cond-mat.str-el 2026-03-30 0

Categorical Time-Reversal Symmetries

The classification of phases using categorical symmetries has greatly expanded the landscape of gapped and gapless phases. So far, however, these developments have largely been restricted to phases with unitary (higher-)categorical symmetries over $\…

Authors: Rui Wen, Sakura Schafer-Nameki

Categorical Time-Reversal Symmetries
Categorical Time-Rev ersal Symmetries Rui W en and Sakura Sc h¨ afer-Nameki Mathematic al Institute, University of Oxfor d, Wo o dsto ck R o ad, Oxfor d, OX2 6GG, Unite d Kingdom The classification of phases using categorical symmetries has greatly expanded the landscap e of gapp ed and gapless phases. So far, ho wev er, these developmen ts hav e largely been restricted to phases with unitary (higher-)categorical symmetries o ver C . In this work, we incorporate anti- unitary symmetries, such as time-reversal symmetry Z T 2 , and show that the relev an t physical struc- tures are naturally described b y fusion categories o v er R . A class of real fusion categories, whic h w e call Galois-r e al fusion c ate gories , provides the correct categorical mo del for anti-unitary symme- tries. A simple example is the time-reversal symmetry Z T 2 itself. W e discuss the basic structures of real fusion categories and present a range of examples, including the group-theoretical categories ( G T ) ω and Rep ( G T ) asso ciated to anti-linear groups G T , as well as non-inv ertible time-reversal symmetries described by a real analogue of T ambara–Y amagami fusion categories. W e then classify gapp ed phases enriched with anti-linear symmetrie s in terms of mo dule categories ov er Galois-real fusion categories. W e furthermore apply the categorical formulation to prov e dualities (i.e. gauge or Morita equiv alences) of anti-linear symmetries generated by gauging subgroups. Complementing this, we also develop a Symmetry T op ological Field Theory (SymTFT) framework, in which Galois- real fusion categories arise as b oundary conditions of a Z T 2 -enric hed SymTFT. Morita equiv alen t an ti-linear symmetries are shown to arise as different boundaries of the same Z T 2 -enric hed SymTFT. CONTENTS I. In tro duction 1 I I. Motiv ating Real F usion Categories from Z T 2 -symmetric Phases 5 A. Z T 2 -T rivial Phase 5 B. Z T 2 SSB-Phase 6 C. H is for Haldane 8 I I I. Basics of Real F usion Categories 10 A. R -Real F usion Categories 11 B. Galois-Real F usion Categories 11 C. Mo dule Categories o v er Real F usion Categories 13 IV. Examples of Real F usion Categories 13 A. The V ec ω G T Categories 13 B. The Rep ( G T ) Categories 15 C. Non-in vertible Time-Rev ersal Symmetry 18 D. The C ⋊ Z T 2 Categories and Time-Rev ersal SSB 18 V. Gapp ed Phases with An ti-Linear Symmetries 20 A. The Setup 20 B. Order P arameters and Drinfeld Cen ter 22 C. Dualit y and Gauging 22 D. Z T 2 Gapp ed Phases Revisited 23 E. General Structure of G T Gapp ed Phases 23 F. Z T 4 -phases 23 G. S T 3 = Z 3 ⋊ Z T 2 26 H. Gauging Finite Subgroups 28 VI. Z T 2 -Enric hed T op ological Orders and Their Boundaries 28 A. T-Enric hment v ersus Gauging 28 B. T-Enric hed Non-chiral T op ological Order 29 C. Alternativ e Description of Z T 2 -SETs 31 D. The Drinfeld Cen ter of Galois-real F usion Categories 32 VI I. SymTFT Quic hes for An ti-Linear Symmetries 33 A. T-Enric hed SymTFT Quic hes 33 B. T-Enric hed Quiche for Z T 2 34 C. T-Enric hed Quiche for Vec Z T 4 and V ec ω Z 2 × Z T 2 35 D. T-Enric hed Quiche for A × Z T 2 and A ⋊ Z T 2 36 E. T-Enric hed Quiche for Galois-real TY-categories 37 VI I I. Conclusions and Outlook 37 A. More on the Semi-direct Product Categories C ⋊ Z T 2 38 B. Pro of of Gauging Finite Subgroup 38 References 40 I. INTR ODUCTION The m o dern understanding of symmetries in quan tum man y-b o dy systems and quan tum field theory has ex- panded well b eyond the traditional setting of groups. This has b een initiated by the deep insight of [ 1 ], that top ological defects are symmetries. One of the key phys- ically relev an t implications of these extensions is that symmetries need not hav e inv erses and form groups, but can form more general, non-inv ertible structures. These are subsumed under the name of categorical symmetries and w ere recen tly review ed in [ 2 – 5 ]. The main fo cus in the literature has b een, more pre- cisely , unitary or at least complex ( C )-linear fusion cat- 2 egories. As is well-kno wn, going back to the w ork of Wigner, symmetry groups in quan tum mechanics can ho wev er also b e an ti-unitary . The main example of an anti-unitary symmetry is time-reversal Z T 2 . Time- rev ersal symmetry plays a crucial role in many phases of matter, starting with the Haldane c hain in (1+1)d, whic h is a Z T 2 -SPT phase [ 6 ]. Other anti-unitary symme- tries suc h as Z T 4 , U (1) ⋊ Z T 2 , U (1) × Z T 2 are also common in topological insulators/sup erconductors [ 7 – 9 ]. The main purp ose of this pap er is to explore the cate- gorical generalization, including identifying a mathemat- ical home for suc h anti-unitary categorical symmetries and studying their implications in the con text of the categorical classification of phases and phase transitions [ 10 – 14 ]. W e sho w that the natural setting for categori- cal anti-unitary symmetries are real fusion categories . The mathematical literature on this is surprisingly scarce [ 15 , 16 ], and w e will be pro viding some of the background on these categories in the following. Tw o t yp es of Real F usion Categories. Unlik e (uni- tary) fusion categories that are defined o v er C , real fusion categories are defined ov er the field of real n umbers R . There are tw o types of real fusion categories, which are distinguished b y the t yp e of endomorphisms of the iden- tit y ob ject: a fusion category o ver R is called R -real if End(1) ∼ = R , and Galois-real if End(1) ∼ = C , in which case it has a Z T 2 -grading C = C 1 ⊕ C T (I.1) with C 1 the linear sector and C T the anti-linear. Cru- cially , we will argue that for a quan tum system, to ha v e a complex Hilbert space as the state space, it is the Galois- real fusion categories that hav e an in terpretation as sym- metry categories – R -real categories, essentially ha ve R acting on the lo cal state space, which w e disregard as a consistent setup for quantum theories. The tw o Z T 2 - graded comp onents of a Galois-real fusion category nat- urally corresp ond to the linear and an ti-linear sectors of a generalized an ti-linear symmetry . In particular, C 1 alone is an ordinary fusion category ov er C . Ho wev er, this does not mean that we should disregard the R -real categories from considerations. These can still arise as the categories of charges, and p erhaps surpris- ingly , can b e gauge (Morita) equiv alent to Galois-real fusion categories. The simplest example of this is the category Rep ( Z T 2 ) = V ec R , which is the category of real v ector spaces, but is also Morita dual to Vec Z T 2 , a Galois- real fusion category . An R -real fusion category has an in- terpretation as the category of defects in a given gapp ed phase, but not as an abstract symmetry category . E.g. Rep ( G T ) is R -real, and thus inadmissible as a symmetry category . Nevertheless Rep ( G T ) app ears naturally as the category of charges in a G T -SPT phase. The Strange W orld of Real F usion Categories. As the theory of real fusion categories is relatively unknown, w e will sp end one section in this pap er, Section I I I , ex- plaining the basics and giving concrete examples for both t yp es of real fusion categories. One class of exam ples are group theoretical, based on the G T an ti-linear group symmetry . Here there is a group homomorphism s : G T → Z T 2 = { 1 , T } , (I.2) suc h that the pre-image of 1 is the linear part and the pre-image of T the anti-linear (time-reversal) part of the symmetry . W e construct the real fusion category Vec ω G T , whic h is Galois-real, and describ e a p ossibly anomalous an ti-unitary G T -symmetry . Morita (i.e. gauge) equiv alence for real fusion cat- egories offers some surprises. Naively “gauging G T in V ec G T ” results in Rep ( G T ), how ev er this is an R -real fu- sion category and thus is inadmissible as a symmetry category – again this is not surprising giv en that this w ould inv olv e gauging time-reversal symmetry , which we b eliev e is not p ossible within the framework of standard quan tum theory 1 . P erhaps even more surprisingly , in the w orld of real- fusion categories the following tw o symmetries are gauge related: let A b e an ab elian group, then V ec A × Z T 2 Morita equiv alen t to Vec A ⋊Z T 2 , (I.3) where in the latter category Z T 2 acts on A as inv ersion and generically the latter is a non-ab elian group. Note that there are v arious versions in general of a group G T dep ending on ( I.2 ), e.g. here we specifically mean the Z T 2 acts as the reflections in the dihedral group. Another surprising prop erty is that the R -real Rep ( A T ) categories for ab elian A can hav e non-in vertible fusion. This is in stark contrast with the unitary/linear category case where Rep ( G ) has non-inv ertible fusion if and only if there are higher than 1d irreps, which requires G to b e non-ab elian. An example that w e will discuss is Rep ( Z T 4 ), whic h has a simple ob ject Q , that realizes a quaternionic represen tation, and has fusion Q ⊗ Q = 4 × 1 . (I.4) Non-In v ertible Time-Rev ersal. Our categorical framew ork allows for a unified description of b oth group- lik e inv ertible anti-unitary symmetry and more exotic non-in vertible time-reversal symmetry . A non-inv ertible time-rev ersal symmetry can arise, for instance, when an inv ertible anti-unitary symmetry is combined with an internal non-inv ertible symmetry . One family of non-in vertible time-reversal symmetries we will consider in this paper are a Galois-real version of T ambara- Y amagami categories, where the Kramers-W annier du- alit y defect is anti-unitary and thus time-reversal sym- metry b ecomes non-inv ertible. These are denoted b y 1 Whether or not time-rev ersal symmetry should or should not b e gauged in systems of quan tum gravit y is an even wider p oint of discussion [ 17 , 18 ]. Here we simply state that time-reversal should not b e gauged in quantum systems, irresp ective of cou- pling to gra vity . 3 Z T 4 -trivial Z T 4 -SPT Z T 4 -SSB Z T 4 → Z 2 partial SSB Rep( Z T 4 ) Rep( Z T 4 ) V ec Z T 4 V ec α Z 2 × Z T 2 V ec V ec Rep( Z 2 ) V ec Z T 2 Rep( Z 2 ) V ec Figure 1. F ull structure of (1 + 1)d gapp ed phases with Z T 4 symmetry . Each vertex is a gapp ed phase. Each arrow is a category of symmetric domain walls. The blue categories are Morita equiv alent symmetry (i.e. Galois-real) categories. The category Rep ( Z T 4 ) in contrast is not an admissible symmetry category even though it is Morita equiv alen t to Vec Z T 4 , as it is an R -real fusion category . TY C ( A ). Unlik e the case of unitary TY categories, here there is a choice of anti-symmetric bic haracter instead of a symmetric one, and there is no F robenius-Sch ur indi- cator. A full classification was giv en in [ 16 ]. Here we will reconsider this from a physical p ersp ective: the SymTFT pro vides a surprisingly simple and intuitiv e classification for this family of non-in vertible time-reversal symmetries. There are also R -real TY R ( A ) categories, whic h w e do not consider as symmetry c ategories for the reason explained b efore. Gapp ed Phases. W e prop ose a mo dule category ap- proac h for the classification of gapped phases with gen- eralized an ti-linear symmetries. This is akin to the uni- tary case studied in [ 10 ]. Gapp ed phases with symmetry are in 1-1 correspondence with mo dule categories ov er the now Galois-real symmetry category , which charac- terize ho w the symmetry category acts on the underlying gapp ed phase. E.g. we can reproduce with this approac h the known twisted group-cohomology classification for in- v ertible G T -symmetries. The mo dule category approac h also provides an explicit description of the categories of defects within each gapp ed phase and b etw een differen t phases in terms of mo dule functors. W e compute the categorical structure of all gapp ed phases with group- lik e symmetry Z T 4 – see Figure 1 – and S T 3 = Z 3 ⋊ Z T 2 – see T able I – including the categories of defects within eac h phase and the categories of domain walls betw een phases. As a consequence of the analysis, we establish the known Morita equiv alence 2 : V ec Z T 4 Morita equiv alen t to V ec α Z 2 × Z T 2 , (I.5) where α is a nontrivial mixed anomaly , and the lesser kno wn and more surprising equiv alences ( I.3 ), which e.g. for A = Z N , b ecome the statement V ec Z N × Z T 2 Morita equiv alen t to Vec D T N . (I.6) The SPT phases with time-reversal symmetry hav e a long history , going back to the Haldane c hain. One of the c hallenges studying these phases is the absence of a lo- cal or string order parameter, for a discussion of this see [ 7 , 20 – 24 ]. The Z T 2 phase can b e detected by computing a partition function on real pro jective space R P 2 , and on a space with b oundary the edge mo des can b e identi- fied with a Kramers doublet. How ever there seems to b e no direct bulk order parameter. Other (1+1)d gapp ed phases with time-reversal symmetry and the interpla y of this with gapp ed phase classification has b een studied in [ 25 – 27 ]. Dual Symmetries. One of the adv an tage of the cate- gorical formulation of symmetries and phases is that it allo ws for direct and rigorous computation of dualities (generalized gauging) b etw een symmetries. W e extend the theory of generalized gauging [ 28 , 29 ] to include anti- linear symmetries, and use the Morita theory for real fusion categories to define dual anti-linear symmetries. As an application of our formalism. W e establish the fol- lo wing family of dualities/gauge-relations generated by gauging central subgroups. Theorem ( Theorem V.8 ) . L et ( G T , s : G T → Z T 2 ) b e an anti-unitary gr oup-like symmetry, and N < ker( s ) b e a unitary c entr al sub gr oup. Denote by K T := G T / N the quotient. Then ther e is Morita e quivalenc e V ec G T ≃ Morita V ec ω b N ⋊ K T , (I.7) wher e G T / N acts on b N := Hom ( N , U (1)) as inversion via s , and ω (( k 1 , γ 1 ) , ( k 2 , γ 2 ) , ( k 3 , γ 3 )) = γ 1 ( e 2 ( k 2 , k 3 )) , (I.8) wher e e 2 ∈ H 2 ( K T , N ) is the extension class of the ex- tension N → G T → K T . Compared with the result of gauging central subgroup for a unitary group, the main new feature here is the semi-direct pro duct b N ⋊ K T . Z T 2 -Enric hed SymTFT. The SymTFT [ 30 – 33 ] has b een a k ey to ol in carrying out the Categorical Landau 2 It is kno wn that Z 2 × Z T 2 with mixed anomaly is related to Z T 4 by gauging [ 19 ]. This relation is usually established through field- theoretic arguments or explicit lattice constructions. T o the best of our knowledge, ho wev er, a pro of in terms of Morita equiv alence has not appeared prior to the presen t work. 4 ( S T 3 , 1) ( S T 3 , ψ ) ( Z T 2 , 1) ( Z T 2 , ψ ) ( Z 3 , 1) (1 , 1) ( S T 3 , 1) Rep( S T 3 ) V ec ⊕ 3 H V ec R V ec H Rep( Z 3 ) V ec ( S T 3 , ψ ) Rep( S T 3 ) Vec H V ec R Rep( Z 3 ) V ec ( Z T 2 , 1) Rep R ( Z 3 ) V ec H ⊕ V ec V ec Vec Z 3 ( Z T 2 , ψ ) Rep R ( Z 3 ) V ec Vec Z 3 ( Z 3 , 1) V ec Z 3 × Z T 2 V ec Z T 2 (1 , 1) V ec S T 3 T able I. (1 + 1)d gapp ed S T 3 -phases and categories of defects among them. The categories in blue are Morita equiv alen t symmetry categories, i.e. those that are Galois-real. P aradigm Program, for gapp ed phases in 1+1d [ 12 , 34 – 39 ] and 2+1d [ 40 – 44 ], including second order, symmet- ric, phase transitions [ 13 , 14 , 45 – 51 ], and mixed phases [ 39 , 52 , 53 ], and most recently contin uous internal sym- metries [ 54 – 58 ]. The fundamen tal idea is to start with a theory with a global symmetry C , coupling this to bac kground fields, and gauge in one dimension higher. This proto col applies to b oth internal symmetries as w ell as c ontinuous spacetime symmetries [ 58 ]. How ev er, for discrete spacetime symmetries such as time-reversal and translations, it was argued in [ 19 ], that instead of gauging these symmetries, the SymTFT for in ternal symmetries is merely enriched by the spacetime symmetries, in the sense of [ 59 , 60 ]. F rom an unoriented T uraev-Viro theory p ersp ective some of these asp ects app eared in [ 61 ]. Boundary Conditions and Symmetry Quic hes. Boundary conditions (BCs), and more generally con- densable algebras, in symmetry enriched SymTFTs or more generally top ological orders hav e app eared b efore in [ 19 , 62 – 65 ]. One fo cus here has b een on symmet- ric b oundary conditions, or at least b oundary conditions that are stable under the enrichmen t symmetry , i.e. the group of condensed any ons are inv ariant under the ac- tion. W e will relax this in order to b e able to incorp o- rate all anti-linear symmetries, which include nontrivial extensions of Z T 2 , and even non-in vertible time-reversal symmetries. F or suc h symmetry considerations, w e only require the SymTFT quic he – the SymTFT with one symmetry gapp ed b oundary condition – rather than the full interv al compactification that classifies phases. If the BC is not Z T 2 -in v arian t, or there is a non-trivial fractionalization class for the condensed any ons on the b oundary , then the b oundary breaks sp ontaneously the Z T 2 symmetry [ 19 ]. F or the purp ose of identifying the symmetry category this is not a hindrance – and has the adv an tage of allowing us to clas sify all p ossible Morita equiv alen t anti-linear symmetry categories. The spontaneous breaking of the enriching Z T 2 - symmetry on the symmetry boundary complicates the form ulation of SymTFT for gapp ed phases. In partic- ular, if we were to use this framew ork to study gapp ed phases, i.e. include a gapp ed physical b oundary condi- tion, then the resulting (1 + 1)d phase will alwa ys break the an ti-linear symmetry do wn to a linear sub-symmetry . Again, w e do not attempt this here, as w e ha v e a succinct and comprehensive wa y to study all gapp ed phases in a pure (1 + 1)d framework using mo dule categories in Sec- tion V . In summary , in this pap er we fo cus on SymTFT quic hes, with a Z T 2 -enric hed SymTFT bulk and a gapp ed symmetry b oundary and sho w that this is a v ery use- ful w ay to c haracterize all Morita equiv alen t anti-linear symmetries. Plan of the P ap er. In Section I I w e motiv ate real fusion categories directly from simple (1 + 1)d systems with time-reversal symmetry . Using a Z T 2 -symmetric spin c hain, we derive the tw o basic categories asso ciated with time-rev ersal, namely Rep ( Z T 2 ) ≃ Vec R and V ec Z T 2 , from the categories of symmetric defects in the trivial and symmetry-brok en phases. W e then revisit the Haldane c hain, explain the app earance of the quaternion alge- bra H and the real category Vec H , discuss the lack of string order parameters for anti-linear symmetries, and form ulate the mo dule-category description of (1 + 1)d Z T 2 - phases. In Section I I I we discuss the general theory of real fusion categories. W e distinguish the tw o types, R - real and Galois-real fusion categories, explain their differ- en t physical roles, and review the theory of mo dule cate- gories and Morita equiv alence ov er real fusion categories. In Section Section IV we illustrate these structures with concrete examples. In particular, w e discuss the group- theoretical categories asso ciated with anti-linear groups G T , the categories Rep ( G T ), non-inv ertible time-reversal symmetries of T ambara–Y amagami type, and the semi- direct product categories C ⋊ Z T 2 . In Section V w e turn to the classification of (1 + 1)d gapp ed phases with general- ized an ti-linear symmetries. W e prop ose that such phases are classified by mo dule categories ov er the Galois-real symmetry category , develop the corresp onding notion of dualit y and gauging, and revisit the basic Z T 2 examples from this persp ective. W e then analyze the general struc- ture of phases with group-like anti-linear symmetry and study in detail the examples of Z T 4 and S T 3 = Z 3 ⋊ Z T 2 , including their defects, domain walls, and Morita dual symmetry categories. W e conclude the section by proving the family of Motira equiv alences generated b y gauging finite unitary subgroups. W e change gear in Section VI and we dev elop the Z T 2 -enric hed (or simply T-enriched) SymTFT framew ork. W e explain the distinction b etw een T-enrichmen t and 5 gauging, formulate the bulk theory in terms of T-enriched top ological order, and identify the relev ant bulk defects with the Drinfeld cen ter of a Galois-real fusion category . In Section VI I w e apply this framework to SymTFT quic hes for an ti-linear symmetries. W e explain how a Galois-real symmetry category arises on a b oundary of a Z T 2 -enric hed SymTFT, and show through examples how Morita equiv alen t anti-linear symmetries app ear as differ- en t symmetry b oundaries of the same bulk. The app en- dices collect some technical material, including additional discussion of the semi-direct pro duct categories C ⋊ Z T 2 and the pro of of the gauging theorem. W e conclude with some op en questions in Section VI I I . I I. MOTIV A TING REAL FUSION CA TEGORIES FR OM Z T 2 -SYMMETRIC PHASES One of the key p oints of this pap er is that real fusion categories are the relev ant mathematical structure under- lying the study of systems with anti-linear symmetries. The goal of this section is to in troduce and motiv ate from a physical picture, how real fusion categories naturally arise in the study of anti-linear symmetries, such as Z T 2 . The most basic example of an anti-linear symmetry is the time-rev ersal symmetry Z T 2 . In this section we will deriv e tw o real fusion categories asso ciated with it, de- noted Vec Z T 2 and Rep ( Z T 2 ), resp ectiv ely , from a concrete 1 spin chain with Z T 2 -symmetry . The notation is delib er- ately chosen to match with the Vec G , Rep ( G ) categories asso ciated with a finite, in ternal, C -linear G -symmetry . The Rep ( Z T 2 ) category is in fact the same as Vec R the category of (finite dimensional) real vector spaces Rep ( Z T 2 ) ≃ V ec R . (I I.1) On the other hand Vec Z T 2 is a less familiar category . It is similar to the C -linear Vec Z 2 category except the gener- ator T ∈ Vec Z T 2 is anti-linear in a precise sense. W e will argue that it is the symmetry category for time-reversal symmetry , just lik e Vec G is the symmetry category for an internal G group symmetry . Recall that Vec G can b e regarded as the category of symmetric defects in a 1D G -SSB phase. Namely the symmetric defects in a G -SSB phase are the G -domain w alls, they are lab eled by the group elements and satisfy the group law under fusion. On the other hand Rep ( G ) is the category of symmetric defects in a 1D G -SPT. Namely the symmetric defects (or excitations) in a G - SPT are G -charges that are lab eled by representations of G . They fuse according to the tensor pro duct of rep- resen tations. Put differently , in a SymTFT realization of a G -SSB phase, we choos e the physical b oundary to b e asso ciated to the Dirichlet BC, which has top ological defects asso ciated to V ec G . In turn for a G -SPT, it is the Neumann BC which has top ological defects given by Rep ( G ). F ollo wing this example, w e in vestigate categories of symmetric defects in a 1D spin chain with Z T 2 -symmetry in the symmetric and SSB phase. Notice in 1D there is a non-trivial Z T 2 -SPT called the Haldane phase [ 6 ]. W e start with the Ising mo del with Hamiltonian H ( J ) = − (1 − J ) X i X i − J X i Z i Z i +1 , (I I.2) where X i , Z i are the Pauli op erators. The Hamiltonian is symmetric under v arious anti-unitary symmetries. One c hoice is U T := K Y i X i = K U , (I I.3) where K means complex conjugation in the basis where Z i are diagonal and U = Q i X i is the usual Z 2 -symmetry generator in the transverse Ising model. The c hoice of the an ti-unitary symmetry action is not unique and will not affect the conclusions in this section. With the choice of generator ( I I.3 ) which we will denote by Z T 2 , the ground state is Z T 2 -symmetric when J < 1 and breaks Z T 2 sp on- taneously when J > 1. In the follo wing w e will consider the system in infi- nite volume and identify defects as follo ws: for H a lo- cal Hamiltonian, a defect is a subspace of the total lat- tice Hilb ert space that lo oks like the ground state space of H near spatial infinity . In other w ords a defect is the ground state space of a Hamiltonian that only differs from H in a lo cal region. When symmetry is inv olv ed, a symmetric defect is the ground state space of a sym- metric Hamiltonian that differs from H in a lo cal region. In this language, the trivial defect is the ground state space of H itself. The physical reason b ehind this defi- nition is that in the IR limit only the ground state space is “visible” for a gapp ed system. A. Z T 2 -T rivial Phase W e will now prov e Theorem I I.1. The c ate gory of symmetric defe cts in a Z T 2 -symmetric trivial phase is Rep ( Z T 2 ) ≃ V ec R . First we will motiv ate this from the spin system. T ake J = 0, then the ground state space of the system is Ω = { z | + · · · + ⟩ ; z ∈ C } , (I I.4) where | + ⟩ is the eigenstate of X with eigenv alue 1. W e iden tify this with the trivial defect. One is tempted to sa y that a defect can b e created by acting with Z k at site k , whic h changes the ground state space to Z k Ω = { z | + · · · + − k + · · · + ⟩ ; z ∈ C } . (II.5) F or the internal Z 2 -symmetry generated by U , this is indeed a non-trivial defect, namely the Z 2 -c harge. How- ev er, we notice that here iZ k is a Z T 2 -symmetric lo cal op erator U T ( iZ k ) = ( iZ k ) U T , (I I.6) 6 that recov ers the original ground state space: ( iZ k ) × Z k × Ω = Ω . (I I.7) A defect/excitation that can b e created or destroy ed by a symmetric lo cal op erator is trivial, hence the spin-flip defect | + · · · − k · · · + ⟩ is in fact trivial as a Z T 2 -symmetric defect. W e see that the category of Z T 2 -symmetric defects in this Z T 2 -symmetric phase has essentially one simple ob ject, namely the ground state space itself. All defects can b e related back to the ground state space by acting with lo cal Z T 2 -symmetric op erators. A slightly more mathematical reasoning is as follows: lo cal op erators in this Z T 2 -symmetric phase should trans- form as represen tations of Z T 2 , where by a representation of Z T 2 w e mean a complex vector space V together with a C -anti-linear op erator T : V → V , T 2 = id V . (I I.8) F or instance, the space generated b y the lo cal op erator Z k is V Z k = { z Z k | z ∈ C } , (I I.9) with the Z T 2 -action: T ( z Z k ) = − z ∗ Z k . (I I.10) If we identify V Z k with the complex plane via ( a + bi ) Z k ← → ai + b, a, b ∈ R , (I I.11) then the Z T 2 -action is simply complex conjugation: T ( ai + b ) = − ai + b . (II.12) W e see that the space spanned by the op erator Z k to- gether with its Z T 2 -action is equiv alen t to the complex plane with the standard complex conjugation action. In fact, any irreducible represen tation of Z T 2 is equiv alen t to the complex plane with Z T 2 acting as complex con- jugation, T . This b ecomes the mathematical statement tha the category Rep ( Z T 2 ) of Z T 2 -represen tations contains only one simple ob ject, ( C , T ), up to equiv alence. Let us consider the endomorphisms of the trivial defect Ω. A morphism of defects Ω → Ω is a linear map that c an b e r e alize d by a symmetric lo c al op er ator. Assume the linear map | + · · · + ⟩ → z | + · · · + ⟩ is realized by a Z T 2 -symmetric lo cal op erator O : O | + · · · + ⟩ = z | + · · · + ⟩ (II.13) Then since U T O U T = O , we hav e z | + · · · + ⟩ = U T O U T | + · · · + ⟩ = U T z | + · · · + ⟩ = z ∗ | + · · · + ⟩ . (I I.14) Therefore only multiplication by real num b ers are al- lo wed as maps of defects. W e conclude that in the cate- gory of symmetric defects, we hav e End( 1 ) ≃ R . Put ev- erything together we see that the category of symmetric defects in the trivial Z T 2 -symmetric phase is a real fusion category generated by a single simple ob ject 1 satisfying End( 1 ) ≃ R . This is nothing but the category Vec R of finite dimensional real vector spaces. In fact Rep ( Z T 2 ) is equiv alen t to V ec R : a complex vector space with anti- unitary Z T 2 -action is the same as a real v ector space b y taking the fixed p oints of the Z T 2 -action: Z T 2 ↷ V ⇔ V R := { v ∈ V | T ( v ) = v } . (I I.15) In summary we obtain Theorem I I.1 . B. Z T 2 SSB-Phase The other phase is an SSB-phase for Z T 2 . W e pro ve the follo wing theorem: Theorem I I.2. The c ate gory of symmetric defe cts in a Z T 2 -SSB phase is V ec Z T 2 := End R ( V ec ) . First let us again motiv ate this from the spin-chain. In the spin system it is obtained b y considering J = 1, then ground state space of the system is Ω = { z | ↑ · · · ↑⟩ + w | ↓ · · · ↓⟩ ; z , w ∈ C } ≃ C ⊕ C , (I I.16) where | ↑⟩ and | ↓⟩ the eigenstates of Z with eigenv alues ± 1. The domain wall defect is created by the symmetric non-lo cal op erator D k := Y j >k X j , (II.17) whic h acts on the ground state space as D k :  | ↑ · · · ↑⟩ 7→ | ↑ · · · ↑ k ↓ k +1 · · · ↓⟩ | ↓ · · · ↓⟩ 7→ | ↓ · · · ↓ k ↑ k +1 · · · ↑⟩ . (I I.18) This m ust b e a non-trivial defect, since any lo cal op erator cannot c hange the behavior of a ground state near spatial infinit y . The lo cation k of the defect is irrelev an t for the top ological sector of the defect, so we will call this defect D . F usion of defects is defined b y concatenation. The fusion D ⊗ D is therefore represented by the space spanned b y the states with tw o domain walls: D ⊗ D ≃ { z | ↑ · · · ↑ j ↓ j +1 · · · ↓ k ↑ k +1 · · · ↑⟩ + w | ↓ · · · ↓ j ↑ j +1 · · · ↑ k ↓ k +1 · · · ↑⟩ | z , w ∈ C } . (I I.19) The Z T 2 -symmetric lo cal op erator Q j k leads to D k → D k : | ↑ · · · ↑ k ↓ k +1 · · · ↓⟩ 7→ O j | ↑ · · · ↑ k ↓ k +1 · · · ↓⟩ = α ∗ | ↑ · · · ↑ k ↓ k +1 · · · ↓⟩ | ↓ · · · ↓ k ↑ k +1 · · · ↑⟩ 7→ O j | ↓ · · · ↓ k ↑ k +1 · · · ↑⟩ = α | ↓ · · · ↓ k ↑ k +1 · · · ↑⟩ . (I I.27) W e see that the left and righ t actions on D k differ b y complex conjugation. W e conclude that the category of symmetric defects in the Z T 2 -SSB phase has the following prop erties. • It is generated by tw o simple ob jects 1 , D . • The fusion rule is D ⊗ D = 1 . • The ob ject 1 satisfies End( 1 ) ≃ C . • The ob ject D satisfies the relation z ⊗ D = D ⊗ z ∗ , z ∈ C . (I I.28) Although End( 1 ) ≃ C , the last relation ab o ve implies that this category is not a C -linear fusion category . It is only a real fusion category . W e will denote this cate- gory as Vec Z T 2 . In the mathematical literature this cate- gory app eared in [ 15 ], where a concrete mo del is given as Bimo d R ( C ), the category of R -linear C -bimo dules. An- other concrete mo del is Vec Z T 2 = End R ( V ec ) the category of R -linear endomorphisms of V ec . W e thus find Theo- rem I I.2 . W e recall that in the internal G -symmetry case, one w ay to define the symmetry category is as the category of defects in the G -SSB phase. This gives the V ec G category for non-anomalous G -symmetry , and V ec ω G for anomalous G -symmetry . Namely , the domain walls in an SSB phase remem b er not only the group law but also the anomaly . Similarly , we should b e able to define the symmetry cat- 8 egory for Z T 2 as the category of defects in the Z T 2 -SSB phase. Hence we will take Vec Z T 2 as the symmetry cate- gory for Z T 2 . In (1 + 1)d, there is no Z T 2 -anomaly since H 3 ( Z T 2 , U (1) T ) = 0. Ho wev er, in other dimensions, there could b e time-reversal anomalies and the symmetry cat- egory should b e able to capture the anomalies. Indeed, the V ec Z T 2 category can b e generalized to higher dimen- sions. In [ 66 ] the authors also constructed the 2-category 2 V ec ω Z T 2 , where ω ∈ H 4 ( Z T 2 , U (1) T ) = Z 2 can be a non- trivial class. In this pap er we will fo cus on (1 + 1)d sym- metries and phases, there can still b e mixed anomalies b et ween Z T 2 and other in ternal symmetries, which should b e captured by symmetry categories generalizing Vec Z T 2 . W e will discuss these categories in the next section. C. H is for Haldane W e hav e discussed the symmetry and charge cate- gories asso ciated with the Z T 2 symmetry . Similar to the case with unitary group G symmetries, if there are tw o SPTs protected by Z T 2 , then the category of symmetric defects is Rep ( Z T 2 ) in either SPT. Hence the category of bulk symmetric defects cannot distinguish differen t SPTs. There are generally tw o approaches to detecting SPT phases. One is via b oundary/edge measurements, the other is via bulk string order parameters. The latter approac h naturally leads to the SymTFT paradigm by realizing that the string order parameters are naturally an yons of a (2 + 1)d top ological order, that end on the ph ysical b oundary but do not end on the symmetry one. In this section we consider suc h approac hes to the non- trivial Z T 2 -SPT called the Haldane chain. The physical prop erties of the Haldane c hain, including its b oundary mo des, are well-kno wn in physics. Our fo cus here is on the categorical characterization of the Haldane c hain. W e will discuss tw o asp ects of the Haldane chain from a cat- egorical p ersp ectiv e. First, w e show that the category of b oundary conditions in the Haldane chain is Vec H , where H is the quaternion algebra. Second, we will show that the Haldane chain do es not hav e a string order pa- rameter, consequently it can not b e described b y any on condensation in a (2 + 1)d SymTFT as for SPTs with unitary G -symmetry . Finally , we will provide a mo dule category approach that provides a complete characteri- zation of (1 + 1)d phases with Z T 2 -symmetry . 1. The Haldane Chain and its Edge-Mo des The Haldane phase with one boundary can b e modeled b y the following Hamiltonian: H 0 = − ∞ X i =0 Z i X i +1 Z i +2 (I I.29) with the Z T 2 -symmetry generated as in ( I I.3 ) by U T = K ∞ Y i =0 X i = K U . (I I.30) The ground state space of H 0 , denoted as Ω, is t wo di- mensional. One chec ks that the op erators X := X 0 Z 1 , Z := Z 0 (I I.31) comm ute with the Hamiltonian, hence they act on the ground state space as a pair of Pauli op erators. W e will write | ↑⟩ 0 , | ↓⟩ 0 for the basis of Ω that are eigenstates of Z . Since we do not hav e explicit ground state wa v e- function, w e will work out the action of U T on the ground state space algebraically . Notice that U T X U T = −X and U T Z U T = −Z . Assume U T | ↑⟩ 0 = e iθ | ↓⟩ 0 , then U T | ↓⟩ 0 = U T X | ↑⟩ 0 = − e iθ | ↑⟩ 0 (I I.32) Next w e consider endomorphisms of the b oundary con- dition Ω. Similar to the case with defects, a morphism of b oundary conditions is a linear map b etw een ground state spaces that can b e realized by symmetric lo cal op- erators. Assume there is a lo cal op erator O such that O | ↑⟩ 0 = α | ↑⟩ 0 + β | ↓⟩ 0 , O | ↓⟩ 0 = γ | ↑⟩ 0 + δ | ↓⟩ 0 . (I I.33) Using U T O = O U T and the action of U T on the ground state space, we obtain that δ = α ∗ , γ = − β ∗ . Hence in the Z -basis the most general form of an endomorphism is  α β − β ∗ α ∗  (I I.34) Suc h matrices form an R -linear algebra, with an R -linear basis given by 1 , i X , i Y , i Z ( X , Y , Z are P auli matrices). Chec king their algebraic relations, we see that the en- domorphism algebra of the b oundary Ω is exactly the quaternion algebra H = R ⟨ 1 , i, j, k ⟩ / ⟨ i 2 = j 2 = k 2 = ij k = − 1 ⟩ . More generally , we observ e that the restric- tion of U T on the ground state space satisfies U 2 T = − 1. This is true not only for the Hamiltonian H 0 but for an y b oundary condition: assume there is a Hamiltonian with one b oundary that differs from H 0 only near the b oundary . Then there exists some k > 0 such that Z i X i +1 Z i +2 = 1 on the ground state space for any i ≥ k . Therefore restricting to the ground state space we can replace U T b y U T Y i ≥ k ( Z i X i +1 Z i +2 ) = X 0 X 1 · · · X k ( Z k Z k +1 ) (II.35) whic h is supp orted near the boundary and squares to − 1. Hence no matter what the Hamiltonian lo oks like near the b oundary , the ground state space will b e acted by an an ti-unitary op erator that squares to − 1. T o formalize the situation let us in tro duce the algebra C [ Z T 2 ] η , which as a linear space is { z + w U T | z , w ∈ C } , and satisfies 9 z U T = U T z ∗ , U 2 T = − 1. Then the ab ov e discussion sho ws any b oundary condition leads to a ground state space that is a representation of the algebra C [ Z T 2 ] η . In fact C [ Z T 2 ] η ≃ H (I I.36) follo ws by identifying i ; i, U T ; j, iU T ; k . (I I.37) W e conclude that the category of b oundary conditions of the Haldane chain is exactly Vec H the category of H - mo dules. The specific boundary condition H 0 can b e though t of as the minimal b oundary condition that cor- resp onds to the simple ob ject H ∈ V ec H . 2. L ack of string or ders F or an SPT protected by a unitary G -symmetry , it is well-kno wn that string order parameters can b e used to detect the SPT phase. Analyzing the mathematical structure of string order parameters is in fact one path to wards the SymTFT paradigm. W e now consider the situation with the Z T 2 -SPT. W e will give arguments that Z T 2 -SPTs do not hav e string order parameters. A string order parameter should be a truncation of the symmetry generator on finite interv als(or half-infinite c hain). This means the string order acts trivially on op- erators aw a y from the interv al and acts the same wa y as the symmetry generator on op erators deep inside the in- terv al. How ev er, since complex conjugation is inv olv ed in the generator U T , there is no sensible truncation of U T to finite interv als. This is b ecause complex conjugation al- w ays acts non-trivially everywhere. W e may write down ( U T ) j k := K Q j

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