A Irreducible Representation of Clifford Algebra

Symmetry Protected Topological States of Interacting Fermions and Bosons

Abstract

We study the classification for a large class of interacting fermionic and bosonic symmetry protected topological (SPT) states, focusing on the cases where interaction reduces the classification of free fermion SPT states. We define a SPT state as whether or not it is separated from the trivial state through a bulk phase transition, which is a general definition applicable to SPT states with or without spatial symmetries. We show that in all dimensions short range interactions can reduce the classification of free fermion SPT states, and we demonstrate these results by making connection between fermionic and bosonic SPT states. We first demonstrate that our formalism gives the correct classification for several known SPT states, with or without interaction, then we will generalize our method to SPT states that involve the spatial inversion symmetry.

pacs:
71.27.+a, 11.10.Kk, 75.40.-s

I Introduction

A symmetry protected topological (SPT) state 1 is usually defined as a state with completely trivial bulk spectrum, but nontrivial (e.g. gapless or degenerate) boundary spectrum when and only when the system including the boundary preserves certain symmetry Chen et al. (2011, 2012, 2013); Wen (2014). The most well-known SPT states include the Haldane phase of spin-1 chain Haldane (1983a, b), quantum spin Hall insulator Kane and Mele (2005a, b), topological insulator Fu et al. (2007); Moore and Balents (2007); Roy (2009), and topological superconductor such as Helium-3 B-phaseBalian and Werthamer (1963); Leggett (1965, 1972, 1975). So far all the free fermion SPT states have been well understood and classified in Ref. Schnyder et al., 2009; Ryu et al., 2010; Kitaev, 2009; Wen, 2012; Morimoto and Furusaki, 2013, and recent studies suggest that interaction can reduce the classification of fermionic SPT states Fidkowski and Kitaev (2010, 2011); Qi (2013); Ryu and Zhang (2012); Gu and Levin (2013); Yao and Ryu (2013); Fidkowski et al. (2013); Wang and Senthil (2014). Unlike fermionic systems, bosonic SPT states do need strong interaction to overcome its tendency to form a Bose-Einstein condensate. Most bosonic SPT states can be classified by symmetry group cohomology Chen et al. (2011, 2012, 2013); Wen (2014), Chern-Simons theory Lu and Vishwanath (2012); Wang et al. (2014) and beyondWang et al. (2014), semiclassical non-linear -model Bi et al. (2013) and cobordism theoryKapustin (2014a, b).

The definition for SPT states we gave above is based on the most obvious phenomenology of the SPT states, and it gives SPT states a convenient experimental signature, which is their boundary state. Indeed, the quantum spin Hall insulator and topological insulator were verified experimentally by directly probing their boundary properties.König et al. (2007); Hsieh et al. (2008, 2009); Chen et al. (2009) However, if a SPT state needs certain spatial symmetry,Fu and Kane (2007); Teo et al. (2008); Turner et al. (2010); Fu (2011); Hughes et al. (2011); Hsieh et al. (2012); Turner et al. (2012); Slager et al. (2013); Chiu et al. (2013) its boundary may be trivial because this spatial symmetry can be explicitly broken by its boundary. In this work we will study SPT states both with and without spatial symmetries, thus in our current work, a SPT state is simply defined as a gapped and nondegenerate state that must be separated from the trivial direct product state defined on the same Hilbert space through one or more bulk phase transitions, as long as the Hamiltonian always preserves certain symmetry.

In this work we study both strongly interacting fermionic and bosonic SPT states. In particular, we focus on the instability of free fermionic SPT (fFSPT) states against interaction, and investigate their interaction reduced classifications. More exotic fermionic SPT states that can not be reduced from free fermion descriptionsWang and Senthil (2014); Gu and Wen (2014); Xu and You (2014) are not discussed in this work. The interaction reduced classification of interacting fermionic SPT (iFSPT) states can be derived by making connection to bosonic SPT (BSPT) states with the same symmetry,2 and we will argue that the classification of BSPT states implies the classification of their fermionic counterparts. More specifically, since BSPT states always need strong interaction, their classification tells us how interaction affects the classification of FSPT states. When describing BSPT states, we will adopt the formalism developed in Ref. Bi et al., 2013, namely we describe a -dimensional BSPT state using an O() nonlinear sigma model (NLSM) field theory with a topological -term, and we only focus on the stable “fixed point” states with . Depending on integer , these fixed points can correspond to either trivial or BSPT state. This formalism fits well with our definition of SPT states: it very naturally tells us whether two “fixed point” states can be connected with or without a phase transition. This is an advantage that we will fully exploit in our work.

We will first demonstrate our method in section II with well-known examples such as Kitaev’s Majorana chain with symmetry Fidkowski and Kitaev (2010, 2011), topological superconductor (TSC) with symmetry Qi (2013); Yao and Ryu (2013); Gu and Levin (2013); Ryu and Zhang (2012), and topological superconductor He-B with symmetry Fidkowski et al. (2013); Wang and Senthil (2014). Previous studies show that although all these states have classification without interaction, their classifications will reduce to , , under interaction. We will demonstrate that these interaction-reduced classifications naturally come from the classification of Haldane spin chain, Levin-Gu paramagnet,Chen et al. (2011); Levin and Gu (2012) and bosonic SPT state with symmetry.Vishwanath and Senthil (2013); Xu (2013) More precisely, the classification of the BSPT states give us a necessary condition for interaction reduced classification of their fermionic counterparts. Moreover, the analysis of BSPT states also leads to a systematic construction of the specific four-fermion interaction that likely gaps out the bulk critical point between free fermion SPT (fFSPT) state and the trivial state in the noninteracting limit. In section III-V we will generalize our method to SPT states that involve the spatial inversion symmetry. By making connection to BSPT states, we will show that interaction reduced classification occurs very generally for inversion SPT states in all dimensions with a systematic pattern.

Ii SPT states without spatial symmetry

ii.1 From Kitaev Chain to Haldane Chain

Lattice Model and Bulk Theory

The Kitaev’s Majorana chain is a free fermion SPT (fFSPT) state protected by the time-reversal symmetry with (symmetry class BDI)Altland and Zirnbauer (1997). For generality, we consider copies of the Majorana chain. The model is defined on a 1d lattice with flavors of Majorana fermions () on each site ,

(1)

with the bond strength alternating along the chain. Each unit cell contains two sites, labeled by and , as shown in Fig. 1. The Hamiltonian is invariant under the time-reversal symmetry , , which flips the sign of the Majorana fermions on the sublattice followed by the complexed conjugation.

Figure 1: Lattice model of Kitaev’s Majorana chain.

Fourier transform to the momentum space and introduce the basis , the Hamiltonian Eq. (1) becomes

(2)

with , and the time-reversal symmetry acts as . In this paper, we use , , to denote the Pauli matrices. When , in the long-wave-length limit (), , so the low-energy effective Hamiltonian reads

(3)

Here we have set and introduced as the topological mass. The time-reversal symmetry operator may be written as , where implements the complex conjugation by flipping the imaginary unit.

On the free fermion level, the BDI class FSPT states are classified,Schnyder et al. (2009); Ryu et al. (2010) as indexed by a bulk topological integer

(4)

where is the fermion Green’s function at zero frequency. Given the model in Eq. (2), the topological number is identical to the fermion flavor number when . Every topological number indexes a distinct fFSPT phase, which, in respect of the symmetry, can not be connected to each other without going through a bulk transition.

Corresponding BSPT state

However with interaction, the classification can be reduced from to , meaning that eight copies of the Majorana chain () can be smoothly connect to the trivial state () without closing the bulk gap in the presence of interaction. This interaction reduced classification was discovered in Ref. Fidkowski and Kitaev, 2010, 2011. But we will show it again by making connection to the boson SPT (BSPT) states, as our approach can be generalized to higher spacial dimensions.

Let us start by showing that four copies of the Majorana chain () can be connected to the Haldane spin chain,Bi et al. (2014) a BSPT state in 1d. In this case, we have four flavors of Majorana fermion per site, denoted (), from which we can define a spin-1/2 object on each site, as in the basis . 3 One can couple the staggered component of the spin to an O(3) order parameter on each site, as . Then the low-energy effective Hamiltonian for four copies of the Majorana chain coupled to the field reads (which we call fermionic -model, or FSM)

(5)

where . The time-reversal symmetry operator necessarily requires to flip the order parameters under the time-reversal. Following the calculation in Ref. Abanov and Wiegmann, 2000, after integrating out the fermion field , we arrive at the effective theory for the boson field , which is a non-linear -model (NLSM) with a topological term at , given by the following action

(6)

describes the remaining dynamics in the bosonic sector. Presumably we work in the large limit, such that the field is deep in its disordered phase. With , the Hamiltonian of Eq. (6) reads (where is the canonical conjugate variable of at each spatial position ), and since flows to under coarse-graining, in the long-wave-length limit the ground state wave function of this theory is a trivial direct product state  Xu (2013) with a fully gapped and nondegenerate spectrum in the bulk and at the boundary (on each coarse grained spatial point, ). However, with , Eq. (6) describes a non-trivial BSPT state for the field, and is equivalent Haldane (1983a, b); Ng (1994) to the Haldane phase of spin-1 chain protected by the spin-flipping time-reversal symmetry . In fact, the spatial boundary of Eq. (6) with is a O(3) NLSM with a Wess-Zumino-Witten (WZW) term at level , and by solving this theory exactly, we can demonstrate explicitly that the ground state of the spatial boundary of Eq. (6) with is doubly degenerate Ng (1994); Bi et al. (2013), which is equivalent to the boundary ground state of four copies of Kitaev’s chain under interaction. In the low-energy limit, the boundary of four copies of the FSPT states is faithfully captured by the bosonic field . Thus we have established a connection between four copies of Majorana chain and a single copy of Haldane chain, bridging the FSPT and BSPT states in .

Using the knowledge of the better understood BSPT states, we can gain insight of the interacting fermion SPT (iFSPT) states. If eight copies of the iFSPT states is a trivial phase, then necessarily the bosonic theory of eight copies of the FSPT states derived using the same method must also be a trivial state. Indeed, because the Haldane phase has a well-known classification Chen et al. (2012), it is expected that two copies of the Haldane chain can be smoothly connected to the trivial state without breaking the symmetry. This can be shown by coupling two layers of the Haldane chain with a large inter-layer anti-ferromagnetic interaction (which preserves the symmetry), as described by the action

(7)

when , and are locked into opposite directions, i.e. . Then the effective NLSM for has due to the cancellation of the angles between the two layers. So two copies of the Haldane chain can be trivialized by the coupling.

Also, when the two Haldane phases in Eq. (7) are decoupled from each other (), both Haldane phases in Eq. (7) are separated from the trivial phase () with a critical point at . However, with , this critical point is also gapped out by the coupling,4 thus with , the entire phase diagram of Eq. (7) has only one trivial phase. This observation already suggests that 8 copies of Kitaev’s Majorana chain is trivial under interaction.

Bulk transition

Now let us carefully investigate the interactions in the fermion model. Two copies of the Haldane chain would correspond to eight copies of the Majorana chain. Recall the relation on the mean-field level, the inter-layer coupling can be immediately ported to the fermion model as an on-site interaction among eight flavors of Majorana fermions

(8)

with . We should expect that eight copies of the Majorana chain can be connected to the trivial state under this interaction, as the same interaction can trivialize the BSPT in the NLSM.

Figure 2: Phase diagram of the interacting Majorana chain at . The bulk criticality at the origin can be avoided if interaction is allowed. Each horizontal chain is four copies of the Majorana chain, equivalent to a Haldane chain. The vertical bound is the on-site spin-spin interaction. and labels the sites in a unit cell. Gray ovals mark out the spin-singlet dimers.

Such an expectation is obvious in the spin sector. In the free fermion limit, depending on the sign of , the fFSPT states may correspond to two different spin-singlet dimerization patterns: the intra-unit-cell dimerization ( trivial state) or the inter-unit-cell dimerization ( SPT state), as shown in Fig. 2. While the strong on-site interaction in Eq. (8) will lead to a third pattern, i.e. the on-site (inter-layer) dimerization, see Fig. 2. The three patterns are connected by the ring exchange of the dimmers. However, it is known that the ring exchange is a smooth deformation and will not close the spin gap, so at least in the spin sector, the and the SPT states can be smoothly connected.

To show that the charge gap also remains open, we can perform an explicit calculation based on the lattice model Eq. (1) in the strong dimerization limit , such that the 1d chain is decoupled into independent two-site segments. In each segment, the interacting fermion system can be exact diagonalized. Then it can be shown that the charge gap indeed persists as is tuned to zero in the present of the interaction , as shown in Fig. 3. So one can smoothly connect the fFSPT state to the fFSPT state in three steps: (i) turn on and turn off , (ii) change the sign of , (iii) turn on and turn off . The bulk gap will never close during this process. Thus the whole phase diagram in Fig. 2 is actually one phase.

Figure 3: Many-body energy levels of the two-site segment by exact diagonalization. From the weak interaction limit (left) to the strong interaction (right) limit, the many-body gap never closes.

In conclusion, we have demonstrated that the classification of the FSPT states with the (BDI class) time-reversal symmetry is reduced from to under interation. We obtain the iFSPT classification by making connection to the BSPT classification. This approach can be readily generalized to higher spacial dimensions in the following. Moreover, the way that the BSPT state can be trivialized in the NLSM naturally provide us the correct fermion interaction that is needed to trivialize the FSPT states, which can be much more general than the currently known Fidkowski-Kitaev type of interaction. And this interaction can gap out the critical point in Eq. (3), which is 8 copies of nonchiral Majorana fermions. This bulk analysis is particularly suitable to study the crystalline SPT states, which may not have symmetry protected physical boundary modes.

ii.2 From TSC to Levin-Gu Paramagnet

Lattice Model and Bulk Theory

Now we turn to the example of the topological superconductor (TSC) protected by a symmetry (symmetric class D)Altland and Zirnbauer (1997). The TSC can be viewed as both layers of the and the TSC’sVolovik (1988); Read and Green (2000); Fu and Kane (2008) stacked together with the symmetry acts only in the layer by flipping the sign of the fermion operator (i.e. the fermion parity transform). On a lattice, the model Hamiltonian can be written as

(9)

where labels the two opposite layers of the chiral TSC’s. The symmetry acts as which prevents the mixing of fermions from different layers, such that the fermion parity is conserved in each layer independently. Depending on the chemical potential , the model has two phases: the strong pairing trivial superconductor phase and the weak pairing topological superconductor phase. Switching to the Majorana basis and in the long-wave-length limit, the effective Hamiltonian reads

(10)

where we have set as our energy unit, and defined the topological mass (assuming ). The symmetry acting on the Majorana fermions as . The trivial () and the topological () phases are separated by the phase transition at where the bulk gap closes. This bulk criticality is protected by the symmetry.

The above TSC is an example of the D class fFSPT states, which are known to be classified,Schnyder et al. (2009); Ryu et al. (2010) and are indexed by the topological numberQi et al. (2006)

(11)

where with is the fermion Green’s function in the frequency-momentum space. The TSC in Eq. (9) corresponds to . While the other topological states in this classification may be realized by considering multiple copies of such TSC’s, which can be described by the following effective field theory Hamiltonian

(12)

with the symmetry protection. -copy TSC would correspond to the topological number .

Corresponding BSPT state

However with interaction, the classification of the TSC is reduced from to  Qi (2013); Ryu and Zhang (2012); Yao and Ryu (2013); Gu and Levin (2013), meaning that eight copies of the TSC () can be smoothly connected to the trivial state () in the presence of interaction. This interaction reduced classification was discussed in Ref. Qi, 2013; Yao and Ryu, 2013; Ryu and Zhang, 2012; Gu and Levin, 2013, but here we will provide another argument for it by making connection to BSPT states.

Let us start by showing that four copies of the TSC () can be connectedLu and Vishwanath (2012); Cheng and Gu (2014) to the Levin-Gu topological paramagnet,Chen et al. (2011); Levin and Gu (2012) a BSPT state in . We first introduce a set of inter-layer -wave pairing terms (with and labeling the 4 copies)

(13)

where ; and couple them to an O(4) order parameter field . The low-energy effective Hamiltonian of this FSM reads

(14)

Because the inter-layer pairing mixes the fermions between the and the TSC’s, they will gain a minus sign under the symmetry transform. To preserve the symmetry, we must require the order parameters to change sign as well, i.e. , under the symmetry action. After integrating out the fermion field , we arrive at the effective theory for the boson field , which is a NLSM with a topological term at , given by the following action ()

(15)

which describes a non-trivial BSPT state for the field Xu and Senthil (2013); Bi et al. (2013), and is equivalent to the Levin-Gu stateBi et al. (2013) protected by the symmetry . This can be understood from the wave function perspective. We first reparameterize where is an O(3) unit vector and . Suppose the system energetically favors (i.e. ), then the wave function for the field in its paramagnetic phase () can be derived from the action Eq. (15) asXu and Senthil (2013)

(16)

which is a superposition of all configurations with a sign factor counting the parity of the Skyrmion number of the field. In the Ising limit where is energetically favored, the Skyrmion number becomes the domain-wall number of the Ising spin , so the wave function becomes the superposition of Ising configurations with the domain-wall sign Xu and Senthil (2013), which is exactly the wave function of the Levin-Gu state Levin and Gu (2012). Thus we have established a connection from four copies of the TSC to a single copy of the Levin-Gu paramagnet, bridging the FSPT and BSPT states in .

Now we can discuss the iFSPT states using the knowledge about the BSPT states: if eight copies of the TSC is trivial, then the bosonic theory derived using the same method above must necessarily be trivial. Indeed, on the BSPT side, we know that two copies of the Levin-Gu paramagnets can be smoothly connected to the trivial state without breaking the symmetry, which can be realized by coupling two layers of the Levin-Gu paramagnet with a large inter-layer anti-ferromagnetic interaction, such that the domain-wall configuration in both layers will become identical, and the domain-wall sign from both layers will cancel out, so that the resulting wave function is just a trivial Ising paramagnetic state. At the field theory level, it can be described by the following action with inter-layer coupling

(17)

It is easy to check that the coupling respects the symmetry. When , and are locked anti-ferromagnetically for their first three components and ferromagnetically for their last components, i.e. () and . Then the effective NLSM for the combined field has due to the cancellation of the angles between the two layers. So two copies of the Levin-Gu paramagnet can be trivialized by the coupling 5. This suggests that eight copies of the original TSC is trivial.

Boundary modes and Bulk transition

Recall the relation () on the mean-field level, the inter-layer coupling can be immediately ported to the fermion model as the following four-fermion interaction (with )

(18)

where is defined in Eq. (13). Without any interaction, eight copies of TSC with the symmetry is separated from the trivial state through a critical point that has 16 copies of massless Majorana fermions in the bulk ( in Eq. (12)). We should expect that the bulk criticality can be gapped out by the interaction Eq. (18), and eight copies of the TSC can be smoothly connected to the trivial state, as the same interaction can trivialize the BSPT in the NLSM.

Admittedly, in (and higher dimensions), it is hard to explicitly demonstrate how the interaction gaps out the gapless bulk fermion at the critical point. Nevertheless we can show that, on an open manifold, the interaction Eq. (18) can gap out the boundary states of eight copies of the TSC () without breaking the symmetry, and hence there should be no obstacle to tune the bulk system smoothly from the state to the state under interaction. The “transition” between and states can be viewed as growing domains inside the state, which is equivalent to sweeping the interface between the two states through the entire bulk (this is essentially the picture of Chalker-Coddington model Chalker and Coddington (1988) for the quantum Hall plateau transition), then as long as the interface is gapped out by interaction, the bulk gap never has to close during this “transition”, namely the bulk phase transition can be gapped out by the interaction. Thus all we need to show here is that the interaction Eq. (18) induces an effective interaction at the boundary, which will gap out the boundary states.

Let us consider a boundary of the system along the axis, i.e. the topological mass changes sign across . For four copies of the TSC as described in Eq. (14), the boundary states are given by the projection operator , such that the effective FSM Hamiltonian along the boundary is given by

(19)

where denotes the Majorana edge modes, and the symmetry acts as . Under a basis transformation , the boundary FSM Hamiltonian can be reformulated as

(20)

which, at the field theory level, is equivalent to four copies of (critical) Majorana chain described by Eq. (5), with the transformed symmetry . is the analogue of the O(3) order parameter of the Majorana chain introduced in the previous section. All these order parameters are forbidden to condense by the symmetry, i.e. , so that the edge is gapless at the free fermion level.

Now we consider the boundary of eight copies of the TSC, which is simply a doubling of Eq. (20). The field theory of this boundary is equivalent to eight copies of the critical Kitaev’s Majorana chain. The bulk interaction Eq. (18) will induce the interaction between Majorana surface modes, which corresponds to the coupling of and at the boundary:

(21)

The term corresponds to exactly the same fermion interaction that trivialized eight copies of Majorana chain in the previous section, and this coupling can gap out the critical point in the previous case. This means that the term can also gap out the boundary of the 8 copies of TSC without degeneracy. Once the boundary is gapped and nondegenerate, a weak term in Eq. (21) will not close the gap of the boundary. Since the boundary coupling Eq. (21) is induced by the bulk interaction Eq. (18), this implies that the interaction in Eq. (18) (with strong enough strength) can gap out the bulk criticality (with 16 copies of massless Majorana fermions) in .

ii.3 From He-B to Bosonic SPT

Lattice Model and Bulk Theory

Let us go one dimension higher, and consider the He superfluid B phaseBalian and Werthamer (1963); Leggett (1965, 1972, 1975) (will be denoted as He-B) which is a TSC protected by the symmetry with (symmetry class DIII)Altland and Zirnbauer (1997). The He-B TSC is described by the following Hamiltonian

(22)

where is the fermion operator for the He atom, and is the -wave pairing strength. The Hamiltonian is invariant under the time-reversal symmetry, which acts as followed by the complex conjugation. He-B TSC corresponds to the topological phase of the model, while for the model describes a trivial superconductor. Switching to the Majorana basis and in the long-wave-length limit (to the first order in ), the effective Hamiltonian reads

(23)

where we have set as our energy unit, and defined the topological mass (which should not be confused with the mass of the He atom ). The time-reversal operator acting on the Majorana basis is given by . The trivial () and the topological () phases are separated by the phase transition at where the bulk gap closes. This bulk criticality is protected by the symmetry.

The He-B TSC belongs to the DIII class fFSPT states, which is known to be classified,Schnyder et al. (2009); Ryu et al. (2010) and are indexed by the topological numberQi et al. (2008); Wang et al. (2010)

(24)

where is the fermion Green’s function at zero frequency . The He-B TSC in Eq. (22) corresponds to . While the other topological states in this classification may be realized by considering multiple copies of the He-B TSC’s, which can be described by the following effective field theory Hamiltonian

(25)

with the symmetry protection (). -copy He-B TSC would correspond to the topological number .

Corresponding BSPT state

However with interaction, the classification of the DIII class FSPT states is reduced from to , meaning that sixteen copies of the He-B TSC () can be smoothly connected to the trivial state () in the presence of interaction. This interaction reduced classification was discussed in Ref. Fidkowski et al., 2013; Wang and Senthil, 2014, but here we will provide another argument for it by making connection to the BSPT states.

Let us start by showing that eight copies of the He TSC () can be connected to the BSPT state with symmetry. Similar to our previous approach in and , here we should introduce five fermion pairing terms and couple them to an O(5) order parameter field , the low-energy effective FSM Hamiltonian reads

(26)

where . It turns out that these order parameters are spin-singlet -wave (time-reversal broken) imaginary pairing among the eight copies of fermions. The particular form of the pairing terms given here is not a unique choice. We only require that the pairing terms anti-commute with each other, and also anti-commute with the momentum and the topological mass terms. However any other set of such pairing terms are related to the above choice by basis transformation among the eight copies of fermions, so we may stick to our current choice without losing any generality.

On this Majorana basis, the time-reversal operator is extended to , from which, it is easy to see that all five -wave pairing terms change sign under . To preserve the symmetry, we must require the order parameters to change sign as well, i.e. , under the transform. After integrating out the fermion field , we arrive at the effective theory for the boson field , which is a NLSM with a topological term at , given by the following action ()

(27)

where is the volume of . Eq. (27) describesBi et al. (2013) a non-trivial BSPT stateVishwanath and Senthil (2013); Xu (2013); Bi et al. (2014) protected by the symmetry . Thus we have established a connection from eight copies of the He-B TSC to a single copy of the BSPT state, bridging the FSPT and BSPT states in .

Now we can discuss the iFSPT states using the knowledge about the BSPT states. On the BSPT side, based on the well-known classification of this state Chen et al. (2012); Vishwanath and Senthil (2013), it is expected that two copies of the BSPT state can be smoothly connected to the trivial state without breaking the symmetry. In our NLSM formalism, this conclusion can be drawn by the following inter-layer coupling

(28)

It is easy to check that the coupling respects the symmetry. When , and are locked anti-ferromagnetically for their first three components and ferromagnetically for their last two components, i.e. () and (). Then the effective NLSM for the combined field has due to the cancelation of the angles between the two layers. So two copies of the BSPT state can be trivialized by the coupling. Again, this is the necessary condition for interaction to reduce the classification for He-B phase to .

Bulk Phase Transition under Interaction

Now we would like to argue that the quantum critical point in the noninteracting limit can be gapped out by interaction for 16 copies of He-B states. We start from the critical point in the FSM Eq. (26), where the bulk gap is closed on the free fermion level. The field theory Eq. (26) at has an extra inversion symmetry (where the space inversion operator sends ), besides the original time-reversal symmetry . Fermion interactions will be generated after integrating out dynamical field . We will argue that in this particular field theory Eq. (26), interaction can gap out the critical point, without driving the system into either or state.

We can first gap out the fermions in the bulk by setting up a fixed configuration of the order parameter field at the cost of breaking the time-reversal symmetry. Then we restore the symmetry by proliferating the topological defects of the field, which is an approach adopted by Ref. Wang and Senthil, 2014; You et al., 2014. Here we consider the point defect, namely the monopole configuration of , which is described by (for ) and near the monopole core. This monopole breaks both and , but it preserves the combined symmetry . After proliferating this monopole, all the symmetries will be restored.

However the potential obstacle is that the monopole may trap Majorana zero modes and is therefore degenerated. Proliferating such defect will not result in a gapped and non-degenerated ground state, and hence fails to gap out the bulk criticality. So we must analyze the fermion modes at the monopole core carefully. By solving the BdG equation for a single copy of the FSM Eq. (26), it can be shown that the monopole will trap four Majorana zero modes, which transforms under as , with , followed by complex conjugation and space inversion. Thus for two copies of FSM Eq. (26), the monopole will trap eight Majorana zero modes, and the symmetry will guarantee the spectrum of the monopole is degenerate at the noninteracting level. Nevertheless the degeneracy can be completely lifted by interactionFidkowski and Kitaev (2010, 2011) without breaking . So after the monopole proliferation, all the symmetries of Eq. (26) are restored, and the system will enter a fully gapped state which still resides on the line . Therefore with two copies of the FSM, the iFSPT state can be smoothly connected to the trivial state via strong interaction, resulting in the classification, which is consistent with the NLSM analysis.

Later we will show that this analysis of bulk phase transition using topological defects can be naturally generalized to all higher dimensions.

Boundary Modes and Bulk transition

Similar to what has been discussed in the and cases, the inter-layer coupling in Eq. (28) can be immediately ported to the fermion model as a four-fermion local interaction (with )

(29)

where are defined for both layers of the fermions. We should expect that sixteen copies of the He-B TSC can be connected to the trivial state under this interaction, as the same interaction can trivialize the BSPT in the NLSM.

Following the same idea of the case, we argue that the interaction can remove the bulk criticality by showing that its boundary states can be symmetrically gapped out under interaction.

Let us consider a boundary of the system along the - plane, i.e. the topological mass changes sign across . For eight copies of the He-B TSC as described in Eq. (26), the boundary states are given by the projection operator , such that the effective FSM Hamiltonian along the boundary is given by

(30)

where denotes the Majorana surface modes, and the symmetry act as on . Under a series of basis transformation as follows

(31)

the boundary Hamiltonian can be reformulated as