Abstract
A peculiar feature of quantum states is that they may embody socalled projective representations of symmetries rather than ordinary representations. Projective representations of space groupsthe defining symmetry of crystalsremain largely unexplored. Despite recent advances in artificial crystals, whose intrinsic gauge structures necessarily require a projective description, a unified theory is yet to be established. Here, we establish such a unified theory by exhaustively classifying and representing all 458 projective symmetry algebras of timereversalinvariant crystals from 17 wallpaper groups in two dimensions189 of which are algebraically nonequivalent. We discover three physical signatures resulting from projective symmetry algebras, including the shift of highsymmetry momenta, an enforced nontrivial Zak phase, and a spinless eightfold nodal point. Our work offers a theoretical foundation for the field of artificial crystals and opens the door to a wealth of topological states and phenomena beyond the existing paradigms.
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Introduction
Symmetry groups and their representations are at the heart of physics. When going from classical to quantum physics, a classical symmetry group G becomes represented in the Hilbert space, where it makes no physical difference if all states are multiplied by a global phase. It follows that the representation allows an extra phase factor, i.e., for g_{1}, g_{2} ∈ G, their representations ρ(g_{1}) and ρ(g_{2}) may satisfy ρ(g_{1})ρ(g_{2}) = ν(g_{1}, g_{2})ρ(g_{1}, g_{2}) with ν(g_{1}, g_{2}) ∈ U(1). This is known as the projective representation of G, and the phase factors ν called the factor system for this representation. As a wellknown example, classifying the projective representations of Poincaré group for elementary particles leads to the two types of particles, bosons and fermions, corresponding to two distinct factor systems^{1}.
The defining symmetries for crystals are the space groups. What is the physical meaning of a projective representation in this context? Consider a spinless quantum particle on a twodimensional (2D) lattice as shown in Fig. 1a. A projective representation for the lattice translations allows (hereafter, we use bold letters to denote the represented symmetry operators) \({{\mathsf{L}}}_{a}{{\mathsf{L}}}_{b}{{\mathsf{L}}}_{a}^{1}{{\mathsf{L}}}_{b}^{1}={e}^{i\theta }\), from which one observes that the extra phase factor corresponds to a gauge flux through the lattice. This shows that projective representations of space groups are associated with gauge fluxes, and somewhat explains the previous ignorance of them in textbooks on solidstate physics^{2}. Because most electronic crystals are free of gauge flux, one can show that their descriptions are restricted to ordinary representations. Nevertheless, it was recognized that rich gaugeflux configurations can emerge in certain strongly correlated spin systems, where projective representations of space groups are needed for their description^{3,4,5,6,7,8}.
The rise of artificial crystals in recent years completely changes the situation. Artificial crystals cover a wide range of systems, such as acoustic, photonic, mechanical, circuit, and coldatom systems^{9,10,11,12,13,14,15,16,17,18,19}. Most artificial crystals intrinsically preserve timereversal (T) symmetry, which allows fluxes 0 and π over the lattices. A salient feature is that these lattice gauge fluxes can be readily engineered. Recent works showed that these fluxes modify the physics in a fundamental way and projective representations are indispensable for understanding artificial crystals^{20,21,22,23,24,25}. This urgently calls for a unified theory of projective representations of symmetries for Tinvariant crystals, which constitutes the foundation of the whole field.
In this work, we develop such a theory and predict its distinguishing consequences. First, we characterize all possible projective symmetry algebras (PSAs) with timereversal symmetry for any space group. This is demonstrated by 458 PSAs—189 of which are algebraically independent—for all 17 wallpaper groups in two dimensions. Then, we show all the 2D PSAs can be systematically realized by lattice models with appropriate gauge fluxes. Finally, we present three signature results of PSAs, including the shift of highsymmetry momenta, an enforced nontrivial Zak phase, and a spinless eightfold nodal point.
Results
Projective symmetry algebras with timereversal invariance
We start by presenting a general result that reduces the problem for systems with time reversal symmetry T. Let G be the space group, then the total symmetry group is \(G\times {{{{{{{{\mathcal{Z}}}}}}}}}_{2}^{T}\), where \({{{{{{{{\mathcal{Z}}}}}}}}}_{2}^{T}=\{E,T\}\) is the twoelement group generated by T. Mathematically, the classification of all possible factor systems for this group corresponds to the second group cohomology \({H}^{2,c}(G\times {{{{{{{{\mathcal{Z}}}}}}}}}_{2}^{T},U(1))\)^{26}. We have proven that due to the antiunitary character of T, \({H}^{2,c}(G\times {{{{{{{{\mathcal{Z}}}}}}}}}_{2}^{T},U(1))\cong {H}^{2}(G,\, {{\mathbb{Z}}}_{2})\times {H}^{2}({{{{{{{{\mathcal{Z}}}}}}}}}_{2}^{T},{{\mathbb{Z}}}_{2})\) with \({{\mathbb{Z}}}_{2}=\{\pm 1\}\) (see Methods). Hence, the computation is simplified to deriving \({H}^{2}(G,\, {{\mathbb{Z}}}_{2})\), since it is known that \({H}^{2}({{{{{{{{\mathcal{Z}}}}}}}}}_{2}^{T},\, {{\mathbb{Z}}}_{2})\cong {{\mathbb{Z}}}_{2}=\{\pm 1\}\) distinguishes integer and half integer spins, respectively. This means we only need to consider G with factors restricted to \({{\mathbb{Z}}}_{2}=\{\pm 1\}\). Our discussion below focuses on spinless systems, which are pertinent to most artificial crystals. It can be directly extended to spinful systems, as we shall comment at the end.
The second group cohomology \({H}^{2}(G,\, {{\mathbb{Z}}}_{2})\) can be derived from the abstract group cohomology theory, e.g., from the twisted tensor product of the cochain complex of the translation subgroup and that of the point group^{27}. Considering the 17 wallpaper groups for 2D, the results are listed in the second column of Table 1. Besides the classification, we also need to know the content of each \({H}^{2}(G,\, {{\mathbb{Z}}}_{2})\), namely, the concrete algebraic relations satisfied by the symmetry operators, which are called the PSAs, because they are directly related to the physics of a system. We have worked out all PSAs in terms of generators of each group, as listed in the fourth column of Table 1. The technical details are given in Supplementary Note 2. The classification is complete, meaning that any Tinvariant crystal system in two dimensions must belong to one of the PSAs listed here.
Interestingly, each \({H}^{2}(G,{{\mathbb{Z}}}_{2})\) is a product of \({{\mathbb{Z}}}_{2}\), i.e., \({H}^{2}(G,{{\mathbb{Z}}}_{2})\cong {{\mathbb{Z}}}_{2}^{n}\). Meanwhile, we find that the corresponding PSA can be captured by a complete set of n\({{\mathbb{Z}}}_{2}\)valued cohomology invariants, which are denoted by σ, α, β, η, and τ in Table 1. The specific meaning of these symbols will be explained in a while. Here, one can easily check that they are indeed cohomology invariants, by noting that they are unchanged when multiplying symmetry operators by arbitrary \({{\mathbb{Z}}}_{2}\) phases ± 1. The ordinary representation just corresponds to the case with all invariants being + 1.
It should be noted that for each G the PSAs classified by \({H}^{2}(G,{{\mathbb{Z}}}_{2})\) have redundancies for their abstract algebraic structures, because often G has nontrivial automorphisms such that two nonequivalent factor systems lead to equivalent algebraic structures. We screen out all nonequivalent algebras, of which the numbers (N_{G}) are listed in the last column of Table 1. We find that there are 189 nonequivalent algebraic structures out of the 458 PSAs.
To illustrate our theory, we take group P2 as an example. The set of generators of P2 consists of two unit translations L_{a}, L_{b} and the rotation R by π (along an outofplane axis). Its group algebras are expressed in terms of the four combinations, R, L_{a}R, L_{b}R and L_{a}L_{b}R, each of which is squared to 1. According to Table 1, \({H}^{2}(P2,\, {{\mathbb{Z}}}_{2})\cong {{\mathbb{Z}}}_{2}^{4}\), so there are four cohomology invariants α_{i} (i = 1, 2, 3, 4), corresponding to the four PSA relations:
Since any permutation of the four twofold rotations above gives an isomorphic PSA, there are only five equivalence classes of PSAs, and each class is specified by how many α^’s equal − 1.
Our result shows that the 17 wallpaper groups together with T symmetry can generate 458 × 2 PSAs (the additional factor 2 is from spin), which is much richer than the case of Poincaré group with only twofold classification. Physically, this is due to the reduced symmetry which allows more gauge flux configurations.
Flux realizations of projective symmetry algebras
After completing the classification, our next task is to develop a construction method to realize each of the PSAs. This is important for two purposes. First, it serves as a validity check for our results in Table 1 and demonstrates that each PSA can indeed be realized in a physical system. Second, it provides guidance for the experimental realization of nontrivial PSAs in artificial crystals.
Our construction is via mapping each PSA in Table 1 to a specific gauge flux pattern. In this process, we distinguish five classes of cohomology invariants in PSAs, corresponding to the five symbols σ, α, β, η, and τ in Table 1. The flux lattices for them are illustrated in Fig. 1c–h, and are elucidated below. The technical details for how these flux lattices represent PSAs can be found in Methods.
(i) The first class refers to \(\sigma={{\mathsf{L}}}_{a}{{\mathsf{L}}}_{b}{{\mathsf{L}}}_{a}^{1}{{\mathsf{L}}}_{b}^{1}\) for the translation subgroup. σ = ± 1 corresponds respectively to flux 0 and π through the plaquette spanned by L_{a} and L_{b}, as illustrated in Fig. 1c.
(ii) The second class concerns cohomology invariants α of symmorphic rotational symmetries. That is, α = R^{n} for an nfold rotation R, where α = ± 1 corresponds to flux 0 or π through the plaquette invariant under the rotation, as in Fig. 1d.
(iii) The third class corresponds to the square of a mirror reflection M, i.e., β = M^{2}. It turns out that M^{2} = − 1 cannot be realized on lattices with only nearest neighbor hoppings within one layer. We propose to realize it either by second neighbor hopping as in Ref. ^{28} or on a bilayer lattice with π flux through the interlayer plaquettes, as shown in Fig. 1e.
(iv) The fourth class (η invariants) is on relations between translations and reflections. For example, \(\eta={{\mathsf{M}}}_{x}{{\mathsf{L}}}_{y}{{\mathsf{M}}}_{x}^{1}{{\mathsf{L}}}_{y}^{1}\), and η = ± 1 corresponds to flux 0 or π through the plaquette in Fig. 1f that preserves L_{y} and M_{x}. Moreover, if \({{\mathsf{M}}}^{2}={{\mathsf{M}}}_{x}{{\mathsf{L}}}_{y}{{\mathsf{M}}}_{x}^{1}{{\mathsf{L}}}_{y}^{1}=1\), we may design the flux pattern as in Fig. 1g.
(v) The fifth class consists of invariants τ that extend the algebraic relations between translations and glide reflections, e.g., \(\tau={{\mathsf{g}}}_{x}{{\mathsf{L}}}_{x}{{\mathsf{g}}}_{x}^{1}{{\mathsf{L}}}_{x}\). As illustrated in Fig. 1h, τ = ± 1 respectively corresponds to flux 0 or π through the area spanned by the translation and glide reflection, which is half of the plaquette spanned by unit translations. It appears only for Pg group in Table 1.
With the above building blocks, we can systematically translate the cohomology invariants into fluxed lattices and obtain models realizing each of PSAs in Table 1. In this way, we have constructed a “canonical” lattice model for each wallpaper group G, in the sense that all PSAs for G can be realized in this single model, by simply varying the 0/π flux distribution in the lattice. In Methods, we categorize the 17 wallpaper groups into five classes to briefly introduce how the canonical lattice models are constructed.
As an example, consider P2 group with four α invariants. The algebraic relation for each α_{i} in Table 1 corresponds to a twofold rotation center in the unit cell, as illustrated in Fig. 2a. Under lattice translation, each α_{i} is associated with a class of translationrelated rotation centers, which are distinguished by four colors in Fig. 2a. Then, the canonical model can be constructed with each plaquette hosting a unique rotation center (see Fig. 2a), corresponding to the dual lattice of the lattice of rotation centers. Each α_{i} = ± 1 can then be realized by inserting flux 0 or π into the corresponding class of plaquettes.
The canonical models for all 17 wallpaper groups are illustrated in Fig. 3, and are explicitly constructed in Supplementary Note 3. For each wallpaper group in Fig. 3, the cohomology invariants correspond to independent fluxes in the lattice model, and we distinguish the fluxes by different colors. This is consistent with the number 2^{n} of PSAs, with n the number of colors in each lattice model.
Physical consequences of projective symmetry algebras
Our revealed PSAs can lead to a wealth of new physics, beyond conventional systems based on ordinary representations. Below, we present three remarkable consequences for demonstration.
(1) Shift of highsymmetry points. In ordinary band structures, highsymmetry points are located either at the center (Γ point) or on the boundary of Brillouin zone (BZ)^{2}. In contrast, with PSAs, the highsymmetry points are redistributed, and they can be at noncentral points in the interior of BZ.
For instance, continue with the example of P2 group. Let’s consider the PSA with α_{1} = α_{2} = 1 and α_{3} = α_{4} = − 1 (see the canonical model realization in Fig. 2b). Clearly, in this case the two translations L_{a} and L_{b} commute as usual, and therefore the BZ is unchanged. However, since \(R{L}_{b}{R}^{1}={L}_{b}^{1}\) is modified to \({\mathsf{R}}{{\mathsf{L}}}_{b}{{\mathsf{R}}}^{1}={{\mathsf{L}}}_{b}^{1}\) by the fluxes, the Rinvariant momenta are transformed from (0, 0), (0, π), (π, 0) and (π, π) to (0, ± π/2) and (π, ± π/2), as illustrated in Fig. 2c (see discussion in Methods).
(2) Enforced nontrivial Zak phase. While ordinary crystal symmetries may protect topological structures of energy bands, we discover that some PSAs can even enforce nontrivial topological structures. That is, once the PSA is realized, certain topological invariant is guaranteed to be nontrivial.
Here, we give one example of this fascinating phenomena, again using the P2 group. Let us consider the PSAs with \({{\mathsf{R}}}^{2}={({{\mathsf{L}}}_{a}{\mathsf{R}})}^{2}=\alpha\) and \({({{\mathsf{L}}}_{b}{\mathsf{R}})}^{2}={({{\mathsf{L}}}_{a}{{\mathsf{L}}}_{b}{\mathsf{R}})}^{2}=\alpha\), which can be realized by the canonical model with the flux configuration in Fig. 2b. The PSAs lead to \({\mathsf{R}}{{\mathsf{L}}}_{b}{{\mathsf{R}}}^{1}={{\mathsf{L}}}_{b}^{1}\) for both α = ± 1. From this relation, one can show that the antiunitary operator RT will shift momentum k to k + G_{b}/2 with G_{b} the reciprocal translation vector corresponding to L_{b} (see Methods).
Now, consider the effect of RT on a single energy band with eigenstates \(\left  {\psi }_{{{{{{{{\boldsymbol{k}}}}}}}}}\right\rangle\). Recall that spacetime inversion symmetry can quantize the Berry phase, also known as the Zak phase, along any periodic path in the BZ to be either 0 or π^{29}. In contrast, here, RT with (RT)^{2} = α exerts a stronger constraint on the Zak phase, i.e., it completely determines the Zak phase θ_{b} along any G_{b}periodic path as
due to the nontrivial action of RT discussed above (see Methods). This result means: if α = − 1, the Zak phase is enforced to be nontrivial. This is confirmed by the concrete model in Fig. 2b. This model has four isolated bands, and each band is enforced to have a nontrivial Zak phase π along k_{b}. Hence, there must be topological edge states within the first and the third energy gaps, as shown in Fig. 2d.
(3) Eightfold degenerate nodal point. Highly degenerate nodal points protected by crystal symmetries have been a hot topic. Without including the twofold degeneracy of spin1/2, the highest degeneracy protected by wallpaper groups is fourfold^{30}. Here, we find that PSA can achieve a degeneracy of eightfold, beyond any ordinary representations.
This is exemplified by the PSA of P3m1 with \({{\mathsf{L}}}_{a}{{\mathsf{L}}}_{b}{{\mathsf{L}}}_{a}^{1}{{\mathsf{L}}}_{b}^{1}=1\) and M^{2} = − 1 (see Table 1). The canonical model is illustrated in Fig. 2e and f. Since L_{a} does not commute with L_{b}, we choose \({{\mathsf{L}}}_{a}^{2}\) and \({{\mathsf{L}}}_{b}^{2}\) to generate an invariant subgroup of P3m1, and the BZ is specified by \({{\mathsf{L}}}_{a}^{2}={e}^{i{{{{{{{\boldsymbol{k}}}}}}}}\cdot {{{{{{{{\boldsymbol{e}}}}}}}}}_{a}}\) and \({{\mathsf{L}}}_{b}^{2}={e}^{i{{{{{{{\boldsymbol{k}}}}}}}}\cdot {{{{{{{{\boldsymbol{e}}}}}}}}}_{b}}\) under the Fourier transform, with e_{a,b} being the translation vectors of \({{\mathsf{L}}}_{a,b}^{2}\). At highsymmetry point Γ, the little cogroup is given by \({{\mathbb{Z}}}_{2}^{2}\rtimes {D}_{3}\times {{{{{{{{\mathcal{Z}}}}}}}}}_{2}^{T}\), where \({{\mathbb{Z}}}_{2}^{2}\) are generated by L_{a,b}. This little cogroup is projectively represented with factors inherited from that of P3m1. We find that it has two 4D irreducible representations and one 8D irreducible representation. The latter gives the eightfold nodal point, which is indeed confirmed via a concrete model as illustrated in Fig. 2g and h.
Discussion
In conclusion, we have established a unified theory for Tinvariant crystals. Particularly, we classified all PSAs of wallpaper groups, developed a general construction method, presented canonical models to realize each PSA, and revealed remarkable physical consequences. The theory can be directly extended to 3D space groups. Our work provides a solid foundation for the study of artificial crystals and opens the door to a wealth of new physics beyond the current paradigm based ordinary symmetry representations.
Notably, although our focus here is on spinless systems (which most artificial crystals belong to), the generalization to spinful systems is straightforward. This is because in the presence of Tinvariance, it is always sufficient to consider \({{\mathbb{Z}}}_{2}\)valued factor systems, as stressed above. Then, in addition to the phases arising from fluxes, one only needs to take care of reflections and rotations of spin1/2 by 2π, which lead to the phase − 1. Hence, all the cohomology invariants in classes (ii) and (iii) are reversed. Formally, we may just replace each α and β by (−1)^{2s}α and (−1)^{2s}β, respectively, with s = 0 and 1/2 for spinless and spin1/2 cases.
Finally, we note that our theory of PSAs is based on two fundamental principles of physics: (a) Physical systems are classified by symmetries (Landau’s paradigm); and (b) Symmetries are projectively represented in a physical system (Wigner’s principle). Hence, the PSAs derived here are general and classify all Tinvariant crystal systems, including not only artificial crystals, but also real materials, strongly correlated spin systems, and beyond.
Methods
Projective representations with timereversal symmetry
In the main text, we emphasized that with T symmetry, the phase factors of space group symmetries can be constrained to be valued in \({{\mathbb{Z}}}_{2}\). Here, we present a proof for this proposition.
Let us enlarge the space group G by including T with T^{2} = 1. Then, each group element can be written as gT^{a} with g ∈ G and a = 0, 1. Suppose that under the projective representation ρ, the phase factor λ arises through
We shall prove that by appropriately modifying the phase of each operator ρ(gT^{a}), we can always transform \(\lambda ({g}_{1}{T}^{{a}_{1}},{g}_{2}{T}^{{a}_{2}})\) into the form,
where ν(g_{1}, g_{2}), \(\omega ({T}^{{a}_{1}},{T}^{{a}_{2}})\in {{\mathbb{Z}}}_{2}\).
We start with observing that for all g ∈ G,
which motivates us to modify the phase of each ρ(g) as
Note that ρ(T) is an antiunitary operator, i.e., ρ(T)c = c* ρ(T) for \(c\in {\mathbb{C}}\). Hence,
We further modify the operators for the other half of group elements as
for all g ∈ G. Note that \(\tilde{\rho }(T)=\rho (T)\). Then, one observes that
Let \(\tilde{\lambda }\) denote the phase factor for \(\tilde{\rho }\). Restricting on G, \(\tilde{\lambda }\) satisfies
for all g_{1}, g_{2} ∈ G. The lefthand side commutes with \(\tilde{\rho }(T)\), so does the righthand side. Hence, \(\nu :=\tilde{\lambda }{}_{G\times G}\in {{\mathbb{Z}}}_{2}=\{\pm 1\}\). On the other hand, \(\tilde{\lambda }(T,T)\) appears in
Clearly, \(\tilde{\lambda }(T,T)\) commutes with \(\tilde{\rho }(T)\), and therefore \(\omega (T,T):=\tilde{\lambda }(T,T)\in {{\mathbb{Z}}}_{2}\).
Finally, it is straightforward to check that
This concludes the proof of our proposition. In the proof, we have repeatedly used the relations: \(\tilde{\rho }(g)\tilde{\rho }(T)=\tilde{\rho }(gT)\) and \(\tilde{\rho }(g)\tilde{\rho }(T)= \tilde{\rho }(T)\tilde{\rho }(g)\).
Projective symmetry algebras and gauge fluxes
Let us consider a set of lattice sites and hopping amplitudes among them, which give rise to a tightbinding model,
Here, \({a}_{i}^{{{{\dagger}}} }\) and a_{j} are the particle creation and annihilation operators at sites i and j, respectively. H_{ij} represents the hopping amplitudes t_{ij} from site j to i if i ≠ j and the onsite energy ϵ_{i} at site i if i = j. H is a Hermitian matrix and called the oneparticle Hamiltonian of the tightbinding model.
Each hopping amplitude t_{ij} may have a phase \({e}^{i{\phi }_{ij}}\) (such that \({t}_{ij}={t}_{ij}{e}^{i{\phi }_{ij}}\)), which is called the gauge connection of the lattice model. Particularly, here we consider the \({{\mathbb{Z}}}_{2}\) gauge connections with ϕ_{ij} ∈ {0, π}. For each closed loop C formed by successive hoppings, one can compute the product W_{C} of the phases of all the hopping amplitudes involved. W_{C} is called the Wilson loop operator of the loop C, and the gauge flux Φ_{C} through the loop C is given by \({W}_{C}={e}^{i{{{\Phi }}}_{C}}\). For the \({{\mathbb{Z}}}_{2}\) gauge field, we have W_{C} ∈ { ± 1} and Φ_{C} = {0, π}.
For each site i, we may change the phase of \({a}_{i}^{{{{\dagger}}} }\) for each i by an arbitrary \({e}^{i{\theta }^{i}}\). Particularly, θ^{i} is valued in {0, π} for the \({{\mathbb{Z}}}_{2}\) gauge field considered here. Accordingly, the hopping amplitudes are transformed as \({t}_{ij}\mapsto {e}^{i{\theta }^{i}}{t}_{ij}{e}^{i{\theta }^{j}}\), which is called a gauge transformation. An immediate result is that \({W}_{C}={e}^{i{{{\Phi }}}_{C}}\) is invariant under any gauge transformation. This can be seen from that the ending site of a hopping is the starting site of the next hopping in a loop C, and therefore all phase changes involved are cancelled out. To summarize, the gauge fluxes are gaugeinvariant quantities, whereas the gauge connections are not.
Only gaugeinvariant quantities are physical. In the current case, the gauge flux configuration completely determines the physics of the model. Hence, a spatial transformation R that leaves the crystal and the gauge flux configuration invariant is regarded as a symmetry of the system. However, R does not necessarily preserve the gaugeconnection configuration A. After the action of R, A is generally changed to another one \({A}^{{\prime} }\). Since the two gaugeconnection configurations A and \({A}^{{\prime} }\) describe the same flux configuration, they are related by a gauge transformation G_{R}. On the lattice, R is represented by a matrix indexed by lattice sites, which we still denote by R. The gauge transformation G_{R} is a diagonal matrix with \({[{{\mathsf{G}}}_{R}]}_{ii}={e}^{i{\theta }_{R}^{i}}\), i.e., with the phase assigned to the ith site. Then, the physical symmetry operator in this case should be the combination
That is, after the spatial transformation R, the gauge transformation G_{R} is needed to recover the original gauge connection configuration A. Notably, it is R = G_{R}R that commutes with the Hamiltonian H, i.e.,
The commutation relation is equivalent to the requirement,
where \({G}_{R}(i)={e}^{i{\theta }_{R}^{i}}\), namely the phase assigned to site i, and R(i) is the site transformed from i by R.
Then, we consider the successive action of two spatial symmetries, \({{\mathsf{R}}}_{1}={{\mathsf{G}}}_{{R}_{1}}{R}_{1}\) and \({{\mathsf{R}}}_{2}={{\mathsf{G}}}_{{R}_{2}}{R}_{2}\). There are two natural operators to implement the action, namely, \({{\mathsf{G}}}_{{R}_{1}}{R}_{1}{{\mathsf{G}}}_{{R}_{2}}{R}_{2}\) and \({{\mathsf{G}}}_{{R}_{12}}{R}_{12}\) with R_{12} = R_{1}R_{2}. Their difference is \({{{\Delta }}}_{{\mathsf{G}}}({R}_{1},{R}_{2})={{\mathsf{G}}}_{{R}_{1}}{R}_{1}{{\mathsf{G}}}_{{R}_{2}}{R}_{1}^{1}/{{\mathsf{G}}}_{{R}_{1}{R}_{2}}\). Δ_{G}(R_{1}, R_{2}) is a diagonal matrix with ith diagonal entry being \({{\mathsf{G}}}_{{R}_{1}}(i){{\mathsf{G}}}_{{R}_{2}}({R}_{1}^{1}(i))/{{\mathsf{G}}}_{{R}_{1}{R}_{2}}(i)\), and therefore represents a gauge transformation. It is clear that Δ_{G}(R_{1}, R_{2}) commutes with all possible symmetrypreserving Hamiltonians. Particularly, let us presume the usual case that H is a connected lattice model, i.e., any two sites are connected by hoppings. The presumption sufficiently leads to the fact that Δ_{G}(R_{1}, R_{2}) is proportional to the identity matrix, namely \({[{{{\Delta }}}_{{\mathsf{G}}}({R}_{1},{R}_{2})]}_{ij}=\nu ({R}_{1},{R}_{2}){\delta }_{ij}\) with \(\nu ({R}_{1},{R}_{2})\in {{\mathbb{Z}}}_{2}\subset U(1)\), i.e., the physical symmetry operators satisfy the PSA,
If ν and \({\nu }^{{\prime} }\) are related by transforming R to \({{\mathsf{R}}}^{{\prime} }=\chi (R){\mathsf{R}}\) with χ(R) ∈ U(1) or \({{\mathbb{Z}}}_{2}\), the two PSAs belong to the same cohomology class. It must be noted that the cohomology class of such a PSA is independent of the choice of gauge connections, and is solely determined by the flux configuration.
Realization of cohomology invariants
Based on the general discussions in the last section, we now show the flux lattices in Fig. 1c–h can realize the five classes of cohomology invariants, respectively.
(i) Let us start with the cohomology invariant \(\sigma={{\mathsf{L}}}_{a}{{\mathsf{L}}}_{b}{{\mathsf{L}}}_{a}^{1}{{\mathsf{L}}}_{b}^{1}\). Since
the algebraic relation is equivalent to
for any lattice site i. For the lattice model in Fig. 1c, we have from (16) the relations
which implies
Here, 1, 2, 3, 4 label the four sites in Fig. 1c, which are counted counterclockwise from the lower left corner. The flux through the rectangle satisfies
Thus, in the presence of flux Φ, L_{a} and L_{b} satisfy \({{\mathsf{L}}}_{a}{{\mathsf{L}}}_{b}{{\mathsf{L}}}_{a}^{1}{{\mathsf{L}}}_{b}^{1}={e}^{i{{\Phi }}}\). This argument can be generalized to other lattices. See Supplementary Figs. 28 and 37.
(ii) For a cohomology invariant \(\alpha={{\mathsf{R}}}_{2\pi /n}^{n}\), n must be even. Here, we have added the subscript 2π/n for R to specify the rotation angle. When n is even, rotating n/2 times is a twofold rotation \({R}_{2\pi /n}^{n/2}={R}_{\pi }\). In general, \({{\mathsf{R}}}_{2\pi /n}^{n/2}=\xi {{\mathsf{R}}}_{\pi }\) with \(\xi \in {{\mathbb{Z}}}_{2}\). No matter whether ξ = ± 1, the cohomology invariant can always be expressed as
Substituting R_{π} = G_{π}R_{π} into the identity above, we see the cohomology invariant can be realized by
for any site i.
Let us label the vertices of the plaquette in Fig. 1d by i = 1, 2, 3, ⋯ 2l with n = 2l. Then, R_{π}(i) = i + l. From (16), the hopping amplitudes satisfy
Then, the flux is found to be
From (24), we conclude that e^{iΦ} = α. Note that all phases are restricted in \({{\mathbb{Z}}}_{2}=\{\pm 1\}\).
(iii) For M^{2} = β, the 2D mirror reflection is interpreted as the twofold rotation through an axis parallel to the 2D plane. Then, it is clear from (ii) that e^{iΦ} = β with Φ the flux through each vertical plaquettes in Fig. 1e.
(iv) The cohomology invariant \(\eta={{\mathsf{M}}}_{x}{{\mathsf{L}}}_{y}{{\mathsf{M}}}_{x}^{1}{{\mathsf{L}}}_{y}^{1}\) is translated as
Hence, we have the identity,
Label the four white sites in Fig. 1f by 1, 2, 3 and 4, which are counted counterclockwise from the lower left site. From (16), we have the identities,
Then, the flux is computed as
(v) The algebraic relation \(\tau={{\mathsf{L}}}_{x}{{\mathsf{g}}}_{x}{{\mathsf{L}}}_{x}{{\mathsf{g}}}_{x}^{1}\) leads to
which is equivalent to
For the lattice in Fig. 1h, the hopping amplitudes satisfy
referring to (16). Here, 1, 2, 3, 4 label the four sites in Fig. 1h, which are counted counterclockwise from the lower left one. Then, the flux through the rectangular plaquette is
Note again that all phases above are either + 1 or − 1.
Construction of canonical models
In the main text, we have elucidated how to realize cohomology invariants σ, α, β, η and τ in Table 1 by lattice flux configurations. Here, we briefly introduce the general procedure for constructing the canonical models for all 17 wallpaper groups, which realize all 458 PSAs. It must be noted that the purpose of these models is to demonstrate the physical realization of all PSAs, so they are made as simple as possible and contain only nearest neighbor hoppings. One can certainly write down more complex models with more complicated lattice structures and hopping processes for a given PSA, just like what one typically does when constructing models based on ordinary representations of space groups.
The illustration for all the canonical models is given in Fig. 3, and the full details for the model construction can be found in the Supplemental Note 3. In this process, we find it useful to categorize the 17 wallpaper groups into five classes.
(a) Groups P1, P3, and Pg are quite simple, since each of them has only one cohomology invariant σ or τ (see Table 1). It is straightforward to design lattice models with the flux patterns as introduced in (i) or (iv).
(b) For groups P2, Pgg, P4, and P6, all cohomology invariants are of type α in (ii), i.e., each α_{i} = R^{n} for some nfold rotation R through a rotation center in the unit cell. Under lattice translations, each rotation center gives a lattice of rotation centers. Accordingly, each α_{i} is associated with such a lattice, and different α_{i}^’s correspond to different lattices. This has been illustrated with our example P2 in the main text. Then, the canonical model is constructed as the dual lattice for the lattice of rotation centers. This means each plaquette in the canonical model hosts a unique rotation center; conversely, each rotation center is the center of a plaquette preserving the rotation symmetry. Then, each α_{i} = ± 1 is realized by inserting flux 0 or π into the corresponding plaquettes.
(c) For groups Cm, P3m1, and P31m, each has two cohomology invariants σ and β. We first construct a onelayer lattice model realizing σ as described in (i). Then, we double the onelayer model into a twolayer model, and introduce the nearestneighbor interlayer hopping amplitudes to realize β as given in (iii) or in Fig. 1e.
(d) Each of groups Pmm, Cmm, P4m, P4g, and P6m has the two types of cohomology invariants α and β in (ii) and (iii), respectively. Here, following (b), we first construct a onelayer model to accommodate all αinvariants. Then, we double the onelayer model into a twolayer model, and appropriately insert fluxes for interlayer plaquettes to realize all βinvariants. Note that according to Fig. 1e, the vertical mirror planes should cross lattice bonds rather than lattice sites, which can always be satisfied.
(e) The remaining two groups are Pm and Pmg, both having η and βinvariants. Therefore, we refer to (iv) and Fig. 1g for constructing twolayer models for them. Since Pmg also has two αinvariants, we first construct the onelayer model according to the αinvariants following (b), and then double it into a twolayer model to accommodate the β and ηinvariants following Fig. 1g.
Shift of highsymmetry points
We derive the shift of highsymmetry points in Fig. 2c. From the PSAs for P2, it is straightforward to derive that
Alternatively, these relations can be derived from the configuration of the canonical model in Fig. 2a. There, the fluxes through the plaquettes colored in red, green, purple, blue correspond to cohomology invarants α_{1}, α_{2}, α_{3}, α_{4}, respectively. If α_{i} = 1(α_{i} = − 1), the corresponding flux is 0 (π).
When α_{1} = α_{2} = α and α_{3} = α_{4} = − α, plaquettes in each row have the same flux, and the flux values alternate across the rows (see Fig. 2b). Accordingly, the third equation above gives \({\mathsf{R}}{{\mathsf{L}}}_{b}{{\mathsf{R}}}^{1}={{\mathsf{L}}}_{b}^{1}\). In momentum space, L_{b} is diagonalized as e^{ik⋅b}. Then, \({\mathsf{R}}{e}^{i{{{{{{{\boldsymbol{k}}}}}}}}\cdot b}{{\mathsf{R}}}^{1}={e}^{i({{{{{{{\boldsymbol{k}}}}}}}}{{{{{{{{\boldsymbol{G}}}}}}}}}_{b}/2)\cdot b}\), with G_{b} the reciprocal lattice vector for b. Hence, we see that k is transformed to − k + G_{b}/2 under R, i.e.,
Thus, the Rinvariant momenta are shifted to ± G_{b}/4 and G_{a}/2 ± G_{b}/4, as shown in Fig. 2c.
Enforced topology by projective symmetry algebras
Here, we provide the details for the nontrivial Zak phase enforced by RT symmetry discussed in the main text. The RT symmetry puts the following constraint for a Hamiltonian:
where U_{RT} is a unitary operator determined by RT. Suppose that \({{{{{{{\mathcal{H}}}}}}}}({{{{{{{\boldsymbol{k}}}}}}}})\) has a single band \(  {\psi }_{{{{{{{{\boldsymbol{k}}}}}}}}}\rangle\) over the G_{b} period from k = 0 to G_{b}. The action of RT on \( {\psi }_{{{{{{{{\boldsymbol{k}}}}}}}}}\rangle\) will give a band eigenstate at k + G_{b}/2, generally differing from \( {\psi }_{{{{{{{{\boldsymbol{k}}}}}}}}+{{{{{{{{\boldsymbol{G}}}}}}}}}_{b}/2}\rangle\) by a k dependent phase, i.e.,
Accordingly, RT relates the Berry connection \({{{{{{{{\mathcal{A}}}}}}}}}_{b}({{{{{{{\boldsymbol{k}}}}}}}})=\left\langle {\psi }_{{{{{{{{\boldsymbol{k}}}}}}}}}\righti{\partial }_{{k}_{b}}\left  {\psi }_{{{{{{{{\boldsymbol{k}}}}}}}}}\right\rangle\) at k and k + G_{b}/2 as
Because of this, the Zak phase \({\theta }_{b}=\oint d{k}_{b}\,{{{{{{{{\mathcal{A}}}}}}}}}_{b}({{{{{{{\boldsymbol{k}}}}}}}})\) over any G_{b}periodic path can be expressed as
Moreover, the PSA relation (RT)^{2} = α requires that \({U}_{{\mathsf{R}}T}{U}_{{\mathsf{R}}T}^{*}=\alpha\), which in turn leads to
Thus, we arrive at \({\theta }_{b}=i\ln \alpha \,{{{{{{\mathrm{mod}}}}}}}\,\,2\pi\) as claimed in the main text. For α = − 1, this Zak phase is guaranteed to be nontrivial.
Eightfold degenerate nodal point
For the group P3m1, when σ = α = − 1, the little coalgebra at the Γ point has two irreducible 4D representations and one irreducible 8D representation. The 8D irreducible representation can be expressed as
Here, K denotes the complex conjugation, and
with \({{{{{{{{\boldsymbol{n}}}}}}}}}_{2}=(1,1,1)/\sqrt{3}\) and \({{{{{{{{\boldsymbol{n}}}}}}}}}_{2}=(0,1,1)/\sqrt{2}\).
Following (c), we can construct the canonical model for P3m1. First, we construct the onelayer lattice that realizes the σinvariant for translations as illustrated in Fig. 2d. Then, we double the onelayer model into the bilayer model as illustrated in Fig. 2e. In order to realize all cohomology invariants, we add flux at all regular hexagons and rectangles. The expression for this lattice model is given in the Supplementary Note 4.
Data availability
The data generated and analyzed during this study are available from the corresponding author upon request.
Code availability
All code used to generate the plotted band structures is available from the corresponding author upon request.
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Acknowledgements
This work is supported by National Natural Science Foundation of China (Grants No. 12161160315 and No. 12174181), Basic Research Program of Jiangsu Province (Grant No. BK20211506), and Singapore MOE AcRF Tier 2 (MOE2019T21001).
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Z.C. and Y.Z. conceived the idea. S.Y. and Y.Z. supervised the project. Z.C., Z.Z., and Y.Z. did the theoretical analysis. Z.C., Z.Z., S.Y., and Y.Z. wrote the paper.
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Chen, Z.Y., Zhang, Z., Yang, S.A. et al. Classification of timereversalinvariant crystals with gauge structures. Nat Commun 14, 743 (2023). https://doi.org/10.1038/s41467023364477
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DOI: https://doi.org/10.1038/s41467023364477
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