CPT Theorem and Classification of Topological Insulators And

arXiv:1406.0307v1 [cond-mat.str-el] 2 Jun 2014 I.Casfiaino PsmercTsi arbitrary in TIs symmetric CP of Classification III. V oooia P hoe n topological and theorem CPT Topological IV. I Dfrinctplgclpae rtce by protected phases topological fermionic 2D II. .Casfiaino DitrcigSTphases: SPT interacting 2D of Classification V. .Introduction I. symmetries dimension topological of T-duality and CPT-equivalence D. .BGsseswt pnU1 n P and U(1) spin with systems BdG D. -arxfruain12 10 formulation K-matrix classes symmetry CPT-equivalent .Otieadmi eut 3 results main and Outline A. .C ymti nuaos4 insulators symmetric CP A. .Casfiainb -hoy7 K-theory by Classification A. .Bl n deKmti hoisincorporated theories K-matrix edge and Bulk A. .Cneto oTsmercisltr 6 insulators symmetric T to Connection C. .Tplgclivrat 8 invariants Topological C. .Casfiaino P hssb K-matrix by phases SPT of Classification C. .BGsseswt pnU1 n P and U(1) spin with systems BdG B. .Drcmdl n P hoe 8 theorem CPT and models Dirac B. .Eg tblt rtrafrSTPae 14 Phases SPT for criteria stability Edge B. .Tplgclcasfiaino other of classification Topological E. P hoe n lsicto ftplgclisltr an insulators topological of classification and theorem CPT 1 ymtis5 symmetries hss neape7 example an phases: ymtis10 8 symmetries symmetries ulSTpae 15 phases SPT and dual phases SPT CPT-equivalent thoeries: ihsmere 12 symmetries with .Eg hoy5 4 theory Edge 2. model tight-binding 1. .Kmti lsicto ffrincSPT fermionic of classification K-matrix 2. bosonic of classification K-matrix 1. eateto hsc,Uiest fIlni tUrbana-Ch at Illinois of University Physics, of Department hss17 16 phases phases SPT non-chiral usdbfr r eae yCTtheorem. CPT ro parti by the related in are clarify show, before l also and cussed is We phases, approach topological interactions. second symmetry-protected the of approa K-mat while effects first the dimensions The incorporates of spatial terms theories. general in bulk in classifi other (2+1)-dimensional K-theory the of and of theory symmetries, terms with in theories one co approaches: and parity, complementary particle-hole, time-reversal, by protected epeetasseai oooia lsicto ffermi of classification topological systematic a present We 2 odne atrTer aoaoy IE,Wk,Saitama, Wako, RIKEN, Laboratory, Theory Matter Condensed CONTENTS hn-s Hsieh, Chang-Tse 1 Ωae:Jn ,2014) 3, June (ΩDated: aaioMorimoto, Takahiro 1 4 7 daaial once o ntepeec faset a of states of trivial not presence are topologically the matter. that conditions, in symmetry matter to, of of connected states adiabatically are (TSCs) conductors inae symmetries, Zirnbauer re sc sTSadPSlse nAltland-Zirnbauer in listed PHS and TRS as (such tries discovered. predicted theoretically been have thereof, combinations and (PHS), symmetry particle-hole I Discussion VI. .Etnlmn pcrmadeffective and spectrum Entanglement A. .Pofo oooia P hoe o interacting for theorem CPT topological of Proof C. .Cluain fsmer rnfrain and transformations symmetry of Calculations B. oooia nuaos(I)adtplgclsuper- topological and (TIs) insulators Topological nmr eetyas nepa ewe nst symme- on-site between interplay years, recent more In bnto fteesmere.W s two use We symmetries. these of mbination mag,11 etGenS,Ubn L61801 IL Urbana St, Green West 1110 ampaign, P hssi -arxtere 21 theories K-matrix in phases SPT o-hrlSTpae ntodmnin 27 References dimensions two in phases SPT non-chiral cnweget 19 symmetries Acknowledgments .Dsuso fSTpae 22 22 phases 21 SPT of Discussion elements 3. identity of operators Equations symmetry 2. of relations Algebraic 1. .Etnlmn pcrm20 with spectrum entanglement of Properties 2. spectrum Entanglement 1. 1,2 ua,tplgclsprodcosdis- superconductors topological cular, eo P hoe ncasfiainof classification in theorem CPT of le ymtis21 symmetries .BsncSTpae ihsmer group symmetry with phases SPT Bosonic b. .FrincSTpae ihsymmetries with phases SPT Fermionic d. .BsncSTpae ihsmer group symmetry with phases SPT Bosonic a. .FrincSTpae ihsymmetry with phases SPT Fermionic c. 2 mtdt w pta iesosbut dimensions spatial two to imited hi pcfi ofe emo theories fermion free to specific is ch ncadbsnctplgclphases topological bosonic and onic aino apdqartcfermion quadratic gapped of cation n hne Ryu Shinsei and Z Z {T 13–20 I n Sscaatrzdb Altland- by characterized TSCs and TIs i hoydsrpino h edge the of description theory rix 2 2 CP T , × CP} .INTRODUCTION I. Z 2 C × 5-18 Japan 351-0198, Z 3 2 P iervra ymty(TRS), symmetry time-reversal superconductors d 1 4–12 n experimentally and CP 23 24 28 19 19 26 25 2 symmetry classes) and non-on-site symmetries (such as T sym. TI CPT-equiv. space group symmetries) has enriched the topological CP sym. TCI phases of matters. A novel class of topological matter (QSHE) characterized (additionally) by non-on-site symmetries, such as topological crystalline insulators (TCIs)21 and topological crystalline superconductors (TSCSs), have T-duality T-duality been discovered.22–29 The topological classification, orig- inally studied in the presence/absence of various on-site symmetries in Altland-Zirnbauer classes,10,11,30 are also Chiral sym. TSC CPT-equiv. P sym.TCSC extended to include the non-on-site symmetries such as with Sz conserv. with Sz conserv. reflection symmetry,31,32 inversion symmetry,33,34 and (crystal) point group symmetries,35–38 recently. (a) Motivated by these recent works, in this paper, we further study TIs and TSCs protected by a wider set of symmetries than symmetries in Altland-Zirnbauer classes by T sym. TSC including, in particular, a parity symmetry (PS), which is a symmetry under the reflection of an odd number of spatial coordinates. One of our focuses is, in addition to the cases where parity is conserved, on situations where a CPT-equiv. CPT-equiv. combination of parity with some other symmetries, such as CP (product of PHS and PS) or PT (product of PS CPT-equiv. and TRS), are preserved. For earlier related works, see, P sym. TCSC T and P sym. TCI* for example, Refs. 30, 39–41. Another issue we will discuss in this paper is the effect of interactions on the classification of those topological (b) phases protected by parity and other symmetries (such as combination of parity and other symmetries). It has FIG. 1. Two sets of topological (crystalline) insulators and su- been demonstrated, in various examples, that there are perconductors related by CPT-equivalence and/or T-duality: phases that appear to be topologically distinct from triv- (a) T symmetric TI (QSHE), CP symmetric TCI, chiral (TC) symmetric TSC with S conservation, and P symmetric ial phases at non-interacting level, which, in fact, can z TCSC with S conservation; (b) T symmetric TSC, P sym- adiabatically be deformable into a trivial state of matter z 42–47 metric TCSC, and T and P symmetric TCI that can support in the presence of interactions. For example, Ref. gapless edge states even in the absence of charge U(1) sym- 47 discusses (2+1)-dimensional [or 2 spatial-dimensional metry. (2D)] superconducting systems in the presence of parity and time-reversal symmetries, which are classified, at the quadratic level, by an integer topological invariant, while systems at long wavelength limit, such as the band topol- once inter-particle interactions are included, states with ogy or the electromagnetic response, can be encoded in an integer multiple of eight units of the non-interacting the so-called topological field theory, which respects the topological invariant are shown to be unstable. Focusing Lorentz symmetry. When these topological properties on 2D bulk topological states that support an edge state are protected (or determined) by some symmetry, TRS, described by a K-matrix theory, i.e., Abelian states, we say, they can also be protected solely by CP symme- will study the stability of the edge state (and hence the try, which is a “CPT-equivalent” partner to TRS. For bulk state) in the presence of parity symmetry or parity example, the magnetoelectric effect in 3D time-reversal symmetry combined with other symmetries. symmetric TIs48 is also expected to be observed in a CP We will also show that, once parity symmetry or parity symmetric TI, because they are both described by the ax- symmetry combined with other discrete symmetries is ion term (effective action for electromagnetic response) included into our consideration, CPT theorem plays an with the same nontrivial (quantized) value of θ angle. important role in classifying topological states of matter. In addition, from the prospect of topological classifica- CPT theorem holds in Lorentz invariant quantum field tion, classifying symmetry-protected topological (SPT) theories, which says, C, P, T, when combined into CPT, phases of a free fermion systems [characterized either by is always conserved, i.e., CPT = 1, schematically. For a gapped Hamiltonian of the (d + 1)-dimensional bulk example, a Lorentz invariant CP symmetric field theory or by a gapless Hamiltonian of d-dimensional boundary also possesses TRS, and vice versa. with symmetries] is equivalent to classifying the corre- In condensed matter systems, however, such relations sponding Dirac operators with symmetry restrictions11. between these discrete symmetries (T, C, and P) do not It is thus natural to associate TIs protected by TRS with arise since we are not to be restricted to relativistic sys- TIs protected by CP symmetry, as the Dirac Hamiltonian tems; symmetries can be imposed independently. Never- has a CPT invariant form. theless, some physical properties of these non-relativistic In this paper, by going through classification prob- 3 lems of non-interacting fermions in the presence of vari- As inferred from the CPT theorem and the existence of ous symmetry conditions, and also microscopic stability time-reversal symmetric topological insulators in two di- analysis of interacting edge theories, we will demonstrate mensions, we will show that there are two topologically explicitly such CPT theorem holds at the level of topolog- distinct classes of insulators with CP symmetry in two ical classification for all cases that we studied. Through dimensions, i.e., Z2 classification. We will present in Sec. this analysis, we can see, for example, that 2D TSCs pro- II A a simple example (tight-binding model) of CP sym- tected by spin parity conservation,45,46 and 2D TSCs by metric band insulators, which is constructed from two parity and time-reversal,47 both of which are classified in copies of two-band Chern insulators with opposite chi- terms of Z8, are related by CPT theorem. ralities. While we defer a systematic classification of CP As mentioned above, CPT theorem (i.e., topological symmetric insulators until Sec. III, we discuss the edge classification problems with different set of symmetries theory of the CP symmetric topological insulators, and related by CPT relations) may largely be expected, for perturbations to it such as mass terms. example, once we anticipate description of SPT phases We will also show, in Sec. II B, that systems with CP by an underlying topological field theories. However, symmetry and charge conservation [charge U(1) symme- perhaps more fundamentally, we will also discuss that try] can also be interpreted as BdG systems that preserve while physical Hamiltonians may not obey CPT theorem, parity and one component of SU(2) spin, Sz, say [spin their entanglement Hamiltonians obey a form of CPT U(1) symmetry]. That is, CP symmetric topological in- theorem.33,49,50 sulators can be realized as topological crystalline super- We will also discuss yet another duality relation, “T- conductors with Sz conservation. In terms of edge the- duality”, for a wide range of topological insulators and ories, the relation between CP symmetric insulators and superconductors. T-duality is a duality that exchanges a P symmetric BdG systems is nothing but T-duality or phase field (φ ∼ ϕL + ϕR) and its dual (θ ∼ ϕL − ϕR) in the Kramers-Wannier duality. To the best of our knowl- the (1+1)-dimensional boson theory or in string theory. edge, this topological superconductor protected by spin [Here, φ and θ are the compact boson fields in the (1+1)- U(1) and parity symmetry has not been discussed in the dimensional boson theory and ϕL/R are their left/right- literature. moving parts.] Similarly to CPT theorem, this duality In Sec. II C, following Ref. 33, we make a further relation (and its proper generalization to K-matrix the- connection between CP symmetric insulators and time- ory with multi-component boson fields) relates topologi- reversal (T) symmetric insulators by considering entan- cal classification of (2+1)-dimensional fermionic systems glement Hamiltonians and effective symmetries thereof. with CP symmetry and charge U(1) symmetry to topo- Then we introduce the ideas of CPT-equivalence and T- logical classification with parity symmetry and spin U(1) duality of topological phases in Sec. II D, taking the symmetry. The latter system is a Bogoliubov-de Genne CP symmetric TI and its related systems [shown in FIG. (BdG) system with conserved parity and spin U(1) sym- 1(a)] as an example that shows such equivalence. metry. Therefore, topological classification of (i) time- In Sec. III, we use K-theory to classify noninteracting reversal symmetric insulators conserving charge U(1) CP symmetric TI in arbitrary dimensions. We found the (the quantum spin Hall effect), (ii) CP-symmetric insula- topological classification of this symmetry class is exactly tors with conserving charge U(1), (iii) parity-symmetric the same as that of the symmetry class possessing TRS. BdG systems with conserved spin U(1), and (iv) TC- An explicit construction of the ”effective” TRS opera- symmetric BdG system with conserved spin U(1), are all tor from the Dirac Hamiltonian with CP symmetry is related (equivalent); all these systems are classified by a given. Then the topological invariants of CP symmetric Z 2 topological number. Such relation is shown pictori- TI are also constructed. On the other hand, using the ally in FIG. 1(a). FIG. 1(b) shows another example for extension problem of the Clifford algebra, the BdG sys- CPT-equivalent insulators and BdG systems with related tems with spin U(1) and P symmetries are also shown symmetries. to fall in the classification equivalent to the symmetry While we were preparing the draft, a preprint that is class possessing TRS in any dimensions. We thus extend 51 related to this paper appeared on arXiv. While our CPT-equivalence and T-duality that we observe in Sec. analysis in terms the K-theory largely overlaps with this II in terms of 2D fermionic systems to all dimensions. preprint, our analysis in terms of K-matrix theories and With the same idea, in Sec. III E, we also study topo- our discussion in terms of CPT-theorem and T-duality logical phases protected by PT symmetry, which are were not discussed therein. CPT-equivalent partners of topological phases protected by PHS. While the latter is usually implemented by a BdG Hamiltonian that breaks charge U(1) symmetry (su- A. Outline and main results perconducting system), it is interesting to find a system of insulator with nontrivial topology protected by The structure of the paper and the main results are PT that manifests the same topological features as TSCs summarized as follows. with PHS. While there is no nontrivial topological phase In Sec. II, we start our discussion by considering 2D protected by PT in 3D and 2D (here we are not inter- fermionic topological phases protected by CP symmetry. ested in the chiral topological phases in 2D), a nontrivial 4

PT symmetric TI, which is characterized by a Z2 class, where the two-component fermion annihilation operator exists in a 1D (and 0D) system. at site r on the two-dimensional square lattice, ψ(r), is In Sec. IV we discuss CPT-equivalence for more gen- given in terms of the electron annihilation operators with T eral symmetry classes. It can be stated as ”topological spin up and down, cr,1/2, as ψ (r) = (cr,1,cr,2), and we CPT theorem” for non-interacting fermionic systems in take t = ∆ = 1. There are four phases separated by three arbitrary dimensions. Furthermore, the complete classi- quantum critical points at µ = 0, ±4, which are labeled ﬁcation of TIs and TSCs (and TCIs and TCSCs if spa- by the Chern number as Ch = 0 (|µ| > 4), Ch = −1 tial symmetries are present) for non-interacting fermionic (−4 <µ< 0), and Ch = +1 (0 <µ< +4). In the systems with T, C , P, and/or their combinations is ob- following, we are interested in the phase with Ch = ±1. tained by considering symmetry classes ”AZ+CPT”: (a) † In momentum space, T = k∈BZ ψ (k)[~n(k) · ~σ] ψ(k), CPT-equivalent symmetry classes ”generated” from AZ classes by a trivial CPT symmetry; (b) Other symmetry −P2∆ sin kx ~n(k)= −2∆ sin ky . (2) classes ”generated” from AZ classes by nontrivial CPT   symmetries. The result is summarized in TABLE II. 2t(cos kx + cos ky)+ µ In Sec. V we use (Abelian) K-matrix theory to classify We will mostly focus on the case of Ch = ±1. 2D interacting topological phases protected by T, C, P A lattice model of the topological insulator with CP the combined symmetries, and/or U(1) symmetries, for symmetry can be constructed by taking two copies of the either bosonic or fermionic systems. The results are sum- above two-band Chern insulator with opposite chiralities. marized in TABLE III and IV. Comparing with the case Consider the Hamiltonian in momentum space, of non-interacting fermions, we also give an interacting † version of ”topological CPT theorem” for 2D interacting H = ψs(k)[~ns(k) · ~σ] ψs(k) bosonic and fermionic topological phases. The key point k∈BZ s=↑,↓ X X is that any perturbations (not necessary Lorentz scalars) = Ψ†(k)H(k)Ψ(k), (3) that can gap the edges of a 2D bulk must be invariant un- ∈ der a ”trivial” CPT symmetry. We also discuss T-duality kXBZ in the K-matrix formalism. Both CPT-equivalence and where s =↑, ↓ represent “pseudo spin” degrees of free- T-duality for 2D interacting topological phases can be dom, Ψ(k) is a four-component fermion ﬁeld, and ~ns(k) seen manifestly in Table III and IV. is given, in terms of ~n as ~n↑(k)= ~n(k), ~n↓(k)= ~n↑(k˜)= ~n(k˜), where k˜ = (−k1, k2). I.e., the single particle Hamil- tonian in momentum space is given in terms of the 4 × 4 II. 2D FERMIONIC TOPOLOGICAL PHASES matrix, PROTECTED BY SYMMETRIES H(k)= nx(k)τzσx + ny(k)τ0σy + nz(k)τ0σz, (4)

In this section, we start our discussion by considering where τ0,z is the Pauli matrix acting on the pseudo spin a simple fermionic tight-binding model which is invariant index. under CP symmetry. We will also note that the fermionic The Hamiltonian is invariant under the following CP system can also be interpreted as a topological supercon- transformations: ductor (BdG system) that conserves the z-component of −1 † spin. Later, we will comment on the connection between UΨ(x)U = UCPΨ (˜x), (5) CP symmetric TIs and T symmetric TIs (QSHE), and in- wherex ˜ := (−x1, x2) and UCP is given either by troduce the ideas of CPT-equivalence and T-duality for U = τ σ U T =+U , (η = +1), topological phases. CP x x CP CP T UCP = τyσx,UCP = −UCP, (η = −1). (6) To ditinguish these two cases, we introduced an index A. CP symmetric insulators η; η = ±1 refers to the ﬁrst/second case. We will also use notation η = e2πiǫ where ǫ = 0, 1/2 for η = 1, −1, 1. tight-binding model respectively. It turns out imposing UCP = τxσx leads to the CP sym- Let us consider the following tight-binding Hamilto- metric topological insulator. This can be seen by looking nian: at the stability of the edge mode that can appear when we terminate the system in y-direction (i.e., the edge is t i∆ T = ψ†(r) ψ(r +ˆx) + h.c. along the x-direction.) One can check, numerically, and i∆ −t r also in terms of the continuum edge theory (see below), X t ∆ UCP = τxσx protects the edge state while UCP = τyσx + ψ†(r) ψ(r +ˆy) + h.c. −∆ −t does not. In the following, these CP transformations will be combined with charge U(1) gauge transformation, and µ 0 + ψ†(r) ψ(r), (1) the corresponding transformation will be denoted by U 0 −µ (See below). 5

2. Edge theory be either column or row vector depending on the context.] The matrix elements obey Ξ = Ξ† (hermiticity) T We now develop a continuum theory for the edge state and ∆ = −∆ (Fermi statistics). The presence of SU(2) that exists when the system is terminated in y direction. spin rotation symmetry is represented by I.e., the edge is along x direction. Let us consider the sa 0 free fermion: H4,Ja =0, Ja := T , (13) 0 −sa v H = dx ψ† i∂ ψ − ψ† i∂ ψ . 2π L x L R x R where a = x,y,z. Z v † ψL With conservation of one component of spin, say, z- = dxΨ i∂xσzΨ , Ψ= . (7) 2π ψR component, we have a U(1) symmetry associated with Z rotation around z-axis. With this Sz conservation, one We consider two types of CP symmetry operation can reduce this BdG Hamiltonian into the following form: −1 iα † c UψL(x)U = e ψR(−x), ξ δ H = c† , c H ↑ , H = ↑ , (14) −1 iα † ↑ ↓ 2 c† 2 † T Uψ (x)U = ηe ψ (−x), (8) ↓ δ −ξ↓ R L where CP transformation is combined with the EM U(1) up to a term which is proportional to the identity matrix. charge twist with an arbitray phase factor α. The sign At the quadratic level, this Hamiltonian is a member of η = ± is +/− for topological/non-topological cases: symmetry class A (unitary symmetry class in AZ classes). [with c → c† , one can “convert” spin U(1) to ﬁctitious +1 topological ↓ ↓ η = e2πiǫ = (9) −1 trivial charge U(1)]. This can be seen as follows. Let us consider BdG Hamiltonians which are invariant under rotations This is CP symmetry with twisting by the charge opera- about the z- (or any ﬁxed) axis in spin space, yielding tor, to the condition [H4,Jz] = 0, which implies that the Hamiltonian can be brought into the form U = e−iαFV CP, (10) a 0 0 b where FV is the total charge operator, 0 a′ −bT 0 H = , a† = a, a′† = a′.(15) 4  0 −b∗ −aT 0  † † † ′T FV := FR + FL = dx ψ ψR + ψ ψL . (11)  b 0 0 −a  R L   Z   There are two fermion mass bilinears that are consis- Due to the sparse structure of H4, we can rearrange the tent with the charge U(1) symmetry: These masses are elements of this 4N × 4N matrix into the form of a 2N × odd under CP when η = +1. We thus conclude that 2N matrix H2 above. the edge theory is at least at the quadratic level stable Let us now consider parity symmetry. For simplicity, (ingappable). we assume orbitals transform trivially under parity, and hence assume the following form:

−iα B. BdG systems with spin U(1) and P symmetries c↑(r) −1 e c↓(˜r) P P = iα . (16) c↓(r) ηe c↑(˜r) In this section, we show that systems with charge U(1) and CP symmetry can be derived from BdG systems with Within the reduced 2N × 2N basis, parity symmetry conserved one component of spin (Sz, say) and parity looks like CP symmetry. To see this, let us write out symmetry. the Hamiltonian in the following form: The system of our interest preserves spin U(1) but not ξ δ c charge U(1). At the quadratic level, this situation is de- H = c† , c ↑ ↑ (17) scribed by the BdG Hamiltonian. Following Altland and ↑ ↓ δ† −ξT c† ↓ ↓ Zirnbauer,3 we consider the following general form of a = c† (r)ξ (r, r′)c (r′) − c (r)ξT (r, r′)c† (r′) BdG Hamiltonian for the dynamics of quasiparticles ↑a ↑ab ↑b ↓a ↓ab ↓b + c† (r)δ (r, r′)c† (r′)+ c (r)δ† (r, r′)c (r′) 1 c ↑a ab ↓b ↓a ab ↑b H = c†, c H , 2 4 c† (summation over repeated indices are implicit). Then, Ξ ∆ H4 = ∗ T , (12) T −∆ −Ξ −1 † ξ↓ −ηδ c↑ PHP = c , c↓ ∗ T † ↑ ′ −ηδ −ξ br˜ ↑ br′,ar c↓ where H4 is a 4N × 4N matrix for a system with N ar˜ † (18) orbitals (lattice sites), and c = (c↑, c↓). [c and c can 6

(The transpose T here acts both a and r). Thus, the a. T-duality (Kramers-Wannier duality) By taking invariance under P implies T-dual or Kramers-Wannier-dual of the above setting, † T ψ (x) → ψ (x), (27) ξ↓ −ηδ ξ↑ δ L L ∗ T = † T (19) −ηδ −ξ↑ ′ δ −ξ↓ ′ we obtain the P symmetric system. In the bosonized ar,b˜ r˜ ar,br language, this amounts to exchange phase ﬁeld φ and its With the transformation or relabeling dual θ. Also, if we decompose the complex fermion ψL in 1,2 1 terms of two real (Majorana) fermions χL , ψL = χL + † iχ2 , the above transformation amounts to χ2 → −χ2 c↑ =: Ψ↑, c↓ =: Ψ↓, (20) L L L while keeping the right moving intact. This is nothing we can write the Hamiltonian as but the Kramers-Wannier duality in the Ising model. The P symmetry, dualized from CP symmetry above, † ′ ′ is given by H = Ψ (r) H2(r, r ) Ψ(r ). (21) r,r′ −1 −iα X UψL(x)U = e ψR(−x) −1 iα Provided the system has translational symmetry, UψR(x)U = ηe ψL(−x). (28) H (r, r′) = H (r − r′), with periodic boundary condi- 2 2 This is P symmetry with Sz twisting, tions in each spatial direction (i.e., the system is deﬁned − on a torus T d), we can perform the Fourier transforma- U = e iαFA P, (29) tion and obtain in momentum space where FA is the axial charge,

† H = Ψ (k) H2(k) Ψ(k), (22) † † FA := FR − FL = dx ψRψR − ψLψL . (30) k∈Bz X Z where the crystal momentum k runs over the first Brillouin zone (Bz), and the Fourier component of C. Connection to T symmetric insulators the fermion operator and the Hamiltonian are given −1/2 ik·r by Ψ(r) = V k∈Bz e Ψ(k) and H2(k) = The CP symmetric model introduced above is in fact e−ik·rH (r), respectively. also time-reversal invariant in the absence of perturba- r 2 P Then the P invariance demands tions. If there is Lorentz invariance, because of CPT P theorem, any perturbation to the model that is CP sym- T ξ↓ −ηδ ξ↑ δ metric is also T symmetric. Hence, within Lorentz invari- ∗ T = † T . (23) ant theories, the same set of perturbations is prohibited −ηδ −ξ↑ ˜ δ −ξ↓ −k k by CP and T symmetries. The topological phase pro- Observing that tected by CP symmetry can thus be also viewed as a T symmetric topological phase. T T However, the above argument based on CPT theorem 0 1 −ξ ηδ 0 η ξ↑ δ ↓ = η of course raises a question as we do not want to be con- η 0 ηδ∗ ξT 1 0 δ† −ξT ↑ ↓ fined to relativistic systems, and Lorentz invariance is ab- (24) sent in the lattice model. Note, however, the following: (i) CPT theorem tells us the presence of antiparticles. we then conclude that, when η = 1, This seems a necessary ingredient to have a topological phase (topologically non-trivial “vacua”). (ii) Topologi- ˜ T T τxH2(−k) τx = −H2(k), τx =+τx (25) cal phases that are characterized by a term of topological origin in the response theory, such as the Chern-Simons whereas when η = −1, term or the axion term for the external (background) U(1) gauge field, are Lorentz invariant. This in particu- ˜ T T τyH2(−k) τy = −H2(k), τy = −τy. (26) lar means CP symmetry dictates the theta angle to be 0 or π (mod 2π), just as TRS does. I.e., the single-particle Hamiltonian H2 is CP symmetric. Finally, while Hamiltonians may violate Lorentz in- It should be noted that P symmetry with η = +1 variance, and hence CPT theorem, a version of CPT is somewhat unusual. When acting twice on spinors, like theorem applies to wavefunctions (= projection op- P2 = +1, whereas we usually expect P2 = −1. This is so erators), or the “entanglement Hamiltonian”. In other since parity should reverse the sign of angular momen- words, wavefunctions or the entanglement Hamiltonian tum, either of orbital or spin origin. The P symmetry have more symmetries than the physical Hamiltonian. with η = 1 can be considered as a composition of a P Due to this, for any CP symmetric system, one can de- symmetry with η = −1 and spin parity (−1)N↑ where fine “effective” time-reversal symmetry for the projector N↑ is the number operator associated to up spins. or the entanglement Hamiltonian. See Appendix A. 7

D. CPT-equivalence and T-duality of topological FIG. I shows some examples about CPT-equivalence phases: an example and T-duality among topological (crystalline) insulators and superconductors. Especially, FIG. 1(a) shows the The above discussion reveals a ”CPT-equivalence” be- connection between T symmetric TIs, CP symmetric TIs, tween CP and T symmetric topological phases. Further- and their dual realizations in BdG systems with Sz con- more, from the fact that, both CP symmetric TI with servations introduced in this section. Another example, (CP)2 = 1 [(CP)2 = (−1)Nf ] and T symmetric TI with as shown in FIG. 1(b), is the CPT-equivalence between T 2 Nf 2 T = (−1) (T = 1), where Nf is the total fermion symmetric TSCs, P symmetric TCSCs, and T and P sym- number operator, possess the same nontrivial Z2 (trivial) metric TCIs that can support gapless edge states even in classification in two dimensions, we expect a specific cor- the absence of charge U(1) symmetry. respondence between these two ”CPT-equivalent” topo- In the following section, we make a more precise dis- logical phases. In general, such correspondence can be cussion for the idea of CPT-equivalence and T-duality observed among topological phases protected by discrete introduced here, focusing on non-interacting fermionic symmetries T, C, P, and/or their combinations. We will CP symmetric TIs in arbitrary dimensions discuss it in the following sections. On the other hand, there is a duality – which we call ”T-duality” in this paper – between topological phases III. CLASSIFICATION OF CP SYMMETRIC of insulating and superconducting systems with corre- TIS IN ARBITRARY DIMENSION sponding symmetries. Imposing a symmetry g on a BdG system with Sz conservation (14) will result a constraint In this section we consider systems of non-interacting on the reduced BdG Hamiltonian H2 by the dual sym- fermions with CP symmetry and classify CP symmetric metry g, which is exactly in the same symmetry class as TIs in arbitrary dimensions using K-theory. a tight-binding Hamiltonian H constrained by the sym- Relevant symmetries are written as constraints on the Hamiltonian matrix H as follows. The particle-hole symmetry ge in a insulating system (with charge conservation implicitly). For example, as we discussed in Sec. II B, metry (PHS) is an anti-unitary operator C that anticommutes with the Hamiltonian as {C, H} = 0, which is the CPe symmetric topological phase can also be realized in a BdG system with P symmetry and Sz conservation. equivalently written using an unitary operator UC as Another known example is that a chiral symmetric topo- ∗ −1 logical phases (class AIII in AZ class) can also be in- UCH (−k1,..., −kd)UC = −H(k1,...,kd). (34) terpreted as a BdG Hamiltonian possessing TRS and Sz conservation.10,52 Interestingly (and expectedly), a 2D T The parity symmetry P is a symmetry that swaps left- symmetric TI, i.e., the QSHE, also has a dual realization handed and right-handed coordinates, which can be im- in a superconducting system – a BdG system with chi- plemented as a mirror symmetry with respect to a particular direction (here we take k1) as ral symmetry and Sz conservation. This can be seen by a similar discussion from Sec. II B. For a reduced BdG −1 P H(−k1, k2,...,kd)P = H(k1, k2,...,kd), (35) Hamiltonian (by Sz conservation) (14), if we impose a ”chiral” symmetry S (which is defined as a combination with a unitary operator P . Combining these two sym- of T and C symmetries) as metries C and P , we define CP symmetry by an unitary † † c e−iαc c operator UCP satisfying S ↑ S−1 = ↓ = e−iατz U ↑ , c iαc† c† ↓ ηe ↑ ! ↓ ! ∗ −1 UCPH (k1, −k2,..., −kd)UCP = −H(k1, k2,...,kd). τ for η =1 (36) U = x (31) iτy for η = −1, A CP symmetric TI is a topological insulator that does then not possess C nor P symmetry but is characterized with c a combined CP symmetry. −1 c† c ↑ −1 H = SHS = S ↑, ↓ H2 c† S ↓ c c† c † ∗ ↑ A. Classification by K-theory = ↑, ↓ U H2U c† (32) ↓ † T In non-interacting fermion systems, CP symmetric TIs (note that S = TC is antiunitary) implies U H U = H2, 2 are classified using K-theory in a way similar to the i.e., the single-particle Hamiltonian H2 is TRS. In conclu- classification of topological defects discussed by Teo and sion, dual symmetries between the tight-binding Hamil- 53 tonian (with charge conservation) and the BdG reduced Kane. Hamiltonian (with S conservation) have the following A TI with CP symmetry (36) is regardedas a TSC with z ˜ ˜ correspondences: PHS C = CP in the d = d − 1 dimensions with momenta k2,...,kd (that are flipped by an action of C˜), containing T ↔TC, P ↔CP. (33) a defect with a co-dimension D = 1 parameterized with 8 k1 (that is not flipped by C˜). When we have PHS C˜ with with a complex conjugation K. Now we can construct 2 2 C˜ = +1 or C˜ = −1, the symmetry class is class D or an effective TRS from CP symmetry as T = γ1UCPK, class C and the associated classifying spaces Rq are given satisfying 11,32 as ∗ −1 γ1UCPH (−k1,..., −kd)(γ1UCP) = H(k1,...,kd). 2 class D C˜ = 1 : R2 (q = 2), (43) 2 class C C˜ = −1 : R6 (q = 6). (37) The existence of γ1 in the Dirac model enables us to convert the CP symmetry into a TRS, which is not the Then the classification for CP symmetric TI is given by case for a general lattice model where a kinetic term 53 a homotopy group along reflected coordinate is not necessarily written by a gamma matrix γ1. πD(Rq−d˜) ≃ π0(Rq−d˜+D)= π0(Rq+2−d). (38) This can be interpreted that the relevant classifying space changes from Rq to Rq+2, which corresponds to the sym- C. Topological invariants metry class AII (R4) orclass AI(R8 ≃ R0), both possessing TRS. Thus CP symmetric TI behaves similar to TR Topological invariants of CP symmetric TIs are con- symmetric TI in terms of topological classification and structed in the same way as those for topological corresponding edge states. This is consistent with the defects.53 For q+2−d =0, 4 in Eq. (38), we have topolog- CPT theorem for Lorentz-invariant systems where CP ical invariants Z. Due to q =2, 6 [Eq. (37)], the topolog- symmetry can be effectively converted into time reversal ical invariants Z are realized in even dimensions d, where symmetry. we can define the Chern number over the Brillouin zone. While we adopted the parity symmetry (35) that flips The Chern number gives the topological invariants, which only one momentum k1, we can generally reverse 2n +1 is written, by putting d =2n, as coordinates for the parity as n 1 2n iF −1 Chn = d k tr , P H(−k1,..., −k2n+1, k2n+2,...,kd)P n! 2π Z = H(k1,...,k2n+1, k2n+2,...,kd). (39) F = dA + A ∧ A, A = huk|d|uki, (44)

The TI with CP symmetry constructed from above P with valence bands |uki and a derivative d with respect can be regarded as a TSC with PHS in the d˜= d − 2n − to momenta k. 1 dimensions with momenta k2n+2,...,kd, containing a Next, first descendant Z2 is given by a Chern-Simons defect with a co-dimension D = 2n + 1 parameterized form, which takes place for q +2 − d = 1 in Eq. (38), with k1,...,k2n+1. Then the classification is given by a so that the dimension d is odd. When we have the first homotopy group descendant Z2, we can choose a continuous gauge A over the entire Brillouin zone, and an integration of the Chern- π (R )= π (R ), (40) D q−d˜ 0 q+4n+2−d Simons form, which is defined for odd dimensions, gives Z where the relevant classifying space looks like Rq+4n+2. topological invariant 2. Z Since we have q = 2 or q = 6, Rq+4n+2 becomes R0 or Second descendant 2 is given by a dimension reduc- Z R4, i.e, the classifying space associated with the symme- tion of the above 2. We consider a one parameter try class with TRS, which is consistent with the CPT family of the Hamiltonian H˜(θ, k) connecting the orig- theorem. inal Hamiltonian H˜(0, k) = H(k) and a reference CP symmetric Hamiltonian H˜(π, k) = H0 with a parameter 0 ≤ θ ≤ π. If we extend a range of θ into −π ≤ θ ≤ π by B. Dirac models and CPT theorem a relation H˜(θ, k , k ,...,k ) While we cannot explicitly construct an anti-unitary 1 2 d ˜∗ −1 operator T for TRS from CP symmetry in general cases, = −UCPH (−θ, k1, −k2,..., −kd)UCP, (45) we can construct T operator from CP symmetry in a ˜ Dirac Hamiltonian we can define a CP symmetric Hamiltonian H(θ, k) over −π ≤ θ ≤ π and k. Then the second descendant Z2 d characterizing the Hamiltonian H(k) is given by an inte- H(k)= mγ + k γ , (41) 0 i i gration of Chern-Simons form for H˜(θ, k). i=1 X where γi’s are anti-commuting gamma matrices, m is a mass, and ki’s are momenta. The CP symmetry (36) D. BdG systems with spin U(1) and P symmetries then leads to relations As we have seen in Sec. IIB, CP symmetric TI can {U K,γ } =0, i =0, 1, CP i be realized by a BdG system with a reflection symme- [U K,γ ]=0, 2 ≤ i ≤ d, (42) CP i try and spin U(1) symmetry (Sz conservation). Here we 9 interpret their equivalence to class AII TIs in terms of unique mass compatible with the CP symmetry. Thus K-theory and Clifford algebras. When we have a unitary Hamiltonians with different signs of the unique mass term operator commuting with the Hamiltonian, we should are topologically distinct. If we try to double the system block-diagonalize the Hamiltonian when we consider a where doubled 2 by 2 degrees of freedom is described by topological classification. When the Sz anti-commutes Pauli matrices ρ, allowed mass terms are not unique since with the PHS C and the reflection symmetry P , the we have σxτxρy and σxτyρy in addition to σz. Then we block-diagonalized Hamiltonian does not possess C nor can adiabatically connect two states in the doubled sys- P any further, while the combined CP still remains as tem by appropriately rotating in the space of mass term, a symmetry of the block Hamiltonian. The situation is which indicates that the classification of CP symmetric summarized as follows, TI in 2D is Z2. Next we show that the above classification for class D {C, H} =0, [Sz, H]=0, {C,Sz} =0, {P,Sz} =0, accompanied with spin U(1) and parity P is equivalent 2 2 to that in class AII, by adopting Clifford algebras classi- Sz =1, P =1, (46) fication for Dirac models. The original classification for along with a parity symmetry (35). [Note: we can choose class D in d-dimensions is given by a Clifford algebra32 2 P = 1 by appropriately multiplying “i”, which may {γ ,C,CJ,Jγ ,...,Jγ } (52) change a commutation/anticommutation relation with 0 1 d C.] and its extension problem with respect to the mass term Now let us look at a topologically non-trivial example γ0 is of this construction. We start with a BdG Hamiltonian Cl → Cl , (53) in class D and two dimensions as d,2 d,3 where the topological index is given by π0(R2−d), espe- H(k)= nx(k)σx + ny(k)σy + nz(k)σz , (47) cially Z for d = 2. The spin U(1) symmetry (Sz) and the parity symmetry (P ), satisfying (46) and [C, P ] = 0, can where ~n(k) is defined in Eq. (2) and we have PHS of C = be included in the Clifford algebra as σxK. A parity symmetry (35) is implemented by taking two copies of the above BdG Hamiltonian (denoted by {γ0,C,CJ,Jγ1,...,Jγd,Jγ1P,γ1PSz}, (54) τ) and a spin U(1) symmetry is implemented by taking for which the extension problem for γ is written as two copies representing spin degrees of freedom (denoted 0 by s), which yields Cld,4 → Cld,5, (55)

and the classification is given by π0(R4−d). In our exam- H(k)= nx(k)σxτzsx + ny(k)σysx + nz(k)σz, (48) ple in d = 2, we have Z2 topological number. Above clas- where we have the spin U(1) symmetry Sz = σzsz. We sification is the same as that for class AII in d-dimensions, have two parity symmetries P (a reflection symmetry which shows that CP symmetric TI and TR symmetric with respect to x-direction) written as TI are equivalent in the level of Dirac models. Indeed, the effective TRS is given from CP symmetry and a kinetic gamma matrix as (43). τxsx, (topological) P = (49) On the other hand, in the case of the parity symmetry τ s . (trivial) ( y x anti-commuting with PHS ({C, P } = 0), the relevant Clifford algebra is A choice of parity P = τxsx, commuting with PHS ([C, P ] = 0), leads to a topologically non-trivial insu- {γ0,C,CJ,Jγ1,...,Jγd,γ1P,Jγ1PSz}, (56) lator, as explained later with Clifford algebras. We can choose either of parity symmetries by adding appropriate and the extension problem for γ0 is terms to the Hamiltonian. Cld+2,2 → Cld+2,3. (57) Block diagonalization with respect to Sz becomes Then the topological invariant is π (R− ), where we have clear, if we change bases as (σxsz, σysz,sx,sy, σzsz) → 0 d a trivial insulator for d = 2 as π (R− ) = 0, that is (σxsx, σysx, σzsx,sy,sz), 0 2 equivalent to class AI in d = 2. This is the reason why

H(k)= −nx(k)σxτzsz − ny(k)σysz + nz(k)σz, we have a trivial insulator if we choose a parity symmetry S = s . (50) P = τysx in Eq. (49). z z While we have so far discussed the system in class D with S and P in two dimensions, we note that a sys- The block Hamiltonian with sz = −1 is given by z tem in three dimensions possesses non-trivial Z2 topo- H(k)= nx(k)σxτz + ny(k)σy + nz(k)σz, (51) logical invariant. This is interesting, since the original class D system in three dimensions is trivial and a block- characterized by a CP symmetry with UCP = τxσx. This diagonalized system with Sz (class A system) is also triv- corresponds to a non-trivial CP symmetric TI given in ial, while the CP symmetry gives rise to a non-trivial Eq. (6) with η = +1. In (51), the mass term σz is the insulator. 10

E. Topological classification of other symmetries CPT operator W = W0 such that the system transforms ”topologically-trivially” under W0. Therefore, we have In a similar manner as CP symmetric TIs, we can de- the following statement: fine PT symmetric TIs. PT symmetry can be defined by Topological CPT theorem for noninteracting fermionic systems: a unitary operator UPT satisfying Let {gi} be a set of symmetries (can be a null set) composed of T , C, P , and/or their ∗ −1 UPTH (k1, −k2,..., −kd)UPT = H(k1, k2,...,kd). (58) combinations. Then for non-interacting fermionic systems there is a ”trivial” CPT operator W = W0, which Then PT symmetric TI is a topological insulator that anticommutes with T and P and commutes with C (from does not possess P nor T symmetry but is characterized which other commutation relations between gi and W0 with a combined PT symmetry. In an analogous way can also be deduced), such that the system with symme- for CP symmetric TI, a classification of PT symmetric tries {gi} and the system of with symmetries {gi, W0} TI in d-dimensions is obtained by considering a system possess the same classifying space or topological classifi- with TRS in d − 1-dimensions containing a topological cation. defect with co-dimension 1. We assume that classifying The proof of the above theorem is straightforward as space for the TR symmetric TI with T˜ = UPTK in 0- we consider the Dirac Hamiltonian (41) (the idea here is 2 dimensions is Rq. We have q = 0 for (UPTK) = +1, similar to the discussion in Sec. III B) : 2 and q = 4 for (UPTK) = −1. Then the classification for PT symmetric TI is given by d H(k)= mγ0 + kiγi, π (R ) ≃ π (R ). (59) i=1 1 q−(d−1) 0 q+2−d X Non-trivial PT symmetric TIs are found in 2-dimensional where γi’s are anti-commuting gamma matrices. The systems in class AI or AII with a reflection symmetry, symmetries T , C, and P (if present) satisfy where TRS and P are broken by a diagonalization with respect to some unitary symmetry but the combined PT [T,γ0]=0, {T,γi=06 } =0, remains, which is characterized by a non-trivial topolog- {C,γ0} =0, [C,γi=06 ]=0, Z ical number . A shift of classifying space by 2 is in- {P,γ1} =0, [P,γi=16 ]=0, (61) terpreted as a change of an effective symmetry class into that with PHS, which is again consistent with the CPT while the CPT symmetry W satisfies theorem. {W, γ0} =0, [W, γi=1]=0, {W, γi=06 ,1} =0. (62)

IV. TOPOLOGICAL CPT THEOREM AND Define M = γ1W , we then have [M,γi]=0 ∀i and thus TOPOLOGICAL CPT-EQUIVALENT M is an unitary symmetry commuting with H: SYMMETRY CLASSES [M, H]=0,M 2 =1. (63) Actually, as we discussed in previous sections, such CPT-equivalence holds ”topologically” for more general For the system with symmetries {gi}, composed of T , symmetry classes (not just for cases discussed in the last C, P , and/or their combinations, if the additional sym- sections). In this section, we discuss the ”topological metry M = M0 commutes with all gi, or equivalently if CPT theorem” and topological CPT-equivalent symme- W = W0 satisfies (if {gi} includes some of the following try classes in noninteracting fermionic systems. symmetries) Combining symmetries T , C, and P , we define the CPT symmetry by an unitary operator W satisfying {W0,T } =0, [W0, C]=0, {W0, P } =0, {W0, CP } =0, [W0, T P ]=0, {W0, T C} =0, (64) W H(k˜)W −1 = −H(k), W 2 =1, (60) we can block diagonalize H with respect to M such that where H(k) is a d-dimensional (d ≥ 1) single particle all symmetries gi are still preserved in each eigenspace of Hamiltonian and k˜ = (−k1, k2,...,kd). Note that here, M. Therefore, the symmetry class and hence the clas- 2 as W is unitary, W can alway be fixed to be 1 by the sification would not change as the symmetry M0 or W0 ′ iα redefinition W = e W with any phase factor (such is added to the original set of symmetries {gi} of the redefinition is also accompanied by changing the com- system. This completes the proof. mutation relations with other existed symmetries at the We would like to point out that, though the topologi- same time). Now, if the system already has some sym- cal CPT theorem is ”proved” (or argued) by considering metries, adding the CPT symmetry constraint [Eq. (60)] the Dirac model (as a representative model of Clifford on H would or would not change the classifying space algebras that capture the topology of classifying spaces), or the topological classification with respect to existing which seems obviously to be invariant under a (trivial) symmetries. That is, in the latter case, there exists a CPT symmetry because of its Lorentz invariance, the 11 same conclusion can be reached by more (mathemati- all have the same classification. In Refs. 31 and 32 the cally) rigorous ways, such as topological K-theory, which first four symmetry classes are denoted respectively as is irrelevant to Lorentz invariance. Actually, this is what classes DIII, D+R+, AII+R−, and DIII+R−+, which we did (in Sec. III) in the discussion for equivalence be- have the same zero-dimensional classifying space R3 and tween T and CP, and C and PT, as part of topological thus the same classification in any dimension (using K- CPT theorem discussed here, using K-theory in a way theory). Moreover, it can be checked that, from the re- similar to topological defects discussed in Ref. 53. sults of Refs. 31 and 32, the topological CPT theorem indeed holds. We note that a natural choice of C, P , and T for spin- Class SM,TC SW,TC Classifying space 1/2 fermions leads to a trivial CPT as expected. This AIII − + C0 can be explicitly seen in the CPT-equivalence of class + ++− Class SM,T or SM,C SW,T or SW,C Classifying space DIII [Γ−+(T, C)] and class DIII+R−+ [Γ−++(T,C,P )]. For spin-1/2 fermions, the symmetry operators are given AI,AII − + C0 as D, C − − C0

Class (SM,T,SM,C) (SW,T,SW,C) Classifying space C = σxK, P = sx,T = isyK, (69)

(−, +) (+, +) Rq → Rq+1 BDI, CII where σi,si are Pauli matrices acting on particle-hole and (+, −) (−, −) Rq → Rq−1 spin degrees of freedom. The parity symmetry P is a (−, −) (+, −) C1 reﬂection along x-direction and involves a π-rotation of (+, −) (−, −) Rq → Rq+1 spin around x-axis, which is denoted by R−+ in classi- DIII, CI (−, +) (+, +) Rq → Rq−1 ﬁcation in Refs. 31 and 32. Then if we consider CPT

(−, −) (+, −) C1 symmetry given as W = −iCPT = σxsz, W satisfies commutation relations in Eq. (64) and is a trivial CPT. TABLE I. Classification of AZ symmetry classes in the Thus an addition of a trivial CPT W changes class DIII presence of a nontrivial CPT symmetry W or an addi- + ++− [Γ−+(T, C)] to class DIII+R−+ [Γ−++(T,C,P )], while it tional unitary symmetry M (commuting with H) in zero does not change the topological classification. dimension. SM,g and SW,g dictate commutation (+) or On the other hand, adding a nontrivial CPT symmetry anticommutation(−) relation of symmetry g, which can be W [which can be represented as a combination of a triv- T , C, or TC. ”Rq” in the last column denotes the original classifying spaces for the corresponding symmetry classes ial CPT symmetry and some (onsite) order-two unitary (before adding M or W ). symmetry commuting with H such that W changes the commutation relations (64)] to a symmetry class would change the original classifying space. Using the result of Ref. 32, we can directly obtain the change of classifying Based on topological CPT theorem, some symmetry spaces for AZ symmetry classes in the presence of extra classes, defined as topological CPT-equivalent symmetry unitary symmetry M = γ W (commuting with H). The classes here, possess the same classification. For example, 1 result is summarized in TABLE I. symmetry classes The complete classification of TIs and TSCs (and TCIs − Γ−(T ), Γ+(CP ), Γ−+(T,CP ), (65) and TCSCs if spatial symmetries such as P or CP are present) for non-interacting fermionic systems with T, C, all have the same classification. Here we use notations P, and/or their combinations, instead of studying these S g1,g2 symmetries separately, can also be obtained by the one ΓSg (g), ΓSg Sg (g1,g2), 1 2 with symmetry classes ”AZ+CPT” [in Refs. 31 and 32 Sg1,g2 Sg2,g3 Sg3,g1 Γ (g1,g2,g3), (66) Sg1 Sg2 Sg3 classification for symmetry classes ”AZ+P (or reflection R)” have been discussed, but some combined symmetries to denote the symmetry classes composed of {g }, with i like CP are not included there]. Two cases are involved: signs S dictating the commutation (+) or anticom- gi,gj (a) CPT-equivalent symmetry classes ”generated” from mutation (−) relation between g and g . (65) can be i j AZ classes by a trivial CPT symmetry; (b) Other sym- deduced from the following CPT-equivalent symmetry metry classes ”generated” from AZ classes by nontrivial classes : CPT symmetries (based on the result in Table I). The ⊜ − − − Γ−(T ) Γ−+(T, W0)=Γ−+(T, W0T )=Γ++(W0, W0T ) result is summarized in Table II. Generally, we can also reverse an odd number of spatial ⊜ Γ+(W0T )=Γ+(CP ), (67) coordinates as the parity P and the corresponding CPT ⊜ where ” ” represents the CPT-equivalence relations for symmetry W , with k˜ = (−k1,..., −k2n+1, k2n+2,...,kd) symmetry classes. Similarly, as another example, sym- in (60). In this situation, the ”effective” unitary sym- n metry classes metry M can be defined as M = i γ1 ··· γ2n+1W , and + + − ++− the commutation relations between the trivial CPT sym- Γ−+(T, C), Γ++(C, P ), Γ−+(T, P ), Γ−++(T,C,P ), − − − metry W0 and other symmetries (64) will also change if Γ++(CP,PT )=Γ−+(TC,PT )=Γ−+(TC,CP ), (68) n is odd (only commutation relations with antiunitary 12

(a)

CPT-equiv. sym. classes ”generated” from AZ classes by trivial CPT CqorRq π0(CqorRq)

”None” (A), Γ+(CPT ) C0 Z − Γ+(TC) (AIII), Γ+(P ), Γ++(TC,P ) C1 0 + Z Γ+(T ) (AI), Γ−(CP ), Γ+−(T,CP ) R0 + − + +−+ + Z Γ++(T,C) (BDI), Γ++(C, P ), Γ++(T, P ), Γ+++(T,C,P ), Γ−+(CP,PT ) R1 2 + Z Γ+(C) (D), Γ+(PT ), Γ++(C,PT ) R2 2 + + − ++− + Γ−+(T,C) (DIII), Γ++(C, P ), Γ−+(T, P ), Γ−++(T,C,P ), Γ++(CP,PT ) R3 0 + Z Γ−(T ) (AII), Γ+(CP ), Γ−+(T,CP ) R4 + − + +−+ + Γ−−(T,C) (CII), Γ−+(C, P ), Γ−+(T, P ), Γ−−+(T,C,P ), Γ+−(CP,PT ) R5 0 + Γ−(C) (C), Γ−(PT ), Γ−−(C,PT ) R6 0 + + − ++− + Γ+−(T,C) (CI), Γ−+(C, P ), Γ++(T, P ), Γ+−+(T,C,P ), Γ−−(CP,PT ) R7 0 (b)

Other sym. classes ”generated” from AZ classes by nontrivial CPT CqorRq π0(CqorRq ) + + + + + Z Γ++(TC,P ), Γ++(T,CP ), Γ−−(T,CP ), Γ+−(C,PT ), Γ−+(C,PT ) C0 ++− +−+ ++− +−+ Γ+++(T,C,P ), Γ−++(T,C,P ), Γ−−+(T,C,P ), Γ+−+(T,C,P ) C1 0 +++ +−− Z Γ+−+(T,C,P ), Γ+++(T,C,P ) R0 +++ +−− Z Γ+++(T,C,P ), Γ−++(T,C,P ) R2 2 +++ +−− Z Γ−++(T,C,P ), Γ−−+(T,C,P ) R4 +++ +−− Γ−−+(T,C,P ), Γ+−+(T,C,P ) R6 0

TABLE II. Classification of TIs and TSCs for non-interacting fermion systems with symmetry classes composed of T , C, P , and/or their combinations in zero dimension. This can be obtained by adding the CPT symmetry (either trivial or nontrivial ones) to the AZ classes: (a) CPT-equivalent symmetry classes ”generated” from AZ classes by trivial CPT; (b) Other symmetry classes ”generated” from AZ classes by nontrivial CPT (based on the result in Table I). In this table we have fixed [Ai,Aj ] = 0 2 and Ui = 1 (other choices are equivalent), where Ai/j and Ui represent antiunitary and unitary symmetries, respectively. Classification in arbitrary dimensions d is given by π0(Cq−d) or π0(Rq−d), as deduced from zero-dimensional classifying spaces Cq or Rq by K-theory. symmetries such as T and C will change). Nevertheless, interacting systems of either fermions or bosons, as the previous discussions on the case for n = 0 (the same for original CPT theorem applies to Lorentz invariant quan- even n) can be straightforwardly applied to the case for tum field theories with interactions. As a simple but odd n. instructive demonstration, in this section we discuss in- As related to the results in this section, a similar but teracting SPT phases (without topological order) in two more general discussion can also be found in Ref. 51. dimensions by using (Abelian) K-matrix Chern-Simons The CPT symmetry defined in (60) here is one kind of theory. order-two spatial symmetries defined there (on a system without defects). Therefore, classification of AZ classes in the presence of either trivial CPT (related by topo- A. Bulk and edge K-matrix theories incorporated logical CPT theorem) or nontrivial CPT (result shifts of with symmetries the classifying spaces shown in TABLE I) discussed here can also be deduced from the general properties of the K- We begin with the bulk K-matrix action Sbulk = 2 0 ex groups for the additional order-two spatial symmetries, dtd x Lbulk, Lbulk = Lbulk + Lbulk: as derived in Ref. 51. R 1 L0 = ǫµνλK a ∂ a , bulk 4π IJ Iµ ν Jλ ex eQI µνλ sSI µνλ V. CLASSIFICATION OF 2D INTERACTING Lbulk = − ǫ Aµ∂ν aIλ − ǫ Bµ∂ν aIλ, (70) SPT PHASES: K-MATRIX FORMULATION 2π 2π

where aµ represents the N-flavors of dynamical Chern- In the previous sections we have discussed topological Simons (CS) gauge fields, Aµ and Bµ are the external phases protected by T, C, P and/or corresponding com- gauge potentials coupling to the electric charges and spin bined symmetries and classification related by topological degrees of freedom (along some quantization axis), K is CPT theorem in non-interacting fermionic systems. Ac- an integer-valued N × N matrix (symmetric and invert- tually, such CPT-equivalence is expected to hold even for ible), and Q and S are integer-valued N-components vec- 13 tors representing electric charges (in unit of the electric a general symmetry group G that has elements as com- charge e) and spin charges (in unit of the spin charge s), binations of T , C, and P, and/or U(1) symmetries, we respectively. The currents in the bulk are have

e s −1 J µ = ǫµνλQ ∂ a , J µ = ǫµνλS ∂ a , (71) GSedgeG = Sedge, ∀G ∈ G, (77) c 2π I ν Iλ s 2π I ν Iλ with the chiral boson fields transformed as where Jc and Js are the total charge and spin currents, −1 respectively. GφG = αGUGφ + δφG, ∀G ∈ G, (78) In the bulk, we have the transformation laws under symmetries such as TRS (T ), PHS (C), and PS (P) in where αG = 1 (−1) represents an unitary (antiunitary) the x-direction [gµν = diag (+, −, −)]: operator G. Specifically, for TRS, PHS, and PS, (77) gives the constraints for the matrices UT, UC, UP, and µ ν µ ν T : Jc → gµν Jc , Js →−gµν Js , (t, x) → (−t, x), charge and spin vectors Q, S µ µ µ µ C : Jc →−Jc , Js →−Js , T TRS : UT KUT = −K, µ ˜µ µ ˜µ µ µ P : Jc → Jc , Js →−Js , x → x˜ , (72) T T IN + UT Q =0, IN − UT S =0, T where we have defined X˜ µ ≡ (X0, −X1,X2)T for any PHS : UC KUC = K, vector Xµ. We assume that the gauge fields aµ (flavor T T IN + UC Q =0, IN + UC S =0, index is suppressed) obey the following transformation T laws: PS: UP KUP = −K, T T IN + UP Q =0, IN − UP S =0, (79) µ −1 ν T a (t, x)T = gµν UTa (−t, x), where I is the N× N identity matrix. Cases of the Caµ(t, x)C−1 = U aµ(t, x), N C combined symmetries like CP are straightforward. µ −1 µ Pa (x)P = UPa˜ (˜x), (73) For the charge and spin U(1) symmetries of the system,

−1 where UT,UC, and UP are integer-valued N ×N matrices, Ucφ(t, x)Uc = φ(t, x)+ δφc, then we can ﬁnd these matrices of transformations by the U φ(t, x)U −1 = φ(t, x)+ δφ , (80) symmetries of the theory. However, the above symmetry s s s transformation law does not fully specify the symmetry 0 0 where U ≡ eiθc R dx jc /e and U ≡ eiθs R dx js /s are the properties of charged excitations.54 c s charge and spin U(1) transformations, respectively, and A convenient way to complete the description of the the corresponding phase shifts are given by symmetries is to consider the action at the edge Sedge = dtdx L , L = L0 + Lex : −1 −1 edge edge edge edge δφc = θcK Q, δφs = θsK S. (81)

R 1 On the other hand, the phases δφ in Eq. (78) are L0 = (K ∂ φ ∂ φ − V ∂ φ ∂ φ ) , edge 4π IJ t I x J IJ x I x J determined by how the local quasiparticle excitations, ex e µν s µν which are described by normal-ordered vertex operators Ledge = ǫ QI ∂µφI Aν + ǫ SI ∂µφI Bν , (74) T T 2π 2π ‡eil φ‡ = ‡eiΛ Kφ‡ ≡‡eiΘ(Λ)‡, with l = KΛ and Λ being integer N-components vectors, under the symme- which is derived from the usual bulk-edge correspondence try transformations. That is, the transformation law for of the bulk Chern-Simons theory (70). Now the currents Θ(Λ) is determined by the algebraic relations of the un- in the edge theory are derlying symmetry operators. To classify these discrete e s Z symmetries for interacting systems (beyond the single- jµ = ǫµν Q ∂ φ , jµ = ǫµν S ∂ φ . (75) 2 c 2π I ν I s 2π I ν I particle picture), we constrain the symmetry operators by the following algebraic relations: Under T , C, and P, the edge currents transform similarly 2 Nf as the bulk currents. The transformation law for the Gi = S , ∀Gi ∈ G; µ Gi bosonic ﬁelds φI is translated from the gauge ﬁelds a G G G−1G−1 = SNf , ∀G , G ∈ G, (82) (73), with additional (constant) phases: i j i j Gi,Gj i j

−1 where S has values ±1. In a bosonic system, the opera- T φ(t, x)T = −UTφ(−t, x)+ δφT, Nf −1 tor S (subscript omitted) is just the identity 1. In a Cφ(t, x)C = UCφ(t, x)+ δφC, fermionic system, SNf can be either the identity 1 or the −1 Nf Pφ(t, x)P = UPφ(t, −x)+ δφP. (76) fermion number parity operator Pf ≡ (−1) (i.e., symmetries are realized projectively) , where Nf is the total The minus sign in front of UT is just a convention for fermion number operator. Since all T , C, and P (and of a antiuntary operator. For the edge theory (74) with course the combined symmetries) commute with Pf , we 14

have SG1,G2 = SG2,G1 for any two symmetry operators where NIJ is the component of some integer matrix. T G1 and G2. For any local quasiparticle excitation ‡ exp iΛ Kφ‡, T In the presence of U(1) symmetries, the algebraic rela- with Λ Kφ = I ΛI (Kφ)I ≡ I θI , the symmetry tions (82) for fermionic systems might be ”gauge equiva- transformation G acts as P P lent” through the redeﬁnition of the discrete symmetry G T − − G‡ eiΛ Kφ ‡ G 1 = G‡ ePI iθI ‡ G 1 to UαG, where Uα can be charge or spin U(1) with some ′ iθ − 1 [iθ ,iθ ] −1 phase α. Denoting G and G the discrete symmetries with = G‡ e I · e 2 PI

{UGi , δφGi } to the inequivalent forms of transformations. For the unitary operator G = Ge (e means ”identity el- b. Statistical phase factors of vertex operators under ement”) that has the form of the identity or the fermion 2 −1 −1 symmetry transformation The edge theory (74) is quan- number parity operator Pf [such as Gi and GiGj Gi Gj −1 tized according to the equal-time commutators in Eq. (82)], we have Geiθj Ge = iθj + const. In this case the phase i∆φΛ vanishes (in the sense of mod [φI (t, x), φJ (t, x′)] = −iπ (K−1)IJ sgn(x − x′)+ΘIJ , Ge 2πi) and thus such Ge is linear on φ. This fact tells us (85) that instead of specifying the transformation properties T −1 Ge(iΛ Kφ)Ge for all local quasiparticle excitations, we where the Klein factor −1 can solely consider the transformations GeiφGe to deter- IJ −1 IK −1 LJ Θ := (K ) [sgn(K − L)(KKL + QK QL)] (K ) mine the phase δφ, as deﬁned in (78), under symmetry (86) transformations. is included to ensure that local excitations satisfy the proper commutation relations when x 6= x′ and I 6= J: B. Edge stability criteria for SPT Phases [(Kφ) (t, x), (Kφ) (t, x′)] I J In this section, we brieﬂy discuss the edge stability and = −iπsgn(I − J)QI QJ +2πiNIJ , (87) criteria for SPT phases.54–56 15

The general terms of interactions (perturbations from phases with T, C, P, the combined symmetries, and/or the tunneling and scattering process of local excitations) U(1) symmetries, for either bosonic or fermionic systems. for the 1+1D edge theory are the bosonic condensations: Here we consider non-chiral SPT phases described by K- matrices with even dimensions (N is even) in the absence bosonic int of topological order (| det K| = 1): the canonical forms Sedge = dtdx of generic K-matrix are Λ Z X T UΛ(t, x)cos Λ Kφ(t, x)+ αΛ(t, x) . (92) K = σx ⊕ σx ⊕···⊕ σx = IN/2 ⊗ σx (95)

T in bosonic systems, and Note that an integer vector Λ = (Λ1, ··· , ΛN ) are T bosonic (i.e. excitation ‡eiΛ Kφ‡ is a boson) if Λ satisfy K = σz ⊕ σz ⊕···⊕ σz = I ⊗ σz (96) πΛT KΛ = 0 mod2π. In the discussion of this paper N/2 we assume the coupling UΛ is a constant (independent in fermionic systems.54,55,58,59 of t and x). In the absence of any symmetry, a collec- Similar to the case of non-interacting fermionic systems tion of bosonic {Λa}, which satisfies Haldane’s null vector discussed in the previous sections, we also have CPT- 57 condition equivalence among interacting bosonic and fermionic T SPT phases with these discrete symmetries. Combin- Λa KΛb =0, ∀a,b =1,...,N/2 (93) ing symmetries T , C, P, we define the CPT symmetry (here N is even since we focus on the K-matrix with equal by an antiunitary operator W, numbers of positive and negative eigenvalues), can con- Wφ(t, x)W−1 = −U φ(−t, −x)+ δφ , (97) dense (be localized) with various (classical) expectation W W int values by adding the corresponding Sedge to Sedge. That which satisfies is, the edge can be gapped by such perturbations (thus −1 the gapless edge modes are unstable), and the phase is WSedgeW = Sedge (topologically) trivial. T ⇒ U KUW = K, Such gapping mechanism might be forbidden by sym- W T T metries, resulting nontrivial SPT phases, which can IN + UW Q =0, IN + UW S =0. (98) not be transformed adiabatically from the trivial phase Imposing W to the system with some existed symmetries within the symmetry constraints, even in the presence would or would not change the classification of the orig- of interactions. That is, if any possible interaction Sint edge inal (interacting) topological phase. As the latter case, can not be added to the edge theory (74) without break- there exists a ”trivial” CPT operator W = W0 such that ing some symmetry, both explicitly and spontaneously, the 1D edge theory with any gapping interactions Sint then the system manifests a nontrivial SPT phase (pro- edge [with a set of Haldane’s null vectors (93)] is invariant un- tected by these symmetries). To be more specific, one der W , and thus the corresponding 2D topological phase wants to check that whether the following conditions are 0 would not be ”additionally” protected by the presence of all satisfied: such trivial CPT symmetry. Therefore, we have the fol- (i) There exists symmetry preserving Sint with a set edge lowing statement: of Haldane’s null vector {Λa}. Topological CPT theorem for interacting (ii) All edge states can be gapped without breaking any fermionic and bosonic non-chiral SPT phases in symmetry spontaneously. This can be checked whether T two dimensions: Let {Gi} be a set of symmetries (can all the elementary bosonic variables {va φ}, with be a null set) composed of T , C, P, the combined symmetries, and/or any (order-two) onsite unitary symme- la va = , ∀a, (94) tries [including U(1) symmetries]. Then for 2D inter- gcd(l ,l ,...,l ) a,1 a,2 a,N acting fermionic or bosonic systems (in the absence of which are generated from any collections of linear combi- topological order) described by K-matrix theories, there 2 T T exists a ”trivial” CPT operator W = W0 , with W be- nations of {la φ = (KΛ)a φ}, condense without breaking 0 any symmetry. ing identity operator and relations to other symmetries −1 −1 T If both conditions are satisfied, the phase is (topologi- (if present) as W0GiW0 Gi = 1 if UGi KUGi = K −1 −1 Nf T cally) trivial. Otherwise, the phase is a SPT phase. and W0GiW0 Gi = (−1) if UGi KUGi = −K for fermionic systems (for bosonic systems the algebraic relations is ”trivial”), such that the topological phases pro- C. Classification of SPT phases by K-matrix tected by {Gi} and the topological phases protected by thoeries: CPT-equivalent SPT phases and dual SPT {Gi, W0} possess the same classification. phases The proof of the above theorem is left to Appendix C. Now, from this theorem we can define CPT-equivalent By studying the stability/gappability of the 1D edge symmetry groups/classes or SPT phases for interact- of K-matrix theories, we can classify 2D interacting SPT ing bosonic and fermionic systems, as we did similarly 16 for non-interacting fermionic systems in Sec. IV. As As symmetries are present, in description (i) {Gi} (de- an example again, the (bosonic or fermionic) topolog- fined to be on aI ) and {Gi} (defined to be on aI ) are ical phases protected by both TRS and charge U(1) identical, while in description (ii) {Gi} and {Gi} ”look” symmetry54,56,60 and the topological phases protected different (e.g. P and CP),e as they act on differente de- by both CP and charge U(1) symmetry61 possess the grees of freedom. However, they both describee the same same classification, even in the presence of interactions ”symmetry of the system”. (to be more precise, we have the CPT-equivalent sym- On the other hand, if we ”rotate” every term in (70) by 2 Nf 2 metry classes {T , Uc| T = (±1) } ⊜ {CP, Uc| CP = X except the gauge fields aI [i.e. remove the tilde of a in Nf ⊜ (∓1) } for interacting fermionic systems, where ” ” (100)], we will obtain a dual theory that describes a dif- represents the CPT-equivalence relations). ferent physical system or SPT phase. For example, if we Through the trivial CPT symmetry W0, any nontrivial take X = K for the K-matrix described by (95) or (96), CPT symmetry W can be expressed as the combination the dual theory, obtained from a theory with symmetries of W and some onsite unitary Z symmetry M. So im- 0 2 {G }, will describe a system with dual symmetries {G } posing a nontrivial CPT symmetry to a system is iden- i i by the correspondence (99 together with charge-spin ex- tical to imposing such unitary symmetry to this system, change. Since the criteria for arguing a SPT phase is which might change the classification of the original SPT e independent of how we choose the gauge X and how we phases with existed symmetries. label the field operators, the dual SPT phase has exactly Besides CPT-equivalence for SPT phases, there are the same classification as the original SPT phase. other ”dualities” between SPT phases: classification In the following subsections, we give a complete classi- of the {G }-protected topological phases and the {G }- i i fication for K-matrix theories with T, C, P, the combined protected phases are the same, where {Gi} and {Gi} are symmetries, and/or U(1) symmetries, for both bosonic dual symmetries (see the discussion later). An examplee is and fermionic systems. We can see that CPT-equivalence the T-duality between the topological phases protectede and T-duality hold exactly through the classification ta- by CP and U(1)c and the topological phases protected bles for 2D interacting SPT phases. by P and U(1)s, as we discussed for the non-interacting fermionic systems in previous sections. In K-matrix formalism, this can be observed by the transformation law 1. K-matrix classification of bosonic non-chiral SPT phases for the two U(1) currents under the discrete Z2 symme- µ tries (72). We can see that the way Jc/s transforms under CP is the same as the way J µ transforms under P. In For bosonic K-matrix theories with T, C, P, the com- s/c bined symmetries, and/or U(1) symmetries, it is suffi- general, we have the following duality between these dis- cient to implement the non-chiral short-range entangled crete symmetries as we exchange charge and spin U(1) states by just considering the 2 × 2 K-matrix with deter- symmetries: minant det(K) = (−1)dim(K)/2 = −1. From the canon- T ↔TC, P ↔CP, ical form of bosonic K-matirx (95) we have K = σx. C↔C, PT ↔PT , CPT ↔CPT . (99) The detail for calculating symmetry transformations and their corresponding SPT phases is left to Appendix B. Actually, for the bulk K-matrix theory (70) [the same Here we summarize the results in TABLE III. prospect for the edge theory (74) by the bulk-edge cor- In TABLE III we show classification of 2D bosonic non- respondence] we can rewrite it as T T chiral SPT phases for {K,Q,S} = {σx, (0, 1) , (1, 0) }, as we focus on cases with vanishing QT KQ and ST KS 1 (so bosonic quantum Hall systems are not included in our L = ǫµνλK a ∂ a − J µA − J µB , bulk 4π IJ Iµ ν Jλ c µ s µ discussion here). There are some remarks for TABLE III: T −1 T T K = X KX, ae = X a, Q =e X Q, e S = X S, e e (i) For each symmetry group listed in TABLE III, X ∈ GL(N, Z), det(X)= ±1. (100) except groups ZC and ZPT, there are multiple choices e e e e 2 2 There are two interpretations for Eq. (100), corre- for physically inequivalent realizations of the symme- sponding to physical equivalent theories described in dif- tries, which are not characterized by the bosonic alge- CP ferent ways [passive or active transformation by X ∈ braic relations. For example, in symmetry group Z2 GL(N, Z)]: the CP symmetries can be represented by {UCP, δφCP} = T T (i) It is nothing but field redefinitions (change of basis); {−σz, (0, 0) } or {−σz, (0, π) }, as they are physically the relabeled gauge fileds aI describe the same degrees inequivalent in the absence U(1) symmetries. All in- of freedom as aI . equivalent choices should be considered in each symme- (ii) The gauge fileds aI eare transformed to the dual try group, and their corresponding nontrivial SPT phases gauge fields aI , which characterize different degrees of will form an Abelian group. freedom (as X is not the identity matrix) as the origi- (ii) When U(1) symmetry [either U(1)c or U(1)s] is nal ones (e.g.e charge-vortex duality). The dual theory present, there might be gauge redundancy among sym- describes the same physical system. metry transformations (as discussed in Sec. V A). For 17

Classiﬁcation of 2D bosonic non-chiral tween related symmetry groups [with the correspondence Sym. group SPT phases (99)] in TABLE III.

No U(1)’s U(1)c is present U(1)s is present T Z2 0 Z2 0 C Z2 0 0 0 2. K-matrix classification of fermionic SPT phases P Z2 0 0 Z2 CP Z Z2 0 2 0 For fermionic K-matrix theories with T, C, P, the com- PT Z2 0 0 0 bined symmetries, and/or U(1) symmetries, we can also TC Z2 0 0 Z2 implement the non-chiral short-range entangled states CPT Z Z2 Z2 Z2 by considering the 2 × 2 K-matrix, except for cases of 2 2 T C C2 = (−1)Nf and (PT ) = (−1)Nf , which must be re- Z2 × Z2 Z2 Z2 Z2 C P alized at least by a 4 × 4 K-matrix. From the canonical Z2 × Z2 Z2 Z2 Z2 P T form of fermionic K-matirx (96) we have K = σz. To dis- Z × Z Z2 Z2 Z2 2 2 cuss the classification of the non-chiral SPT phases with CP PT Z Z Z Z2 × Z2 2 2 2 symmetries specified by fermionic algebraic relations, for T CP Z2 Z2 Z Z2 × Z2 2 2 2 convenience we use the following notation C PT Z2 × Z2 Z2 Z2 Z2 S P TC Z2 Z Z2 G1,G2 Z2 × Z2 2 2 2 GSG (G), GSG (Uc/s, G), G (G1, G2), SG1 SG2 T C P Z4 Z2 Z2 Z2 × Z2 × Z2 2 2 2 SG1,G2 SG1,G2 SG2,G3 SG3,G1 G (Uc/s, G1, G2), G (G1, G2, G3), SG1 SG2 SG1 SG2 SG3

TABLE III. Classification of 2D interacting bosonic non-chiral SG1,G2 SG2,G3 SG3,G1 G (Uc/s, G1, G2, G3) (101) SPT phases with symmetry groups generated by T , C, P, SG1 SG2 SG3 and/or U(1) symmetries. Each nontrivial SPT phase in this table is implemented by a 2 × 2 K-matrix: {K,Q,S} = to denote fermionic symmetry groups, where Gi are the T T {σx, (0, 1) , (1, 0) }. Classification shown in this table has symmetry operators, signs SGi and SGi,Gj are defined removed U(1) gauge redundancy. in Eq. (82), and Uc/s is the charge/spin U(1) symmetry. The detail for calculating symmetry transformations and their corresponding SPT phases is left to Appendix B. the example in (i), the two representations of CP are Here we summarize the results in Table IV. gauge equivalent when U(1)s is present, as we can rede- In TABLE IV we show classification of 2D ′ fine (CP) = Us(α) · CP with a phase α = π to change fermionic non-chiral SPT phases for {K,Q,S} = T T one representation to another. The classification shown {σz, (1, −1) , (1, 1) }. Here we focus on deconfined in TABLE III has removed such U(1) gauge redundancy. fermionic SPT phases obtained from perturbing non- (iii) CPT-equivalence: At first glance CPT-equivalence interacting fermions. Classification of confined fermionic seem violated in TABLE III. For example, SPT phases SPT phases with bosonic degrees of freedom (such as CPT Z with Z2 symmetry are characterized by a 2 instead bosonic Cooper pairs formed by fermions) can be de- of a trivial classification, which is resulted in (non-chiral) scribed by the bosonic SPT phases discussed in the last SPT phases without any symmetries. Actually, both subsections. There are some remarks for TABLE IV: trivial and nontrivial CPT symmetries can be realized (i) As discussed in Sec. V A, the nontrivial statistical CPT W0M from Z2 = Z2 , where W0 is the trivial CPT and phase factors might be present when symmetries act on M is some onsite unitary Z2 symmetry. The nontrivial the bosonized fields of fermions (due to Fermi statistics). Z2 SPT phase here is protected by the nontrivial CPT Using Eqs. (89) and the commutations relations of chiral symmetry W0 ·M with M represented by {UM, δφM} = bosons (87) we can determine the statistical phase fac- T CPT M {I2, (π, π) }. Therefore, Z2 is CPT-equivalent to Z2 , tors for different symmetries on local quasiparticle excita- 55 T which identically gives the Z2 classification. As im- tions ‡ exp iΛ Kφ‡. For example, for bosonic excitations T plied from the topological CPT theorem, adding W0 Λ± ≡ (1, ±1) we have to some symmetry group G will not change the clas- Λ± Λ± Λ± sification of SPT phases by G. Similar argument ap- ∆φT = ∆φC = ∆φTC =0 mod2π; (102) plies for other symmetry groups that result nontrivial ∆φΛ± = ∆φΛ± = ∆φΛ± = ∆φΛ± = π mod 2π. CPT symmetries (by combining the symmetries), such P CP PT CPT T CP T M as Z2 × Z2 , which is CPT-equivalent to Z2 × Z2 and We must be careful about these extra phase factors CP M Z2 Z2 × Z2 (both have 2 classification; the former case (which might cause sign changes) when we analyze the is discussed in Ref. 55). On the other hand, if the invariance of condensed (local) bosonic field variables un- symmetry groups related by CPT relations do not pos- der symmetry transformations. It would affect the way T CP sess nontrivial CPT symmetries, such as {Z2 ,Z2 } and we determine the SPT phases with correct fermionic al- T C C P P T CP PT {Z2 × Z2 ,Z2 × Z2 ,Z2 × Z2 ,Z2 × Z2 }, they must gebraic relations [signs S’s in Eq. (101)]. have the same classification. (ii) Contrast to bosonic systems, for each symmetry (iv) Finally, we can also see T-duality holds exactly be- group listed in TABLE IV, except groups G±(CPT ) and 18

Symmetry groups for 2D nontrivial fermionic non-chiral SPT phases Top. Sym. Non-int. No U(1)’s U(1)c is present U(1)s is present class.

T - G−(Uc, T ) - Z2 Γ−(T ) C - - - - -

P - - G+(Us, P) Z2 ‡Γ+(CP )

CP - G+(Uc, CP) - Z2 Γe+(CP ) PT - - - - -

TC - - G−(Us, TC) Z2 ‡Γ−(T )

CPT G+(CPT ) G+(Uc, CPT ) G+(Us, CPT ) Z4 Γ+e(CPT ) + + + Z + T , C G−+(T , C) G−+(Uc, T , C) G−+(Us, T , C) 2 Γ−+(T,C) + + + Z + C, P G++(C, P) G++(Uc, C, P) G++(Us, C, P) 2 Γ++(C, P ) − − − Z − P, T G+−(P, T ) G+−(Uc, P, T ) G+−(Uc, P, T ) 2 Γ−+(T, P ) + + + Z + CP, PT G++(CP, PT ) G++(Uc, CP, PT ) G++(Us, CP, PT ) 2 Γ++(CP,PT ) + + Z + G−+(T , CP) G−+(Uc, T , CP) ↓ 2 Γ−+(T,CP ) + + T , CP + + G++(Uc, T , CP), + Z Γ++(T,CP ), G++(T , CP), G−−(T , CP) + G++(Us, T , CP) 4 + G−−(Uc, T , CP) Γ−−(T,CP ) + + + Z + C, PT G++(C, PT ) G++(Uc, C, PT ) G++(Us, C, PT ) 4 Γ++(C,TP ) + + Z + G+−(P, TC) ↓ G+−(Us, P, TC) 2 Γ−+(T,CP ) + e P, TC + + + G++(Us, P, TC), + G++(P, TC), G−−(P, TC) G++(Uc, P, TC) + Z Γ++(TC,P ) G−−(Us, P, TC) 4 −++ −+− G+++(T , C, P), G−++(T , C, P), − −− − − ++− ++ + ++ ++ Z2 Γ− (T,C,P ) G−++(T , C, P), G−++(T , C, P), G−++(Uc, T , C, P) G−++(Us, T , C, P) ++ T , C, P +−+ G−+−(T , C, P) +++ −−− +++ +++ +++ G+++(T , C, P), G++− (T , C, P), G+++(Uc, T , C, P), G+++(Us, T , C, P), Z Γ+++(T,C,P ), −−− +++ +−− −−− 4 +−− G−++(T , C, P), G−+−(T , C, P) G−++(Uc, T , C, P) G++−(Us, T , C, P) Γ−++(T,C,P )

TABLE IV. Classification of 2D interacting nontrivial fermionic non-chiral SPT phases with symmetry groups generated by T , 2 Nf C, P, the combined symmetries, and/or U(1) symmetries. In this table we do not consider the case for SC = −1 [C =(−1) ] 2 Nf and SPT = −1 [(PT ) = (−1) ], so each nontrivial SPT phase in this table can be implemented by a 2 × 2 K-matirx: T T {K,Q,S} = {σz, (1, −1) , (1, 1) }. Symmetry groups shown in this table have removed U(1) gauge redundancy. Classification is obtained for deconfined fermionic SPT phases with perturbative interactions. The last column shows relevant symmetry classes represented in single particle Hamiltonians (non-interacting fermionic systems) from TABLE II; symmetry groups with additional U(1)s can be realized in BdG systems with Sz conservation, and here we list Γ+(CP ) and Γ−(T ) (noted by ‡) as examples (Γ indicates ”T-dual” to Γ). e e e

± G++(C, PT ), there is only one physically inequivalent re- try [either U(1)c or U(1)s]. We can remove such alization of the set of symmetries, which has been char- gauge redundancy based on the algebraic relations be- acterized (or fixed) by fermionic algebraic relations. For tween discrete symmetries and U(1) symmetries (83). S + + T,CP For example, groups G (Us, T , CP), G (Us, T , CP), example, in symmetry group GS S (T , CP) the symme- ++ +− T CP + + tries are represented by G−+(Us, T , CP), G−−(Us, T , CP) are all U(1)s gauge equivalent. T UT = σx, δφT = π(0, ηT) ; (iv) CPT-equivalence: From the topological CPT the- T ηT,CP ηT,CP orem, classification of fermionic SPT phases with symme- UCP = −σx, δφCP = π n + ,n + ηCP + ; 2 2 try groups generated by {Gi} and by {Gi, W0} are equivalent. Here the trivial CPT symmetry W (with W2 = 1) ηT, ηCP, ηT,CP, n =0, 1. 0 0 satisfies the fermionic algebraic relations SW0,Gi = ǫ for As the parameters η’s specify the fermionic symmetry −1 UGi KUGi = ǫK, when it is included to a symmetry group, the ”internal” parameter n can always be fixed group. Some examples can be found in TABLE IV: (to 0, say) by redefining (CP)′ = (−1)Nf ·CP, since the ⊜ ⊜ − fermion parity is conserved. On the other hand, there G−(Uc, T ) G+(Uc, CP) G−+(Uc, T , CP), + ⊜ + ⊜ − are two physically inequivalent realizations of symmetries G−+(T , C) G++(C, P) G+−(P, T ) (specified by some ”internal” parameters) in G±(CPT ) ⊜ + ⊜ ++− ± G++(CP, PT ) G−++(T , C, P), (103) and G++(C, PT ). (iii) There is gauge redundancy for specifying fermionic where ⊜ represents the CPT-equivalence relations for symmetry groups in the presence of U(1) symme- symmetry groups and the last symmetry group in the 19

+ first line is U(1)c gauge equivalent to G−+(Uc, T , CP) in they have representation in terms of parity symmetric TABLE IV. On the other hand, since a nontrivial CPT BdG systems with Sz conservation, which may be more can be represented by by W = W0·M for some onsite uni- realizable. On the other hand, with fine tuning, CP sym- tary Z2 symmetry M, we also have the CPT-equivalence metric systems may be realized in electron-hole coupled among symmetry groups generated by {Gi, M} and by systems (like excitons); As seen in this example, CPT- {Gi, W}. This happens when symmetries in a group can equivalence and T-duality allow us to explore topologi- combine to nontrivial CPT symmetries. For example, cal phases not listed in conventional Altland-Zirnbauer NL M can be the spin parity (chiral Z2 parity) (−1) or classes. Other interesting examples to explore are insu- (−1)NR so that we have lators with T P or {T, P } symmetry, which are dual to known topological superconducting phases. ⊜ G+(CPT ) G+(M), Another issue which we have not discussed is a relation + ⊜ − ⊜ − of these topological classification to quantum anomalies. G++(T , CP) G++(T , M) G++(CP, M) ⊜ + ⊜ − ⊜ − The boundary (edge) theories that we discussed in an- G−−(T , CP) G−+(T , M) G−+(CP, M). (104) alyzing topological classification are not possible to gap out in the presence of symmetry conditions (i.e., “pro- Topological phases protected by these symmetry groups tected” by the symmetries). These theories should not are all characterized by Z classification (while they are 4 exist as an isolated system but should be realized only as all characterized by Z classification for non-interacting a boundary of a bulk topological system. In other words, fermions). Therefore, due to CPT-equivalence, impos- these theories should be, in the presence of an appropri- ing the nontrivial CPT symmetry effectively enforces the ate set of symmetry conditions, anomalous or inconsis- spin parity, resulting the same classification of either non- tent. We plan to visit possible anomalies that pertain interacting or interacting SPT phases. This provides con- to topological phases discussed in this paper in a forth nections between TSCs protected by (nontrivial) CPT coming publication. For the cases of topological phases and TSCs protected by spin parity discussed in Ref. 46, protected by CP symmetry and charge U(1) symmetry, and also between TSCs protected by T and P (the same partial discussion on a quantum anomaly that underlies as by T and CP since C is trivial for Majorana fermions) the topological classification is given in Ref. 61. and TSCs protected by T and spin parity, as discussed in Refs. 47 and 45, respectively. (Actually, all the above TCSs possess Z instead Z classification. The differ- 8 4 ACKNOWLEDGMENTS ence comes from the fact that Majorana edge modes of TCSs have half-integer center charge, while the edges of K-matrix Chern-Simons theory we study here have inte- We thank Gil Young Cho for discussion and collabora- ger center charge. Nevertheless, the above argument is tions in closely related works. in a consistent and reasonable way, as indicated in Ref. 55.) (v) As a comparison, for each symmetry group with Appendix A: Entanglement spectrum and effective symmetries nontrivial SPT phases in TABLE IV, we also list the relevant symmetry classes represented in the single-particle Let us consider a tight-binding Hamiltonian, Hamiltonians from TABLE II. As each Z2 classification is unchanged, each (non-chiral) Z classification changes † αα′ ′ ′ ′ to Z4 classification from non-interacting to interacting H = ψα(r) H (r, r ) ψα (r ) ′ ′ topological phases. Xr,r α,αX (vi) Like bosonic theories, T-duality also holds exactly † = ψI HII′ ψI′ , (A1) between related symmetry groups [with the correspon- I,I′ dence (99)] in TABLE IV. X where ψα(r) (α = 1,...,Nf ) is an Nf -component fermion annihilation operator, and index r = VI. DISCUSSION (r1, r2,...,rd) labels a site on a d-dimensional lattice. In the second line in Eq. (A1), we have used a more com- We have gone through topological classification prob- pact notation with the collective index I = (r, α), etc. lems in the presence of parity symmetry with emphasis Each block in the single particle Hamiltonian H(r, r′) on duality (equivalence) relations among various topo- is an Nf × Nf matrix, satisfying the hermiticity con- logical phases. dition H†(r′, r) = H(r, r′), and we assume the total size One issue which we did not discuss is possible phys- of the single particle Hamiltonian HII′ is Ntot × Ntot = ical realizations of these topological systems considered Nf V ×Nf V , where V is the total number of lattice sites. in this paper. While we leave detailed discussion on this The components in ψ(r) can describe, e.g., orbitals or issue for the future, a few comments are in order; CP spin degrees of freedom, as well as different sites within a symmetric systems are rather exotic in condensed mat- crystal unit cell centered at r. With a canonical transfor- ter context, but we have shown that, through T-duality, mation, the Ntot × Ntot Hamiltonian can be diagonalized 20 as which has ±1 as its eigenvalues. For the total system divided into two subsystems L † U HU = diag(EA), A =1, ··· ,Ntot, and R, we introduce the following block structure,

with U = (~u1,...,~uNtot ), (A2) CL CLR † C = , CRL = CLR. (A9) where ~uA is the A-th eigenvector with the eigenenergy CRL CR ! E . Under this canonical transformation, the fermionic A Then the set of eigenvalues {ξ } of C is the entangle- operator ψ = ψ (r) can be expressed into fermionic ν L I iα ment spectrum. Similarly, the set of eigenvalues of operator χA as QL := 1 − 2CL (A10) Ntot

ψI = UIAχA. (A3) is {1−2ξν}, with −1 ≤ 1−2ξν ≤ +1. We refer CL and QL A=1 as the entanglement Hamiltonian. (These terminologies X may not be entirely precise, since ln(1/ξµ −1) may better Through out the paper, we consider situations where be entitled to be called the entanglement energy.) there is a spectral gap in the single particle Hamiltonian We now derive the algebraic relations obeyed these and the fermi level is located within the spectral gap. blocks, by making use of C2 = C.33,49 Then, Then the ground state |ΨGi at zero temperature can be 2 expressed as CL − CL = −CLRCRL, QLCLR = −CLRQR, Nocc † CRLQL = −QRCRL, |ΨGi = χA|0i, (A4) A=1 2 Y CR − CR = −CRLCLR, (A11) where we assume the eigenvalues EA for A = where QR := 1 − 2CR. This algebraic structure, inher- 1,...,Nocc

Then, the Q-matrix satisfies a chiral symmetry Combining O and the chiral symmetry Λ, † † ∗ QΛ= −ΛQ. (A17) QΛO =ΛO Q . (A23) In particular, This effective chiral symmetry can be combined with † † ∗ other physical symmetries. E.g., CP symmetry. QLCLROLR = CLROLRQL, † † ∗ QRCRLORL = CRLORLQR. (A24) 2. Properties of entanglement spectrum with These show that the Q-matrix and its diagonal blocks symmetries QL,R obey an effective time-reversal symmetry.

We now focus on the case where the dimensions of the two Hilbert spaces L and R are the same. (In this case, in Appendix B: Calculations of symmetry general, there is no zero mode of S expected from SUSY.) transformations and SPT phases in K-matrix In addition, we will consider the cases where there is a theories (discrete) symmetry which relates (or: “intertwines”) the two Hilbert spaces. As in the case of the symmetry pro- 1. Algebraic relations of symmetry operators tected topological phases, such cases arise when there is a (discrete) symmetry in the total system before bipar- As mentioned in the text, we consider a symmetry titioning, and when the bipartitioning is consistent with group generated by Z2 symmetry operators (T , C, P, the symmetry. and the combined symmetries in our discussion) with the Let us consider a symmetry operation O that acts on following algebraic relations: the fermion operator as follows: 2 Nf Gi = SGi , ∀ discrete Gi ∈ G; † † − − Nf OψI O = OIJ ψJ , (A18) G G G 1G 1 = S , ∀ discrete G , G ∈ G, (B1) i j i j Gi,Gj i j where O is an Nf V ×Nf V unitary matrix. The system is where S has values ±1. In a bosonic system, the oper- † invariant under the symmetry operation when OHO = ator SNf (subscript omitted) is just the identity 1. In H, i.e., a fermionic system, SNf can be the identity 1 or the Nf fermion number parity operator Pf ≡ (−1) , where O†HT O = −H. (A19) Nf is the total fermion number operator. Note that S = S for any two symmetry operators G and This symmetry property of the Hamiltonian is inherited G1,G2 G2,G1 1 G . by the correlation matrix, 2 On the other hand, algebraic relations between Z2 O†Q∗O = −Q, (A20) symmetries and U(1) symmetries are described as follows. The total charge operator Defining a block structure as in Eq. (A9), we have e N ≡ dxj0 = Q dx∂ φ (t, x)= eN . (B2) c c 2π I x I f OL OLR Z Z O = . (A21) and the corresponding charge U(1) transformation U ≡ O O c RL R ! eiθcNc/e satisfy the following relations: −1 Our focus below is the case where the symmetry oper- GNcG = Nc, for G = T , P, TP; ation intertwines the L and R Hilbert spaces. In other GN G−1 = −N , for G = C, TC, CP, CPT ; words, we can naturally categorize discrete symmetries c c −1 into two groups; Firstly, there are symmetry operations GUcG = Uc, for G = P, TC, CPT ; which act on L and R Hilbert spaces independently. If −1 −1 GUcG = U , for G = T , C, CP, TP. (B3) the bipartitioning is done in the manner that respects c the locality of the system, these include local symme- Similarly, the total spin operator try operations such as time-reversal symmetry, and spin- s N ≡ dxj0 = S dx∂ φ (t, x) (B4) rotation symmetry, etc. On the other hand, certain spa- s s 2π I x I tial symmetries such as reflection, inversion, and (dis- Z Z crete) spatial rotations can exchange (intertwines) the and the corresponding spin U(1) transformation Us ≡ iθ N /s two sub Hilbert spaces. Focusing on the latter situations, e s s satisfy we thus assume the following off-diagonal form: −1 GNsG = Ns, for G = TC, CP, TP; −1 0 O GNsG = −Ns, for G = T , C, P, CPT ; O = LR , O O† = O O† =1. LR LR RL RL GU G−1 = U , for G = T , CP, CPT ; ORL 0 ! s s −1 −1 (A22) GUsG = Us , for G = C, P, TC, TP. (B5) 22

Again for the bosonic systems we just take all η = 0. Note that in the above (and the following) equations all Sym. Transformations (boson: K = σx) phases are mod 2π.

T UT = σz, δφT = nTπv The constraints including the U(1) symmetry (B3) and (B5) also become C UC = −I2, δφC = 0

P UP = σz, δφP = nPπu −1 ±1 ∓1 2 Nf GUAG = UA or GUA GUA = G = SG : CP UCP = −σz, δφCP = nCPπv (IN + αGUG)δφG + (IN ∓ αGUG)δφA = ηGπtN PT UPT = I2, δφPT = 0 ⇒ (IN ∓ αGUG)δφA =0, (B8) TC UTC = −σz, δφTC = nTCπu CPT UCPT = −I2, δφCPT = nCPTπu + mCPTπv where the label A represents charge or spin U(1) symme- UT = σz, δφT = nTπu + mTπv tries. As examples, for T, C, and P symmetries we have T , C UC = −I2, δφC = 0 the corresponding U(1) symmetries (if present) satisfying

UC = −I2, δφC = 0 C, P TRS : (IN − UT) δφc =0, (IN + UT) δφs =0, UP = σz, δφP = nPπu + mPπv PHS : (IN + UC) δφc =0, (IN + UC) δφs =0, UP = σ , δφP = nPπu P, T z UT = σz, δφT = nTπv PS :(IN − UP) δφc =0, (IN + UP) δφs =0, (B9)

UCP = −σz, δφCP = nCPπu + mCPπv −1 CP, PT respectively. Identifying δφc = θcK Q and δφs = UPT = I2, δφPT = 0 −1 θsK S, (B9) exactly corresponds to (79), which are UT = σ , δφT = nTπv T , CP z equations of symmetry constraints for gauged K-matrix UCP = −σz, δφCP = nCPπu + mCPπv Chern-Simons theories (coupled to external gauge fields). UC = −I2, δφC = 0 C, PT In this appendix, we use these constraint equations to UPT = I2, δφPT = nPTπu + mPTπv find the way that the chiral boson fields transform under

UP = σz, δφP = nPπu T , C, P, the combined symmetries, and/or the U(1) sym- P, TC metries. Note that we also have the gauge equivalence UTC = −σz, δφTC = nTCπu + mTCπv for the forms of these symmetry transformations: UT = σz, δφT = nTπu + mTπv

T , C, P UC = −I2, δφC = 0 −1 −1 {UG, δφG} →{X UGX, X (δφG − αG∆φ + UG∆φ)}, UP = σ , δφP = nPπu + mPπv z if X ∈ GL(N, Z), XT KX = K, (B10) TABLE V. (Gauge inequivalent) symmetry transformations where αG = 1 (−1) if G is an unitary (antiunitary) oper- in different classes for bosonic K-matrix theories with K = σx. In this table: u = (1, 0)T , v = (0, 1)T , and all n’s and m’s ator. This means we can choose some X and ∆φ to fix have the value 0 or 1. {UGi , δφGi } to the inequivalent forms of transformations. Here we just consider the case of 2 × 2 K matrix (K = σx for a bosonic system and a K = σz for fermionic system). All gauge inequivalent solutions for discrete symmetry transformations in different classes are sum- 2. Equations of identity elements marized in TABLE V for bosonic systems, and in TABLE VI for fermionic systems, respectively. In the basis of chiral boson fields, the constraints (B1) give the following equations of identity elements (for 3. Discussion of SPT phases bosonic systems we just take all SNf = 1 in the following expressions): Using the criteria for SPT phases developed in Sec. 2 Nf 2 V B, we can find nontrivial SPT phases and thus topolog- G = S : U = IN , (IN + αG UG )δφG = ηG πtN ; i Gi Gi i i i i ical classification for both bosonic and fermionic systems, −1 −1 Nf 2 Nf based on the symmetry transformations described in TA- GiGj Gi Gj = SGi,Gj or (GiGj ) = SGi SGj SGi,Gj : BLEs V and VI. The results are summarized in TABLEs (U U )2 = I , (B6) Gi Gj N VII and VIII, which directly derive TABLEs III and IV

(IN + αGi αGj UGj UGi ) δφGj + UGj δφGi = ηGi,Gj πtN , in the text, respectively. To accomplish TABLEs VII and VIII, we give explicit where αG = 1 (−1) represents an unitary (antiunitary) calculations for SPT phases with some examples. In T operator G, tN ≡ (1, ··· , 1) is an N-component vector, the following discussions, bosonic systems are realized T T and the numbers ηG and ηG1,G2 via the relations to SG by {K,Q,S} = {σx, (0, 1) , (1, 0) }, while fermionic sys- T T and SG1,G2 : tems are realized by {K,Q,S} = {σz, (1, −1) , (1, 1) }. The discussion for ﬁnding group structures of SPT phases iπηG iπηG1,G2 e = SG, e = SG1 SG2 SG1,G2 . (B7) follows Ref. 55. 23

Sym. Transformations (fermion: K = σz)

T UT = σx, δφT = ηTπv

C UC = −I2, δφC = 0

P UP = σx, δφP = ηPπv

CP UCP = −σx, δφCP = ηCPπv

PT UPT = I2, δφPT = 0

TC UTC = −σx, δφTC = ηTCπv ηCPT ηCPT CPT UCPT = −I2, δφCPT = n + 2 πu + m + 2 πv ηT−ηT,C ηT+ηT,C UT = σ , δφT = n + πu + n − πv T , C x 2 2 UC = −I2, δφC = 0

UC = −I2, δφC = 0 C, P − U σ δφ n ηP ηC,P πu n ηP+ηC,P πv P = x, P = + 2 + + 2 UP = σ , δφP = ηPπv P, T x UT = σx, δφT = ηTπv ηCP+ηCP,PT ηCP−ηCP,PT UCP = −σ , δφCP = n + πu + n − πv CP, PT x 2 2 UPT = I2, δφPT = 0

UT = σ , δφT = ηTπv T , CP x ηT,CP ηT,CP UCP = −σx, δφCP = n + πu + n + ηCP + πv 2 2 UC = −I2, δφC = 0 C, PT ηC,PT ηC,PT UPT = I2, δφPT = n + πu + m + πv 2 2 UP = σ , δφP = ηPπv P, TC x ηP,TC ηP,TC UTC = −σx, δφTC = n + 2 πu + n + ηTC − 2 πv − U σ δφ n ηT ηT,C πu n ηT+ηT,C πv T = x, T = + 2 + − 2 T , C, P UC = −I2, δφC = 0 − U σ δφ n ηP ηC,P πu n ηP+ηC,P πv P = x, P = + 2 + + 2

TABLE VI. (Gauge inequivalent) symmetry transformations in diﬀerent classes for fermionic K-matrix theories with K = σz. In this table: (i) u = (1, 0)T , v = (0, 1)T , and all η’s [deﬁned in (B7)], n’s, and m’s have the value 0 or 1 ; (ii) here we do not consider cases for ηC = 1 and ηPT = 1, which must be realized in the theory with a 4 × 4 K-matrix at least; (iii) ηP,T =0 in symmetry classes {P, T } and {C, P, T }, as deduced from the equations of identity element (B6).

CP a. Bosonic SPT phases with symmetry group Z2 say, hφ1i has expectation value 0 or π (depending on how s.p. we choose Sedge). Thus there are just trivial phases in From Table V, the symmetry transformations for CP the absence of U(1) symmetries. are given by In addition to CP, if we now include the charge U(1) T symmetry UCP = −σz, δφCP = π(0,nCP) , nCP =0, 1. (B11) −1 T R Without U(1) symmetries, the symmetry invariant Uc : δφc = θcK Q = qcθc(1, 0) , θc ∈ , (B15) perturbations can be either (for any values of n ) CP then only the charge preserved perturbations (bosonic variable φ ) are allowed to add to the system (to con- Sc.p. = A dtdx cos (2lφ + α ) (B12) 2 edge l 2 l dense). Since ∈Z Xl Z or CP hφ2i −→ hφ2i + nCPπ mod 2π, (B16) s.p. Sedge = Bl dtdx cos(lφ1 + klπ) , (B13) we ﬁnd that nCP = 1 (nCP = 0) corresponds to a non- l∈Z X Z trivial (trivial) SPT phase, as the edge can not (can) R Z c.p. where Al, Bl, αl ∈ , kl ∈ , and c.p. (s.p.) stands be gapped by Sedge without breaking CP. Denoting for ”charge preserved” (”spin preserved”) perturbations, [nCP] the phase with corresponding CP symmetry, we i.e., invariant under U(1)c [U(1)s]. In this case, we can have [1] ⊕ [1] = [0], i.e., putting (adding) two copies of always condense φ without breaking CP: k k 1 nCP = 1 phases {φ1 , φ2 , k = 1, 2} together we can gap 1 2 1 2 CP out the edge by condensing {φ1 − φ1, φ2 + φ2} without hφ1i −→ −hφ1i mod 2π, (B14) spontaneously breaking CP and charge U(1) symmetries. 24

Nontrivial bosonic non-chiral SPT phases and their classiﬁcation Sym. group Parameters (from TABLE V) No U(1)’s U(1)c is present U(1)s is present T Z2 [nT] - 0 [1] Z2 - 0 C Z2 - - 0 - 0 - 0 P Z2 [nP] - 0 - 0 [1] Z2 CP Z2 [nCP] - 0 [1] Z2 - 0 PT Z2 - - 0 - 0 - 0 TC Z2 [nTC] - 0 - 0 [1] Z2 CPT Z2 [nCPT,mCPT] [1, 1] Z2 [1, 1] Z2 [1, 1] Z2 T C Z2 × Z2 [nT,mT] [1, 1] Z2 [1, 1] Z2 [1, 1] Z2 C P Z2 × Z2 [nP,mP] [1, 1] Z2 [1, 1] Z2 [1, 1] Z2 P T Z2 × Z2 [nP, nT] [1, 1] Z2 [1, 1] Z2 [1, 1] Z2 CP PT Z2 × Z2 [nCP,mPT] [1, 1] Z2 [1, 1] Z2 [1, 1] Z2 T CP [0, 1, 1], [1, 1, 0], 2 [0, 1, 1], [1, 1, 0], 2 Z Z2 × Z2 [nT, nCP,mCP] Z2 Z2 [0, 1, 1] 2 [1, 1, 1] [1, 1, 1] C PT Z2 × Z2 [nPT,mPT] [1, 1] Z2 [1, 1] Z2 [1, 1] Z2 P TC [0, 1, 1], [1, 0, 1], Z2 Z [0, 1, 1], [1, 0, 1], Z2 Z2 × Z2 [nP, nTC,mTC] 2 [0, 1, 1] 2 2 [1, 1, 1] [1, 1, 1] 15 nontrivial phases [0, 0, 1, 1], [0, 1, 1, 0], [0, 0, 1, 1], [1, 0, 0, 1], T C P generated by Z4 Z2 Z2 × Z2 × Z2 [nT,mT, nP,mP] 2 Z2 2 [0, 0, 1, 1], [0, 1, 1, 0], 2 [0, 1, 1, 1] [1, 0, 1, 1] [1, 0, 0, 1], [1, 1, 0, 0]

TABLE VII. Nontrivial bosonic non-chiral SPT phases and their classification with symmetry groups generated by T , C, P, the T T combined symmetries, and/or U(1) symmetries, implemented by a 2 × 2 K-matrix: {K,Q,S} = {σx, (0, 1) , (1, 0) }. Results are based on symmetry transformations described in TABLE V. Here we use [n, m, · · · ] to label SPT phases (parameters n’s and m’s shown here are not specified explicitly in symmetry groups in bosonic systems). Classification shown in this table has removed U(1) gauge redundancy. TABLE III is constructed directly from this table.

⋊ CP c.p. s.p. Therefore, bosonic SPT phases with U(1)c Z2 are clas- gapped, as either Sedge (B12) or Sedge is added to the siﬁed by a Z2 group: [nCP = 0] and [nCP = 1] correspond system, without breaking any symmetry for some choices to (two) elements of this Z2 group. of n’s. The following lists the nontrivial SPT phases that On the other hand, if we include the spin U(1) sym- are denoted as [nT,mT,nP,mP]: metry instead [no charge U(1)]

−1 T Us : δφs = θsK S = qsθs(0, 1) , θs ∈ R, (B17) then only the the spin preserved perturbations (bosonic [0, 0, 1, 1], [0, 1, 1, 0], [0, 1, 1, 1], variable φ1) are allowed to add to the system (to con- [1, 0, 0, 1], [1, 1, 0, 0], [1, 1, 0, 1], dense). In this case, there are just trivial phases as we [1, 0, 1, 1], [1, 1, 1, 0], [1, 1, 1, 1]. (B19) can alway condense φ1 (B14). b. Bosonic SPT phases with symmetry group ZT × ZC × ZP 2 2 2 Putting two identical copies of each phase above to- 2 gether will result a trivial phase, i.e., [nT,mT,nP,mP] = From Table V, the symmetry transformations for [trivial]. On the other hand, all phases above are inequiv- {T , C, P} are given by alent [we can not obtain a trivial phase by putting any U = σ , δφ = π(n ,m )T , n , m =0, 1; two diﬀerent elements in the set (B19) together]. How- T z T T T T T ever, phases shown in (B19) are not all possible nontrivial UC = −I2, δφC = 0; T C P SPT phases with symmetries Z2 × Z2 × Z2 ; putting dif- T UP = σz, δφP = π(nP,mP) , nP, mP =0, 1. (B18) ferent phases above together might result other nontrivial SPT phases that are not listed above. To determine the Without U(1) symmetries, the symmetry invariant group structure of these phases, we observe that putting c.p. perturbations can be either (for any values of n’s) Sedge three phases in any vertical or horizontal line of the array s.p. (B12) or Sedge (B13). However, the edge can not be (B19) together will result a trivial phase, or equivalently, 25

Sym. groups for nontrivial fermionic non-chiral SPT phases Top. Sym. Parameters (from TABLE VI) No U(1)’s U(1)c is present U(1)s is present class.

T ηT - 1 - Z2 C - - - - -

P ηP - - 0 Z2

CP ηCP - 0 - Z2 PT - - - - -

TC ηTC - - 1 Z2

CPT ηCPT 0 0 0 Z4

T , C (ηT, ηT,C) (1, 1) (1, 1) (1, 1) Z2

C, P (ηP, ηC,P) (0, 0) (0, 0) (0, 0) Z2

P, T (ηP, ηT) (0, 1) (0, 1) (0, 1) Z2

CP, PT (ηCP, ηCP,PT) (0, 0) (0, 0) (0, 0) Z2

(1, 0, 1) (1, 0, 1) ↓ Z2 T , CP (ηT, ηCP, ηT,CP) (0, 0, 0), (1, 1, 0) (0, 0, 0), (1, 1, 0) (0, 0, 0) Z4

C, PT ηC,PT 0 0 0 Z4

(0, 1, 1) ↓ (0, 1, 1) Z2 P, TC (ηP, ηTC, ηP,TC) (0, 0, 0), (1, 1, 0) (0, 0, 0) (0, 0, 0), (1, 1, 0) Z4 (0, 0, 1, 0), (1, 0, 0, 0), Z (1, 0, 1, 0), (1, 0, 1, 1), (1, 0, 1, 0) (1, 0, 1, 0) 2 T , C, P (ηT, ηP, ηT C, ηC P) , , (1, 1, 1, 0) (0, 0, 0, 0), (0, 1, 1, 0), (0, 0, 0, 0), (0, 0, 0, 0), Z4 (1, 0, 0, 1), (1, 1, 1, 1) (1, 0, 1, 1) (0, 1, 1, 0)

TABLE VIII. Symmetry groups for nontrivial fermionic non-chiral SPT phases and topological classification, by considering T T T , C, P, the combined symmetries, and/or U(1) symmetries for a 2 × 2 K-matrix: {K,Q,S} = {σz, (1, −1) , (1, 1) }. Results are based on symmetry transformations described in TABLE VI. Parameters η’s shown here can be transferred to the signs S’s that characterize symmetry groups in fermionic systems by Eq. (B7) (note that ηC, ηPT, and ηP,T, which are all zero, are not specified as parameters). Symmetry groups and classification shown in this table have removed U(1) gauge redundancy. TABLE IV is constructed directly from this table. we have On the other hand, if we include the spin U(1) symmetry (B17) instead [no charge U(1)], then only the spin [0, 0, 1, 1] ⊕ [0, 1, 1, 0] = [0, 1, 1, 1] preserved perturbations (bosonic variable) are allowed to ⊕ ⊕ ⊕ add to the system (to condense). Similar to the discus- [1, 0, 0, 1] ⊕ [1, 1, 0, 0] = [1, 1, 0, 1] sion for charge U(1), the U(1)s gauge inequivalent non-

= = = trivial phases can be [0, 0, 1, 1], [1, 0, 0, 1], and [1, 0, 1, 1], which form a Z2 group. [1, 0, 1, 1] ⊕ [1, 1, 1, 0] = [1, 1, 1, 1]. (B20) 2 From this fact, we know that all nontrivial phases can be generated by a speciﬁc set of four phases, say, c. Fermionic SPT phases with symmetry CP {[0, 0, 1, 1], [0, 1, 1, 0], [1, 0, 0, 1], [1, 1, 0, 0]} (there are 3 × 3 = 9 equivalent choices for these four group gen- From Table VI, the symmetry transformations for CP erators); there are total ﬁfteen nontrivial SPT phases, are given by which form a Z4 group. 2 U = −σ , δφ = π(0, η )T , η =0, 1. (B21) In addition to T , C, and P, if we now include the charge CP x CP CP CP U(1) symmetry (B15), then only the charge preserved Without U(1) symmetries, the symmetry invariant perturbations (bosonic variables) are allowed to add to perturbations can be either (for any values of ηCP) the system (to condense). Besides those nontrivial phases c.p. discussed in the case without U(1) symmetries, there are Sedge = Al dtdx cos [2l(φL − φR)+ αl] (B22) other nontrivial phases protected by U(1) additionally. l∈Z c X Z Some of the nontrivial phases are U(1)c gauge equivalent or to each other, and we can just consider the inequivalent Z2 Ss.p. = B dtdx cos [2l(φ + φ )+ k π] , (B23) set [0, 0, 1, 1], [0, 1, 1, 0], and [0, 1, 1, 1], which form a 2 edge l L R l ∈Z group. Xl Z 26

R Z s.p. where Al, Bl, αl ∈ and kl ∈ . Note that under CP (B22) or Sedge (B23). Note that under T /CP the boson the boson ﬁelds lT φ ≡ φ ± φ transform as T ± L R ﬁelds l±φ ≡ φL ± φR transform as

− l± T 1 T − l± (CP)(l±φ)(CP) = l± (UCPφ + δφCP) + ∆φ , (B24) T 1 T CP T (l±φ)T = l± (−UTφ + δφT) + ∆φT , l l T −1 T ± ± (CP)(l±φ)(CP) = l± (UCPφ + δφCP) + ∆φCP, (B30) with the statistical phase ∆φCP = π mod 2π (while statistical phases in a bosonic K-matrix theory is always l± trivial; see discussions in Sec. V A). In this case, we can with the statistical phases ∆φT = 0 mod2π and l± always condense l+φ without breaking CP: ∆φCP = π mod 2π, respectively. Label the phases with symmetry transformations CP hl+φi −→ −hl+φi + (ηCP + 1)π mod 2π, (B25) (B29) as [ηT, ηCP, ηT,CP ,n]. Then we want to study the group structure of these phases protected by symmetry sT,CP say, hl+φi has expectation value (ηCP ±1)π/2 (depending group Gs s . As we are interested, the nontrivial SPT s.p. T CP on how we choose Sedge). Thus there are just trivial phases, where the edge cannot be gapped out without phases in the absence of U(1) symmetries. spontaneously breaking T or CP symmetries, are found In addition to CP, if we now include the charge U(1) to be: symmetry (1) Z2 classes: When (ηT, ηCP, ηT,CP) = (1, 0, 1), + T which is the symmetry group G−+(T , CP), neither l+φ = U : δφ = θ K−1Q = q θ (1, 1)T , θ ∈ R, (B26) T c c c c c c φL + φR or l−φ = φL − φR can condense to be invariant under both T and CP. Thus [1, 0, 1,n] 6= 0 (n can then only the charge preserved perturbations (bosonic be 0 or 1) are nontrivial SPT phase. However, for two variable l−φ) are allowed to add to the system (to con- k k copies of this theory with variables {φL, φR, k =1, 2}, the dense). Since 1 2 1 2 edge can be gapped by condensing {φL − φR, φR − φL} CP without spontaneously breaking any symmetries. So we hl−φi −→ hl−φi + (ηCP + 1)π mod 2π, (B27) have [1, 0, 1,n]2 = [1, 0, 1,n] ⊕ [1, 0, 1,n] = 0. On the other hand, it is easy to show [1, 0, 1, 0] ⊕ [1, 0, 1, 1] = 0, we find that ηCP = 0 (ηCP = 1) corresponds to a non- which means [1, 0, 1, 0]−1 = [1, 0, 1, 1], and thus we have trivial (trivial) SPT phase, as the edge can not (can) c.p. [1, 0, 1, 0] = [1, 0, 1, 1], i.e., these two phases correspond be gapped by Sedge without breaking CP. Moreover, to the same nontrivial SPT phase. Therefore, the topo- SPT phases with η = 0 [group G (U , CP)] form a Z + Z CP + c 2 logical classification for G−+(T , CP) forms a 2 group, group. This can be seen if we put two copies of ηCP =0 generated by the element [1, 0, 1, 0] = [1, 0, 1, 1]. k k phases {φL, φR, k = 1, 2} together and then gap out the (2) Z4 classes: When (ηT, ηCP, ηT,CP) = edge by condensing independent bosonic variables, say, (0, 0, 0) [G+ (T , CP)] or (1, 1, 0) [G+ (T , CP)], the 1 2 1 2 ++ −− {φL − φR, φR − φL}, without spontaneously breaking CP family of condensed bosonic fields [or the set of in- and charge U(1) symmetries. dependent elementary bosonic variables (94)] are not On the other hand, if we include the spin U(1) sym- all invariant under both T and CP for one, two, metry instead [no charge U(1)] and three copies of the edge theory. Only when we consider four copies of the theory the edge can be −1 T R Us : δφs = θsK S = qsθs(1, −1) , θs ∈ , (B28) gapped out without breaking any symmetry. Thus both these symmetry groups have Z classification. To be then only the the spin preserved perturbations (bosonic 4 more specific, let us consider the case for symmetry variable l+φ) are allowed to add to the system (to con- group G+ (T , CP). For G+ (T , CP) it is easy to dense). In this case, there are just trivial phases as we ++ ++ show [0, 0, 0,n] for n = 0, 1 are both nontrivial SPT can alway condense l+φ (B25). phases. Now consider two copies of the edge theory with k k variables {φL, φR, k = 1, 2}, and extended K-matrix K = K ⊕ K = σ ⊕ σ . Then we find that (either n = d. Fermionic SPT phases with symmetries {T , CP} 2 z z 0 or 1), for any independent Haldane null vectors l1 and T −1 T −1 T −1 l2 (which satisfy l1 K2 l1 = l2 K2 l2 = l1 K2 l2 = 0), From Table VI, the symmetry transformations for there exists an elementary bosonic variable {T , CP} are given by T T 1 2 T va φ = la φ/gcd(la,1,la,2,la,3,la,4), where φ = (φ , φ ) T and la ≡ a1l1 + a2l2 is some linear combination UT = σx, δφT = π(0, ηT) ; T of l1 and l2, such that va φ cannot condense to T ηT,CP ηT,CP be invariant under both T and CP. This means U = −σ , δφ = π n + ,n + η + ; CP x CP 2 CP 2 [0, 0, 0,n]2 for n = 0, 1 are also nontrivial SPT phases. ηT, ηCP, ηT,CP, n =0, 1. (B29) Then, if we put four [0, 0, 0,n] states with edge k k variables {φL, φR, k = 1, 2, 3, 4} together, the edge Without U(1) symmetries, the symmetry invariant can be gapped out without breaking any symme- c.p. perturbations can be either (for any values of η’s) Sedge try, by localizing the following independent bosonic 27

1 2 3 4 1 2 3 variables {φL + φL + φR + φR, φR + φR + φL + (i) W0 preserves 4 1 1 3 4 1 1 3 4 φL, φL + φR + φL + φR, φL + φR + φR + φL}, i.e., we have [0, 0, 0,n]4 = 0 for n = 0, 1. On the other bosonic hand, we also have [0, 0, 0, 0] ⊕ [0, 0, 0, 1] = 0, which Sint = U dtdx cos ΛT Kφ + α (C3) means [0, 0, 0, 0]−1 = [0, 0, 0, 1], and thus we have edge Λ Λ 3 Λ∈ZN Z [0, 0, 0, 0] = [0, 0, 0, 1], from the above result. Therefore, X + Z all different phases of G++(T , CP) form a 4 group, −1 generated by the element [0, 0, 0, 0] = [0, 0, 0, 1] . with any collections of bosonic vectors {Λa} (i.e., + T Similar analysis can be applied to G−−(T , CP). πΛ KΛ = 0 mod2π) satisfying Haldane’s null vector In addition to T and CP, if we now include the charge condition (93), and U(1) symmetry (B26), then only the the charge preserved (ii) edge states are gapped without breaking W0 spon- perturbations (bosonic variables) are allowed to add to taneously: all elementary bosonic variables {vT φ} [de- the system (to condense). In this case, we find topolog- a + fined in (94)] are invariant under W0. ical phases protected by (1, 0, 1) [G−+(Uc, T , CP)] − For a bosonic system with generic K-matrix K = and by (1, 0, 0) [G−+(Uc, T , CP)] are clas- IN/2 ⊗ σx, a trivial CPT operator W0 can be chosen sified by Z2, while those protected by + + as {ηW0 ,UW0 , δφW0 } = {0, −IN , 0}. Since for a bosonic (0, 0, 0) [G++(Uc, T , CP)], by (1, 1, 0) [G−−(Uc, T , CP)], Λ − system the statistical phase factor is trivial (∆φW0 = 0 by (0, 0, 1) [G (Uc, T ,byCP)], and by ++ mod 2π), we have W ΛT Kφ(t, x)W−1 =ΛT Kφ(−t, −x) (1, 1, 1) [G− (U , T , CP)] are classified by Z , as 0 0 −− c 4 for any Λ ∈ ZN and thus any interactions Sint and the we apply similar argument from previous cases. We edge T can check the gauge equivalence among these symmetry associated {va φ} are invariant under W0. groups in the presence of U(1)c symmetry (See TABLE For a fermionic system with generic K-matrix K = VIII). IN/2 ⊗ σz, a trivial CPT operator W0 can be cho-

On the other hand, if we include the spin U(1) sym- sen as {ηW0 ,UW0 , δφW0 } = {0, −IN ,tN/2 ⊗ χL} or T metry (B28) instead [no charge U(1)], then only the spin {0, −IN ,tN/2 ⊗ χR}, where χL ≡ (π, 0) and χR ≡ preserved perturbations (bosonic variables) are allowed (0, π)T . Due to the nontrivial statistical phase factor to add to the system (to condense). In this case, there that may arise in a fermionic system, under W0 we have − is only Z4 classiﬁcation for nontrivial SPT phases, corre- T 1 T T W0Λ Kφ(t, x)W0 = Λ Kφ(−t, −x)+Λ KδφW0 + + Λ sponding to symmetry groups (0, 0, 0) [G++(Us, T , CP)], ∆φW0 . Now we show that, for any bosonic vectors Λ sat- + + T Λ (0, 1, 1) [G+−(Us, T , CP)], (1, 0, 1) [G−+(Us, T , CP)], and isfying Haldane’s null vector criterion, Λ KδφW +∆φ + 0 W0 (1, 1, 0) [G−−(Us, T , CP)], respectively. Again, there is is a multiple of 2π. Considering the case δφW0 = gauge equivalence among these symmetry groups in the tN/2 ⊗ χL, we have presence of U(1)s symmetry (See TABLE VIII).

T Λ KδφW0 = π(Λ1 +Λ3 + ··· +ΛN−1)

Appendix C: Proof of topological CPT theorem for = π ΛI . (C4) interacting non-chiral SPT phases in two dimensions oddXI

The CPT symmetry W satisﬁes [Eqs. (97) and (98)] On the other hand, the statistical phase factor associated with Λ is given by Eq. (89):

−1 Wφ(t, x)W = −UWφ(−t, −x)+ δφW, Λ 1 −1 −1 T T T ∆φW0 ≡ ΛI ΛJ [W0(iKφ)I W0 , W0(iKφ)J W0 ] UWKUW = K, IN + UW Q =0, IN + UW S =0, 2i I

1 M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 14 D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, (2010). and M. Z. Hasan, Nature (London) 452, 970 (2008). 2 X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 15 Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, (2011). X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. 3 A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142 Zhang, I. R. Fisher, Z. Hussain, and Z.-X. Shen, Science (1997). 325, 178 (2009). 4 C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 16 D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, (2005). J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. 5 C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 Hor, R. J. Cava, and M. Z. Hasan, Science 323, 919 (2009). (2005). 17 Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Ban- 6 B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science sil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, 314, 1757 (2006). Nat. Phys. 5, 398 (2009). 7 L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 18 Y. Wada, S. Murakawa, Y. Tamura, M. Saitoh, Y. Aoki, 106803 (2007). R. Nomura, and Y. Okuda, Phys. Rev. B 78, 214516 8 J. E. Moore and L. Balents, Phys. Rev. B 75, 121306 (2008). (2007). 19 S. Murakawa, Y. Tamura, Y. Wada, M. Wasai, M. Saitoh, 9 R. Roy, Phys. Rev. B 79, 195321 (2009). Y. Aoki, R. Nomura, Y. Okuda, Y. Nagato, M. Yamamoto, 10 A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud- S. Higashitani, and K. Nagai, Phys. Rev. Lett. 103, wig, Phys. Rev. B 78, 195125 (2008). 155301 (2009). 11 A. Y. Kitaev, AIP Conf. Proc. 1134, 22 (2009). 20 S. Murakawa, Y. Wada, Y. Tamura, M. Wasai, M. Saitoh, 12 G. E. Volovik, Universe in a helium droplet (Oxford Uni- Y. Aoki, R. Nomura, Y. Okuda, Y. Nagato, M. Yamamoto, versity Press, 2003). S. Higashitani, and K. Nagai, J. Phys. Soc. Jpn. 80, 13 M. Konig, S. Wiedmann, C. Brune, A. Roth, H. Buhmann, 013602 (2011). L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 21 L. Fu, Phys. Rev. Lett. 106, 106802 (2011). 318, 766 (2007). 22 J. C. Y. Teo, L. Fu, and C. L. Kane, Phys. Rev. B 78, 29

045426 (2008). (2013). 23 C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. B 41 C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. 86, 115112 (2012). Lett. 112, 106401 (2014). 24 T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and 42 L. Fidkowski and A. Kitaev, Phys. Rev. B 81, 134509 L. Fu, Nat. Commun. 3, 982 (2012). (2010). 25 S.-Y. Xu, C. Liu, N. Alidoust, M. Neupane, D. Qian, I. Be- 43 L. Fidkowski and A. Kitaev, Phys. Rev. B 83, 075103 lopolski, J. D. Denlinger, Y. J. Wang, H. Lin, L. A. Wray, (2011). B. Landolt, J. H. Slomski, J. H. Dil, A. Marcinkova, E. Mo- 44 A. M. Turner, F. Pollmann, and E. Berg, Phys. Rev. B rosan, Q. Gibson, R. Sankar, F. C. Chou, R. J. Cava, 83, 075102 (2011). A. Bansil, and M. Z. Hasan, Nat. Commun. 3, 1192 (2012). 45 X.-L. Qi, New J. Phys. 15, 065002 (2013). 26 Y. Tanaka, Z. Ren, T. Sato, K. Nakayama, S. Souma, 46 S. Ryu and S.-C. Zhang, Phys. Rev. B 85, 245132 (2012). T. Takahashi, K. Segawa, and Y. Ando, Nat. Phys. 8, 47 H. Yao and S. Ryu, Phys. Rev. B 88, 064507 (2013). 800 (2012). 48 X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 27 P. Dziawa, B. J. Kowalski, K. Dybko, R. Buczko, A. Szczer- 78, 195424 (2008). bakow, M. Szot, E. Lusakowska, T. Balasubramanian, 49 T. L. Hughes, E. Prodan, and B. A. Bernevig, B. M. Wojek, M. H. Berntsen, O. Tjernberg, and T. Story, Phys. Rev. B 83, 245132 (2011). Nat. Mater. 11, 1023 (2012). 50 P.-Y. Chang, C. Mudry, and S. Ryu, ArXiv e-prints 28 F. Zhang, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. (2014), arXiv:1403.6176 [cond-mat.str-el]. 111, 056403 (2013). 51 K. Shiozaki and M. Sato, ArXiv e-prints (2014), 29 Y. Ueno, A. Yamakage, Y. Tanaka, and M. Sato, Phys. arXiv:1403.3331 [cond-mat.mes-hall]. Rev. Lett. 111, 087002 (2013). 52 P. Hosur, S. Ryu, and A. Vishwanath, Phys. Rev. B 81, 30 S. Ryu, A. P. Schnyder, A. Furusaki, and A. Ludwig, New 045120 (2010). J. Phys. 12, 065010 (2010). 53 J. C. Y. Teo and C. L. Kane, Phys. Rev. B 82, 115120 31 C.-K. Chiu, H. Yao, and S. Ryu, Phys. Rev. B 88, 075142 (2010). (2013). 54 M. Levin and A. Stern, Phys. Rev. B 86, 115131 (2012). 32 T. Morimoto and A. Furusaki, Phys. Rev. B 88, 125129 55 Y.-M. Lu and A. Vishwanath, Phys. Rev. B 86, 125119 (2013). (2012). 33 A. M. Turner, Y. Zhang, R. S. Mong, and A. Vishwanath, 56 T. Neupert, L. Santos, S. Ryu, C. Chamon, and C. Mudry, Phys. Rev. B 85, 165120 (2012). Phys. Rev. B 84, 165107 (2011). 34 Y.-M. Lu and D.-H. Lee, ArXiv e-prints (2014), 57 F. Haldane, Phys. Rev. Lett. 74, 2090 (1995). arXiv:1403.5558 [cond-mat.mes-hall]. 58 J. Wang and X.-G. Wen, ArXiv e-prints (2012), 35 R.-J. Slager, A. Mesaros, V. Juricic, and J. Zaanen, Nat. arXiv:1212.4863 [cond-mat.str-el]. Phys. 9, 98 (2013). 59 Y.-M. Lu and A. Vishwanath, ArXiv e-prints (2013), 36 C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. B arXiv:1302.263 [cond-mat.str-el]. 87, 035119 (2013). 60 M. Levin and A. Stern, Phys. Rev. Lett. 103, 196803 37 J. C. Y. Teo and T. L. Hughes, Phys. Rev. Lett. 111, (2009). 047006 (2013). 61 C.-T. Hsieh, O. M. Sule, G. Y. Cho, S. Ryu, and R. Leigh, 38 W. A. Benalcazar, J. C. Y. Teo, and T. L. Hughes, ArXiv ArXiv e-prints (2014), arXiv:1403.6902 [cond-mat.str-el]. e-prints (2013), arXiv:1311.0496 [cond-mat.supr-con]. 62 I. Peschel, Journal of Physics A Mathematical General 36, L205 (2003) 39 Y. C. Hu and T. L. Hughes, Phys. Rev. B 84, 153101 cond-mat/0212631. (2011). 40 T. Mizushima and M. Sato, New. J. Phys. 15, 075010