Boundary supersymmetry of 1+1 d fermionic SPT phases
BBoundary supersymmetry of 1+1 d fermionic SPT phases
Abhishodh Prakash ∗ and Juven Wang † International Centre for Theoretical Sciences (ICTS-TIFR),Tata Institute of Fundamental Research, Shivakote, Hesaraghatta Hobli, Bengaluru 560089, India Center of Mathematical Sciences and Applications,Harvard University, Cambridge, MA 02138, USA
We prove that the boundaries of all non-trivial 1+1 dimensional intrinsically fermionic symmetry-protected-topological phases, protected by finite group on-site symmetries (unitary or anti-unitary),are supersymmetric quantum mechanical systems.
Topological phases of quantum matter are exotic insu-lating phases which fall outside the paradigm of spon-taneous symmetry breaking. The simplest class of thesephases are symmetry-protected-topological (SPT) phaseswhich are almost indistinguishable from the trivial phasein the bulk but have marvellous properties on the bound-aries [1–3] such as protected gaplessness and persistentordering [4], which are protected by symmetries. Well-known examples of SPT phases are topological insula-tors [1, 2], topological superconductors [2, 3] and theHaldane spin chain [5]. SPT phases are a subset of themore general invertible phases whose anomalous bound-aries may survive even if symmetries are broken. Fora particular dimension of space and global symmetries,invertible phases of bosons or fermions form an
Abeliangroup of which SPT phases are a subgroup. The programof classifying invertible and SPT phases [6, 7] has uncov-ered fascinating algebraic structures present in the spaceof quantum many-body systems through a vibrant collab-oration between areas of theoretical physics and mathe-matics. The non-trivial nature of the boundaries of SPTphases arises from an anomalous [8] realization of globalsymmetries on the boundary degrees of freedom [9]. Thisis manifested in the form of emergent, extended symme-tries [10] and fractional quantum numbers [11]. In onespatial dimensions, the anomalous boundary manifestsitself by symmetries being represented projectively [12–14]. In this letter, we present a universal property ofnon-trivial 1+1 d intrinsically fermionic
SPT phasesi.e. which cannot be interpreted as bosonic SPT phasesstacked with gapped fermions protected by finite on-sitesymmetries. We prove that as a consequence of projec-tively represented fermionic symmetries, the boundaryHamiltonians of any member of any such phase is a su-persymmetric quantum mechanical system [15–17].Originally introduced as a resolution to certain issuesin fundamental and particle physics [18], supersymme-try (SUSY) has served as a powerful theoretical tool touncover aspects of quantum field theories [19] as wellas a way to elegantly prove mathematical theorems us-ing techniques from physics [20]. Although signatures ofspace-time SUSY in particle interactions has not yet beendetected in particle colliders [21], proposals exist for theiremergence and detection in condensed matter [22–29] and cold-atomic systems [30–33]. To the best of our knowl-edge, all these proposals require some kind of fine tun-ing of parameters. Our work provides a generic settingfor the emergence and detection of the simplest versionof SUSY- supersymmetric quantum mechanics (SUSYQM) [15–17] without any fine-tuning . This is becausethe emergent SUSY in our case is a property of entirephases of matter and consequently can thus be observedin any member of any of these phases.In the first part of this letter, we demonstrate theemergence of boundary SUSY using the example of1+1 d time-reversal invariant topological superconduc-tors [13, 14]. In the second, we place the example ina general setting by proving a theorem which establishesthat boundary SUSY is a generic feature of all non-trivial1+1 d intrinsically fermionic SPT phases with finite on-site symmetries.
Boundary SUSY of topological superconductors :In refs [13, 14], Fidkowski and Kitaev established thatinvertible phases of 1+1 d interacting topological su-perconductors with time-reversal symmetry T satisfying T = + form a Z group. Of these, the phases labeledby even numbers, ν = 0 , , ,
6, forming a Z subgroup,are SPT phases i.e. require symmetry protection and aretrivial in its absence. Let us consider the phase corre-sponding to ν = 2. A representative model in this phaseis two copies of Kitaev’s Majorana chain [34, 35]. Usinga local change of basis [36], for a system of size L , theHamiltonian and symmetry operators (time-reversal, T and fermion parity, P f ) can be written as H = − i L (cid:88) j =1 ( γ ↑ ,j γ ↓ ,j +1 − ¯ γ ↑ ,j ¯ γ ↓ ,j +1 ) , (1) T = K L (cid:89) j =1 ( γ ↓ ,j ¯ γ ↑ ,j ) , P f = L (cid:89) j =1 (i¯ γ ↓ ,j γ ↓ ,j ) (i¯ γ ↑ ,j γ ↑ ,j ) . The Hilbert space on each site consists of two complexfermions (labeled σ = ↑ , ↓ ) and represented using fourMajorana operators ( γ σ , ¯ γ σ ). We work with a basis suchthat γ σ are real and symmetric, and ¯ γ σ are imaginaryand antisymmetric [37]. Under symmetries, the Majo- a r X i v : . [ c ond - m a t . s t r- e l ] N ov rana operators transform as T : γ σ (cid:55)→ τ zσσ (cid:48) γ σ (cid:48) , ¯ γ σ (cid:55)→ τ zσσ (cid:48) ¯ γ σ (cid:48) , i (cid:55)→ − i (2) P f : γ σ (cid:55)→ − γ σ , ¯ γ σ (cid:55)→ − ¯ γ σ , i (cid:55)→ i , (3)where, τ z is the Pauli-Z matrix. The symmetry operatorssatisfy T = P f = , T P f = P f T . (4)and generate a Z T × Z f group. With open boundaryconditions, we get a pair of unpaired Majorana modeson each end as shown in fig. 1 which results in a four-fold ground state degeneracy. We can now consider thetime-reversal and fermion parity operators restricted tothe Hilbert space of one of the boundaries (say left):ˆ T = K γ, ˆ P f = i ¯ γγ. (5)We have relabeled the boundary modes γ ↓ , ≡ γ , ¯ γ ↓ , ≡ ¯ γ for convenience. Throughout this letter, we use operatorswith a hat to label boundary operators (symmetry andHamiltonian) to distinguish them from those in the bulk.It can be checked that the boundary symmetry genera- FIG. 1. Schematic representation of the Hamiltonianin eq. (1) with Z T × Z f symmetry. tors satisfy ˆ T = ˆ P f = , ˆ T ˆ P f = − ˆ P f ˆ T , (6)and thus form a non-trivial projective representation [38]of the bulk Z T × Z f group eq. (4). The action of sym-metries on the boundary operators isˆ T : γ (cid:55)→ + γ, ¯ γ (cid:55)→ +¯ γ, i (cid:55)→ − i (7)ˆ P f : γ (cid:55)→ − γ, ¯ γ (cid:55)→ − ¯ γ, i (cid:55)→ i , (8)and the only possible boundary Hamiltonian consistentwith symmetries is ˆ H = c . (9)Where, c is a real number which we choose to be positivewithout any loss of generality. Observe that the spec-trum of eq. (9) has a Bose-Fermi degeneracy consistingof one bosonic ( ˆ P f = +1) and one fermionic ( ˆ P f = − E = c ). This is a con-sequence of the fact that the boundary time-reversal op-erator anti-commutes with fermion parity. We can definetwo Hermitian supercharges , ˆ Q + ≡ √ cγ and ˆ Q − ≡ √ c ¯ γ which satisfy the N = 2 SUSY QM algebra, { ˆ Q α , ˆ Q β } = 2 ˆ Hδ αβ , [ ˆ H, ˆ Q α ] = { ˆ P f , ˆ Q α } = 0 . (10) The action of time-reversal on the supercharges can alsobe determined and is best seen on the complex super-charges, Q ≡ ˆ Q + + i ˆ Q − and ¯ Q ≡ ˆ Q + − i ˆ Q − ,ˆ T : (cid:18) Q ¯ Q (cid:19) (cid:55)→ (cid:18) ¯ QQ (cid:19) . (11)Remarkably, this emergent SUSY is a feature of any member of the ν = 2 phase, not just the model of eq. (1).To see this, let us consider a model in the same phase witha more complex boundary Hilbert space. The Z classifi- FIG. 2. Schematic representation of the Hamiltonianin eq. (12) with Z T × Z f symmetry. cation [13, 14] tells us that we can stack an additional 4 N copies of the Hamiltonian of eq. (1) onto the original onewithout changing the phase of matter. Doing this givesus the following Hamiltonian and symmetry operators: H = − i L (cid:88) j =1 4 N +1 (cid:88) a =1 ( γ ↑ ,a,j γ ↓ ,a,j +1 − ¯ γ ↑ ,a,j ¯ γ ↓ ,a,j +1 ) , (12) T = K N +1 (cid:89) a =1 L (cid:89) j =1 ( γ ↓ ,a,j ¯ γ ↑ ,a,j ) , (13) P f = N +1 (cid:89) a =1 L (cid:89) j =1 (i¯ γ ↑ ,a,j γ ↑ ,a,j ) (i¯ γ ↓ ,a,j γ ↓ ,a,j ) . (14)The action of symmetry on the Majorana operators is thesame as shown in eqs. (2) and (3). With open boundaryconditions, as shown in fig. 2, the Hilbert space on the leftend now consists of 8 N + 2 unpaired Majorana operators( γ , . . . , γ N +1 , ¯ γ , . . . , ¯ γ N +1 ). We have again relabeledthe boundary modes γ ↓ ,a, ≡ γ a , ¯ γ ↓ ,a, ≡ ¯ γ a for conve-nience. The boundary symmetry generators as well astheir action on boundary Majorana operators are:ˆ T = K N +1 (cid:89) a =1 γ a , ˆ P f = N +1 (cid:89) a =1 i¯ γ a γ a , (15)ˆ T : γ a (cid:55)→ + γ a , ¯ γ a (cid:55)→ +¯ γ a , i (cid:55)→ − i , (16)ˆ P f : γ a (cid:55)→ − γ a , ¯ γ a (cid:55)→ − ¯ γ a , i (cid:55)→ i . (17)It can be checked that ˆ T and ˆ P f form the same projec-tive representation as eq. (6). However, the symmetry-allowed boundary Hamiltonian can be more complex andcan include any operator consisting Majoranas coupledin multiples of four (quartic, octonic, . . . ):ˆ H = (cid:88) a,b,c,d J abcd γ a γ b γ c γ d + J abcd ¯ γ a γ b γ c γ d + . . . (cid:88) a,b,c,d,e,f,g,h K abcdefgh γ a γ b γ c γ d γ e γ f γ g γ h + . . . (18)With only quartic couplings and J αabcd chosen from a ran-dom distribution, this is the Sachdev-Ye-Kitaev (SYK)model [39–41]. The projective boundary symmetry rep-resentation eq. (6) again enforces the spectrum to haveBose-Fermi degeneracy. To see this, observe that if | µ, + (cid:105) is some bosonic eigenstate with a non-zero eigenvalue E µ ,eq. (6) ensures that | µ, −(cid:105) = T | µ, + (cid:105) is a fermionic eigen-state with the same eigenvalue. This means, in the ab-sence of any other accidental degeneracies (ensured by J αabcd etc. being sufficiently random), the Hamiltonian ofeq. (18) can be written in diagonal form asˆ H = (cid:88) µ E µ ( | µ, + (cid:105)(cid:104) µ, + | + | µ, −(cid:105)(cid:104) µ, −| ) , (19)ˆ H | µ, ±(cid:105) = E µ | µ, ±(cid:105) , ˆ P f | µ, ±(cid:105) = ±| µ, ±(cid:105) . If we shift energies so as to ensure all E µ are positive,as shown by Behrends and B´eri [42], we can again definetwo superchargesˆ Q + = (cid:88) µ ˆ Q µ + ˆ Q † µ , ˆ Q − = (cid:88) µ i (cid:16) ˆ Q µ − ˆ Q † µ (cid:17) , (20)where, ˆ Q µ = (cid:112) E µ | µ, + (cid:105)(cid:104) µ, −| , which generate the same N = 2 SUSY QM algebraas eq. (10) as well as the action of time-reversal sym-metry on the supercharges shown in eq. (11). A similarstory also exists for the ν = 6 phase generated by 4 N + 3copies of Hamiltonian eq. (1) whose boundary also corre-sponds to a N = 2 SUSY QM system but with a differentaction of symmetries.The fact that the SYK model can serve as a boundaryHamiltonian for topological superconductors was pointedout by You, Ludwig and Xu [39] and the SUSY natureof the ‘standard’ SYK model was recently studied byBehrends and B´eri [42]. The main result of this workis a proof that this is true of all systems belonging to all non-trivial 1+1 d intrinsically fermionic SPT phasesprotected by any finite on-site symmetry, irrespective ofthe choice of boundary Hamiltonians. Proof of the general theorem:
We now prove themain theorem of this letter:
Any boundary Hamiltonian of a system belonging to anon-trivial 1+1 d SPT phase protected by finite on-site unitary or anti-unitary symmetries can be expressed as asupersymmetric quantum mechanical system if and onlyif the SPT phase is intrinsically fermionic.
Let us first carefully state the setting we are consider-ing and clarify some terminology. By non-trivial 1+1 dSPT phases, we mean classes of Hamiltonians in one spa-tial dimensions which have a unique ground state withclosed boundary conditions but have ground state degen-eracies in the presence of any open boundary conditionspreserving symmetry. If the global symmetries are explic-itly broken, the boundary can be gapped and the phasebecomes trivial. We are interested in fermionic systemswhose total symmetry group G f consists of fermion parity P f ≡ ( − ) N f which commutes with all other symmetryoperators. In other words, Z f ∼ = { , P f } is in the center of the group G f [43]. We also define the bosonic symme-try group as the quotient G b ∼ = G f / Z f [44]. We assumethat G f forms a finite group and all operators in G f ,which can be unitary or anti-unitary, are on-site [12] i.e.can be written as a product of operators which act ona finite number of local degrees of freedom, possibly fol-lowed by complex conjugation. The boundary symmetryoperators of such an SPT phase, ˆ G f forms a projectiverepresentation of G f and is classified by the second groupcohomology group H ( G f , U (1) T ) [13, 45, 46] which inturn also classifies the 1+1 d fermionic SPT phase [47].An SPT phase is intrinsically fermionic unless the classi-fication reduces to H ( G b , U (1) T ) [12] in which case thephase can be thought of as a non-trivial bosonic SPTphase stacked with trivial gapped fermions.To begin, let us consider a Hamiltonian H belongingto a non-trivial SPT phase protected by total symmetry G f . This means that the boundary symmetry operatorsˆ G f are projectively realized which forbids any bound-ary Hamiltonian ˆ H invariant under ˆ G f from acquiring aunique ground state invariant. We prove the main theo-rem in two steps: (I) We prove that an SPT phase is not intrinsicallyfermionic if and only if ˆ P f commutes with all elementsof ˆ G f . (II) We prove that if ˆ P f does not commute with all ele-ments of ˆ G f , then ˆ H is supersymmetric. Proof of (I) : We first prove this using the formal resultsof classification of fermionic SPT phases [45, 46, 48–50]and then provide a physical interpretation.The classification of fermionic SPT phases, given by H ( G f , U (1) T ) can be specified by two pieces of data- α and β (see refs [45, 46] for details). If and only if β ∈ H ( G b , Z ) is trivial, α reduces to α ∈ H ( G b , U (1) T )and the phase is not intrinsically fermionic. Now, β ,which assigns a Z element (either 0 or 1) to each elementof G b also encodes the action of fermion parity on eachoperator ˆ V g ∈ ˆ G f [46].ˆ P f ˆ V g ˆ P − f = ( − β ( π ( g )) ˆ V g . (21)Here, π is the surjective map from the elements of thegroup g ∈ G f to the quotient G b ∼ = G f / Z f . Now, if β is trivial i.e. assigns 0 to every element of G b , wesee from eq. (21) that this precisely means that fermionparity commutes with all elements of ˆ G f . Furthermore,the argument also works in reverse- if ˆ P f commutes withall the elements of ˆ G f , eq. (21) tells us that β must betrivial and the SPT phase is not intrinsically fermionic.Let us now understand what this means physically. Ifthe boundary fermion parity ˆ P f commutes with all otherelements of ˆ G f , we can simply add ˆ P f to the boundaryHamiltonian, ˆ H without breaking any symmetries to getˆ H ( λ ) = ˆ H − λ ˆ P f . (22)Now, we can take λ to be large and positive so as toensure that the low energy states are all bosonic i.e.have P f = +1. The effective representation of sym-metry generators on this low energy bosonic subspaceis ˆ G b ∼ = ˆ G f / Z f . Since the SPT is assumed to be non-trivial, the projection of ˆ H ( λ ) to the P f = +1 sectorcannot have a unique ground state. This means ˆ G b isa projective representation classified by some non-trivialelement of H ( G b , U (1) T ). Thus, the SPT phase is notintrinsically fermionic. Proof of (II) : Recall that the fermion parity ˆ P f gradesoperators as bosonic or fermionicˆ P f ˆ O b ˆ P − f = + ˆ O b , ˆ P f ˆ O f ˆ P − f = − ˆ O f . (23)We will assume that the symmetry operators of ˆ G f all have definite fermion parity i.e. are either bosonicor fermionic. This is indeed the case for projectiverepresentations of fermionic symmetries as shown ineq. (21) [45, 46]. If ˆ P f does not commute with all sym-metry operators and there exist some symmetry oper-ators ˆ V g that anti-commutes with ˆ P f , the eigenstatesof any Hamiltonian ˆ H invariant under ˆ V g with non-zero eigenvalues has Bose-Fermi degeneracy similar toeqs. (9) and (18). To see this, observe that if | µ (cid:105) is abosonic eigenstate with non-zero eigenvalue E µ , ˆ V g | µ (cid:105) isa fermionic eigenstate with the same eigenvalue. In di-agonal form, ˆ H can be written asˆ H = (cid:88) µ,a µ E µ ( | µ, a µ , + (cid:105)(cid:104) µ, a µ , + | + | µ, a µ , −(cid:105)(cid:104) µ, a µ , −| ) , ˆ H | µ, a µ , ±(cid:105) = E µ | µ, a µ , ±(cid:105) , ˆ P f | µ, a µ , ±(cid:105) = ±| µ, a µ , ±(cid:105) . The label a µ keeps track of additional degeneracies of E µ arising from the specific nature of the symmetries (e.g.: Kramers degeneracy, non-Abelian symmetries etc.). Af-ter shifting energies to make all E µ positive, we can al-ways define atleast two superchargesˆ Q + = (cid:88) µ ˆ Q µ + ˆ Q † µ , ˆ Q − = (cid:88) µ i (cid:16) ˆ Q µ − ˆ Q † µ (cid:17) , (24)where, ˆ Q µ = (cid:88) a µ (cid:112) E µ | µ, a µ , + (cid:105)(cid:104) µ, a µ , −| , which satisfy the N = 2 SUSY algebra of eq. (10). Westress that we cannot rule out the possibility of a largernumber of supercharges.Contraposing ( I ), we can establish that an SPT phase isintrinsically fermionic if and only if the boundary fermionparity ˆ P f does not commute with all boundary symmetryoperators. Combined with (II) , this implies that theboundary Hamiltonian, ˆ H is a supersymmetric quantummechanical system. This completes the proof. Summary and outlook:
We have described a generalsetting for the emergence of supersymmetric quantummechanics. Since this applies to a large class of physicalsystems belonging to several phases with no fine tun-ing, we expect it is a promising avenue for experimen-tal detection of SUSY in cold-atom as well as condensedmatter systems. Our work also opens up several ques-tions for theoretical investigation. First, we expect thatsome version of emergent SUSY should also be present onthe boundaries of invertible phases which are non-trivialeven in the absence of global symmetries. For exam-ple, the ν = 1 member of 1+1 d time-reversal invarianttopological superconductors [13] to which Kitaev’s Ma-jorana chain [34] belongs. However, the unusual bound-ary Hilbert space of such phases demands a more carefulinvestigation. Next, it is interesting to see how theseresults generalize to higher dimensions. While we havemanaged to establish the emergence of boundary SUSYin several higher dimensional examples [51], a generalproof is currently lacking. It is also interesting to con-nect these results to the symmetry-extension frameworkof ref [10] as well as unwinding of SPT phases [36] whichwill be reported elsewhere [52]. Finally, the emergence of space-time SUSY on the boundary of topological phaseshas been investigated, first by Grover, Sheng and Vish-wanath [23] and subsequently others [27–29], when theboundary is tuned to criticality. It would be interestingto see in what way their results are related to ours. Weleave these questions for future work.
Acknowledgments : We are grateful to N.S. Prab-hakar and R. Loganayagam for helpful comments anddiscussions. AP acknowledges funding from the Si-mon’s foundation through the ICTS-Simons prize post-doctoral fellowship. JW is supported by the NSF GrantDMS-1607871 “Analysis, Geometry and MathematicalPhysics” and by Center for Mathematical Sciences andApplications at Harvard University. ∗ [email protected] † [email protected][1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010).[2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011).[3] M. Sato and Y. Ando, Reports on Progress in Physics , 076501 (2017).[4] Z. Komargodski, A. Sharon, R. Thorngren, and X. Zhou,SciPost Physics (2019), 10.21468/scipostphys.6.1.003.[5] I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys.Rev. Lett. , 799 (1987).[6] C. Z. Xiong, Journal of Physics A: Mathematical andTheoretical , 445001 (2018).[7] D. Gaiotto and T. Johnson-Freyd, Journal of High En-ergy Physics (2019), 10.1007/jhep05(2019)007.[8] X.-G. Wen, Phys. Rev. D , 045013 (2013).[9] D. V. Else and C. Nayak, Phys. Rev. B , 235137 (2014).[10] J. Wang, X.-G. Wen, and E. Witten, Phys. Rev. X ,031048 (2018).[11] A. Vishwanath and T. Senthil, Phys. Rev. X , 011016(2013).[12] X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B ,235128 (2011).[13] L. Fidkowski and A. Kitaev, Phys. Rev. B , 075103(2011).[14] L. Fidkowski and A. Kitaev, Phys. Rev. B , 134509(2010).[15] F. Cooper, A. Khare, and U. Sukhatme, Physics Reports , 267–385 (1995).[16] E. Witten, Nuclear Physics B , 513 (1981).[17] E. Witten, Nuclear Physics B , 253 (1982).[18] E. Witten, “Introduction to supersymmetry,” in TheUnity of the Fundamental Interactions , edited byA. Zichichi (Springer US, Boston, MA, 1983) pp. 305–371.[19] N. Seiberg, Nuclear Physics B , 129–146 (1995).[20] E. Witten, J. Diff. Geom. , 661 (1982).[21] C. Autermann, Progress in Particle and Nuclear Physics , 125–155 (2016).[22] S.-S. Lee, (2010), arXiv:1009.5127 [hep-th].[23] T. Grover, D. N. Sheng, and A. Vishwanath, Science , 280 (2014).[24] T. H. Hsieh, G. B. Hal´asz, and T. Grover, Phys. Rev.Lett. , 166802 (2016).[25] X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang,Phys. Rev. Lett. , 187001 (2009).[26] S.-K. Jian, Y.-F. Jiang, and H. Yao, Phys. Rev. Lett. , 237001 (2015).[27] Z.-X. Li, A. Vaezi, C. B. Mendl, and H. Yao, ScienceAdvances (2018), 10.1126/sciadv.aau1463.[28] S.-K. Jian, C.-H. Lin, J. Maciejko, and H. Yao, Phys.Rev. Lett. , 166802 (2017).[29] Z.-X. Li, Y.-F. Jiang, and H. Yao, Phys. Rev. Lett. ,107202 (2017). [30] M. Tomka, M. Pletyukhov, and V. Gritsev, ScientificReports , 13097 (2015), arXiv:1407.5213 [quant-ph].[31] M. Lahrz, C. Weitenberg, and L. Mathey, Phys. Rev. A , 043624 (2017).[32] Y. Yu and K. Yang, Phys. Rev. Lett. , 090404 (2008).[33] B. Bradlyn and A. Gromov, Phys. Rev. A , 033642(2016).[34] A. Y. Kitaev, Physics-Uspekhi , 131 (2001).[35] Z.-C. Gu, Phys. Rev. Research , 033290 (2020).[36] A. Prakash, J. Wang, and T.-C. Wei, Phys. Rev. B ,125108 (2018).[37] “Notes on clifford algebra and spin(n) representa-tions,” http://hitoshi.berkeley.edu/230A/clifford.pdf , accessed: 2019-07-10.[38] Briefly, a projective representation of a group G is an as-signment of a matrix V ( g ) for each element g i ∈ G thatsatisfies group composition upto an overall complex phase V ( g ) V ( g ) = e iω ( g ,g ) V ( g .g ). The projective repre-sentation is non-trivial if the complex phases e iω ( g ,g ) cannot be removed by re-phasing V ( g ) (cid:55)→ e iθ ( g ) V ( g ).The distinct possible projective representations form anAbelian group H ( G, U (1) T ).[39] Y.-Z. You, A. W. W. Ludwig, and C. Xu, Phys. Rev. B , 115150 (2017).[40] A. Y. Kitaev, “A simple model of quantum holography,”(Feb. 12, April 7, and May 27, 2015), kITP strings sem-inar and Entanglement 2015 program.[41] V. Rosenhaus, Journal of Physics A: Mathematical andTheoretical , 323001 (2019).[42] J. Behrends and B. B´eri, Phys. Rev. Lett. , 236804(2020).[43] M. Hamermesh, Group theory and its application to phys-ical problems (Addison-Wesley, Reading, MA, 1962).[44] G b could be a subgroup of G f or not. e.g.: T = time-reversal symmetry has G f ∼ = Z f × Z T and T = P f time-reversal symmetry has G f ∼ = Z f,T . Both the groupshave G b ∼ = Z T but the former contains G b as a subgroupwhile the latter does not.[45] A. Kapustin, A. Turzillo, and M. You, Phys. Rev. B ,125101 (2018).[46] A. Turzillo and M. You, Phys. Rev. B , 035103 (2019).[47] Invertible phases that are non-trivial even in the absenceof symmetries are not included in the H ( G f , U (1) T )classification. For example, the the ν = 1 member oftime-reversal invariant topological superconductors with T = of which Kitaev’s Majorana chain is a mem-ber [34]. See [45, 46] for more details on the classificationof such phases.[48] Z.-C. Gu and X.-G. Wen, Phys. Rev. B , 115141(2014).[49] Q.-R. Wang and Z.-C. Gu, Phys. Rev. X , 011055(2018).[50] Q.-R. Wang and Z.-C. Gu, Phys. Rev. X10