Fermion masses without symmetry breaking in two spacetime dimensions

I study the prospect of generating mass for symmetry-protected fermions without breaking the symmetry that forbids quadratic mass terms in the Lagrangian. I focus on 1+1 spacetime dimensions in the hope that this can provide guidance for interacting fermions in 3+1 dimensions. I first review the SO(8) Gross-Neveu model and emphasize a subtlety in the triality transformation. Then I focus on the"m = 0"manifold of the SO(7) Kitaev-Fidkowski model. I argue that this theory exhibits a phenomenon similar to"parity doubling"in hadronic physics, and this leads to the conclusion that the fermion propagator vanishes when p = 0. I also briefly explore a connection between this model and the two-channel, single-impurity Kondo effect. This paper may serve as an introduction to topological superconductors for high energy theorists, and perhaps as a taste of elementary particle physics for condensed matter theorists.


I Introduction
In an effort to demonstrate the interdisciplinary value of the study of topological superconductors, let me begin with a problem in elementary particle physics in 3+1 spacetime dimensions. In the Standard Model (SM) of particle physics, all fundamental fermions are massless at energies above the electroweak scale, v = 246 GeV. This is because one imposes the gauge symmetry This representation is chiral and hence does not admit a G SM -invariant mass term for any fundamental fermion field. The simplest way to give the fermions a mass at energies below the scale v is to posit the existence of a spin-0 Higgs field transforming as (1, 2, − 1 2 ) and then to write a gauge-invariant Yukawa interaction. When the Higgs condenses, the electroweak part of the gauge group, G EW = SU (2) weak ×U (1) hypercharge , is broken to U (1) EM , and the fermions obtain mass.
It is by now widely accepted that the quarks and leptons of the SM obtain masses in this way. The recent experimental discovery of the Higgs boson [1,2] strongly reinforces the expectation that the fermions should be massless in the G EW -symmetric phase and massive in the G EW -broken phase.
As a matter of theoretical interest, it is worth emphasizing that the above picture is based on weak coupling perturbation theory. One might instead consider nonperturbatively large interactions and ask the following question: Is it possible for the fundamental fermions of the SM to obtain mass in the G EW -symmetric phase? This is exactly the type of question that condensed matter theorists ask when they speak of "reducing the classification of topological superconductors" [3,4,5,6,7]. It turns out that there are strong physical indications that, if one includes a gauge-singlet antineutrino per generation, then all physical excitations in the SM can be fully gapped without breaking any part of the SM gauge group [8,9,10]. (The relationship between the reduced classification of topological superconductors and the anomaly matching condition was discussed in [11].) It might be thought that the phenomenon studied in condensed matter physics is simply an artifact of the lattice and should not have a continuum description. However, there are recent numerical results which support the conjecture that the transitions in question are second order and should be described by an interacting quantum field theory [12,13].
This paper focuses on this type of problem in 1+1 spacetime dimensions within the framework of "symmetry protected topological" (SPT) phases [14,15]. In this context, the symmetry group G is a global symmetry of the model, but it is often a useful theoretical device to gauge that symmetry by the usual minimal coupling procedure [7].
An SPT phase in d spatial dimensions with global symmetry G is a zero-temperature state of quantum matter whose three defining phenomenological properties are: 1) In a system without spatial boundaries, the ground state is unique and all excitations above the ground state are gapped. (The bulk is said to be "trivial".) 2) In a system with spatial boundaries, the ground state is degenerate or there exist gapless excitations. (The boundary is said to be "nontrivial".) 3) The boundary theory cannot be defined self-consistently as an independent quantum theory in (d − 1) spatial dimensions.
If the global symmetry G is broken (either spontaneously or by an explicit G-breaking term in the Lagrangian), then the formerly gapless boundary excitations become gapped, and the theory flows to a trivial gapped state at low energy.
The simplest field theoretic example is the continuum limit of the 1d Kitaev chain [16]. (For a review, see Appendix A.) Consider a (1+1)-dimensional relativistic theory of a massless Majorana fermion, 1 N = η iη , (I. 1) coupled to a time-independent, spatially-dependent, semiclassical background scalar field φ(x) (here x stands for the spatial coordinate only). The Lagrangian is: Consider the infinitesimally thin kink profile: For x > 0 there is a free Majorana fermion with a physical mass m, and for x < 0 there is also a free Majorana fermion with physical mass m. (By the value m being the "physical mass" I mean that the fermion transforms as the Poincaré representation However, at x = 0 there is a time-independent real fermion stuck to the core of the kink. To see this [17], write N = N + + N − , where γ 5 N ± = ±N ± . The equations of motion δL /δN + = 0 and δL /δN − = 0 admit a solution of the form where c is a real operator. In condensed matter theory, such real fermion operators are called Majorana operators. The number of spinor components has been cut in half, and there is a zero-energy fermion localized in the vicinity of x = 0. Let m F be the coefficient of 1 2N N in the Lagrangian of Eq. (I.2). In a system without spatial boundaries, one typically assumes that the m F = +m phase and the m F = −m phase describe the same quantum state, because the sign can be compensated by a transformation N → γ 5 N . The existence of the Majorana mode at the kink core means that, for a system with spatial boundaries, these phases are different: at the interface between the two states, there is an additional degree of freedom [18]. The sign of the fermion mass term will play a crucial role throughout this paper. Now imagine a scalar field profile of the following form: (I.5) At x = L there is a real fermion, c, and at x = −L there is another real fermion, c . These can be paired up into a complex fermion annihilation operator, If |0 is the vacuum with energy E 0 , then the state f † |0 has an energy 2 E 1 −E 0 ∼ e −mL .
In the limit L → ∞ (namely, the thermodynamic limit), the state f † |0 becomes degenerate with the vacuum. So if one thinks of these states as belonging to the boundaries of the m F > 0 phase while considering the m F < 0 phase as the "ordinary" gapped phase, then this profile models a topologically nontrivial 1d system of length 2L. But this 1d system of length 2L is not yet an SPT state, because the gaplessness of the excitation is protected by the thermodynamic limit, not by the imposition of a global symmetry. To emphasize this point, consider two flavors of the above setup, indexed by a label a = 1, 2. Then it is possible to write the local interactions ic 1 c 2 and ic 1 c 2 at x = +L and x = −L, respectively. All excitations above the ground state are gapped, and this is a trivial phase (in the sense described earlier).
For this two flavor system, impose a flavor-independent antiunitary discrete symmetry, which may as well be called a peculiar version of time reversal that squares to +1: This transformation leaves c 1 c 2 and c 1 c 2 unchanged, but it flips the sign of the prefactor i (whose presence in the Hamiltonian is required for hermiticity). Hence if this Z T 2 is imposed on the Lagrangian, then all fermion bilinears at the x = ±L boundaries will be forbidden. This is true for an arbitrary number of flavors, n ∈ Z. Each value of n defines a distinct phase. So this setup describes a 1d SPT phase which is classified by an integer that labels the number of gapless edge modes. 3 It is crucial to observe that the flavor-diagonal transformation in Eq. (I.7) leaves the bilinearN a N b invariant, so the bulk remains gapped.
The possible free-fermion SPT phases in various dimensions and with various global symmetries have already been enumerated [19,20]. The question that connects this to the particle physics problem described earlier is whether those systems with integer classification are stable to interactions. In condensed matter physics one is typically concerned only with time reversal, SU (2) spin symmetry, and particle-hole symmetry (or its incarnation as an artificial redundancy in superconducting theories). But if the 2 This is because the two ends at x = ±L have to talk to each other in order to form a term in the Hamiltonian of the form icc . The interior is fully gapped and admits only local interactions, so the amplitude for the two ends to interact is exponentially suppressed for L m −1 . 3 The reader may want to verify that this setup satisfies the three conditions described earlier. Conditions (1) and (2) are obviously fulfilled. Condition (3) is fulfilled because the action of time reversal as in Eq. (I.7) cannot be implemented self-consistently on an independent 0d quantum system. Define the annihilation operator for a fermionic oscillator at x = +L by a ≡ c 1 + ic 2 . The Z T 2 transformation flips the sign of i but leaves c 1 and c 2 invariant. Therefore, Z T 2 : a → a † , and time reversal does not commute with (−1) F when acting on physical states. transition between the trivial superconducting phase and the SPT phase is continuous, then it admits a field theoretic description, and the results obtained in that description hold for any system described by the same low-energy effective Lagrangian.
Just as the electron of the SM is protected by SU (2) weak × U (1) hypercharge , here in the 1d Kitaev chain the 0d edge fermions are protected by Z T 2 . By turning on local interactions for a system with n flavors, is it possible to gap out these symmetry-protected edge modes without breaking Z T 2 spontaneously? Kitaev and Fidkowski (KF) [3] showed that the answer is yes, if and only if n = 8k, k ∈ Z. (The reader who is unfamiliar with this result should not worry: it will be discussed thoroughly in the body of this paper.) One says that the interactions "reduce the classification" from Z to Z 8 .
The purpose of this paper is to explore in greater detail the "m = 0" manifold of the KF model purely within the continuum field theory description, with an eye toward extracting general lessons for interacting field theories in higher dimensions. Just as 1+1 interacting systems have proved insightful for studying confinement in higher dimensions, I hope that a thorough analysis in 1d will provide guidance for interacting fermions in 3+1 dimensions.
The layout is as follows. First, in Sec. II, I will review the SO(8) Gross-Neveu model (GN). The purpose of this is to provide necessary background material, to establish notation, and to point out a subtlety in the "triality" invariance of the Lagrangian. Then, in Sec. III, I will discuss the "m = 0" manifold of the SO(7) KF model with an emphasis on the fermion propagator. In particular, I will argue that an analog of "parity doubling" occurs, and that the leading term in the spectral decomposition is simply proportional to p µ . In Sec. IV.1, I will attempt to relate the KF model to physical conduction electrons in the context of impurity scattering. In Sec. V, I will summarize the results and suggest possible directions for future work.

II Eight Majorana fermions with SO(8) symmetry
The goal is to study the effects of interactions on the mass gap and excitation spectrum for a theory of eight relativistic Majorana fermions, The free massless Lagrangian is: From now on, the standard repeated index summation convention will be used.
defines the SO(8) Gross-Neveu (GN) model [25]. It has a global chiral Z 2 symmetry: However, this symmetry is spontaneously broken at low energy: the coupling gets strong and the fermion mass bilinear forms an SO(8)-invariant condensate, Perturbing around a fixed choice of minimum, one finds nonzero fermion masses for the fluctuations described by the primed fields: This part of the story is the well-known analysis of the SO(N ) GN model at large N (see, for example, [27]) and is not unique to the value N = 8.

II.1 Bosonization and triality
The "triality" of SO(8) is a cyclic permutation of the three real 8-dimensional representations, which are the vector (denoted by 8 v ) and the two chiral spinors (denoted by 8 + and 8 − ) [21]. Although the SO(8) group possesses this outer automorphism, the physics of the SO(8) GN model is a little more subtle. This is the same subtlety which occurs in the Ising model: a theory with two ground states cannot be equivalent to a theory with one ground state, so the Ising duality transformation must be accompanied by the introduction of a topological Z 2 gauge theory [28,29]. This will be reviewed in Sec. II.4, but first let me proceed with the SO(8) theory.
A physically clear way to implement the triality operation is to use abelian bosonization [26,30,31,32]. (For a discussion of triality in non-abelian bosonization, see [33].) First bosonize the Majorana fermions in pairs: Define the non-chiral bosons Then the SO(8)-invariant fermion mass term is: The triality transformation from the 8-vector to the 8 + -spinor is defined by the following special orthogonal transformation in the space of bosons (I use the conventions of Ludwig and Maldacena [30]): (II.11) This change of basis defines the bosons Θ I , I = 1, ..., 4. (To dispel any potential confusion, I should note that I will not use the notation "Θ" for the dual of Φ. For the dual of Φ I will writeΦ ≡ ϕ A +φ A .) By straightforward algebra, one obtains: (II.13) By the subscript "REN" I mean that this equality holds after the renormalization procedure of subtracting the cosine-squared terms from both sides. The reason for doing this is because the quantum theory possesses the unusual relation (see the appendix of [34]): (II.14) So these terms actually contribute to a renormalization of the boson kinetic terms and should not be considered as part of the interactions.
In analogy with the definition Φ A = ϕ A −φ A , now define the chiral bosons θ I and θ I via Θ I ≡ θ I −θ I andΘ I ≡ θ I +θ I . These new chiral bosons can be fermionized: 5 (II.15) So after the subtraction described above, the following equality is obtained: This process can be repeated starting from a modified version of Eq. (II.11): Again it is useful to define chiral bosons via Ξ X = ξ X −ξ X and fermionize them: Therefore [26,35,36]: (II.21) The kinetic terms also satisfy an analogous equality, so the whole Lagrangian takes the same form whether written in terms of the η, the ψ, or the χ fermions. These fields are nonlocally related to each other, but the Lagrangian written in terms of a given representation is local. This is what is usually considered the physical manifestation of the group-theoretic triality symmetry of the SO(8) GN model. The equality of the fourth-order polynomials in Eq. (II.21) was just derived explicitly above, so this part of the usual story remains unchallenged. I simply wish to point out a subtlety in the analysis if one studies the system in terms of the ψ-variables: the discrete "γ 5 " transformation ψψ → −ψψ is actually a gauge symmetry.

II.2 Global Z 2 symmetry and emergent Z 2 gauge symmetry
Consider the global chiral Z 2 symmetry defined back in Eq. (II.4). This corresponds to a shift for all A = 1, 2, 3, 4 simultaneously. The goal is to determine how this transformation affects the fields ψ ∼ 8 + and χ ∼ 8 − .
Recall the transformations in Eqs. (II.11) and (II.17), which I repeat below for convenience: (II.23) I have written this relation in terms of the left-moving chiral bosons, because it is these which are shifted by the chiral Z 2 transformation. The matrices S and T satisfy S 2 = I and T 2 = I, and hence S = S −1 , T = T −1 . Therefore, in terms of the original chiral bosons {ϕ A } 4 A = 1 , the defining relations above imply: Upon refermionization as in Eqs. (II. 15) and (II.20), I conclude that the physical Z 2 symmetry acts as follows on the 8 ± fermions: Therefore, the 8 + mass bilinear ψ iψj is even and hence is not an order parameter for the Z 2 symmetry. It may self-consistently obtain an expectation value without spontaneously breaking Z 2 .
On the other hand, the 8 − mass bilinear χ xχy is odd and hence cannot obtain an expectation value if the Z 2 transformation is to remain a symmetry of the low-energy theory. In this way, perhaps counterintuitively, the two SO(8) spinors are not created equal: it is not possible to use the 8 − in order to connect the trivial and topological phases.
The point about the triality transformation is to consider the analogous chiral sign flip for the 8 + variables, which I will denote by Z 2 . This operation is defined as The goal is now to determine how this transforms the fields η ∼ 8 v and χ ∼ 8 − . To do this, it is necessary to express the chiral bosons ϕ A and ξ X in terms of the 8 + bosons θ I , which transform as Recalling the triality transformation in Eq. (II.23), one finds (also recall that T −1 = T ): The product of the two transformation matrices, contains an odd number of minus signs per row. Meanwhile, the matrix S has an even number of minus signs per row. Therefore, the operation in Eq. (II.27) shifts the other two sets of bosons as (II.31) Since e iπ = e −iπ = −1, the relative sign in the transformation for the ξ X is immaterial, and I conclude: Therefore, the alternative chiral reflection defined by Eq. (II.27) leaves the original fermion fields η a totally unaffected. This transformation is invisible in terms of the original fields in the Lagrangian and hence should be thought of as an emergent gauge symmetry.
Finally, one should consider the theory written in terms of the 8 − variables and define a third Z 2 transformation which acts as (χ x ,χ y ) → (−χ x ,χ y ). I will not give this operation a new name because it turns out to be equivalent to the original global Z 2 symmetry. The by-now-familiar triality transformations give: The product of S and T in this order contains an even number of minus signs per row, while the matrix T contains an odd number of minus signs per row. Therefore, the shift ξ X → ξ X + 1 2 results in the shifts This is exactly the same transformation as the one described by Eqs. (II.22) and (II.25).
It is convenient to summarize this situation in terms of the mass bilinears. There are two Z 2 transformations, one global and one gauged. The physical Z 2 global symmetry acts as The artificial Z 2 gauge symmetry acts as So, strictly speaking, the physics of the SO(8) Gross-Neveu model is not quite invariant under triality: the description in terms of the 8 + variables requires coupling to a topological Z 2 gauge theory. This gauging procedure does not add any additional local degrees of freedom, but it projects out sectors of the state space which are not invariant under the transformation in Eq. (II.37).

II.3 Fermion parity
In addition to the chiral Z 2 transformation N a → γ 5 N a , it is also interesting to consider the transformation N a → −γ 5 N a . The analysis goes through exactly as before, except with the barred chiral fields playing the role of the unbarred chiral fields. The product of both of these transformations is fermion parity, Therefore, the conclusions of the previous section imply that the fields Ψ i = ψ i iψ i are even under fermion parity, while the fields This presents a puzzle: if one wishes to describe the original theory of η variables in terms of ψ variables, how is it possible to recover the sector of the original Hilbert space which contains an odd number of fermions? It is clear that additional non-local data is required, and I do not yet have a complete solution to this problem.
Furthermore, it is also interesting to consider the "artificial" fermion parity, This too is a gauge symmetry and should be modded out in the ψ-description of the original theory.

II.4 Z 2 transformations in the Ising model
It is useful to recall various properties of the 2d Ising model [37,38,47]. (The second "spatial" direction in this context should be thought of as Euclidean time.) I will work in the extreme anisotropic limit, which admits a description in terms of a transfer matrix (formally equivalent to deriving the path integral formulation by cutting up the total time interval into a large number of arbitrarily small steps).
In the transfer matrix description, there is a 1d lattice labeled by sites On each site lives a "spin" variable σ s which can be up or down, denoted by +1 and −1 respectively. In operator language, I choose this to be an eigenvalue of the third Pauli operator,σ z s . The states | N s = 1 σ s ≡ |σ 1 ⊗ |σ 2 ⊗ ... ⊗ |σ N satisfy: The Hamiltonian is:Ĥ Free boundary conditions have been chosen in the spatial direction. The low temperature phase is described by λ 1, and the high temperature phase is described by λ 1. The critical point is λ = 1.
The Hamiltonian has the following reflection symmetry: for all s = 1, ..., N simultaneously. There are two possible ground states: all spins are aligned, and they all point either up or down: The transformation of Eq. (II.43) exchanges these states: The total spin operator, or "magnetization" (normalized by the number of sites), has a nonzero vacuum expectation value: In either case, the system is ordered (or "magnetized"). The global symmetry Z spin 2 is broken spontaneously.
A local excitation above one of the two ground states is given by the flip of a single spin. One can also consider a non-local type of excitation, in which all spins to the left of a specified site, say r, are flipped. This excitation is called a kink (or domain wall), and is formally created by the following operator: Comparison of this with Eq. (II.42) reveals that the bulk energy spectrum obeys E(λ) = λE(1/λ), showing the equivalence between the high temperature and low temperature phases. This is well-known, but I wish to emphasize the following subtlety regarding this description in terms of the µ variables [28,29].
In analogy with the original description, define states in terms of "dual" spins, meaning eigenvalues ofμ z r :μ At 1/λ = 0, there appear to be two possible ground states: But the transformation λ ↔ 1/λ exchanges the high and low temperature phases, and the high temperature (disordered) phase of the original Ising model is unique. Therefore, the dual of Eq. (II.43), namely the transformatioñ  Equivalently, if one describes the original Ising model in terms of the dual variables, one must impose a "Gauss's law" constraint on the physical states, |Ψ phys : whereQ is the operator defined in Eq. (II.55). This implies: The physical ground state of the dual model is then theQ-invariant superposition The orthogonal combination, is not gauge invariant and hence is projected out of the Hilbert space. The total dual spin, or "disorder parameter,"K has nonzero expectation value in each of the two gauge-variant states: It is in this sense that the Z spin 2 -symmetric disordered phase of the Ising model is recovered by the condensation of kinks. However, strictly speaking, the kink operator has zero expectation value in the physical ground state: This is consistent with the general principle that gauge symmetries can never actually be broken spontaneously [39,40]. But, as indicated for example by Eq. (II.61), it is often extremely convenient to fix a gauge and to use the terminology which is more correctly reserved for the spontaneous breaking of global symmetries.
In summary, the Kramers-Wannier (KW) duality transformation in terms of unconstrained variables [28]. That the Hamiltonian written in terms of those fermion variables is local and quadratic constitutes the miracle of the Ising model [38].
Define the operatorsσ Define the real and imaginary parts of the fermionic operators in Eq. (II.66): where a, b ∈ {1, ..., 2N }. Then the Hamiltonian takes the form of the Kitaev chain with λ = J 1 /J 2 (see Appendix A): The continuum limit is described by the Lagrangian for a free Majorana fermion: The KW duality exchanges λ > 1 with λ < 1 and hence changes the sign of the fermion mass term: This is the "γ 5 " transformation that emerges when the fermion mass term is tuned to zero.

II.5 Ground state degeneracy in GN
In preparation for a later discussion of the SO(7) Kitaev-Fidkowski model (Sec. III.2), it will be important to establish that the ground state of the SO(8) GN model is twofold degenerate. The potential written in terms of the bosons for the original Majorana fermion fields (the η a ,η a ) is: This potential is invariant under the simultaneous sign flip of all cos(2πΦ A ). (This is just the physical Z 2 symmetry that I have already discussed at length). The minima occur when all cosine terms equal +1 or −1. Following the terminology of Shankar [26], I will call these "positive vacua" and "negative vacua" respectively.
Recall that the bosons Φ A were defined by the relations in Eq. (II.8), so each Φ A is defined only modulo 1. Therefore, all of the positive vacua correspond to a single state |0 > in the Hilbert space labeled by the configuration Φ A = 0 for all A ∈ {1, 2, 3, 4}: For the same reason, all of the negative vacua also correspond to a single state |0 < labeled by the configuration Φ A = 1 2 for all A ∈ {1, 2, 3, 4}: There are two degenerate ground states, corresponding to the spontaneous breaking of the global Z 2 symmetry that interchanges them.

II.6 Kinks and the 8 + basis
In the SO(8) GN model, the chiral Z 2 symmetry is spontaneously broken by an SO (8) Triality suggests that it should be possible to arrive at the conclusion of the previous section by studying the GN model in terms of these kink fields. In the 8 + basis, the SO(8) GN model has the Lagrangian (II.76) As discussed, it is understood that this should be coupled to a topological Z 2 gauge theory. Up to this subtlety, this Lagrangian looks formally equivalent to L (η,η), so the local dynamics are the same: at low energy the theory forms an SO (8) Suppose the Z 2 transformation were not gauged. Then the two choices of sign in Eq. (II.77) would correspond to different ground states, just like the two minima of a standard double-well potential. Denote these two ground states by |0 + and |0 − . The Z 2 transformation exchanges these two states: The physical implication of gauging the Z 2 symmetry is that the ground state is in fact the gauge-invariant linear superposition of the two possible configurations: The orthogonal combination, |0 unphys = 1 √ 2 (|0 + − |0 − ), is not gauge invariant and hence is projected out of the Hilbert space.
But the conclusion of Sec. II.5 was that the two possible choices of sign in the ηη condensate do correspond to different physical ground states. How can one arrive at this conclusion from studying the ψ variables?
For this purpose it is useful to think of the ψ 1 , ..., ψ 8 as eight Ising fermions [43,44,45,46]. Then, from the bosonization rules, one finds [47]: and finds a nonzero mass term for the fluctuations: The mass parameter for an Ising fermion is proportional to T − T c : For the "+" sign, the corresponding Ising models are in their disordered phase: σ = 0. For the "−" sign, they are in their ordered phase: σ = 0. Either way, the SO(8) symmetry requires all eight Ising models to be in the same phase, either ordered or disordered, so at low energy one always has: The Z 2 operation which flips the sign of this bilinear is a physical symmetry (not a gauge redundancy) and transforms a given ground state into another inequivalent ground state. This is one way to arrive at the conclusion of Sec. II.5 from the 8 + basis.
For the SO(8) GN model, this argument was needlessly complicated: one could have just analyzed the theory in terms of the original η variables and arrived at the correct conclusion directly. The purpose of this exercise was to show that the formation of a condensate in ψψ does not necessarily imply ground state degeneracy. The ground state may or may not be unique, irrespective of whether ψψ = 0.

III Eight Majorana fermions with SO(7) symmetry
After this long but necessary preliminary discussion of the SO (8)  Therefore, the appropriate course of action is to add an additional four-fermion interaction to the GN model which singles out a direction in the 8 + representation [3,4].
int , where the additional interaction term is: (III.1) The symbols Γ a denote the gamma matrices for SO (8): there are 8 of these, and each is a matrix of size 16×16. The state |S is defined as a particular element of the 8 + ; in the Wilczek-Zee notation [48,49], the choice in this paper (and in [3]) is: The basis of field coordinates ψ 1 , ..., ψ 8 is chosen so that ψ 8 corresponds to the state |S . (In other words, I could self-consistently choose the notation ψ S ≡ ψ 8 .) The square brackets around the indices a, b, c, d denote complete antisymmetrization.
The interaction in Eq. (III.1) explicitly breaks the SO(8) symmetry but conserves the SO(7) subgroup which rotates among the 7 remaining states of the 8 + representation (namely those states which are orthogonal to |S ). Therefore, as explained in [3], a triality transformation (understood in the sense discussed previously) to the ψ-fermion basis must result in a local polynomial of the form: I will take A > 0 and B < 0. It is easiest to focus on the region 0 < |B| A .
First set B = 0. At low energy an SO(7)-invariant fermion condensate will form: This will induce an effective mass parameter for the first seven fermions: Upon turning on a small negative B, one also induces a small mass parameter for the eighth fermion: In this region, the lowest-lying excitation above the ground state is this eighth fermion, and the mass gap of the theory is |m 8 |. Let me emphasize that the parameters m 1 , ..., m 8 should not be confused with the ηη mass parameter "m" which is forbidden by the chiral Z 2 symmetry in Eq. (II.36).

III.1 Absence of spontaneous symmetry breaking
In the introduction, I defined the bulk of an SPT phase to be invariant under a symmetry G and to have a unique ground state. The goal of this section is to argue, purely within the low-energy field theory, that the Z 2 symmetry of Eq. (II.36) is not spontaneously broken. The goal of the next section will be to show that the ground state is non-degenerate.
I showed in Sec. II.2 that the bilinears iψ iψj are even under the physical chiral Z 2 symmetry. In contrast, the bilinears in the original fermion fields, iη aηb , are odd under the chiral Z 2 symmetry (by definition). So the goal is first to argue that η aηb = 0.
Recall the relationship in Eq. (II.80) between ηη and the ψ order/disorder parameters, repeated below for convenience: In the SO (8) On the other hand, if 7 i = 1 iψ iψi < 0, then m 1 = ... = m 7 < 0 and m 8 > 0. The first seven Ising models are disordered, but the eighth one is ordered: Either way, a small negative B allows the phase of the eighth Ising model to be anticorrelated with the phase of the first seven, and one always concludes: In the SO(7) theory, the eight η a still transform as an 8-dimensional representation, so this implies η aηa = 0 (no sum on a) for each a = 1, ..., 8.
It is also necessary to check that 8 x = 1 χ xχx = 0, since the χ mass bilinear is also odd under the Z 2 operation. From the bosonization transformations, one finds the relation: (III.11) So the same argument will show that 8 x = 1 χ xχx = 0 as well. Therefore, the physical Z 2 symmetry remains unbroken at low energy.
To emphasize the special nature of this particular model, now treat the original Majorana fermion fields, η 1 , ..., η 8 , as Ising fermions, and consider adding and subtracting the analogs of Eqs. (III.7) and (III.11): (III.12) Here I have dropped the unimportant overall numerical factor common to both equations. In the usual critical Ising model, one has a gapless theory whose Lagrangian is invariant under the exchange σ a . Here, however, one has ψψ = 0, χχ = 0, and σ (η) for all a, b = 1, ..., 8. (Remember now these are the order/disorder operators for the η a variables, which still transform as an 8-dimensional representation in the SO(7) model.) The SO(7) model on the "m = 0" manifold has a form of Kramers-Wannier invariance while still being a gapped theory. This is very different from a garden-variety Ising model.

III.2 Uniqueness of the ground state
The ground state of the SO(8) GN model is two-fold degenerate, as discussed in Sec. II.5. In the SO(7) KF model, however, since η aηb = 0, then the physical Z 2 symmetry is unbroken and the ground state is unique. To determine the ground state, I study the potential for the KF model: (III.13) The bosonization rules in Sec. II.1 allow the bilinear of a single Majorana fermion to be expressed in terms of a non-chiral boson and its dual: iψ 2I−1ψ2I−1 = 1 2 cos(2πΘ I ) − cos(2πΘ I ) , iψ 2Iψ2I = 1 2 cos(2πΘ I ) + cos(2πΘ I ) .
I already argued that in the KF theory a simultaneous change in sign of all these cosines is a gauge symmetry. Therefore, there is only one ground state. In a fixed gauge, say choosing option (3) above, this ground state can be expressed as: In conclusion, a continuous tuning from m > 0 (topological phase) to m < 0 (trivial phase) does not pass through a point which breaks Z 2 , and it does not pass through a point for which the ground state is degenerate. This is consistent with the claim that there is no bulk phase transition between the two situations. It is also consistent with the corresponding study of the (0+1)d fermions at the spatial boundaries, which would amount to a repeat of the lattice analysis in [3].

III.3 Propagator
The standard Lehmann-Källén spectral decomposition for the N a propagator in a translationally invariant system is [50,51]: (III.20) Here α, β are SO(1, 1) Dirac spinor indices, m 2 stands for the squared mass of the excitations for which the N a (x) are good interpolating fields, m th is the threshold energy at which the multiparticle continuum begins. In the SO(8)-invariant Gross-Neveu model, the quantity |m| is the 8 v -fermion mass, which is the same as the kink mass (as required by triality).
In the SO(7)-invariant KF model, it is clear that the quantity |m| above should correspond to the rest energy of the excitation which creates a kink in the value of ε i ≡ ψ iψi , i = 1, ..., 7. Denote this rest energy by m kink ≡ |m|. The N a have the correct quantum numbers to annihilate those kinks, so the leading term in the expansion for iD(p) should have an isolated single-particle pole at p 2 = −m 2 kink .
The situation of interest is when the mass term for N a is absent in the Lagrangian. As emphasized previously, the low-energy theory remains invariant under the chiral Z 2 transformationN a N b → −N a N b : this symmetry is not spontaneously broken in the IR. The factor of m in the single-particle contribution to the propagator would break this chiral Z 2 invariance and therefore cannot appear in Eq. (III.20).
How can this apparent contradiction be reconciled? From an arithmetic point of view, the simplest resolution would simply be to cross out the m in the numerator while keeping the denominator equal to p 2 + m 2 kink − iε. This peculiar prescription actually seems to be the correct answer. This requires careful consideration of the steps leading to the spectral decomposition in Eq. (III.20).
The decomposition follows from inserting a resolution of the identity between the two fields N a (x) andN b (0) in the definition of the propagator. The vacuum gives zero contribution. The single-particle states give the first nonvanishing contribution, which is proportional to 1/(p 2 + m 2 ) for the appropriate choice of m 2 . It is this single-particle contribution which requires further scrutiny.
As remarked back in the introduction, in a (1+1)-dimensional field theory without spatial boundaries one normally thinks of the m > 0 Lagrangian and the m < 0 Lagrangian as equivalent descriptions of the same physical theory. One writes the Dirac Lagrangian as L =Ψ(i ∂ − mI)Ψ with a fixed sign of the mass parameter (say m > 0) and derives the equation of motion (i ∂ − mI)Ψ = 0. (Here Ψ is a generic Dirac spinor with no relation to the ψ variables discussed previously.) This equation of motion is to be thought of as projecting out half of the degrees of freedom of the spinor Ψ(x).
But for a free theory on a compact space, the two choices of sign for m define different phases. So it makes perfect sense that, if one is interested in the resolution of the identity in the form "1 = states |state state|" for a particular phase, then one fixes a particular sign of m and includes only that corresponding single-particle state in the sum.
For the special case of n = 8k flavors, for example in the case k = 1 studied above, the whole point is that the m > 0 theory and the m < 0 theory can be adiabatically deformed into each other without going through any phase transition. So in this model, in the resolution of the identity, one should sum over both possible signs of the mass parameter. This removes the chirality violating term from the propagator without removing the pole at p 2 = −m 2 with |m| = m kink . Therefore, the propagator for the N a (x) fields on the "m = 0" manifold of the SO(7)invariant KF model should take the form: This peculiar expression shows that the excitations described by N a (x) propagate with an ordinary relativistic massive dispersion relation but nevertheless do not ever flip chirality. This is unfamiliar, but there is nothing wrong with it.
The amplitude for a left-handed fermion to flip chirality and turn into a right-moving fermion is still proportional to the mass parameter m. But in this case there is a doubling of the number of degrees of freedom, one on-shell fermion for each sign of the mass parameter, and the amplitude for a chirality flip is proportional to m + (−m) = 0. I refer to this phenomenon as "parity doubling" in analogy with an effect in nuclear physics [52].
Along the "m = 0" manifold between the "trivial" and "topological" phases of the free-fermion theory, indeed it is the case that the fermion propagator vanishes linearly with p µ as p µ → 0 [53,54]. The remaining issue is to identify the origin of the extra "parity-conjugate" states.

III.4 Kinks in KF
Recall that in the SO(8)-invariant situation, the lowest-lying physical excitations transform as one of three distinct 8-dimensional representations, namely 8 v , 8 + , or 8 − . The explicit breaking of the symmetry to SO(7) was defined by the decomposition Under this decomposition, the two other 8-dimensional representations remain 8-dimensional representations: I intentionally do not distinguish between the two instances of "8" above: the group SO(7) has only one spinor representation. While the 8 v and 8 − were distinct in SO (8), these degrees of freedom transform as the same representation of SO(7) and therefore can mix in the low-energy theory.
One might worry that the nontrivial Z 2 charge of the χ variables [recall Eq. (II.37)] might preclude this possibility. Another way to say this is that, in terms of the ψ description, the theory contains "even" kinks (the η a ∼ 8 v ) and "odd" kinks (the χ x ∼ 8 − ). But this Z 2 is broken (better to say "Higgsed") by a nonzero condensate ψ iψj . So, in a fixed gauge, one should be able to think of the η particles and χ particles propagating together. The χ particles contribute the additional degrees of freedom required to realize the form in Eq. (III.21) for the η propagator.
Although this explicit identification of the appropriate states came from the study of a particular 1d model, it seems that this phenomenon should generalize to more complicated systems in higher dimensions. Consider a fermionic SPT phase classified by an integer n whose classification can be reduced by interactions to n ∼ n + k for some k. My general conjecture is that the Hilbert space of the theory must be enlarged to include states corresponding to the opposite sign of the mass parameter.
If this is correct, then along the "m = 0" manifold, the "m > 0" fermions and "m < 0" fermions should propagate together with a dispersion relation p 2 = −m 2 * for some m * = 0. The Green's function will be proportional to 1 2 [( p + m * ) + ( p − m * )] = p below the multiparticle threshold. This appears to be the only possibility that is consistent with all of the known results about Green's functions for symmetry protected topological phases. Unfortunately, I do not yet know how to check this proposal more explicitly.

III.5 A remark about ψ 8
In the previous sections, the KF model was studied for 0 −B A . In the limit B = 0, the Lagrangian describes one massless Majorana fermion (ψ 8 ), seven degenerate massive Majorana fermions (ψ 1 , ..., ψ 7 ), and massive kinks. The form of the Lagrangian in this limit seems to indicate that the field ψ 8 is totally decoupled from ψ 1 , ..., ψ 7 , so one might ask whether one could just "delete" ψ 8 altogether and study the SO (7) Gross-Neveu model [55].
Instead of performing any detailed calculations, let me argue based on general principles that this cannot be done. The argument rests on the observation that the Z 2 transformation which flips the sign of ψ iψj is a gauge symmetry.
In 1+1 spacetime dimensions, it is possible to construct a Lagrangian formulation of this Z 2 gauge theory by embedding Z 2 into U (1) and writing a "BF" theory with 0-form potential B and 2-form field strength F = dA, where A = A µ dx µ is the U (1) gauge potential. (See Appendix B for a brief review.) The Z 2 transformation of interest is chiral, in that it only rephases the left-moving fermions ψ i while leaving the right-moving fermionsψ i unchanged. So the only way for this gauge theory to be non-anomalous is for the U (1) charges to sum to zero. For example, the fermions can be paired up as ψ 2I−1 + iψ 2I and assigned the U (1) charges (−1) I , with I = 1, ..., 4.
In Dirac notation, the U (1) current for this embedding would be: The field ψ 8 cannot be deleted from the Lagrangian, because an odd number of real fermions cannot all be charged under U (1) .

IV Relation to electronic systems
From the perspective of experimental condensed matter physics, the Kitaev-Fidkowski interaction may seem somewhat foreign, in that it singles out "half" a fermionic degree of freedom. Although I do not propose any explicit experimental realization of this interaction, I do feel it would be useful to relate the model to another physical system with the same symmetries.
The basic required ingredients are four complex fermionic degrees of freedom. Two of these are already provided by spin, so one is interested in a problem with two degenerate "channels," or "flavors," of spinful fermions.
One such system is the two-channel Kondo effect [56,57], where itinerant conduction electrons scatter off a two-state impurity localized at the origin. It turns out that this system is not quite appropriate, because it possesses only the SO(5) × SO(3) subgroup of SO(8) instead of the larger SO (7) subgroup. However, the one-channel two-impurity Kondo effect [58,59] is another system with the correct number of degrees of freedom, and in this model the global symmetry is exactly the desired SO(7) subgroup which leaves a component of the 8 + fixed [60,30].

IV.1 Two-channel, one-impurity Kondo effect
It is conceptually simplest to begin with the two-channel, one-impurity Kondo problem. One has two channels, or flavors, of conduction electrons in three spatial dimensions, labeled by an index i = 1, 2. Each electron also has spin, labeled by α = ↑, ↓. The impurity is taken to have spin- 1 2 and is localized at the origin.
The scattering of the conduction electrons on the impurity is dominated by the = 0 angular momentum mode ("s-wave"), and hence can be reduced to a problem purely in the radial direction. Upon integrating over the angular variables, one is left with an effective 1+1 dimensional action on the half-line. In the low-energy theory, the residual effect of the impurity is to provide a boundary condition for the electronic degrees of freedom.
The complex left-handed fermion fields which describe the two flavors of electrons will be denoted as follows: (IV.1) There are 8⊗ A 8 = 28 different left-handed currents which generate infinitesimal SO(8) L transformations: In principle these SO(8) L currents can be written in terms of η a ∼ 8 v , ψ i ∼ 8 + , or χ x ∼ 8 − . The corresponding expressions j ab , j ij , and j xy are related by a triality transformation. I have chosen to work in the ψ i ∼ 8 + basis because, as will be seen shortly, if the physical conduction electrons are described as in Eq. (IV.1), then working in the ψ i ∼ 8 + basis will effect a generalized "spin-charge separation." Since iψ 2I−1 ψ 2I = i∂ x θ I and the θ I are related to the ϕ A in the same way as before, the N I can be expressed in terms of the conduction electrons as follows: Since this problem turns out to be very closely related to the scattering of 3+1 dimensional relativistic fermions from an SU (5) magnetic monopole [61,62] (see also [63]) -and in fact the two-impurity single-channel case is identical [30] -I will take the liberty of calling this fourth quantum number "baryon number." Thus for this problem the rotation to the 8 + basis describes the separation of charge, spin, flavor, and baryon number.
The effect of the impurity is to impose the following boundary condition on the scattering of a left-moving fermion into a right-moving fermion at the physical boundary From this it is clear that the symmetry of the problem is reduced from SO (8) to To make a connection with the Kitaev-Fidkowski model, the desired symmetry is the larger subgroup SO (7), and in particular that SO(7) which is defined by 8 + → 7 ⊕ 1. It turns out that this is exactly the symmetry group for the one-channel, two-impurity problem.

IV.2 One-channel, two-impurity Kondo effect
In this case, one has only a single channel of physical conduction electrons, again labeled by spin α = ↑, ↓. There are now two spin-1 2 impurities, distributed symmetrically about the origin, say at locations x = ± 1 2 R for some fixed constant vector R. Because of this spatial separation of the two impurities, the long-distance description is one of a single effective spin-1 impurity, which couples differently to the different parities of the conduction electrons. Linear combinations of parity-even and parity-odd eigenstates provide the "flavor" label, i = 1, 2.
Again the long distance theory reduces to an effective 1+1 dimensional theory of 8 Majorana fermions with a boundary condition at x = 0. In this case, the boundary condition is:  Hence the continuous global symmetry of the low-energy theory is SO (7). Since these boundary conditions are written, intentionally, in terms of the (ψ i ,ψ i ) variables, indeed the correct choice of SO (7) subgroup is singled out. This is exactly what happens for the KF interaction (up to a trivial relabeling of ψ 7 ↔ ψ 8 ): the U (1) rotations in the (7, 8)-plane are explicitly broken by the g term, which singles out the 8th component of the 8 + .
To relate these two systems literally would require a physical implementation of the interactions in Eq. (III.1). In terms of electronic degrees of freedom on the lattice, these can arise from a Hubbard-Heisenberg interaction (see, for example, [12]). Writing the interaction in this manner has the advantage of being expressed in terms of familiar physical variables, but it has the disadvantage of obscuring the SO(7) symmetry.
It is enlightening to observe that, in terms of the χ x ∼ 8 − fermions, one has: where I have defined new fields (IV.8) Evidently these new fields have the same charge, spin, and flavor quantum numbers as the original electron fields, but their baryon number is flipped: theẽ αi are the "antibaryons" of the e αi . So if U (1) B is broken by the interactions, then the e αi and theẽ αi will carry exactly the same quantum numbers. As discussed previously, these are the states which combine together to form a propagator proportional to p/(p 2 +m 2 * ) for some m 2 * = 0.

V Discussion
In this paper I have emphasized a subtlety in the triality invariance of the SO(8) Gross-Neveu model (Sec. II) and studied the SO(7) Kitaev-Fidkowski model along the "m = 0" manifold (Sec. III). The purpose was to obtain a more thorough understanding of the latter model in the continuum limit and to extract lessons for interacting relativistic quantum field theories in higher dimensions. The main observation was that the fermion propagator should exhibit a form of "parity doubling" for which states of equal and opposite mass parameters conspire to give a numerator proportional to γ µ p µ while maintaining the single-particle pole at p 2 = −m 2 kink = 0. This was motivated by the known spectrum of the SO(8) GN model and the conclusion that the physical Z 2 symmetry is not broken spontaneously.
Since I cannot imagine any other possibility consistent with the known results for SPT phases as well as with the principles of relativistic quantum field theory, I conjecture that the fermion propagator in Eq. (III.21) should describe the general situation: when a topological superconductor with Z classification can be reduced by interactions to some Z k , then the Hilbert space of the theory along the "m = 0" must be doubled. It seems necessary to include states for both projections in the Dirac operator, one for a mass term +m and one for a mass term −m. That is how a relativistic fermion can obtain mass without breaking any symmetry which forbids the mass terms in the Lagrangian.
I will conclude by proposing a novel application of the reduced classification of SPT phases to the study of elementary particle physics. It was already recognized that this phenomenon could be considered as a way to evade certain fermion doubling theorems on the lattice [67]. However, there exists at least one example in which unwanted fermion doubling occurs purely within a field theoretic framework without any reference to a discretization of space.
In an attempt to unify not only the nuclear and electromagnetic forces in a grand unified SO(10) theory [68,69,70], but also to combine the three generations of fundamental fermions into a single representation of a larger gauge group, a grand unified theory of families was proposed based on the gauge group SO(18) [48,71]. In this model, all known fermions could fit into a single 256-dimensional chiral spinor representation of SO (18), and there was some hope that the peculiar repetitive family structure of the SM could be explained by group theory.
However, the desired property that spinors of SO(2n + 2m) contain spinors of SO(2n) also proved to be the main phenomenological flaw of this approach: under the breaking of SO (18) to SO(10), the chiral spinor of SO(18) splits into the desired families as well as "mirror" families with the opposite quantum numbers (see, for example, [72]). The problem was to explain why this mirror matter is not observed at low energy.
Since the study of interacting symmetry protected topological superconductors has suggested that the SO(10) mirror fermions can likely be gapped out without generating a mass for the ordinary fermions, it is possible that this new insight from condensed matter theory may revive the SO(18) model. This would be an interesting problem to work out in detail, but it is clearly beyond the scope of this paper.
To capture the correct low-energy physics, it is necessary to expand in the vicinity of both of these points. Therefore, the appropriate Fourier decomposition is: For the c j to be hermitian, the η operators must satisfy: From this it is clear that the "m > 0" and "m < 0" phases can be realized from the appropriate tuning of J 2 relative to J 1 . In particular, the "m = 0" manifold is realized when J 2 = J 1 .
The Z n subgroup of U (1) consists of those U (1) transformations for which θ = 2π/n. Therefore: Z n : H → e in( 2π n ) H = H , (B.4) or, in terms of the amplitude and phase, Z n : ρ → ρ , ϕ → ϕ + 2π . (B.5) In this notation it is clear that H can obtain a nonzero vacuum expectation value without breaking the Z n subgroup of U (1).
The amplitude mode will obtain mass and decouple from the low energy theory. One can write ρ ≈ v and hence: Now "dualize" the periodic scalar, ϕ. What this means is to perform the following three steps: 1) Replace the 1-form "field strength" dϕ by a new 1-form f subject to the constraint df = ν defect . The 2-form ν defect measures the number density of singular configurations on which partial derivatives do not commute.
3) Replace f by its equation of motion to arrive at a new Lagrangian written purely in terms of B(x) and the U (1) gauge potential A(x) = A µ (x) dx µ .
The result of this procedure is: In the low-energy limit 7 v → ∞, one finds a "BF" theory, with B a 0-form and F = dA.