Lattice Homotopy Constraints on Phases of Quantum Magnets
Hoi Chun Po, Haruki Watanabe, Chao-Ming Jian, Michael P. Zaletel
LLattice Homotopy Constraints on Phases of Quantum Magnets
Hoi Chun Po,
1, 2
Haruki Watanabe, Chao-Ming Jian,
4, 5 and Michael P. Zaletel Department of Physics, University of California, Berkeley, CA 94720, USA Department of Physics, Harvard University, Cambridge MA 02138, USA Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan. Station Q, Microsoft Research, Santa Barbara, California, 93106, USA. Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California, 93106, USA Department of Physics, Princeton University, Princeton, NJ 08544, USA.
The Lieb-Schultz-Mattis (LSM) theorem and its extensions forbid trivial phases from arising in certain quan-tum magnets. Constraining infrared behavior with the ultraviolet data encoded in the microscopic lattice ofspins, these theorems tie the absence of spontaneous symmetry breaking to the emergence of exotic phaseslike quantum spin liquids. In this work, we take a new topological perspective on these theorems, by ar-guing they originate from an obstruction to “trivializing” the lattice under smooth, symmetric deformations,which we call the “lattice homotopy problem.” We conjecture that all LSM-like theorems for quantum magnets(many previously-unknown) can be understood from lattice homotopy, which automatically incorporates thefull spatial symmetry group of the lattice, including all its point-group symmetries. One consequence is thatany spin-symmetric magnet with a half-integer moment on a site with even-order rotational symmetry must bea spin liquid. To substantiate the claim, we prove the conjecture in two dimensions for some physically relevantsettings.
Quantum magnets arise naturally in Mott insulators, wherestrong Coulomb repulsion freezes the position of electronsand leaves behind their spin degrees of freedom. With strongfrustration, quantum fluctuations can suppress spin orderingand lead to symmetric, quantum-entangled phases of matterthat survive down to zero temperature. Quantum spin liquids,the spin analogues of fractional quantum Hall states, representone of the most sought-after phases arising in this context [1].They possess intrinsic topological order with emergent frac-tionalized excitations, which have been proposed as a usefulresource for robust quantum computation [2, 3].Detecting whether a quantum magnet is a spin liquid, amany-body problem, is notoriously hard. Conventionally, theabsence of symmetry breaking is regarded as an indicator forspin-liquid physics [3, 4]. However, a symmetric quantummagnet could also be in a symmetry-protected topological(SPT) phase, like the spin-1 Haldane chain [5, 6] and its gen-eralizations [7], which does not support fractionalized excita-tions despite a nontrivial degree of entanglement. Conceptu-ally, there is a sharp distinction between these phases: spinliquids are long-range entangled (LRE), and are necessarilyeither gapless or topologically ordered, while SPT phases areonly short-range entangled (SRE). Experimentally, however,such distinction is subtle and one must rely on additional cri-teria to rule out all symmetric SRE (sym-SRE) phases beforeclaiming discovery of a spin liquid.Fortunately, it is possible to rule out all sym-SRE phases incertain quantum magnets on purely theoretical grounds. Thisline of reasoning was pioneered by Lieb, Schultz, and Mattis(LSM), who proved that any one-dimensional quantum mag-net with both lattice-translation and spin-rotation symmetriescannot be sym-SRE if each unit cell contains a half-integraltotal spin [8]. Multiple generalizations of the LSM theoremhave since been made, covering systems in higher dimensionsand with less stringent physical assumptions [9–17]. We willcollectively refer to these results as “LSM-like theorems.” Thecommon denominator of these generalizations is a constraint between the microscopic details of the system, specificallythe lattice and the symmetry-transformation properties of thesites’ Hilbert spaces, and the degree of ground-state degener-acy. Since any sym-SRE phase is expected to have a gapped,unique ground state on any thermodynamically large spacewithout defects or boundaries, all sym-SRE phases are ruledout whenever the degeneracy is constrained to be nontrivial.In this sense, the LSM-like theorems are “no-gos” for sym-SRE phases.One important direction for generalization is to make fulleruse of the spatial symmetries of the system. This was partiallyaddressed in Refs. [15–17], which showed that combinationsof nonsymmorphic symmetries like glides and screws, being“fractions” of the lattice translations, can lead to stronger no-gos. Ideally, to expose the strongest constraints one wouldattempt to utilize all spatial symmetries of the problem. How-ever, the nonsymmorphic generalizations in Refs. [15–17] ig-nore all point-group symmetries (e.g., rotations), which fix atleast one point in space. New techniques are required for thedesired extension.In this work, we address the problem of incorporatingall spatial symmetries in deriving stronger LSM-like no-gos,which are operative even when all earlier theorems are not.We will rely on two key insights: First, the presence of a no-go should be insensitive to a smooth, symmetric deformationof the underlying lattice. We will refer to the study of such de-formations as the “lattice homotopy problem;” second, thereis a strong sense of locality in sym-SRE phases due to the lim-ited range of entanglement, and therefore, compared to moreexotic phases like spin liquids, they respond in a more conven-tional manner when fluxes are inserted into the system. Com-bining these observations, we conjecture that a quantum mag-net which is nontrivial under lattice homotopy is obstructedfrom being sym-SRE.In the following, we will elaborate on the conjecture, whichencompasses all earlier LSM-like theorems for quantum mag-nets, and then offer a physical argument for its proof restrict- a r X i v : . [ c ond - m a t . s t r- e l ] S e p ing to 2D systems with an internal symmetry group G be-ing either finite Abelian or SO(3) . As an example, we willshow that sym-SRE phases are forbidden whenever a half-integer spin, carrying a projective representation of
SO(3) ,sits at an even-order rotation center. Intuitively, this is be-cause any symmetric deformation brings in an even numberof spins, which cannot screen the half-integer moment at thecenter.
Statement of the conjecture. – Consider a quantum magnetwith Hamiltonian ˆ H defined on a lattice Λ . For simplicity, wewill first assume ˆ H is symmetric under the group G = SO(3) of spin rotations, and later discuss how the ideas apply tomore general on-site symmetry groups. We are interested inwhether ˆ H can be in a sym-SRE phase. As demonstratedby the LSM-like theorems, the microscopic data encodedin Λ may present an obstruction. The key ingredient in ourargument will be the spatial distribution of half-integer vs.integer spins. Therefore, as far as obstructions are concerned,we view Λ as a lattice of black and white circles, denotinghalf-integer and integer spins respectively (Fig. 1). Noobstruction is expected on a lattice composed only of integerspins, and we say such lattices are “trivial.” In addition, thepresence of obstructions should be insensitive to a smoothdeformation of the lattice, provided that the deformationrespects all spatial symmetries (Fig. 1e). This motivates thefollowing conjecture: Conjecture : A sym-SRE phase is possible only when Λ issmoothly deformable to a trivial lattice. Let us make precise what is meant by a “smooth defor-mation.” We suppose the magnet is symmetric under a spacegroup S . By deformation, we refer first to a collective, S -symmetric movement of sites. Second, when sites collide they“fuse;” since we only keep track of the integer vs. half-integernature of the sites, the fusion follows a Z -rule (Fig. 1 a-d).In this process, an even number of half-integer sites may an-nihilate. Generally, when a collection of sites are symmet-rically collapsed at a point, the number of sites involved isdetermined by the degree of the point-group symmetry. Wealso allow the inverse of fusion, in which half-integer spinsare created in pairs.A sequence of such deformations defines an equivalencerelation between lattices, and we refer to the enumeration ofthe resulting equivalence classes [Λ] as the “lattice homotopy”problem. { [Λ] } naturally forms an Abelian group under stack-ing, with the empty (trivial) lattice the identity element. Theconjecture is that a sym-SRE obstruction is present whenevera lattice belongs to a nontrivial class. We note that all thepreviously-known LSM-like theorems feature nontrivial lat-tices [8–17].Thanks to its geometrical nature, lattice homotopy can of-ten be computed in an intuitive manner. For instance, con-sider a 1D translation and mirror symmetric spin chain. Spinsat generic positions can be smoothly brought to the mirrorplanes, where they will annihilate pairwise. Since this cannotchange the color on the mirror plane, the only lattice invariantis the color at the two inequivalent mirror planes in a unit cell, ===+ =+ (e)(j) (f)(g)(h)(i)(a)(b)(c)(d) = = FIG. 1.
Lattice homotopy. (a-d) Representations of the rotationgroup
SO(3) fuse following a Z rule. Open and filled circles respec-tively denote the representations of integer and half-integer spins. (e)A smooth deformation of a lattice (circles) symmetric under mirrorplanes (dashed lines) and three-fold rotations (about the stars). (f-i)There are two inequivalent sites (big and small circles) in each unitcell (shaded) of a mirror-symmetric 1D lattice. Under lattice homo-topy, there are four distinct lattice classes. (j) Assuming the symme-tries of (e), a honeycomb lattice of half-integer spins is equivalent toa kagome lattice of integer spins, as demonstrated by the depictedsmooth deformation. giving a [Λ] ∈ Z × Z classification (Fig. 1f-i). In fact, ano-go for the three nontrivial elements was already proven inRef. [17]. Together with the original LSM theorem invokingonly translations, this proves the conjecture for the two 1Dspace groups.As another example, a square lattice of spin-1/2 moments isnontrivial, but that of spin-1 is trivial. This is consistent withthe known LSM-like theorems for the former [9–13], and theexistence of sym-SRE phases for the latter [18]. A more in-triguing example is a honeycomb lattice of half-integer spins,which is symmetric under both three-fold rotations and mir-rors. As shown in Fig. 1j, the lattice is smoothly deformableto a kagome lattice of integer spins, and therefore belongs tothe trivial class. Interestingly, this picture is consistent with arecent construction of sym-SRE wave-functions [18, 19].It is conceptually revealing to generalize the internal sym-metry group G in the discussion above beyond SO(3) spin-rotations. We assume that the total symmetry group is adirect product of the internal and space group symmetries, G × S , where S acts by permuting the sites. (The casewith “spin-orbit coupling” is an interesting future direction.)The role of “half-integer” vs. “integer” spin is now played bythe Abelian group of distinct projective representations of G , H [ G, U (1)] . The Z -fusion of spins generalizes to groupmultiplication in H [ G, U (1)] , and the above conjecture nat-urally carries over to this more general setting.The resulting group of lattice homotopy classes depends on G . For instance, suppose G is such that the projective repre-sentations have Z fusion, and consider again the 1D latticewith reflection symmetry. If two copies of a projective repre-sentation [ ω ] approach a mirror plane, they do not annihilate,since [ ω ] = [ ω ] − in Z . Consequently, the projective rep- TABLE I. The lattice homotopy classification for the 17 wallpapergroups, assuming Z projective representations, as in the case forspin-rotation invariant quantum magnets.Lattice homotopy Wallpaper group No. [20] Z
1, 4, 5, 13, 14, 15 ( Z )
3, 8, 12, 16, 17 ( Z )
7, 9, 10, 11 ( Z )
2, 6 resentation on a mirror plane is not conserved, and the latticehomotopy classification collapses down to Z .Computing the lattice classification can be automated by areduction to the properties of high-symmetry points (Wyckoffpositions). We relegate details Appendix A. In Table I, we tab-ulate the lattice classification results for all 2D space-groups.Here, we present the case relevant to spins, H [ G, U (1)] = Z – the general form, which extends readily to any finiteAbelian H [ G, U (1)] , is tabulated in Supplementary Table I.
Proof of conjecture in 2D. – We now sketch a physical ar-gument supporting the conjecture for quantum magnets sym-metric under any of the 17 2D space groups, assuming G =SO(3) or is finite Abelian. The logic proceeds by first deriv-ing three concrete conditions on Λ , each implying a no-go forsym-SRE phases: • Bieberbach no-go. A “fundamental domain” D is a re-gion which tiles the plane under the action of translationand glide symmetries. If the total projective representa-tion in D is nontrivial, [ ω ] D = (cid:81) r ∈ D [ ω ] r (cid:54) = 1 , then asym-SRE phase is forbidden [17]. • Mirror no-go. Let (cid:96) be a mirror-line parallel to a trans-lation T (cid:107) . We define the projective representation perunit length of (cid:96) , [ ω ] (cid:96) = (cid:81) r ∈ (cid:96) (cid:48) [ ω ] r , by letting the prod-uct runs over a unit-length interval (cid:96) (cid:48) of (cid:96) as definedby T (cid:107) . If [ ω ] (cid:96) does not have a “square-root,” i.e., if no ζ ∈ Z n satisfies [ ω ] (cid:96) = ζ , then a sym-SRE phase isforbidden [17]. • Rotation no-go. Let r be a site with rotational point-group symmetry C m and projective representation [ ω ] r .If [ ω ] r does not have an “ m -th root”, i.e., if no ζ ∈ Z n satisfies [ ω ] r = ζ m , then a sym-SRE phase is forbid-den.We then show that these no-gos forbid a sym-SRE phase in a2D lattice Λ if and only if [Λ] (cid:54) = 1 . Both the Bieberbach andmirror no-gos were derived in an earlier work [17], so here wefocus on illustrating the key ideas behind the derivation of the“rotation no-go” – the key missing piece for establishing theconjecture in 2D – with further details given in Appendix B. Derivation of the rotation no-go. – For simplicity, we will il-lustrate the ideas using systems symmetric under G = SO(3) and C rotation. Roughly speaking, we will modify theHamiltonian by inserting a pair of C -related spin fluxes, andshow that when a half-integer moment lies on a C -invariant point, the system has a symmetry-protected degeneracy. Wewill then argue that, despite the presence of fluxes, such de-generacy remains impossible in sym-SRE phases, and therebyarriving at a no-go.We begin with the following observation: While a sym-SRE phase has a gapped, unique ground state on R d , it maypossess symmetry-protected ground-state degeneracy in thepresence of defects or boundaries (a notable example beingthe edge states of the AKLT chain). In contrast to LRE phases,however, the degeneracies in a sym-SRE phase should be“localized” to the defect regions (for example, each edge ofthe AKLT chain carries an independent two-fold degeneracy).Physically, this arises because a sym-SRE phase can only re-spond to local data, defined with respect to the correlationlength ξ , so it should not be possible to “share” a degener-acy between two distant defect regions (note we are only con-sidering bosonic models; certain fermionic SPTs violate thisassumption [21]).To formalize this intuition we introduce the notion of “ de-generacy localization ” (Appendix C). A “defect region” is aregion in which the Hamiltonian is not local-unitarily equiv-alent to the Hamiltonian of the bulk [17] (examples could in-clude an impurity spin, dislocation, or external flux), and welet { R ( i ) : i = 1 , . . . , N D } be a collection of defect regions offinite extent, which are separated from each other on distances r (cid:29) ξ . We say the system exhibits degeneracy localization if each R ( i ) can be modeled as an emergent d i -dimensional,degenerate, degree of freedom, so that the total ground-statesubspace H (cid:48) GS is (cid:16)(cid:81) N D i =1 d i (cid:17) -dimensional. This implies thatif ˆ U is a local operator taking the ground-state subspace H (cid:48) GS into itself (e.g., a symmetry), then its projection into H (cid:48) GS canbe “factorized” as ˆ U | GS = (cid:78) N D i =1 U ( i ) + O ( e − r/ξ ) for some d i -dimensional matrix U ( i ) acting only on the degeneracy lo-calized at the region R ( i ) . In other words, degeneracy local-ization passes the locality structure from the full Hilbert spaceonto H (cid:48) GS . The discussed intuition about sym-SRE phases canthen be summarized by the following physical assumption: Abosonic sym-SRE phase exhibits degeneracy localization.
We now use this assumption to prove the C -rotation no-go with G = SO(3) . Recall that a (projective) representationof SO(3) is classified by [ ω ] r ∈ Z = { , − } , which en-codes the phase factor for the commutator of two orthogonal π -rotations at site r , say ˆ X r ˆ Z r = [ ω ] r ˆ Z r ˆ X r , where ˆ X, ˆ Z are π -rotations about ˆ x, ˆ z . Let the C -invariant point be the ori-gin. Clearly, the no-go condition is unmet whenever [ ω ] = 1 ,and hence it suffices to prove a no-go with [ ω ] = − .To this end, we modify the Hamiltonian by introducinga pair of X -fluxes at the C -related points ± r X for somearbitrarily large | r X | (Fig. 2a). An “ X -flux” is analogousto a twist in boundary condition, and is microscopically de-fined as follows [22]. We choose a line segment γ connect-ing ± r X , and for each local term ˆ h = (cid:80) j ˆ O j L ˆ O j R in theHamiltonian intersecting γ , where ˆ O j L and ˆ O j R are respec-tively localized to the left and right of γ , we replace it by ˆ h (cid:48) ≡ (cid:80) j ˆ O j L (cid:16) ˆ X ˆ O j R ˆ X † (cid:17) to obtain ˆ H (cid:48) . Note that, while theflux insertion points ± r X are fixed and correspond to defects (a) (b) FIG. 2.
Flux insertion. (a) A C symmetric lattice with a pairof X -fluxes (crosses) inserted at ± r X , which leads to “defect re-gions” (shaded) near the fluxes. Far away from ± r X , flux insertionamounts to choosing a defect line (dash-dot) and twisting the localHamiltonian by X along the line. (b) As X − = X , the systemretains a twisted C (cid:48) symmetry, since the transformed defect line canbe brought back to the original by applying a gauge transformationon the region A . in the system, the choice of γ is arbitrary, and one can deform γ → γ (cid:48) by applying the gauge transformation (cid:81) r ∈ A ˆ X r inthe region A enclosed by γ (cid:48) − γ . Also, the orientation of γ isimmaterial as X is an order-two symmetry.Though the two fluxes are C -related, the choice of thedefect line γ naively spoils the C symmetry. However,the change γ → C ( γ ) can be removed by a gauge trans-formation (Fig. 2b). Consequentially, ˆ H (cid:48) is symmetric un-der a twisted- C operation: ˆ C (cid:48) = (cid:16)(cid:81) r ∈ A ˆ X r (cid:17) ˆ C , where ∂A = γ − C ( γ ) . In addition, ˆ Z ≡ (cid:81) r ˆ Z r remains a symme-try of ˆ H (cid:48) . Computing the commutation relation between thetwo symmetries, one finds ˆ C (cid:48) ˆ Z ˆ C (cid:48)− ˆ Z − = (cid:89) r ∈ A ˆ X r ˆ Z r ˆ X − r ˆ Z − r = (cid:89) r ∈ A [ ω ] r = [ ω ] , (1)where in the last equality we used the fact that A is C -symmetric, and by symmetry [ ω ] r = [ ω ] − r . Since both ˆ Z and ˆ C (cid:48) are symmetries of ˆ H (cid:48) , they leave the ground space H (cid:48) GS in-variant. We can therefore project Eq. (1) into H (cid:48) GS , and obtainthe corresponding relation ˆ C (cid:48) | GS ˆ Z | GS = [ ω ] ˆ Z | GS ˆ C (cid:48) | GS .When [ ω ] = − , there is (at least) a two-fold degener-acy that we will now show is impossible in a sym-SRE phase,provided the degeneracy localization assumption holds. Ifthe system was sym-SRE, degeneracy localization implied ˆ Z | GS = ˆ Z | (+)GS ⊗ ˆ Z | ( − )GS , where ± denotes the fluxes at ± r X .In addition, as C exchanges the two fluxes, the local degen-eracies satisfy d + = d − , and without loss of generality we canchoose a basis in which C (cid:48) | GS is simply ( ˆ C (cid:48) | GS ) | α + α − (cid:105) = | α − α + (cid:105) , where α ± denotes the independent degenerate states“trapped” at ± r X . In this basis, the commutation relationreads ˆ C (cid:48) | GS ˆ Z | GS ˆ C (cid:48) | † GS = ˆ Z | ( − )GS ⊗ ˆ Z | (+)GS = − ˆ Z | (+)GS ⊗ ˆ Z | ( − )GS .A solution to this requires ˆ Z | ( − )GS = ν ˆ Z | (+)GS for some ν ∈ U(1) satisfying − ν = ν , leading to a contradiction. Hence theclaim. In closing, we remark that our no-gos are circumvented ifthe system becomes LRE. An example is discussed in Ap-pendix D. Discussion and outlook. – In conclusion, we have conjec-tured that all LSM-like theorems for quantum magnets, wheremicroscopic degrees of freedom forbid symmetric short-rangeentangled phases, can be understood intuitively as topologi-cal obstructions to smoothly deforming the underlying latticeinto a trivial one. We proved the conjecture in 2D for quan-tum magnets that are either spin-rotation invariant, or possesson-site unitary finite-Abelian symmetries.Our 2D arguments, in fact, cover all 80 layer groups, whichare symmetries of 2D lattices embedded in 3D (Appendix E).They also extend to some genuinely 3D lattices – in partic-ular, the three no-gos remain true, where mirror lines and C m rotation-invariant points in 2D become planes and linesin 3D. Such extensions can have immediate implications onspin-liquid candidates. As an example, we note that both theBieberbach and mirror no-gos are silent for the pyrochlorequantum spin ice Yb Ti O [23], but the C -rotation no-goremains active if we model the system as a spin-rotation in-variant quantum magnet. Yet, we caution that spin-orbit cou-pling is strong in the actual material [23], and so this idealiza-tion is not immediately justified.A closer inspection, however, reveals that these three no-gos only prove the conjecture for some but not all of the 2303D space groups (Appendix F includes simple examples forwhich the current set of no-gos are insufficient.) New tech-niques will be required, and we describe some partial resultsin Appendix G. We also note that it would be most useful ifonly time-reversal T was required in the no-gos, with the roleof projective representation played by the Kramers degener-acy from T = − . However, it is not clear how to extendour flux-insertion proof to this case. In addition, actual ma-terials are composed of itinerant fermions carrying spin andthe quantum-magnet description is often an approximation. Itwould be useful to know if our results extend to this moregeneral case. With a Mott gap, it naively seems that thereshould be a sharp notion of “where” the spins of the electronslie (at least up to the lattice equivalence relations), but certainexamples suggest this may not be the case [24, 25]. Finally,we note that our conjecture has interesting connection to thestudy of crystalline SPTs [26–32], which we comment brieflyin Appendix H. ACKNOWLEDGMENTS
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Appendix A: Systematic computation of lattice homotopy usingWyckoff positions
Here we will develop a systematic method to compute lat-tice homotopy for lattices (as summarized in Table I in themain text) symmetric under a space group S (in any spatialdimension d ). In the following, we will always let η be a gen-erator (e.g. e πi/n ) of H [ G, U (1)] = Z n . The more generalcase of H [ G, U (1)] = (cid:81) i 2, 6 Z n × (cid:0) Z gcd( n, (cid:1) 3, 8 Z n × Z gcd( n, 7, 9 Z n × (cid:0) Z gcd( n, (cid:1) 10, 11 Z n × Z gcd( n, × Z gcd( n, Z n × Z gcd( n, 13, 14 Z n × (cid:0) Z gcd( n, (cid:1) Z n × Z gcd( n, 16, 17 Z n × Z gcd( n, × Z gcd( n, a tuning parameter one can tune to bring a collection of sitesto a higher symmetry point, and thereby enhancing the site-symmetry group from S x to S y . In this case, we can smoothlydeform (cid:104) ( x ; η ) (cid:105) to (cid:104) ( y ; η p ) (cid:105) by fusing p ≡ |S y | / |S x | points at y . As such, one can always trivialize the projective represen-tation contained in a reducible Wyckoff position by “pushing”them to the irreducible ones. Therefore, the lattice homotopyproblem is reduced to a study of trivial elements of Z N IWP n ,where N IWP denotes the number of “irreducible” Wyckoffpositions for S , and the notion of “lattice stacking’ is simplygiven by the group addition in Z N IWP n . We label irreducibleWyckoff positions by α = 1 , , · · · , N IWP (in the same or-dering as in Supplementary Ref. [20]) and express a latticeas ( η q , η q , · · · , η q N IWP ) ∈ Z N IWP n when the total projec-tive representations on the irreducible Wyckoff position W α is η q α .While any manifestly non-projective lattices will corre-spond to (1 , , . . . , ∈ Z N IWP n , some other entries of Z N IWP n may also be trivial under lattice homotopy, as for the examplediscussed in the main text, concerning the (cid:104) (0; η ) , (1 / η ) (cid:105) lattice (or equivalently, ( η, η ) ∈ Z N IWP =2 n =3 ) for the case of a1D mirror and translation symmetric spin chain with projec-tive representations in Z . The trivial lattices among Z N IWP n form a group under addition, and in particular they will cor-respond to a subgroup K ≤ Z N IWP n . To find a set of gen-erators for K , we simply note the following: Let x and x belong to two different irreducible Wyckoff positions W and W . Consider another point x ∈ W , and suppose W is “re-ducible” into both W and W by setting some free parame-ters to special values. Physically, this describes moving somelattice sites in W to lattice sites in W via lower-symmetrypoints in W : (cid:104) ( x ; η p ) (cid:105) ∼ (cid:104) ( x ; η ) (cid:105) ∼ (cid:104) ( x ; η p ) (cid:105) where p i ≡ |S x i | / |S x | . Thus one sees that (cid:104) ( x ; η − p ) , ( x ; η p ) (cid:105) ,which can be written as ( η − p , η p , , · · · , ∈ Z N IWP n , istrivial in the lattice homotopy sense, and therefore is a gener-ator of K . All generators of K can be found by checking ifsuch reduction is possible in any pairs of irreducible Wyckoffpositions. Once this is achieved, the lattice homotopy classi-fication is given by the quotient Z N WP n /K . As the Wyckoffpositions are well tabulated in crystallographic references likeSupplementary Ref. [20], the computation can be easily auto-mated.In Supplementary Table II, we tabulate the lattice homo-topy classification for the 17 wallpaper groups assuming Z n projective representations. This is a generalized version ofTable I in the main text, where we restricted to Z projectiverepresentations. Note that Supplementary Table II generalizesreadily to any H [ G, U (1)] = (cid:81) j Z n j . In this case, it is easyto see that the lattice homotopy problem factorizes over j , i.e.,we just take a product over the classification for each Z n j . Appendix B: General proof of conjecture in 2D for finiteAbelian G Here we will prove (up to the degeneracy localization as-sumption) a more general version of the conjecture in 2D,where instead of considering spin-rotation invariant systemswith G = SO(3) , we let the internal symmetry group take theform G = Z n × Z n , which has H [ G, U (1)] = Z gcd( n ,n ) .In fact, for the most general finite Abelian group G = (cid:81) i Z n i ,the projective representations are classify by H [ G, U (1)] = (cid:81) i Let S be a space group in 2D (one of the 17wallpaper groups) and let the internal symmetry group be G = Z n × Z n . A quantum magnet, defined on a lattice Λ and symmetric under S × G , can be in a sym-SRE phase onlyif the lattice homotopy class [Λ] = 1 .The proof takes the same structure as the simplified versionpresented in the main text, in which we will again assumesym-SRE phases exhibit degeneracy localization. First, wewill derive the “Bieberbach,” “mirror,” and “rotation” sym-SRE no-gos for G = Z n × Z n . Second, we will connectthe no-gos to lattice homotopy, and show at least one of thesym-SRE no-gos we derived is present if and only if [Λ] (cid:54) = 1 .Before we move on to the derivation, we first discuss therelevant structure of G that leads to the no-gos. We denote thegenerators of the two factors of G = Z n × Z n by X and Z , such that X n = Z n = 1 . While X and Z commute asgroup elements, their corresponding unitary operator satisfy ˆ X ˆ Z ˆ X − ˆ Z − = ω ( X, Z ) ω ( Z, X ) ≡ [ ω ]( X, Z ) , (B1)implying they do not commute whenever [ ω ] is nontrivial.Hence, we will write ˆ X r ˆ Z r = [ ω ] r ˆ Z r ˆ X r for local opera-tors at site r , where [ ω ] r ∈ H [ G, U (1)] = Z n and n ≡ gcd( n , n ) . Note that by restricting the following discussionto Z × Z , we recover the proof for the spin-rotation invariantmagnets by viewing G as the π -rotation subgroup. 1. Deriving the three sym-SRE no-gos In the following, we will restate and then derive the threeno-gos. First, recall that the mirror no-go and the C m rota-tion no-go are silent when [ ω ] r is an “ m -th root” ( m = 2 for the mirror no-go), i.e., if [ ω ] r = ζ m for some ζ ∈H [ G, U (1)] = Z n . If we express [ ω ] r = η q ( q ∈ Z ) usinga generator η of H [ G, U (1)] (e.g. e πi/n ), we can rephrasethis condition as q = 0 mod gcd( n, m ) . (B2)To see this, note that there always exist integers s and t suchthat sn + tm = gcd( n, m ) . Hence, if Eq. (B2) holds, ∃ l suchthat [ ω ] r = η q = η l gcd( n,m ) = η l ( sn + tm ) = ζ m , (B3)with ζ = η lt ∈ Z n . Conversely, if [ ω ] r = ζ m for ζ = η k ∈ Z n , then [ ω ] r = ζ m = η km = η l gcd( n,m ) = η q , (B4)with l ≡ km/ gcd( n, m ) ∈ Z and q ≡ l gcd( n, m ) , andEq. (B2) holds. a. Bieberbach no-go Bieberbach no-go: A “fundamental domain” D is a regionwhich tiles the plane under the action of translation and glidesymmetries. If the total projective representation in D isnon-trivial, [ ω ] D ≡ (cid:81) r ∈ D [ ω ] r (cid:54) = 1 , then a sym-SRE phase isforbidden [17].The original LSM-like theorems in SupplementaryRefs. [8–14] translates the net count of [ ω ] in the primitiveunit cell to a sym-SRE obstruction. In the presence of non-symmorphic symmetries, which, like translations, move everypoint in space, the notion of “unit cell” can be generalized toa smaller region of space D called the “fundamental domain,”which tessellates space under the action of a nonsymmorphicsubgroup (more precisely, the maximal fixed-point-freesubgroup) of S [17]. We will now show that a sym-SREno-go is present whenever [ ω ] D ≡ (cid:81) r ∈ D [ ω ] r (cid:54) = 1 . The fullargument, which is applicable to a much more general classof models, was presented in in Supplementary Ref. [17]. Inthe following we sketch the key ideas involved.We begin by discussing the LSM-like theorems that uti-lize only the lattice translation symmetry. We take D asthe primitive unit cell D . Thanks to translation symme-try, the system can be consistently defined on any torus ofsize L × L × · · · × L d . Let T dV denote a torus of size V ≡ (cid:81) di =1 L i , where we have set the size of the primitiveunit cell to . The system defined on T dV is symmetric underboth X and Z , which as quantum operators satisfy ˆ X ˆ Z ˆ X † ˆ Z † = (cid:89) r ∈ T dV [ ω ] r = [ ω ] VD . (B5)Writing [ ω ] D = η q D with an integer q D , we see that ˆ X and ˆ Z do not commute whenever q D V (cid:54) = 0 mod n . Choosing V to be co-prime with n , this implies the ground state of thesystem must be degenerate unless q D = 0 mod n, (B6)despite V can be arbitrarily large. For a sym-SRE phase, how-ever, the ground state should be gapped and unique in the ther-modynamic limit, and hence we arrive at a contradiction.The generalization to more general fixed-point-free (non-symmorphic) symmetries proceeds in a similar manner [17].The only difference is that, instead of defining the system onthe torus, we define it on certain flat compact manifolds M (known as the Bieberbach manifolds). The Bieberbach man-ifolds are tied to the fixed-point-free subgroup of S that tes-sellates the entire space upon acting on the fundamental do-main D . Focusing on a 2D system, the only nonsymmorphicsymmetry is a glide mirror, which has the Klein bottle as theassociated Bieberbach manifold. One can check that, in thepresence of a glide symmetry, the entire lattice can be definedby only specifying the content of half of the primitive unit cell D , so that D can be identified as half of D . The volume ofthe Bieberbach manifold V M is generally quantized in units of V D , the volume of D . There a similar ground state degeneracyis exposed unless q D = 0 mod n , since one can always choose V M (measured in V D ) to be co-prime with n . Importantly, asym-SRE phase should not be able to detect the topology of M , as the Hamiltonian ˆ H M defined on M is locally indistin-guishable from that on the infinite space (see SupplementaryRef. [17] for details). Therefore the ground state degeneracyagain leads to a sym-SRE obstruction. b. Mirror no-go Mirror no-go. Let (cid:96) be a mirror line parallel to a translation T (cid:107) . We define the projective representation per unit lengthof (cid:96) , [ ω ] (cid:96) ≡ (cid:81) r ∈ (cid:96) (cid:48) [ ω ] r , by letting the product run over aunit-length interval (cid:96) (cid:48) of (cid:96) as defined by T (cid:107) . If [ ω ] (cid:96) does nothave a “square-root,” i.e., if no ζ ∈ Z n satisfies [ ω ] (cid:96) = ζ ,then a sym-SRE phase is forbidden [17].This was again proven in Supplementary Ref. [17], but wewill provide here an alternative flux-insertion argument as awarm-up to the rotation no-go. When n is odd, one can al-ways find a ζ ∈ Z n such that [ ω ] (cid:96) = ζ , as for any gen-erator η of Z n , η is also a generator. Hence, it suffices toconsider G of the form Z a × Z b , which gives [ ω ] r ∈ Z n with n = 2 gcd( a, b ) . By definition ¯ a ≡ a/ gcd( a, b ) and ¯ b ≡ b/ gcd( a, b ) are coprime, and without loss of generalitywe take ¯ a to be odd.We will first argue in a 1D setting, which generalizes read-ily to 2D. Consider a finite but arbitrarily large ring of spinswith a mirror m centered at the site x = 0 . Now imagine in-serting an X a flux through the system by twisting the bound-ary condition (BC) between sites γ and γ + 1 . The choice of γ is immaterial, as twisting instead at γ (cid:48) < γ is equivalent to applying the gauge transformation (cid:81) x ∈ [ γ (cid:48) +1 ,γ ] ˆ X ax . By thedegeneracy localization assumption, the flux-inserted Hamil-tonian ˆ H (cid:48) should still have a gapped, unique ground state, aswe have only inserted a pure flux and there are no defects totrap degeneracies.We will now argue a contradiction to the sym-SRE assump-tion when the mirror invariant is nontrivial. The action of ˆ m on ˆ H (cid:48) has two effects: first, it moves the BC twist to that be-tween sites − γ and − γ − ; second, it reverses orientation andtherefore the flux inserted is inverted (Supplementary Fig. 3).However, by construction X − a = X a , and as argued the BCtwist can be brought back to γ by a gauge transformation. Thisimplies (cid:104) ˆ m (cid:48) , ˆ H (cid:48) (cid:105) = 0 , where ˆ m (cid:48) ≡ (cid:16)(cid:81) x ∈ [ − γ,γ ] ˆ X ax (cid:17) ˆ m . As ˆ Z = (cid:81) x ˆ Z x remains a symmetry, we consider ˆ m (cid:48) ˆ Z ( ˆ m (cid:48) ) † ˆ Z † = (cid:89) x ∈ [ − γ,γ ] [ ω ] ax = [ ω ] a , (B7)where we used the symmetry constraints to reduce [ ω ] ax [ ω ] a − x = [ ω ] ax = 1 . Writing [ ω ] = η q for some inte-ger q , one finds [ ω ] a = η n ¯ aq/ . Recalling ¯ a is odd, we seethat ˆ m (cid:48) and ˆ Z commute iff q is even. This implies ˆ H (cid:48) has aground state degeneracy whenever q is odd. Since [ ω ] (cid:54) = ζ for any ζ ∈ Z n is equivalent to q being odd, the claim follows.To generalize to the 2D case, we simply take a torus geom-etry with L (cid:107) unit cells in the direction parallel to the mirrorline. The net count of [ ω ] on the mirror line is [ ω ] L (cid:107) (cid:96) , where [ ω ] (cid:96) is the count within one unit cell. While the BC twist isnow promoted from a point to a closed line, it remains invisi-ble to a sym-SRE phase. The 1D argument goes through withthe effective 1D invariant [ ω ] L (cid:107) (cid:96) . As we are free to choose anodd L (cid:107) , we conclude a sym-SRE phase is forbidden wheneverthe 2D mirror invariant is nontrivial. c. Rotation no-go Rotation no-go. Let r be a site with rotational point-groupsymmetry C m and projective representation [ ω ] r . If [ ω ] r does not have an “ m -th root,” i.e., if no ζ ∈ Z n satisfies [ ω ] r = ζ m , then a sym-SRE phase is forbidden.As we have already discussed the proof for the special caseof G = Z × Z in the main text, here we only discuss therequired generalization to G = Z n × Z n . First, we notethat the condition for the no-go to be active is a directly gen-eralization of the mirror case, where instead of an order-twosymmetry (mirror) we now consider an order- m one ( C m ro-tation).We define the integer a ≡ n / gcd( n , m ) and b ≡ n / gcd( n , m ) . As before, let η denote the generator of H [ G, U (1)] . Then we can rewrite [ ω ] r as [ ω ] r = η q fora certain integer q . The condition for [ ω ] r to have an “ m -th root” is identical to the condition that q is divisible by gcd( n, m ) as discussed in Eq. (B2).Generalizing the construction in the main text, we can insert m of the X a -fluxes at the vertices of a C m -symmetric poly-gon A centered at the origin, which requires a choice of de-fect line such as the one shown in Supplementary Fig. 3. Notethat consistency requires X am = 1 , which is true by con-struction. The flux-inserted system retains a C m -symmetry ˆ C (cid:48) m ≡ (cid:16)(cid:81) r ∈ A ˆ X a r (cid:17) ˆ C m . We then have ˆ C (cid:48) m ˆ Z ( ˆ C (cid:48) m ) † ˆ Z † = (cid:89) r ∈ A [ ω ] a r = [ ω ] a = η aq , (B8)where we again used the symmetry condition [ ω ] C m ( r ) =[ ω ] r . If aq = 0 mod n , the no-go is silent in this configu-ration. Similarly, we can as well consider inserting m of the Z b -fluxes at the vertices of a C m -symmetric polygon A cen-tered at the origin. Again, the consistent condition Z bm = 1 isautomatically satisfied. This flux-inserted system also retainsa C m -symmetry ˆ C (cid:48)(cid:48) m ≡ (cid:16)(cid:81) r ∈ A ˆ Z b r (cid:17) ˆ C m . We then have ˆ C (cid:48)(cid:48) m ˆ X ( ˆ C (cid:48)(cid:48) m ) † ˆ X † = (cid:89) r ∈ A [ ω ] b r = [ ω ] b = η bq . (B9)In this configuration, the no-go is silent when bq = 0 mod n .For the no-go to be completely silent in both the configurationwith X a -fluxes and that with Z b -fluxes, we would simultane-ously need qan = q gcd( n, m ) N ∈ Z , (B10) qbn = q gcd( n, m ) N ∈ Z , (B11)where N ≡ n gcd( n,m ) n gcd( n ,m ) and N ≡ n gcd( n,m ) n gcd( n ,m ) . Since byconstruction N and N are co-prime integers, the necessaryand sufficient condition for Eqs. (B10) and (B11) is q = 0 mod gcd( n, m ) . [To see N and N are co-prime, note that n /n ( n /n ) is divisible by N ( N ) and that n /n and n /n are co-prime.] As discussed above, this is equivalent with theabsence of “ m -th root” for [ ω ] r .Without loss of generality, we can just study the case when [ ω ] a (cid:54) = 1 . As before, we must show this degeneracy cannotbe localized. This follows from a direct generalization of theargument in the main text. Suppose on the contrary that thedegeneracies are localized. Then we can project the symmetryoperators ˆ C (cid:48) m and ˆ Z into the ground space, and with a properchoice of basis obtains ˆ Z | GS = ˆ Z | (1)GS ⊗ ˆ Z | (2)GS ⊗ · · · ⊗ ˆ Z | ( m )GS ;ˆ C (cid:48) m | GS = ˆ P (1 , ,...,m ) , (B12)where ˆ P (1 , ,...,m ) cyclically permutes the m defect regions.Computing ˆ C (cid:48) m | GS ˆ Z | GS ˆ C (cid:48) m | † GS in two ways, we demand [ ω ] a ˆ Z | (1)GS ⊗ ˆ Z | (2)GS ⊗ · · · ⊗ ˆ Z | ( m )GS = ˆ Z | ( m )GS ⊗ ˆ Z | (1)GS ⊗ · · · ⊗ ˆ Z | ( m − , (B13)which implies ˆ Z | (1)GS ∝ ˆ Z | (2)GS ∝ · · · ∝ ˆ Z | ( m )GS , where ∝ heremeans that the two sides are identical up to a U(1) phase. We FIG. 3. A C symmetric lattice with all the spins living on the sites(orange and red). The rotation center is specified by the red site. X a fluxes are inserted at 4 locations related to each other by the C rotation. The dotted lines are the branch cut associated to the fluxes.The symmetry action associated with each dotted line is given by X a (e.g. that for the tripped dotted line is X a ). The consistencycondition requires X a = 1 . can therefore write ˆ Z | ( j )GS = ν j ˆ Z | (1)GS , where ν j ∈ U(1) for j =2 , . . . , m . Substituting this into the two sides of Eq. (B13),one finds [ ω ] a ν ν . . . ν m = ν ν . . . ν m , (B14)an impossibility as [ ω ] a (cid:54) = 1 . Therefore, we proved that for G = Z n × Z n , the absence of an “ m -th root” for the pro-jective representation [ ω ] r is the necessary and sufficient con-dition for the existence of a no-go theorem, via flux insertedconfigurations, to a sym-SRE phase. 2. No-gos for the most generic finite Abelian group G For the most general finite Abelian group G = (cid:81) i Z n i ,the analysis on the no-gos to sym-SRE follows directly fromthe case with G = Z n i × Z n j for each pair of ( i, j ) . Hence,the absence of an “ m -th root” for the projective representa-tion [ ω ] r is again the necessary and sufficient condition forthe existence of a no-go theorem to a sym-SRE phase. 3. The technical obstacle for a general G In this work, we only consider the case of a finite Abeliangroup G . For a generic group G , we can only rely on ouranalysis of no-gos for any finite Abelian subgroup of G . Anactive no-go for any finite Abelian subgroup definitely impliesa no-go for the full symmetry group G . However, we are un-able to prove that the silence of no-gos for all finite Abeliansubgroups implies the silence of no-gos for the full symmetrygroup G , which presents an obstacle for extending our generalresult to a general group G .0The essence of the proof for Abelian G was to examine eachprojective commutation relation ˆ X ˆ Z ˆ X − ˆ Z − = [ ω ]( X, Z ) ,which, for Abelian G , completely characterizes the projectiverep. The case with a general group G is much more compli-cated, since, if the pair does not commute, the commutator isnot a gauge-invariant quantity.For a general projective representation [ ω ] ∈ H [ G, U (1)] ,we can still study the projective commutation relations forevery commuting pairs of group elements g, h ∈ G , i.e. gh = hg : ˆ g ˆ h = [ ω ]( g, h ) ˆ h ˆ g, (B15)where ˆ g and ˆ h are the operators in the projective represen-tation [ ω ] associated to the group elements ˆ g and ˆ h , and [ ω ]( g, h ) ∈ U(1) is a phase factor. This relation is a directgeneralization of the Abelian case. Supplementary Ref. [33]points out that [ ω ]( g, h ) as a U(1) -valued function on all thecommuting pairs of group elements ( g, h ) fully specifies theprojective representation [ ω ] ∈ H [ G, U (1)] for any finitegroup G .For a general group G , each commuting pair of group el-ements g, h ∈ G allow us to define an Abelian subgroup A g,h ⊆ G generated by g and h , and one can view [ ω ]( g, h ) as a projective representation in H [ A g,h , U (1)] . The analy-sis for the rotation no-go theorem of sym-SRE with Abeliansymmetry group Z n × Z n can be directly applied to the sub-group A g,h . This is because a sym-SRE with global symme-try G can also be viewed as a sym-SRE with global symmetry A g,h ⊆ G . If [ ω ] has an m -th root in H [ G, U (1)] , [ ω ]( g, h ) naturally has an m -th root in H [ A g,h , U (1)] for each com-muting pairs ( g, h ) . In this case, all the rotation no-gos, whichare derived from the configurations with fluxes inserted, forall the Abelian subgroup symmetry A g,h are silent. On theother hand, if, for certain commuting pairs ( g, h ) , [ ω ]( g, h ) does not admit an m -th root in H [ A g,h , U (1)] , a sym-SREphase under A g,h is forbidden, which then implies a no-go forsym-SRE phase with global symmetry G .However, we cannot prove that, if [ ω ] does not have an m -th root in H [ G, U (1)] , there must exist a pair ( g, h ) suchthat [ ω ]( g, h ) does not have an m -th root in H [ A g,h , U (1)] ,or equivalently there must exist an Abelian subgroup sym-metry A g,h that provides a no-go. This is a purely math-ematical question (which we would appreciate help with :).Nevertheless, the fact that the projective representation [ ω ] ∈H [ G, U (1)] can be completely specified by the [ ω ]( g, h ) with commuting pair ( g, h ) for all finite group G , combinedwith the fact that when [ ω ] = ζ m for some ζ ∈ H [ G, U (1)] the rotation no-go is silent, is suggestive that the absence ofthe m -th root of [ ω ] is tied to the existence of a rotation no-go. 4. Relating sym-SRE no-gos to lattice homotopy Thus far, we have defined the lattice homotopy equivalencerelation ∼ LH , shown how to compute the resulting classes [Λ] ,and independently derived a set of sym-SRE no-gos. We nowshow that a no-go is present if an only if [Λ] (cid:54) = 1 . To this end, we note that the no-gos themselves suggest an equiva-lence relation: Given a set of sym-SRE no-gos, we say twolattices Λ ∼ NG Λ if the lattice Λ − Λ is no-go-free, wherethe minus sign “ − ” denotes the operation of inverting all theprojective representations in Λ and then stacking it with Λ .Note that ∼ NG can be defined for any set of sym-SRE no-goswhich inherit a natural group structure from lattice stacking.In general, when fewer no-gos are employed a coarser classi-fication is obtained. Here, we will clarify the relation between ∼ LH and ∼ NG for the case H [ G, U (1)] = Z n and the threeno-gos we derived.We will again work in terms of irreducible Wyckoff posi-tions. We showed in Sec. A of this supplementary note thatit is sufficient to consider only lattices defined on irreducibleWyckoff positions Z N IWP n to compute the lattice homotopy. Inparticular, the lattice homotopy classification can be achievedby finding the subgroup K LH of Z N IWP n that includes all triviallattices under lattice homotopy: K LH = { Λ ∈ Z N IWP n | Λ ∼ LH (1 , , · · · , } . (B16)By the same token, the lattice classification under ∼ NG canbe computed by finding the subgroup K NG that correspondsto lattices without any of the three sym-SRE no-gos we havederived: K NG = { Λ ∈ Z N IWP n | Λ ∼ NG (1 , , · · · , } . (B17)Note that K LH ⊆ K NG (B18)automatically follows in general regardless of the dimension-ality of the space or the set of sym-SRE no-gos, as long asone assumes the stability of no-gos against smooth deforma-tions of a lattice. Namely, if Λ is smoothly deformable into Λ while respecting all symmetries, any no-gos applicable to Λ should also be applicable to Λ and vice versa. Thus, Λ ∼ LH Λ indicates Λ ∼ NG Λ . Therefore, to establish ∼ LH = ∼ NG it remains to show K LH = K NG based on thethree no-gos derived above. Note that these no-gos are notenough to show K LH = K NG for all space groups in 3D (Sec.F), and one needs to add more to achieve the equality. a. K LH Let us determine K LH by deriving the conditions on whena lattice Λ = ( η q , η q , · · · , η q N IWP ) ∈ Z N IWP n is trivial in thelattice homotopy sense.As discussed before, if there exists a Wyckoff position W reducible into both W α and W β , we have (cid:104) ( x α ; η p α ) (cid:105) ∼ LH (cid:104) ( x β ; η p β ) (cid:105) where p α |W α | = p β |W β | . Here, |W α | denotesthe number of sites per unit cell of a lattice belonging tothe Wyckoff position W α . It is easy to show that |S x α | ×|W α | always coincides with the order of the point group |S /T | . Moreover, if there is also a relation (cid:104) ( x β ; η p β ) (cid:105) ∼ LH (cid:104) ( x γ ; η p γ ) (cid:105) , then one can combine the two relations to get (cid:104) ( x α ; η p α ) (cid:105) ∼ LH (cid:104) ( x γ ; η p γ ) (cid:105) . By repeating this process, one1can move sites in Wyckoff position W α to W β , and subse-quently to W γ and so forth.Now let us fix α > and find the minimum of p α in theserelations. We denote the minimum by n α . Here, n α can beeither , , , or . By studying Supplementary Ref. [20], wefind the following nice properties in 2D: (i) The minimum isachieved when the destination is the most-symmetric Wyckoffposition W : (cid:104) ( x α ; η n α ) (cid:105) ∼ LH (cid:104) ( x ; η n α |W α | / |W | ) (cid:105) or ( η − n α |W α | / |W | , , · · · , , η n α , , · · · , ∈ K LH . (B19)(ii) The set of Eq. (B19) for each α > generates K LH .Namely, there is no element of K LH that cannot be obtainedby combining the relations of the form of Eq. (B19).Armed with these facts in 2D, we now come back to thequestion of when a lattice Λ = ( η q , η q , · · · , η q N IWP ) ∈ Z N IWP n is trivial in the lattice homotopy classification. If q α ( α > ) is an integer multiple of gcd( n, n α ) , one can com-pletely trivialize the projective representations at W α by mov-ing them to W . After moving all projective representations to W , the lattice becomes (cid:104) ( x ; η (cid:80) N IWP α =1 q α |W α | / |W | ) (cid:105) , whichis trivial only when the exponent is an integer multiple of n .Thus a lattice is trivial (i.e., an element of K LH ) if and only ifthe following two conditions are simultaneously met C1: q α = 0 mod gcd( n, n α ) for α = 2 , , . . . , N IWP . C2: (cid:80) N IWP α =1 q α |W α | / |W | = 0 mod n .The lattice homotopy classification Z N IWP n /K LH is thus givenby Z n × Z gcd( n,n ) × Z gcd( n,n ) × · · · × Z gcd( n,n N IWP ) , (B20)which leads to Supplementary Table II. Note again that thisexpression is not generally valid in higher dimensions, al-though the lattice homotopy classification will always be afinite abelian group (assuming Z n projective classes). b. K NG Let us move on to K NG . We ask when a lattice Λ =( η q , η q , · · · , η q N IWP ) ∈ Z N IWP n is free from the Bieberbach,mirror and rotation sym-SRE no-gos.Let us start with the Bieberbach no-go. Recall that wehave a projective representation η q α on each site of IWP W α and there are |W α | such sites in a primitive unit cell. There-fore, the total projective representations in a primitive unit cell D is given by (cid:81) r ∈ D [ ω ] r = η (cid:80) N IWP α =1 q α |W α | . However, inthe presence of the glide symmetry the true fundamental do-main D is reduced by the factor of / |W | from D (notethat this may not hold in 3D [17]). Hence, the total projec-tive representation in D , [ ω ] D ≡ (cid:81) r ∈ D [ ω ] r = η q D , satisfies q D = (cid:80) N IWP α =1 q α |W α | / |W | . The “Bieberbach” no-go is ap-plicable whenever q D (cid:54) = 0 mod n , see Eq. (B6).Next, let us discuss the mirror and rotation no-gos. Wewill introduce another integer m α = 2 , , , or for eachIWP, and we say a mirror plane is “effective” when the mirror plane includes only one IWP and contains an odd number ofsites. Now we make three observations: (i) When an IWP ison a C m rotation axis but is not on any effective mirror plane, m α = m ; (ii) When an IWP is on an effective mirror planebut is not symmetric under any rotation, m α = 2 ; (iii) Whenan IWP is on a C m rotation axis and is also on at least one ef-fective mirror plane, m α = lcm ( m, . We list m α and |W α | for each IWP for all 17 wallpaper groups in SupplementaryTable III. The mirror and rotation no-gos are effective when q α (cid:54) = 0 mod gcd( n, m α ) for some α , see Eq. (B2). Even if q α = 0 mod gcd( n, m α ) for every α , it might still be possiblethat an ineffective mirror leads to no-go. However, as we willsee shortly we do not have to worry about this possibility.To summarize, the necessary conditions to go around thethree sym-SRE no-gos are C1’: q α = 0 mod gcd( n, m α ) for α = 2 , , . . . , N IWP . C2: (cid:80) N IWP α =1 q α |W α | / |W | = 0 mod n .Here we intentionally dropped the case α = 1 from C1’ . Let K (cid:48) NG be the set of lattices satisfying C1’ , C2 . These condi-tions may not be sufficient because we dropped the case α = 1 and also because we do not take into account ineffective mir-rors. Hence, K (cid:48) NG gives only an “upper bound” of K NG : K NG ⊆ K (cid:48) NG (B21) c. Relation between K LH and K NG Staring at the above expressions, one sees that conditions C1 , C2 and C1’ , C2 are almost the same. We see that C1 and C1’ differ only by the fact that n α in C1 is replaced by m α in C1’ . By studying Supplementary Ref. [20], we found that,in fact, n α = m α for all wallpaper groups and IWPs except for the wallpaper group No. 12. Therefore, for all wallpapergroups but one, we have K LH = K (cid:48) NG . (B22)Combining Eqs. (B18), (B21), and (B22), we establish K LH = K NG ( = K (cid:48) NG ) and hence ∼ LH = ∼ NG .For the wallpaper group No. 12, one needs (i) q = 0 mod gcd( n, , (ii) q = 0 mod gcd( n, , and (iii) q + q =0 mod n to go around the three no-gos. It is easy to checkthat conditions (i)–(iii) recover C1 and C2 , and hence K LH = K NG by the same logic. Appendix C: Discussions on degeneracy localization Here we discuss the details of the notion of “degeneracylocalization,” and demonstrate how it enables us to pass thelocality structure of the full Hilbert space to the ground-statesubspace.2 TABLE III. IWP data for each of the 17 wallpaper groups.WG No. ( m , |W | ) ( m , |W | ) ( m , |W | ) ( m , |W | ) (1 , – – –2 (2 , 1) (2 , 1) (2 , 1) (2 , (2 , 1) (2 , – –4 (1 , – – –5 (2 , – – –6 (2 , 1) (2 , 1) (2 , 1) (2 , (2 , 2) (2 , 2) (2 , –8 (2 , 2) (2 , – –9 (2 , 1) (2 , 1) (2 , –10 (4 , 1) (4 , 1) (2 , –11 (4 , 1) (4 , 1) (2 , –12 (4 , 2) (2 , – –13 (3 , 1) (3 , 1) (3 , –14 (3 , 1) (3 , 1) (3 , –15 (6 , 1) (3 , – –16 (6 , 1) (3 , 2) (2 , –17 (6 , 1) (3 , 2) (2 , – 1. Entanglement structure of H (cid:48) GS For concreteness, let ˆ H be the original, defect-free Hamil-tonian defined on a boundaryless geometry. Now consider aHamiltonian ˆ H (cid:48) identical to ˆ H except for the presence of de-fects at two well-separated finite regions, A and B (see themain text for the definition of “defect region”). For simplicity,we consider the case of two defect regions in the following,although the arguments apply directly to the case of multipleregions.Pictorially, a sym-SRE phase is incapable of detecting thepresence of the defects at length scales much larger than thecorrelation length ξ , and therefore if ˆ H (cid:48) exhibits ground-statedegeneracy due to the presence of defects, these degeneraciescannot be “shared,” i.e. they should be independently trappedwithin each defect region. This is the idea behind the “degen-eracy localization” assumption, which we make precise be-low.Let H (cid:48) GS be the ground-state subspace of ˆ H (cid:48) , which due tothe presence of defects is generally not -dimensional. We saythe system exhibits degeneracy localization if(i) There exists an orthonormal basis H (cid:48) GS = span {| α, β (cid:105) : α = 1 , . . . , d A ; β = 1 , . . . , d B (cid:105) ; (C1)and(ii) For any normalized d A -component vector ψ , we definea state in H (cid:48) GS | ψ, β (cid:105) ≡ d A (cid:88) α =1 | α, β (cid:105) ψ α . (C2) Then for any ψ and a “reference” basis state labeled by α , there exists a local unitary operator ˆ U A ( ψ,α ) , actingnontrivially only in A , such that | ψ, β (cid:105) = ˆ U A ( ψ,α ) | α , β (cid:105) . (C3)Note that α on the right-hand side is not summed over.We also assume the corresponding conditions for B .In words, (i) formalizes the notion that the ground-state de-generacy of ˆ H (cid:48) arises from that trapped at A and B , and inparticular dim H (cid:48) GS = d A d B ; (ii) formalizes the idea that thedegeneracies in A and B are independent, in the sense thatone can freely rotate between the individual degeneracies us-ing local unitary operators. Note that in general the unitaryoperators ˆ U A ( ψ,α ) are only exponentially localized in A withthe localization length ξ . In principle, we can enlarge the sizeof A and B to maximize the “localization,” but the sizes ofthe defect regions are ultimately limited by r , the separationbetween A and B . Hence, the discussion below carries anexponentially small correction O ( e − r/ξ ) , which we will notaddress carefully.Intuitively, the notion of “degeneracy localization” suggests H (cid:48) GS (cid:39) H A ⊗ H B , where dim H A,B = d A,B . While thisserves as a useful mental picture, this factorization cannot be(immediately) taken literally, since H (cid:48) GS , being a subspaceof the full Hilbert space, has more structure than this sim-ple formula entails. (More accurately, the only subtlety lies inthe physical interpretation of a “cut” separating H A and H B ,since up to isomorphism there is only one Hilbert space of thefinite dimension d A d B . The concern here is that the tensorproduct ⊗ does not automatically coincide with the physicalone defined using the underlying lattice.)To see this more explicitly, consider an entanglement cutwith respect to an arbitrarily large region R , with linear size l R ∼ r (cid:29) ξ , containing A but not B . The Schmidt decompo-sition for a particular basis state | α , β (cid:105) ∈ H (cid:48) GS reads | α , β (cid:105) = (cid:88) i | α ; i (cid:105)| β ; i (cid:105) s i , (C4)where (cid:104) α ; i (cid:48) | α ; i (cid:105) = (cid:104) β ; i (cid:48) | β , i (cid:105) = δ i (cid:48) ,i .By the degeneracy localization assumption, any other states | ψ, ϕ (cid:105) of H (cid:48) GS will have the same entanglement spectrum { s i } . This can be seen from the reduced density matrix (ˆ ρ ψ,ϕ ) R ≡ Tr ¯ R ( | ψ, ϕ (cid:105)(cid:104) ψ, ϕ | )= Tr ¯ R (cid:16) ˆ U B ( ϕ,β ) | ψ, β (cid:105)(cid:104) ψ, β | ˆ U B † ( ϕ,β ) (cid:17) = Tr ¯ R (cid:16) ˆ U A ( ψ,α ) | α , β (cid:105)(cid:104) α , β | ˆ U A † ( ψ,α ) (cid:17) = ˆ U A ( ψ,α ) Tr ¯ R ( | α , β (cid:105)(cid:104) α , β | ) ˆ U A † ( ψ,α ) = ˆ U A ( ψ,α ) (ˆ ρ α ,β ) R ˆ U A † ( ψ,α ) , (C5)where in going to the third line we have used the cyclic prop-erty of the trace, and in going to the fourth we used the as-sumption that ˆ U A ( ψ,α ) | ¯ R = ˆ1 . Since (ˆ ρ ψ,ϕ ) R and (ˆ ρ α ,β ) R are unitarily-related, they have the same spectrum, i.e. all3the states | ψ, ϕ (cid:105) share the same entanglement spectrum { s i } .This is expected, since for these states the entanglement spec-trum w.r.t. R should be indistinguishable from the sym-SREground state of the original Hamiltonian ˆ H , which we denoteby | (cid:105) , up to an exponential accuracy O ( e − l R /ξ ) .Now consider a state | ψ, β (cid:105) . By Eqs. (C2) and (C4), weget | ψ, β (cid:105) = (cid:88) i (cid:32)(cid:88) α | α ; i (cid:105) ψ α (cid:33) | β ; i (cid:105) s i . (C6)From the discussion in Eq. (C5), we see that the Schmidtstates of | ψ, β (cid:105) in ¯ R coincide with that of | α , β (cid:105) . By theorthonormality of the Schmidt states, we then see that theSchmidt states in R are given by | ψ ; i (cid:105) = (cid:80) α | α ; i (cid:105) ψ α . Onthe other hand, Eq. (C5) shows that the reduced density ma-trices (ˆ ρ ψ,β ) R and (ˆ ρ α ,β ) R , and hence the Schmidt states,are related by the unitary ˆ U A ( ψ,α ) . This therefore implies | ψ ; i (cid:105) = ˆ U A ( ψ,α ) | α ; i (cid:105) = (cid:88) α | α ; i (cid:105) ψ α , (C7)which basically allows us to “carry over” the similar relationfrom the original many-body state to the Schmidt states.Being Schmidt states, | ψ ; i (cid:105) for different i ’s are orthonor-mal. This has an important consequence, which sharpens ournotion of “independent degeneracies.” Consider δ i,j = (cid:104) ψ ; i | ψ ; j (cid:105) = (cid:88) α,α (cid:48) ψ ∗ α (cid:48) (cid:104) α (cid:48) ; i | α ; j (cid:105) ψ α = ψ † M i,j ψ, (C8)where we defined ( M i,j ) α (cid:48) ,α ≡ (cid:104) α (cid:48) ; i | α ; j (cid:105) . Now we extractthe Hermitian parts by defining M ( i,j ) ≡ (cid:0) M i,j + M j,i (cid:1) ; M [ i,j ] ≡ i (cid:0) M i,j − M j,i (cid:1) . (C9)Observe that Eq. (C8) and the same equation with i and j interchanged imply that ψ † M ( i,j ) ψ = δ i,j , ψ † M [ i,j ] ψ = 0 (C10)for any ψ . Since the matrices involved are Hermitian, we canchoose ψ to be any of their eigenvectors. This forces all eigen-values of M ( i,j ) to be δ i,j and those of M [ i,j ] to be , so that M ( i,j ) = δ i,j , M [ i,j ] = 0 , (C11)where M i,j = δ i,j 1, i.e. (cid:104) α (cid:48) ; i | α ; j (cid:105) = ( M i,j ) α (cid:48) ,α = δ i,j δ α (cid:48) ,α . (C12)In particular, for any pair of d A -dimensional vectors ψ and φ , we have (cid:104) ψ ; i (cid:48) | φ ; i (cid:105) = (cid:88) α (cid:48) ,α ψ (cid:48) α (cid:104) α (cid:48) ; i (cid:48) | α ; i (cid:105) ψ α = ( ψ, φ ) δ i,i (cid:48) , (C13) where ( ψ, φ ) denotes the usual inner product. This sharpensour notion of the defect region A “trapping” a local Hilbertspace of dimension d A : If there is only one Schmidt state (sin-gle s i = 1 ), the “factorization” H A ⊗ H B is literal, and suchinner-product structure is automatic. The above computationshows that in the presence of entanglement in | (cid:105) , as is in thegeneral case, each “channel,” labeled by i , acts as if the fac-torization holds, and as we will see this allows us to establisha notion of locality in H (cid:48) GS .Our next task is to expose the entanglement structure within H (cid:48) GS . So far, we have only focused on states of the form | ψ, ϕ (cid:105) , which are the analog of “product states” between thedefect regions (despite, as we have discussed, the many-bodystate still generally possesses nonzero entanglement). A gen-eral state in H (cid:48) GS would be more entangled. Intuitively, weexpect that the entanglement should have two components:(i) that already present in | (cid:105) , and (ii) additional contributionfrom entangling the defect states. In the remaining of this sub-section we quantify this intuition.Let | Ψ (cid:105) ∈ H (cid:48) GS be defined by | Ψ (cid:105) ≡ d A (cid:88) α =1 d B (cid:88) β =1 Ψ α,β | α, β (cid:105) , (C14)where normalization is given by (cid:80) α,β Ψ ∗ α,β Ψ α,β = 1 . Weconsider again the Schmidt decomposition of | Ψ (cid:105) with respectto the region R , and we will show that the entanglement spec-trum of | Ψ (cid:105) factorizes as { s i } ⊗ { σ j } , where { s i } is just thatinherited from | (cid:105) , and { σ j } comes from the entanglement in Ψ α,β .To see this, consider the singular-value decomposition of Ψ α,β : Ψ α,β = (cid:88) j σ j W α,j V ∗ β,j , (C15)where (cid:80) α W ∗ α,j W α,j (cid:48) = (cid:80) β V ∗ β,j V β,j (cid:48) = δ j,j (cid:48) . Note that, foreach j , W j and V ∗ j are respectively d A and d B dimensionalvectors. As such, we see that | Ψ (cid:105) = (cid:88) j,α,β σ j W α,j V ∗ β,j | α, β (cid:105) = (cid:88) i,j s i σ j (cid:32)(cid:88) α W α,j | α ; i (cid:105) (cid:33) (cid:88) β V ∗ β,j | β ; i (cid:105) = (cid:88) i,j s i σ j ( | Ψ A ; i, j (cid:105) ) ( | Ψ B ; i, j (cid:105) ) , (C16)where in going to the second line we again used the observa-tion that all basis states have the same entanglement spectrum { s i } , and | Ψ A,B ; i, j (cid:105) in the last line are defined as the corre-sponding parenthesis. Now check that, by Eq. (C13), (cid:104) Ψ A ; i (cid:48) , j (cid:48) | Ψ A ; i, j (cid:105) = (cid:104) W j (cid:48) ; i (cid:48) | W j ; i (cid:105) =( W j (cid:48) , W j ) δ i,i (cid:48) = δ i,i (cid:48) δ j,j (cid:48) , (C17)and similarly for {| Ψ B ; i, j (cid:105)} . By the uniqueness of Schmidtdecomposition (i.e. SVD), we see that the Schmidt weights of4 | Ψ (cid:105) are { s i σ j } = { s i } ⊗ { σ j } . (C18)This verifies our claim. In particular, | Ψ (cid:105) has the same entan-glement spectrum as | ψ, ϕ (cid:105) if and only if { σ j } is a singleton,meaning that Ψ is a “product state” Ψ α,β = W α, V ∗ β, . Moregenerally, the entanglement entropy is given by the sum ofthat from { s i } and that from { σ j } , and therefore the “productstates” | ψ, ϕ (cid:105) are the only states in H (cid:48) GS with minimal entropy. 2. Symmetry operations in H (cid:48) GS Next, we utilize the discussion above on entanglementstructure to constrain the form of symmetry representations in H (cid:48) GS , showing that they indeed factorize in the way describedin the main text.Let ˆ g be an on-site unitary operator leaving H (cid:48) GS invariant,and let N D denotes the number of defect regions. The projec-tion of ˆ g into the ground space is a (cid:16)(cid:81) N D i =1 d i (cid:17) -dimensionalunitary matrix ˆ g | GS , with the entries determined by the ex-pansion ˆ g | α , . . . , α N D (cid:105) = (cid:88) β i (ˆ g | GS ) α ,...,α ND β ,...,β ND | β , . . . , β N D (cid:105) . (C19)Similar to before, we imagine an entanglement cut isolatinga defect region R ( i ) , and we compare the entanglement spec-trum obtained from the two sides of Eq. (C19). As ˆ g is on-site, and that | β , . . . , β N D (cid:105) can be rotated into | α , . . . , α N D (cid:105) using the local unitary operators ˆ U ( i )( α i ,β i ) , the entanglementspectrum obtained from ˆ g | α , . . . , α N D (cid:105) and | β , . . . , β N D (cid:105) must be identical (up to exponential accuracy controlled bythe localization of ˆ U ( i )( α i ,β i ) ). This implies the matrix ˆ g | GS cannot generate any entanglement between different defectregions. From our earlier discussion, the only states in H (cid:48) GS with the same (minimal) entanglement entropy as the basisstates are those described by a factorizable Ψ α ,...,α ND . Theform of ˆ g | GS is therefore strongly constrained, and in generalwe can write [34] ˆ g | GS = (cid:32) N D (cid:79) i =1 ˆ g | ( i )GS (cid:33) P g , (C20)where P g is a permutation of the defect regions, and ˆ g | ( i )GS is a d i -dimensional unitary matrix corresponding to the rotation oflocal degenerate states at the defect region R ( i ) . In addition,for an on-site ˆ g the permutation P g must be trivial. To seethis, we simply note that the action of ˆ g | GS in a defect region R ( i ) cannot depend on that of any other defect regions, as onecan independently alter the local degeneracies d i by creating g -symmetric defects. Therefore, an on-site unitary symmetry ˆ g is represented by ˆ g | GS = (cid:78) N D i =1 ˆ g | ( i )GS in the ground space upto an exponentially small correction, i.e. ˆ g | GS also acts locallyin each defect region. More generally, the symmetry ˆ g could by itself be spatialand hence involve site permutations. As long as ˆ g does notgenerate entanglement between different defect regions, sayfor ˆ g being a pure site permutation followed by an on-site uni-tary transformation, the form Eq. (C20) holds, with P g deter-mined by the permutation of defect regions dictated by ˆ g .Finally, we show that when all the defect regions are relatedto each other by a spatial symmetry g , there exists a choice ofbasis for which ˆ g | GS = P g . Let m be the order of ˆ g , i.e. ˆ g m =ˆ1 , and label the regions according to P g ( R ( i ) ) = R ( i +1) ( mod m ). This implies d = d = · · · = d m , and the matricesin Eq. (C20) satisfy ˆ g | ( m )GS . . . ˆ g | (2)GS ˆ g | (1)GS = . (C21)Under a basis change given by a unitary matrix u ( i ) for theregion R ( i ) , we have ˆ g | ( i )GS (cid:55)→ u ( i ) (cid:16) ˆ g | ( i )GS (cid:17) u ( i − † . (C22)To achieve our goal, we simply choose u ( i ) = ˆ g | (1) † GS ˆ g | (2) † GS . . . ˆ g | ( i ) † GS , (C23)where in particular we have u ( m ) = Appendix D: Toric code The toric code ( Z gauge theory) is one of the simplestmodels with topological order. In this appendix, we first ex-plain that the topological order of the toric code is, in fact,guaranteed by our no-go theorem (provided we know themodel is gapped and does not spontaneously break symme-try). We then derive the representation of ˆ C (cid:48) | GS and ˆ Z | GS forthis particular model, to see concretely how the “contradic-tion” in our argument [discussed in the section “Derivation ofthe rotation no-go” in the main text] is circumvented in thepresence of the topological order. We thank M. Cheng for adiscussion on this subject. 1. The model In the toric code, a spin-1/2 lives on each bond of the squarelattice (Supplementary Fig. 4a). The Hamiltonian reads ˆ H = − J e (cid:88) vertex v (cid:89) i ∈ v ˆ X i − J m (cid:88) plaquette p (cid:89) i ∈ p ˆ Z i (D1)( J e > , J m > ). Since each term contains an evennumber of X i ’s and Z i ’s, the Hamiltonian has the internal G = Z × Z symmetry represented projectively by spin-1/2’s. In addition to this internal symmetry, as the simplestexample, here we assume the spatial symmetry of wallpapergroup No. 2, generated by the C rotation and translation sym-metries.5The wallpaper group fixes the positions of spins to be thecenter of the bonds corresponding to irreducible Wyckoff po-sitions w = a and b of this wallpaper group. Then this lat-tice of spins falls into a nontrivial lattice in the lattice homo-topy classification so that the system cannot be in a sym-SREphase. Hence, knowing that the ground state is gapped anddoes not spontaneously break symmetries, we can concludethat the ground state has a topological order.If we put the system on a torus by taking the periodicboundary condition specified in Supplementary Fig. 4(a), forexample, then the ground states have four-fold degeneracy andthey can be distinguished by the values of the Wilson loops (cid:81) i ∈ horizontal loop ˆ X i and (cid:81) i ∈ vertical loop ˆ X i . 2. Flux insertion When we insert X -fluxes in a C -symmetric way (Sup-plementary Fig. 4 b), an odd number of plaquette terms flipsign in the Hamiltonian. We marked them by orange di-amonds in Supplementary Fig. 4(b). For those plaquettes, (cid:81) i ∈ p ˆ Z i = − is energetically favored. However, becauseof the global constraints (cid:81) all plaquettes (cid:81) i ∈ p ˆ Z i = +1 , theremust be one “unhappy” plaquette somewhere in the systemin the ground state, and there will be a ground state degener-acy originating from the choice of the position of the unhappyplaquette (Supplementary Fig. 4c) in addition to the four folddegeneracy distinguished by two Wilson loops. This degener-acy is extensive for the model Hamiltonian, but with genericsymmetry-preserving perturbations the degeneracy will be re-duced, though must preserve at least a two-fold degeneracydue the global anti-commutation relation between C (cid:48) and ˆ Z ,as discussed in the main text. Roughly speaking, the two re-maining states correspond to whether the unhappy plaquette islocalized near one, or the other, X -flux, with the degeneracybetween them guaranteed by C (cid:48) . Such an implementation ofthe degeneracy is precisely the opposite of degeneracy local-ization, since the two states are related by acting with string-like operator which transports an emergent magnetic flux (i.e.,an unhappy plaquette) between the two X -flux.To see how the global commutation relation is then imple-mented, recall the flux-inserted Hamiltonian is still symmetricunder ˆ C (cid:48) = ( (cid:81) i ∈ A ˆ X i ) ˆ C . A pair of ground states inter-change under ˆ C (cid:48) (Supplementary Fig. 4c ). The two pairedground states have the opposite eigenvalue of ˆ Z = (cid:81) i ˆ Z i .Therefore, ˆ C (cid:48) and ˆ Z are respectively represented by σ and σ within each pair, and thus ˆ C (cid:48) | GS = ⊕ all pairs of GS σ , (D2) ˆ Z | GS = ⊕ all pairs of GS σ . (D3)As required, ˆ C (cid:48) | GS and ˆ Z | GS do anticommute. However, thereis no contradiction in this case because ˆ C (cid:48) | GS is not a SWAPoperator. To see this, note that the trace of the SWAP opera-tor for localized degeneracies {| α , α (cid:105) : α , α = 1 , . . . , d } must be d , while the ˆ C (cid:48) | GS derived here is traceless. This means that the degeneracy localization assumption is violated,as expected for long-range entangled phases. (a)(c) (b) FIG. 4. (a) The toric code model. Arrows indicate periodic bound-ary conditions. (b) A configuration of C -symmetric X -fluxes. Theorange diamonds represent plaquette flipped by the X fluxes. (c) Il-lustration of the ground states after inserting the X -fluxes. The bluecircle means that the plaquette has (cid:81) i ∈ p ˆ Z i = − . Those on the top(bottom) row has +1 ( − ) eigenvalue of ˆ Z = (cid:81) i ˆ Z i . Appendix E: Layer Groups In this section we explain that our 2D argument for wallpa-per groups actually covers all 80 layer groups. Layer groupsare symmetries of 2D lattices embedded in 3D. Each ele-ment g ∈ S maps ( x, y, z ) to ( x (cid:48) , y (cid:48) , z (cid:48) ) , where ( x (cid:48) , y (cid:48) ) = p g ( x, y ) + t g ( p g is a O (2) matrix and t g is a two compo-nent vector) and z (cid:48) = + z or − z depending on g . In partic-ular, the translation subgroup of S is only in the 2D plane.When one talks about spinful electrons with significant spin-orbit couplings, it is important to use layer groups instead ofwallpaper groups, since it is not sufficient to know how x and y transform to identify the transformation of the spin degreeof freedom, as (the physical electron) spins are a projectiverepresentation of O (3) , not O (2) .In this paper, however, we assume that the spatial symmetry S acts only on the spatial coordinate and leaves the spin de-gree of freedom (more generally, the projective representationof G ) intact. In this case, one can regard the spatial coordinate z as an additional internal degree of freedom. If one com-pletely forgets about the transformation of z , the 2D part ofthe transformation ( x (cid:48) , y (cid:48) ) = p g ( x, y ) + t g defines one of the17 wallpaper groups, which we denote by S (cid:48) . SupplementaryTable IV summarizes the correspondence.Now, let ( x, y, z ) be a point that belongs to an IWP of S .Then the symmetry orbit of ( x, y, z ) under S does not include ( x, y, − z ) , since if this was not the case one could “reduce”the Wyckoff position by setting z → . Moreover, the 2D partof the coordinate ( x, y ) belongs to an IWP of S (cid:48) . Therefore,on each site of the corresponding IWP of S (cid:48) , the additionalinternal degree of freedom z is “frozen,” i.e., it can only beeither one of z or − z , or simply , and, after all, it is notreally a “degree of freedom” per se. Therefore one can always6set z = 0 , which makes the correspondence between S and S (cid:48) clearer. TABLE IV. Correspondence between layer groups and wallpapergroups. The layer groups in bold face act trivially on z . The asteriskindicates that, if the transformation of z is completely neglected, thetranslation subgroup of the layer group is enhanced.Wallpaper group Layer group1 , 4, 5 ∗ , 6, 7 ∗ , 27, 28 ∗ , 30 ∗ , 31 ∗ , 36 ∗ , 29, 33 ∗ , 32 ∗ , 34 ∗ , 356 14, 19, , 37, 38 ∗ , 41 ∗ , 48 ∗ , 40, 43 ∗ , 45 ∗ , 449 18, 22, , 47, 39 ∗ , 42 ∗ , 46 ∗ , 50, 51, 52 ∗ 11 53, , 57, 59, 61, 62 ∗ , 64 ∗ 12 54, 58, 60, , 6313 , 7414 67, , 7815 68, , 7916 66, , 7517 71, 72, 76, , 80 In this way, one can reduce the problem of a layer group S to that of the corresponding wallpaper group S (cid:48) . The latticehomotopy classification and the proof of no-gos for S are allidentical to those for S (cid:48) . Appendix F: The conjecture in 3D Here, we present some examples of space groups whichdemonstrate how our current set of no-gos applies to 3D, andalso illustrate how they are insufficient to prove the conjecturein general.The first example is space group ( P ), generatedby three orthogonal π -rotations C ,α ( α = x, y, z ) and lat-tice translations. The space group has eight IWPs: r =( r x , r y , r z ) where r α = 0 or / [20]. For an internal symme-try group G giving H ( G, U(1)) = Z n , the lattice homotopycan be readily computed and the result is Z n × ( Z gcd( n, ) .Suppose n = 2 , which gives = 256 distinct lattices un-der lattice homotopy. Whenever the net representation on theintersection of a C rotation axis and the primitive unit cellis projective, the “rotation no-go” is applicable and thereforethis nontrivial lattice (in the lattice homotopy sense) indeedforbids any sym-SRE phase. To see this, consider putting thesystem on a large but finite torus with an odd circumferencealong the C axis of interest, which we suppose is along ˆ z . Wecan then formally view the system as two-dimensional, whereeach “point” is interpreted as a loop in the z -direction. Our 2D argument will then go through, except that the fluxes arenow promoted to a flux loops and the defect line is promotedto a defect surface (with the topology of a cylinder).On the other hand, when every IWP is occupied by the non-trivial projective representation [ ω ] r = − , we are unable toprove an obstruction for sym-SRE phases using the no-gos wederived, although this configuration is a nontrivial element inthe lattice homotopy classification. In fact, the derived no-gosrule out sym-SRE phases in all but one of the nontriviallattices. This also illustrates the fact that, despite we are un-able to prove the full conjecture in 3D, our results can still beapplied to many 3D lattices, or even to realistic 3D materials.(More precisely, if a lattice is nontrivial under the “no-go”classification discussed in Sec. B 4, defined using the threeno-gos we derived, a sym-SRE phase is forbidden.)As an other example, we consider the space group ( P ¯1 ).This space group has the inversion symmetry in addition tolattice translations. The IWPs and the lattice homotopy clas-sifications are identical to that of , and the presence/ ab-sence of no-gos also follows from the preceding discussion.Note that, however, in applying the dimension reduction de-scribed above, the z -coordinate formally becomes an internaldegree of freedom, which is “flipped” under the action of the3D inversion. As discussed in Sec. E, insofar as the symmetrygroup is S × G , where S acts only on the spatial coordinates,this does not affect the arguments. Appendix G: New no-gos in 3D While we have not yet derived the full set of sym-SRE no-gos required to prove the conjecture in 3D, we discuss herean interesting extension of the no-go arguments. In particu-lar, this discussion can shed some light on cases for which theBieberbach argument presented in Supplementary Ref. [17]does not lead to a provably-tight bound on the filling condi-tion.We begin by reviewing the “tight no-go problem” describedin Supplementary Ref. [17], which studied a more generalsetting for which the electrons can be delocalized and spin-orbit coupled, assuming instead only time-reversal symme-try. Generally, a sym-SRE no-go detects some, but not all,obstructions to realizing sym-SRE phases in a system. Forinstance, consider the original Lieb-Schultz-Mattis theorem,which states that a sym-SRE phase is forbidden whenever ν ,the electron filling per primitive unit cell, satisfies ν (cid:54)∈ N .This implies sym-SRE phases are not forbidden when ν iseven, but depending on the symmetries of the systems therecan be further no-gos that are not detected by this single cri-terion. We say a set of no-go is “tight” if a sym-SRE phase ispossible whenever the no-gos are silent. Proving tightness isgenerally nontrivial, since it involves explicit constructions ofsym-SRE phases for all instances permitted by the no-gos.In Supplementary Refs. [17], the tightness of the Bieber-bach no-go is only proven for 218 of the 230 space groups.Such tightness was established by studying the possible fill-ings of band insulators [24, 35]. The 12 exceptional SGs cor-respond to cases where at certain fillings band insulators are7disallowed despite the Bieberbach no-go being silent. (As atechnical remark, we note the number of exceptional SGs de-pends on whether spin-orbit coupling and / or TR are assumedor not, as the set of band-insulator fillings can be modified bythe symmetry settings [35]. Here, “12” refers to case of TRsymmetric system with negligible spin-orbit coupling.)The tightness of the Bieberbach no-gos for these 12 excep-tional SGs is still an open question. Here, we study some as-pect of this problem by extending the new no-gos developedin the present work to these SGs. In particular, recall that herewe are focusing on localized spin problems with on-site uni-tary symmetries in the limit of vanishing spin-orbit coupling.This is a much more restrictive setting than the one consideredin Supplementary Ref. [17], and as such it may not be too sur-prising if one can expose further obstructions, as we will nowdemonstrate.For concreteness, we focus on only one of the 12 excep-tional SGs, , and consider the most physically-relevantsetting for which the internal-symmetry group is taken to be G = SO(3) , appropriate for crystals with negligible spin-orbitcoupling. The Bieberbach no-go forbids any sym-SRE phaseunless the electron filling ν ∈ N [17], whereas band insula-tors are possible only for ν ∈ N [35]. The mismatch betweenthe two, ν = 8 n − for any n ∈ N , corresponds to cases forwhich the tightness of the interacting bounds is unclear. Here,we show that within the class of localized spin models weconsidered here, a no-go is in fact present and obstructs anysym-SRE phases when ν = 8 n − .To proceed, we first discuss the symmetries of SG . TheSG is primitive tetragonal, and we take the primitive latticevectors as t = (1 , , , t = (0 , , and t = (0 , , . Itcan be viewed as being generated by the following two non-symmorphic symmetries together with the lattice translations: S z = T / R z,π/ ; G x = T / T / m y , (G1)where R z,θ denotes a rotation by angle θ about the positive z axis, m y denotes the mirror y (cid:55)→ − y , and T i is the lat-tice translation by t i . Note that S z is a screw, mean-ing that S z = T R z,π is a symmorphic C rotation. Whilethe point group is of order , sites can sit on the C rota-tion axis and hence one can find higher-symmetry Wyckoffpositions with only sites per unit cell. This can be seenfrom Supplementary Ref. [20], which lists three Wyckoff po-sitions with |W a | = |W b | = 4 and the generic position |W c | = 8 . Both W a and W b are irreducible, and froma lattice-homotopy point of view the lattices are classified by Z × Z (for Z projective representations).A general lattice in SG can be viewed as a stack of n w copies of the minimal lattices in W w for w = a , b or c , andsuch a lattice will contain n a + n b ) + 8 n c sites in the prim-itive unit cell. For strongly-localized electronic systems, eachsite in the lattice represents a spin-1/2 electron, and thereforethe electron filling is simply given by the number of sites. Thefillings of interest, ν = 8 n − , therefore corresponds to lat-tices with n a + n b being odd.To show that a sym-SRE is forbidden (in this localized-spin,spin-orbit-coupling-free) whenever ν = 8 n − , it sufficesto consider the minimal realizations of lattices in W a and FIG. 5. New no-go in 3D. The flux-insertion argument for sym-SREno-go can be generalized to higher dimensions with more compli-cated geometries. Here we show a compact space obtained by iden-tifying the two marked faces by a screw and the other pairs bylattice translations. A pair of symmetry-related line fluxes (greenlines) are inserted, and the system maintains a C rotation symmetryabout the dashed axis. The defect surface (shaded) and its C partner(not shown) together enclose a single site (sphere). W b . First, we consider a minimal lattice in W a , whichcontains four sites in the primitive unit cell with coordinates r = (0 , , z ); r = (0 , , z + 1 / r = (1 / , / , z ); r = (1 / , / , z + 1 / , (G2)where z is a free parameter. Note that all sites are invariant un-der a C rotation, and are all symmetry-related to each other,e.g. S z ( r ) = r , and G x ( r ) = r .As set up, the Bieberbach no-go is silent. In addition, any C axis intersects exactly two sites in each unit cell, and there-fore a direct generalization of the C no-go to the 3D settingwill also be silent. The key idea for deriving a no-go is that the C no-go can become active if we identify the two S z -relatedsites in each unit cell sitting on the same rotation axis, i.e. ano-go is exposed once we combine the Bieberbach ideas withthe rotation no-gos.More precisely, we identify points in R that are related byany elements in the group Γ ≡ (cid:104) t , t , S z (cid:105) . As Γ is fixed-point free, R / Γ is a Bieberbach manifold [17] and we con-sider the corresponding system defined on this compact space.Note that the volume of R / Γ is only half of the primitive unitcell, which is far away from the thermodynamic limit. This,however, is only a minor technical subtlety, as one can alwaystake a “scaled-up” version of Γ and obtain a manifold that isas large as one pleases. All the following discussions will beunaffected by such scaling-up.Importantly, C satisfies the condition listed in Supple-mentary Ref. [35] and therefore is a “remnant symmetry,”i.e. the modded-out system defined on R / Γ remains C sym-metric. Now imagine introducing a pair of defect loops inthe modding-out procedure, and we consider a topologically-nontrivial configuration for which the loop closes itself with a S z twist (Supplementary Fig. 5). In particular, we can positionthe defect loops such that they form a C -related pair. Nowwe insert X -fluxes into the system. The flux insertion proce-dure is then similar to the 2D case, except that now we twistthe local Hamiltonian along a defect surface bounded by thetwo defect loops. As can be seen from Supplementary Fig. 5,under a C rotation the defect surface is transformed to its C W b through a shift of origin, and in fact to any lattices with n a + n b odd. (Actually, we achieved even more, since theargument presented already shows ∼ LH = ∼ NG for this par-ticular SG.) This establishes a no-go for any spin-rotation-symmetric quantum magnets in SG with ν = 8 n − . Inaddition, for each filling ν ∈ N one can design a symmetry-preserving valence-bond solid with the desired filling. Thiscan be achieved by putting sites in the generic position andpinning the center of the valence bonds to either W a or W b . This implies the tight filling constraints for a sym-SREphase in SG is ν ∈ N in the spin-orbit-free quantum-magnet setting.We note that when spin-orbit coupling is negligible and thesystem is in addition TR invariant, we have established thesym-SRE constraint ν ∈ N in both the free [35] and infinite,on-site repulsion limits. If a sym-SRE phase is indeed possi-ble for ν = 8 n − , it will require either changing the sym-metry setting, or require interaction strengths that are neces-sarily “moderate.” Somewhat surprisingly, in SupplementaryRef. [25] it was established that if time-reversal is absent butspin-rotation symmetry remains intact, it is possible to realizea band insulator at the filling of ν = 4 . (Note that Supple-mentary Ref. [25] provides a band insulator example at filling ν = 2 for spinless particles. Restoring spin but assuming spin-rotation invariance, this corresponds to a filling ν = 4 bandinsulator.) For such problems, if an on-site (density-density)repulsion U is incorporated, the system necessarily undergoesan interaction-driven phase transition from a sym-SRE phaseto either a symmetry broken or a more exotic phase as U istuned from → ∞ . It is currently unclear whether the phasetransition happens in the spin or charge sector, and it is aninteresting open problem to understand the full landscape offilling constraints for this SG in the different symmetry set-tings. In closing, we also remark that similar extension of the no-gos apply to some, if not all, of the exceptional SGs, andsurveying all of them is another interesting problem that weleave for future studies. Appendix H: Connection to crystalline symmetry-protectedtopological orders First, we note our conjecture has a simple interpretation interms of “valence bond” patterns, which are commonly usedin the discussion of SPTs arising in quantum magnets, as inthe pictorial description of the AKLT chain. In this language,our conjecture states that a sym-SRE is possible only whenthere could be a symmetric valence-bond type state. To seethis, consider a lattice of spins Λ which is trivial, i.e., Λ ∼ ,and let γ denote the path which annihilates the lattice. For-mally, we can view Λ as an S -invariant 0-chain (a collectionof points), and we can think of γ as an S -invariant 1-chain(a collection of paths), where the points and lines take val-ues in Z . The statement of homotopy can then be phrasedin terms of the boundary of the chain, Λ = ∂γ . If two sitesare connected by a path in γ , we can view the connection as avalence-bond which projects them into the singlet state. The S -invariance of γ implies the resulting valence-bond patternrespects the spatial symmetries.In addition, there is also an intriguing similarity betweenour conjecture and recent results on the classification of point-group SPT phases [30, 31]. In the case of translation sym-metry, it was realized that LSM-constraints in d -dimensionshave a deep connection to the physics of SPT phases in d + 1 -dimensions [28]. The d -dimensional boundary of a d + 1 -dimensional SPT must be either symmetry-broken, or LRE– just like the LSM-constraint. It was pointed out that a 2Dspin-1/2 magnet can actually be considered as the surface ofthe 3D AKLT phase, which is an SPT. Using the SPT bulk-boundary correspondence [26, 27], one can argue not only theusual LSM-constraint, but a generalized constraint on the pre-cise set of topological orders that can emerge. Thus it is natu-ral to suspect that our space-group LSM will relate to the clas-sification of space-group SPTs, a subject of current investiga-tion. Presumably, each of the lattice equivalence classes [Λ][Λ]