Interaction effects on 3D topological superconductors: surface topological order from vortex condensation, the 16 fold way and fermionic Kramers doublets.
Three dimensional topological superconductors with time reversal symmetry (class DIII) are indexed by an integer , the number of surface Majorana cones, according to the free fermion classification. The superfluid B phase of He realizes the phase. Recently, it has been argued that this classification is reduced in the presence of interactions to . This was argued from the symmetry respecting surface topological orders of these states, which provide a non-perturbative definition of the bulk topological phase. Here, we verify this conclusion by focusing on the even index case, , where a vortex condensation approach can be used to explicitly derive the surface topological orders. We show a direct relation to the well known result on one dimensional topological superconductors (class BDI), where interactions reduce the free fermion classification from Z down to Z. Finally, we discuss in detail the fermionic analog of Kramers time reversal symmetry, which allows semions of some surface topological orders to transform as .
Following the initial success in the prediction, experimental discovery and classification of free fermion topological phasesFranz and Molenkamp (2013), recent theoretical attention has turned to the effect of interactions. It has been now understood that the essential character of these states can be defined in a wider context that includes strongly interacting systems (see Ref. Senthil, 2014; Turner and Vishwanath, 2013 for reviews). In the process, three qualitatively new features were identified that are absent in the non-interacting problem:
First, new topological phases appear that necessarily require the presence of interactions. In fact, it is now understood that free fermion topological insulators and superconductors are examples of so-called“symmetry protected topological” (SPT) phases of matter. These phases have a bulk gap, and as their name suggests, are defined only in the presence of symmetry; once the symmetry is broken, a non-trivial SPT phase can be continuously connected to a local product state. This means that SPT phases have no “intrinsic topological order,” i.e. they have only non-fractional excitations in the bulk. However, the non-triviality of an SPT phase is manifested by the presence of exotic edge states. Note that without interactions only SPT phases of fermions are possible, but in the presence of interactions SPT phases of bosons (or spins) also exist. While examples of such bosonic SPT phases were known in 1D Haldane (1983); Chen et al. (2011), a recent breakthrough has been the identification of SPT phases in 2D and 3D.Chen et al. (2012, 2011, 2013); Levin and Gu (2012); Lu and Vishwanath (2012); Vishwanath and Senthil (2013); Metlitski et al. (2013a); Bi et al. (2013); Kitaev ; Kapustin and Thorngren (2014); Kapustin (2014)
Second, qualitatively new kinds of surface states may appear for SPT phases in 3D. In particular, the surface may be fully gapped while preserving all symmetries, if it realizes an appropriate form of intrinsic topological order. The symmetry action on the anyons of the topological order is of a form that is impossible to realize in a purely 2D systemVishwanath and Senthil (2013). Examples of such surface states for SPT phases of bosons were discussed in Refs. Vishwanath and Senthil, 2013; Burnell et al., 2013; Wang and Senthil, 2013; Metlitski et al., 2013a; Bi et al., 2013; Cho et al., 2014; Chen et al., 2014. Progress has also been made in identifying the corresponding surface topological order (STO) for SPT phases of fermions, as well, including the “conventional” non-interacting topological insulators and superconductors. Here, the symmetry respecting, gapped topologically ordered surface state is obtained by turning on strong interactions on the surface, while keeping the bulk of the system non- (or weakly) interacting. An SO(3) non-Abelian surface topological order was proposed for the topological superconductor in Ref. Fidkowski et al., 2013. Shortly after, the surface topological order for topological insulators was determined Metlitski et al. (2013b); Wang et al. (2013); Bonderson et al. (2013); Chen et al. (2014); Wang et al. (2014); Senthil (2014), which was also found to be necessarily non-Abelian.
Third, the free fermion classification of topological phases may be reduced in the presence of interactions. Two phases that appear to be topologically distinct at the level of free fermions, may in fact be essentially the same phase on including interactions. That is, although on tuning from one phase to the other one always encounters a phase transition if the fermions are free, they can be adiabatically connected in the presence of interactions, and hence are the same phase. Indeed, it was shown that in 1D, topological superconductors (TSc’s) in class BDI (superconductors with time reversal symmetry, where ), which are labelled by integers according to the free fermion classification, are reduced to just eight distinct phases, i.e. Z Z with interactions Fidkowski and Kitaev (2010, 2011); Turner et al. (2011); You et al. (2014a). In the absence of interactions, the edge of a 1D TSc with label supports stable Majorana zero modes. However, with interactions, the edge can be fully gapped. Similar results hold for various 2D phases of fermionsQi (2013); Ryu and Zhang (2012); Gu and Levin (2013); Yao and Ryu (2013) including non-chiral topological superconductors with a global Z symmetry. In this last case, while a free fermion classification indicates that any number of counter propagating Majorana edge modes with opposite Z charge will be stable, adding interactions reduces this down to a Z classification Lu and Vishwanath (2013); Gu and Levin (2013). In the above examples, the effect of interactions may be treated perturbatively to check for the stability of edge states. Moreover, the edge states are and dimensional systems and hence amenable to theoretical analysis even at strong coupling. One may ask a similar question regarding the stability of three dimensional topological phases.
A particularly important example is provided by topological superconductors protected by time reversal symmetry (class DIII, where time reversal acts as , with - the fermion parity). This is, indeed, the physical case of time reversal symmetric superconductors with spin-orbit interactions. At the level of free fermion classification, there is an integer classification of topological superconductors in 3D, characterized by Majorana surface cones. Opposite signs of refer to Majorana cones with left or right chirality, which is well defined in the presence of time reversal symmetry. The B-phase of superfluid He, where the fermionic atoms pair to form an atomic superfluid, realizes the topological phase. A key question then is: are distinct values of truly different phases, or can they be smoothly connected in the presence of interactions? An elementary argument establishes that at least a Z subgroup must survive interactions Wang et al. (2011), i.e. the phases with odd integer labels cannot be trivial. A perturbative analysis of Majorana cone surface states shows that interactions are irrelevant and hence the free fermion classification can survive weak interactions. However, this conclusion may be changed with strong interactions: perhaps, by turning on the interactions beyond a certain threshold, the surface can be driven into a trivial gapped, symmetry respecting phase. Answering this question seems formidable since it appears to require a non-perturbative treatment of a 2+1D system. Initially, anomaly based argumentsWang et al. (2011); Ryu et al. (2012); Qi et al. (2013) concluded that the entire Z classification was stable to interactions. However, more recently it was realized that the free fermion classification collapses to Z Kitaev ; Fidkowski et al. (2013). This was shown in Ref. Fidkowski et al., 2013 by using symmetry respecting surface topological order as a non-perturbative definition of the 3D topological phase, and identifying four distinct surface topological orders that lead to exactly phases and a Z classification. Arguments for connecting these topological orders with specific were also given.
Here we will verify this conclusion by explicitly constructing the surface topological orders for even topological superconductors starting with the free fermion surface states and showing that they form a Z group (combined with the odd this gives Z). We note that during the completion of this work, other groups have used a similar vortex condensation approach in the context of topological superconductors with various symmetries Wang and Senthil (2014) and in different dimensionsYou et al. (2014b). The results derived here agree broadly with these works and with Ref. Fidkowski et al., 2013, although additional details of the surface topological order and its symmetry transformation properties are reported here.
The interacting generalizations of the free fermion classes AIII (superconductors with time reversal and conservation of -component of electron spin) and CI (superconductors with time reversal and full spin-rotation symmetry) are also reduced using similar arguments from Z to Z and Z, as discussed in Ref. Wang and Senthil, 2014. An alternate way to view the reduction of topological phases is that interactions allow us to beat certain fermion doubling theorems that are derived assuming free fermions. Within a purely 2D free fermion model with time reversal symmetry, it is impossible to mimic the surface of the topological superconductor, i.e. to realize a state with 16 Majorana cones of the same ‘chirality’. However, this can be realized in an interacting system: one concrete recipe is to take a slab of the topological superconductor and gap one surface with interactions while preserving all symmetries. Now, reducing the system to two dimensions gives the desired state. Such a viewpoint was recently advocated in Ref. You et al., 2014b on extending to 4+1D topological phases, and interesting consequences for the lattice regularization of the standard model of particle physics were derived.
Our construction of surface topological orders for even 3D topological superconductors will be based on the following strategy. We note that when is even, the system admits an enlarged ‘flavor’ symmetry. We first imagine driving the surface of the topological superconductor into a ‘superfluid’ phase where this symmetry is spontaneously broken and the surface Majorana cones are gapped. This surface phase also breaks the time reversal symmetry, however, a combination of time reversal and a discrete rotation in the group survives. Unlike the physical time reversal symmetry , this remnant anti-unitary symmetry satisfies . We then imagine quantum disordering the surface superfluid by proliferating vortex defects. However, it turns out that for general the vortices are non-trivial: a vorticity defect traps ‘chiral’ Majorana zero modes in its core, and so resembles the edge of a 1D topological superconductor in class BDI. As we already noted,Fidkowski and Kitaev (2010, 2011) interactions can fully lift the degeneracy associated with the Majorana zero modes only when they come in multiples of . Thus, for a topological superconductor the elementary vortex is trivial. Its proliferation restores symmetry, while preserving , and thereby also restores , giving rise to a topologically trivial symmetry-preserving gapped surface state. This proves that 3D TSc is trivial. For smaller even , the elementary vortex is non-trivial and cannot proliferate without breaking , however, vortices with are trivial. Their proliferation gives rise to a symmetry respecting topologically ordered surface state, whose anyon content we explicitly determine. Our results are summarized in table 5.
Some of these surface phases support semion excitations with unusual quantum numbers under time reversal symmetry. Usually, we are only able to define the action of time reversal symmetry locally and assign an anyon a precise value of when its statistics is unaffected by . Here, although a semion is converted into an anti-semion under , we are still able to define a generalized local time reversal symmetry, since the semion and the anti-semion differ by a local electron excitation. This leads to a generalization of Kramers doublets, where the two -partners within the doublet have different fermion parity. We term such -partners, ‘fermionic Kramers doublets.’ We will discuss various examples of fermionic Kramers doublets in Section V. In addition to surface anyons, we will also recall more familiar contexts such as the edges of 1D topological superconductors and vortex defects of 2D topological superconductors, where a similar action of time reversal symmetry occurs.
This paper is organized as follows. We begin in Section II with a brief introduction to topological superconductors in 3D with time reversal symmetry (class DIII) and point out that the phases labeled by even integers admit an enlarged U(1) symmetry that will prove useful. In Section III this symmetry is exploited to define vortices and map the problem to a well known prior result on the stability of 1D topological superconductors (class BDI). In section IV we deduce the symmetry respecting topologically ordered surface phases via vortex proliferation. Section V is devoted to studying fermionic Kramers doublets in more detail.
Ii Topological Superconductors in 3D
Topological superconductors protected by time reversal symmetry with (class DIII) in 3D have gapless Majorana cones on their 2D surfaces. For example, the topological superconductor, realized by superfluid He B, has gapless surface states described in the simplest case by the low energy dispersion:
where is a two component Majorana field () and are the Pauli matrices in the usual representation. Time reversal symmetry acts via:
Clearly when acting on the fermions. The mass term is forbidden by time reversal symmetry.
In the following we will consider topological superconductors where is an even integer. The reason is that we will be able to introduce an artificial U(1) symmetry in those cases that will greatly aid the analysis. Eventually, we will break down the symmetry back to the physically relevant time reversal symmetry.
First consider the case of , when a pair of Majorana cones is present. Assuming that they have the same velocity and are centered in the same spot of the Brillouin zone, we have:
We can, therefore, enlarge the symmetry to include an O(2) (=U(1)) group, that involves rotations of the Majorana operators between the two flavors . Combining these flavors into a complex fermion:
In these variables, the Hamiltonian (2) takes the form:
and is identical to the Hamiltonian of a topological insulator surface with a single Dirac cone and the chemical potential pinned to the Dirac point. However, there is an important distinction in the way in which time reversal acts. Thus, while transforms under the symmetry as,
under time reversal it has the following unusual transformation where the U(1) charge is reversed:
Note that . However, commutes with rotations, i.e. the total symmetry is , unlike for the topological insulator where it is . In other words the U(1) ‘charge’ here behaves like a component of spin as far as its time reversal properties are concerned. A corollary is that a vortex of this U(1) will remain a vortex under time reversal. In contrast, the vortex of the usual charge U(1) is converted into an anti-vortex under time reversal.
Now consider spontaneously breaking the flavor U(1) symmetry by inducing a ‘superfluid’ on the surface, via the condensate of . At the mean-field (Bogolioubov-de Genne) level this condensation can be described by adding a term,
to the Hamiltonian. This will gap the Majorana cones. Clearly, since the Majorana cones do not require the flavor U(1) for stability, this condensate must also break -symmetry. Indeed, under ,
Intuitively this is clear: we mentioned that in terms of the interplay with , the flavor U(1) symmetry is like a ‘spin’ rotatation, and so is akin to a ferromagnetic order parameter, whose orientation is reversed by .
However, we can combine time reversal with a rotation by in the U(1) group. The resulting transformation is a symmetry of the system and may be viewed as a modified time reversal since it remains an anti-unitary operator:
hence, the superfluid phase respects . Note, however, that in contrast to the original time reversal symmetry, we have and so the gapped fermionic Bogolioubov quasiparticle of the surface superfluid is a Kramers singlet. Below, we will study in detail the vortices of this surface superfluid.
Surface states with any even number of Majorana cones can be treated in a similar manner, introducing an artificial symmetry by grouping them in pairs. This results in a surface state with Dirac fermions , (), where each transforms as (5) under the common symmetry and as (6) under . The Dirac cones are stable at the free fermion level due to the presence of time reversal symmetry. One can then similarly spontaneously break the symmetry by a condensate of . This condensate again simultaneously breaks , but leaves the combination preserved.
Iii Vortices and the Eight Fold Way
Here, we introduce vortices into the surface superfluid of a topological superconductor with even index , and deduce their transformation properties under the modified time reversal symmetry .
iii.1 The surface with strength vortices.
Let us begin with the surface. Imposing a static vortex configuration with vorticity in Eq. (7), , and solving the Bogolioubov-de Gennes equation, one finds Majorana zero modes , , localized in the vortex core.Jackiw and Rossi (1981); Fu and Kane (2008) Crucially, a vortex maps into a vortex rather than an anti-vortex under , so one can study the transformation properties of the Majorana modes under . As we show in appendix A, the Majoranas transform in a chiral manner,
Thus, a vortex resembles the end of a 1D topological superconductor in class BDI, with playing the role of a time reversal symmetry. As noted in the introduction, in the absence of interactions, such 1D TSc’s are labeled by an integer , which counts the number of chiral Majorana zero modes at the edge. A vortex of strength is, thus, like an edge of a 1D TSc with . The mass terms are prohibited by , so in the absence of interactions the Majorana modes are stable. However, as was shown in Refs. Fidkowski and Kitaev, 2010, 2011 interactions can fully lift the degeneracy associated with the Majorana modes if they come in multiples of 8, breaking down the classification of 1D TSc’s to Z. Let us review this result in the present context.
For an elementary vortex, there is a single Majorana mode in the vortex core. As is well-understood, the stability of this zero mode is purely topological and does not rely on any symmetry. Indeed, the fermionic Hilbert space associated with a single Majorana mode is not well defined. However, if we have two vortices, each possessing a Majorana in its core then the two Majoranas , , form a two-dimensional Hilbert space, corresponding to the complex fermion mode being occupied or empty. If the vortices are far apart, these occupied and empty states are nearly degenerate as only a non-local fermion tunneling term can split them. The presence of the Majorana mode endows the vortex with non-Abelian braiding statistics, as we will review shortly.
For a vortex, one finds a pair of Majorana modes in the vortex core. This pair of modes is not topologically protected, since a local mass term lifts the degeneracy. However, the degeneracy is protected by . Indeed, the mass term breaks , furthermore, is precisely the ‘local’ fermion parity , counting whether the complex fermion mode in the vortex core is empty or filled. Now, . Thus, each vortex state must have a degenerate partner under with opposite fermion parity. We will refer to such defects as fermionic Kramers doublets, and will study them in more detail in section V. Representing , , , , we see that for the vortex, . In contrast, for the vortex, choosing an identical representation of ’s, the opposite chirality of Majorana modes in Eq. (11) gives, , so that . Thus, for such fermionic Kramers doublets the value of is not tied to the degeneracy. Note that the two states in each fermionic Kramers doublet differ by a Bogolioubov quasiparticle with , so they have the same value of . One may ask if the value of is even physically meaningful in the present case. Indeed, a definite value of (or ) can only be assigned when the state and its time reversal partner differ by a local excitation. We are used to this excitation being a local boson, in which case one either has (ordinary Kramers singlet) or (ordinary Kramers doublet). However, an electron (Bogolioubov quasiparticle) is also local excitation, which leads to the possibility of a fermionic Kramers doublet with a well-defined value of (see section V for more details).
Next, consider the vortex. The only perturbation in the ground state subspace allowed by is,
is nothing but the fermion parity of the zero energy subspace. Therefore, the four-fold degenerate zero energy subspace splits into two doublets with opposite fermion parity. These doublets are actually (ordinary) Kramers doublets under and cannot be further split. (Note that in the present case preserves the fermion parity). Indeed, let us represent ’s by matrices: , , , . We see that can be implemented by and . The fermion parity commutes with . Note that even though each of the vortices has , when we fuse them together we obtain a vortex with . Likewise, fusing a vortex and a vortex, which have and , respectively, gives a zero vorticity state with . As we will show in section V, the fusion product of two fermionic Kramers doublets , , actually satisfies , consistent with the present observations.
Finally, let’s consider . Represent as above and , , , . We can take . We see that and commutes with the fermion parity . Therefore, the degeneracy will generally be fully lifted and the vortex transforms trivially under .
Thus, we see that the transformation properties of vortices under remain invariant under a shift of vorticity . So, we only need to discuss values . It remains to discuss the vortex, which carries three Majorana modes. Again, one of these Majoranas is protected topologically just as in the case. However, all three modes are protected by . The presence of these three Majorana modes is manifested in the following way. Imagine a pair of distant vortices with and . The total of six resulting Majoranas will give rise to an 8-fold degenerate state. Out of this degeneracy, a 2-fold degeneracy is associated with opposite values of the overall fermion parity of the two-vortex state. The remaining four-fold degeneracy can be attributed to a (standard) local two-fold Kramers degeneracy of each vortex. In this sense, we may say that the vortex (and the vortex) has . (We make this notion more precise in section V.) On the other hand, if we have a pair of distant vortices with and then the system has only a two-fold degeneracy associated with the overall fermion parity, so the vortices carry no local Kramers degeneracy. We, thus, say that the (and the ) vortex has .
We list values deduced for all the vorticity sectors in table 1 (bottom).
iii.2 Vortices on the surface of a TSc.
It is simple to generalize the above discussion to a 3D TSc with a general . Considering a defect with vorticity on the superfluid surface, each flavor of Dirac cones supports Majorana modes in the vortex core, transforming as in Eq. (11) under . Thus, the vortex carries a total of chiral Majorana modes, and transforms under as an end of a 1D TSc with . In particular, for the elementary vortex switches fermion parity under , for it is a (standard) Kramers doublet, and for it is a trivial Kramers singlet. Therefore, for the elementary vortex can proliferate, while preserving . This proliferation restores the symmetry and hence, simultaneously restores . One, thus, obtains a gapped, symmetry respecting surface phase with no intrinsic topological order, demonstrating the triviality of the 3D TSc phase. This establishes the deep connection between 1D TSc’s in class BDI and 3D TSc in class DIII, and the respective breakdown of the non-interacting classification, (1D TSc’s) and (3D TSc’s). For smaller values of , the elementary vortex transforms non-trivially under and so cannot proliferate without breaking , however, vorticity defects with are trivial under and can proliferate, giving rise to a symmetry respecting, gapped surface with intrinsic topological order, which we will discuss in the next section.
Iv Surface Topological Order from Vortex Condensation
In this section, we will construct symmetry respecting topologically ordered surface states for 3D TSc’s with . We will obtain these by quantum disordering the surface superfluid via vortex proliferation, closely following the discussion of Metlitski et al.Metlitski et al. (2013b) (see also Wang et al.Wang et al. (2013)), who derived a STO for the fermionic topological insulators. We have described how vortices transform under the symmetry in the previous section; this, in particular, tells us which vortices can condense without breaking . Another important property of the vortices that we will need to establish is their statistics - only vortices with bosonic statistics can proliferate. Vortex statistics can be inferred using the ‘slab trick’ introduced in Ref. Metlitski et al., 2013b and reviewed below. We will find that the smallest -trivial vortex (one with strength , where ) actually has bosonic statistics. We imagine condensing this vortex. This restores the symmetry and preserves , thereby also restoring . If , the resulting gapped surface state supports topological order. Finally, we will imagine a further surface phase transition, where the ‘artificial’ symmetry is broken (but is preserved), exposing the surface topological order relevant to the topological superconductors. We will follow this procedure in turn for each even . The case is discussed in detail below; and are relegated to appendix C; the remaining cases are discussed in appendix B.
iv.1 The STO of the topological superconductor (single Dirac cone) from vortex condensation
iv.1.1 Vortex Statistics
Let us first tackle the case of the TSc’s, that most closely parallels the discussion of the TI in Ref. Metlitski et al., 2013b (see also Ref. Wang and Senthil, 2013). Let us begin by deducing the vortex statistics. Recall that to make the notion of vortex statistics (particularly, the Abelian part of the statistics) precise, we need to gauge the symmetry by coupling the system to a weakly fluctuating gauge field . Vortices of strength now carry magnetic flux . Next, consider a slab of the TSc with the opposite faces separated by a thickness much larger than the confinement length of the surface states. Thus, opposite surfaces are effectively decoupled. Imagine driving the top surface of the slab into a supefluid phase by a condensate of , Eq. (7). In contrast, break on the bottom surface by a -preserving term,
This results in a fully gapped bottom surface whose response to the gauge field is characterized by a Hall conductivity . Considering a vortex on the top surface, the associated magnetic flux will be confined to the vortex core near the top surface but will spread out in the bulk of the slab and on the bottom surface.
By solving the Bogolioubov-de Gennes equation one finds that the 1D interface on the TSc surface between the superfluid phase and the phase supports a single chiral Majorana mode with central charge . The edge of our slab is precisely such an interface. Thus, viewing the entire slab as a 2D system, the presence of the chiral Majorana edge mode allows us to identify it with a superconductor. The statistics of vortices in a superconductor are known to be described by the Ising anyon theory. This theory has three topologically distinct sectors: . Vortices with odd vorticity carry an odd number of Majorana zero modes and belong to the non-Abelian sector, while vortices with even vorticity have an even number of Majorana zero modes and belong to sectors , where is the trivial bosonic sector and - the Bogolioubov quasiparticle. The topological spins of the sectors are .
It is clear that the topological spins of the vortices piercing the slab are sensitive to the way in which is broken on the bottom surface: if one breaks in the opposite way, the bottom surface has a Hall conductivity , the edge of the slab carries a Majorana mode moving in the opposite direction, and the slab is identified with a superconductor. As a result, the odd strength vortices will have a topological spin . We are not actually interested in the statistics of the vortices associated with the entire slab. Rather, we would like to deduce the intrinisic contribution to the statistics coming from the top (supefluid) surface. To isolate these intrinsic statistics, we decompose the action of the slab as . The slab as a whole is described by Ising anyon theory, . Note that during the vortex motion the bulk and the bottom surface are only affected via the magnetic flux emanating from the vortex, and so their contribution can be described by the effective actions , . The bulk action, does not contribute to the vortex statistics. On the other hand, the Hall response of the bottom surface gives,
Therefore, we deduce
where the gauge field is constrained to follow the vortex motion via , with - the vortex current. This constraint allows us to rewrite the second term in Eq. (15) in terms of a dynamical Chern-Simons gauge field at level coupled to the vortex current,
Thus, the intrinsic vortex statistics is described by an anyon theory. This theory has topological sectors , where denotes the charge in the Ising sector and the subscript denotes the charge in the sector, which coincides with the vorticity. Note that only a subset of all anyon types in the theory is realized by the vortices, since vortices of odd strength necessarily carry charge in the Ising sector and vortices of even strength carry charges or . The resulting allowed topological sectors together with the corresponding topological spins, derived from , are listed in table 1 (top). Following Ref. Chen et al., 2014, we call such a restriction of the anyon theory - the T-Pfaffian. Note that the topological properties are invariant under shifting the vorticity . This coincides with the periodicity of the vortex transformation properties under the time reversal-like symmetry , discussed in section III. Let us check that the vortex statistics deduced above are consistent with the transformation properties under .
Under , the vorticity is left invariant (which is a significant point of difference from the case of the topological insulator surface). As under the topological spin, , we must have, and . This is consistent with our discussion in section III, where we found that the and vortices change their fermion parity under . All the other topological sectors are mapped into themselves under , again consistent with section III. Next, observe that is a fusion product of and , which are exchanged by and have mutual semionic statistics. A very general argumentMetlitski et al. (2013b); Wang et al. (2013); Chen et al. (2014) then implies that must be a fermion, again in agreement with section III.
Before proceeding to the vortex condensation to obtain the STO, we first note that the vortex statistics obtained here are identical to those derived by Metlitski et al. for the superconducting surface of a TI, except that time reversal acts very differently here, leaving fluxes invariant. Second, imagine a magnetic monopole of the gauge field in the bulk of the phase. From the Hall conductivity of the -broken surface state we conclude that the monopole must carry a half-odd-integer charge, i.e. the bulk electromagnetic response is characterized by , as in the TI. We can study the passage of a magnetic monopole excitation from the vacuum into the bulk of the 3D system. This leaves behind a Dirac string flux of , which we identify with the vortex. The semionic/anti-semionic statistics of this vortex is intimately tied to the half-integer charge of the monopole as discussed in Ref. Wang et al., 2013.
iv.1.2 Vortex condensation: topological order
Now, we consider condensing the strength 8 vortices . These are trivial both topologically and in terms of their transformations under . Thus, their proliferation restores the global symmetry, while preserving . When the vortices condense, the vortices with , survive as -neutral anyon excitations, preserving their statistics. In addition to these neutral vortex descendants, the resulting surface phase also possesses charged boson excitations , with fractional charge , . (The ‘charge quantum’ is dual to the flux of the condensing vortex; ). The excitation can be identified with a charge Cooper pair , so the topologically distinct excitations have . All anyons of the surface phase can be obtained by fusing a charged boson with a vortex descendant, and will be denoted as . The charged boson and the vortex descendant have a ‘charge-flux’ mutual statistics . Thus, the topological spin of their fusion product is . The resulting topological order has 96 distinct anyons, and is identical to in Ref. Metlitski et al., 2013b. All anyons have trivial mutual statistics with a charge fermion, , which is identified with the electron .
Before we proceed further, it is instructive to discuss how to drive a phase transition from the obtained topologically ordered surface phase back to the superfluid phase. This occurs via the condensation of the charge boson , which spontaneously breaks the symmetry. Due to the non-trivial mutual statistics between and , the condensation (logarithmically) confines all the anyons: go back to being strength vortex defects of the superfluid. To see this, note that in the presence of a gauge field , after condensation, will bind magnetic flux so that the statistical phase acquired by upon going around is cancelled by the electromagnetic Aharonov-Bohm phase.
We now discuss the transformation properties of the STO under the restored time reversal symmetry . Let us first discuss how the various anyon sectors map into each other under . Since all anyons carry a definite charge, if sends anyon to anyon , so does . The vortex descendants preserve the transformation properties they had in the superfluid phase: , , and all the other particles map into themselves. The transformations of the charged bosons are determined by the requirement that the charge under . Thus, .
Next, we come to the issue of assignments. For an anyon with charge , the assignment can be obtained from the assignment,
where we’ve used the fact that and commute (we give a more careful proof of the result (17) in section V). We, however, must remember that and assignments are only meaningful for anyons that transform locally, i.e. or , where is the electron. The first case, is familiar: here can be a Kramers-singlet ( or a Kramers-doublet (). The second case, is that of fermionic Kramers doublets, already mentioned in section III and discussed in more detail in section V. Thus, among the charged bosons the only anyon that can be assigned and is the charge particle, : the time reversal partners and differ by a local Cooper pair . Recalling that the transition to the -preserving superfluid phase proceeds via condensation of , we conclude that must have and hence, by Eq. (17), . More generally, an anyon preserves its value (if defined) after the transition into the superfluid phase, where the charge part ‘melts’ and reduces to a vortex . (From this point of view melts into the zero vorticity sector after the transition and so, indeed, carries ). The assignments of superfluid vortices computed in section III are listed in table 1 (bottom). Thus, from Eq. (17), the neutral vortex descendants have , i.e. , have ; , have ; have ; have . Fusing the neutral fermion with the charge boson , we obtain the physical electron with , as required.
Note that the neutral vortex descendants and ( and ) can not be assigned a or value in the topologically ordered phase, despite having a well-defined () in the superfluid. Indeed, in the superfluid the -partners and differ by the Bogolioubov quasiparticle , which is just the local electron. Thus, in the superfluid they form a fermionic Kramers doublet. However, in the topologically ordered phase becomes a non-local anyon excitation, which is distinct from the electron , so and do not have a well-defined assignment. On the other hand, the charge semion and the charge anti-semion differ by the electron and do form a fermionic Kramers doublet, . Upon transition to the supefluid phase, these reduce to vortices and and so inherit the assignment . Therefore, from Eq. (17) we find has and has . Such unusual assignements were first introduced in Ref. Fidkowski et al., 2013 and are discussed further in section V. Note that and have opposite values of consistent with them differing by a electron . Also note that () and () fuse to an (ordinary) Kramers-doublet (). This is consistent with the general rule proved in section V: if anyon has fermionic Kramers parity and anyon has fermionic Kramers parity then the (ordinary) Kramers parity of the fusion product satisfies .
Repeating the above argument, for , we find a fermionic Kramers pair, , where has and has .
In contrast to the case of the TI, it is not possible to further simplify this topological order without breaking symmetry. We now do so to recover the STO of the TSc with only symmetry. To this end we condense the charge boson, . The resulting topological order, which we name , has 24 particles, since this condensate confines all particles except those with , and particles that differ by the condensate are considered equivalent. Thus, the resulting anyons are with and . The electron becomes identified with . (Note that this topological order is different from the 24 anyon STO obtained for TIs in Refs. Metlitski et al., 2013b; Wang et al., 2013).
It is convenient to divide the anyons into two sets: and . The first set originate from the neutral vortex descendants and form the T-Pfaffian topological order, with topological spins listed in table 1 (top). The anyons in the second set can be written as , where is a semion (topological spin ). Note that , and has trivial mutual statistics with all anyons . Thus, we may write the STO as a direct product of two sectors,
We now discuss the action of time reversal symmetry on . maps the T-Pfaffian sector onto itself, as summarized in table 2. We use the notation of Ref. Fidkowski et al., 2013, which introduced two variants of the T-Pfaffian topological order, denoted (T-Pfaffian) with and . These two variants differ by the Kramers parity of non-Abelian quasiparticles: for , have , while have ; for , the values are reversed. With this notation, the T-Pfaffian sector of the above STO for the TSc is (T-Pfaffian). As shown in appendix B, the TSc has an identical surface topological order, Eq. (18), but with the (T-Pfaffian) variant.
Next, consider the action of on the sector. We find . Thus, and form a fermionic Kramers doublet, with carrying and carrying . Note that in the decomposition (18), the electron belongs to the T-Pfaffian sector. However, since the electron is local with respect to all anyons, we may equivalently write,
where the semion-fermion topological order SF is defined as SF. Thus, the action of time reversal on factors into its action on the T-Pfaffian and SF sectors. For future reference, we have introduced two variants of the semion-fermion topological order, labeled by the index . The assignement of the semion is , and of the anti-semion is . While the STO of the TSc contains the SF variant, as discussed in appendix B, the STO of the TSc contains the SF variant.
Next, we discuss how to further simplify the STO for the TSc. However, for this purpose, it proves convenient to discuss the topological superconductors together.
iv.2 STO for other topological superconductors; 16-fold way
The construction presented above can be easily followed to derive the STO for , and topological superconductors. The details are given in appendix B and the result is shown in table 5. We find that and respectively have (T-Pfaffian)SF and (T-Pfaffian)SF surface topological order. On the other hand, and respectively have (T-Pfaffian)SF and (T-Pfaffian)SF surface topological order.
It has been argued in Ref. Chen et al., 2014 that the two (T-Pfaffian) states are connected via a surface phase transition either to a trivial state or to the STO of the 3D bosonic SPT phase with symmetry. The latter STO is the ordinary toric code , with , - bosons, and - a fermion. Following Ref. Wang and Senthil, 2013, we refer to a toric code with such unusual assignments as the eTmT state. Note that once physical electrons are present the eTmT state is equivalent to a 3 fermion state , where , , have the same fusion rules as , , in the toric code, but instead are all fermions. Indeed, with the identification , , . Thus, we will use the labels eTmT and FFF interchangeably. Note that two copies of eTmT can be connected via a surface phase transition to a trivial state.
Furthermore, the two (T-Pfaffian) states were shown to differ by precisely the eTmT state.Chen et al. (2014) Hence, one of them is connected via a surface phase transition to a trivial state and the other - to the eTmT state. Thus, either the STO of and can be reduced to and , respectively, while and can be reduced to and , respectively; or vice-versa. Unfortunately, at the present time it is unclear for which value of the T-Pfaffian is connected to a trivial phase, so we cannot say which of the above two possibilities is realized. In appendix C.2, we will show that the topological superconductor has precisely the eTmT surface topological order. This is consistent with STO of and ( and ) phases differing by eTmT.
Also observe that can be driven into a trivial phase, consistent with () being trivial.Fidkowski et al. (2013) Indeed, write . is a semion with , and is a semion with . Under , , so . By the rule for calculating the (ordinary) Kramers parity of the fusion product of two fermionic Kramers anyons, . So is a fermion and is a boson. Condensation of this boson confines all the anyons and gives a trivial phase.
On the other hand, cannot be made trivial.Fidkowski et al. (2013) In section C.1 of the appendix, we will show that precisely coincides with the surface topological order of the phase, derived via vortex condensation.
By combining two topological superconductors, we obtain a TSc, whose topological order is a product of four SF sectors: . Now with is a fermion, so is a boson. Condensing this boson reduces the STO to
Letting , , , we see that are fermions with that realize the three-fermion toric code topological order FFF, which as we showed above is equivalent to the eTmT topological order in the presence of physical electrons. Thus,
In section C.2 of the appendix, we will show that this is precisely the topological order of a TSc deduced from vortex condensation.
The deduced STOs of all even topological superconductors are summarized in table 5.
Finally we note that combining two of the surface topological orders leads to a trivial surface state, implying that the bulk topological phase is actually the same as in the presence of interactions. This implies that interactions reduce the free fermion classification down to .
V Fermionic Kramers doublets and time reversal action
In section IV, we have encountered an example of anyons that transform under time reversal as , where is the electron. Furthermore, we have claimed that such anyons can be assigned a definite value of : one of the anyons , carries and the other . As the two anyons differ by a electron , their opposite values of appear consistent. We will call such anyon pairs , - fermionic Kramers doublets. Such unusual doublets were first discussed in Ref. Fidkowski et al., 2013; here we further elaborate on this phenomenon. Our treatment closely follows Ref. Levin and Stern, 2012 where the notion of a local (ordinary) Kramers degeneracy was rigorously defined.
Let us first recall what it means for a many-body state to have a local Kramers degeneracy. Take a many-body state with short-range correlations, i.e. one where for any two operators , localized at widely separated points. Assume has an even number of electrons. Imagine that under time reversal,
where and are bosonic operators localized near distant points and . These two points can be locations of anyon excitations or of classical defects (such as vortices or edges of a 1D system). Assume the normalization