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FIG. tpbc n ics h olwn w iwonson viewpoints Bais by two proposed following condensation the boson discuss the and back of step theory a develop and conden- reasonable condensation. fermion fermion perfectly which in is context work, sation physical this a in conden- propose however, fermion we counterintuitive; glance, first be a might At sation system? other the some in with anyons statistics self-fermion braiding condense nontrivial to have possible inter- which it particularly Is A is: bTO. question a esting in self-bosons condensing to oooia re ecie yteqatmdouble quantum the by described order one topological example, For a bTO. new consider closes a the gap may to energy in rises the giving parameter that reopens, some such and bTO, tuning the imagine of us Hamiltonian let hand, one ecndieaHgstasto ycnesn certain condensing group by gauge the transition that such Higgs excitations a charge drive can We onto down pc” mgn bTO “real in a transitions Imagine 1(a))—phase space”. 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B fe odnigcranself- certain condensing After . B Orders l (b) D [ H tal et .However, ]. G F nthe On . sbroken is D [ G s, ]. 2 densation can be viewed as a special type of GDWs be- in bTOs. tween B and B′, where every anyon in B′ can penetrate With these rules, we study properties of fermion con- the GDW and transform into some anyon in B, rather, densation. In particular, we reach a formula between the some anyons in B may never be able to penetrate the total quantum dimension of the parent bTO and that GDW (i.e., they are confined in B′ in the sense of param- of its child bTO due to certain fermion condensation. eter space phase transition). The self-bosons that con- The previously known formula for boson condensation13 dense in B transform into the trivial boson 1 B′, when appears to be a special case of our new formula. In addi- they cross the GWD. The connection between∈ GDWs in tion, we find that there is an equivalence between fermion bTOs and self-boson condensation has been thoroughly condensation in bTOs and boson condensation in fTOs. studied recently.13,19–25 It has been found that GDWs This equivalence in turn corroborates our rules of fermion between bosonic topological orders can be classified by condensation. We work out explicit examples to illustrate anyon condensation13,21. the results described above. Finally in the discussion sec- That being said, we are now ready to explore self- tion, we take an outlook into certain future directions. fermion condensation in bTOs. By self-fermion conden- We believe that fermion condensation can also be de- sation, we mean some kind of “condensation transition” scribed by Frobenius algebras that are of a type different through which certain self-fermions in a bTO become lo- from the type of Frobenius algebras characterizing boson cal excitations after the condensation. If we imagine such condensation. Nevertheless, in this work, we would not a phase transition in parameter space, one finds that the dwell on the abstract math of fermion condensation but original bosonic system turns into a fermionic system af- instead focus on the physical content and consequences ter the condensation—because, with the current knowl- of fermion condensation by minimizing the mathemat- edge of topological orders, only fermionic systems, where ics. While we shall report the results of our ongoing the fundamental degrees of freedom are fermions, support research of the full mathematics of fermion condensation local fermionic excitations. In bTOs, all self-fermions are elsewhere, in the appendix we would briefly review the nonlocal. It is however still mysterious how can one turn type of Frobenius algebras describing boson condensation a bosonic system into a fermionic system through certain and show what constraints are relaxed on bosonic Frobe- phase transition if without involving, say, for example, a nius algebra to obtain a fermionic Frobenius algebra. background layer of fermionic system. The study of fermion condensation can in turn help us On the other hand, the GDW picture makes perfect understand fTOs. Although the most common condensed sense for discussing about self-fermion condensation. We matter systems—electron systems—are fermionic, fTOs just need to consider a GDW between a bTO and a are less understood than bTOs. Currently, a mathe- fermionic topological order (fTO)26–28, whose fundamen- matically rigorous and complete description of fTOs is tal degrees of freedom are fermions (Fig. 1(b)). From the yet unavailable. It is therefore worthwhile of under- GDW picture of self-boson condensation, we expect that standing fTOs indirectly from other perspectives. With a self-fermion condensation has the following properties: fermion condensation, we can construct new fTOs based (1) all excitation in F can pass through the wall and on bTOs, and thereby extract a proper description of become some excitations in B; (2) the condensed self- fTOs from the data of bTOs. fermions in B becomes local fermions in F when they pass As a warning of terminology abusing: Throughout the through the wall; (3) certain anyons in B cannot pass paper, we refer to self-bosons (fermions) simply by bosons through the wall. In general, we expect that self-bosons (fermions) wherever no confusion would arise; We often and self-fermions can condense simultaneously, and we do not differentiate an anyon type from an anyon if the have (4) the condensed self-bosons in B becomes the triv- context is clear, otherwise we refer to an anyon type as ial boson in F. Of course, these properties only give a a topological sector; We may also use bTOs and unitary general intuitive picture of self-fermion condensation. modular tensor categories interchangeably. The goal of this paper is to establish more precise rules It is worth of note that simple-current fermion con- densation, which is equivalent to our minimal fermion on self-fermion condensation, or simply put, fermion con- 29 densation, with the GDW picture in mind. To this end, condensation, was briefly studied in Refs . During the we propose a physically and mathematically reasonable preparation of the manuscript, a more detailed study of simple current fermion condensation was done by Gaiotto ansatz for fermion condensation. Based on this ansatz, 28 we prove a Hierarchy Principle of fermion condensation et al . In this work, we consider general fermion conden- in bTOs, in the sense that any well-defined fermion con- sation, and propose a realization through GDWs between densation, however complicated, can be decomposed into bTOs and fTOs. two steps: First, we can collect all self-bosons in the con- densate and condense them alone; Second, what remains is only to condense a single self-fermion, which we call II. BOSONIC AND FERMIONIC minimal fermion condensation. We then find the rules TOPOLOGICAL ORDERS of minimal fermion condensation, which, combined with the known rules of boson condensation, give rise to a full We begin with a brief review of the basics of bTOs set of rules of performing generic fermion condensation and fTOs. A bTO is a 2D quantum many-body system 3 with an energy gap, and the excitations are generally angles of anyons in each pair satisfy θaf = θa, following anyons. The microscopic degrees of freedom underlying the fact that 1f is a fermion and is transparent.− An fTO a bTO are bosons, such that the vacuum state (also called that contains only 1 and 1f is regarded as a trivial fTO f the trivial boson) is a true boson and unique, which has and denoted by F0 = 1, 1 . A trivial fTO can be real- trivial mutual statistics with every anyon in the system. ized in gapped free fermion{ } systems. Note that for any Anyons in a bTO are believed to be fully characterized fTO F, F F0. The anti-anyona ¯ is defined in the same by a unitary modular tensor category (UMTC) B, where way as in⊇ bTOs, such that 1 a a¯. The anti-anyon is the modularity refers to that the trivial boson is the only unique and the trivial boson 1∈ occurs× in a a¯ only once. topological sector with trivial mutual statistics with all × B topological sectors of the topological order. A bTO Fermionic topological orders can be divided into two is composed of the following topological data. First, a types, namely primitive fTOs and non-primitive ones27. finite set of anyons 1,a,b,... equipped with quantum { } Primitive fTOs are those that cannot be obtained dimensions 1, da, db,... , where da 1 are real num- { } ≥ from stacking two other topological orders, while non- bers. The unitarity demands the positivity of quantum primitive ones can. There are two ways to obtain fTO’s dimensions. Here, 1 denotes the trivial anyon and d1 = 1. from stacking: (1) stack an fTO with a bTO and (2) stack B The dimension of the UMTC or the total quantum di- two fTO’s. The simplest kind of non-primitive fTOs are B 2 mension of the bTO is defined by DB = a∈B da, those obtained by stacking a layer of bTO B with a layer where the sum includes the trivial boson 1 as well. Sec- pP of trivial fTO F0. We denote this stacking operation ond, the fusion interaction between the anyons, namely by B ⊠ F . It is shown that any Abelian fTO admits c 0 a b = c Nabc, where the sum is over all anyons includ- this kind of layer decomposition27. One can also stack × c ing the trivial anyon 1, and N are nonnegative integers. F F F ⊠ F P ab two different fTOs 1 and 2 together by 1 F0 2, The fusion is associative, and 1 is the identity of fusion. the symbol ⊠F means that the F0 F1 and F0 F2, Each anyon a in B has an anti-anyon a¯ B, such that 0 ⊂ ⊂ two copies of F0, must be identified as a single F0. Cer- 1 a a¯. The anti-anyona ¯ of a is unique,∈ and the ∈ × tainly, one can stack any finite number of TOs in this vacuum 1 can only appear exactly once in a a¯, i.e., manner to construct more complicated TOs, bosonic or 1 × Naa¯ 1. Following Bais’ convention, we also call this fermionic. As such, a primitive fTO does not admit any ≡ 10 feature the unitarity condition , which should not be nontrivial stacking structure, other than the trivial stack- confused with the unitarity of UTMC. An anyon a may F F ⊠ F ing = F0 0. The topological properties of the TOs be self-dual in the sense that a =¯a. Third, a modular T obtained by such stacking operations can be straightfor- matrix, T = Diag 1,θa,θb,... where θa = exp(i2πha) { } wardly obtained from those of individual layers. The is the self-statistical angle of a with ha the topological anyons in some B⊠F are the pairs (a,a′) a B, a′ F. B spin of a. That is a unitary also implies that θa = θa¯. ′ ∀ ∈′ ∈ The quantum dimensions read d(a,a ) = dada . Topolog- Fourth, a modular S matrix encoding the braiding be- ical spins are sums, h(a,a′) = ha + ha′ . The modular S tween the anyons, whose matrix elements are matrix of B⊠F is the tensor product of that of B and that 1 θ of F. We will bring up an important difference between S = N c c d , (1) ab D ab θ θ c primitive and non-primitive fTOs in Section IV D. Xc a b 2 B where the total quantum dimension D = a da. This Finally, we make comments on the trivial bTO 0 = F f definition of S-matrix implies that S1a = pdaP/D. 1 and trivial fTO 0 = 1, 1 . The above discussions An fTO, however, has not only a trivial boson but also {focus} on the anyonic content{ of} topological orders. Two a trivial fermion (also known as a transparent fermion) topological orders with exactly the same anyonic content that has trivial mutual statistics with all topological sec- may still be distinct, in the sense that they cannot be tors in the fTO. Hence, unlike a bTO, an fTO does not smoothly deformed to each other without closing the en- respect modularity. The existence of a trivial fermion is ergy gap. It is believed that to fully characterize a topo- a result of the fermionic microscopic degrees of freedom logical order, one needs an extra piece of data, the chiral underlying the topological order. The trivial fermion central charge c of the edges modes (some conformal field has unit quantum dimension too. Although an fTO is theory) living on the boundary of the 2D topologically or- not described by a UMTC, it is still a fusion category; dered system. Therefore, there are various bTOs/fTOs hence, its total quantum dimension takes the same defi- with the same anyonic content but different values of c. It nition as that of a bTO. A general categorical theory of is believed that bTOs with B0 can be obtained by stack- 30 fTOs is still unclear but certain features of fTOs, such ing the E8 states which has c = 8, and fTO’s with F0 as the non-modularity, are captured by what are known can be obtained by stacking p + ip states3132 which has as premodular tensor categories27. Nevertheless, several c =1/2. In the case of gapped domain walls, c is always properties of fTO’s are understood from physics point of the same on the two sides of the wall. So, we will not view. In general, an fTO F contains a finite set of anyons emphasize on c any further in the sequel. In fact, we will 1, 1f ,a,af ,b,bf ,... , where 1 is the trivial boson and 1f and have already abused to certain extent the notions of {the trivial fermion.} Topological sectors always come in a TO and its anyonic content. We will ignore the variety pairs, a and af , with af = a 1f . The self-statistical in c, and equate a TO with its anyonic content. × 4

III. FROM BOSON CONDENSATION TO FERMION CONDENSATION

B B′ B ⊠ B′ vac In this section, we write down our ansatz for fermion condensation in bTOs, which consists of several necessary conditions for fermion condensation to satisfy. To moti- vate this ansatz, we retrospect to and quickly go through (a)Gapped domain wall (b)Folding trick the basic idea of boson condensation and the GDWs be- tween bTOs. The core notion arises there is the mutual FIG. 2. (a) A bTO B and a B′ connected by a GDW via locality between condensed bosons. As to be seen, we can condensing some bosons in B. Via the folding trick along the naturally extend boson condensation to fermion conden- GDW, this picture is equivalent to (b) A bTO B ⊠ B′ sharing sation and write down a reasonable ansatz. a gapped boundary with the vacuum. Actually, it is plausible that one can generalize the idea to condensation of arbitrary anyons, which we would briefly touch upon in the discussion Section VI. versa. Meanwhile, an anyon (a,b) B ⊠ B′ that cannot disappear at the gapped boundary∈ in the folded picture corresponds to the situation that there is no way (opera- A. Boson Condensation and GDWs tor) to turn a into b at the GDW in the unfolded picture. In general, there exists certain a B such that a can- Consider two (not necessarily different) bTOs B and B′ not turn into any b B′, and there∈ exists certain b′ B′ that share a GDW in between (Fig. 2)33. To understand such that b′ cannot∈ turn into any a′ B, through∈ the the GDW between B and B′, we have two equivalent GDW. A special case, which we call strict∈ condensation ways. On the one hand, we can fold the system along wall, is: For every b B′, there exist ways (operators) the GDW and turn it into the configuration in Fig. 2(b). at the GDW to turn ∈b into some a B. That is, every The bTO B ⊠ B′ is a simple stacking of B and B′, where anyon in B′ can pass though a strict∈ condensation wall B′ means that the spins and braiding statistics should be and becomes a B. Note that a may not be unique. In a mirror reflection of those of B′. We can visualize the particular, let A∈ be a subset of anyons in B, which are GDW (or called a gapped boundary in this case) between the anyons that can turn into 1 B′. The set A always B ⊠ B′ and the vacuum as follows. The anyons in B ⊠ B′ contains 1 B. On the other∈ hand, not every anyon take the form (a,b), where a B and b B′. Imagine in B can pass∈ through the GDW. In this sense, strict ∈ ∈ ′ one creates an anyon (a,b) in the bulk of B ⊠ B′ and condensation walls are asymmetric between B and B . moves it to the gapped boundary. Since the boundary is It was shown that strict condensation walls are com- gapped, certain such anyons would have to be destroyed pletely classified by boson condensation11,13,14,21, in the at the boundary by local operators acting on the bound- following sense. The anyons in A are those that are con- ary and become part of the vacuum. In other words, densed in B, and B′ is the TO after condensing A in they are condensed on the boundary. Condensing such B. After condensation, all anyons in A are mapped to anyons at the boundary results in a gapped boundary, 1 B′, corresponding to the fact that anyons in A can in a way analogous to the Higgs mechanism or Cooper pass∈ through the GDW and turn into 1 B′. It is shown pair condensation. But not all anyons in B ⊠ B′ can con- that if A contains nontrivial anyons, B has∈ more types of dense at the boundary and disappear into the vacuum. anyons and larger total quantum dimension than B′, or One may trade this picture of gapped boundary with a mathematically, B′ is a subcategory of B.11,13,14,21 The phase transition in certain parameter space, in which one anyons in B that cannot cross the GDW correspond to can physically trigger a Higgs transition from B ⊠ B′ to those that will be confined in B′. Since both B and B′ the vacuum by condensing those anyons that can disap- are bTOs, the condensation must involve only bosons, in pear at the boundary. In this scenario, those anyons of fact self-bosons, of B. Note that the above is a over sim- B ⊠ B′ that cannot disappear at the boundary would be- plified story of boson condensation but this suffices for come confined through the phase transition and are not our purpose in this section. More detailed rules of boson physical excitations in the vacuum. Condensability and condensation will be reviewed in Section IV B. confinement are tied to the notion of mutual locality to One subtle case is that B = B′ and A = 1 . In this be discussed shortly. case, every anyon in B can move across the{ GDW} and On the other hand, one can unfold the gapped bound- turn into some anyon in B′, and every anyon in B′ can ary picture in Fig. 2(b) back to the original picture of pass through the GDW and turn into some anyon in B GDW in Fig. 2(a). This way, one can see that an anyon (i.e., the wall is symmetric). The trivial case is a uni- (a,b) in B ⊠ B′ that can disappear at the gapped bound- form TO B = B′ everywhere. Nevertheless, there still ary corresponds to two anyons, a B and b B′, such exist nontrivial GDWs that permute the topological sec- that they can meet at the GDW between∈ B and∈ B′ and tors in the topological order. As shown in Ref.13,19,21, a disappear simultaneously. Alternatively, it can be viewed nontrivial GDW between B and itself always implements as that a can cross the GDW and turn into b, and vice a global symmetry on B, in the sense that the wall acts as 5 linear map of anyons of B that carry (not necessarily all) by the S-matrix element Sab. After appropriate nor- different representations of the global symmetry group. malization, what we actually measure is the monodromy We thus dub such a GDW a symmetry wall, which may matrix11,35–39 be realized by a defect line in a lattice model or a real 13,21,34 c system . In the unfolded picture, a symmetry wall SabS11 c Nabdcθc Mab = = . (2) shall not be called a condensation wall, because the only Sa1Sb1 Pdaθadbθb “condensed” boson is 1. In this paper, we always consider the case that A contains anyons other than 1. Hence, the Two anyons a and b, not necessarily different, are mutu- subtlety of symmetry wall is not significant for us. ally local, i.e., winding one of them around the other has Now, back to the folded picture. One can see that a trivial effect on the state, if and only if Mab = 1. Keep gapped boundary between B⊠B′ and the vacuum is also in mind this is the definition in bTOs. Generic anyons a strict condensation wall. Hence, through the folding a and b may have more than one fusion channels, and c trick, we see that general GDWs, not limited to strict the monodromy matrix (2) is a weighed average of Mab; condensation walls, can be understood as boson conden- hence, we can define a weaker notion of mutual locality: sation in bTOs. two anyons a and b are mutual-local via c if there exists a fusion channel c a b, such that ∈ ×

B. Mutual Locality c θc Mab = =1. (3) θaθb With the above briefing of boson condensation, we now Two anyons mutual-local via certain fusion channel are discuss about for a given bTO B, what anyons can con- also said to be partially mutual-local11. dense and what are confined. The unconfined anyons form a new bTO B′. The crucial notion condensability We are now enabled to define that in a bTO, an anyon a and confinement is mutual locality. is mutual-local “with respect to” the trivial boson 1 if and only if M 1 =1. If a is self-dual, then we need M 1 = 1. In Cooper pair condensation that leads to supercon- aa¯ aa Furthermore, if two distinct anyons a and b, which are not ductivity, the condensed particles form the new vacuum anti-anyons of each other, can condense simultaneously, of the system after the condensation. It is natural to ex- we would also require a and b being mutual-local via pect in any kind of anyon condensation, the condensed some anyon c a b, and c better condense as well. anyons would become the new vacuum of the condensed This is reasonable∈ because× a and b will both become the phase. But what is a vacuum depends on the underlying new vacuum after condensation, and the vacuum is so degrees of freedom of the system under consideration. trivial that a and b should better be at least partially In the case with GDWs between bTOs, the underlying mutual-local. degrees of freedom are purely bosonic. As such, a well- 1 Clearly, the partially mutual-local condition Maa¯ = 1 defined vacuum of such a system would better be de- 2 implies that θaθa = θ = 1 θa = 1. Namely, what scribed by a bosonic theory, and in such a vacuum the ¯ a ⇒ ± only well-defined topological sector would have to be a can condense is either a self-fermion or a self-boson but trivial boson 1. Having agreed on this, since the conden- nothing else if we demands such a definition of mutual- sation describing a GDW between bTOs is to become locality, with respect to a bosonic vacuum. Nevertheless, a new purely bosonic vacuum that again contains only this condition is not strong enough for classifying GDWs the trivial topological sector, the condensed anyons are between bTOs, because if a self-fermion condensed, it expected to be mutual local with respect to the trivial would become a trivial fermion that exists only in an boson 1. The question now is: What does it mean by fTO. In other words, fermion condensation would result being mutual-local with respect to 1? Note that here in a fermionic vacuum rather than a purely bosonic one. we does not say “with 1” but instead “with respect to Therefore, having been restricted to classifying GDWs 1”. The reason is that the trivial boson 1 is indeed triv- between bTOs, one has to impose further constraints, ial and thus has trivial exchange statistics with anything such as, only self-bosons can condense. The mathemat- else. That is, “with respect to 1” has a particular mean- ical formulation of such further constraints are reviewed ing, which can be understood if we know how one usually in Section IVB and not to be repeated here. 1 measures the topological spin of an anyon or braiding of Now that the condition Maa¯ = 1 seems leaving us some two anyons. room for condensing fermions, as argued above, to con- To measure the braiding effect of two anyons, say a dense fermions in bTOs, we need to consider the GDWs 1 and b, we create a and its anti-anyona ¯, and b and ¯b on between a bTO and an fTO. Note that Mff¯ = 1 for any a surface, then let a and b move on the surface such that self-fermion, including the trivial fermion 1f in any fTO. they exchange their positions twice, and finally annihi- Moreover, 1f and 1f always fuse to 1. Hence, the mutual- ¯ 1 late respectively a witha ¯ and b with b. This procedure locality condition Maa¯ = 1 is well suited but at most is equivalent to measuring the the interference between suited for fermionic vacua. So, GDWs between bTOs the state of the system after the braiding a with b and and fTOs provide prefect systems for us to consider con- the state without braiding a and b, which is captured densing self-fermions. 6

C. Fermion Condensation and GDWs one seems always able to work in the regime of boson condensation in the folded picture, which is true in prin- Before we propose our ansatz for fermion condensa- ciple. Nevertheless, we choose to start from scratch and tion, let us provide a correspondence between fermion understand fermion condensation in bTOs by itself. Con- condensation and the GDWs between bTOs and fTOs, versely, the equivalent picture of boson condensation in which is analogous to the correspondence between boson fTOs can serve a double check of the principles of fermion condensation and GDWs between bTOs. condensation in bTOs to be found in the sequel. We imagine condensing certain self-fermions in a bTO B and ending up with a topological order with a trans- parent (trivial) fermion besides a trivial boson, i.e., end- D. Ansatz of Fermion Condensation ing up with an fTO F F0. Although we do not know whether such mechanism⊇ of fermion condensation in cer- In this subsection, we write down our ansatz of fermion tain order parameter space may exist or not, we do have condensation, which can be generalized to more general a real space picture of the phase transition: the bTO B anyon condensation in Appendix C. We take a formalism and fTO F are connected by a GDW (Fig. 3(a)), which similar to that characterizing boson condensation, which belongs to some kind of strict condensation walls between is reviewed in Section IV B. bTOs and fTOs. The properties of such strict conden- B sation walls actually “define” what we mean by fermion Ansatz. Fermion condensation in a bTO is a com- posite object A = naa nbb B, where a,b,... are condensation and lead to our ansatz for fermion conden- B ⊕ ⊕···∈ sation, discussed in the next subsection. anyons in and na the multiplicity (number of occur- Discussed in Sec. III A, it is equivalent to look at the rences) of a in A, that meet the following conditions. GDWs between two TOs via the folding trick.13,21,23 As 1. Unit: B’s vacuum—trivial boson—1 A, n1 =1. in the case of boson condensation in bTOs, one can fold ∈ the system in Fig. 3(a) along the GDW between B and 2. Self-dual: For any a A, the anti-anyon a¯ A. F. The result is the system in Fig. 3(b) as stacking a ∈ ∈ 1 layer of B over a layer of F, which is the time reversal of 3. Self-locality: For any a A, Maa¯ =1. f ∈ F. Since the trivial fTO F0 = 1, 1 F, condensing ′ B { } ⊆ 4. Closure: For any a,b A, there exists at least a self-fermions f is equivalent to condensing the pairs c∈ ′ f ∈ c a b, such that M =1 and c A. (f , 1 ) B ⊠ F, which are effectively self-bosons. ∈ × ab ∈ ∈ 5. Trivial associativity: (A A) A = A (A A). × × × × This ansatz is physically reasonable because the con- densation would form the new fermionic vacuum of the F B F B ⊠ F 0 system after the condensation, and a vacuum needs to have trivial statistics in order that the quasiparticle exci- tations have self and mutual statistics well defined with (a)Gapped domain wall (b)Folding trick respect to the vacuum. A vacuum must also be closed under the fusion of its constituents. The trivial associa- B F tivity condition is physically transparent but technically FIG. 3. (a) A bTO and an fTO connected by a GDW via highly nontrivial. That an anyon may have to appear condensing some fermions in B. Via the folding trick along the GDW, (a) is equivalent to (b) A topological order B ⊠ F. more than once in A is also a consequence of trivial as- sociativity, an example of which can be found in boson sharing a gapped boundary with F0, a fermionic vacuum. condensation in Ref.23. We shall look at this condition more closely upon necessity. The picture of equivalence delineated above not only As discussed in Section III B about mutual-locality, the helps understand fermion condensation in a bTO but also self-locality condition in the ansatz forbids anything but offers a physical realization of fermion condensation in a self-bosons and self-fermions to condense. We repeat the bTO. That is, in order to condense certain self-fermions argument here for clarity: M 1 = 1 θ θ = 1; how- f ′ B, one can stack a layer of a trivial gapped fermion aa¯ a a¯ ever, since we know θ = θ , we have ⇒θ2 =1 θ = 1. system∈ F with B to form B ⊠ F and then condense the a a¯ a a 0 0 Anyon condensation in a two-layer system⇒ always± fall effective self-bosons (f ′, 1f ) in this bilayer system (the into two scenarios: 1) the condensation is separate in each trivial gapped fermion system would have chiral central individual layer, and 2) some composite anyons made of charge c = 0 such that stacking it with B does not change F⊠ F those in each layer condense, such that the condensa- the total c). The result would be F0 0. But according F ⊠ F F tion can be viewed as occurring in a single-layer system. to the definition of stacking, F0 0 = . One thus ′ ∼ Anyon condensation in multi-layer systems follow like- condensef realizes the transition B F. We will see an wise. Our Ansatz thus suffices in any bTOs. It is worth −−−−−−−→ example of this equivalence in Section V B. of mention that in classifying GDWs between bTOs via One may however wonder given such an equivalence, anyon condensation, the first scenario is usually neglected 7 because it is rather trivial, as the two layers are com- certain anyons may force some other anyons to condense pletely uncoupled, independent systems.40 As such, we as well. Suppose two self-bosons (not necessarily differ- shall ignore the first scenario too. ent) a and b condense simultaneously, if a b c with ×c ∋ Although condensable anyons in a bTO can be either c a self-boson too, then the monodromy Mab 1. So, self-bosons or self-fermions, there exists anyon conden- c potentially may condense; however, whether≡c would sation that involves only self-bosons. Such cases are be forced to condense or not further depends on the de- called (pure) boson condensation and have been thor- tail of the consequences of condensing a and b, which is oughly studied.9–11,13,21–23,34 In boson condensation, all irrelevant for our discussion here. On the other hand, c condensates must be self-bosons and at least partially if c is a self-fermion instead, then Mab 1 = 1 for mutual-local with one another, which form what is known sure, which is true from any self-bosons a≡and − b6. Thus, a twist-free commutative separable Frobenius algebra13 condensing a and b cannot force c to condense, and no (CSFA), also referred to as a condensable algebra21. For matter what other self-bosons are fed to condense to- simplicity, we shall refer to the Frobenius algebras de- gether with a and b, c would not be obliged to condense. scribing boson condensation bosonic Frobenius algebras, Unless certain fermion d that can condense along with a where ‘bosonic’ encodes the extra conditions on top of and b is mandated to condense by hand (i.e., by turning Frobenius algebras. on a particular interaction to make d condense), and d Fermion condensation is more general than boson con- can fuse with some condensed boson, say, a for example, c densation. In fact, as we will show shortly, by our produces c. If so, because Mda 1, c may be forced ansatz—in particular the trivial associativity condition— to condense now if its condensing≡ does not violate other when a self-fermion f condenses, if f f may have to conditions in the ansatz. Therefore, if one would like to b × contain a self-boson b, such that Mff = 1 and b con- condense self-bosons only, one can leave the self-fermions denses along with f. Thus, fermions that condense may alone without violating the ansatz. be accompanied by bosons that have to condense simul- Nevertheless, condensing self-fermions may force cer- taneously. In other words, boson condensation is a spe- tain self-bosons to condense too. Suppose two self- cial case of fermion condensation. One thus can reason- fermions a and b (not necessarily different) condense, and ably expect that the rules governing fermion condensa- there is a fusion channel c a b that is a self-boson. c ∈ × tion contain the rules of boson condensation as a subset, Because Mab 1 in this case, c may be demanded to as to be seen. Since we always include the trivial boson condense, which≡ depends on further detail of condensing in the set of condensed anyons for consistency, there does a and b. We now show that in fermion condensation in not exist pure fermion condensation. a bTO, if there are two or more distinct self-fermions condense together, there must be concurrent self-boson condensation. IV. FERMION CONDENSATION IN bTOs Consider a bTO B that supplies sufficient distinct self- fermions and self-bosons for our need. Let us condense The main purpose of this section is to study properties only a single self-fermion f B. Since the trivial boson 1 of fermion condensation in bTOs, with the ansatz intro- is always included in any condensation,∈ we denote the set F duced in the previous section and with the GDW picture of condensates temporarily by A0 =1 f and see if it ful- in mind. We establish a hierarchy principle for fermion fills the ansatz. Immediately, we see that⊕ f has to be self- condensation in bTOs. At the end of this section, we also dual, i.e., f f 1; otherwise, by the ansatz f¯ = f must discuss about boson condensation and fermion condensa- also condense.× Now∋ that f is self dual, we have6 at least 1 tion in fTOs. Mff 1, and thus f can indeed condense. All conditions ≡ F in the ansatz are satisfied. Hence, A0 is a well-defined fermion condensation; in fact it is the smallest possible A. Principles of Fermion Condensation in bTOs fermion condensation, and the only fermion condensa- tion that does not contain any nontrivial self-boson. We F Having had an expedition in the physics and ansatz dub A0 minimal fermion condensation. We will justify of fermion condensation, we are good to go back to the the meaning of this minimality further in Lemma 2. An abstract picture Fig. 3(a) to understand the rules and example is AF = 1 ψ in the Ising topological order 0 ⊕ consequences of fermion condensation in bTOs. BIsing = 1, σ, ψ , where ψ is the only self-fermion. { } Recall in the previous section we find that there can Before we build larger anyon condensation consisting be pure boson condensation but fermion condensation in of more distinct anyons, let us gain better understand- general involves boson condensation as well. Actually, ing of the trivial associativity condition (A A) A = we have a stronger relation between boson condensation A (A A) in the ansatz. Recall that condensation× × A is and fermion condensation. First let us have a better un- an× object× of the UMTC B describing a bTO, namely, it derstanding why boson condensation can always stand is a composite anyon. If A condenses, it would become alone without invoking any fermions to condense. The the vacuum of the the new topological order after the reason roots in our fundamental ansatz. In order that condensation. In contrast to nontrivial anyons—nonlocal all the conditions in the ansatz are satisfied, condensing objects—in a topological order, the vacuum sector is lo- 8

B F cal. In the standard picture of topological orders, non- has nothing to do with that in . This A0 will play a trivial anyons are created by nonlocal string operators key role in fermion condensation. To that end, let us add and thus are sitting at the ends of strings. The vacuum, more condensable anyons to build larger fermion conden- however, is associated with local operators and hence sation. There are the following three ways of doing this not attached to any string or equivalently speaking is enlargement, a consistent discussion of which requires us attached to a trivial string. Fusion between nontrivial not to assume f being a simple current. anyons can be viewed as interaction between the strings We again take a single self-fermion f. First, we can the anyons are attached to. Fusion is associative, i.e., for assume that f is not self-dual. So, there exists an f¯ = f three anyons a,b, and c, a (b c) = (a b) c. The B ¯ 6 × × ∼ × × in . The ansatz requires f to condense with f. But this isomorphism ∼= indicates that its two sides, a (b c) causes an issue: since f is not self-dual, f f 1; hence, and (a b) c are two Hilbert spaces in general.× × But b× 6∋ 2 × × we need another channel b f f with Mff = θb/θf = 1, isomorphism is not equality, as reconnecting the strings such that f meets the ansatz∈ × and can continue to con- involved in the fusion interaction to go from a (b c) × × dense. It is clear that θb = 1, i.e., b is a nontrivial self- to (a b) c changes the basis of the isomorphic Hilbert boson. As such, by the ansatz our condensation becomes spaces.× Such× a basis change is a nontrivial linear trans- F ¯ F A1 = 1 f b f. Such that this A is closed under formation. fusion, we⊕ need⊕ ⊕ the fusion rules f f b, f b f¯, Equipped with this understanding, the trivial associa- b b 1, f¯ b f, f¯ f¯ b, and f× f¯ ∋ 1. The× anyons∋ tivity in our ansatz simply states that for condensation that× are∋ neglected× ∋ in these× ∋ fusion products× ∋ are irrelevant A, to become the new vacuum, the linear transformation F because they do not have to condense to make A1 well from (A A) A to A (A A) is trivial—an identity. × × × × defined. We demand b self-dual also to avoid introducing It is however highly nontrivial to prove that an arbitrar- more anyons to condense. But we refrain from discussing ily constructed A, which clearly satisfies all but not ob- whether these condensable anyons are simple currents or viously the trivial associativity condition, actually does F not, which is case dependent. Examples of A1 can be satisfy the trivial associativity. For this, in general one found in Abelian bTOs. would need the full topological data of the bTO where Second, we can still let f being self-dual but add extra A belongs to. Nevertheless, fortunately, if an A does not anyons to enlarge the condensation. For our purposes, let meet the trivial associativity, it is rather simple to show. us add exactly one more self-fermion f ′ B to condense. In fact, as proven in the Appendix B, trivial associativity This implies that f ′ is self-dual too.∈ Again, an issue of A leads to a necessary criterion for judging whether A would arise if we only condensed f and f ′. Since f = f ′ may be well-defined condensation. For clarity, we repeat and they are both self-dual, f f ′ 1. We thus need6 this necessary condition here. × ′ 6∋ b at least another channel b f f with M ′ = 1 for ∈ × ff Lemma 1. For any nontrivial anyon a A, A being f and f ′ can simultaneously condense, which forces b well-defined anyon condensation, if a is a nonsimple∈ cur- being a self-boson. Moreover, AF must be closed under rent, i.e., da > 1, then there exists at least a nontrivial fusion. So, either b or some other self-boson similar to self-boson b a a¯, such that b A. b must condense. To limit the size of the condensation, ∈ × ∈ we simply make b condense and self-dual. We then have Note that in the lemma, b is not necessarily different F ′ ′ A1 =1 f b f with each element self-dual and extra from a ora ¯. Also note that the anyons a under considera- fusion rules⊕ ⊕f ⊕b f ′, f f ′ b, and f ′ b f. An tion are either self-bosons or self-fermions. The nontrivial ×′ ∋ × ∋ × ∋ example is AF = 1 ψ1 ψψ¯ 1ψ¯, where b = ψψ¯, in anyon b must be a self-boson no matter what a is because 1 the doubled Ising topological⊕ ⊕ order⊕ B ⊠ B . The it is the only way of guaranteeing M b = 1. Lemma 1 Ising Ising aa¯ stacking in this example is nontrivial because condensing leads to an important result. Let us look at the minimal ψψ¯ requires coupling the ψ and ψ¯ respectively of the two- fermion condensation AF =1 f we brought up earlier, 0 layers. That is, B ⊠ B is treated as a single-layer where we only require that f ⊕ f 1; however, Lemma Ising Ising × ∋ F ′ 1 readily demands that df = 1 and hence f f = 1, as system. Condensing A1 will appear to be rather simple F × otherwise, A0 must contain at least another self-boson as soon as we complete developing the rules of fermion for it to be well-defined condensation. We thus have the condensation. following lemma. Third, instead of adding more self-fermions to con- dense along with the f, we can try to add only one more Lemma 2. The fermion f in minimal fermion conden- self-boson, say, b B to the condensation. We need f sation AF =1 f must be a simple current, i.e., it has 0 and b being both self-dual∈ so as not to involve any more quantum dimension⊕ d = 1, and therefore satisfies the f anyons to condense. It is straightforward to check that fusion rule f f =1. F B × the fermion condensation defined by A2 =1 b f F satisfies the ansatz if b and f, apart from being⊕ self-dual,⊕ ∈ Note that the self-fermion f in A0 is not a universal particle that is the same in all bTOs; rather, it is a partic- are also subject to the fusion rules f b f, f f b, B b b b, and f b b. We will show× an∋ example× of∋AF ular self-fermion in any given bTO , which has certain × ∋ × ∋ 2 braiding with other anyons in B. Given a different bTO condensation in Section V C. B′, f is certainly some particular self-fermion of B′ and Without further enlarging AF , the above discourse 9

F F B F has already shown clearly that any A larger than A0 densing A A would alter the anyon content of the must contain nontrivial self-bosons. Such self-bosons original bTO⊂B and hence the self-fermions in the AF can surely condense simultaneously, and as argued ear- but we need not to know any such detail to proceed. Ac- lier, their condensing does not force the self-fermions cording to the theory of boson condensation11,13,14,17, we in the same AF to condense although the converse is know that condensing AB turns B into a smaller bTO B′, true. These self-bosons are included in the condensa- the details of which is irrelevant for now. As such, the ′ tion to make fermion condensation well-defined. In other AF AB would become some AF as a composite object F ⊃ ′ words, the self-bosons in an A form legal boson con- in B′. Our claim is that AF AF . This can be proven 13 0 densation on their own. According to Ref. , boson con- by contradiction as follows. ≡ ′ densation in a bTO is specified by a bosonic Frobenius There are three steps. 1) The object AF must contain algebra AB whose elements are simultaneously condens- ′ at least one self-fermion. Suppose not, i.e., AF = 1, since able self-bosons, including the trivial boson 1. Although boson condensation does not induce any fermion conden- we lack a completely rigorous mathematical account of sation, this supposition would imply that there were not the algebraic structure of AF , our ansatz and the discus- any self-fermion involved in the AF anyway in the first sion above conjectures that AF would still be a Frobe- place, which is beyond our consideration. 2) There is ex- nius algebra but not a bosonic one because the twist-free F ′ condition and commutativity condition seem relaxed, as actly one self-fermion in A . If not, according to our discussed in Appendix A. Let us then from now on call discussion before, there has to be certain self-bosons re- F ′ F ′ AF a fermionic Frobenius algebra. The long exposition maining in A in order that A is well-defined, which above readily establishes the following lemma. is contradictory to that we have already condensed all the self-bosons in AF . 3) By Lemma 2, fermion conden- F F Lemma 3. Any fermionic Frobenius algebra A char- sation of exactly one self-fermion must be A0 , namely, ′ acterizing fermion condensation in a bTO contains a AF AF . These establish the first half of the following B F 0 bosonic Frobenius algebra A A as a subalgebra. theorem.≡ Importantly, the self-fermions in⊂ an AF cannot form a F closed algebra under fusion on their own. Theorem 4. Any fermion condensation A in a bTO B always admits the decomposition B F The trivial case is that A0 = 1 A0 . Note that B ⊂ F B ⊠ F an A is not only a Frobenius algebra but a twist-free A = A −→A0 , (4) CSFA. Twist-free requires AB contain only self-bosons. Commutativity is the usual one for algebra multiplica- where AB is the bosonic part of AF . Such condensation tion. Separability admits a direct sum presentation of turns B into an fTO F, such that the dimensions of B, B B F F A in the bTO B, which is a UMTC. That is, A =1 bb , and A satisfy the relation as a single (but not simple) object in B. We take these⊕ B DB DB conditions for granted and simply call an A a Frobenius F √ D = 2 F = . (5) algebra to avoid clutter. We also remark that although dim A √2 dim AB the fermions in an AF do not form a closed algebra, some- In this theorem, particularly in Eq. (4), we use the times, certain simple current fermions together with the ⊠ trivial boson of AF may form a condensable subalgebra notation −→, which is similar to a stacking operation in of AF . We shall see an example of this in Section V B. only one direction. We use this notation because we still have not fully understood the rigorous algebraic structure Lemma 3 has a crucial consequence; it leads to our first F F and main principle as follows of doing fermion condensa- of A and the representation category of A , which we tion in bTOs. call a fermionic Frobenius algebra. Fortunately we find that the effect of AF condensation in B is equivalent to Hierarchy Principle. To do fermion condensation B F F B condensing a twist-free CSFA A A followed by min- specified by a fermionic Frobenius algebra A A in a F ⊂ B ⊇ imal fermion condensation A0 , also a minimal fermionic bTO , one can always first perform the boson conden- B sation AB entirely, applying the rules of boson conden- Frobenius algebra. Condensing A first results in an in- B′ sation, and then condense the necessary self-fermions. termediate bTO , the simple current self-fermion f in F B′ B the A0 is an anyon of rather than an anyon of . This principle needs more elaboration, and interest- F ing results will follow. First, if AF = AF , there is no The relation between this f and the self-fermions in A 0 B F nontrivial boson condensation involved, and one shall di- is nontrivial and depends on the details of and A , rectly proceed to condense the only self-fermion in AF . which we shall address this in an example later. We sim- 0 ply denote such a two-step procedure of condensing AF We will get to the rules of such condensation later. F B F F B by Eq. (4), as stacking A0 on top of A . We also use Second, if A = A0 , A is not trivial. One can then 6 B ⊠ F proceed to condense A first. What is next? The answer −→ to emphasize that A0 is not a fermionic Frobenius ′ is surprising and simple: Having condensed AB AF , algebra in B but in B ; otherwise, we would have simply what remains to condense is merely an AF , i.e. the⊂ previ- used the usual tensor product . 0 ⊗ ously defined minimal fermion condensation, which con- Conversely, any hierarchical fermion condensation B ⊠ F tains only a simple current self-fermion. Certainly, con- A −→A0 can be lifted to a nontrivial fermion condensa- 10 tion AF . We will com back and elaborate on this scenario If an anyon a B appears in more than one simple in Section IV D. objects of T, we say∈ that a splits

We now prove the second half of Theorem 4, namely m Eq. (5), which is the relation between the total quan- i a n ai, (7) tum dimensions of the original bTO B, that of the fTO → a Xi=1 F after the condensation, and the dimension of the con- densation AF . For a boson condensation AB, which is i Z where na + is the multiplicity of species ai that are a Frobenius algebra, its dimension has a canonical def- simple objects∈ of T. Practically, a splits if a is a fixed point B 13,14,17 F inition dim A = a∈AB da. Here, since A is of its fusion with any nontrivial condensed self-boson, i.e., F B still a Frobenius algebra,P we have dim A = a∈AF da. if a c a for any c A . Splitting preserves quantum F × ∋ ∈ Then, we have dim A0 = 2. Since the decompositionP dimension: F B ⊠ F A = A −→A0 is indeed analogous to stacking, we nat- m urally have dim AF = dim AB dim AF = 2 dim AB. Ac- i 0 da = nadai . (8) 13 cording to the theory of boson condensation , the di- Xi=1 mensions of B and B′, due to boson condensation AB, B are related by DB′ = DB/ dim A . Hence, what remains Besides, splitting and fusion commute: to show is that i j c k naai nbbj = Nabnc ck, (9) DB′ DB′  ×  F √ Xi Xj Xc,k D = 2 F = . (6) dim A0 √2 which directly follows from that F is a tensor functor. We defer proving this until a little later when we fully Clearly, identification and splitting generally can hardly F understand A0 condensation. Before that, let us have a be two separated, independent phenomena. quick review of the rules of boson condensation. Nevertheless, T is not the final result of AB condensa- B tion. Since 1T [ A is the new vacuum, a well-defined ← new anyon has to be mutual local with respect to 1T. B. Review of Boson Condensation Among all simple objects of T, only those coming from the anyons of B that are mutual local with the algebra B A may be mutual local with 1T, and the rest simple For a full account of boson condensation, please see objects of T are said to be confined. In general, because for example Ref.11,13 for details. A boson condensation of the complexity of AB and the splitting of anyons, it is in a parent bTO B is described by a bosonic Frobenius not easy to apply this criterion of confinement. A con- algebra, i.e., a twist-free CSFA AB , which is a composite venient trick is offered by Bais et al11: Consider t T object in B. This AB fulfills all the conditions in our and t = a b . . . , where a,b, B, the t is confined∈ ansatz, and on top of this, each constituent of AB must if any two⊕ constituents⊕ in this···∈ combination have topo- be a self-boson, which is the twist-free condition. Besides, logical spins differ by a noninteger, say, for example, if AB is commutative (see appendix). Having condensed h h Z. This is physically reasonable because a new AB, the entire algebra AB is projected to the new trivial a b anyon− must6∈ have well-defined topological spin. Topolog- boson—new vacuum. The intermediate result is a tensor ical spins that differ by merely an integer are regarded category T. The simple objects of T are defined with B the same, as they differ by a true boson. respect to the new vacuum 1T [ A and are generally Given a UTMC B, after condensing AB B, all composite objects of the original←B. unconfined anyons form a child UTMC and∈ hence a So, the next step is to find out all the simple objects bTO B′. The relation between various dimensions is of T. To do so, note that there exists a functor map B DB′ = DB/ dim A , as aforementioned. One sees that def F : B T, such that for any anyon a B, F (a) === this relation between dimensions is merely a special case B → ∈ a A T. Nevertheless, not every simple object of T is of our formula (5). The confined sectors do not just dis- × ∈ mapped from an anyon of B by the functor F . It turns appear but become defects in B′ and in many cases gen- ′ out that for any a B, if da < 2, F (a) T is a simple erate a global symmetry acting on B . Hence, the child ∈ ∈ object. Using this, one can then list out F (a), a B topological order is in fact a symmetry-enriched topolog- ∀ ∈ and search for the simple objects of T recursively. ical order22,34. Typically, a global symmetry on B′ arises If a simple object t T takes the form t = F (a) for when the condensed self-bosons are all simple currents ∈ some a B, then dt = da; if not, say, for instance t = because simple currents form a group under fusion, so ∈ B a b = F (c), c B, then dt = (da + db)/ dim A . Keep do the confined sectors due to the condensation. Con- ⊕ 6 ∀ ∈ in mind that dt is the quantum dimension of t defined in densation of nonsimple currents may also yield a global T. The phenomenon that t T is a direct sum of certain symmetry on B′ but this is not guaranteed because the anyons of B is called identification∈ , in the sense that the confined sectors due to nonsimple current condensation anyons appearing in the direct sum are identified as the do not form a group under fusion in general. The con- same simple object in T. fined sectors can also be viewed as symmetry fluxes in 11

B′. Gauging the global symmetry on B′ takes B′ back Verlinde formula4142 to its parent bTO B. e B B ⊠ B ⊠ SabSac = da NbcSae, (11) A typical example is = Ising Ising = 1, σ, ψ { } eX∈B 1, σ,¯ ψ¯ and AB = 1 ψψ¯ B. After condensing this { B } B′ ⊕ ∈ Z c SaeSbeSce¯ A , we have = 1,e,m,ǫ , which is the 2 toric code Nab = , (12) { } S1e bTO. Here, the splitting occurs as σσ¯ e + m, and eX∈B ψ1 and 1ψ¯ are identified to be ǫ. The global→ symmetry Z Z where a,b,c,e B. Setting in Verlinde formula (11) generated on the 2 toric code is a 2 symmetry whose ∈ action exchanges e and m. Gauging this Z2 symmetry b = c = f, the condensed self-fermion, which enforces e = f f = 1, we have takes the toric code back to the doubled Ising. × 2 2 da 2 da (Saf ) = (Sa1) = ( ) = Saf = . (13) DB ⇒ ±DB C. Minimal Fermion Condensation That is, all anyons in B are graded by the simple current fermion f B into two disjoint sets of anyons, Having recalled the protocol of boson condensation, ∈ we are now good to study minimal fermion condensa- B UB = 1,f,u1,u2,... ui ,Suif = dui /DB , tion. Consider a generic bTO B that contains a minimal { | ∈ } (14) CB = c ,c ,... ci B,Sc f = dc /DB . fermionic Frobenius algebra AF = 1 f B. We em- { 1 2 | ∈ i − i } 0 ⊕ ∈ phasize that this f has the properties f f = 1, df = 1, Now clearly, the set CB will be confined, whereas set UB × and θf = 1. Like boson condensation, after condensing will be unconfined, after AF condensation. In the picture F F − 0 A0 , A0 would be projected to the new vacuum. The of GDWs, the confined anyons are those cannot penetrate only and key difference here is that in an fTO F, the the GDW between B and F, while the unconfined ones vacuum contains two trivial sectors, a trivial boson and can cross the GDW from B and become (possibly after a trivial fermion, which are both mutual local with any recombination) well-defined anyons in F. other anyons in the fTO. In other words, there is a triv- An important result follows from the above grading. F f F ial fTO 0 = 1, 1 , as defined earlier, embedded in . In Verlinde formula (12), let a = c = 1 and b = f, and { } In general, such an embedding is not trivial. In other since 1 f = f, we have words, generally F does not admit a stacking decompo- × F F ⊠ B′ B′ F 1 sition = 0 for any bTO . This 0 has total 0= N1f = S1eSfe quantum dimension DF0 = √2 and a canonical modular eX∈B S-matrix, which is however degenerate, = S1eSfe + S1eSfe eX∈UB eX∈CB 1 1 1 F 2 2 S 0 = . (10) d d √2 1 1 e e = 2 2 , DB − DB eX∈UB eX∈CB Hence, F0 is not a modular tensor category. F B F F Condensing A0 means to identify A0 with 0 in where use of the grading (14) is made. Thus we obtain the resultant fTO F∈ after the condensation. All nontriv- 2 2 ial anyons in F must be defined with respect to F0. Sim- de = de. (15) ilar to boson condensation, identification and splitting of eX∈UB eX∈CB B anyons can happen too in fermion condensation, so do confinement and unconfinement. Since the anyons in the set UB will become (possibly after recombination) anyons in the fTO F after AF condensa- Let us first focus on confinement. For AF condensa- 0 0 tion in B, we obtain tion, the scenarios are much simpler. Because f f = 1, × it can neither split or be identified with anything else DB DB B f DF = = √2 . (16) in . That is, f must unanimously become the 1 in √2 dim AF F. Moreover, since f is a simple current, the fusion 0 B B F between f and any anyon of produces exactly one Since is an arbitrary bTO that supplies an A0 , we anyon of B. Hence, to see whether a B anyon a will conclude that Eq. (6) is correct43, and so is Eq. (5). be confined due to f condensation, it suffices to check This completes the proof of Theorem 4. b the mutual monodromy Maf = Maf , where b is the Now let us study splitting and identification of anyons sole fusion channel b = a f. By Eq. (2), we have through AF condensation in B. In boson condensation, × 0 Maf = Saf /S1a = DBSaf /da. Hence, the monodromy splitting of an anyon may occur only when the anyon is Maf is completely determined by the S-matrix element a fixed point of its fusion with any nontrivial condensed Saf . Clearly, Maf = 1 only if Saf = da/DB. Before self-boson. In fermion condensation, we expect the sim- F we make this claim formally, however, let us check what ilar phenomenon, and in particular for A0 condensa- Saf can be most generally. For this, we shall deploy the tion, we can address splitting straightforwardly. Suppose 12 a B satisfies a f = a, i.e., a is a fixed point. The a super pair. In contrast, in a primitive fTO, the Hilbert definition∈ of S-matrix× (1) implies that space of a generic super pair, say, (b,bf ) is rather non- trivial and admits no decomposition into fermion parity θada da even and odd subspaces. This is because the fusion b b Saf = . × θaθf DB ≡−DB in general can produce both anyons and their super part- ners as well. So do the fusion bf bf and b bf . Thus, a CB and is confined no matter it splits or not. × F × ∈ Now that any fermion condensation A can be decom- As to the phenomenon of identification, consider two posed into a boson condensation AB followed by solely distinct anyons a,b B, such that a f = b. Again the AF condensation, the Zf symmetry is guaranteed to exist S-matrix (1) implies∈ that × 0 2 after the condensation. The boson condensation AB may F θada θada also generate a global symmetry Gs on . If this hap- Saf = = Zf θbθf DB −θbDB pens, the total symmetry would be certain 2 -extension Zf d of Gs. A trivial extension would be 2 Gs. a Z F × = Saf = iff ha hb +/2. The last aspect of A condensation in B regards the ⇒ DB | − | ∈ 0 fusion rules in the child fTO F. By the commutativity Because a f = b also implies b f = a, as f f = 1, between splitting/identification and fusion, the fusion be- an anyon may× only be identified× with exactly one× other tween unconfined, namely anyons in F cannot yield any anyon. The derivation above shows that any two iden- confined anyons, also as the latter are excluded from the tified anyons whose topological spins differ by a half in- spectrum of F in the first place. Since there is no real B F teger will be unconfined and become anyons in the final identification of anyons through A0 condensation, all fTO F. Such identification in fermion condensation has unconfined B anyons remain as individual anyons in F a subtlety. despite being grouped into super pairs. Therefore, the Recall that in boson condensation, if two anyons are fusion rules of the anyons in F are simply the same as B F identified and unconfined, their topological spins differ those of these anyons in except that the f A0 that ∈ f by merely an integer, or in other words, they differ by a appear in B’s fusion rules must now be replaced by 1 in trivial boson that is identified with the condensed self- the corresponding fusion rules of F. boson. If we are not concerned with the global sym- For clarity, we can summarize the above discussion in F metry generated on the unconfined anyons by the boson the following five (not mutually exclusive) rules of A0 condensation,44 the anyons that are identified are truly condensation in B. the same anyon in the resultant topological order. 1. Any a B with Saf = da/DB is unconfined and Back in the fermion condensation under consideration, ∈ F otherwise confined. the new vacuum after condensing A0 is not anything structureless but consists of two superselection sectors 2. Any a B with a f = a is confined. that differ by fermion parity. The vacuum of a physi- ∈ × cal fermionic system respects locality would be in either 3. Any a,b B with a f = b are unconfined if but not both of the two superselection sectors, such that ∈ × ha hb Z+/2 and otherwise confined. A pair of fermion parity is a good quantum number. Hence, two |such−a and| ∈ b becomes super pairs in F. anyon excitations that differ by fermion parity, i.e. that F F B their topological spins differ by a half integer, can be well 4. The fTO due to A0 condensation in contain distinguished by local experiments and thus would not be all the unconfined sectors. Dimensions of F and B F truly identified. Rather, in A0 condensation, two anyons satisfy Eq. (16). a and b related by a f = b may be considered as a doublet in which they× have opposite fermion parity. In 5. Fusion rules in F are the same as those in B with Zf f replaced by 1f . other words, there is a 2 —fermion parity symmetry— F F generated on due to A0 condensation. Actually, in any fTO, the anyons are grouped in “super pairs”, and in each super pair the two anyons are “super partners” D. Other GDWs between fTOs of each other.45 For non-prime fTOs, the Hilbert space of a super pair We have studied fermion condensation in a bTO B that admits a simple decomposition into subspaces of even and takes B to an fTO F. Such a picture is equivalent to a odd fermion parities respectively. Consider a non-prime GDW between B and F on which the condensed anyons fTO Fnp = Bp ⊠ F0 for some bTO Bp. Then the anyon become excitations that can be created and annihilated content of F takes the form (a,af ) a B . Since by local operators. The confined anyons in the conden- np { | ∈ p} Bp is a well-defined bTO, the fusion of any two anyons sation picture are those B anyons that cannot go from B a,b B never produces neither of their super partners, through the GDW into the fTO F, while the unconfined ∈ p and vice versa. The fusion of a Bp anyon with its super one are those who can move between B and F through the partner always yields the super partners of Bp anyons. GDW, which acts as a transformation on the unconfined This manifests the decomposition of the Hilbert space of anyons. Bearing this and the discussion so far in mind, 13 we now elaborate on an important remark on Theorem One may also think of the GDWs between two fTOs 4 we made in Section IV A. F and F′. Such GDWs can correspond to two kinds of Suppose a bTO B admits a boson condensation AB anyon condensation. The first is boson condensation in and breaks into a child bTO B′. Assume that this B′ F that takes F to F′. In this case, boson condensation F happens to bear a minimal fermion condensation A0 and follows the same rules as those of boson condensation in can break into an fTO F. This guarantee that B admits bTOs, while the super partners of the condensed self- a nontrivial AF that directly breaks it into F. In other bosons are treated as usual anyons. We shall see an ex- words, there exists an AF B that can be decomposed ample in Section V B. The second is fermion condensa- ∈ ′ as AB−→⊠AF . We shall see an example of this in Section tion in F. The result is certainly also an fTO, say, F . 0 ′ V B. The following figure depicts this scenario from the Via the folding trick along the GDW between F and F , perspective of GDWs. seen in Fig. 5, we obtain an equivalent configuration, F ⊠ F′ F0 , with a gapped boundary separating the vac- uum. By the same token as in the remark below Fig. 3, such fermion condensation is equivalent to boson con- F ⊠ F F B B′ F B F densation in F0 0 = . Hence, boson condensation and fermion condensation in fTOs are actually the same thing.

WBB′ WB′F WBF = WBB′ WB′F (a) (b)

′ FIG. 4. (a) A GDW between B and B and one between ′ ′ F F F ⊠F F F0 B′ and F imply a GDW between B and F directly. B′ is 0 compressible between B and F.

The above scenario is physically sound. On the one (a)GDW due to fermion (b)Folding trick hand, the entire system in Fig. 4(a) is gapped, includ- condensation ing the domains walls and B′. In fact, one can view B′ FIG. 5. (a) Two fTOs F and F′ connected by a GDW via together with the two GDWs WBB′ and WB′F as a gen- condensing some fermions in F. Via the folding trick, this eralized, 2-dimensional GDW. Hence, one can imagine picture is equivalent to (b) A non-prime fTO F ⊠F F′. shrinking the width of B′ to zero, in which limit the two 0 gapped domain walls WBB′ and WB′F would become a single GDW, i.e., Fig. 4(b). On the other hand, as al- luded to earlier, a GDW serves as a map of the anyons in one topological order to those in another. Such maps can V. EXAMPLES certainly be concatenated into a single map. In purely bosonic cases, a GDW is represented by a rectangular In this section, we apply our principles of fermion con- matrix, and the above process of combining two GDWs densation developed in previous sections to a few exam- is merely a matrix multiplication23,27. A GDW between ples explicitly. a bTO and an fTO would also be represented by a matrix; however, in this paper we shall not address this for the B Z following reason. In purely bosonic cases, a matrix rep- A. = 2-Toric Code resenting a GDW between two bTOs is either the mass matrix of a nonchiral RCFT or the branching matrix of We first go through a simple example, the Z2 toric certain vertex algebra embedding.13 As such, boson con- code. This is an Abelian bTO with four anyons, 1, the densation corresponds to modular invariants of RCFTs. trivial boson, e,m, which are self-bosons, and a self- In the cases with GDWs between bTOs and fTOs, how- fermion ǫ. All anyons here are self-dual and simple cur- ever, we do not yet know the role of modular invariants rents. The nontrivial fusion rules are e m = ǫ, e ǫ = m. × × played here because not only the usual CFTs but also The Z2 toric code thus have a unique minimal fermion F fermionic CFTs are involved. We thus would like to de- condensation A0 = 1 ǫ. According to the rule of con- fer the investigation of how fermion condensation may finement listed in the previous⊕ section, because e ǫ = m × corresponds to fermionic RCFT to future work. and θe θm = 0, e and m will be confined and do not ap- There is another subtlety in the purely bosonic case as pear in− the child fTO F, which in this example is simply f f well: If there is a global symmetry on the relevant bTOs, F0 = 1, 1 . Clearly, the ǫ becomes 1 and no nontriv- { } F i.e., if one is actually dealing with SETs, the compress- ial anyons survive the A0 condensation. In other words, ibility of a bTO between two other bTOs may not be there is a gapped boundary between the Z2 toric code and guaranteed because the global symmetry may also im- the vacuum, e and m cannot cross the boundary into the pose extra conditions on boson condensation. Similar vacuum. In this example, since DB = 2, and DF0 = √2, subtlety may also arise in our current considerations. the dimension formula (6) is obviously satisfied. 14

√ √ √ √ B. B = SU(2)4l−2 dan 1 1 1+ 3 1+ 3 2+ 3 2+ 3 1 1 1 1 ha 0 0 n 2 6 − 3 2 f f f We then look at a family of examples, which are spec- 0 0 2 2 4 4 f f f ified by the representation categories of the affine Lie al- 0 0 0 2 2 4 4 Z 0f 0f 0 2f 2 4f 4 gebra SU(2)k for k =4l 2, l +. Boson condensation f f f f f f f − ∈ 13 2 2 2 0+2+4 0 + 2 + 4 2+4+4 4 + 4 + 2 in this family of bTOs have been studied thoroughly , f f f f f f f f and now we study AF condensation in these bTOs. 2 2 2 0 + 2 + 4 0 + 4 2 +4+4 2+4+4 0 4 4 4f 2+4+4f 2f +4+4f 0+2+2f +4+4f 0f +2+2f +4+4f In an SU(2) , the anyons a are labeled by integers, f f f f f f f f f f 4l−2 4 4 4 4 + 4 + 2 2+4+4 0 +2+2 +4+4 0+2+2 +4+4 a =0, 1, 2,..., 4l 2. The quantum dimensions and topo- − logical spins are respectively TABLE I. Quantum dimensions, topological spins, and fusion F f f rules of SU(2)10 , where a0 and a0 are renamed to 0 and 0 . sin[(a + 1)π/4l] d = , a sin(π/4l) (17) This two-step condensation process can be combined a(a + 2) F ha = (mod 1). into a single fermion condensation of A . First, the ψ 16l in here in BIsing descends from the anyons 4 and 10 of B B Clearly, the anyon a =4l 2 for any l is a simple current SU(2)10 through condensing the A . Thus, if one would − B F self-fermion, i.e., d4l−2 1 and h4l−2 = (2l 1)/2. The combine the A and A0 here as a nontrivial fermion ≡ − B fusion rule for any a and b is condensation in SU(2)10 , one would have to invoke the condensation of 4 and 10, which is legitimate because this a b = cab +(cab +2)+ +min a+b, 8l 4 a b , (18) does not violate any of conditions of the ansatz. In fact, × · · · { − − − } B ⊠ F this two step condensation (A =0 6)−→(A0 =1 ψ) where cab = a b . So, all anyons are self-dual. Hence, ⊕ ⊕ | − | can be promoted to the nontrivial fermion condensation we have for any such bTO with a given l a minimal F A = 0 6 4 10 SU(2)10, whose condensation fermion condensation AF = 0 (4l 2). Condensing ⊕ ⊕ ⊕ ∈ 0 directly breaks SU(2)10 into F0. The discussion above AF results in an fTO F ⊕. To− find the anyons in 0 SU(2)4l−2 simply manifests the hierarchy principle of condensing F we need the unconfined anyons of SU(2) F SU(2)4l−2 4l−2 this A . due the condensation. For any a, its fusion with the Yet, an alternative route exists to achieve the final condensed self-fermion 4l 2 yields exactly one anyon, f trivial fTO F0. The 4 in FSU is a self-boson. It a (4l 2) = (4l 2 a−), by the fusion rule (18). If (2)10 × − − − is actually the anyon 6 in SU(2)10 before condensing ha h4l−2−a Z+/2, a and its super partner 4l 2 a F | − | 6∈ − − the A0 . In SU(2)10, the anyon 6 is allowed to con- will be confined. Using Eq. (17), by elementary arith- 10,13 dense individually, as studied in Ref. . In FSU , metics, we obtain for each l value an fTO: (2)10 4f can condense as well, and its condensation follows the F f same rules of boson condensation in bTOs. The result is SU(2)4l−2 = an,an an =2n,n =0,...,l 1 , (19) { | − } again the trivial fTO F0. These two steps of condensa- f tion can also be combined into the fermion condensation where an =4l 2 an, in terms of the SU(2)4l−2 labels, F − − A =0 6 4 10 SU(2)10. What we just did is con- is the super partner of an. The trivial boson is a0 = 0, ⊕ F ⊕ ⊕ ∈ F f densing A0 =0 10 first, which led to SU(2)10 , followed and trivial fermion is a0 =4l 2. . We emphasize that ⊕B′ f f − f by condensing A = 1 4 FSU(2) (note that 4 is in our notation, the superscript f in an does not imply ⊕ ∈ 10 f f a boson). Therefore, we have the following commutative that an is a fermion at all, as it only indicates that an f f diagram of anyon condensation. differs from an by a fermion parity because a = an a . n × 0 This is no more than a mere choice of convenience. The 0 ⊕ 6 F SU(2)10 SO(5)1 anyons in SU(2)4l−2 fuse in exactly the same way as if F they were SU(2)4l−2 anyons. 0 ⊕ 10 A 1 ⊕ ψ . Among this family of examples, let us focus on the FSU(2) F0 case with l = 3, i.e., SU(2)10, to illustrate the Hierarchy 10 1 ⊕ 4f Principle discussed in the previous section. To this end, F we tabulate the topological data of SU(2)10 as follows. Seen from the commutative diagram above, the The case with l = 3 also illustrates the scenario shown fermion condensation AF =0 6 4 10 here also ad- B ⊕ ⊕ ⊕ in Fig. 4. We again begin with SU(2)10 . This bTO mits a different hierarchical decomposition, namely the B F contains a boson condensation A = 0 6, which has minimal fermion condensation A0 = 0 10 followed by B ⊕ B′ f ⊕ been shown to break SU(2)10 to the SO(5)1 topological the boson condensation A = 1 4 . This is because B ⊕ order SO(5)1 = 1, σ, ψ , where ψ is a self-dual simple minimal fermion condensation involves merely a simple { } B current self-fermion. The bTO SO(5)1 is a sibling of current fermion, whose condensation does not cause any B Ising and has a minimal fermion condensation AF0 = boson condensation in general. Hence,one can first con- F 1 ψ, as mentioned earlier in the work. Condensing the dense the A0 and then do the residual boson condensa- ⊕ B F AF0 breaks SO(5)1 into the trivial fTO 0. tion in the resulted fTO. Clearly, such a decomposition 15 is possible only if certain fermion condensation directly F e1 e1 = 1 e1 e2 = e2 contains some A0 . One will see a counterexample in the × × e2 e2 =1+ e1 + e2 next example. × l l + − − + Finally, as we mentioned previously, fermion conden- e1 m = m e1 m = m e1 m = m × 2 2 × 3 3 × 3 3 sation in bTOs is equivalent to boson condensation in l j k e2 m = m + m , l = j = k fTOs, via the folding trick. In the current case, the × 2 2 2 6 6 + + − − + − F B e2 m3 = m3 + m3 e2 m3 = m3 + m3 A = 0 6 4 10 condensation in SU(2)10 can × × ⊕ ⊕ ⊕ l l l l j k be equivalently understood as condensing certain bosons m m =1+ e1 + m m m = e2 + m , l = j = k B ⊠ F 2 × 2 2 2 × 2 2 6 6 in the fTO SU(2)10 0, where anyons are again la- l + + − l − + − m2 m = m + m m2 m = m + m beld by 0, 0f , 1, 1f ,..., 10, 10f . If we condense AB = × 3 3 3 × 3 3 3 + + l f f F m m =1+ e2 + m 0 6 4 10 , we end up the trivial fTO 0. 3 × 3 l=0,1,2 2 ⊕ ⊕ ⊕ + − l m m = e1 + e2 +P m 3 × 3 l=0,1,2 2 − − P l B m3 m3 =1+ e2 + m2 C. = D[D3] × l=0,1,2 P 11,46 This is another example with nonsimple current self- TABLE III. Fusion Rules of D[D3]. fermion condensation, which certainly also induces self- boson condensation. The parent bTO is the quantum double D[D3] of the dihedral group D3 that is isomor- phic to the permutation group S3 of three elements. We propose the fermion condensation—a fermionic Frobe- F − need not to get into the detail of how the mathematical nius algebra—AD[D3] = 1 e2 m3 . This condensa- F ⊕ ⊕ structure of the quantum double D[D3] is constructed tion A satisfies the ansatz, all conditions of which D[D3] and how the anyon content is obtained. For our purpose, are obviously fulfilled except the trivial associativity. It we simply tabulate in Table II all the eight anyons of is highly nontrivial and requires the F -symbols to show D[D3] and their properties and work on the condensa- that AF is trivially associative. So, we would not tion from there. D[D3] do the proof in this paper. On the other hand, AF D[D3] clearly satisfies the necessary condition of trivial asso- 1 e e ml=0,1,2 m+ m− 1 2 2 3 3 ciatively, namely the condition in Lemma 1. Therefore, we shall just proceed to condense AF , which is a per- d 112 2 33 D[D3] h 0 0 0 l 0 1 fect example to exhibit the hierarchy principle of fermion 3 2 condensation. TABLE II. The eight anyons of the quantum double D[D3]. The fermionic Frobenius algebra AF has a bosonic The total quantum dimension is D = 6. D[D3] D[D3] B subalgebra A = 1 e2. Then, let us first condense D[D3] ⊕ AB . This boson condensation has been studied in Because all the quantum dimensions are integers any- D[D3] 11 way in a quantum double of any finite group, for detail in Ref. . After this boson condensation, the child B′ B Z mnemonics, the subscript (an integer) in each anyon label bTO of D[D3] is the 2 toric code, which contains explicitly indicates the quantum dimension of the anyon. a simple current self-fermion ǫ (see Section V A). This − self-fermion ǫ is in fact a descendent of the m− AF The anyon m3 is a self-fermion, and to see whether it 3 ∈ D[D3] can condense, we need to look at the fusion rules as fol- that splits into two parts due to the condensation of e2. lows. We shall neglect the trivial fusion rule between the More explicitly, with condensing e2, we have the splitting − − − trivial boson and other anyons. m3 m3,1 + m3,2. To understand this splitting, we can − → The self-fermion m3 is a nonsimple current, as it has apply the rules of boson condensation reviewed in Section quantum dimension 3; hence, according to Lemma 2, it IV B and the fusion rules in Table III: cannot condense individually but has to induce certain boson condensation. Staring at the last row of Table − 2 0 m− AB = (m−) (m+ m−), III, the self fusion of m3 , since e and m2 are both self- 3 D[D3] 3 3 3 − 0 × ⊕ ⊕ bosons, m3 is self mutual-local via e2 and m2. Thus + B + + − − 0 m3 AD[D3] = (m3 ) (m3 m3 ). condensing m3 might force e2 and/or m2 to condense × ⊕ ⊕ too. But in the fourth row of Table III, the fusion be- 0 tween e2 and m says that they are not partially mutual 2 It is then not hard to see that in the child bTO (i.e., local and thus cannot condense simultaneously. B − the Z2 toric code) after the A condensation, m For our purposes, we choose e2 to condense along with D[D3] 3 m−. The second row of Table III implies that e ’s con- and m+ m− are both simple objects that are not of 3 2 3 ⊕ 3 densing might result in e1’s condensing. Nevertheless, the form of F (a) for any a D[D3]. We thus do the − + − − − ∈ − − + because e m = m , e and m are not partially mu- redefinition m = m B′ and m = (m m )B′ , where 1 3 3 1 3 3,1 3 | 3,2 3 ⊕ 3 tual local and× thus cannot condense together. We then the subscript B′ is clearly a restriction. We can obtain 16

− the quantum dimensions of the two parts of m3 as SET order. Fermion condensation in a bTO also leads to global symmetries on the child fTO of the bTO. On − d − dimB m m3 the one hands, the boson condensation contained in the − 3 dm = B = =1, 3,1 dim A 1+ de fermion condensation would result in a usual global sym- D[D3] 2 metry on the child fTO as it does in the case with pure − + d − + d + dimB(m m ) m3 m3 − 3 3 boson condensation. Likewise, the minimal fermion con- dm = ⊕ = =2, 3,2 dim AB 1+ d D[D3] e2 densation after the boson condensation would yield the fermion parity symmetry on the child fTO. It is impor- where the conservation of quantum dimension through − tant and reasonable to expect that fermion condensation splitting is evident: d − = d − + d − . Since m de- m3 m3,1 m3,2 3,2 can also lead to a mapping between the Hilbert space of − + a parent bTO and that of the child fTO of the parent scends from the identification of m3 and m3 that have − one. Such a mapping is plausibly projective because the topological spins different not by an integer, m3,2 must be confined after condensing AB . But m− descends vacuum of an fTO is not trivial but consists of both triv- D[D3] 3,1 − ial bosons and trivial fermions and thus has an internal solely from m3 , it is thus unconfined and inherits the − space. In Appendix B, we slightly touch upon the search topological spin 1/2 of m3 and become a self-fermion in of such mapping by restricting to the vacua of the par- B′ Z , namely the ǫ in the 2 toric code. We are not con- ent bTO and the child fTO only. By doing so, we prove cerned of the rest of the AB condensation, which can D[D3] Lemma 1. Nevertheless, because of the potential sub- be found in Ref.11, but continue to finish the hierarchical tleties caused by the trivial fermions in an fTO, we are condensation of AF . D[D3] still looking for a rigorous and appropriate description ′ Now that we have B being the Z2 toric code, we nat- of the full Hilbert space of an fTO. We shall report our urally have the decomposition AF = AB ⊠AF , progress along this direction elsewhere. D[D3] D[D3]−→ 0 F In Ref.13, it is found that boson condensation is in where A0 = 1,ǫ . Hence, the next and final step is { } one-to-one correspondence with vertex operator algebra to condense the ǫ in the Z2 toric code. This minimal fermion condensation is discussed in Section V A, and embedding and hence with modular invariants in RCFTs. It would be interesting to see if fermion condensation the result is the trivial fTO F0. Our dimension formula (5) is verified: would have a similar correspondence with the modular invariants in fermionic RCFTs. DD[D3] 6 In the end, we would also like to touch upon the √2 = √2 = √2= DF . (20) dim AF 1+2+3 0 D[D3] possibility of condensing anyons more exotic than self- fermions, e.g., semions, etc. Such discourses are how- ever way beyond not only the focal range of the cur- VI. DISCUSSION rent work but also our rudimental understanding of the consequences. Hence, we would rather note down our Following the ansatz of fermion condensation we lay preliminary thoughts of general anyon condensation in down, we obtain a hierarchy principle of fermion con- Appendix C. densation, which permits decomposing arbitrary fermion condensation into certain boson condensation followed by minimal fermion condensation. The rules of mini- mal fermion condensation we then find and the previ- ACKNOWLEDGMENTS ously known rules of boson condensation combine into a full prescription of performing fermion condensation The authors appreciate Davide Gaiotto for his deep in bTOs. Fermion condensation, now in a more general insights into the problem of fTOs, helpful discussions, sense by including boson condensation as special cases, and comments on the manuscript. YW is grateful to naturally corresponds to GDWs between topological or- Jurgen Fuchs for helpful answers to his questions re- ders, both bTOs and fTOs. Our results are supported by garding Frobenius algebras and helpful comments on the the explicit examples worked out in the section above. manuscript. YW also thanks Yuting Hu for helpful dis- Boson condensation in a parent bTO gives rise to a cussions. YW is supported by the John Templeton foun- linear mapping between the Hilbert spaces of the parent dation No. 39901. CW thanks the Aspen Center for bTO and that of the child bTO obtained from the par- Physics for hospitality, where part of this work is com- ent one via the condensation, which are studied in detail pleted. The Aspen Center for Physics is supported by in Ref.22,47. This mapping enables one to express the National Science Foundation grant PHY-1066293. This topological quantities, such as the modular S and T ma- research was supported in part by Perimeter Institute trices, of the child bTO in terms of those of the parent for Theoretical Physics. Research at Perimeter Institute bTO47. This mapping also offers a toolkit for extracting is supported by the Government of Canada through the the global symmetry action on the child bTO, which is Department of Innovation, Science and Economic Devel- an effect of the confined anyons due to the boson conden- opment Canada and by the Province of Ontario through sation involved. In this sense, the child bTO is in fact an the Ministry of Research, Innovation and Science. 17

Appendix A: Review of Frobenius algebra isfying the criterion

Rep A = X T (A A X)RXARAX = A A X , (A1) We briefly review in physical terms the concept 0 { ∈ | ⊗ ⊗ } of twist-free commutative separable Frobenius algebra C with the fusion A defined with respect to A. The details (CSFA) of a UMTC . This mathematical structure has can be found in⊗ Ref.17,49 and not repeated here. been used for classifying the quantum subgroups of quan- Apparently, a twist-free CSFA cannot describe any tum groups, the vertex operator algebra embedding in 2- fermion condensation. In order to do so, it seems that dimensional rational conformal field theories, and domain the twist-free condition and commutativity would be re- walls between 3-dimensional topological field theories.17 laxed. Unfortunately, the mathematically rigorous conse- Thorough studies of twist-free CSFAs can be found in quences of relaxing these two conditions are yet unclear, Ref.14,17,21 for example. which deserves further investigation. Frobenius algebra48. A UMTC C admits a special type of objects—Frobenius algebra objects. A Frobenius algebra A in this context is generally a direct sum of Appendix B: A Necessary Condition of the Trivial certain simple objects of C. For a bTO described by Associativity of Condensation C, the simple objects are the distinct elementary anyon types. This object A is a Frobenius algebra because 1) We show how we obtain Lemma 1, namely a neces- it is endowed with a product µ : A A A by the sary condition of the trivial associativity condition in our fusion of the simple objects of C, 2)⊗ an inclusion→ ι : A ansatz of fermion condensation. ,1C ֒ A, where 1C is the unit object or vacuum of C → In cases of boson condensation, the Hilbert space of such that 1C is also the unit of A, 3) the product µ is a parent bTO B can be linearly mapped to that of the associative, and 4) haploid: the unit is unique, namely child phase B′ of B via certain boson condensation, and dim HomC(1C, A) = 1. A commutative Frobenius algebra vice versa. The Hilbert space of B is a fusion space, is one whose product µ commute with the braiding of A. decomposed into subspaces, each of which is specified by This means µ R = µ, where R is the R matrix AA AA the number of anyon excitations and the types of anyons. of A in C. This◦ commutativity is physically sound in All such subspaces can be built on top of the basic fusion the case with boson condensation because A is going to space H , which involves three anyons only. An H is become the new vacuum when it condenses. In the case of 3 3 spanned by the basis vectors Ψab ,50 where a,b,c B fermion condensation, since R1 = 1 for any condensed c ff and Ψab the spatial wavefunction| ofi this state. Note∈ that fermion f, the commutativity may− cease to hold. c the state vector of a single anyon excitation a is a special One may formally write A = 1 Υ, where Υ is the basis vector Ψa1 = Ψ1a H . Since, attaching a ⊕ C a a 3 direct sum of the nontrivial simple objects of that com- trivial boson| 1 toi an anyon| i is ∈ somewhat tautological, we prise A. An A describing anyon condensation is necessar- may also denote a single anyon state by Ψa whenever it ily self-dual, also called rigid in Kirillov17. This means | i ab is necessary to do so. Canonically, the basis vector Ψc mathematically there is a non-degenerate projection from refers to the particular fusion channel c a b.| Thei A A to 1C. Physically, viewed as a composite anyon H H∗ ∈ c × fusion space 3 has a dual space 3 = Ψab a,b,c ⊗C ∗ {| i| ∈ in , A is the anti-anyon of itself, consistent with that A B c a¯¯b , where Ψab = Ψc¯ is the wavefunction relation. A would become the new vacuum after its condensation. }c Ψab may be viewed as a and b fuse into c. Representation category RepA. There exists a | Herei is an important remark. In the definition of the representation category RepA over A. One can define dual space H3, we still use kets rather than bras. So, the the twist θA of A as the self-statistical angle of A, as A duality lies in the wavefunction normalization is an object in C. If A is twist-free, θA = idA, then RepA is a tensor category. The twist-free condition is c a˙ b˙ c1 Ψ Ψ ′ = δ ′ d d /d Ψ , (B1) equivalent to one that any simple object in A is a self- a˙ b˙ c cc a b c c p boson. Physically, this means that A specifies certain where we usedx ˙ to indicate the internal anyons. In other boson condensation, and after its condensation, A would words, we write all the state vectors in terms of the cor- become the new trivial boson, or vacuum. For RepA to responding wavefunctions. We take such an unusual con- be also semisimple, A needs to be separable, which gives vention to avoid the potential ambiguity51 of representing rise to well-defined simple objects in RepA. The tensor a transparent fermion by a line in fTOs. As such, we can products of the simple objects of RepA can be written as build the states in a Hilbert space of four anyons, e.g., direct sums of the simple ones. These simple objects are a md˙ a md˙ Ψbm˙ Ψc = Ψbm˙ Ψc , without summing over the the superselection sectors to be identified as anyons or internal| fusioni | channeli ⊗ | m.i Note that the ˙ inm ˙ is not defects after condensing A. A trivial example of a twist- C part of the index labeling the anyon but merely indicates free CSFA is A =1C in any , meaning no actual anyon that m is internal. The associativity of fusion reads condensation. ′ a md˙ ad m ad m˙ In general, RepA is not braided but has a braided sub- Ψ Ψ = [F ] ′ Ψ ′ Ψ , (B2) | bm˙ c i bc m | m˙ bc i category Rep0A that consists of the objects in RepA sat- Xm′ 18 where the complex coefficients are the F -symbols. The We are now ready to look at the trivial associativity of Definition and properties of F -symbols can be easily AF . As explained in Section III C, the trivial associativ- found in any standard introduction to topological orders ity of AF means that after AF condenses and becomes or tensory categories. Here we need only the usual nor- a fermionic vacuum, the F -symbols relating the vacuum malization of F -symbols states are an identity. In particular, Lemma 1 follows from the following two cases ab 0 dc c [Fab ]c = Nab. (B3) 1 11˙ 11 1˙ rdadb Ψ Ψ = Ψ Ψ , (B8) | 11˙ 1 i | 1˙ 11i we expect that such a description of Hilbert spaces still 1f 11˙ f 1f 1f 1˙ Ψ1f 1˙ Ψ1f = Ψ1˙ Ψ1f 1f . (B9) applies to fTOs. Nonetheless, since we will restrict our- | i | i selves to the vacuum of any fTO in this appendix, we where the dotted ones are the internal trivial may leave any subtleties of the Hilbert space of an fTO bosons/fermions. Let us focus on Eq. (B8) first and for future studies. lift its both sides. We have

In cases with boson condensation, the Hilbert spaces am˙ ∗ md˙ a md˙ B B′ φb φc ΨbmΨc of and are related by a linear map, which, when | ˙ i boiled down to H3 reads a,b,c,d,mX∈AF ′ ′ ′ ′ ′ a b 2 a d m˙ αβ αβ γ ab B′ = φm˙ ′ Ψm˙ ′ Ψb′c′ , Ψγ = a b c Ψc , α,β,γ , (B4) | | | i | i | i ∀ ∈ a′,b′,c′X,d′,m′∈AF a,b,cX∈B  a,b,cX which holds term by term. Boths sides of the equation where αβ γ are complex coefficients, which are called a b c above now refers to the parent bTO B; hence, applying 22,47 vertex lifting coefficients (VLC) because we “lift” a the usual associativity (B2) to the LHS leads to αβ ab fusion vertex Ψγ of the child TO to fusion vertices Ψc ′′ ad m am˙ ∗ md˙ ad m˙ of the parent TO by the linear map. The VLCs satisfy [F ] ′′ φ φ Ψ ′′ Ψ 22,47 bc m b c | m˙ bc i certain consistency conditions. But we do not need a,b,c,d,mX∈AF m′′X∈AF any such details here. ′ ′ ′ ′ ′ a b 2 a d m˙ = φ ′ Ψ ′ Ψ ′ ′ , Back in cases with fermion condensation, because | m˙ | | m˙ b c i fermionic vacua are nontrivial, we would expect the map a′,b′,c′X,d′,m′∈AF between a parent bTO B and its child fTO F to be at where the F -symbols are those of B. Now setting in the least projective rather than linear. We shall report de- above a = b = a′ = b′ = x, c = d = c′ = d′ =x ¯, and tailed studies of the map elsewhere. In this appendix, ′ ′′ m = m =1B, we obtain our goal is to find a necessary condition of the trivial as- F sociativity of any fermion condensation A B, we need xx¯ m xm˙ ∗ m˙ x¯ xx¯ 1˙ xx¯ 2 xx¯ 1˙ ∈ [Fxx ] φx φx Ψ˙ Ψxx = φ˙ Ψ˙ Ψxx , only to focus on the relation between the vacuum of F ¯ 1 ¯ | 1 ¯i | 1 | | 1 ¯i mX∈AF and AF . Hence, restricted to the vacuum space of F, (B10) its relation with the Hilbert space of B restricted to AF where the subscript in 1B is omitted for simplicity. Here would still be linear. As such, we can rewrite the VLC comes the crucial step in proving Lemma 1. Let us as- equation (B4) as sume that x is a nonsimple current, i.e., dx > 1, and x Ψαβ = φab Ψab , α,β,γ 1, 1f F, andx ¯ do not produce any fusion channel that is also in | γ i c | c i ∀ ∈{ }⊆ AF except the trivial boson 1. Then the summation in a,b,cX∈AF a,b,cX Eq. (B10) is gone, i.e., m = 1B only, and together with (B5) ab the constraints (B7), we would have where φc are the VLCs simplified when restricted to the αβ vacuum of F. Note that any vacuum state Ψ in the ˙ ˙ | γ i [F xx¯]1 Ψxx¯Ψ1 = Ψxx¯Ψ1 , (B11) vacuum of F has even fermion parity because 1 1f =1f xx¯ 1| 1˙ xx¯i | 1˙ xx¯i f f × and 1 1 = 1. Because d1 = d1f = 1, the normalization But by the F -symbol normalization (B3), of vacuum× wavefunctions, similar to Eq. (B1), is xx¯ 1 xx¯ 1˙ 1 xx¯ 1˙ xx¯ 1˙ 1 1˙1˙ 1 [Fxx¯ ]1 Ψ1˙ Ψxx¯ = Ψ1˙ Ψxx¯ = Ψ1˙ Ψxx¯ , (B12) Ψ1˙1˙ Ψ1 =Ψ , | i dx | i 6 | i f f 1 1˙ 1˙ 1 (B6) Ψ1˙ f 1˙ f Ψ1 =Ψ , obviously violating the trivial-associativity. Thus, the 1f 1˙ f 1˙ 1f assumption is wrong. The fusion of x andx ¯ must contain Ψ f Ψ f =Ψ , 1˙ 1˙ 1 at least one nontrivial anyon that also condenses in order where we emphasize on the one-particle states by Ψ1 and that the sum in the LHS of Eq. (B10) can turn out to f Ψ1 . Now lift both sides of the equations in (B6) follow- be unity. One can do similar analysis for Eq. (B9) and ing the defining equation (B5), we obtain the following obtain the same result. Therefore, we can conclude the constraints. validity of Lemma 1. A remark is that in the definition of fermion conden- φ11 = φa1 = φ1a = φaa¯ =1, a AF . (B7) sation in our fundamental ansatz, a condensable boson 1 a a 1 ∀ ∈ 19 or fermion may have a multiplicity, i.e., it may appear level after the condensation, the vacuum of the system multiple times in the Frobenius algebra describing the becomes anyonic, namely an anyonic liquid. To cater to condensation. In our discussion in this appendix, we anyonic vacua, we would have to modify the notion of assume for simplicity all multiplicities being one. But mutual locality. this assumption causes no loss of generality because one For example, imagine we had a semionic vacuum made s can simply treat all occurrences of an anyon, had it have of trivial semions 1 , with θ1s = i; however, a semionic a nonunit multiplicity, as virtually distinct anyons with vacuum would automatically contain trivial fermions and unit multiplicity. trivial bosons too, as a fermionic vacuum contains both 1 The trivial associativity in our ansatz of fermion con- trivial fermions and trivial bosons. Since, M s s = 1, 1 1 − densation may have more implications but we shall leave to account for a semionic vacuum, for a condensable them for future work. anyon a in a bTO, we would impose the mutual local- 1 ity Maa¯ = 1. In a similar fashion, we can allow more exotic anyons± to condense in a bTO by further relaxing Appendix C: Some preliminary thoughts of general the notion of mutual locality. This logic is justified be- anyon condensation cause the topological sectors in a topological order may only be well-defined up to the vacuum structure. If the vacuum is fermionic, semionic, etc, the topological spin One may ask the question: is fermion condensation the of a topological sector may be defined only up to fermion most general anyon condensation? Certainly nature has parity, semion type, etc. It turns out that we need only not displayed any physical system whose fundamental de- to relax the condition on mutual-locality in our funda- grees of freedom are semions or more exotically, anyons. mental ansatz of fermion condensation to accommodate It is however likely that certain highly nontrivial FQHS general anyon condensation. Let us write this down. may be interpreted as a result of condensing anyons in less nontrivial FQHS (possibly multi-layer)52. More- Ansatz. General anyon condensation of order N in a 5–8 over, consider the idea of anyon superconductivity , in single-layer bTO B is a composite object A = naa nbb ⊕ ⊕ which certain anyons may group into a composite anyon B, where a,b,... are anyons in B and na the multi- that behaves like a self-boson and can condense. It is ···∈plicity (number of occurrences) of a in A, that must meet in principle plausible to stack layers of trivial anyonic the following conditions. gases/liquids to certain bTO and condense a self-bosonic B composite anyon made of certain anyon in the bTO and 1. Unit: ’s vacuum—trivial boson—1 A, na =1. ∈ appropriate trivial anyons in the other layers. Neglect- 2. Self-dual: For any a A, the anti-anyon a¯ A. ing the other layers but focusing on the bTO only, one ∈ ∈ can then study the principles and consequences of con- 1 N 3. Self-locality: For any a A, (Maa¯) =1. densing the nontrivial anyon in the bTO, in a fashion ∈ similar to fermion condensation. This is in fact a jus- 4. Closure: For any a,b A, there exists at least a ∈c N tification via the folding trick, similar to that for boson c a b, such that (M ) =1 and c A. ∈ × ab ∈ condensation in Fig. 2. To date, we are not sure about the physical barriers of condensing arbitrary anyons or 4. Mutual-locality: For any a,b A, if c A and c a b, a and b are mutual-local∈ via c. ∈ the existence of fundamentally anyonic systems. But ∈ × in theory or mathematically, there seems no a priori an 5. Trivial associativity: (A A) A = A (A A). obstruction preventing one from studying general anyon × × × × condensation. Topological orders may be a playground We leave any further studies on the consequences of for condensing generic anyons, such that at the effective this ansatz to future work.

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