups obtained for rigidbody translations τ with the same coordinates in are isosymbolic; their actual representations in space are scaled according to the length and the rotation .
of the bilayer does not depend on the value of the coincidence angle3.3. A simple lowsymmetry example
We consider two bilayers A and B with coincidence lattices built from structures of symmetry class m_{x} with space groups for A and for B. In both cases, the is made of the identity for the intersection group and m_{y} (original m_{x} rotated by plus because of the inversion) for the exchange set: with translation group . This group has three strata expressed in the of : (0,y), (1/2,y) and (x,y) with little groups, respectively, m_{y}, m_{y} and 1. The translation expressed in the of generates the group for the structure A and for structure B, both of translation group , and vice versa for the translation (Fig. 10).
4. Conclusion
To summarize, we find that infinitely many coincidence lattices generically exist down to the rectangle symmetry provided that the ratio ρ of the lengths of the unitcell vectors is the square root of a rational number: . They are generated by specific coincidence rotations of angle α of the form where n and m are coprime integers and can be written as
where and .
With each coincidence lattice is associated a union lattice
homothetic to and which is the translation group of the τ. Both coincidence and union lattices share at least the symmetry class of the original layer. For square and hexagonal systems, the three lattices , Λ and are twobytwo homothetic in the linear ratio .
of the equivalent translations of the rigidbody translationThe complete set of possible coincidence lattices characterized by the rotation angle α and the unitcell size Σ of the corresponding coincidence lattice form a diagram in onetoone correspondence with the socalled set of points visible from the origin and can be analyzed using Farey sequences. They are distributed on branches, each characterized by a geometric invariant relating the sinus of the rotation angle to the square root of the unitcell size.
In the case where a coincidence lattice exists, the τ between the two layers. There are as many different symbolic names of space groups as there are strata in the group of the equivalent translations τ to a given one. These symbolic names do not depend on the value of the rotation α.
of the bilayer depends on the value of the rigidbody translationBecause the group has as translation τ becomes a nonpertinent parameter – analogous to a phason field in quasicrystals – for twisted bilayers with very small rotation.
the unitcell size of which tends to zero for rotations tending to zero, the rigidbody translationA subsequent work will discuss the case of general bilayers where is a module of rank 4 in connection with the notion of 0lattice which is independent of the possible existence of a coincidence lattice.
APPENDIX A
The specific example of graphene bilayers
Graphene has a 2D periodic structure of group with a carbon atom at special position 2b 3m (1/3,2/3). It is described in the complex plane by the primitive hexagonal lattice Λ () defined by
with a carbon atom at position z_{1} = (1+2j)/3 [and equivalently z_{2} = (2+j)/3] as shown in Fig. 11(a). [It turns out that the commonly used notation in the physics community of graphene is to take the hexagonal reference frame with the acute angle instead of the crystallographic definition that uses the angle . This corresponds to choosing as reference frame instead of (1,j). Noting thus that a node z can be equivalently written as we obtain n = N+M and m = N and thus .] The unitcell parameter equal to a = 0.2456 nm is chosen here as the unit length.
The mm are generated by the rotation of located at the origin and transforming (the six hexagonal rotations are ) z into and the mirror along x transforming z into . The orbit G_{z} of a generic point z has thus 12 elements per
elements of 6A1. Graphene bilayers with coincidence lattices
Twisted graphene bilayers are certainly among the most studied materials in the world [see, for instance, the recent review by Geim (2009)], often created, for example, in epitaxial graphene growth on the Cterminated face of Si–C (see Campanera et al., 2007; Hass et al., 2008; Varchon et al., 2008; Bistritzer & MacDonald, 2011). These twisted bilayers are the superimposition of two single graphene sheets slightly twisted with respect to each other by a small angle α of a few degrees or less. It was seen a couple of years ago that these twisted graphene bilayers have remarkable electronic structures (see Trambly de Laissardière et al., 2010). As already mentioned, from the geometric point of view discussed here, the two graphene sheets are considered as infinitely thin and located on the same plane.
Twist rotations of angle α leading to coincidence lattices are infinitely many (see Feuerbacher, 2021). They are characterized by the rotations α that superpose a representative of a given orbit of nodes z = n+jm on top of another point of the same orbit G_{z}. As previously mentioned, because of the high symmetry of the hexagonal system, it is enough to examine the rotation around the origin that transforms the lattice point into as shown in Fig. 11, where n and m are positive coprime integers with :
or
where and , as application of equation (7) to the hexagonal system.
A2. Hexagonal and rectangular coordinates
The connection between the rectangular ctype lattice with reference frame with coordinates both integers or both halfintegers and the hexagonal lattice reference frame both integers is given by
It is easily verified that σ is an integer for both and being halfintegers,
and that
{The relation , found for the rectangular system, is based on = = and leads to – and thus as expected – when is not a multiple of 3, but to when and thus which seems contradictory to the present result. In fact, because , the sum is a multiple of 3 and the actual coincidence
reduces to , which is indeed three times smaller.}A3. Coincidence and union lattices
and
The unit vectors of are T_{1} = n+jm and T_{2} = jT_{1} = ; because of relation (33), this translates into and since T_{2} = jT_{1}:
The coincidence lattice is an hexagonal lattice deduced from the original lattice Λ of the layer by a rotation of δ with unitcell parameter . The calculation of the union lattice leads to the same expression as in equation (10):
which is the original hexagonal lattice rotated by δ with a unitcell parameter linearly shrunk by .
A4. The coincidence pattern
The coincidence pattern is shown in Fig. 12, generated by one unique representative out of the 12 equivalents of the rotation, with the corresponding Σ plotted on a logarithmic scale. As already discussed in Section 2.3, the coincidence points are distributed on branches converging asymptotically to specific rotation values when . Putting = and from equation (14), we obtain
with . The basic invariance relations (13) for each branch are in hexagonal coordinates:
where Cte stands for a constant value. Here, again, the optimal branches are those where for running drawn in cyan and red in Fig. 12. Of greatest importance are the optimal branches associated with the nodes of the initial Farey sequence since they cover the entire angular definition domain of generating the smallest Σ values. We find
The same δ angle is found between the two branches for with . Indeed, since then n_{k,1} = we see that n_{k,1}+m_{k,1} = and, of course, = are both multiples of 3, leading to a three times smaller. This shows that the asymptotic branch is the optimal solution leading to the smallest unit cells:
A5. Symmetry groups of twisted graphene bilayers
Our last task is to analyze the overall symmetry of the graphene twisted bilayers with coincidence lattices. The group is easily found as of translation group whatever the value of the coincidence angle δ in . The group p6mm contains six symmetry strata listed in Table 3. There are thus only six different possible space groups for the bilayer according to the coordinates of τ expressed in units of the union lattice as shown in Table 3 and exemplified in Fig. 13.

The case of very small rotations deserves some attention. Rotations decreasing to zero are associated with coincidence lattices with larger and larger unit cells and therefore to shorter unit cells for the union lattices. As noticed in the body of the text, this leads to rigidbody translations tending to zero and therefore losing any physical pertinence. Small rotations can indeed be locally described as translations between the two almostparallel layers, as shown in Fig. 14(a). The at is the group of the p6mm, scaled by and rotated by δ: the bilayer has exactly the same symmetry properties as the initial graphene but magnified to mesoscopic scales. It can be roughly described as an hexagonal tiling made of three main microscopic highsymmetry structures occurring at each special point of the large hexagonal coincidence unit. In that renormalizationlike view, the centers of the initial hexagons are replaced by the structure of 6mm, usually designated as AA, that is the graphene itself, the carbon atoms by the structure of 3m, called AB(BA), as in the and the binding between carbon atoms, designated as SP, by the structure 2mm shown in Fig. 14(b) of c2m with cell parameters and two carbon atoms at positions (0,1/3) and (1/2,1/3). These three basic highsymmetry structures, 6mm, 3m and 2mm, correspond to the first three special positions of dimension 0 as shown in Table 3 and Fig. 13. This exhausts all the symmetry possibilities: there are no other kinds of structures, for any and any (here, meaningless) values of τ.
APPENDIX B
Heterophase bilayers with coincidence lattices
B1. Homogeneous dilatation–rotation coincidence lattices for heterophase bilayers
Heterophase bilayers are formed by two layers of different structures; they show very similar geometrical properties to the homophase bilayers. However, our present context of using complex numbers allows us to treat here only those heterophase bilayers where the lattices Λ and of the layers can be deduced from each other by a dilatation–rotation, i.e. when
Here, is the dilatation coefficient and α the rotation from Λ to . This kind of transformation is the general case when the lattices belong both to either the square or the hexagonal systems as examplified in Fig. 15.
We choose here to discuss heterophase bilayers in the square system for simplicity.
Let Λ and be the two square lattices of lattice parameters a = 1 for Λ and for :
A coincidence lattice exists if two pairs of integers (n,m) and exist with such that
in which case the coincidence lattice is {T_{1} = n+im, expressed on the of Λ or equivalently , expressed on the of . Explicitly
or by putting , , we obtain
Therefore, two square lattices (it can be easily verified that the same property applies for the case of hexagonal lattices) of different sizes can share a coincidence lattice only if the ratio of the unitcell lengths is the square root of a rational number in which case the area of the coincidence Σ of the area of the of the first lattice and another integer multiple, , of the of the second lattice in the ratio of the rational number .
is simultaneously an integer multipleLet ϕ and be the rotation angles between the of the coincidence lattice and those of, respectively, Λ and (see Fig. 15); we have
and therefore
Also, because :
or
which is consistent with the expression of α given by relation (38).
The union lattice is given by
It is easily checked that, in the homophase case, where , , this relation simplifies to (10).
Fig. 15 gives a simple example of a rotation–dilatation transformation between two square lattices where the node (n,m) = (2,1) of Λ (in red) superimposes on the node of (in blue) by a rotation α and a dilatation . Here and . Thus, we find and . We have with , and with and .
APPENDIX C
Homophase bilayer under mechanical deformation
The two layers can in general be of two different structures of space groups and with lattices, respectively, Λ and . We still designate by the transformation from Λ to ,
as exemplified in Fig. 16. For simplicity, we treat here the case of the initial lattice Λ belonging to the square system.
 Figure 16 ; (right) a shear with coincidence . 
C1. Pure shear deformation
Here results from a pure shear deformation of the square lattice Λ of parameter a = 1 in the direction of angle α with the x axis of Λ and of intensity η:
with the shear direction .
If we choose: (i) the angle α among those generating a coincidence lattice,
as discussed in the body of this article, using , and
(ii) the shear intensity η rational with respect to the square lattice parameter
then the transformation is written as a matrix with rational coefficients,
which generates a coincidence lattice defined by the unit vectors
where .
Acknowledgements
Special thanks are due to Guy Trambly de Laissardière, Vincent Renard, Florie Mesple, Hakim Amara, Bertrand Toudic, Sylvie LartiguesKorinek and Olivier Hardouin Duparc for very helpful discussions during the writing of the present paper. The very impressive and careful proofreading by one of the referees and their many suggestions were essential for improving the quality of the paper.
Funding information
Funding for this research was provided by: Agence Nationale de la Recherche (project ANR FLATMOI 21CE30002904G).
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