Irreducible Projective Representations and Their Physical Applications

Jian Yang1, 2 and Zheng-Xin Liu3, ∗ 1International Center for Quantum Materials, School of Physics, Peking University, Beijing, 100871, China 2Department of physics, Tsinghua University, Beijing 100084, China. 3Department of physics, Renmin University, Beijing 100876, China. (Dated: December 19, 2017) An eigenfunction method is applied to reduce the regular projective representations (Reps) of ﬁnite groups to obtain their irreducible projective Reps. Anti-unitary groups are treated specially, where the decoupled factor systems and modiﬁed Schur’s lemma are introduced. We discuss the applications of irreducible Reps in many-body physics. It is shown that in symmetry protected topological phases, geometric defects or symmetry defects may carry projective Rep of the symmetry group; while in symmetry enriched topological phases, intrinsic excitations (such as spinons or visons) may carry projective Rep of the symmetry group. We also discuss the applications of projective Reps in problems related to spectrum degeneracy, such as in search of models without sign problem in quantum Monte Carlo Simulations. PACS numbers:11.30.Er, 02.20.-a, 71.27.+a

Contents C. Anti-unitary groups: decoupled cocycles 26 1. Type-I anti-unitary groups: T 2 = E 27 2m I. Introduction 1 2. Type-II anti-unitary groups: T = E, m ≥ 2 28

II. Regular Projective Reps 3 D. Classiﬁcation of some simple anti-unitary A. Factor systems and 2-cocycles3 groups 30 T B. Regular projective Reps5 1. Direct product group H × Z2 30 T 2. Semidirect product group H o Z2 with −1 III. CSCO and irreducible projective Reps 6 T g = g T 30 A. Brief review of the reduction of regular linear a. When H = ZN 30 Reps for unitary groups7 b. When H is a direct product of cyclic groups32 B. Reduction of regular projective Reps8 3. The case of complex cocycles 33 1. Unitary groups8 2. Anti-unitary groups9 E. Proof of some statements 33 C. Some examples 13 1. The Dimension of irreducible projective Rep is D. CG coeﬃcients for projective Reps 14 a multiple of the period of its factor system 33 2. Completeness of irreducible bases 33 IV. Applications of projective Reps 14 3. Two relations between class operators of A. 1-D SPT phases 14 anti-unitary groups 34 B. Defects in 2D topological phases 20 F. Reduction of regular projective Reps 35 1. Vortices in topological superconductors 1. Unitary groups 35 (fermionic SPT phases) 20 2. Anti-unitary groups 37 2. Monodromy defects in bosonic SPT phases 20 a. Abelian anti-unitary group Z × Z × ZT 37 3. 2D topological order/SET phases 21 2 2 2 b. Non-Abelian anti-unitary group ST 40 C. Sign problems in Quantum Monte Carlo 21 4 D. Space groups, Spectrum degeneracy and other References 40 applications 23

V. Conclusion and Discussion 23 I. INTRODUCTION arXiv:1605.05805v3 [cond-mat.str-el] 16 Dec 2017 Acknowledgments 23 Group theory has wide applications in contemporary A. Group cohomology 24 physics, including particle physics, quantum field theory, gravitational theory and condensed matter physics. B. Canonical gauge choice and solutions of There are mainly two types of groups in physics, the cocycles 24 symmetry groups and the gauge groups, where the group elements correspond to global or local operations respectively. The linear Rep theory of groups is one of the fun- damental mathematical tools in quantum physics. For ∗Electronic address: [email protected] example, the Hilbert space of orbital (or integer spin) 2 angular momentum forms linear Rep space of SO(3) ro- where Ub is a unitary operator and K is the complex- tational symmetry group, while charged particles carry conjugate operator satisfying Reps of U(1) gauge group. On the other hand, projec- ∗ tive Reps of groups were less known to physicists [1–12]. Ka = a K (1) In projective Reps, the representing matrices of group for any complex number a. Anti-unitary elements also elements obey the group multiplication rule up to U(1) differ from unitary ones by their nontrivial module as phase factors. These U(1) phase factors are called factor defined later in Eq.(11). In the case that an anti-unitary systems of the corresponding projective Rep (seeIIA). group has the same group table with that of a unitary In some sense, the theory of projective Reps has closer group, one can distinguish them by checking the existence relationship to quantum theory since quantum states are of anti-unitary elements (such as the anti-unitary time- defined up to global U(1) phase factors. The well known reversal group ZT = {E,T } and unitary spatial-inversion Kramers degeneracy owning to time reversal symmetry 2 group ZP = {E,P }). The simplest anti-unitary group is actually a typical example of projective Rep[13, 14]. 2 is the two-element group ZT = {E,T }, where the time When acting on operators, the square of time-reversal 2 reversal operator T is anti-unitary. operator is equal to identity T 2 = E and thus defines T Anti-unitary groups play important roles in quantum a two-element group Z2 = {E,T }; however, when acting on a Hilbert space of odd number of electrons, the theory. For example, the Schrödingerequations and square of (the Rep of) the time reversal operator is no the Dirac equations for free particles are invariant under the time-reversal symmetry group ZT if there are longer identity: Tb2 = −1. This nontrivial phase factor 2 T no magnetic fields. Under time-reversal transforma- −1 stands for a nontrivial projective Rep of Z2 , which tion T , in addition to t → −t, the equations should guarantees that each energy level is (at least) doubly de- be transformed into their complex conjugation. The generate. Another typical example is that half-integer importance of anti-unitary groups can also be seen spins carry projective representations of SO(3) group. from the well known CPT theorem[27]. In condensed Projective Rep is a natural tool to describe symme- matter physics, anti-unitary symmetry groups are also try fractionalization and is widely used in recently dev- important. For instance, the magnetic point groups oleped theories such as Symmetry Protected Topological and magnetic space groups are anti-unitary groups[28]; (SPT) phases[15–17] and Symmetry Enriched Topologi- the well studied topological insulators[29–34] are SPT cal (SET) phases [18, 19]. These exotic quantum phases phases protected by U(1) charge conservation symme- are beyond the Laudau symmetry breaking paradigm be- try and time reversal symmetry. Many properties of cause they exhibit no long-range correlations of local or- unitary groups cannot be straightforwardly generalized der parameters. The SPT states are short range en- to anti-unitary groups. For example, the classification tangled and thus carry trivial topological order, while of SPT phases with anti-unitary groups in three dimen- the SET states are long-range entangled[20] and thus sions is different from that with unitary groups[35, 36]. carry nontrivial topological orders. In these novel quan- Comparing to unitary groups [37], it is also difficult tum states, certain defects (such as boundaries or sym- to gauge a global anti-unitary symmetry into a local metry fluxes) or elementary excitations (called anyons) symmetry[38]. Since anti-unitary groups are special, a carry projective Reps of the symmetry group. Especially, systematic method of obtaining their linear Reps (also one-dimensional SPT phases with an on-site symmetry called co-representations [26]) and projective Reps (also group G are characterized by projective Reps [21] of G called co-ray representations[39]) is urgent. and therefore classified by the second group cohomology 2 Similar to the linear Rep theory, only irreducible pro- H (G, U(1)) [22, 23]. jective Reps are of physical interest. So it is an important Most of the groups in quantum physics are unitary. issue to find a systematic way to obtain all the irreducible A group is unitary if every group element stands for a projective Reps. The irreducible projective Reps of a unitary operator which keep the inner product of any group G can be obtained from the linear Reps of its cov- two states hψ1|ψ2i invariant, namely, ering groups (also called representation groups[40]). The hgψ |gψ i = hψ |ψ i. covering groups are central extensions of G and are also 1 2 1 2 classified by the second group cohomology. For example, the double point group in spin-orbit coupled systems is where |gψ1,2i =g ˆ|ψ1,2i for any group element g. On the other hand, a group is called anti-unitary (also called a covering group of the corresponding point group. For ‘antiunitary’[24, 25] or ‘nonunitary’[26]) if it contains at a given factor system, the corresponding covering group least one anti-unitary element g (such as the time reversal can be constructed in the following steps: (1) transform operator T ) which transforms the inner product of two the factor system into its standard form where all the states into its complex conjugate, U(1) phase factors are the Nth roots of 1, here N is the period of the factor system (see Sec.E 1); (2) split each ∗ 2 N hgψ1|gψ2i = hψ1|ψ2i = hψ2|ψ1i. group element g in G into N elements, ηg, η g, ..., η g, where ηN = 1; (3) taking into account the factor system, Generally, the anti-unitary operatorg ˆ corresponding to the new elements obey a new multiplication rule under the anti-unitary element g can be written asg ˆ = UKb , which the G × N elements (here G is the order of group 3

0 0 G) form a new group G satisfying G /ZN = G. This ex- and the regular projective Reps of finite groups. In sec- tended group G0 is the covering group corresponding to tionIII, we apply the eigenfunction method to reduce the given factor system. Then from the irreducible linear the regular projective Reps into irreducible ones. Uni- Rep of G0 and the many-to-one mapping from G0 to G, tary groups and anti-unitary groups are discussed sep- one can obtain irreducible projective Rep of the quotient arately and the results of some finite groups are listed group G with the given factor system. Since the covering in TableI. Readers who are more interested in the ap- group G0 is larger than G, this method is very indirect plications of projective Reps in concrete physical prob- and will not be discussed in detail. lems can skip this part and go to sectionIV directly, Alternatively, we can obtain the irreducible projective where symmetry fractionalizations in topological phases Reps from the regular projective Reps without referring of matter, sign problems in quantum Monte Carlo simu- to the covering group. Useful knowledge can be learned lations, and other topics related to spectrum degeneracy from linear Reps of groups, where all irreducible Reps are discussed. SectionV is devoted to the conclusion and can be obtained by reducing the regular Reps. A re- discussion. markable eigenfunction method was introduced by J.-Q. Chen[41, 42] to reduce the regular Reps. In this approach the Rep theory is handled in a physical way. The main II. REGULAR PROJECTIVE REPS idea is to label the irreducible bases by non-degenerate quantum numbers, namely, the eigenvalues of a set of A. Factor systems and 2-cocycles commuting operators. So the main task is to find the Complete Set of Commuting Operators (CSCO) of the Projective Reps and the factor systems. We first Rep space. By making use of the class operators of the consider U(1)-coefficient projective Reps of a finite uni- group G and those of the canonical subgroup chain of G, tary group G. Later we will also discuss A-coefficient together with the class operators of the canonical sub- projective Reps, where A is a finite subgroup of U(1). ¯ group chain of the intrinsic group G, the regular Reps of A projective Rep of G is a map from the element g ∈ finite groups and the tensor Reps of compact Lie groups G to a matrix M(g) such that for any pair of elements (such as U(n)) are successfully reduced. Chen [42] ap- g1, g2 ∈ G, plied this method to obtain irreducible Reps of space iθ2(g1,g2) groups, and as a byproduct, part of the projective Reps of M(g1)M(g2) = M(g1g2)e , (2) point groups were obtained. In this paper, we will gen- iθ2(g1,g2) eralize the eigenfunction method to reduce the regular where the U(1) phase factor ω2(g1, g2) = e is a projective Reps of finite groups, especially anti-unitary function of two group variables and is called the factor groups. The factor systems of projective Reps are classi- system. If ω2(g1, g2) = 1 for any g1, g2 ∈ G, then above fied by group cohomology and can be obtained by solving projective Rep is trivial, namely, it is a linear Rep. the 2-cocycle equations. For anti-unitary groups we show The associativity relation of matrix multiplication that the factor system can be decoupled into two parts, yields constraints on the factor system. For any three T one contains the information of the quotient group Z2 elements g1, g2, g3 ∈ G, while other part contains the information of the unitary normal subgroup. The introduction of decoupled factor M(g1)M(g2)M(g3) = [M(g1)M(g2)]M(g3) systems is an important step which greatly simplifies the iθ2(g1,g2) iθ2(g1g2,g3) = M(g1g2g3)e e calculations. The regular projective Reps are then obtained by acting the group elements on the group space = M(g1)[M(g2)M(g3)] iθ2(g2,g3) iθ2(g1,g2g3) itself, and the matrix elements are the 2-cocycles. After = M(g1g2g3)e e , finding out the CSCO and their eigenfunctions of the regular projective Reps, all the irreducible projective Reps which requires that are obtained. In this approach, we only need to treat matrices with maximum dimension G, so it is much simpler ω2(g1, g2)ω2(g1g2, g3) = ω2(g2, g3)ω2(g1, g2g3), (3) than the covering group method. or equivalently The physical applications of projective Reps are then discussed. Some new viewpoints are presented, for in- θ (g , g ) + θ (g g , g ) = θ (g , g ) + θ (g , g g ), stance, the Majorana zero modes in topological super- 2 1 2 2 1 2 3 2 2 3 2 1 2 3 conductors are explained as projective Reps of some sym- where the equal sign means equal mod 2π. The factor metry groups; some models which are free of sign prob- system of a projective Rep must satisfy equation (3). lem under quantum Monte Carlo simulations are inter- Conversely, all the solutions of above equation corre- preted as the properties of projective Reps of anti-unitary spond to the factor system of a projective Rep. groups, and generalizations (i.e. possible new classes of If we introduce a ‘gauge transformation’ to the projec- sign-free models) are proposed. tive Rep, The paper is organized as follows. In sectionII, we introduce the projective Reps and the factor systems, M 0(g) = M(g)eiθ1(g), (4) 4

iθ1(g) where the phase factor Ω1(g) = e depends on a sin- (C) if g1 is unitary while g2 is anti-unitary, then the gle group variable, then result is

0 0 iθ2(g1,g2) iθ1(g1)+iθ1(g2) iθ2(g1,g2) M (g1)M (g2) = M(g1g2)e e M(g1)M(g2)K = M(g1g2)e K; 0 0 iθ (g1,g2) = M (g1g2)e 2 , (D) if g1 is anti-unitary while g2 is unitary, then the where the factor system transforms into result is

∗ iθ2(g1,g2) iθ1(g1)+iθ1(g2) M(g )KM(g ) = M(g )M (g )K = M(g g )e K. iθ0 (g ,g ) iθ (g ,g ) e 1 2 1 2 1 2 e 2 1 2 = e 2 1 2 , eiθ1(g1g2) If we deﬁne namely, 1, if g is unitary, 0 s(g) = ω2(g1, g2) = ω2(g1, g2)Ω2(g1, g2), (5) − 1, if g is antiunitary, with and deﬁne the corresponding operator Ks(g) as

Ω1(g1)Ω1(g2) Ω2(g1, g2) = (6) I, if s(g) = 1, Ω1(g1g2) K = s(g) K, if s(g) = −1,

iθ1(g) 0 where Ω1(g) = e . Since M (g) can be adiabati- cally transformed into M(g) by continuously adjusting then the above four cases (A)∼(D) can be uniﬁed as a single equation the phase Ω1(g), the two projective Reps are considered to be equivalent. Generally, if the factor systems of any 0 iθ2(g1,g2) two projective Reps M(g) and M (g) are related by the M(g1)Ks(g1)M(g2)Ks(g2) = M(g1g2)e Ks(g1g2). relation (5), then M(g) and M 0(g) are considered to belong to the same class of projective Reps (even if their Substituting above results into the associativity rela- dimensions are diﬀerent). tion of the sequence of operations g1 × g2 × g3, similar to Now we consider anti-unitary groups. If a group G (3), we can obtain is anti-unitary, then half of its group elements are anti- unitary, and the remaining unitary elements form a nor- M(g1)Ks(g1)M(g2)Ks(g2)M(g3)Ks(g3) mal subgroup H, = M(g1g2g3)ω2(g1, g2)ω2(g1g2, g3)Ks(g1g2g3) s(g ) T 1 = M(g1g2g3)ω2(g1, g2g3)ω2 (g2, g3)Ks(g1g2g3), G/H ' Z2

T 2 namely, with Z2 = {E, T} and T = E. There are two types of anti-unitary groups. In Type-I anti-unitary groups, T s(g1) can be chosen as an element of G, in this case the group G ω2(g1, g2)ω2(g1g2, g3) = ω2 (g2, g3)ω2(g1, g2g3). (7) T is either a product group G = H × Z2 or a semi-product T Eq.(7) is the general relation that the factor systems group G = HoZ2 . In type-II anti-unitary groups, T ∈/ G and the period of any anti-unitary elements in G is at of any ﬁnite group (no matter unitary or anti-unitary) least 4. More rigorous deﬁnitions of type-I and type-II should satisfy. anti-unitary groups are given in appendixC. In this work, Similar to Eq.(4), if we introduce a gauge transfor- 0 we will mainly focus on type-I anti-unitary groups. Only mation M (g)Ks(g) = M(g)Ω1(g)Ks(g), then the factor some simple fermionic groups of type-II will be discussed. system changes into Supposing g is a group element in G, then it is repre- 0 sented by M(g) if g is a unitary element and represented ω2(g1, g2) = ω2(g1, g2)Ω2(g1, g2), (8) by M(g)K if g is anti-unitary, where K is the complex- conjugate operator shown in Eq.(1).The multiplication of with projective Reps of g1, g2 depends on if they are unitary s(g1) or anti-unitary. There are four cases: Ω1(g1)Ω1 (g2) Ω2(g1, g2) = . (9) (A) both g1, g2 are unitary, then we obtain Ω1(g1g2)

iθ2(g1,g2) M(g1)M(g2) = M(g1g2)e ; The equivalent relations (8) and (9) define the equivalent classes of the solutions of (7). The number of equivalent which is the same as Eq.(2); classes for a finite group is usually finite. nd (B) both g1, g2 are anti-unitary, then the result is The 2 group cohomology and the 2-cocycles. Actually, the equivalent classes of the factor systems asso- ∗ iθ2(g1,g2) M(g1)KM(g2)K = M(g1)M (g2) = M(g1g2)e ; ciated with the projective Reps of group G form a group, 5 called the second group cohomology. The group coho- However, above canonical gauge condition doesn’t com- mology {Kernel d/Image d} is defined by the coboundary pletely fix the coboundaries and there are still infinite operator d (for details see appendixA), number of solutions. In later discussion, we will select one of the 2-cocycle solutions in each class as the factor (dωn)(g1, . . . , gn+1) system of the corresponding projective Rep. In that case, (−1)n+1 the gauge degrees of freedom (i.e. the 2-coboundaries) = [g1 · ωn(g2, . . . , gn+1)]ωn (g1, . . . , gn) × n are completely fixed. Y i ω(−1) (g , . . . , g , g g , g , . . . , g ). (10) The case where A is a finite abelian group. n 1 i−1 i i+1 i+2 n+1 Mathematically, the second group cohomology H2(G, A) i=1 with A-coefficient classifies the central extensions of G where g1, ..., gn+1 ∈ G and the variables ωn(g1, . . . , gn) by A. In the default case, A = U(1) and the projective take value in an Abelian group A [usually A is a subgroup Reps of G are classified by H2(G, U(1)) and correspond of U(1)]. The set of variables ωn(g1, . . . , gn) is called a to linear Reps of U(1) extensions of G. A-cochain. The module g· is defined by Physically, we are also interested in the cases where A is a finite abelian group, such as ZN . For example, the s(g) g · ωn(g1, . . . , gn) = ωn (g1, . . . , gn). (11) topological phases with Z2 topological order and symme- 2 try group G are (partially) classified by H (G, Z2) (see With this notation, Eq. (7) can be rewritten as section IV B 3). The classification of ZN coefficient projective Reps is usually different from the U(1) coefficient (dω2)(g1, g2, g3) = 1, projective Reps (see appendixB), for instance, certain U(1) coboundaries may be nontrivial ZN cocycles. the solutions of above equations are called 2-cocycles with Once the factor systems are obtained by solving the co- A-coefficient. Similarly, Eq. (9) can be rewritten as cycle equaitons and coboundary equations (see appendix B), the method of obtaining A-coefficient irreducible pro- Ω2(g1, g2) = (dΩ1)(g1, g2), jective Reps is the same with the cases of U(1)-coefficient. In the following we will discuss the method of obtaining where Ω1(g1), Ω1(g2) ∈ A and thus defined Ω2(g1, g2) U(1) coefficient irreducible projective Reps by reducing 0 are called 2-coboundaries. Two 2-cocycles ω2(g1, g2) and the regular projective Reps. ω2(g1, g2) are equivalent if they differ by a 2-coboundary, see Eq. (8). The equivalent classes of the 2-cocycles 2 ω2(g1, g2) form the second group cohomology H (G, A). B. Regular projective Reps A projective Rep whose factor system is a A-coefficient 2-cocycle is called A-coefficient projective Rep. By de- Regular projective Reps. For a given factor sys- fault, projective Reps are defined with U(1) coefficient, tem, we can easily construct the corresponding regu- namely, A = U(1). In this case, ω2(g1, g2) ∈ U(1) so lar projective Rep (the regular Rep twisted by the 2- iθ2(g1,g2) we can write ω2(g1, g2) = e , where θ2(g1, g2) ∈ cocycles) using the group space as the Rep space. The [0, 2π). The cocycle equations (dω2)(g1, g2, g3) = 1 can group element g is not only an operatorg ˆ, but also a be written in terms of linear equations, basis |gi. The operatorg ˆ1 acts on the basis |g2i as the following, s(g1)θ2(g2, g3) − θ2(g1g2, g3) + θ2(g1, g2g3) iθ2(g1,g2) −θ2(g1, g2) = 0. (12) gˆ1|g2i = e Ks(g1)|g1g2i, (14)

iθ1(g1) or in matrix form Similarly, if we write Ω1(g1) = e and Ω2(g1, g2) = eiΘ2(g1,g2), then the 2-coboundary (9) can be written as gˆ1 = M(g1)Ks(g1), (15) Θ2(g1, g2) = s(g1)θ1(g2) − θ1(g1g2) + θ1(g1). (13) with matrix element

The equal sign in (12) and (13) means equal mod 2π. iθ2(g1,g2) M(g1)g,g2 = hg|gˆ1|g2i = e δg,g1g2 . (16) From these linear equations, we can obtain the classification and the cocycles solutions of each class (for details For any group element g3, we have see appendixB). The classification of the factor system h i iθ2(g2,g3) also gives a constraint on the dimensions of correspond- gˆ1gˆ2|g3i =g ˆ1 e Ks(g2)|g2g3i ing projective Reps, as discussed in appendixE1. h i iθ2(g1,g2g3) iθ2(g2,g3) Gauge fixing. For each class of 2-cocyles satisfying = e Ks(g1) e Ks(g2)|g1g2g3i (12), there are infinite number of solutions which differ = eiθ2(g1,g2g3)eis(g1)θ2(g2,g3)K |g g g i, by 2-coboundaries. We will adopt the canonical gauge s(g1g2) 1 2 3 by fixing and

ω (E, g) = ω (g, E) = 1. g g |g i = eiθ2(g1g2,g3)K |g g g i. 2 2 d1 2 3 s(g1g2) 1 2 3 6

Comparing with the 2-cocycle equation (12), it is easily On the other hand, iθ2(g1,g2) obtained thatg ˆ1gˆ2 = e gd1g2. In matrix form, this relation reads gˆ2g¯ˆ1|g3i =g ˆ2ω2(g3, g1)|g3g1i = ωs(g2)(g , g )ω (g , g g )K |g g g i. iθ2(g1,g2) 2 3 1 2 2 3 1 s(g2) 2 3 1 M(g1)Ks(g1)M(g2)Ks(g2) = e M(g1g2)Ks(g1g2). (17) Comparing with the cocycle equation (dω2)(g2, g3, g1) = 1, we have Eq.(17) indicates that M(g)Ks(g) is indeed a projective Rep of the group G. For the trivial 2-cocycle where g¯ˆ1gˆ2 =g ˆ2g¯ˆ1. iθ2(g1,g2) e = 1 for all g1, g2 ∈ G, M(g)Ks(g) reduces to the regular linear Rep of G. In matrix form, above equation reads Intrinsic Regular projective Reps. In order to re- M(¯g )M(g )K = M(g )K M(¯g ). duce the regular projective Rep of G, we will make use of 1 2 s(g2) 2 s(g2) 1 the intrinsic group G¯. Each element g in G corresponds ¯ to an intrinsic group elementg ¯ in G, and the intrinsic III. CSCO AND IRREDUCIBLE PROJECTIVE elements obey right-multiplication rule withg ¯1g2 = g2g1 REPS andg ¯1g¯2 = g2g1. Obviously, the intrinsic group G¯ commutes with G since (¯g g )g = (g g¯ )g = g g g . Cor- 1 2 3 2 1 3 2 3 1 Supposing a group element g ∈ G is represented by responding to G¯, we can define the intrinsic regular pro- M(g)K , we assume that all the Rep matrices M(g) jective Rep, s(g) are unitary (for finite groups all the Reps can be transformed into this form). If a projective Rep can not be g¯ˆ |g i = eiθ2(g2,g1)|g g i, (18) 1 2 2 1 reduced by unitary transformations into direct sum of lower dimensional Reps, then it is called irreducible pro- or in matrix form g¯ˆ1 = M(¯g1) with jective Rep. If two irreducible projective Reps M(g)Ks(g) 0 iθ (g ,g ) and M (g)K can be transformed into each other by a M(¯g ) = e 2 2 1 δ . (19) s(g) 1 g,g2 g,g2g1 unitary matrix U, The complex-conjugate operator K is absent in (18), M 0(g)K = U †M(g)K U, (21) indicating that the intrinsic projective Reps for anti- s(g) s(g) unitary groups are essentially unitary. 0 then M (g)Ks(g) and M(g)Ks(g) are said to be the same For any group element g3, we have projective Rep; otherwise, they are two different Reps. On the other hand, if two different irreducible projec- ˆ ˆ ˆ iθ2(g3,g2) g¯1g¯2|g3i = g¯1e |g3g2i tive Reps have the same (class of) factor systems, they iθ2(g3,g2) iθ2(g3g2,g1) = e e |g3g2g1i belong to the same class. Obviously, any one-dimensional Rep of a group must be a trivial projective Rep since it and is gauge equivalent to the identity Rep (except for the cases where the coefficient group A is a finite group). iθ2(g3,g2g1) gd2g1|g3i = e |g3g2g1i If M(g)Ks(g) is a projective Rep of group G with factor system ω2(g1, g2), then its complex conjugate Comparing with the 2-cocycle equation (12), we find that ∗ M (g)Ks(g) is also a projective Rep whose factor system ∗ −1 is ω2 (g1, g2) = ω (g1, g2). ˆ ˆ is(g3)θ2(g2,g1) 2 g¯1g¯2|g3i = gd2g1e |g3i, (20) It is known that all the irreducible linear Reps of a finite group G can be obtained by reducing its regular which means that if the state |g3i is anti-unitary (uni- Rep. Similar idea can be applied for projective Reps. In tary), then the coefficients of the operators on its left sectionII, we have constructed the regular projective Rep will take a complex conjugate (remain unchanged). In with a given factor system(or 2-cocycle). In this section matrix form, above relation can be written as we will generalize the eigenfunction method to reduce the h i regular projective Reps into irreducible ones. iθ2(g2,g1) −iθ2(g2,g1) M(¯g1)M(¯g2) = M(g2g1) e Pu + e Pa , As mentioned, we will completely fix the 2-coboundary by selecting one solution in each class of 2-cocyles (for where Pu(Pa) is the projection operator projecting onto anti-unitary groups we will transform the 2-cocyles into the Hilbert space formed by unitary (anti-unitary) bases. the decoupled factor system as defined in AppendixC). Now we show that the regular projective Rep of G com- Suppose that we choose a different 2-coboundary mutes with its intrinsic regular projective Rep. On the [see(8) and (9)], then the corresponding regular projec- one hand, tive Rep is given byg ˆ1 = W (g1)Ks(g1) with

s(g ) g¯ˆ1gˆ2|g3i = g¯ˆ1ω2(g2, g3)K |g2g3i 1 s(g2) Ω1(g1)Ω1 (g2) W (g1)g,g2 = δg,g1g2 ω2(g1, g2) . (22) = ω2(g2g3, g1)ω2(g2, g3)Ks(g2)|g2g3g1i. Ω1(g1g2) 7

Above W (g1)Ks(g1) is equivalent to M(g1)Ks(g1) deﬁned A. Brief review of the reduction of regular linear in (15) since they are related by a unitary transformation Reps for unitary groups U followed by a gauge transformation Ω1(g1), First we introduce Schur’s lemma and its corollary for † W (g1)Ks(g1) = Ω1(g1)[U M(g1)Ks(g1)U], (23) unitary groups without proving them. They are also valid for irreducible projective Reps of unitary groups. where U is a diagonal matrix with entries Ug,g0 = Schur’s lemma: If a nonzero matrix C commutes (ν) δg,g0 Ω1(g). Similarly, the intrinsic regular projective Rep with all the irreducible Rep matrices M (g) of a unitary (19) becomes g¯ˆ1 = W (¯g1) with group G, namely,

† ∗ (ν) (ν) W (¯g1) = U M(¯g1)U [Ω1(g1)Pu + Ω1(g1)Pa] . (24) CM (g) = M (g)C,

Therefore, we can safely select one 2-cocyle in each class for g ∈ G, then C must be a constant matrix C = λI. to construct the regular projective Rep. Corollary: If a unitary operator Cb commutes with a Some tools used in linear Reps, such as class opera- unitary group G with [ˆg, Cb] = 0 for all g ∈ G, then an tors and characters, can be introduced for the projective eigenspace of C (or the surporting space of Jordan blocks (ν) with the same eigenvalue) is a Rep space of G. Reps. For unitary groups, the character χi of an irreducible (projective) Rep is a function of class operators In most cases, the operator C (such as the class oper- (each conjugate class gives rise to a class operator) ators discussed below) is diagonalizable and the Hilbert space can be reduced as direct sum of its eigenspaces. X However, it is possible that C cannot be completely di- C = g−1g g, (25) i i agonalized, in this case C can be transformed into its g∈G Jordan normal form[43] and we can just replace each here we have ignored a normalization constant. On the ‘eigenspace’ by a ‘supporting space of Jordan blocks with the same eigenvalue’. other hand, for a given class operator Ci the character (ν) Reducing the regular Rep is equivalent to identifying χi is a function of irreducible (projective) Reps (ν) since all the bases of the irreducible Reps. In Ref. 42, the au- the unitary transformation (21) will not change the char- (ν) thors use a series of quantum numbers, i.e. the eigenval- acter. The characters χi form complete bases for both ues of the complete set of commuting operators (CSCO), the class space {Ci} and the irreducible (projective) Rep to distinguish the irreducible bases. This idea comes from space {(ν)}. As a result, the number of different irre- quantum mechanics. For example, for spin systems we ducible (projective) Reps is equal to Nc, the number of use two quantum numbers |S, mi to label a state, where independent class operators. Group theory also tells us S is the main quantum number labeling the irreducible that for a finite group G, a nν dimensional irreducible Rep and m is the magnetic quantum number labeling Rep appears nν times in the reduced regular Rep. Con- different bases. Noticing that S(S + 1) and m are eigen- sequently, 2 2 values of the operators (Sˆ , Sˆz) respectively, where Sˆ is the invariant quantity (Casimir operator) of SO(3) and N Xc ˆ n2 = G, Sz is that of its subgroup SO(2). The two commuting ν ˆ2 ˆ ν=1 operators (S , Sz) form the CSCO of the Hilbert space of a spin, so we can use their eigenvalues to label different where G is the order of the group G. This result also bases. holds for the projective Reps of unitary groups. Similarly, for any unitary group G, we can use invariant However, we should carefully use these tools for anti- quantities (that commute with the group G) to provide unitary groups. For example, the ‘character’ of the anti- the main quantum numbers to label different irreducible unitary element g may be changed under unitary trans- Reps. The class operators defined in (25) are ideal can- formation didates since they commute with each other and commute with all the group elements in G. According to the Tr [M(g)K] 6= Tr [U †M(g)KU] = Tr[U †M(g)U ∗] corollary of Schur’s lemma, every irreducible Rep space (ν) is an eigenspace of all the class operators Ci and can for arbitrary unitary matrix U. Similarly, the two unitary be labeled by the eigenvalues (it can be shown that the elements g andg ˜ = T −1gT , which belong to the same eigenvalues are proportional to the corresponding char- (ν) class, may have different characters. Therefore we need acters χi ). It turns out that the Nc independent class to redefine the conjugate classes and the class operators operators can provide exactly Nc sets of different quan- such that the number of different irreducible Reps (of the tum numbers and can completely identify the Nc different same class) is still equal to the number of independent irreducible Reps. The reason is that the Nc linearly in- class operators. Similarly, some other conclusions of uni- dependent class operators form an algebra since they are tary groups should be modified for anti-unitary groups closed under multiplication. The natural Rep of the class (see Sec.III B 2). algebra provides Nc different sets of eigenvalues (corre- 8 sponding to the eigenstates in the class space). These use the complete set (C,C(s), C¯(s)) to lift the degener- Nc different eigenvalues are nothing but the quantum acy. Then a unitary transformation matrix U can be numbers labeling the irreducible Reps. Generally, these formed via the common eigenvectors of (C,C(s), C¯(s)) quantum numbers are degenerate and can not distinguish and the regular linear Rep matrices of all group elements each of the irreducible bases. Therefore, the class opera- can be block diagonalized simultaneously. We only give tors of G form a subset of the commuting operators and the results of two generators: are called the CSCO-I.   Actually, we can linearly combine the Nc independent −1 0 0 0 0 0 P 0 −1 0 0 0 0 class operators to form a single operator C = i kiCi    0 0 1 0 0 0  (where ki are some real constants) as the CSCO-I, as U †M(P )U =   , (27)  0 0 0 −1 0 0  long as the operator C has Nc different eigenvalues.   Secondly, we need some ‘magnetic’ quantum num-  0 0 0 0 1 0  bers to distinguish different bases in one copy of irre- 0 0 0 0 0 1 ducible Rep. We will make use of the chain of sub-  −1 0 0 0 0 0  √ groups G(s) = G1 ⊃ G2 ⊃ ..., the set of their CSCO-I  0 1 3 0 0 0  C(s) = (C(G ),C(G ), ...) commute with the CSCO-I of  √2 2  1 2  0 3 − 1 0 0 0  G and provide the required quantum numbers. The set U †M(Q)U =  2 2 √  . (28)  0 0 0 1 3 0  (C,C(s)) formed by C (the CSCO-I of G) and C(s) is  √2 2   3 1  called the CSCO-II [42].  0 0 0 2 − 2 0  Finally, since the nν dimensional irreducible Rep (ν) 0 0 0 0 0 1 appears nν times in the regular Rep, the quantum numbers of CSCO-II are still degenerate. To lift this degener- From above results, we find two inequivalent one- acy, we need more quantum numbers (i.e. the multiplic- dimensional Reps occur only once, but a two-dimensional ity quantum numbers). Noticing that the intrinsic group Rep occurs twice. The quantum numbers of C¯(s) are G¯ commutes with G, we can use the CSCO-Is of the used to distinguish the two equivalent irreducible Reps. subgroup chain G¯(s) = G¯1 ⊃ G¯2 ⊃ ..., namely, C¯(s) = (C¯(G1), C¯(G2), ...). The operators C¯(s) commute with the CSCO-II of G and can completely lift the degener- B. Reduction of regular projective Reps acy. So we obtain the complete set (C,C(s), C¯(s)), called the CSCO-III. The CSCO-III provides G sets of non- Now we generalize the eigenfunction method to reduce degenerate quantum numbers which completely label all the regular projective Reps. Unitary groups and anti- the irreducible bases in the regular Rep space. unitary groups will be discussed separately. As an example, we apply this method to reduce the regular Rep of the permutation group S3. For S3, there are 6 group elements and the canonical 1. Unitary groups subgroup chain is S3 ⊃ S2. The group elements P = (12),Q = (23),R = (13) belong to the same class. So First of all, we need to define the class operators. For a we may choose class operator C = (12) + (23) + (13), regular projective Rep of any unitary group G, the class the class operator of subgroup S2 C(s) = (12) and the operator C1 corresponding to g1 ∈ G is defined as [41] class operator of intrinsic subgroup S2 C¯(s) = (12) to construct CSCO. In the regular Rep space (group space), X −1 X −1 C1 = gˆ2 gˆ1gˆ2 = M (g2)M(g1)M(g2). (29) the Rep matrices of these class operators can be written g2∈G g2∈G as,respectively The class operator C commutes with all the regular     1 0 1 1 1 0 0 0 1 0 0 0 0 projective Rep matrices since  1 0 0 0 1 1   1 0 0 0 0 0   1 0 0 0 1 1   0 0 0 0 0 1  X −1 C =   ,C(s) =   , C1gˆ3 = gˆ3[ˆg2gˆ3] gˆ1gˆ2gˆ3  1 0 0 0 1 1   0 0 0 0 1 0      g ∈G  0 1 1 1 0 0   0 0 0 1 0 0  2 X −1 0 1 1 1 0 0 0 0 1 0 0 0 =g ˆ3 gd2g3 gˆ1gd2g3  0 1 0 0 0 0  g2∈G =g ˆ C  1 0 0 0 0 0  3 1  0 0 0 0 1 0  C¯(s) =   . (26) 0 −1  0 0 0 0 0 1  for all g3 ∈ G. It can be shown that if g1 = g g1g   then C 0 ∝ Cg , namely, each conjugate class gives at  0 0 1 0 0 0  g1 1 0 0 0 1 0 0 most one linearly independent class operator. Since a nontrivial projective Rep M(g) is not a faithful Rep of The eigenvalues of C are 3, 0, −3, in which 3 and −3 G, the class operators for some conjugate classes may be are non-degenerate but 0 is four-fold degenerate. We may zero. Consequently, the number of linearly independent 9 class operators is generally less than the number of classes However, the class operators C¯(s)[C¯(s) can be ob- of the group G. Since character is still a good quantity, tained from (30) by replacing g withg ¯] of the chain of the number of different irreducible projective Reps (of the subgroups G¯(s) = G¯1 ⊃ G¯2 ⊃ ... are different from same class) is still equal to the number of independent C(s) and can lift the degeneracy of the eigenvalues of class operators. Due to the corollary of Schur’s lemma, CSCO-II. Now we obtain the complete set of class oper- we can use the eigenvalues of the class operators to label ators C,C(s), C¯(s), called CSCO-III. The eigenvalues different Rep spaces. Therefore, the class operators (or of C¯(s) are the same as those of C(s), consequently the P their linear combination C = i kiCi) form CSCO-I of number of times an irreducible projective Rep (ν) occurs the regular projective Rep. P 2 is equal to its dimension nν , and ν nν = G is still valid The eigenvalues of CSCO-I are degenerate. To dis- for a given class of irreducible projective Reps. tinguish the bases of the same irreducible Rep, we can For unitary groups, the common eigenvectors of the op- make use of the class operators of the subgroup chain erators in CSCO-III are the orthonormal bases of the irre- G(s) = G1 ⊃ G2 ⊃ .... For instance, the class operators ducible projective Reps, and each eigenvector has unique of the subgroup G1 are defined as ‘quantum numbers’. These eigenvectors form a unitary X matrix U which block diagonalizes all regular projective C (G ) = M −1(g )M(g )M(g ), g ∈ G , (30) 1 1 2 1 2 1 1 Rep matrices M(g) simultaneously. g2∈G1 Now we summarize the steps to obtain all the irre- P and C(G1) = i kiCi(G1) is the CSCO-I of G1. Repeat- ducible projective Reps with the eigenfunction method: ing the procedure we obtain the set of CSCO-Is for the 1, Solve the 2-cocycle equations, obtain the solutions subgroup chain C(s) = (C(G1),C(G2), ...). The opera- and their classification (see appendixB); tor set (C,C(s)) is called CSCO-II, which can be used to 2, Select one solution of a given class as the factor distinguish all the bases if every irreducible Rep occurs system and obtain the corresponding regular projective only once in the reduced projective Rep. Rep (see sectionIIB); However, a nν -dimensional irreducible projective Rep 3, Construct the CSCO-III with C,C(s), C¯(s), where (ν) may occur more than once and this will cause de- C,C(s), C¯(s) are class operators of G, of the subgroups generacy in the eigenvalues of CSCO-II. To lift this de- G(s) ⊂ G, and of the subgroups G¯(s) ⊂ G¯ respectively. generacy we need more quantum numbers to label the Then reduce the regular projective Reps into irreducible mutiplicity. Similar to the linear Reps, we can use the ones; class operators of the intrinsic group G¯. It can be shown ¯ 4, Change the class and repeat above procedure, until that the class operators of G is identical to those of G: the irreducible projective Reps of all classes are obtained. for any g3 ∈ G, we have " # ¯ X ˆ ˆ ˆ−1 ˆ C1|g3i = g¯2g¯1g¯2 g¯3|Ei 2. Anti-unitary groups g2 X −1 −iθ2(g2,g2 ) d−1 = e (g¯ˆ2g¯ˆ1)g2 g¯ˆ3 |Ei Now denote the time reversal conjugate of g asg ˜, −1 T g2 namely, T gT =g ˜ for g ∈ G. Since G/H = Z2 , any X −1 −1 group element in G either belongs to the unitary nor- −iθ2(g2,g2 ) iθ2(g1,g2) iθ2(g2 ,g1g2) = e e e mal subgroup H or belongs to its coset TH = HT . This g2 means that any anti-unitary group element must be writ- −1 iθ2(g3,g2 g1g2) −1 ˜ ˜ ×e |g3g2 g1g2i, (31) ten in forms of hT or T h with h, h ∈ H. Some conclusions of unitary groups should be modi- −1 d−1 iθ2(g2,g2 ) −1 where we have used g2 = e g¯ˆ2 . On the other fied for anti-unitary groups. We first generalize Schur’s hand, lemma to anti-unitary groups. " # Generalized Schur’s lemma: If a nonzero matrix C X −1 commutes with the irreducible (projective or linear) Rep C1|g3i = gˆ gˆ1gˆ2 gˆ3|Ei 2 (ν) of an anti-unitary group G with Cgˆ =gC ˆ , namely, g2 " # X −1 −1 (ν) (ν) −iθ2(g2,g2 ) CM (g)Ks(g) = M (g)Ks(g)C, =g ˆ3 e gd2 gˆ1gˆ2 |Ei g2 X −iθ (g ,g−1) iθ (g ,g ) iθ (g−1,g g ) for all g ∈ G, then C has at most two eigenvalues which = e 2 2 2 e 2 1 2 e 2 2 1 2 are complex conjugate to each other, including the special g2 case that C is a real constant matrix C = λI with a single −1 iθ2(g3,g2 g1g2) −1 real eigenvalue λ. ×e |g3g2 g1g2i, (32) The proof is simple. No matter C is completely diag- −1 −1 iθ2(g2,g2 ) −1 ¯ where gd2 = e gˆ2 . Therefore, C1 = C1, onalized or not, C has at least one eigenstate |λi with namely, the class operators of G¯ do not provide any new invariant quantities. C|λi = λ|λi. 10

For any unitary element h ∈ H and anti-unitary element Since the discussions for the two types of anti-unitary T h, we have groups are similar, we will mainly focus on type-I anti- unitary groups in the following. Chˆ|λi = hCˆ |λi = λ hˆ|λi , Class operators and CSCO-I. In order to reserve the relation between irreducible representations and ∗ CTc h|λi = Tc hC|λi = Tc h(λ|λi) = λ Tc h|λi . classes, we should redefine the classes and class operators for an anti-unitary group G. The class corresponds to Above relations indicate that unitary operators reserve unitary element gi is defined as the eigenvalue of C, while anti-unitary operators switch the eigenspace of λ to the eigenspace of its complex con- −1 −1 −1 Class(gi) = {h gih, (T h) gi (T h); h ∈ H}, jugate λ∗. If (ν) is irreducible, then C is completely diagonalizable and has at most two eigenvalues λ and while for anti-unitary element giT , the conjugate class is λ∗. Specially, if C is Hermitian, then it must be a real defined as constant matrix C = λI with a single real eigenvalue λ −1 ˜ −1 −1 2 ˜ [26]. The generalized Schur’s lemma yields the following Class(giT ) = {h (giT )h, (T h) (˜giT ) T (T h); h ∈ H}. corollary: Corollary: If a linear operator Cb commutes with an Obviously, {T } forms a class itself. anti-unitary group G with [ˆg, Cb] = 0 for all g ∈ G, Before defining class operators, we define the following namely, operators,

−1 M(C)M(g)Ks(g) = M(g)Ks(g)M(C), X ˆ−1 ˆ −1 Dgi = h gîh + Tc h gî Tc h , then a direct sum of two eigenspaces whose eigenvalues h∈H −1 are mutually complex conjugate is a Rep space of G (If X ˆ−1 −1ˆ Dg−1 = h gî h + Tc h gîTc h , C is not completely diagonalizable, we can transform it i h∈H into Jordan normal form and replace each ‘eigenspace’ −1 by a ‘supporting space of Jordan blocks with the same X ˆ−1 ˜ˆ −1 2 ˜ DgiT = h gdiT h + Tc h g˜diT Tb Tch , eigenvalue’). h∈H Our aim is to use the eigenvalues of a set of linear −1 X −1 −1 2ˆ −1 ˆ ˜ c˜ operators to label the irreducible bases of anti-unitary D(giT ) = h g˜diT Tb h + Tc h gdiT T h . groups. h∈H Decoupled factor systems. For type-I anti-unitary (34) groups, we can adopt the following decoupled factor sys- 2 tem (see AppendixC for details of tuning the cobound- For type-I anti-unitary groups T = E, and ω2(gi,T ) = ary), which satisfy ˆ ω2(T, g˜i) = 1, we haveg îTb = Tbg˜i. Therefore,

ω2(E, g1) = ω2(g1,E) = 1, X ˆ−1 ˆ ˆ−1 −1ˆ H H D = h gˆ h + h g˜ˆ h = C + C −1 , gi i i gi ω2(T,T ) = ±1, g˜i h∈H ω2(T, g1) = ω2(g1,T ) = 1, X ˆ−1 −1ˆ ˆ−1 ˆ H H −1 D −1 = h gˆ h + h g˜ˆ h = C −1 + C , g i i g˜i ω2(T g1, g2) = ω2 (g1, g2), i gi h∈H ω2(g1, T g2) = ω2(g1, g˜2), X ˆ−1 ˆ ˆ−1ˆ−1ˆ DgiT = h gîh + h g˜ h Tb = Dgi T,b ω2(T g1, T g2) = ω2(˜g1, g2)ω2(T,T ), (33) i h∈H for any g1, g2 ∈ H. The first equation is the canonical X −1 −1 −1 −1 ˆ ˆ ˆ ˆ ˆ −1 D(giT ) = h gî h + h g˜ih Tb = Dg T,b gauge condition. When above relations are satisfied, it i can be shown that h∈H (35) ω2(g1, g2)ω2(˜g1, g˜2) = 1 where CH , CH , CH and CH denote the class opera- gi g˜−1 g−1 g˜i is automatically satisfied. From (33), only the cocycles i i tors of the normal subgroup H corresponding to unitary ω2(T,T ) and ω2(g1, g2) with g1, g2 ∈ H are important, −1 −1 elements gi,g ˜i , gi andg ˜i respectively. therefore we only need to focus on Tb and the normal It is easy to prove that subgroup H. Once the representation matrices of T and the unitary elements in H are reduced, we then obtain −1 −1 Tb Dgi Tb = Dg , the irreducible Rep for the full group G. i For type-II anti-unitary groups, above discussion of de- −1 coupled factor system should be slightly modified. In Ap- or Dgi Tb = TDb . Namely, the operators Dgi do not gi pendixC2, we provide the procedure to obtain the decou- commute with all the group elements. We can define the pled factor system in the fermionic anti-unitary groups. following class operators to solve this problem: 11

−1 1) if gi and gi belong to different classes, namely, if are redundant (The two Reps are non-equivalent when −1 gi 6= gi and gi 6=g ˜i, then we define two class operators one considers Z2 coefficient projective Reps). On the −1 2 corresponding to the classes of gi and gi , other hand, if Tb = −1, the bases form Kramers doublets and T cannot reduce to one dimensional Reps. In other H H H H b C + = D + D −1 = C + C −1 + C −1 + C , (36) i gi g gi g˜i i g˜i gi words, the operator Tb has no ‘eigenvalues’ at all. In H H H H this case, the class operator T doesn’t contribute any C − = i(D − D −1 ) = i(C + C −1 − C −1 − C ). b i gi g gi g˜ g g˜i i i i quantum numbers to label different Reps. (37) Since Ci±T = Ci± Tb = TCb i± , the eigenvalues of Ci±T , if exist, are equal to the product of eigenvalues of Ci± Obviously C ± commute withg ˆ for any g ∈ H, C ± gˆ = i j j i j and T . As a result, all the anti-unitary class opera- gˆ C ± . Since b j i tors do not contribute useful quantum numbers. There- TCH = CH T, TCH = CH T, fore, the number of different irreducible projective Reps b gi g˜i b b g˜i gi b is determined by the unitary class operators C = {Ci± }. H H H H TCb −1 = C −1 T,b TCb −1 = C −1 T,b Namely, C = {Ci± } are the CSCO-I of the anti-unitary gi g˜i g˜i gi group G. Ci± commute with the time reversal operator, It can be shown that unitary elements belonging to the same class have the same character in an irreducible H H H H TC + = (C + C −1 + C −1 + C )T = C + T, b i g˜i gi b i b gi g˜i projective Rep. Therefore, the characters of the unitary H H H H group elements as functions of the irreducible Reps, are TC − = −i(C + C −1 − C −1 − C )T = C − T. b i g˜i gi b i b gi g˜i also functions of the unitary class operators. As a consequence, the number of different irreducible projective Therefore, Ci± commute with all the operators in G. −1 Reps is equal to the number of linearly independent uni- 2) if gi and gi belong to the same class, namely, if −1 tary class operators in {Ci± }. This result is similar to the gi = gi or gi =g ˜i, then the class operator corresponding unitary groups. Since all the class operators correspond- to g is C = C + = D (obviously C − = 0 in this case). i i i gi i ing to unitary classes are Hermitian, their eigenvalues are It can be easily checked that above class operators Ci± all real numbers, each set of eigenvalues corresponds to are Hermitian if the Rep is unitary. an irreducible Rep space. Owing to relation (35), the class operators for anti- CSCO-II and CSCO-III. The class operators C(s) unitary elements g T and (g T )−1 are i i of the subgroups G(s) ⊂ H are defined as usual unitary groups. Then we obtain the CSCO-II (C,C(s)), where C + = Dg T + D −1 i T i (giT ) the quantum numbers of C(s) can be used to distinguish H H H H = (C + C + C −1 + C −1 )T = C + T, (38) gi g˜i b i b different bases of one copy of irreducible Rep. gi g˜i Since an irreducible Rep may occur more than once, C − = i(Dg T − D −1 ) i T i (giT ) we need to make use of the class operators of the intrinsic H H H H = i[(C − C ) − (C −1 − C −1 )]T = C − T. gi g˜i g g˜ b i b group to label the multiplicity. i i −1 (39) For the unitary elementsg ¯i andg ¯i , we define the following operators This gives a one-to-one correspondence between the anti- X ¯ˆ−1 ¯ˆ ¯ˆ−1ˆ−1¯ˆ H H unitary class operators Ci±T and the unitary class oper- D¯ = h g¯ˆ h + h g˜¯ h = C¯ + C¯ −1 , gi i i gi g˜i ators Ci± , where the unitary class operators can be ob- h¯∈H tained solely from the subgroup H by the equations (36) X ¯ˆ−1 −1¯ˆ ¯ˆ−1ˆ ¯ˆ H H D¯ −1 = h g¯ˆ h + h g˜¯ h = C¯ −1 + C¯ and (37). However, it can be shown that the anti-unitary g i i g˜i i gi class operators do not provide any meaningful quantum h¯∈H numbers for U(1) coefficient projective Reps. (40) To see why the anti-unitary class operators do not correspond to more irreducible Reps, we focus on the class For the anti-unitary elements T gi the corresponding operator Tb first. Noticing that M(T ) is a real matrix, so conjugate class is defined as 2 2 2 Tb = [M(T )] = ω2(T,T ) = ±1. If Tb = 1, then M(T ) + − ¯−1 ˜¯ −1 −1 2˜ has eigenvalues ±1 with eigenstates |φ i and |φ i sat- Class(T gi) = {h T gih, hT T g˜i T hT ; h ∈ H}, isfying Tb|φ±i = ±|φ±i respectively. Under the unitary transformation |φ+i0 = i|φ+i, we have

+ 0 + + 0 −1 −1 2 Tb|φ i = −i|φ i = −|φ i , ¯ X ¯ˆ−1 ˜¯ˆ ˜c DT gi = h Td gih + hTc Tdg˜i Tb hT 0 which gives M + (T ) = −1 = M −(T ). So the two one- h¯∈H dimensional Reps (+) and (−) are equivalent, and the X ¯ˆ−1ˆ ¯ˆ ¯ˆ−1¯ˆ−1¯ˆ ¯ = h g¯ih + h g˜i h Tb = Dgi T,b quantum numbers corresponding to the class operator Tb h¯∈H 12

−1 2 −1 T X ˆ−1 ¯ˆ Z ), so generally M(T ) 6= M(T ). This implies C¯ + 6= ¯ −1 ¯ ˜ ˜c 2 i T D(T gi) = h Tdg˜i Tb h + hTc Td gihT C + , namely, the anti-unitary class operator is generally ¯ i T h∈H different from its intrinsic partner. This is a difference X ¯ˆ−1 −1¯ˆ ¯ˆ−1ˆ ¯ˆ = h g¯ˆ h + h g˜¯ih Tb = D¯ −1 T.b between unitary and anti-unitary groups. i gi h¯∈H Since G is anti-unitary while all the operators in (41) CSCO-II are obtained from the unitary normal subgroup H (and its subgroups), we need to use at least one anti- −1 0 The class operators corresponding tog ¯i, g¯i ∈ H , T gi unitary class operator of G¯ (or its subgroup chain G¯(s )) −1 and T gi are defined following (36) and (37) as to construct CSCO-III. In most cases, we can adopt the class operator Tb as a member of C¯(s0) in CSCO-III (an ¯ + ¯ ¯ −1 Ci = Dgi + D gi exception is given in AppendixF2a). If an intrinsic sub- H H H H ¯ ¯ ¯ ¯ T = C¯ + C¯ −1 + C¯ −1 + C¯ , (42) group G ∈ G(s) is anti-unitary, and G /H = Z where gi g˜ g g˜i 1 1 1 2 i i ¯ ¯ ¯ H1 is the unitary normal subgroup with H1 ⊂ H, then ¯ − ¯ ¯ −1 Ci = Dgi − D S the corresponding class operators are given by gi H H H H = (C¯ + C¯ −1 − C¯ −1 − C¯ )S, (43) gi g˜ g g˜i H H H i i C¯H1 + C¯ 1 + C¯ 1 + C¯ 1 , gi −1 −1 g˜i g˜i gi where Eq. (20) has been used, and S = i(P − P ) = u a (C¯H1 + C¯H1 − C¯H1 − C¯H1 )S, (46) gi −1 −1 g˜i iI 0 g˜i gi (here I stands for a H-dimensional identity 0 −iI matrix, H is the order of normal subgroup H) is a di- where C¯H1 (or C¯H1 ) is the class operator of H¯ corre- gi g˜i 1 agonal matrix with entries i for unitary bases or −i for sponding to the elementg ¯ (or g˜¯ ). On the other hand, anti-unitary bases, which satisfies S∗ = −S, S2 = −1 and i i if an intrinsic subgroup G¯2 ∈ G¯(s) is unitary, then its ∗ (intrinsic) class operators are defined in the same way as M(gi)S = SM(gi),M(giT )S = SM(giT ), ∗ usual unitary groups, the only constraint is that all these M(¯gi)S = SM(¯gi),M(giT )S = SM(giT ). class operators should commute with all the class oper- Above relations can be easily verified since M(g ) and ators in CSCO-II and should be mutually commuting. i ¯ 0 A 0 After carefully choosing the operators of C(s ) such that M(¯g ) are block diagonalized with the form , i 0 B the degeneracy of the quantum numbers are completely ¯ 0 where the blocks A and B are H × H nonzero matri- lifted, we obtain the CSCO-III C,C(s), C(s ) , where C = {Ci± }. ces, while M(giT ) and M(giT ) are block-off-diagonalized 0 A Before going to examples, we summarize some special with the form . B 0 properties of anti-unitary groups which are different from Furthermore, it can be checked (see AppendixE) that unitary groups: 1, The anti-unitary class operators do not contribute ¯ ¯ Ci+ = Ci+ , Ci− = Ci− , any meaningful quantum numbers to CSCO-I. The sim- ¯ ¯ ¯ ¯ plest example is ZT , which has only one 1-dimensional Ci+T = Ci+ T,b Ci−T = Ci− T,b (44) 2 linear Rep 1 and one 2-dimensional irreducible projective

¯ + ¯ ¯ −1 ¯ − ¯ where Ci T = DT gi + D(T gi) and Ci T = S(DT gi − Rep. ∗ ¯ −1 ¯ ¯ −1 D(T gi) ) = (DT gi − D(T gi) )S . 2, An irreducible Rep (after lifting the multiplicity) Again, Tb forms a class operator itself. And for any may be either labeled by a real quantum number, or la- g ∈ G we have beled by a pair of complex conjugating quantum numbers. Since we redeﬁned the conjugate classes for anti- Tbg¯ˆ = g˜¯ˆT,b Tbgˆ =g ˆT,b unitary groups and all the class operators in CSCO-I are Hermitian, the main quantum numbers are real. But or equivalently, the multiplicity quantum numbers are generally complex numbers, so an irreducible Rep is generally labeled by a M(T )M(¯g) = M(g˜¯)M(T ), pair of complex conjugating quantum numbers. M(T )M(g)Ks(g) = M(g)Ks(g)M(T )

= M(g)M(T )Ks(g), (45)

1 T When acting on a Hilbert space, the 1-dimensional Reps of Z2 where M(T ) is a real matrix. Tb doesn’t contain complex- 1 T is classified by H (Z2 ,U(1)) = Z1, so there is only one 1- conjugate operator K and commutes with all the Rep dimensional Rep. However, when acting on Hermitian opera- 1 T matrices M(g) of G. tors, then the 1-dimensional Reps is classified by H (Z2 ,Z2) = Z2, namely, there are TWO different Reps characterized by Noticing that Tb|gi = |T gi, Tb|gi = |gT i = |T g˜i, and T OTˆ −1 = ±Oˆ, where Oˆ is an Hermitian operator. This result g˜ is generally different from g (except that G = H × can be generalized to any anti-unitary groups. 13

C. Some examples ω2(PT,PT )). The coboundary variables (12, 15) are set to be 1. T In this section, we list the nontrivial irreducible projec- Z2 × Z2 × Z2 = {E,P } × {E,Q} × {E,T } with tive Reps of a few finite groups. The group elements, the three generators P, Q, T . The classification is labeled by generators, the classification labels and the coboundary ω2(PT,PT ), ω2(P T, QT ), ω2(QT, QT ), ω2(P QT, P QT ). variables are given below. The results of the projective The coboundary variables (48, 54, 56, 61 ∼ 63) are set to Reps are shown in TableI. For simplicity, we only list be 1. T the irreducible projective Rep with the lowest dimension (Z2 × Z2) o Z2 = ({E,P } × {E,Q}) o {E,T } with in each nontrivial class, and only the Rep matrices of the PQ = QP, T P = PT and TQ = P QT . This group generators are given. can be generated by two generators Q, T since (TQ)2 = 2 Unitary groups: (QT ) = P . The classification is labeled by ω2(PT,PT ), 2 2 ω (QT, Q). The coboundary variables (48, 60 ∼ 64) are Z2 × Z2 = {E,P } × {E,Q} with P = Q = E, QP = 2 PQ. There are two generators P,Q. The classifica- set to be 1. T 2 3 tion is labeled by ω2(Q, P Q). The coboundary variables Z4 × Z2 = {E,P,P ,P } × {E,T }, there are two (11, 15, 16) are set to be 1. generators P and T . The classification is labeled by 2 2 3 2 Z2 × Z2 × Z2 = {E,P } × {E,Q} × {E,R} with ω2(P T,P T ), ω2(P T,P ). The coboundary variables three generators P, Q, R. The classification is labeled (56, 60 ∼ 64) are set to be 1. T 2 3 m by ω2(P R, QR), ω2(P R, P QR), ω2(QR, P QR).The Z4 o Z2 = {E,P,P ,P } o {E,T } with P T = coboundary variables (46, 54, 55, 61 ∼ 64) are set to be 1. TP 4−m, there are two generators P and T . The classifi- 2 2 2 2 3 3 Z3 ×Z3 = {E,P,P }×{E, Q, Q } with two generators cation is labeled by ω2(P T,P T ), ω2(P T,P T ). The 2 2 2 P,Q. The classification is labeled by ω2(PQ ,P Q ). coboundary variables (53, 54, 56, 61 ∼ 63) are set to be 1. T T 2 The coboundary variables (70, 71, 76 ∼ 81) are set to be Z3 × (Z3 o Z2 ) ' (Z3 × Z3) o Z2 = {E,P,P } × 1. ({E, Q, Q2} o {E,T }) with TP = PT,TQ = Q2T,PQ = 2 2 2 Z3 × Z3 × Z3 = {E,P,P } × {E, Q, Q } × {E,R,R } QP , there are three generators P, Q, T . The classifica- 2 2 with three generators P, Q, R. The classification is tion is labeled by ω2(P QT, P Q T ). The coboundary 2 2 2 2 2 2 2 2 2 labeled by ω2(P QR ,PQ R ), ω2(P QR ,P Q R ), variables (264, 270, 309, 311, 312, 314 ∼ 324) are set to be 2 2 2 2 2 ω2(PQ R ,P Q R ). The coboundary variables 1. T T (642, 645, 695, 696, 698, 699 ∼ 701, 712 ∼ 729) are set to S4 = A4 o Z2 , there are three generators P = be 1. (123),Q = (124) and T = (12)K with P 3 = Q3 = T 2 = 2 3 2 2 2 2 Z4 × Z8 = {E,P,P ,P } × E and TP = P T,TQ = Q T,PQ = Q P . The classi- 2 2 {E, Q, Q2,Q3,Q4,Q5,Q6,Q7}, with two generators fication is labeled by ω2(P QT, QP T ), ω2(P QT, P Q). 2 7 3 7 P,Q. The classification is labeled by ω2(P Q ,P Q ). The coboundary variables (548 ∼ 552, 557, 559 ∼ The coboundary variables (989, 990, 991, 997 ∼ 1024) are 561, 564 ∼ 576) are set to be 1. T set to be 1. A4 × Z2 , there are three generators P = (123),Q = 3 3 2 A4 = {E, (123), (132), (124), (142), (134), (143), (234), (124) and T with P = Q = T = E and TP = (243), (12)(34), (13)(24), (14)(23)}, the normal subgroup PT,TQ = QT, P Q = Q2P 2. The classification is labeled 2 2 2 of S4 = {E, (123), (132), (124), (142), ..., (12), (13), by ω2(P ,P ), ω2(P ,Q T ). The coboundary variables (23), (14), (24), ...} formed by even-parity permutations (34, 35, 37, 45, 84, 93, 148, 152, 260, 286, 296, 334, 351, 373, with S4/A4 ' Z2 = {E, (12)}. The group A4 390, 399, 408, 427, 459, 477, 537, 549) are set to be 1. has generators P = (123),Q = (124) with P 3 = Fermionic Anti-unitary groups: 3 2 2 T 4 2 Q = E,PQ = Q P . The classification is la- Z4 = {E,Pf ,T,Pf T } with T = E and T = Pf . 2 2 2 2 beled by ω2(P Q ,P Q ). The coboundary vari- There is only one generator T . The classification is la- ables (104, 105, 107, 108, 128, 129, 136, 138, 140, 141, 143) beled by ω2(Pf T,Pf ). The coboundary variables (15, 16) are set to be 1. are set to be 1. The projective Reps can also be charac- 2 3 m 4 Z4 o Z2 = {E,P,P ,P } o {E,Q} with P Q = terized by the invariant [M(T )K] = ±1 [see eqs. (C15) QP 4−m, there are two generators P and Q. The classi- and (C16)]. 2 T fication is labeled by ω2(P Q, Q). The coboundary vari- Z2 × Z4 = {E,P } × {E,Pf ,T,Pf T } with two gen- ables (54 ∼ 56, 61 ∼ 64) are set to be 1. erators P and T . The classification is labeled by Z2 × Z2 × Z2 × Z2 = {E,P } × {E,Q} × ω2(Pf T,Pf T ), ω2(PPf T,Pf ). The coboundary variables {E,R} × {E,S} with four generators (56, 60 ∼ 64) are set to be 1. T 2 3 3 P, Q, R, S. The classification is labeled by Z2nZ4 = {E,P }n{E,T,T ,T } with TP = PT is a 2 ω2(P QS, P RS), ω2(P QS, QRS), ω2(P QS, P QRS), type-I anti-unitary group since (TP ) = E. It is easy to T T T ω2(P RS, QRS), ω2(P RS, P QRS), ω2(QRS, P QRS). verify that Z2nZ4 is isomorphic to D2d ' (Z2×Z2)oZ2 . + The coboundary variables (205, 221, 222, 234, 237 ∼ G−(Z4,T ) with two generators P and T where Z4 = 2 3 2 2 239, 249 ∼ 256) are set to be 1. {E,P,P ,P } and P = T = Pf , PT = TP . Since 2 4 + T Anti-unitary groups: (TP ) = E and P = E, G−(Z4,T ) ' Z4 × Z2 , they T Z2 × Z2 = {E,P } × {E,T } with two generators have the same representations. − P and T . The classification is labeled by (ω2(T,T ), G−(Z4,T ) with two generators P and T where Z4 = 14

2 3 2 2 m 4−m {E,P,P ,P } and P = T = Pf , P T = TP . The IV. APPLICATIONS OF PROJECTIVE REPS classification is labeled by ω2(Pf T,Pf T ). The coboundary variables (56, 60 ∼ 64) are set to be 1. In this section, we will summarize some physical appli- Remarks: cations of irreducible projective Reps. 1) Some of the above groups are isomorphic to point groups. For example,Z2 × Z2 ' D2; Z2 × Z2 × Z2 ' D2h; Z4 o Z2 ' C4v (or D2d); A4 ' T (the symmetry A. 1-D SPT phases group of tetrahedron); S4 ' Td (or O). For anti- unitary groups, we interpret the operations containing The well studied spin-1 Haldane phase[44–46] was mirror reflection as anti-unitary elements, for example, known for its disordered gapped ground state and its we regard the horizontal mirror reflection in the group spin-1/2 edge states at the open boundaries[47, 48]. T T Z2 × Z2 ' C2h or the vertical mirror reflection in the The exotic properties of the Haldane phase is protected group (Z × Z ) ZT ' T , as the generator T of ZT . 2 2 o 2 D2d 2 by Z2 × Z2 spin rotation symmetry or time reversal It should be noted that although C4v ' D2d, the anti- symmetry[15, 21]. The edge states vary projectively un- T T unitary groups C4v and D2d are NOT isomorphic since der the action of the symmetry group. The Haldane their unitary normal subgroups are different. phase was later generalized to other 1D SPT phases 2) When solving the cocycle equations (12) to obtain with different symmetries classified by the second group the factor systems, we have set some coboundary vari- cohomology[22, 23]. In Ref.17, topological nonlinear ables to be 1. In order to label these variables, we first sigma model (NLSM) was generalized to finite symmetry label the G group elements as 1, 2, ..., G. For direct prod- group G in discrete space-time to describe SPT phases uct (or semi-direct product) groups G1 × G2, the group in any spatial dimensions. In the following, based on elements are sorted by the coset of the first group G1, for irreducible projective Reps we will give a minimal field example, for the Z2 ×Z2 group the elements are sorted by theory description of 1D SPT phases. {E,P } × {E,Q} = {{E,P }, {E,P }Q} = {E, P, Q, P Q}. Traditionally, the spin-1 Haldane phase is described Then we sort the G×G variables of the 2-cocycle with the by the O(3) NLSM with the following topological θ-term order ω2(1, 1), ω2(1, 2), ..., ω2(1,G), ω2(2, 1), ..., ω2(G, G) [44, 45], and further label them by numbers 1, 2, ..., G2. The values of the classification labels and the coboundary vari- i L [n(xx, τ)] = θ n · (∂ n × ∂ n), ables completely fix the factor system (see appendixB). θ 4π x τ 3) For anti-unitary groups, we adopt the decoupled ˆ factor system (33) by multiplying a coboundary Ω1(g) where θ = 2π and |ni is the spin coherent state hn|S|ni = (see AppendixC). After the reduction, we divide the ir- n which varies under rotation in the following way, reducible Rep matrices M(g) by the coboundary Ω (g) 1 ˆ iϕ to go back to the original factor system. R|ni = e |Rnni, 4) In TableIII in the appendix, we list all the irre- the phase factor eiϕ depends on the axis of the rotation ducible linear Reps of several anti-unitary groups. We R and the gauge choice of the bases |ni. The collection also give the number of independent unitary class oper- of the end points of the vector n form a sphere, i.e. the ators and the multiplicity of each irreducible Rep. From symmetric space of the SO(3) group S2 = SO(3)/SO(2). the table we can see that different from unitary groups, The Lagrange density iθ n · (∂ n × ∂ n)dxdτ describes the number of times an irreducible Rep occurs in the 4π x τ the ‘Berry phase’ of a spin-1/2 particle evolving among regular Rep is not always equal to its dimension. three states at (x, τ), (x + dx, τ) and (x, τ + dτ). If the space-time is closed, then the action amplitude R e− dxdτLθ [n(xx,τ)] equals 1. If spacial boundary condition D. CG coefficients for projective Reps is open, then the Berry phase on the boundary explains the existence of spin-1/2 edge state[48]. The direct product of two projective Reps (ν ) and The spin-1/2 edge state varries as M[Rm(θ)] = 1 −imm· σ θ (ν ) of group G is usually a reducible (projective) Rep. e 2 under SO(3) spin rotation of angle θ along di- 2 2 Once the direct product Rep is reduced into irreducible rection m. Since M[Rm(π)] = M[Rm(2π)] = −1 and (projective) Reps, the minus sign can not be gauged away, the edge state carries a nontrivial projective Rep of the symmetry group M SO(3). †M (ν1)(g) ⊗ M (ν2)(g)K = M (ν3)(g)K C s(g)C s(g) Above picture can be generalized even if the symmetry ν3 is a finite group G. Similar to the spin coherent state, we introduce the following group element labeled bases we can obtain the corresponding Clebsch-Gordan (CG) coefficients . The eigenfunction method can also be C |gri =g ˆr|α(ν)i, applied to calculate the CG coefficients. Here we will 1 l l (ν∗) not give details of the calculations. |g i =g ˆ |β1 i, 15

TABLE I: Irreducible projective Reps of some simple ﬁnite groups, we only give the lowest dimensional Rep, and only list the i 2π i 2π 1 i π representation matrices of the generators. The symbols σx,y,z denote the Pauli matrices, and ω = e 3 , Ω = e 9 , ω 2 = e 3 . Group Rep of generators label of classiﬁcation Z2 × Z2 ' D2 PQ −iσy σz −1 Z2 × Z2 × Z2 ' D2h PQR −iσz σx iσz (+1, +1, −1) −σz iσy −iσy (+1, −1, +1) −iσz −iσz σx (+1, −1, −1) −σz σx −iσy (−1, +1, +1) iσz I −σx (−1, +1, −1) I iσz σx (−1, −1, +1) −iσz −iσx iI (−1, −1, −1) Z3 × Z3 PQ  0 0 1   Ω 0 0  2 7 Ω  1 0 0   0 Ω 0  ω 0 1 0 0 0 Ω4  0 0 1   Ω8 0 0  7 2 2 Ω  1 0 0   0 Ω 0  ω 0 1 0 0 0 Ω5 Z3 × Z3 × Z3 PQR  Ω5 0 0   0 0 ω2   0 0 Ω  2 2 4  0 Ω 0   ω 0 0   Ω 0 0  (1, 1, ω) 0 0 Ω8 0 ω2 0 0 Ω7 0  Ω4 0 0   0 0 ω   0 0 Ω8  5 2  0 Ω7 0   ω 0 0   Ω 0 0  (1, 1, ω ) 0 0 Ω 0 ω 0 0 Ω2 0  1 0 0   0 0 ω2   0 ω 0   0 ω 0   ω 0 0   0 0 ω  (1, ω, 1) 0 0 ω2 0 ω2 0 ω2 0 0  Ω8 0 0   Ω2 0 0   0 Ω2 0  5 8 8  0 Ω 0   0 Ω 0   0 0 Ω  (1, ω, ω) 0 0 Ω2 0 0 Ω5 Ω5 0 0  Ω4 0 0   0 ω 0   ω 0 0  2 2  0 Ω7 0   0 0 ω   0 ω 0  (1, ω, ω ) 0 0 Ω ω2 0 0 0 0 ω  1 0 0   0 0 ω   0 ω2 0  2 2 2  0 ω 0   ω 0 0   0 0 ω2  (1, ω , 1) 0 0 ω 0 ω 0 ω 0 0  Ω5 0 0   0 ω2 0   ω2 0 0  2 2 2  0 Ω 0   0 0 ω   0 ω 0  (1, ω , ω) 0 0 Ω8 ω 0 0 0 0 ω2  Ω 0 0   Ω7 0 0   0 Ω7 0  4 2 2  0 Ω 0   0 Ω 0   0 0 Ω  (1, ω , ω ) 0 0 Ω7 0 0 Ω4 Ω4 0 0  ω 0 0   0 0 ω   0 ω2 0   0 1 0   ω 0 0   0 0 1  (ω, 1, 1) 0 0 ω2 0 ω 0 ω 0 0  Ω2 0 0   0 ω 0   Ω7 0 0  8  0 Ω 0   0 0 ω   0 Ω 0  (ω, 1, ω) 0 0 Ω5 ω 0 0 0 0 Ω4  Ω4 0 0   ω 0 0   0 0 ω2  2  0 Ω 0   0 ω 0   1 0 0  (ω, 1, ω ) 0 0 Ω7 0 0 ω 0 1 0  1 0 0   Ω5 0 0   0 Ω5 0  2 2  0 1 0   0 Ω 0   0 0 Ω  (ω, ω, 1) 0 0 1 0 0 Ω8 Ω5 0 0  Ω5 0 0   0 0 Ω4   0 0 Ω4  2  0 Ω 0   Ω 0 0   Ω 0 0  (ω, ω, ω) 0 0 Ω8 0 Ω 0 0 Ω 0 16

Z3 × Z3 × Z3 PQR label of classiﬁcation (continue)  Ω7 0 0   0 0 1   0 0 Ω4  2  0 Ω 0   ω 0 0   Ω 0 0  (ω, ω, ω ) 0 0 Ω4 0 ω 0 0 Ω4 0  ω 0 0   0 0 ω   0 1 0  2 2 2  0 ω 0   ω 0 0   0 0 ω  (ω, ω , 1) 0 0 1 0 ω2 0 1 0 0  Ω8 0 0   Ω7 0 0   0 ω2 0  5 2 2  0 Ω 0   0 Ω 0   0 0 ω  (ω, ω , ω) 0 0 Ω2 0 0 Ω4 ω2 0 0  Ω 0 0   0 Ω 0   Ω4 0 0  4 4 2 2  0 Ω 0   0 0 Ω   0 Ω7 0  (ω, ω , ω ) 0 0 Ω7 Ω7 0 0 0 0 Ω  ω2 0 0   0 0 ω2   0 ω 0  2 2  0 1 0   ω 0 0   0 0 1  (ω , 1, 1) 0 0 ω 0 ω2 0 ω2 0 0  Ω5 0 0   ω2 0 0   0 0 ω  8 2 2  0 Ω 0   0 ω 0   1 0 0  (ω , 1, ω) 0 0 Ω2 0 0 ω2 0 1 0  Ω7 0 0   0 ω2 0   Ω2 0 0  2 8 2 2  0 Ω 0   0 0 ω   0 Ω 0  (ω , 1, ω ) 0 0 Ω4 ω2 0 0 0 0 Ω5  ω2 0 0   0 0 ω2   0 1 0  2  0 ω 0   ω2 0 0   0 0 ω  (ω , ω, 1) 0 0 1 0 ω 0 1 0 0  Ω8 0 0   0 Ω8 0   Ω5 0 0  5 5 2 2  0 Ω 0   0 0 Ω   0 Ω 0  (ω , ω, ω) 0 0 Ω2 Ω2 0 0 0 0 Ω8  Ω 0 0   Ω2 0 0   0 ω 0  4 8 2 2  0 Ω 0   0 Ω 0   0 0 ω  (ω , ω, ω ) 0 0 Ω7 0 0 Ω5 ω 0 0  1 0 0   Ω4 0 0   0 Ω4 0  7 2 2  0 1 0   0 Ω7 0   0 0 Ω  (ω , ω , 1) 0 0 1 0 0 Ω Ω4 0 0  Ω2 0 0   0 0 1   0 0 Ω5  8 2 8 2 2  0 Ω 0   ω 0 0   Ω 0 0  (ω , ω , ω) 0 0 Ω5 0 ω2 0 0 Ω5 0  Ω4 0 0   0 0 Ω5   0 0 Ω5  8 8 2 2 2  0 Ω7 0   Ω 0 0   Ω 0 0  (ω , ω , ω ) 0 0 Ω 0 Ω8 0 0 Ω8 0 Z4 o Z2 ' D2d ' C4v PQ i 3π ! e 4 0 i 5π σx −1 0 e 4

Z4 × Z8 PQ  i π  0 e 4 0 0 −i 5π ! −i 7π e 8 0  0 0 e 8 0  σ ⊗   +i z i 7π  i 3π  0 e 8  0 0 0 e 8  −i π e 4 0 0 0 i 5π i 3π e 4 σz e 4 σx −1  −i π  0 e 4 0 0 i 5π ! i 7π e 8 0  0 0 e 8 0  σ ⊗   −i z −i 7π  −i 3π  0 e 8  0 0 0 e 8  i π e 4 0 0 0 A4 ' T P = (123) Q = (124)√ √ −i π ω 0 3 e√ 6 2i − 2 i π −1 0 ω 3 2i e 6 17

Z2 × Z2 × Z2 × Z2 PQRS label of classiﬁcation iσz σy σy σx (+1, +1, +1, +1, +1, −1) −σz iσy σz σx (+1, +1, +1, +1, −1, +1) −iσz iσz −σx I (+1, +1, +1, +1, −1, −1) −σz σx −I iσy (+1, +1, +1, −1, +1, +1) iσz I σx −iσz (+1, +1, +1, −1, +1, −1) I −iσz σy iσz (+1, +1, +1, −1, −1, +1) −iσz iσy σx iσx (+1, +1, +1, −1, −1, −1) −σz −σz iσy σx (+1, +1, −1, +1, +1, +1) iσz −σx −iσz I (+1, +1, −1, +1, +1, −1) −σz −iσy iσy −I (+1, +1, −1, +1, −1, +1) iσz iσz iσz σx (+1, +1, −1, +1, −1, −1) −σz ⊗ I σx ⊗ σz −iσy ⊗ I σz ⊗ iσy (+1, +1, −1, −1, +1, +1) −iσz ⊗ II ⊗ σz −iσz ⊗ σx σy ⊗ iσy (+1, +1, −1, −1, +1, −1) σz ⊗ I σz ⊗ iσz −iσy ⊗ σx −iσx ⊗ σz (+1, +1, −1, −1, −1, +1) −iσz ⊗ I −iσy ⊗ II ⊗ iσz I ⊗ iσy (+1, +1, −1, −1, −1, −1) −σz I −σx iσy (+1, −1, +1, +1, +1, +1) iσz −σx I iσz (+1, −1, +1, +1, +1, −1) −σz ⊗ I −iσy ⊗ I −σx ⊗ σz −σz ⊗ iσy (+1, −1, +1, +1, −1, +1) iσz ⊗ σz iσz ⊗ I σx ⊗ σy iI ⊗ σx (+1, −1, +1, +1, −1, −1) −σz σx σx −σz (+1, −1, +1, −1, +1, +1) iσz −I −I σy (+1, −1, +1, −1, +1, −1) −σz ⊗ σz iσz ⊗ I σx ⊗ II ⊗ σx (+1, −1, +1, −1, −1, +1) −iσz ⊗ σz −iσx ⊗ I σz ⊗ I σy ⊗ σx (+1, −1, +1, −1, −1, −1) I −σz iσy iσy (+1, −1, −1, +1, +1, +1) −iσz −σy iσx −iσy (+1, −1, −1, +1, +1, −1) I ⊗ σz iI ⊗ σx iσz ⊗ I −iσy ⊗ σx (+1, −1, −1, +1, −1, +1) −iσz ⊗ I iI ⊗ σz −iσx ⊗ σx −I ⊗ iσy (+1, −1, −1, +1, −1, −1) I ⊗ σz −σx ⊗ σx −iσz ⊗ I −I ⊗ σx (+1, −1, −1, −1, +1, +1) −iσz ⊗ I σz ⊗ σz −iσx ⊗ I −σz ⊗ σy (+1, −1, −1, −1, +1, −1) −I iσz iσz σx (+1, −1, −1, −1, −1, +1) iσz iσx iσx σx (+1, −1, −1, −1, −1, −1) −I σz σx iσy (−1, +1, +1, +1, +1, +1) iI ⊗ σz σy ⊗ σy σx ⊗ σy −iσz ⊗ σx (−1, +1, +1, +1, +1, −1) −σz −iσy I iσy (−1, +1, +1, +1, −1, +1) −iσz ⊗ I iσz ⊗ σz σx ⊗ σx I ⊗ iσy (−1, +1, +1, +1, −1, −1) −σz −σx σz σx (−1, +1, +1, −1, +1, +1) −iσz ⊗ σz −σz ⊗ I −σy ⊗ I σy ⊗ σx (−1, +1, +1, −1, +1, −1) I iσz I σx (−1, +1, +1, −1, −1, +1) −iσz ⊗ I −iσy ⊗ σx σx ⊗ σy −σx ⊗ σz (−1, +1, +1, −1, −1, −1) −σz I iσy iσy (−1, +1, −1, +1, +1, +1) −iσz ⊗ I −σy ⊗ σy −iσz ⊗ σz iI ⊗ σx (−1, +1, −1, +1, +1, −1) −σz −iσy iσx −iσz (−1, +1, −1, +1, −1, +1) iσz ⊗ I −iσz ⊗ σz −iσz ⊗ σx −iσy ⊗ σy (−1, +1, −1, +1, −1, −1) −I ⊗ σz −σz ⊗ σy −I ⊗ iσy σy ⊗ I (−1, +1, −1, −1, +1, +1) iσz I −iσz σy (−1, +1, −1, −1, +1, −1) −σz ⊗ I iI ⊗ σz −iσy ⊗ σx σy ⊗ σz (−1, +1, −1, −1, −1, +1) iσz −iσy iσz σz (−1, +1, −1, −1, −1, −1) −σz −σz −σx σx (−1, −1, +1, +1, +1, +1) iσz ⊗ I σx ⊗ σx I ⊗ σz −I ⊗ σx (−1, −1, +1, +1, +1, −1) −σz ⊗ σz I ⊗ iσy σz ⊗ σy σx ⊗ σy (−1, −1, +1, +1, −1, +1) iσz −iσz I −σy (−1, −1, +1, +1, −1, −1) σz −σy −σx iI (−1, −1, +1, −1, +1, +1) iσz ⊗ σz −σz ⊗ I −σx ⊗ σy iσx ⊗ σz (−1, −1, +1, −1, +1, −1) −σz ⊗ σz −iI ⊗ σz −σx ⊗ σz −iI ⊗ σx (−1, −1, +1, −1, −1, +1) −iσz iσy −I −iI (−1, −1, +1, −1, −1, −1) I I iσz σx (−1, −1, −1, +1, +1, +1) −iI ⊗ σz −σy ⊗ σx σx ⊗ iσy −I ⊗ σy (−1, −1, −1, +1, +1, −1) −σz ⊗ σz iσy ⊗ I iσz ⊗ I σz ⊗ σy (−1, −1, −1, +1, −1, +1) −iσz −iσz iσx σz (−1, −1, −1, +1, −1, −1) I ⊗ σz −σx ⊗ σx −iσz ⊗ σz I ⊗ iσy (−1, −1, −1, −1, +1, +1) iσz I iσy −iI (−1, −1, −1, −1, +1, −1) I −iσz iσy iI (−1, −1, −1, −1, −1, +1) −iσz ⊗ σz I ⊗ iσy −iI ⊗ σx iσx ⊗ σz (−1, −1, −1, −1, −1, −1) 18

Anti-unitary group Rep of generators label of classiﬁcation T Z2 T iσyK −1 T T Z2 × Z2 ' C2h TP σxK iσz (+1, −1) σyK iσz (−1, +1) σyKI (−1, −1) T T Z2 × Z2 × Z2 ' D2h TPQ σyK iσz −iσz (+1, +1, +1, −1) σyK iσz I (+1, +1, −1, +1) σxK I iσz (+1, +1, −1, −1) σyK σz σx (+1, −1, +1, +1) IK iσz iσx (+1, −1, +1, −1) IK iσz σy (+1, −1, −1, +1) σy ⊗ σxKI ⊗ σz iI ⊗ σx (+1, −1, −1, −1) σyK I iσz (−1, +1, +1, +1) σxK iσz I (−1, +1, +1, −1) σxK iσz iσz (−1, +1, −1, +1) σyKII (−1, +1, −1, −1) IK σy iσz (−1, −1, +1, +1) σy ⊗ σxK iσz ⊗ σz σz ⊗ σx (−1, −1, +1, −1) σy ⊗ σxK iσz ⊗ σz iI ⊗ σy (−1, −1, −1, +1) σx ⊗ σxKI ⊗ σz σz ⊗ σy (−1, −1, −1, −1) T T (Z2 × Z2) o Z2 ' D2d TPQ −i π ! e 4 0 i π K −iσz iσy (+1, −1) 0 e 4 σyK −I iI (−1, +1) −i π ! e 4 0 σy ⊗ i π K iI ⊗ σz −I ⊗ σy (−1, −1) 0 e 4 T T Z4 × Z2 ' C4h TP −i π ! e 4 0 σyK i π (+1, −1) 0 e 4 σyKI (−1, +1) −i π ! e 4 0 σxK i π (−1, −1) 0 e 4 T T Z4 o Z2 ' C4v TP −i π σxK e 4 σz (+1, −1) i 5π σyK e 4 σz (−1, +1) σyK iI (−1, −1) T Z3 × (Z3 o Z2 ) T ' (Z3 × Z3) o Z2 TPQ  Ω8 0 0   0 0 1   Ω 0 0  5 7 1 iσy ⊗  0 Ω 0  K −I ⊗  1 0 0  −I ⊗  0 Ω 0  ω 2 0 0 Ω2 0 1 0 0 0 Ω4  Ω 0 0   0 0 1   Ω8 0 0   0 Ω4 0  K  1 0 0   0 Ω2 0  ω 0 0 Ω7 0 1 0 0 0 Ω5 3 iσyK −I −I ω 2  Ω8 0 0   0 0 1   Ω 0 0   0 Ω5 0  K  1 0 0   0 Ω7 0  ω2 0 0 Ω2 0 1 0 0 0 Ω4  Ω 0 0   0 0 1   Ω8 0 0  4 2 5 iσy ⊗  0 Ω 0  K −I ⊗  1 0 0  −I ⊗  0 Ω 0  ω 2 0 0 Ω7 0 1 0 0 0 Ω5 T T T A4 Z ' S ' T T = (12)KP = (123) Q = (124) o 2 4 d √ √ −i π ! ω 0 3 e 6 2i IK 2 √ i π (+1, −1) 0 ω 3 2i e 6 σ K −II (−1, +1) y √ √ −i π ! ω 0 3 e 6 2i σy ⊗ IK −I ⊗ 2 I ⊗ √ i π (−1, −1) 0 ω 3 2i e 6 T A4 × Z2 TP = (123) Q = (124) σ K −II (+1, −1) y √ √ −i π ! ω 0 3 e 6 2i σy ⊗ σyKI ⊗ 2 I ⊗ √ i π (−1, +1) 0 ω 3 2i e 6 √ √ −i π ! ω 0 3 e 6 2i σyK − 2 √ i π (−1, −1) 0 ω 3 2i e 6 19

Fermionic anti-unitary groups Reps label of classiﬁcation T Z4 T i π 0 e 4 −i π K −1 e 4 0 T Z2 × Z4 TP i π 0 e 4 −i π KI (+1, −1) e 4 0 σyK −iσz (−1, +1) i π 0 e 4 −i π K iσz (−1, −1) e 4 0 − G−(Z4,T ) TP σyK iσz −1

(ν) (ν∗) (A) (B) where |α1 i (or |β1 i) is one of the irreducible bases of ∗ g g3 projective Rep (ν) [or (ν ), the complex conjugate Rep k of (ν)], gr and gl are diﬀerent group elements with

r r r gˆ|g1i = ω2(g, g1)|gg1i, −1 −1 gi gk g g gˆ|gl i = ω−1(g, gl )|ggl i. j k 1 2 1 1 g2 r l The states {|g i} (or {|g i}) are not necessarily orthog- g0

gi gj onal, but they form over complete bases for the Rep (ν) −1 g g g1 [or (ν∗)] since (see appendixE2) i j

nν FIG. 1: (A)Aharonov-Bohm phase as the topological term; X r r X (ν) (ν) |g ihg | ∝ I = |αi ihαi |. (B)Quantized topological action amplitude on closed space- g i=1 time. So summing over |gri is equivalent to summing over all (ν) (ν) the nν bases α1 , ..., αnν of the irreducible Rep space (ν) of freedom and can be used to describe the low energy (and similar result holds for |gli). effective field theory of the system. The physical degrees of freedom at site i are combina- Neglecting dynamic terms, we obtain the fixed point l r tions of the bases |gii and |gi i partition function of the corresponding SPT phase, l r p l l −1 r l r |g , g i = ω2(g , (g ) g )|g , g i, i i i i i i i X −S({gi}) X Y sijk Z ∝ e = ϕ (|gii, |gji, |gki) l r p such that |gi, gi i varies under group action in a way {gi} {gi} {ijk} similar to |ni varies under rotation (the difference is that (47) there is no phase factor eiϕ here), r where s is the orientation of the triangle (ijk), which l r p l l −1 r ω2(g, gi ) l r l r p ijk gˆ|gi, gi i = ω2(gi, (gi) gi ) l |ggi, ggi i = |ggi, ggi i . is equal to 1 if it is pointing outside and −1 otherwise. ω2(g, g ) i Similar to the θ-term of the O(3) NLSM, the total In the bond between neighboring sites i and i+1, the de- action amplitude is normalized if the space-time is closed r l grees of freedom gi and gi+1 are locked by the constraint (the simplest case is the surface of a tetrahedron, see r l gi = gi+1 owning to strong interactions. Fig.1(B)), r We discretize the space-time and put a variable gi at l s012 s013 each space-time point i and omit gi since it is the same as ϕ (|g0i, |g1i, |g2i)ϕ (|g0i, |g1i, |g3i) r gi−1. (Without causing confusion, we can eliminate the ×ϕs023 (|g i, |g i, |g i)ϕs123 (|g i, |g i, |g i) superscript r in the following.) The action amplitude for 0 2 3 1 2 3 a space-time unit (ijk) is defined as the Aharonov-Bohm = 1 phase of the basis |gii evolving between the two paths −1 −1 −1 owning to the cocycle equation (gj gk) (gi gj)|gii and (gi gk)|gii, see Fig.1(A),

−1 −1 −1 −1 −1 −1 h −1 i (dω2)(g0 g1, g1 g2, g2 g3) = 1. ϕ(|gii, |gji, |gki) = M(gj gk)M(gi gj) M(gi gk)

s(gi) −1 −1 The partition function can be regarded as imaginary = ω (g gj, g gk). 2 i j time evolution operator Z = U(0, τ) = |ψτ ihψ0|, where The total action amplitude stands for the topological |ψτ i is the ground state at time τ. After some calcula- phase factor (the ‘Berry phase’) of the physical degrees tion we can write out the ground state (under periodic 20 boundary condition) as where O is an arbitrary operator. Obviously,

X Y −1 r r l r l r l r p Γˆ1(γ1) = γ1γ1γ1 = γ1, |ψi = B (|Ei, |gi i, |gj i)|gigi gjgj gkgk...i ˆ {gigj gk} {ijk} Γ1(γi) = γ1γiγ1 = −γi, for any hγ1|γii = 0. X Y −1 r r −1 r l r l r l r p = ω2 (gi , (gi ) gj )|gigi gjgj gkgk...i . Since Γˆ1(H) = γ1Hγ1 = H, the operation Γˆ1 is a ‘sym- {gigj gk} {ijk} metry operation’ of the system. The eigenvalues of Γˆ1 are ±1 and Γˆ2 = I, so Γˆ generates a Z ‘symmetry group’. Noticing that in 1D j = i + 1, k = j + 1, ...., the 1 1 2 −1 r r −1 r −1 l l −1 r It should be mentioned that the form of Γî ( or γ1) wave function ω2 (gi , (gi ) gj ) = ω2 (gj, (gj) gj ) is the CG coefficient that fractionalize the physical degrees depends on the details of H, this Z2 ‘symmetry group’ of freedom into two projective Reps is not a symmetry in the usual sense. The existence of this special Z2 symmetry is a consequence of the non- l r −1 l l −1 r l r p |gjgj i = ω2 (gj, (gj) gj )|gjgj i , trivial winding number of the p + ip superconductor and the presence of topological defect (i.e. vortex) where the the ground state wave function can also be written in Majorana mode γ1 locates[50]. forms of product of dimers If γ2 is another majorana zero mode of the Hamilto- nian, then it defines another Z symmetry group gener- X l r l r l r 2 |ψi = |...gi)(gi gj)(gj gk)(gk...i, ated by l r {gigi ...} Γˆ2(O) = γ2Oγ2, r l r l where gi = gj, gj = gk, ... and each bracket means a ˆ ˆ ˆ ˆ singlet (or a dimer) on a bond between neighboring sites. and it is easily checked that Γ1Γ2(O) = Γ2Γ1(O), namely, The dangling degrees of freedom on the ends stand for the Γˆ1Γˆ2 = Γˆ2Γˆ1. edge states which carry projective Reps of the symmetry group. So, if H contains two Majorana zero modes γ1 and γ2, From the above ground state wave function of SPT then it has a Z2 × Z2 ‘symmetry group’. The two ma- phase, it is easily seen that the fixed point parent jorana zero modes result in two-fold degeneracy of the Hamiltonian is constructed by projector onto the bond ground states. In the ground state subspace, the opera- ˆ ˆ singlets[17]. If we further project the physical degrees of tors Γ1, Γ2 act projectively and their Rep matrices are freedom to its subspace, then we can obtain an AKLT- Γˆ → M(Γ ) = σ , Γˆ → M(Γ ) = σ , type state, and the parent Hamiltonian can also be con- 1 1 x 2 2 y 2 2 structed using projection operators[49]. with M(Γ1) = M(Γ2) = 1 and the fermionic anti- commuting relation {M(Γ1),M(Γ2)} = 0. Now suppose the system has four Majorana zero modes B. Defects in 2D topological phases γ1, γ2, γ3, γ4, then the corresponding symmetry operations Γˆ1, Γˆ2, Γˆ3, Γˆ4 generate a Z2 ×Z2 ×Z2 ×Z2 symmetry Except for 1D SPT phases, projective Reps also have group. The degenerate ground states carry an irreducible applications in 2D topological phases, including SPT projective Rep of the class (−1, −1, −1, −1, −1, −1) (up phases and intrinsic topological phases. In the follow- to a gauge transformation comparing with TableI) ing we will give several examples. M(Γ1) = σz ⊗ σz,

M(Γ2) = I ⊗ σy,

1. Vortices in topological superconductors (fermionic SPT M(Γ3) = I ⊗ σx, phases) M(Γ4) = σx ⊗ σz,

It is known that the vortices of p + ip topological which satisfy the relation {M(Γj),M(Γk)} = 2δjk. superconductor[50] carry Majorana zero modes and the Similar results can be generalized to the cases with degeneracy of the wave function depends on the number more majorana zero modes, and the degeneracy of the of spatially separated vortices. In the following, we will ground states can be understood as the existence of pro- show that if we interpret each Majorana zero mode as a jective Rep of the corresponding ‘symmetry group’. ‘symmetry’ operation, then the set of symmetry opera- Above picture is still valid even if there are interactions tions form an Abelian group, and the degeneracy of the in the Hamiltonian H. Majorana Hilbert space can be understood as the projective Rep of this group. 2. Monodromy defects in bosonic SPT phases Supposing the majorana zero mode γ1 is an eigen state of a Hamiltonian H with [γ1,H] = 0, then we can deﬁne In 2D fermionic topological insulators, a magnetic π- an operation Γˆ1 corresponding to the majorana mode γ1, ﬂux (a symmetry defect) gives rise to extra degeneracy. Γˆ1(O) = γ1Oγ1, Actually, this picture also holds for bosonic SPT phases. 21

Generally, symmetry defects can not be created locally The physical meaning of above result is that in the if they carry fractional symmetry flux. These defects SPT phases corresponding to nonzero pIII , the symme- are the end points of strings and are called the Mon- try fluxes of ZN1 carry nontrivial projective Reps (thus odromy defects in literature[51]. In some SPT phases, give rise to degeneracy of the energy spectrum) of the monodromy defects may carry projective Reps of the group ZN2 × ZN3 [52, 53]. Similarly, the ZN2 (or ZN3 ) symmetry group[51–54]. fluxes carry projective Reps of the group ZN1 × ZN3 (or

Bosonic topological insulators. Bosonic topolog- ZN1 × ZN2 ). Similar conclusion also holds for certain T ical insulators are SPT phases protected by U(1) o Z2 SPT phases with non-Abelian symmetry groups. symmetry[55]. In 2D bosonic topological insulators, a π-flux (gauging the U(1) symmetry) carries projective T Rep of the remaining symmetry Z2 which gives rise to 3. 2D topological order/SET phases Kramers’ degeneracy. This nontrivial result comes from the fact that in the bosonic topological insulator (which We have shown that in SPT phases, Monodromy de- is a nontrivial SPT phase), the vortex creation operator fects may carry projective Reps of the symmetry group −1 † reverses its sign under time reversal Tb bvTb = −bv, then (or subgroup of the symmetry). If the system carries intrinsic topological order, then point-like excitations, such ˆ Tb|πi = | − πi = bv|πi, as spinons (‘electronic’ excitations) or visons (‘magnetic’ excitations), can carry projective Reps of the symmetry group. Topologically ordered phases enriched by sym- ˆ −1 † Tb| − πi = TbbvTb Tb|πi = −bv| − πi = −|πi, metry are called symmetry enriched topological (SET) phases. so An example is the projective symmetry group 2 (PSG)[57, 58] in quantum spin liquid with lattice space Tb = −1, group symmetry. In the parton construction of spin-1/2 T quantum spin liquids, the system has an SU(2) gauge which is a projective Rep of Z2 . On the other hand, in −1 † symmetry. At mean field level, this gauge symmetry the trivial bosonic insulator Tb bvTb = b , which yields v may break down to U(1) or Z (called invariant gauge a linear Rep T 2 = 1. In this case creating π fluxes will 2 b group, or IGG in brief) owning to paring between the not cause degeneracy. spinons. Generally the space group is no longer the sym- Z × Z × Z SPT phases. Another example N1 N2 N3 metry of the mean field ground state, but the mean field is the 2D bosonic SPT phases with symmetry group state is invariant under space group operation followed by G = Z × Z × Z (where N ,N ,N are integers) N1 N2 N3 1 2 3 a gauge transformation. The symmetry elements, which with generator h , h , h respectively. The SPT phases 1 2 3 are combinations of space group operations and the corre- are classified by the third group cohomology H3(G, U(1)) sponding gauge transformations, form the PSG. In PSG, and each nontrivial phase corresponds to a nontrivial the multiplication rule of two space group operations are class of 3-cocycles[16, 17]. We will focus on the type- twisted by gauge transformations in the IGG. As a result, III SPT phases described by the type-III cocycles [56]. the PSG can be classified by the second group cohomol- If we label a group element A as A = (a1, a2, a3) = 2 a1 a2 a3 ogy H (SG, IGG) (for details, see Refs. 58, 59). (h1 , h2 , h3 ) with 0 ≤ ai ≤ Ni − 1, then a type-III cocycle takes the following form[56], PSG actually describes how the spinons (the ‘electronic’ excitations) carry fractional charge of the sym-

pIII metry group. On the other hand, visons (the ‘magnetic’ 2πi a1b2c3 III ω3(A, B, C) = e N123 , 1 ≤ p ≤ N123, excitations), or bond states of vison and spinon, may also carry projective Reps of the symmetry group[60, 61]. For where N123 is the greatest common devisor of N1,N2,N3. example, in the SO(3) non-abelian chiral spin liquid, a From above 3-cocyle, one can construct a 2-cocyle from vison not only carries non-Abelian anyon, but also carries the slant product iA: spin-1/2 degrees of freedom, which is a projective Rep of the symmetry group SO(3). As a result, a more com- χA(B,C) = (iAω3)(B,C) −1 plete classification of SET phases with symmetry group = ω3(A, B, C)ω3 (B, A, C)ω3(B,C,A) G is H2(G, A)[19, 62], where A is the fusion group of pIII 2πi (a1b2c3−b1a2c3+b1c2a3) abelian anyons in the raw topological order. A subtle = e N123 . issue is that obstructions may exist in realizing all the phases classified by H2(G, A) in pure 2D[19, 62]. Since iA commutes with the coboundary operator d, χA(B,C) is a 2-cocycle: (dχA)(D,B,C) = diAω3(D,B,C) = iAdω3(D,B,C) = 1. Furthermore, III C. Sign problems in Quantum Monte Carlo if p 6= 0 then χA(B,C) is a nontrivial 2-cocyle since it agrees with the analytic solution in Eq. (B13). Spe- cially, if A ∈ ZN1 with a2 = a3 = 0, then χA(B,C) is a QMC simulations. The generalized Schur’s lemma nontrivial 2-cocyle of the subgroup ZN2 × ZN3 . corresponding to projective Reps of anti-unitary groups 22 can also be applied to search for models which are free of sign problems in quantum Monte Carlo (QMC) TABLE II: Real orthogonal projective Reps of the group G = Z × ZT = {E,P,T,PT } have classification H2(G, Z ) = 3, simulations[43, 63–66]. By introducing the Hubbard- 2 2 2 Z2 which is labeled by (ω2(T,T ), ω2(PT,PT ), ω2(T,PT )) respec- Stratonovich field ξ(x, τ), interacting fermion models can tively. We have fixed ω2(TP,T ) = 1, therefore ω2(T,PT ) = be mapped to free fermion Hamiltonians. The Boltzmann ±1 means that Tb and TPd are commuting/anti-commuting, weight ρ(ξ) of the auxiliary field configuration ξ can be respectively. obtained by tracing out the fermions. Sign problem oc- Z × ZT TPTP classification label curs when ρ(ξ) < 0, in which case one have trouble to 2 2 IK iσy iσyK (+1, −1, +1) simulate the quantum system in a Markov process. iσyK iσy IK (−1, +1, +1) When the fermion number is conserved in the iσyK −I iσyK (−1, −1, +1) Hubbard-Stratonovich Hamiltonian σzK iσy −σxK (+1, +1, −1) σxK −σz −iσyK (+1, −1, −1) † H(ξ) = C H (ξ)C (48) −iσyK σz σxK (−1, +1, −1) iσy ⊗ IK σz ⊗ iσy −σx ⊗ iσyK (−1, −1, −1) T where C = (c1, ..., cN ) are fermion bases, then the Boltzmann weight in the QMC simulation is given by representations are required to be real orthogonal matri- − R β H(ξ)dτ ρ(ξ) = Tr [Te 0 ] = det[I + B(ξ)], ces such that after the group action the bases are still Majorana fermions. Under this constraint, the projec- R β − H (ξ)dτ 2 3 where B(ξ) = Te 0 , T means time ordered in- tive Reps are classified by H (G, Z2) = Z2, which con- tegration, I is an identity matrix, 1/β is the temperature tains 7 nontrivial classes as shown in TableII (it should and ξ is space-time dependent Hubbard-Stratonovich be noted that real orthogonality is a sufficient but not field. If the matrix H , and then the matrix (I + B), are necessary condition for Z2 coefficient projective Reps, commuting with the projective representation of an anti- namely, real orthogonal projective Reps are classified by T 2 2 unitary symmetry group Z2 = {E,T } with Tb = −1, H (G, Z2) but a Z2 coefficient projective Rep is not nec- then according to the generalized Schur’s lemma and essarily real orthogonal). In the class (−1, +1, −1) irre- its corollary, the eigenvalues of (I + B) are either pairs ducible projective Rep where {T,b PTd} = 0 and Tb2 = −1, of complex conjugate numbers, or even-fold degenerate (PTd)2 = 1, one cannot find a 2-dimensional (diagonal) (the Kramers degeneracy) real numbers, or a mixture block of H¯ which satisfies the following conditions: it is of them. As a consequence, the determinant must be skew-symmetric and commutes with the Rep of G. The a non-negative number[43], i.e. ρ(ξ) is non-negative and smallest block satisfying above conditions is at least 4- the model is free of sign problem under QMC simulation. dimensional, i.e. it spans the direct sum space of at least If fermion pairing terms exist after the Hubbard- two block of irreducible projective Reps I ⊗ M(g)Ks(g). Stratonovich decomposition, one can use Majorana The 4-dimensional block takes the form iσy ⊗ (aI + biσz) bases to write the decoupled Hamiltonian as H(ξ) = which has eigenvalues b ± ia and −b ± ia, where a, b are T T Γ H (ξ)Γ, where Γ = (γ1, ..., γ2N ) are Majorana bases real numbers. This will not cause sign problem even if T and H is a skew symmetric matrix H = −H at half- all the eigenvalues are purely imaginary. Another way filling (consequently all the nonzero eigenvalues of H to see the absence of sign problem is to use the eigen- appear in pairs of ±λ)[63]. In this case, the Boltzmann states of P to block diagonalize H¯ into two parts that weigh is given by are mutually complex conjugate to each other[64]. The class (+1, −1, −1) gives the same result since it defers −1 1 ρ(ξ) = [det(B + B)] 2 , from the class (−1, +1, −1) only by switching the roles of T and PT . These symmetry classes are called the −2 R β (ξ)dτ −β ¯(ξ) where B(ξ) = Te 0 H = e H , H¯(ξ) is Majorana class[64]. a skew symmetric matrix. The Boltzmann weigh can ¯ In the class (−1, −1, −1) where {T,b PTd} = 0 and also be written in forms of the eigenvalues λk of H as 2 2 Q 2βλk −2βλk Tb = −1, (PTd) = −1, the Rep is 4-dimensional. It can ρ(ξ) = (e + e ), where any pair of (λk, −λk) k Q ¯ be reduced into 2-dimensional irreducible Reps by unitary are counted only once in k. If H (and then H ) T transformations but cannot be reduced by real orthog- commute with the projective Rep of Z2 symmetry with 2 ∗ ∗ onal transformations. Since Pb is skew-symmetric and Tb = −1, both (λk, −λk) and (λk, −λk) are eigenvalues ¯ anti-commutes with Tb, we can use the eigenstates of P of H when λk is not purely imaginary. However, if λk is ∗ to transform the Hamiltonian in forms of (48). It is free purely imaginary, then λ = −λk and it is possible that k of sign problem owning to the T 2 = −1 symmetry. This only (λk, −λk) appear in the eigenvalues. In that case, b sign problem appears. To avoid this problem, one can is called the Kramers class[64]. enlarge the symmetry group. Above results can be generalized. If particle num- The simplest case is to enlarge the symmetry group as ber is conserved in H(ξ), the condition Tb2 = −1 can T G = Z2 × Z2 = {E,P,T,PT } and let H commute with be relaxed. The essential point of avoiding the sign a non-trivial projective Rep of G. In Majorana bases, the problem is that the Hubbard-Stratonovich Hamiltonian 23

H (ξ) commutes with a nontrivial projective Rep of an More explicitly, anti-unitary group G, where all the irreducible projective Reps of this class are even-dimensional. Even if for Rx(Sx) = Sx,Rx(Sy) = −Sy,Rx(Sz) = −Sz, any anti-unitary operator T g (where g ∈ G is unitary) Ry(Sx) = −Sx,Ry(Sy) = Sy,Ry(Sz) = −Sz. 2 [M(T g)K] 6= −1, the sign problem can still be avoided. If the system contains an odd number of spins, then the Here we give two examples. For the type-I anti-unitary operators carry linear Rep of the symmetry group Z ×Z groups, the class (+1, −1) irreducible projective Rep of 2 2 T T but the total Hilbert space forms a (reducible) projective the group (Z2 × Z2) o Z2 ' D2d in TableI is an even Rep. As a consequence all the energy levels are at least dimensional Rep, where [M(T )K]2 = [M(PT )K]2 = 1 2 2 doubly degenerate as long as the Z2 × Z2 symmetry is and [M(QT )K] = −[M(P QT )K] = −iσz. Espe- not broken explicitly. cially, as an example of the type-II anti-unitary groups, the nontrivial irreducible projective Rep of the group T Z4 is an even dimensional Rep and is characterized by V. CONCLUSION AND DISCUSSION [M(T )K]4 = −1. Similar even-dimensional projective Reps can also be found in other anti-unitary groups. In summary, we generalized the eigenfunction method If particle number is not conserved in H(ξ), the real to obtain irreducible projective Reps of finite unitary or orthogonal condition of the Reps of the symmetry group anti-unitary groups by reducing the regular projective G (which is imposed in the Majorana class and Kramers Reps. To this end we first solved the 2-cocyle equations symmetry class) can be relaxed. If an anti-unitary sym- of a group G to obtain the factor systems and their clas- metry group G has a 4-dimensional (or 4n-dimensional, sification, specially if G is anti-unitary we introduced the n ∈ Z) irreducible unitary projective Rep, and if H(ξ) decoupled factor systems for convenience. Therefore reg- commutes with it, then the model is free of sign prob- ular projective representations are obtained. Then us- lem. For example, the group Z × Z × ZT ' T has 2 2 2 D2h ing class operators we constructed the complete set of four classes of 4-dimensional irreducible projective Reps commuting operators, and using their common eigen- (see TableI). If ¯ [or ( −1 + )] commutes with one of H B B functions we transformed the regular projective repre- these projective Reps, the degeneracy of the eigenvalues sentations into irreducible ones. Anti-unitary groups are of ( −1 + ) is increased: if an eigenvalue Λ is real, then B B generally complicated than unitary groups in the reduc- it is 4-fold degenerate; if Λ is complex, then both Λ and tion process, where modified Schur’s lemmas were used. Λ∗ are 2-fold degenerate. Resultantly, the Boltzmann We applied this method to a few familiar finite groups weight is non-negative. and gave their irreducible Reps. These generalizations provide hint to find new classes We then discussed applications of projective Reps of models which are free of sign problem in quantum in many-body physics, for instance, the edge states of Monte Carlo simulation. one-dimensional Symmetry Protected Topological (SPT) phases, symmetry fluxes in 2-dimensional SPT phases D. Space groups, Spectrum degeneracy and other and anyons in Symmetry Enriched Topological (SET) applications phases carry projective Reps of the symmetry groups. We also showed that recently discovered models which are free of sign problem under quantum Monte Carlo sim- Projective Reps can also be applied in many other ulations are commuting with nontrivial projective Reps fields, such as space groups, spectrum degeneracy and of certain anti-unitary groups. Other applications related so on. to spectrum degeneracy were also summarized. Non-symmorphic space groups. In the Rep the- Our approach can be generalized to obtain irreducible ory of non-symmorphic space groups, the ‘little co-group’ projective Reps of infinite groups, such as space groups, (subgroup of the point group) is represented projectively magnetic space groups and Lie groups. Furthermore, it at some symmetric wave vectors, where the factor system may shed light on the reduction of symmetry operations comes naturally from fractional translations. The repre- corresponding to higher order group cocycles, namely, in sentation theory of space groups was thoroughly stud- the boundary of 2D SPT phases which are classified by ied in literature, for instance in Ref. 41 and references the third group cohomology. therein, so we will not discuss in more detail here. Spectrum degeneracy. Similar to the Kramers degeneracy in time reversal symmetric systems with odd Acknowledgments number of electrons, in some cases projective Reps can also explain the degeneracy of the ground states or ex- cited states. For example, consider a spin-1/2 system We thank Zheng-Cheng Gu, Xie Chen, Meng Cheng, respecting Z × Z = {E,R } × {E,R } spin-rotation Peng Ye, Xiong-Jun Liu, Shao-Kai Jian, Zhong-Chao 2 2 x y Wei, Cong-Jun Wu, Tao Li and Li You for helpful dis- symmetry, where the two Z2 subgroups are generated by π rotation along x and y directions respectively, cussions. We especially thank Xiao-Gang Wen for discussion about the numerical calculation of group cohomol- −iπSx iπSx −iπSy iπSy Rx(Sm) = e Sme ,Ry(Sm) = e Sme . ogy, and thank Hong Yao and Zi-Xiang Li for comments 24 and helpful discussion about the application of projective we find Reps in QMC. This work is supported by NSFC (Grant s(g1) Nos. 11574392), the Ministry of Science and Technology 1 ω1 (g2)ω1(g1) Z (G, U(1)) = {ω1 | = 1}, (A6) of China (Grant No. 2016YFA0300504), and the Fun- ω1(g1g2) damental Research Funds for the Central Universities, and the Research Funds of Renmin University of China and

(No. 15XNLF19). J.Y. is supported by MOST (No. s(g1) 1 ω0 2013CB922004) of the National Key Basic Research Pro- B (G, U(1)) = {ω1 | ω1(g1) = }. (A7) gram of China, by NSFC (No. 91421305) and by National ω0

Basic Research Program of China (2015CB921104). 1 1 Z (G,U(1)) H (G, U(1)) = B1(G,U(1)) is the set of all inequivalent 1D Reps of G. Appendix A: Group cohomology From

ωs(g1)(g , g )ω (g , g g ) In this section, we will briefly introduce the group co- (d ω )(g , g , g ) = 2 2 3 2 1 2 3 , (A8) 2 2 2 1 2 3 homology theory . ω2(g1g2, g3)ω2(g1, g2) Let ωn(g1, . . . , gn) be a U(1) valued function of n n we find group elements. In other words, ωn : G → U(1). n Let C (G, U(1)) = {ωn} be the space of such func- s(g1) n 2 ω2 (g2, g3)ω2(g1, g2g3) tions. Note that C (G, U(1)) is an Abelian group Z (G, U(1)) = {ω2 | = 1} 00 ω2(g1g2, g3)ω2(g1, g2) under the function multiplication ωn(g1, . . . , gn) = 0 (A9) ωn(g1, . . . , gn)ωn(g1, . . . , gn). We define a map d from Cn(G, U(1)) to Cn+1(G, U(1)): and

s(g1) (dω )(g , . . . , g ) 2 ω1(g1)ω1 (g2) n 1 n+1 B (G, U(1)) = {ω2 | ω2(g1, g2) = }. (−1)n+1 ω1(g1g2) = [g1 · ωn(g2, . . . , gn+1)]ωn (g1, . . . , gn) × (A10) n 2 i 2 Z (G,U(1)) Y (−1) The 2-cohomology of group H (G, U(1)) = 2 ωn (g1, . . . , gi−1, gigi+1, gi+2, . . . , gn+1),(A1) B (G,U(1)) i=1 can classify the projective Reps of symmetry group G. Correspondingly, it can classify 1-dimensional SPT where g · ωn(g2, . . . , gn+1) = ωn(g2, . . . , gn+1) if g is uni- phases with on-site symmetry group G. −1 tary and g · ωn(g2, . . . , gn+1) = ωn (g2, . . . , gn+1) if g is anti-unitary. In other words, g · ωn(g2, . . . , gn+1) = s(g) ωn (g2, . . . , gn+1), where s(g) = 1 if g is unitary and Appendix B: Canonical gauge choice and solutions s(g) = −1 otherwise. From above definition (A1), it is of cocycles easily checked that d2 ≡ 1. Let Once the cocycle equations and the coboundary functions are written in terms of linear equations (see section n n−1 B (G, M) = {ωn | ωn = dωn−1 | ωn−1 ∈ C (G, M)} IIA), the classification of cocycles is equivalent to solv- (A2) ing some linear algebra. In the following, we will focus and on 2-cocycles. For a general group G, the 2-cocycle ω (g , g ) = Zn(G, M) = {ω | dω = 1, ω ∈ Cn(G, M)} (A3) 2 1 2 n n n eiθ2(g1,g2) has G2 components (G is the order of group) Bn(G, M) and Zn(G, M) are also Abelian groups which which satisfy the following equations (g1, g2, g3 ∈ G): satisfy Bn(G, M) ⊂ Zn(G, M). The nth-cohomology of s(g )θ (g , g ) + θ (g , g g ) − θ (g g , g ) − θ (g , g ) = 0. G is defined as 1 2 2 3 2 1 2 3 2 1 2 3 2 1 2

Hn(G, M) = Zn(G, M)/Bn(G, M). (A4) Notice that ‘=0’ in above equation means mod 2π = 0 (the same below). Now we give some examples. From Above equations can be written in matrix form as X ω (g )ωs(g1)(g ) (Cmn) θn = 0, (B1) (dω )(g , g ) = 1 1 1 2 , (A5) 2 1 1 2 n ω1(g1g2) where C is an G3 × G2 matrix (the matrix elements are 2 given above), and θ2 is a G -component vector. We do 2 For an introduction to group cohomology, see wiki not specify the coefficient space at the beginning, and http://en.wikipedia.org/wik/Group cohomology and Romyar will go back to it when discussing the classification of Sharifi, AN INTRODUCTION TO GROUP COHOMOLOGY the cocycles. 25

iΘ2(g1,g2) The coboundaries Ω2(g1, g2) = e are given by: As an example, we consider the group D2 = Z2 ×Z2 = {E, P, Q, P Q} where P 2 = Q2 = E. If we adopt the Θ2(g1, g2) = s(g1)θ1(g2) + θ1(g1) − θ1(g1g2). canonical gauge condition, there are 9 nonzero variables The θ1 contains G variables. Above equations can also in θ2(g1, g2) and 3 nonzero ones in θ1(g1). The cocycle be written in matrix form equations and the coboundary functions correspond to two matrices C (27 × 9, with rank 6) and B (9 × 3, with m X mn n Θ2 = (B ) θ1 , (B2) rank 3), respectively: n  0 0 0 0 0 0 0 0 0  where B is an G2 × G matrix.  −1 1 1 0 0 0 0 0 0    The C matrix in (B1) defines a set of equations while  −1 1 1 0 0 0 0 0 0    the B matrix in (B2) defines a set of functions. All the  0 −1 1 1 0 0 −1 0 0    functions defined by B are solutions of the equations  0 −1 0 0 1 0 0 −1 0   1 −1 0 0 0 1 0 0 −1  defined by C. If one can impose some constraints on    0 1 −1 −1 0 0 1 0 0  both C and B, the classification will be easier to obtain.    1 0 −1 0 −1 0 0 1 0  For instance, in the extreme case, if there exists a cer-    0 0 −1 0 0 −1 0 0 1  tain ‘gauge’ condition such that there is only one trivial    1 0 0 −1 0 0 −1 0 0  function defined by B (namely, all the 2-coboundaries    0 1 0 −1 0 1 0 −1 0    Θ2(g1, g2) equal zero), then each solution of C will stand  0 0 1 −1 1 0 0 0 −1    for an equivalent class, in which case, the classification  0 0 0 1 −1 1 0 0 0    reduces to solving the equations under the special gauge C =  0 0 0 0 0 0 0 0 0  , (B6)   condition.  0 0 0 1 −1 1 0 0 0    Actually, we can really fix some gauge degrees of free-  −1 0 0 0 1 −1 1 0 0    dom. It has been proved that the canonical gauge[17] is  0 −1 0 1 0 −1 0 1 0    a valid gauge condition by setting  0 0 −1 0 0 −1 0 0 1     1 0 0 −1 0 0 −1 0 0  θ (E, g ) = θ (g ,E) = 0, (B3)   2 1 2 1  0 1 0 0 −1 0 −1 0 1     0 0 1 0 0 −1 −1 1 0  where E is the unit of group G. Under this convention,   2  −1 0 0 1 0 0 0 −1 1  there are only (G−1) components remaining nonzero in  0 −1 0 0 1 0 0 −1 0  3   θ2(g1, g2) and the size of the new matrix C is (G − 1) ×  0 0 −1 0 0 1 1 −1 0  2   (G − 1) . Similarly, for coboundaries, we have  0 0 0 0 0 0 1 1 −1     0 0 0 0 0 0 1 1 −1  Θ2(E, g1) = θ1(E) = 0. (B4) 0 0 0 0 0 0 0 0 0 As a result, we only have (G − 1)2 nonzero variables for coboundaries and the size of the new matrix B is  2 0 0  (G − 1)2 × (G − 1). Although this canonical convention 1 1 −1 does not fix all the ‘gauge’ degrees of freedom, it does sim-    1 −1 1  plify the calculation to solve the cocycle equations. After    1 1 −1  this simplification, the rank of 2-cocycle equations (B1)   B =  0 2 0  . (B7) is denoted as R , and R is the rank of 2-coboundary   C B  −1 1 1  equations (B2). For finite groups, the identity    1 −1 1  2   (G − 1) − RC = RB (B5)  −1 1 1  is always satisfied. 0 0 2 As usual linear problems, the cocycle equations can After some linear operations as mentioned above, the be solved by elimination method. To this end, we need matrices C and B can be written in the Smith normal to transform the matrix C (and B) into partially diago- form nal form C0 ( and B0) by linear operations, namely, by adding or subtracting multiple of one row (or column) to  1 0 0 0 0 1 −1 1 −1  some other row (or column). However, since the equa-  0 1 0 0 0 0 −1 1 0   0 0 1 0 0 1 0 0 −1  tion is defined mod 2π, in each step of linear operations, C0 =   , (B8)  0 0 0 1 0 −1 −1 2 0  the coefficients must be integer numbers. As a result, in   the final matrix all the entries are integer numbers. The  0 0 0 0 1 0 0 1 −1  (partially) diagonal matrix with integer entries is called 0 0 0 0 0 2 0 −2 0 Smith normal form3. The diagonal entries of C0 and B0  1 −1 −1  0 completely determine the classification of cocycles. B =  0 2 0  . (B9) 0 0 2

Notice that the number of variables (number of columns) 3 See wiki http://en.wikipedia.org/wik/Smith normal form in (B8) is greater than the number of equations (number 26 of rows), and the difference is 3. If the coefficient space From the Smith normal form, we can also obtain the is U(1), the solutions form a 3-dimensional continuous solutions of the cocycles. For example, in the Smith space U(1) × U(1) × U(1). Furthermore, the last row normal form (B8) of the cocycle equations of group of (B8) indicates the two variables corresponding to 2 is D2 = Z2 × Z2 = {E, P, Q, P Q}, the columns correspond 2mπ determined upto 2 (m = 0, 1), which gives an extra Z2 to variables θ2(P,P ), θ2(P,Q), θ2(P,PQ), θ2(Q, P ), group. So the solution space of θ2(g1, g2) is U(1)×U(1)× θ2(P Q, P ), θ2(Q, P Q), θ2(Q, Q), θ(P Q, Q), θ2(P Q, P Q), U(1) × Z2,while the coboundary space is U(1) × U(1) × respectively. We note these variables as X1,X2, ..., X9. U(1). The quotient gives the classification of cocycles: The last three variables are coboundary degrees of freedom and can be fixed as X7 = X8 = X9 = 0. Un- 2 H (D2,U(1)) = Z2. (B10) der this condition, the last equation becomes 2X6 = 0 mod 2π, which has two solutions X6 = 0 and X6 = π. If the coefficient space is Z, then the common factor 2 The first solution is trivial, from which we can obtain in the last row of (B8) can be eliminated, and the solu- that all other components are zero; the second solution space is Z × Z × Z. On the other hand, the common tion is nontrivial, from (B8) we can easily obtain that factor 2 in the last two rows of (B9) indicates that the X1 = −π, X2 = 0,X3 = −π, X4 = π, X5 = 0. Equiv- coboundary space is Z × 2Z × 2Z. Therefore, the classi- alently, the nontrivial solution is given by ω2(P,P ) = 2 fication is H (D2,Z) = Z2 × Z2. ω2(P,PQ) = ω2(Q, P ) = ω2(Q, P Q) = −1, other com- If the coefficient space is Z2, then the solution space ponents are equal to 1. of θ2(g1, g2) is Z2 × Z2 × Z2 × Z2. The last two rows In this paper we solve all the 2-cocycle equations us- of (B9) give trivial functions, so the coboundary space ing above method. However, for abelian unitary groups is Z2. The quotient gives the classification of cocycles: there exist analytic cocyle solutions. For example, for 2 3 H (D2,Z2) = Z2. the group G = ZN1 × ZN2 × ... × ZNk with genera- Similarly, if the coefficient space is ZN , then the clas- tors h1, ..., hk, if we label the group elements as A = 2 a1 a2 ak a1 a2 ak sification of cocycles is H (D2,ZN ) = Z(N,2) × Z(N,2) × (a1, a2, ..., ak) = (h1 , h2 , ..., hk ) = h1 h2 ...hk , then Z(N,2), where (N, 2) stands for the greatest common di- the analytic 2-cocycle solutions take the following form: visor of N and 2. The first (N,2) is owing to the factor Z P pij 0 2πi i 2. ω2(T, gT ) = ω2(T,T ). (C5) ω2(T, g˜)

1. Type-I anti-unitary groups: T 2 = E On the other hand, (dω2)(T, g, T ) = 1 gives

ω2(T, gT ) This kind of groups can be written in forms of direct = ω2(T, g)ω2(g, T ). (C6) T ω2(T g, T ) product group G = H ×Z2 or semi-direct product group T G = H o Z2 . In the following we will note the generator Comparing (C4),(C5) and (C6), we obtain T of Z2 as T = T. We can decouple the cocycles into two parts, namely, ω2(T, g)ω2(g, T ) = ω2(T, g˜)ω2(˜g, T ). T the Z2 part ω2(T,T ) and the H part ω2(g1, g2), and illustrate that they determine the classification. The cross- Owing to this relation, the following gauge fixing equations have solutions: ω0 (T, g) = ω (T, g)(dΩ )(T, g) = ing terms, such as ω2(T g1, g2), ω2(g1, T g2), ω2(T g1, T g2), 2 2 1 1, ω0 (T, g˜) = ω (T, g˜)(dΩ )(T, g˜) = 1, can be either fixed or expressed in terms of ω2(T,T ) and 2 2 1 ω0 (g, T ) = ω (g, T )(dΩ )(g, T ) = 1, ω0 (˜g, T ) = ω2(g1, g2). This can be shown in the following four steps: 2 2 1 2 ω (˜g, T )(dΩ )(˜g, T ) = 1, where (1) No gauge degrees of freedom in ω2(T,T ) = 2 1 ±1. Let us see ω2(T,T ) first. From the cocycle equation −1 −1 (dΩ1)(T, g) = Ω1 (g)Ω1 (T g)Ω1(T ), (dω )(T,T,T ) = ω−1(T,T )ω−1(E,T )ω (T,E)ω−1(T,T ) −1 −1 2 2 2 2 2 (dΩ1)(T, g˜) = Ω1 (˜g)Ω1 (gT )Ω1(T ), = 1, −1 (dΩ1)(˜g, T ) = Ω1(T )Ω1 (T g)Ω1(˜g), 2 (dΩ )(g, T ) = Ω (T )Ω−1(gT )Ω (g), we get ω2(T,T ) = 1, so 1 1 1 1

ω2(T,T ) = ±1, (C1) and the solutions are

−1 −1 here we have adopted the canonical gauge. Since Ω1(˜g) = Ω (g)ω2(T, g)ω (˜g, T ), −1 −1 1 2 (dΩ1)(T,T ) = Ω1 (T )Ω1 (E)Ω1(T ) = 1, there are no −1 Ω1(T g) = Ω1(T )Ω1 (g)ω2(T, g), (C7) gauge degrees of freedom for ω2(T,T ), ω2(T,T ) = 1 and Ω1(gT ) = Ω1(T )Ω1(g)ω2(g, T ), ω2(T,T ) = −1 stand for different classes. (2) Tune coboundaries to fix ω2(T, g) = 1 and where Ω1(g) is an U(1) variable and Ω1(T ) can be set as ω2(g, T ) = 1. Further, for unitary elements g ∈ H, we 1. Once the gauge fixing condition (C2) is satisfied, then will show that the following gauge can be fixed: in the remaining gauge degrees of freedom for each pair

ω2(T, g) = ω2(T, g˜) = 1, of g,˜g withg ˜ 6= g, ω (g, T ) = ω (˜g, T ) = 1, (C2) 2 2 Ω1(g)Ω1(˜g) = 1 where T g =gT ˜ . is required and there remains one free coboundary vari- Ifg ˜ = g, then we can tune the coboundary able Ω1(g). −1 −1 In later discussion we will ﬁx ω (T, g) = ω (g, T ) = 1 (dΩ1)(T, g) = Ω1 (g)Ω1 (T g)Ω1(T ), 2 2 −1 for any g ∈ H. (dΩ1)(g, T ) = Ω1(T )Ω1 (gT )Ω1(g), (3) The values of ω2(T g1, T g2) are determined. 0 such that ω2(T, g) = ω2(T, g)(dΩ1)(T, g) = 1, and From the equation ω0 (g, T ) = ω (g, T )(dΩ )(g, T ) = 1. This can be done 2 2 1 −1 −1 −1 by setting (dω2)(T, T, g) = ω2 (T, g)ω2 (E, g)ω2(T, T g)ω2 (T,T ) = 1, 1/2 we get ω (T, T g) = ω (T, g)ω (T,T ) = ω (T,T ). Sim- ω2(T, g) 2 2 2 2 Ω1(g) = σ , (C3) ilarly, from (dω2)(T, g, T ) = 1 we have ω2(T g, T ) = ω2(g, T ) ω2(˜gT, T ) = ω2(T,T ). That is to say, ω2(T, T g) and 1/2 Ω1(gT ) = σΩ1(T )[ω2(T, g)ω2(g, T )] . ω2(T g, T ) can be ﬁxed as real where σ = ±1 is a Z variable and Ω (T ) is a U(1) vari- 2 1 ω2(T, T g) = ω2(T g, T ) = ω2(T,T ) = ±1, able which can be set as 1. In the square root, −1 should be treated as eiπ such that p(−1) × (−1) = −1 and for any g ∈ H. p(−1)/(−1) = 1. Furthermore, from the cocy- Ifg ˜ 6= g, then the illustration of the gauge choice cle equation (dω2)(T g1, T, g2) = −1 −1 −1 (C2) is a little complicated. From the cocycle equation ω2 (T, g2)ω2 (˜g1, g2)ω2(T g1, T g2)ω2 (T g1,T ) = 1 (dω2)(˜g, T, T ) = 1, we have with g1, g2 ∈ H, we obtain

ω2(T g, T )ω2(˜g, T ) = ω2(T,T ). (C4) ω2(T g1, T g2) = ω2(˜g1, g2)ω2(T,T ). (C8) 28

2 (4) The values of ω2(T g1, g2) and ω2(g1, T g2) are Ω1(Pf ) = 1, under the gauge fixing (C14), ω2(Pf ,Pf ) is determined. Finally, considering the following cocycle completely fixed without any gauge degrees of freedom. equations In other words, different values of ω2(Pf ,Pf ) stand for different classes of 2-cocycles. (dω2)(T, g1, g2) From (dω2)(T,T,Pf ) = 1 we have −1 −1 −1 = ω2 (g1, g2)ω2 (T g1, g2)ω2(T, g1g2)ω2 (T, g1) −1 −1 ω2(T,TPf ) = ω2(Pf ,Pf )ω2(T,T ). = ω2 (g1, g2)ω2 (T g1, g2) = 1, (C9) On the other hand, (dω2)(TPf ,T,TPf ) = 1 gives (dω2)(g1, T, g2) −1 −1 = ω2(T, g2)ω2 (g1T, g2)ω2(g1, T g2)ω2 (g1,T ) ω2(TPf ,T )ω2(T,TPf ) = 1; −1 = ω (g1T, g2)ω2(g1, T g2) = 1, (C10) 2 while (dω2)(T,Pf ,T ) = 1 yields (dω2)(g1, g2,T ) −1 −1 ω2(T,TPf ) = ω2(TPf ,T ). = ω2(g2,T )ω2 (g1g2,T )ω2(g1, g2T )ω2 (g1, g2) −1 = ω2(g1, g2T )ω2 (g1, g2) = 1, (C11) Comparing these relations, we obtain 2 2 we obtain the following relations [ω2(Pf ,Pf )ω2(T,T )] = ω2(Pf ,Pf ) = 1,

−1 namely, ω2(Pf ,Pf ) = ±1. Here we have used ω2(T g1, g2) = ω (g1, g2), 2 2 ω2(T,Pf ) ω2(T,T ) = ω (P ,T ) = 1, which results from ω2(g1, T g2) = ω2(g1, g˜2), (C12) 2 f (dω2)(T,T,T ) = 1. We have shown previously that and an important constraint ω2(Pf ,Pf ) has no gauge degrees of freedom, therefore ω2(Pf ,Pf ) = 1 (trivial) and ω2(Pf ,Pf ) = −1 (nontriv- ω2(g1, g2)ω2(˜g1, g˜2) = 1, (C13) ial) stand for two different classes. In the following we will show that all other components are dependent on where g1, g2 ∈ H. ω2(Pf ,Pf ) or can be fixed to be 1. We call the cocycles satisfying relations (C1), (C2), From (dω2)(Pf ,Pf ,T ) = 1 and (dω2)(Pf ,T,Pf ) = (C8),(C12) as ‘decoupled cocyles’ or decoupled factor sys- 1 we obtain that ω2(Pf ,TPf ) = ω2(TPf ,Pf ) = tems, in which case only ω2(T,T ) and ω2(g1, g2) are im- ω2(Pf ,Pf ). 2 −1 portant. Since ω2(T,T ) contributes Z2 classification to Since ω2(T,T ) = 1 and (dΩ1)(T,T ) = Ω1(Pf ) = 2 H (G, U(1)), in appendixD we will see how the con- σ = ±1, the value of ω2(T,T ) only depends on the straint (C13) influences the classification of ω2(g1, g2). gauge degree of freedom σ. We can tune σ such that ω2(T,T ) = ω2(Pf ,Pf ) and consequently ω2(T,TPf ) = ω2(TPf ,T ) = 1. Finally, the value of ω2(TPf ,TPf ) = 1 2m 2. Type-II anti-unitary groups: T = E, m ≥ 2 can be derived from (dω2)(TPf ,TPf ,T ) = 1. Above we showed that the value of ω2(Pf ,Pf ) = ±1 2 T We focus on the case m = 2, which corresponds to completely determines the classification H (Z4 ,U(1)) = the anti-unitary symmetry group of fermions with half- Z2. The nontrivial factor system is given as integer spin. T ω2(Pf ,Pf ) = −1, Fermionic time reversal group Z4 . The simplest example of the type-II anti-unitary group is the fermionic ω2(Pf ,TPf ) = ω2(TPf ,Pf ) = ω2(Pf ,Pf ) = −1, T 4 time reversal group Z4 = {E,Pf ,T,Pf T } with T = E ω2(T,T ) = −1, 2 and T = Pf , Pf is the fermion parity. The unitary T f and others are fixed to be 1. normal subgroup of Z4 is H = Z2 = {E,Pf } with 2 T T Similar to the invariant [M(T )K] = ±1 for the pro- Z4 /H = Z2 = {E, T}. Obviously T is not an element of T T T jective Reps of Z2 , the projective Reps of Z4 also have Z4 . In the following we will figure out the classification T an invariant of 2-cocyles of Z4 by gauge fixing procedure. Firstly we adopt the canonical gauge by setting [M(T )K]4 = ±1. (C15) ω2(E, g) = ω2(g, E) = 1. We still define the time reversal −1 −1 4 2 conjugation as T gT =g ˜, since T = TPf , we have Noticing [M(T )K] = ω2(T,T ) ω2(Pf ,Pf ), it is in- 0 −1 −1 variant under gauge transformation M (g)K = T gT˜ = T gT˜ = g, namely, g˜ = g. Since P˜f = Pf , s(g) similar to (C3) we can fix M(g)Ω1(g)Ks(g) since 0 4 0 2 0 [M (T )K] = ω (T,T ) ω (Pf ,Pf ) ω2(T,Pf ) = ω2(Pf ,T ) = 1. (C14) 2 2 2 2 = ω2(T,T ) ω2(Pf ,Pf )Ω2(T,T ) Ω2(Pf ,Pf ) After above gauge fixing, from (C3), there are still a Z 2 Ω (P )2 degree of freedom σ and an U(1) degree of freedom Ω (T ). 2 1 f 1 = ω2(T,T ) ω2(Pf ,Pf ) 2 Ω1(Pf ) If we set Ω1(T ) = 1, then there remains a Z2 gauge degree 4 of freedom Ω1(Pf ) = σ = ±1. Since (dΩ1)(Pf ,Pf ) = = [M(T )K] . (C16) 29

General fermionic anti-unitary groups. In the Similar to (C4), (C5), (C6), from the cocycle equa- following, we will discuss general fermionic groups Gf . tions (dω2)(g, T, T ) = 1, (dω2)(T, T, g) = 1, and Since physical operations usually conserve fermion parity (dω2)(T, g,˜ T ) = 1, we have (the fermion creation or annihilation operators do not conserve fermion parity, but they are not allowed in the ω2(g, T )ω2(T, g) −1 = σg = σg˜ = σg˜ = ω2(˜g, Pf ).(C17) Hamiltonian), we assume that [Pf , g] = 0 for any g ∈ Gf . ω2(T, g˜)ω2(˜g, T ) f Therefore, the subgroup Z2 = {E,Pf } is a center of Gf and we have In the following we fix the values of ω2(T, g) and ω2(g, T ) and decouple the factor system as discussed in f Gf /Z2 = Gb, sectionC1. (1), Ifg ˜ = g, namely, T g = gT , then σg = 1. Similar where Gb is the bosonic (namely, type-I) anti-unitary to (C3), we can fix group. If we note Hf as the unitary normal subgroup of Gf ω2(T, g) = ω2(g, T ) = 1 T such that Gf /Hf = Z2 , and note the unitary normal T by tuning the values of Ω (g) and Ω (T g). subgroup of Gb as Hb such that Gb/Hb = Z2 , then 1 1 f (2), Ifg ˜ 6= g, the gauge fixing depends on the value of Hf /Z2 = Hb, there are two cases: (1)Hf is a direct prod- f f σg: uct group of Hb and Z2 , Hf = Hb × Z2 , then the group T T A) σg = 1. In this case, the discussion in equations Gf can be written as Gf = Hb ×Z4 or Gf = Hb oZ4 ; (2) f (C4), (C5), (C6) is still valid, we can simultaneously fix Hf is not a direct product group of Hb and Z2 , then the ± ω2(T, g) = ω2(g, T ) = ω2(T, g˜) = ω2(˜g, T ) = 1 by adopt- group can be noted as Gf = G−(Hf ,T ), where the sub- 2 ing the gauge in (C7). script − means T = Pf , and the superscript ± means ±1 B) σg = −1. Owing to (C17), the values of that T h = h T for h ∈ Hf . ω2(T, g), ω2(g, T ), ω2(T, g˜), ω2(˜g, T ) cannot be simultane- After slight modification, the discussion about decou- ously set to be 1 (in this sense the factor system is pled factor system in sectionC1 can also be applied for not completely decoupled), and the coboundary in (C7) the fermionic anti-unitary groups. T should be slightly modified. We can set For the cocycles of the Z4 subgroup of G, we can adopt the previous gauge fixing procedure. In the following ω2(T, g) = ω2(T, g˜) = 1, we try to decouple the factor system as we have done previously. and set one of ω2(g, T ), ω2(˜g, T ) to be 1. For example, Supposing g ∈ Hf , from the cocycle equations we can choose (dω2)(Pf ,Pf , g) = 1 and (dω2)(g, Pf ,Pf ) = 1, we get ω2(˜g, T ) = 1, ω2(g, Pf ) ω2(Pf ,Pf g) = . ω2(g, T ) = σg. ω2(Pf , g) ω2(gPf ,Pf ) Since g andg ˜ are mutually time-reversal conjugate, there On the other hand, from the cocycle equation is still a gauge degree of freedom in above equation. (dω2)(Pf , g, Pf ) = 1 we have Above gauge fixing can be carried out by adopting the ω (g, P ) ω (P g, P ) following coboundary 2 f = 2 f f . ω2(Pf , g) ω2(Pf , gPf ) −1 −1 Ω1(˜g) = Ω1 (g)ω2(T, g)ω2 (˜g, T ), −1 Since gPf = Pf g, comparing above two equations we Ω1(T g) = Ω1(T )Ω1 (g)ω2(T, g), obtain Ω1(gT ) = σgΩ1(T )Ω1(g)ω2(g, T ). ω (g, P )2 2 f = 1, We further discuss the gauge fixing of other compo- ω2(Pf , g) nents. The equations (C8),(C12) should be modified as follows: ω2(g,Pf ) namely, = ±1 is a Z2 variable, we note it as ω2(Pf ,g)

ω2(T g1, T g2) = ω2(g1,T )ω2(˜g1, g2)ω2(T,T )σg˜1 σg2 σg˜1g2 , ω2(g, Pf ) = σg, g 6= Pf . ω2(Pf , g) (here we have used ω2(˜g1Pf , g2) = ω2(˜g1, g2)σg˜1 σg2 σg˜1g2 ) and We can ﬁx the coboundary Ω1(Pf g) = 0 −1 ω2(Pf , g)Ω1(g)Ω1(Pf ) such that ω2(Pf , g) = ω2(T g1, g2) = ω2 (g1, g2), ω2(Pf , g)(dΩ1)(Pf , g) = 1. After this gauge ﬁxing, −1 ω2(g1, T g2) = ω2(g1, g˜2)ω2 (˜g2,T )ω2(g1g˜2,T ). we have (g 6= Pf ) And the constraint (C13) becomes ω2(Pf , g) = 1,

ω2(g, Pf ) = σg. ω2(g1, g2)ω2(˜g1, g˜2) = σg1 σg2 σg1g2 . 30

It can be proved that σg1 σg2 σg1g2 = 1 and so we have

σg˜1 σg2 σg˜1g2 = 1, so σg carries a Z2 valued linear rep- g1g2 = g2g1, resentation of Hf . After the factor system being decoupled, the reduction of regular projective Reps can be namely, the subgroup H is essentially Abelian. performed following the procedure of type-I anti-unitary We ﬁrst show that ω2(g, g˜) must be real. From equa- groups (slight modiﬁcation may be necessary). tion

−1 −1 (dω2)(˜g, g, g˜) = ω2(g, g˜)ω2 (E, g˜)ω2(˜g, E)ω2 (˜g, g) = 1 Appendix D: Classiﬁcation of some simple anti-unitary groups we obtain ω2(g, g˜) = ω2(˜g, g). On the other hand, from 2 (C13), ω2(g, g˜)ω2(˜g, g) = 1, so we have ω2(g, g˜) = 1, In this section we only discuss type-I anti-unitary which yields the constraint, groups. In subsectionsD1 andD2, we discuss the groups ω2(g, g˜) = ±1. (D3) where all the 2-cocycles can be set as real numbers. In subsectionD3, we give some examples where the 2- Notice that there are no gauge degrees of freedom cocycles cannot be set as real numbers. for ω2(g, g˜), because the value of the coboundary (dΩ1)(g, g˜) = Ω1(g)Ω1(˜g) = 1 has been ﬁxed after (C7). Furthermore, from the relations T 1. Direct product group H × Z2 M(g1)M(g2) = M(g1g2)ω2(g1, g2),

T M(˜g2)M(˜g1) = M(g1g2)ω2(˜g2, g˜1) If the group is G = H × Z2 , then T g = gT and g g˜ = g. and M(g)M(˜g) = ω2(g, g˜), we can obtain From (C13) we have ω2(g1, g2)ω2(˜g2, g˜1) = ω2(g1, g˜1)ω2(g2, g˜2)ω2(g1g2, g1g2). 2 g ω2(g1, g2) = 1 Comparing with (C13), we have ω2(˜g1, g˜2) = which constraints ω2(˜g2, g˜1)ω2(g1, g˜1)ω2(g2, g˜2)ω2(g1g2, gg1g2), or equivalently ω2(g1, g2) = ±1. ω2(g1, g2) = ω2(g2, g1)ω2(g1, g˜1) Owing to the relations (C9) ∼ (C11), we have ×ω2(g2, g˜2)ω2(g1g2, gg1g2). (D4) ω2(T g1, g2) = ω2(g1, T g2) = ω2(T g1, T g2) = ω2(g1, g2), Since H is abelian, it must be a cyclic group or a direct so all the cocycles (and the remaining coboundary de- product of cyclic groups. We will consider the two cases grees of freedom) are fixed as real numbers. separately. T Consequently, the classification of H × Z2 are determined by two parts, ω2(T,T ) = ±1 which is given by 2 T a. When H = ZN H (Z2 ,U(1)), and ω2(g1, g2) = ±1 which is given by 2 H (H,Z2). In other words, We first consider the case H is a cyclic group ZN gen- 2 T 2 T 2 N T T H (H × Z2 ,U(1)) = H (Z2 ,U(1)) × H (H,Z2) erated by g with g = E. Since Z2 o Z2 = Z2 × Z2 , we 2 assume N ≥ 3. = Z2 × H (H,Z2) Suppose that in the projective Rep of H ZT , the Rep = n, with n ≥ 1. o 2 Z2 matrix of g is M(g). We tune the phase factor of M(g) such that T −1 2. Semidirect product group H o Z2 with T g = g T [M(g)]N = 1.

T Obviously, M(gn) is proportional to [M(g)]n by a phase The general case of semidirect product groups H o Z2 is a little complicated. Here we only consider the simple factor. Suppose −1 case where H is abelian and T g = g T , or equivalently M(gn) = [M(g)]nµ(n), g˜ = g−1. Noticing that where µ(n) is an U(1) phase factor and obviously µ(0) = µ(N) = 1. From T g˜1g˜2 = T (˜g1g˜2) = T gg2g1 = (g2g1)T, (D1) M(gm)M(gn) = M(gn)M(gm) on the other hand, = M(g)m+nµ(m)µ(n) m+n T g˜1g˜2 = g1T g˜2 = g1g2T = (g1g2)T, (D2) = M(g )µ(m)µ(n)/µ(m + n) 31 we have owing to (C13) we have µ(m)µ(n) 2m 2m ω (gm, gn) = ω (gn, gm) = . (D5) ω2(g , g˜ ) = 1. 2 2 µ(m + n) Further, from (D5), since gN = E and gN/2 = m n n m N/2 2 N/2 N/2 From ω2(g , g ) = ω2(g , g ) and (D4) we have g˜ , µ(N) = µ(N/2) /ω2(g , g˜ ) = 1, we 2 2 m m n n m+n m+n have µ (N/2) = µ(2Q) = 1(here we have used ω2(g , g˜ )ω2(g , g˜ )ω2(g , g˜ ) = 1 (D6) N/2 N/2 2Q 2Q ω2(g , g˜ ) = ω2(g , g˜ ) = 1), which yields m m From (D3) and (D6), we can see that ω2(g , g˜ ) form a µ(N/2) = µ(2Q) = ±1. Z2 Rep of the group H, namely, they are real numbers 1 and are classified by H (H,Z2). Since there is remaining Z2 coboundary degrees of free- 2Q 2Q 2Q In the following we discuss N=even and N=odd sepa- dom for Ω1(g ) after fixing ω2(T, g ) = ω2(g ,T ) = 1 2Q −1 rately. (see (C3)), we can choose Ω1(g ) = µ (2Q) and rede- 0 2Q 2Q 2Q 2Q 1), N is odd: N = 2Q + 1,Q ∈ Z fine M (g ) = M(g )Ω1(g ) = M(g) such that 1 0 2Q Since H (Z2Q+1,Z2) = Z1, there are no nontrivial so- µ (2Q) = µ(2Q)Ω1(g ) = 1. m m lutions for ω2(g , g˜ ), we can safely set Similar to the discussion in case (1), we can introduce

m m the following conboundary to gauge away µ(m), here we ω2(g , g˜ ) = 1 constrain −2Q < m < 2Q: in the following discussion. 0 m m m M (g ) = M(g )Ω1(g ), Since µ(m)µ(−m) = ω (gm, g˜m) = 1, for each pair of 2 M 0(g−m) = M(g−m)Ω−1(gm) gm, g˜m, only one µ(m) is free. Remembering that for 1 each pair of gm, g˜m we have a free coboundary degrees m −1 with Ω1(g ) = µ (m) such that of freedom, so we can further gauge µ(m) away by intro- 0 m ducing µ (m) = µ(m)Ω1(g ) = 1, 0 −1 m m m 0 m m m µ (−m) = µ(−m)Ω1 (g ) = ω2(g , g˜ ), M (g ) = M(g )Ω1(g ), 0 0 0 −m −m −1 m µ (0) = µ (4Q) = 1, M (g ) = M(g )Ω1 (g ) µ0(2Q) = 1, m −1 with Ω1(g ) = µ (m) for 1 ≤ m ≤ Q, such that then µ0(m) = µ(m)Ω (gm) = 1, 1 0 m 0 n 0 m+n 0 m n 0 −1 m m m M (g )M (g ) = M (g )ω2(g , g ), µ (−m) = µ(−m)Ω1 (g ) = ω2(g , g˜ ) = 1, µ0(0) = µ(0) = 1, where 0 µ (2Q + 1) = µ(2Q + 1) = 1, 0 m n 0 0 0 ω2(g , g ) = µ (m)µ (n)/µ (m + n) then we have with µ0(m ± 4Q) = µ0(m). M 0(gm)M 0(gn) = M 0(gm+n)ω0 (gm, gn), Until now, all the cocycles are set to be real and all 2 the coboundary degrees of freedom are fixed (except for Ω1(T ), which is not important since it does not change 0 m n 0 0 0 ω2(g , g ) = µ (m)µ (n)/µ (m + n), any cocycle values, so it can be set as 1). This means that 2 0 m n 0 m the H part of the 2-cocycles are classified by H (H,Z2). where ω2(g , g ) = 1 is a trivial cocycle. So M (g ) Notice that all the nontrivial 2-cocycles are related to form a linear Rep of H = Z2Q+1. 1 ω2(g, g˜), which is classified by H (H,Z2). So we conclude Until now, all the coboundary degrees of freedom are T that for the group Z4Q o Z2 , fixed (except for Ω1(T ), which is not important since it does not change any cocycle values, so it can be set as 1). 2 T 2 T 2 H (H o Z2 ,U(1)) = H (Z2 ,U(1)) × H (H,Z2) So we conclude that the classification of projective Reps 1 T 2 T = Z2 × H (H,Z2) (D7) of Z2Q+1 o Z2 is purely determined by H (Z2 ,U(1)) = = 2. Z2. Namely, Z2 2 T 2 T H (Z2Q+1 Z ,U(1)) = H (Z ,U(1)) = 2. Remark: When ω2(T,T ) = 1, the nontrivial sign o 2 2 Z m m ω2(g , g˜ ) = −1 stands for a Kramers doublet. The 2), N is even: N = 4Q, Q ∈ Z. reason is the following. Notice that (T gm)2 = E and Since m m m m −1 m m ω2(T g , T g ) = ω2(T g , g˜ T ) = ω2 (g , g˜ ) = −1, M(gm)M(gm)M(˜gm)M(˜gm) m m m m 2m 2m we have = ω2(g , g )ω2(˜g , g˜ )ω2(g , g˜ ) m m 2 m 2 m 2 = ω2(g , g˜ ) = 1, [M(T g )K] = −M[(T g ) ] = −1. 32

m m So the nontrivial sign ω2(g , g˜ ) = −1 corresponds to a Noticing that Kramers doublet. 3), N is even: N = 4Q + 2,Q ∈ Z. f(θ)f(−θ) ∗ ω2(θ, −θ) = = f (θ)f(θ) = 1, The discussion is similar to case (2), except that f(0) µ(N/2) = µ(2Q + 1) is not necessarily ±1, it can be which is under the same condition with the group complex: Z2Q+1.So the classiﬁcation is purely determined by 2 2Q+1 2Q+1 2 T µ (2Q + 1) = ω2(g , g˜ ) = ±1. H (Z2 ,U(1)) = Z2. T However, for the group U(1)×Z2 , we have TUθ = UθT But the conclusion is the same, since we can introduce and the following coboundary for −Q ≤ m ≤ Q: with ∗ 2m+1 −1 2m iNθ −iN θ ∗ Ω1(g ) = µ (2m + 1)µ(2Q + 1) and Ω1(g ) = M(T )Ke f(θ) = M(T )e f (θ)K µ−1(2m) such that = eiNθf(θ)M(T )K,

0 2m+1 µ (2m + 1) = µ(2m + 1)Ω1(g ) = µ(2Q + 1), which means 0 −1 2m+1 µ (−2m − 1) = µ(−2m − 1)Ω1 (g ) ∗ ∗ 2m+1 2m+1 −1 −M(T )N = NM(T ) , f (θ) = f(θ). = ω2(g , g˜ )µ (2Q + 1), 0 0 2m 0 f(θ)f(θ ) µ (2m) = µ(2m)Ω1(g ) = 1, So f(θ) is real f(θ) = ±1, and ω2(θ, θ ) = f(θ+θ0) = 0 −1 2m 2m 2m µ (−2m) = µ(−2m)Ω1 (g ) = ω2(g , g˜ ), ±1. This is consistent with the discussion in section 0 0 0 D 1,where it is shown that all the cocycles ω2(θ, θ ) can µ (0) = µ (4Q + 2) = 1, 2 0 be fixed as real and is classified by H (U(1),Z2). The µ (2Q + 1) = µ(2Q + 1), T complete classification of U(1) × Z2 is then given by then 2 T 2 T 2 2 H (U(1)×Z2 ,U(1)) = H (Z2 ,U(1))×H (U(1),Z2) = Z2. 0 m 0 n 0 m+n 0 m n (D10) M (g )M (g ) = M (g )ω2(g , g ), where

0 m n 0 0 0 ω2(g , g ) = µ (m)µ (n)/µ (m + n) b. When H is a direct product of cyclic groups 0 0 m n with µ (m±(4Q+2)) = µ (m). Notice that all ω2(g , g ) 2 Now suppose the group H = ZM × ZN has more than are real numbers which are classified by H (H,Z2). So, similar to case (2), we conclude that, for the group one generator, noted as h1, h2, .... For each cyclic sub- T group, the argument in the previous section applies, and Z4Q+2 o Z2 , m n m n all cocycles ω2(h1 , h1 ) and ω2(h2 , h2 ) are constrained m 2 T 2 T 2 to be ±1 by introducing gauge transformation Ω1(h ) H (H o Z2 ,U(1)) = H (Z2 ,U(1)) × H (H,Z2) 1 and Ω (hm) as discussed previously. = × H1(H,Z ) (D8) 1 2 Z2 2 In the following we will discuss about the cocycles = 2. m n Z2 ω2(h1 , h2 ). In the coboundary Remark: When approaching U(1) by taking the limit n m m n −1 n m (dΩ1)(h2 , h1 ) = Ω1(h1 )Ω1(h2 )Ω1 (h2 h1 ), limN→∞ ZN = U(1), the result depends on if N is even 2 n m n m n m or odd. Actually, careful analysis shows that H (U(1) o if we set Ω1(h2 h1 ) = Ω1(h2 )Ω1(h1 )ω2(h2 , h1 ), where T n m Z2 ,U(1)) = Z2. The reason is the following. the values of Ω1(h2 ) and Ω1(h1 ) have been fixed in previ- n m Suppose that the generator of projective Rep of U(1) is ous gauge fixing processes, then ω2(h2 , h1 ) can be fixed: an Hermitian matrix N, then we can assume 0 n m n m n m ω2(h2 , h1 ) = ω2(h2 , h1 )(dΩ1)(h2 , h1 ) = 1. iNθ M(Uθ) = e f(θ) (D9) From Eqs. (D4) and (D3), we have where Uθ, θ ∈ [0, 2π) is an element in U(1), and |f(θ)| = m n n m m ˜m n ñ 1 to ensure that M(Uθ) is unitary. Obviously f(0) = ω2(h1 , h2 ) = ω2(h2 , h1 )ω2(h1 , h1 )ω2(h2 , h2 ) 1. We further assume that T is represented by M(T )K, m n m n ·ω2(h1 h2 , h^1 h2 ) then the gauge condition ω2(T, θ) = ω2(−θ, T ) = 1 and m ˜m n ñ m n m n TUθ = U−θT indicate that = ω2(h1 , h1 )ω2(h2 , h2 )ω2(h1 h2 , h^1 h2 ) ∗ = ±1. M(T )KeiNθf(θ) = M(T )e−iN θf ∗(θ)K −iNθ m n = e f(−θ)M(T )K. Thus all the components with the form ω2(h1 , h2 ) and m n ω2(h2 , h1 ) are set to be real numbers. Above equation is true for arbitrary θ, so we have m n p Owing to the cocycle equations (dω2)(h1 , h2 , h2) = n p m n+p ∗ ∗ ω2(h2 ,h2 )ω2(h1 ,h2 ) m n p M(T )N = NM(T ) , f (θ) = f(−θ). m n p m n = 1, we have ω2(h1 h2 , h2) = ω2(h1 h2 ,h2 )ω2(h1 ,h2 ) 33

n p m n+p ω2(h2 ,h2 )ω2(h1 ,h2 ) m n , which is a real number. Simi- which satisfies ω2(h1 ,h2 ) larly, owning to cocycle equations, all the components q p 0 D m n ω2(g1, g2) = 1. (E3) ω2(h1 h2 , h1h2) take values ±1 under above gauge fixing. Now it is clear that the cocycles ω2(g1, g2) (where 2 g1, g2 ∈ H) are classified by H (H,Z2). This conclusion Now we define the period P of a factor system can be generalized to the case where H is a direct product ω2(g1, g2) as the smallest positive integer number such P of more cyrclic groups H = ZM × ZN × .... that ω2(g1, g2) is a trivial class of 2-cocycle. P can be In conclusion, the complete classification is given by read out from the group structure of the classification of 2-cocycles, i.e. the second group cohomology of G[54]. 2 T 2 T 2 H (H o Z2 ,U(1)) = H (Z2 ,U(1)) × H (H,Z2) For example, if the second group cohomology of G is 2 H2(G, U(1)) = × , and the cocycle ω (g , g ) be- = Z2 × H (H,Z2) (D11) Zm Zn 2 1 2 longs to the (a, b)th class, where a, b ∈ Z and 1 ≤ a ≤ m and 1 ≤ b ≤ n and the (m, n)th class stands for the 3. The case of complex cocycles trivial class. Then the period of cocycle ω2(g1, g2) is

T [a, m] [b, n] For a general group G = H oZ2 , the cocycles may not P = , , 2 be tuned into real numbers. For example, H (Z3 × (Z3 o a b T Z2 ),U(1)) = Z6, 2 classes of the cocycles can be set as real numbers, and the remaining 4 classes are complex where [a, m] is the least common multiple of a and m. numbers. From the relation (E3), we conclude that the dimension T It can be shown that for G = Zm × (Zn o Z2 ), D of any projective Rep with factor system ω2(g1, g2) must be integer times of P , 2 2 2 H (G, U(1)) = H (Zm,Z2) × H (Zn,Z2) 2 2 T ×H (H,U(1)) × H (Z2 ,U(1)), D = NP, where H = Zm × Zn. where N ∈ Z and N ≥ 1.

Appendix E: Proof of some statements 2. Completeness of irreducible bases 1. The Dimension of irreducible projective Rep is a multiple of the period of its factor system Since

nν For a symmetry group G, its irreducible projective Rep (ν) X (ν) |gri =g ˆ|α i = M(g) K |α i, (E4) matrix is M(g), then i ji s(g) j j=1 M(g )M (g ) = ω (g , g )M(g g ). (E1) 1 s(g1) 2 2 1 2 1 2 P r r the complete relation g |g ihg | ∝ I can be written as We denote the determinant of Rep matrix as following nν det[M(g )] = ε , X X ν ν † 1 1 M(g)ki|αkihαj |M (g)ij ∝ 1, (E5) det[M(g2)] = ε2, g k,j=1 det[M(g g )] = ε . 1 2 12 which is equivalent to D From (E1) we have ε1 · (ε2)s(g1) = ω2 (g1, g2)ε12, where ∗ X † (ε2)s(g1) = ε2 if g1 is unitary and (ε2)s(g1) = ε2 if g1 is M(g)kiM (g)ij ∝ δkj, (E6) anti-unitary. This yields g

1 D i 2πn ε1 · (ε2)s(g1) where we assume M(g) represents equivalent irreducible ω2(g1, g2) = e D × , ε12 unitary projective Reps. To prove (E6), we introduce the matrix where D is the lowest dimension of the irreducible projective Reps corresponding to the factor system ω (g , g ) X † 2 1 2 Y = M(g)Ks(g)XKs(g)M (g) and n = 0, 1, ··· ,D − 1. We may choose a cobound- g 1 h i D ε12 X † ary Ω2(g1, g2) = to get a diﬀerent factor = M(g)XM (g), (E7) ε1·(ε2)s(g1) system g

2πn 0 i D ω2(g1, g2) = Ω2(g1, g2)ω2(g1, g2) = e , (E2) where the matrix X is real. 34

X −is(g )θ (g−1,g ) is(g g )θ (g ,g ) For any gn ∈ G, from (17), the matrix Y satisﬁes = e 3 2 2 2 e 2 3 2 1 2

g2∈G M(gn)Ks(g )Y −1 −1 n is(g3)θ2(g2 ,g1g2) iθ2(g3,g2 g1g2) −1 ×e e |g3g2 g1g2i, X † = M(gn)Ks(gn)M(g)Ks(g)XKs(g)M (g) −1 −1 −1 g d is(g3)θ2(g2 ,g2) here we have used g2 |g3i = e g¯ˆ2 |g3i. X † On the other hand, from = M(gng)ω2(gn, g)Ks(gng)XKs(g)M (g)Ks(gn) g −1 −1 −1 iθ2(g ,g2) gdgˆ2 = M(g )M (g2) = e 2 , gˆ2 = M(g2)K , † 2 2 s(g2) s(g2) ·M (gn)M(gn)Ks(gn) X we get = M(gng)ω2(gn, g)Ks(gng)X −1 −1 −1 −iθ2(g ,g2) −1 g gˆ = M (g2)K = e 2 M(g )K 2 s(g2) s(g2) 2 s(g2) † ·[M(g )K M(g)K ] M(g )K −iθ (g−1,g ) −1 n s(gn) s(g) n s(gn) = e 2 2 2 gd. X 2 = M(g g)ω (g , g)K X n 2 n s(gng) So, we have g   † ·[ω2(gn, g)M(gng)Ks(gng)] M(gn)Ks(gn) X −1 C1|g3i =  gˆ2 gˆ1gˆ2 gˆ3|Ei X ∗ = M(gng)ω2(gn, g)Ks(gng)XKs(gng)ω2 (gn, g) g2∈G g   † X ·M (gng)M(gn)Ks(g ) −1 n =g ˆ3  gˆ2 gˆ1gˆ2 |Ei X † g2∈G = M(gng)Ks(gng)XKs(gng)M (gng) · M(gn)Ks(gn) g   X −iθ (g−1,g ) −1 =g ˆ e 2 2 2 gdgˆ gˆ |Ei = YM(gn)Ks(gn), (E8) 3  2 1 2 g2∈G which means [ˆgn,Y ] = 0. X −1 −1 −is(g3)θ2(g2 ,g2) iθ2(g3,g2 g1g2) Supposing X is a diagonal matrix with only one = e e g2∈G nonzero element Xii = 1, then from Eq. (E7), Y is an −1 is(g3)θ2(g2 ,g1g2) is(g2g3)θ2(g1,g2) −1 Hermitian matrix with ×e e |g3g2 g1g2i.

X † Therefore, C1 = C¯1. Ykj = M(g)kiM (g)ij, (2) i(CH −CH ) = (C¯H −C¯H )S, where S = i(P −P ). g i ˜i i ˜i u a Under the gauge transformation Ω1(g) = π i (δg,g −δg,g˜ ) e 2 1 1 (g1 is unitary), from the relations Since the projective representation M(gn)Ks(gn) is ir- (23) and (24), the regular projective Reps will be reducible, according to the generalized Schur lemma, Y changed into must be a constant matrix, i.e. Y = λI, where λ is a real † number. Therefore, Ykj = λδkj, and the relation (E6) is M(g1) → U iM(g1)U, † proved. M(˜g1) → −U iM(˜g1)U, † † M(¯g1) → U M(¯g1)[i(Pu − Pa)]U = U M(¯g1)SU, M(g˜¯ ) → U †M(g˜¯ )[−i(P − P )]U = −U †M(g˜¯ )SU, 3. Two relations between class operators of 1 1 u a 1 † anti-unitary groups M(g2) → U M(g2)U, g2 6= g1, g˜1 † M(¯g2) → U M(¯g2)U, g2 6= g1, g˜1 Here we prove the following two relations between the where Ugi,gj = δgi,gj Ω1(gi) is a diagonal matrix. class operators. H ¯ Consequently, the class operators C1 = C1 + (1) Ci = Ci for anti-unitary groups, where Ci is the CH , C¯ = C¯H + C¯H are transformed into 1˜ 1 1 1˜ class operator corresponding to the unitary element gi. † H H Suppose G is anti-unitary, g3 ∈ G is an arbitrary ele- C1 → U i(C1 − C1˜ )U, ment, and C1 is a class operator corresponding to a unitary elementg ˆ1. Then ¯ † ¯H ¯H C1 → U (C1 − C1˜ )SU.   After the gauge transformation, above two quantities X C¯ |g i = g¯ˆ g¯ˆ g¯ˆ−1 g¯ˆ |Ei are still equal, namely, U †i(CH − CH )U = U †(C¯H − 1 3  2 1 2  3 1 1˜ 1 H g2∈G C¯ )SU. This yields 1˜ X −is(g )θ (g−1,g ) −1 3 2 2 2 ˆ ˆ d ˆ H H ¯H ¯H = e (g¯2g¯1)g2 g¯3 |Ei i(C1 − C1˜ ) = (C1 − C1˜ )S. g2∈G 35

i 2π TABLE III: Irreducible linear Reps of some anti-unitary groups. The symbols σx,y,z are Pauli matrices, and ω = e 3 , Nuc is the number of independent unitary class operators, namely, half of the total number of class operators. The multiplicity is the number of times each irreducible Rep occurs in the regular Rep. T Z2 × Z2 TP multiplicity (Nuc = 2) 1K 1 2 1K −1 2 T Z4 T (Nuc = 2) 1K 2 σyK 1 T Z4 × Z2 TP (Nuc = 3) 1K 1 2 1K −1 2 σxK iσz 2 T Z4 o Z2 TP (Nuc = 4) 1K i 2 1K −1 2 1K −i 2 1K 1 2 T Z3 × (Z3 o Z2 ) ' TPQ T (Z3 × Z3) o Z2 (Nuc = 6 ) 1K 1 1 2 1K 1 ω 2 1K 1 ω2 2 ω 0 σ K I 2 x 0 ω2 ω 0 σ K ωI 2 x 0 ω2 ω 0 σ K ω2I 2 x 0 ω2 T T T A4 o Z2 ' S4 ' Td T = (12)KP = (123) Q = (124) (Nuc = 4 ) 1K 1 1 2 1K ω2 ω 2 1K ω ω2 2  1 − 1   1 0 0   1 0 0  −1 2ω 2 2ω 2 2 1 1 − 1 0 1 0 K 0 ω 0  2 2  6     3  2ω ω 2  − 1 1 0 0 1 0 0 ω 2ω 2 2 ω 2

(ν) Appendix F: Reduction of regular projective Reps where χg is the character of element g ∈ G. This result also holds for projective Reps. In this section, we use the procedure in sectionIIIB In the following, we will illustrate the reduction of reg- to reduce the regular projective Reps of some important ular projective Reps via an example Z2 × Z2 × Z2. The groups, including anti-unitary groups. We only present Abelian group Z2 × Z2 × Z2 has eight group elements, its results of generators. canonical subgroup chain is Z2 ×Z2 ×Z2 ⊃ Z2 ×Z2 ⊃ Z2. From 2-cocycle and 2-coboundary equations (12),(13), we ﬁnd there are seven classes of 8-dimensional non-trivial regular projective Reps of Z2 ×Z2 ×Z2. We only present 1. Unitary groups how to reduce one class (+1, −1, −1) . The nontrivial cocycle solutions are

For unitary groups, it is known that if a Rep (ν) is ω (P,P ) = ω (P,Q) = ω (P,R) = ω (P, P QR) = −1, irreducible, then its characters must satisfy 2 2 2 2 ω2(Q, P ) = ω2(Q, Q) = ω2(Q, R) = ω2(Q, P QR) = −1,

X (ν) 2 ω2(PR,P ) = ω2(PR,Q) = ω2(PR,R) |χg | = G, g∈G = ω2(P R, P QR) = −1, 36

  ω2(QR, P ) = ω2(QR, Q) = ω2(QR, R) 0 0 −4 0 0 0 0 0  0 0 0 4 0 0 0 0  = ω2(QR, P QR) = −1.    4 0 0 0 0 0 0 0  The regular projective Rep matrices of three generators  0 −4 0 0 0 0 0 0  C¯(s ) =   , of the group are 1  0 0 0 0 0 0 −4 0     0 0 0 0 0 0 0 4   0 −1 0 0 0 0 0 0     0 0 0 0 4 0 0 0   1 0 0 0 0 0 0 0    0 0 0 0 0 −4 0 0  0 0 0 1 0 0 0 0   0 0 −1 0 0 0 0 0   0 −2 0 0 0 0 0 0  M(P ) =   ,  0 0 0 0 0 1 0 0   2 0 0 0 0 0 0 0       0 0 0 0 −1 0 0 0   0 0 0 2 0 0 0 0     0 0 −2 0 0 0 0 0   0 0 0 0 0 0 0 −1  C¯(s ) =   . 2  0 0 0 0 0 −2 0 0  0 0 0 0 0 0 1 0    0 0 0 0 2 0 0 0     0 0 −1 0 0 0 0 0   0 0 0 0 0 0 0 2  0 0 0 0 0 0 −2 0  0 0 0 1 0 0 0 0     1 0 0 0 0 0 0 0   0 −1 0 0 0 0 0 0  M(Q) =   ,  0 0 0 0 0 0 1 0    Practically, we can diagonalize the linear combination  0 0 0 0 0 0 0 −1  ¯ ¯   Ob = C + aC(s1) + bC(s2) + cC(s1) + dC(s2), here a,b,c  0 0 0 0 −1 0 0 0  and d are arbitrary nonzero real constants ensuring all 0 0 0 0 0 1 0 0 the eigenvalues of Ob are non-degenerate. From the or-  0 0 0 0 1 0 0 0  thonormal eigenvectors (column vectors) of Ob, we can  0 0 0 0 0 1 0 0  obtain a unitary transformation matrix    0 0 0 0 0 0 1 0   0 0 0 0 0 0 0 1  M(R) =   .  1 0 0 0 0 0 0 0   1 1 1 1    2 0 2 i 0 − 2 i 0 0 2 i  0 1 0 0 0 0 0 0  1 1 1 1    − 2 i 0 − 2 0 2 0 0 2   0 0 1 0 0 0 0 0   1 1 1 1   2 i 0 2 0 2 0 0 2  0 0 0 1 0 0 0 0  1 1 1 1   − 2 0 − 2 i 0 − 2 i 0 0 2 i  U =  1 1 1 1  . (F1) From Eqs. (29) and (30), the matrices of all the class  0 2 i 0 − 2 i 0 2 2 i 0   1 1 1 1  operators are given below:  0 2 0 2 0 2 i 2 0   0 − 1 0 − 1 0 1 i 1 0   0 0 0 1 0 0 0 0   2 2 2 2  0 − 1 i 0 1 i 0 1 1 i 0  0 0 1 0 0 0 0 0  2 2 2 2    0 1 0 0 0 0 0 0   1 0 0 0 0 0 0 0  C =   ,  0 0 0 0 0 0 0 1    The above three generators can be transformed  0 0 0 0 0 0 1 0  into,respectively:    0 0 0 0 0 1 0 0  0 0 0 0 1 0 0 0  0 0 −4 0 0 0 0 0   i 0 0 0 0 0 0 0   0 0 0 4 0 0 0 0   0 −i 0 0 0 0 0 0       4 0 0 0 0 0 0 0   0 0 −i 0 0 0 0 0   0 −4 0 0 0 0 0 0   0 0 0 i 0 0 0 0  C(s ) =   , U †M(P )U =   , (F2) 1  0 0 0 0 0 0 4 0   0 0 0 0 −i 0 0 0       0 0 0 0 0 0 0 −4   0 0 0 0 0 i 0 0       0 0 0 0 −4 0 0 0   0 0 0 0 0 0 −i 0  0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 i  0 −2 0 0 0 0 0 0   −i 0 0 0 0 0 0 0   2 0 0 0 0 0 0 0   0 i 0 0 0 0 0 0       0 0 0 2 0 0 0 0   0 0 i 0 0 0 0 0   0 0 −2 0 0 0 0 0   0 0 0 −i 0 0 0 0  C(s ) =   , U †M(Q)U =   , (F3) 2  0 0 0 0 0 2 0 0   0 0 0 0 −i 0 0 0       0 0 0 0 −2 0 0 0   0 0 0 0 0 i 0 0       0 0 0 0 0 0 0 −2   0 0 0 0 0 0 −i 0  0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 i 37

 0 i 0 0 0 0 0 0  the projective Rep of class (+1, −1, +1, +1) of the group −i 0 0 0 0 0 0 0   Z × Z × ZT is reducible though the eigenvalues of Tb  0 0 0 −1 0 0 0 0  2 2 2   are complex numbers.  0 0 −1 0 0 0 0 0  U †M(R)U =   . (F4)  0 0 0 0 0 i 0 0     0 0 0 0 −i 0 0 0  T   a. Abelian anti-unitary group Z2 × Z2 × Z2  0 0 0 0 0 0 0 1  0 0 0 0 0 0 1 0 T As for the Abelian group Z2 × Z2 × Z2 = {E,P } × {E,Q} × {E,T }, where T 2 = E. Solving 2-cocycle and 2. Anti-unitary groups 2-coboundary equations (12),(13), there are 15 classes of 8-dimensional projective Reps of this group. Because the group contains anti-unitary time reversal transformation, For anti-unitary groups, as mentioned in section we have the following mapping: III B 2, there is a one-to-one correspondence between the anti-unitary class operator Ci±T and the unitary class T → M(T )K,P → M(P ) ,Q → M(Q), (F5) operator Ci± , where the unitary class operators can be obtained solely from the unitary normal subgroup H by where K is the complex-conjugate operator deﬁned in the relation (36) and (37). Moreover, the anti-unitary sectionI. The normal subgroup H is D2 = Z2 × Z2. −1 class operators do not provide meaningful quantum num- Since T giT =g ˜i = gi, it is clear that Ci− = 0 from P bers. So the independent class operators include only (37). Therefore, we can only use C = i kiCi+ to con- the unitary class operators Ci+ and Ci− . We may use struct CSCO-I. The CSCO-II are (C,C(s)), where C(s) P 0 are class operators of the subgroup Z ⊂ H. We discuss their linear combination C = i(kiCi+ + kiCi− ) to form 2 0 CSCO-I of the regular projective Rep, where ki and ki only two classes which can be reduced into 2-dimensional are arbitrary nonzero real constants. Rep. For the class (+1, +1, +1, −1) in TableI, the non- Notice that the characters for the unitary subgroup H trivial cocycle solutions in the decoupled factor systems are still good quantities. Depending on the Rep matrices (33) are of the normal subgroup H, there are three diﬀerent sit- uations for the irreducible Rep (ν) of anti-unitary group ω2(P,P ) = ω2(P,Q) = ω2(P,PT ) = ω2(P, QT ) = −1, G: ω2(Q, P ) = ω2(Q, Q) = ω2(Q, P T ) = ω2(Q, QT ) = −1,

1) the induced Rep of H in (ν) [we will note it as (νH )] ω2(T,T ) = ω2(T,PT ) = ω2(T, QT ) = ω2(T, P QT ) = −1, is irreducible. In this case, the characters of H satisfy, ω2(PT,P ) = ω2(PT,Q) = ω2(PT,T )

X (ν) 2 = ω2(P T, P QT ) = −1, |χg | = H, g∈H ω2(QT, P ) = ω2(QT, Q) = ω2(QT, T )

= ω2(QT, P QT ) = −1, where H is the order the normal subgroup H; 2) the induced Rep of H in (ν) is a direct sum of the ω2(P QT, T ) = ω2(P QT, P T ) = ω2(P QT, QT ) induced irreducible linear Rep (νH ) and its complex con- = ω2(P QT, P QT ) = −1. ∗ ∗ jugate (νH ), where (νH ) is not equivalent to (νH ), then the characters of H satisfy, The regular projective Rep matrices of these generators are given below: X (ν) 2 |χg | = 2H;  0 0 0 0 −1 0 0 0  g∈H  0 0 0 0 0 −1 0 0     0 0 0 0 0 0 −1 0  3) the Rep matrices of H in (ν) is a direct sum of two  0 0 0 0 0 0 0 −1  M(T )K =   K, copies of the induced irreducible Rep (νH ) [in this case  1 0 0 0 0 0 0 0    all the characters of (νH ) are real], then the characters  0 1 0 0 0 0 0 0    of H satisfy,  0 0 1 0 0 0 0 0  X (ν) 2 0 0 0 1 0 0 0 0 |χg | = 4H. g∈H  0 −1 0 0 0 0 0 0  In case 3), the eigenvalues of the unitary class oper-  1 0 0 0 0 0 0 0    ators in CSCO-I are real, and there exists at least one  0 0 0 1 0 0 0 0  intrinsic class operator of anti-unitary elements in CSCO-  0 0 −1 0 0 0 0 0  M(P ) =   , III whose eigenvalues are complex numbers. It should be  0 0 0 0 0 −1 0 0    carefully checked if there exists another set of CSCO-III  0 0 0 0 1 0 0 0    where all the eigenvalues are real; if true, then (ν) can  0 0 0 0 0 0 0 1  be further reduced. For example, as shown in Sec.F 2 a, 0 0 0 0 0 0 −1 0 38

 0 0 −1 0 0 0 0 0  The above Rep matrices of three generators can be trans-  0 0 0 1 0 0 0 0  formed into:  1 0 0 0 0 0 0 0     0 −i 0 0 0 0 0 0   0 −1 0 0 0 0 0 0  M(Q) =   . i 0 0 0 0 0 0 0  0 0 0 0 0 0 −1 0       0 0 0 −i 0 0 0 0   0 0 0 0 0 0 0 1       0 0 i 0 0 0 0 0  0 0 0 0 1 0 0 0 U †M(T )KU =   K,    0 0 0 0 0 −i 0 0  0 0 0 0 0 −1 0 0    0 0 0 0 i 0 0 0    When gi = E, P, Q, P Q, we can get 2 independent class  0 0 0 0 0 0 0 −i  P2 0 0 0 0 0 0 i 0 operators Ci+ , then C = i=1 kiCi+ . The class operator of subgroup Z2 = {E,P } ⊂ H can be calculated from (F7) Eq.(30):  i 0 0 0 0 0 0 0   0 −1 0 0 0 0 0 0   0 −i 0 0 0 0 0 0   0 0 −i 0 0 0 0 0   1 0 0 0 0 0 0 0       0 0 0 i 0 0 0 0   0 0 0 1 0 0 0 0  U †M(P )U =   , (F8)  0 0 −1 0 0 0 0 0   0 0 0 0 i 0 0 0  C(s) = 2   .    0 0 0 0 0 −1 0 0   0 0 0 0 0 −i 0 0       0 0 0 0 1 0 0 0   0 0 0 0 0 0 −i 0     0 0 0 0 0 0 0 1  0 0 0 0 0 0 0 i 0 0 0 0 0 0 −1 0  −i 0 0 0 0 0 0 0   0 i 0 0 0 0 0 0  As emphasized in section IIIB2, we can adopt Tb and    0 0 i 0 0 0 0 0  the class operator of intrinsic subgroup Z2 = {E, P } as  0 0 0 −i 0 0 0 0  U †M(Q)U =   . (F9) members of C(s0) because of the accidental symmetry  0 0 0 0 i 0 0 0     0 0 0 0 0 −i 0 0  [T,b Pb] = 0:    0 0 0 0 0 0 −i 0   0 −1 0 0 0 0 0 0  0 0 0 0 0 0 0 i  1 0 0 0 0 0 0 0    Now we consider another class (+1, −1, +1, +1), under  0 0 0 1 0 0 0 0   0 0 −1 0 0 0 0 0  the decoupled factor systems (33), the nontrivial cocycle C = 2   , P  0 0 0 0 0 −1 0 0  solutions are    0 0 0 0 1 0 0 0    ω2(P,P ) = ω2(P,PQ) = ω2(P,PT ) = ω2(P, P QT ) = −1,  0 0 0 0 0 0 0 1  ω (Q, P ) = ω (Q, Q) = ω (Q, P T ) = ω (Q, QT ) = −1, 0 0 0 0 0 0 −1 0 2 2 2 2 ω (P Q, Q) = ω (P Q, P Q) = ω (P Q, QT )  0 0 0 0 −1 0 0 0  2 2 2 = ω2(P Q, P QT ) = −1,  0 0 0 0 0 −1 0 0    ω (T,T ) = ω (T,PT ) = ω (T, QT )  0 0 0 0 0 0 −1 0  2 2 2  0 0 0 0 0 0 0 −1  M(T ) =   . = ω2(T, P QT ) = −1,  1 0 0 0 0 0 0 0    ω (PT,P ) = ω (PT,PQ) = ω (PT,T )  0 1 0 0 0 0 0 0  2 2 2    0 0 1 0 0 0 0 0  = ω2(P T, QT ) = −1, 0 0 0 1 0 0 0 0 ω2(QT, P ) = ω2(QT, Q) = ω2(QT, T ) We need to diagonalize the linear combination of above = ω2(QT, P QT ) = −1, ω2(P QT, T ) = ω2(P QT, P T ) = ω2(P QT, Q) class operators Ob = C + aC(s) + bCP + cM(T ), where a, b, c are arbitrary nonzero real constants. The eigenval- = ω2(P QT, P Q) = −1. ues of operator Ob are non-degenerate, thus the orthonor- The regular projective Rep matrices of these generators mal eigenvectors constitute the unitary transformation can be obtained: matrix  0 0 0 0 −1 0 0 0   −i i −i i −1 −1 −i i   0 0 0 0 0 −1 0 0   −1 −1 1 1 i −i 1 1   0 0 0 0 0 0 −1 0      √  1 1 −1 −1 i −i 1 1   0 0 0 0 0 0 0 −1  2  i −i i −i −1 −1 −i i  M(T )K =   K, U =   . (F6)  1 0 0 0 0 0 0 0   1 1 1 1 −i i 1 1    4    0 1 0 0 0 0 0 0   −i i i −i −1 −1 i −i       0 0 1 0 0 0 0 0   i −i −i i −1 −1 i −i  0 0 0 1 0 0 0 0 −1 −1 −1 −1 −i i 1 1 39

 0 −1 0 0 0 0 0 0  However, we can choose another set of class operators  1 0 0 0 0 0 0 0  such that the eigenvalues are all real numbers. Noticing  0 0 0 −1 0 0 0 0    P[ QT anti-commutes with M(PT ), we adopt the follow-  0 0 1 0 0 0 0 0  M(P ) =   , ing ‘class’ operators:  0 0 0 0 0 −1 0 0     0 0 0 0 1 0 0 0     0 0 0 0 0 1 0 0   0 0 0 0 0 0 0 −1   0 0 0 0 −1 0 0 0  0 0 0 0 0 0 1 0    0 0 0 0 0 0 0 −1   0 0 −1 0 0 0 0 0   0 0 0 0 0 0 1 0  M(PT ) =   , 0 0 0 1 0 0 0 0  0 −1 0 0 0 0 0 0       1 0 0 0 0 0 0 0   1 0 0 0 0 0 0 0       0 −1 0 0 0 0 0 0  0 0 0 1 0 0 0 0 M(Q) =   .    0 0 0 0 0 0 −1 0    0 0 −1 0 0 0 0 0  0 0 0 0 0 0 0 1     0 0 0 0 0 0 0 i   0 0 0 0 1 0 0 0   0 0 0 0 0 0 −i 0  0 0 0 0 0 −1 0 0    0 0 0 0 0 i 0 0   0 0 0 0 −i 0 0 0  When g = E, P, Q, P Q, we can get only one inde- SM(P QT ) =   i  0 0 0 i 0 0 0 0  pendent class operator C + = E. The class operator   i  0 0 −i 0 0 0 0 0  of subgroup Z = {E,P } ⊂ H can be calculated from   2 0 i 0 0 0 0 0 0 Eq.(30):   −i 0 0 0 0 0 0 0  0 −1 0 0 0 0 0 0  as members of C¯(s0), where S = i(P − P ) =  1 0 0 0 0 0 0 0  u a   diag(i, ...i, −i, ... − i) is a diagonal matrix. Due to the  0 0 0 −1 0 0 0 0   0 0 1 0 0 0 0 0  |{z} | {z } C(s) = 2   . G/2 G/2  0 0 0 0 0 −1 0 0    accidental symmetry [SP[ QT , PTd] = 0 , the above ‘class’  0 0 0 0 1 0 0 0    operators commute with each other (the accidental in-  0 0 0 0 0 0 0 −1  0 0 0 0 0 0 1 0 variant quantity SP[ QT is not a true class operator since it does not commute with Tb). The eigenvalues (quan- 0 As for the class operators of intrinsic subgroup C(s ), if [ we follow the original procedure to use the class operator tum numbers) of aM(PT ) + bSP QT with a = 1, b = 2 are (−1, −1, −3, −3, 1, 1, 3, 3). Thus we obtain four 2- Tb and class operator CP of intrinsic unitary subgroup Z2 dimensional irreducible Reps. Diagonalizing the linear as members of C(s0): combination Ob = Ci+ + aC(s) + bM(PT ) + cSP[ QT with  0 0 0 0 −1 0 0 0  a, b, c arbitrary nonzero real constants, we obtain the unitary transformation matrix,  0 0 0 0 0 −1 0 0   0 0 0 0 0 0 −1 0     −i i 1 −1 i i 1 1   0 0 0 0 0 0 0 −1  M(T ) =   , −1 −1 −i −i −1 1 −i i  1 0 0 0 0 0 0 0       −1 1 i −i 1 1 i i   0 1 0 0 0 0 0 0  √     2  i i 1 1 i −i 1 −1   0 0 1 0 0 0 0 0  U =   .  1 1 −i −i −1 1 i −i  0 0 0 1 0 0 0 0 4    −i i −1 1 −i −i 1 1      0 −1 0 0 0 0 0 0  i i −1 −1 −i i 1 −1   1 0 0 0 0 0 0 0  1 −1 i −i 1 1 −i −i    0 0 0 1 0 0 0 0   0 0 −1 0 0 0 0 0  C = 2   , The above Rep matrices of three generators can be trans- P  0 0 0 0 0 −1 0 0    formed into irreducible Reps with U:  0 0 0 0 1 0 0 0     0 0 0 0 0 0 0 1   0 −i 0 0 0 0 0 0  0 0 0 0 0 0 −1 0  i 0 0 0 0 0 0 0     0 0 0 −i 0 0 0 0  the eigenvalues (quantum numbers) of aM(T )+bC with  0 0 i 0 0 0 0 0  P U †M(T )KU =   K, a = 2, b = 0.5 are (−i, −i, i, i, −3i, −3i, 3i, 3i). From  0 0 0 0 0 i 0 0    these quantum numbers, it seems that there are two 4-  0 0 0 0 −i 0 0 0    dimensional irreducible Reps, one is labeled by −i, −i, i, i  0 0 0 0 0 0 0 −i  and the other by −3i, −3i, 3i, 3i. 0 0 0 0 0 0 i 0 40

 i 0 0 0 0 0 0 0  permutation operations (including identity element E)  0 −i 0 0 0 0 0 0  form a unitary normal subgroup H = A4.    0 0 i 0 0 0 0 0  By solving 2-cocycle and 2-coboundary equations  0 0 0 −i 0 0 0 0  U †M(P )U =   , (12),(13), we ﬁnd there are 3 classes nontrivial projective  0 0 0 0 −i 0 0 0    Reps of this group. Under the decoupled factor systems  0 0 0 0 0 i 0 0    (33), the 24-dimensional regular projective Rep matrices  0 0 0 0 0 0 i 0  of all the group elements can be obtained. 0 0 0 0 0 0 0 −i The unitary independent class operators Ci+ and Ci−  0 −i 0 0 0 0 0 0  can be calculated from (36) and (37), where gi ∈ H = A4. −i 0 0 0 0 0 0 0 P 0   The linear combination C = (k C + +k C − ) is exactly   i i i i i  0 0 0 i 0 0 0 0  the CSCO-I. Together with the class operators C(s) of  0 0 i 0 0 0 0 0  U †M(Q)U =   . the subgroup Z3 generated by P = (123), we obtain the  0 0 0 0 0 i 0 0    CSCO-II (C,C(s)).  0 0 0 0 i 0 0 0    For the intrinsic subgroups, we adopt the anti-unitary  0 0 0 0 0 0 0 −i  0 0 0 0 0 0 −i 0 class operator Tb and the class operators [of the form (46)] T T of the intrinsic subgroup S3 = Z3 o Z2 with two generators P = (123), T = (12)K. It can be veriﬁed that all T b. Non-Abelian anti-unitary group S4 the class operators commute with each other. With above CSCO-III, we can completely reduce the T T T For the group A4 o Z2 ' S4 ' Td , all the odd- regular irreducible Reps. The results about three gener- permutation operations are anti-unitary, and all the even- ators are given in TableI.

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