Commuting-projector Hamiltonians for 2D topological insulators: edge physics and many-body invariants
Jun Ho Son Jason Alicea






Department of Physics, Stanford University, Stanford, CA 94305, USA


Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA




Inspired by a recently constructed commuting-projector Hamiltonian for a two-dimensional (2D) time-reversal-invariant topological superconductor [Wang et al., Phys. Rev. B 98, 094502 (2018)], we introduce a commuting-projector model that describes an interacting yet exactly solvable 2D topological insulator. We explicitly show that both the gapped and gapless boundaries of our model are consistent with those of band-theoretic, weakly interacting topological insulators. Interestingly, on certain lattices our time-reversal-symmetric models also enjoy 𝒞𝒫 symmetry, leading to intuitive interpretations of the bulk invariant for a 𝒞𝒫-symmetric topological insulator upon putting the system on a Klein bottle. We also briefly discuss how these many-body invariants may be able to characterize models with only time-reversal symmetry. 



Commuting-projector Hamiltonians for 2D topological insulators: edge physics and many-body invariants
    Jason Alicea
    December 30, 2023
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§ INTRODUCTION
 
 
 Commuting-projector Hamiltonians consist of sums of local terms that commute with each other. In these models, both ground states and excited states can be obtained simply by finding simultaneous eigenstates of all local terms—typically implying exact solvability despite strong interactions among microscopic degrees of freedom.   
Commuting-projector models have yielded great insights into interacting, gapped topological phases of matter. 
 Historically, the two now-canonical commuting-projector models—Kitaev quantum double <cit.> and Levin-Wen string-net models <cit.>—laid important cornerstones for the study of bosonic topologically ordered phases. More recently, various commuting-projector models for fermionic topologically ordered states and symmetry-protected topological phases <cit.> have been constructed, establishing concrete lattice models of topological phases predicted by more abstract formalisms. 
 
 Despite the remarkable recent progress, commuting projector models for fermionic topological phases with both antiunitary symmetry and continuous on-site symmetry have, to our knowledge, so far eluded construction. The first goal of the paper is to build a commuting-projector model for the most well-known example of such phases: a two-dimensional (2D) topological insulator with time-reversal symmetry 𝒯 and U(1) particle conservation, also known as a quantum spin Hall system (we use both names interchangeably in this paper). Topological insulators are one of the first-discovered symmetry-protected topological phases and have been extensively explored via band theory.  Nevertheless, it has remained unclear how such states can emerge in a local lattice model outside of band-theoretic frameworks. 
 
 We will show that decorating Ising-spin domain walls with two Kitaev chains <cit.> in a time-reversal symmetric and particle-conserving manner (instead of a single Kitaev chain as done in Ref. Wang2017) allows one to construct commuting-projector models for 𝒯-symmetric topological insulators. Our model, though strongly-interacting, possesses the same symmetry and edge properties of band-theoretic quantum spin Hall insulators. In particular, to study gapless edge states of our model, we derive a strictly one-dimensional (1D) Hamiltonian whose low-energy physics is identical to that of fully symmetric quantum-spin-Hall edge states.  This Hamiltonian constitutes a generalization of a 1D model that recently appeared in Ref. jones2019. 

 
 

 
 We further point out that the exactly solvable models for  𝒯-invariant topological superconductors and insulators also possess 𝒞𝒫 symmetry when defined on certain lattices (here 𝒞 and 𝒫 respectively denote charge conjugation and spatial reflection symmetries). Recent studies have explored 𝒞𝒫-protected topological phases <cit.>.  While 𝒞𝒫 symmetry is rather unnatural in realistic condensed-matter setups, many topological phases are described by Lorentz-symmetric field theories in the infrared even though they emerge from non-relativistic settings; moreover, in relativistic theories 𝒞𝒫 is equivalent to 𝒯 due to the 𝒞𝒫𝒯 theorem. Hence, studying 𝒞𝒫-invariant topological phases may shed light on how to understand 𝒯-invariant topological phases as well. We will show that our 𝒞𝒫-symmetric models can be defined on a Klein bottle, upon which many-body invariants that characterize 𝒞𝒫-protected topological phases obtain intuitive interpretations. We also comment on possible applications of these ideas to 𝒯-symmetric topological insulators without any exact 𝒞𝒫 symmetry built in.
 
The rest of the paper is organized as follows.  Section <ref> briefly reviews the commuting-projector model for 𝒯-invariant topological superconductors <cit.> and then generalizes the construction to 2D topological insulators. Sections <ref> and <ref> explore gapped and gapless edge phases in our commuting-projector topological-insulator Hamiltonians, demonstrating consistency with band-theoretic phenomenology for weakly interacting topological insulators.  We explain how to define our models on a Klein bottle and how to compute topological invariants in Sec. <ref>.  Finally, concluding remarks appear in Sec. <ref>.
 


§ THE MODEL





 §.§ Symmetry actions in a Majorana representation




 
 In this paper it is convenient to express models in terms of Majorana fermions rather than complex fermions.  Hence we review how U(1), 𝒯, and 𝒞𝒫 symmetries act on Majorana operators.
We start with the first two symmetries, which are relevant to time-reversal-symmetric topological insulators. 
Let f_i,s denote fermion operators for spatial index i and spin s = ↑,↓.  
Symmetries U_α∈U(1) and 𝒯 act on these operators as follows:

    U_α :  f^†_i,s→ e^iαf^†_i,s,    f_i,s→ e^-iαf_i,s


    𝒯 :     f^†_i,↑→ f^†_i,↓,    f_i,↓^†→ -f_i,↑^†
        f_i,↑→ f_i,↓,    f_i,↓→ -f_i,↑^†.


When acting on individual fermion operators, time-reversal and charge conservation symmetries satisfy

    𝒯^2 = -1,    U_α𝒯 = 𝒯U_-α.

Non-commutation between 𝒯 and U(1) encoded in the second relation reflects antiunitarity of 𝒯.   
Hence, a topological insulator is often denoted as protected by U(1)⋊𝒯, where the semidirect product emphasizes the above relation. 
 
 Now we define two Majorana operators associated with each f_i,s operator via

    γ_1,i,s = f_i,s+f_i,s^†/2,     γ_2,i,s = f_i,s^†-f_i,s/2i.

One can then show that U_α and 𝒯 transform Majorana operators according to 

    U_α : [ γ_1,i,↑; γ_2,i,↑; γ_1,i,↓; γ_2,i,↓ ]→   M_α[ γ_1,i,↑; γ_2,i,↑; γ_1,i,↓; γ_2,i,↓ ]
       𝒯 : [ γ_1,i,↑; γ_2,i,↑; γ_1,i,↓; γ_2,i,↓ ]→ M_𝒯[ γ_1,i,↑; γ_2,i,↑; γ_1,i,↓; γ_2,i,↓ ]
 
where

    M_α = [  cosα -sinα     0     0;  sinα  cosα     0     0;     0     0  cosα -sinα;     0     0  sinα  cosα ]
       M_𝒯 = [  0  0  1  0;  0  0  0 -1; -1  0  0  0;  0  1  0  0 ].

The matrices M_α and M_𝒯 satisfy relations analogous to Eq. (<ref>), i.e.,

    M_𝒯^2 = -1,   M_α M_𝒯 = M_𝒯 M_-α.

Thus, one may take the above relations as  defining properties—that is, the SO(2) transformation matrix on Majorana operators M_α and time-reversal transformation matrix M_𝒯 encode the correct symmetries of a 2D topological insulator provided Eq. (<ref>) holds.


 
 One can similarly encapsulate symmetries of a U(1) ⋊𝒞𝒫 topological insulator in real matrices representing the symmetry action on Majorana operators. Complex fermion operators transform under 𝒞𝒫 as

    𝒞𝒫 :  f^†_i,s→ f_-i,s,    f_i,s→ f_-i,s^†;

here -i in the subscript denotes the spatial index obtained from reflecting site i with respect to some axis of our choice.  


The U_α and 𝒞𝒫 symmetries satisfy

    (𝒞𝒫)^2 = 1,    U_α𝒞𝒫 = 𝒞𝒫U_-α.

Here non-commutativity between 𝒞𝒫 and U_α comes from the fact that 𝒞𝒫 exchanges particles and holes, which acquire opposite U(1) phases.  



In the Majorana representation 𝒞𝒫 sends

    𝒞𝒫 : [ γ_1,i,↑; γ_2,i,↑; γ_1,i,↓; γ_2,i,↓ ]→   M_𝒞𝒫[ γ_1,-i,↑; γ_2,-i,↑; γ_1,-i,↓; γ_2,-i,↓ ]
 
with


    M_𝒞𝒫 = [  0  0  1  0;  0  0  0 -1;  1  0  0  0;  0 -1  0  0 ].

As expected, M_α and M_𝒞𝒫 satisfy relations akin to Eq. (<ref>),

    M_𝒞𝒫^2 = 1,    M_αM_𝒞𝒫 =  M_𝒞𝒫 M_-α,

which may be taken as the defining property of the symmetry transformations.
 





 §.§ Review of the commuting-projector Hamiltonian for 𝒯-invariant topological superconductors


 We briefly review the exactly solvable model for 𝒯-invariant 2D topological superconductors introduced in Ref. Wang2017; our commuting-projector Hamiltonian for 2D topological insulators naturally extends this model as we will see in Sec. <ref>.  

For brevity and ease of generalization to 𝒞𝒫-protected topological phases, throughout the main text we focus on models constructed on the honeycomb lattice. We emphasize, however, that the Hamiltonians can be easily generalized into arbitrary trivalent lattices.
 
The degrees of freedom in the 2D topological superconductor model are two spinful Majorana fermions per honeycomb-lattice edge and one spin-1/2 per plaquette.  See Fig. <ref>(a) for an illustration.  In an equivalent picture that we will frequently exploit, one can modify the honeycomb lattice by replacing each vertex with a small triangle, generating the Fisher lattice sketched in Fig. <ref>(b); the two Majorana fermions at each honeycomb-lattice edge can then be viewed as living on Fisher-lattice vertices.  We label Pauli operators for the spin at plaquette p by σ^x_p and σ_p^z, and denote the Majorana operators at Fisher-lattice vertex v by γ_v,s, where s = ↑,↓ labels spin. 
 

 
Due to subtleties with global fermion parity, defining the model consistently requires specifying a Kasteleyn orientation on the the Fisher lattice. Kasteleyn orientations are defined as a choice of arrows on the lattice that satisfies the following condition, often denoted as the `clockwise-odd rule': Around any closed clockwise cycle, there are an odd number of clockwise-oriented arrows. 
(See Refs. Tarantino2016 and Ware2016 for a detailed discussion.)  We adopt the following convention:


  * As in Ref. Wang2017, let `long edges' denote Fisher-lattice edges derived from the original honeycomb lattice.  Moreover, label the two honeycomb sublattices by A and B [colored red and green, respectively, in Fig. <ref>(b)].  
Arrows on all long edges point from the A sublattice to B sublattice.  

  * Let `short edges' denote the edges of the small triangles in the Fisher lattice.  Arrows on short edges orient clockwise on going around any small triangle. 

Figure <ref>(b) illustrates the resulting Kasteleyn orientation. 


The Hamiltonian,

    H_TSC = -∑_t A_t -∑_p B_p,

consists of vertex terms A_t defined for each vertex t of the honeycomb lattice (corresponding to a small triangle on the Fisher lattice, hence the subscript t) along with plaquette terms B_p. Before explaining these terms in detail, we define a sense of `pairing' of Majorana fermions belonging to Fisher-lattice vertices v and v' linked by either a long or short edge: γ_v,s and γ_v',s' are paired in a state |ψ⟩ if iγ_v,sγ_v',s'|ψ⟩ = g_vv'|ψ⟩.  Here g_vv' = 1 if the Kasteleyn arrow points from v' to v; otherwise g_vv' = -1. 
 
 

 
 The role of A_t is to give an energy advantage whenever Ising spin configurations around the vertex (i.e., σ_p^z values around the vertex) and Majorana pairings are consistent with the local rules illustrated in Fig. <ref>. Some key features of these local rules are the following:
 

  * The local rules enforce Majorana pairing along short edges only when there is a domain wall of Ising spin configuration along it. Away from such domain walls, Majoranas always pair along long edges. One can alternatively understand short-edge pairings as decorating Kitaev chains along domain walls. 

  * Majorana fermions with the same spin pair along long edges.  Meanwhile, short-edge Majorana pairings are always between opposite spins.

  * The Majorana spins involved in short-edge pairings also depend on the adjacent Ising spins.  In particular, the upper and lower panels for each sublattice in Fig. <ref> are related by an Ising-spin flip, and have Majoranas with opposite spin indices paired.
 
After defining the projector 

    P_vs,v's' = 1+ig_vv'γ_v,sγ_v',s'/2,

which projects onto the state where γ_v,s and γ_v',s' are paired, one can explicitly express A_t as

    A_t = ∑_u_t𝒜_u_t𝒮_u_t
       𝒜_u_t = ( ∏_(vs,v's') ∈𝒫_u_t P_vs,v's') .

Here u_t denotes an Ising spin configuration around the honeycomb vertex t; the u_t sum runs over all eight possible Ising spin configurations. The factor 𝒮_u_t projects onto states with Ising spin configuration u_t. And 𝒫_u_t is a set whose elements are pairings (vs,v's') enforced by the Ising spin configuration u_t according to the local rules. Simply stated, A_t =1 if the state is consistent with the local rules illustrated in Fig. <ref>, while A_t=0 on all other states.

Plaquette terms B_p allow Majorana pairings and the plaquette spin to fluctuate in a way that preserves the local rules if they are satisfied initially.  These terms read 

    B_p = ∑_u_pℬ_u_pσ_p^x𝒮_u_p
       ℬ_u_p = 1/√(2)( ∏_(v_1s_1,v_2s_2) ∈𝒫_u_p√(2) P_v_1s_1,v_2s_2) 
       ( ∏_(v_3s_3,v_4s_4) ∈𝒫_u_p P_v_3s_3,v_4s_4) .

In the above equations u_p denotes an Ising spin configuration for plaquette p together with the six neighboring plaquettes. Moreover, 𝒮_u_p projects onto Ising spin configuration u_p; 𝒫_u_p denotes a set of Majorana pairings consistent with u_p; and 𝒫_u_p is a modified set of Majorana pairings consistent with the spin configuration in which σ^z_p is flipped.  
Thus, the bosonic part σ_p^x𝒮_u_p flips the plaquette spin, and the two strings of Majorana projectors in ℬ_u_p project onto the state with Majorana pairings consistent with the new Ising spin configuration. Finally, some integer powers of √(2) are added to normalize B_p to have an eigenvalue ± 1 and 0 on entire Hilbert space. See Fig. <ref> for an illustration of the action of B_p.



 One can explicitly show that all A_t and B_p operators commute with each other. Also, B_p is Hermitian and unitary on the subspace in which A_t=1. Hence, on this subspace, B_p^2 = 1. The ground state can be easily constructed by diagonalizing each term, and corresponds to an equal superposition of each Ising spin configuration accompanied with correct Majorana pairings.  

 Finally, let us discuss symmetry of the Hamiltonian.  Denote a vertex of the Fisher lattice adjacent to sublattice A and B of the honeycomb lattice as v_A and v_B, respectively.  We can define an antiunitary symmetry 𝒯_TSC that acts on Majorana operators and Ising spins according to  

    𝒯_TSC :    σ_p^z→ - σ_p^z,   σ_p^x→σ_p^x,
    
       [ γ_v_A,↑; γ_v_B,↑; γ_v_A,↓; γ_v_B,↓ ]→[  0  0  1  0;  0  0  0 -1; -1  0  0  0;  0  1  0  0 ][ γ_v_A,↑; γ_v_B,↑; γ_v_A,↓; γ_v_B,↓ ]

Notice that the action on Majorana fermions is identical to how 𝒯 is defined in Eqs. (<ref>) and (<ref>). One can prove that the Hamiltonian preserves 𝒯_TSC. The following observation comes in handy for the proof: for all possible Majorana pairs γ_v,s and γ_v's' that can be paired due to the local rules, only one acquires a minus sign under time-reversal. An additional minus sign from complex conjugation results in a transformation of iγ_v,sγ_v',s' to iγ_v,sγ_v',s', where the overline denotes an opposite spin index.





 §.§ Extension to 𝒯-invariant topological insulators



Next we will generalize the model reviewed in the previous subsection to construct a commuting-projector Hamiltonian for the celebrated quantum spin Hall insulator. As a first step we double the number of Majorana fermions per Fisher-lattice vertex from two to four.  The corresponding Majorana operators on vertex v are denoted by γ_v,s,a, the subscript a=1,2 being a `layer index'.  Essentially, we will just construct a Hamiltonian in which the ground state is decorated with two layers of Kitaev chains instead of one as in the topological superconductor case. 
 
The Hamiltonian once again takes the form 

    H_ TI = -∑_t A_t - ∑_p B_p.

Vertex and plaquette terms are straightforwardly modified to reflect that there are now two layers of Majorana fermions that pair identically according to the same local rules summarized in Fig. <ref>.
To this end, define the projector

    P_vs,v's'^TI = 1+ig_vv'γ_v,s,1γ_v',s',1/21+ig_vv'γ_v,s,2γ_v',s',2/2,

which is a product of two projectors in Eq. (<ref>), one for layer 1 and another for layer 2.
We then have 

    A_t = ∑_u_t𝒜_u_t𝒮_u_t
       𝒜_u_t = ∑_u_t( ∏_(vs,v's') ∈𝒫_u_t P_vs,v's'^TI) 𝒮_u_t.

and

    B_p = ∑_u_pℬ_u_pσ_p^x𝒮_u_p
       ℬ_u_p = 1/2∑_u_p( ∏_(v_1s_1,v_2s_2) ∈𝒫_u_p 2 P_v_1s_1,v_2s_2^TI) 
       ( ∏_(v_3s_3,v_4s_4) ∈𝒫_u_p P_v_3s_3,v_4s_4^TI) .

Aside from normalization factors in ℬ_p, the vertex and plaquette terms are identical to those in the topological superconductor model, but with P_vs,v's'→ P_vs,v's'^TI.  

















The Hamiltonian H_ TI exhibits the following three symmetries:


  * The time-reversal transformation 𝒯_TSC inherited from the original topological superconductor model applied to plaquette spins and Majorana fermions on both layers.  We will soon find out that this operation does not coincide with physical time-reversal symmetry of an electronic topological insulator—which we hereafter denote by 𝒯_TI.  

  * A ℤ_2 layer interchange symmetry ℒ, which transforms γ_v,s,1↔γ_v,s,2.  The Hamiltonian obviously preserves this symmetry since the layers are treated identically.  

  * A U(1) symmetry with elements U_α that acts on Majorana operators as

    [ γ_v,s,1; γ_v,s,2 ]→[  cosα -sinα;  sinα  cosα ][ γ_v,s,1; γ_v,s,2 ].

To see that this is actually a symmetry of the Hamiltonian, we observe that P_vs,v's'^TI is invariant under the above U(1) transformation due to a simple corollary of a lemma presented in Appendix <ref>.  Since all fermionic terms in the Hamiltonian appear in the form of P_vs,v's'^TI, invariance of P_vs,v's'^TI implies U(1) invariance of the whole Hamiltonian.


The transformations specified above imply that U_αℒ = ℒU_-α, which is reminiscent of the second relation satisfied by time-reversal symmetry in Eq. (<ref>).  Nevertheless, ℒ clearly is not the time-reversal symmetry appropriate for a quantum spin Hall insulator since it is unitary and obeys ℒ^2 = 1.  
Furthermore, although 𝒯_TSC is antiunitary and obeys 𝒯_TSC^2 =-1, this operation can not be the desired time-reversal symmetry either since it commutes with U(1). Instead, the correct time-reversal symmetry is 

    𝒯_TI = 𝒯_TSCℒ.
  
One can explicitly show that 𝒯_TI and U_α satisfy the relations given in Eq. (<ref>), and equivalently in Eq. (<ref>).



We close this subsection with two remarks.  First, defining time-reversal symmetry as layer interchange followed by a symmetry inherited from the topological superconductor may appear somewhat artificial. However, we will see in Secs. <ref> and <ref> that this choice allows one to see topological properties transparently, indicating that our choice is indeed physically correct. Second, our model has an extra ℤ_2 layer-interchange symmetry (or equivalently, 𝒯_TSC symmetry) that one may suspect plays a crucial role in topological properties.  It does not.  This `accidental' symmetry can be broken without altering the topological properties of the model as discussed later on.  

 



 §.§ 𝒞𝒫 symmetry









Both the 𝒯-symmetric topological insulator and topological superconductor models additionally possess 𝒞𝒫 symmetry.  The dashed line in Fig. <ref>(b) shows our choice of the reflection axis associated with 𝒫.  We note two important features of this reflection: First, it maps maps sublattice A onto sublattice B and vice versa. 
Second, in the preceding sections we adopted a `special Kasteleyn orientation' for which all arrows flip under this reflection.  

 
 For simplicity, let us first consider the topological superconductor model. The second point above suggests that naive parity symmetry is absent. However, one can consider the following operation:

    𝒞𝒫_TSC :    σ_p^z→ -σ_-p^z
       γ_v_A,↑→γ_-v_A, ↓
       γ_v_A,↓→ -γ_-v_A, ↑
       γ_v_B,↑→ -γ_-v_B, ↓
       γ_v_B,↓→γ_-v_B, ↑.

As in Sec. <ref>, the minus sign on the vertex and plaquette indices refers to the reflection operation. Note that since -v_A and -v_B respectively belong to sublattice B and A, (𝒞𝒫_TSC)^2 = 1 despite some minus signs picked up by Majorana fermions in the transformation. Equation (<ref>) essentially flips the plaquette spins and Majorana spins, spatially reflects, and adds some minus signs. 

 To see why 𝒞𝒫_TSC is a symmetry of the Hamiltonian, first observe that the vertex rules illustrated in Fig. <ref> are symmetric under spatial reflection and simultaneous flipping of plaquette spins and Majorana spins. This property is not sufficient, however, because we earlier observed that the spatial reflection flips the Kasteleyn-orientation arrows. The minus signs added in the transformation rule for Majorana fermions in Eq. (<ref>) remedy the issue, which can be seen as follows.  Long-edge pairing always pairs Majoranas with the same spin but opposite sublattice. The transformation rule dictates that precisely one of those Majorana fermions acquires an extra minus sign, thereby correcting the Kasteleyn-orientation flip from the spatial reflection. Similarly, short-edge pairing always pairs Majoranas on the same sublattice but with opposite spin; here too only one such Majorana fermion acquires a minus sign, again correcting the reversed Kasteleyn orientation.  
 
The topological-insulator model, which has doubled Majorana degrees of freedom and exhibits U(1) and ℒ layer-interchange symmetry, straightforwardly inherits 𝒞𝒫_TSC symmetry.  As we already encountered for time-reversal symmetry, however, 𝒞𝒫_TSC is not the symmetry appropriate for the quantum-spin-Hall setting—this operation satisfies neither Eq. (<ref>) nor the equivalent Eq. (<ref>).  We can nevertheless construct 𝒞𝒫_TI, the 𝒞𝒫 symmetry relevant to the 𝒞𝒫-protected topological insulator, by combining ℒ with 𝒞𝒫_TSC:

    𝒞𝒫_TI = 𝒞𝒫_TSCℒ.

Note the similarity to Eq. (<ref>).
One can explicitly show that 𝒞𝒫_TI^2 =1, and that the U(1) and 𝒞𝒫_TI action on Majorana degrees of freedoms satisfies Eq. (<ref>).



 
Let us briefly comment on the fate of 𝒞𝒫 symmetry beyond the honeycomb-lattice setting. Recall that the existence of 𝒞𝒫 symmetry above relied on the bipartite and reflection-symmetric nature of the lattice. We detail in Apppendix <ref> how 𝒞𝒫 symmetry may survive in models defined on any reflection-symmetric lattice (not necessarily bipartite) as well.  Conversely, if 𝒯-symmetric topological phases are defined on a non-parity-symmetric lattice, there cannot be any strict 𝒞𝒫 symmetry. We will, however, see in Sec. <ref> that some bulk diagnostic of non-trivial topology of 𝒞𝒫-symmetric phases still remains in such models despite the lack of 𝒞𝒫 symmetry. 
 
 




§ GAPPED, SYMMETRY-BREAKING EDGE PROPERTIES OF THE 𝒯-INVARIANT TOPOLOGICAL INSULATOR MODEL
 

 Quantum spin Hall insulators are protected by the interplay of U(1) and 𝒯 symmetries.  Hence, breaking either symmetry (e.g., via introduction of superconductivity or magnetism) suffices to gap out the edge, in turn enabling multiple inequivalent gapped edge phases.  Domain walls separating incompatibly gapped regions of the edges bind interesting zero-dimensional modes that have been widely studied, typically by assuming that the edge is governed by an effective model for helical 1D Dirac fermions.  In this section we show that our commuting-projector model reproduces precisely the same physics.  Interestingly, due to the exact solvability, these zero modes can be observed explicitly on the lattice level without resorting to an analysis of an effective low-energy edge description.



 

 


 §.§ Construction of broken-symmetry gapped edges



First, we discuss how one can break 𝒯 symmetry [but preserve U(1)] at the boundary without losing exact solvability or generating spurious gapless degrees of freedom.  Consider the `dangling-bond edge termination' shown in Fig. <ref>(a) in which boundary plaquettes are incomplete. In such a geometry, one can define vertex terms A_t precisely as before since all vertices remain trivalent.  Likewise, one can define B_p as before for all complete `bulk' plaquettes. For incomplete edge plaquettes indexed by p̃, we violate 𝒯 by polarizing edge spins with a Zeeman term 

    H_ Zeeman = h ∑_p̃σ_p̃^z.

Since H_ Zeeman clearly commutes with all other Hamiltonian terms, exact solvability is retained and the spectrum remains gapped. The ground state is readily constructed by freezing the edge spins to minimize the Zeeman term, then allowing bulk spins to fluctuate and Majorana pairings to follow the constraint given by A_t's. See Fig. <ref>(b) for an illustration.
  
 

 
 
 
We can also break U(1) at the edge while preserving time-reversal symmetry.  To do so we employ the same lattice termination as above, but now we remove the spins from all incomplete plaquettes instead of polarizing them via a Zeeman field.  Vertex and plaquette terms that invoked these eliminated spins in their definition must then be replaced.  We do so by imposing the following Majorana-pairing rule at the boundary: 
Majorana fermions on layer 1 pair as if the eliminated spins point down, whereas Majorana fermions on layer 2 pair as if the eliminated spins point up. Figure <ref>(c) shows an example of Majorana pairings consistent with this rule.
 
Since Majorana fermions in layers 1 and 2 couple differently, the Hamiltonian explicitly breaks both the layer-mixing U(1) symmetry and the layer-interchange ℒ symmetry. The Hamiltonian also breaks 𝒯_ TSC since (i) this symmetry acts separately on Majorana fermions within each layer and (ii) the boundary Majorana fermions behave as if the eliminated spins were polarized.  The modified rule above does, however, preserve 𝒯_TI = 𝒯_TSCℒ, emphasizing that this somewhat-arbitrary-looking composite operation is indeed the correct antiunitary symmetry that protects the topological insulator!

Note that we can alternatively construct a time-reversal-invariant, gapped boundary by pairing Majorana fermions as if the eliminated edge spins point up for layer 2 and down for layer 1 (opposite to what we described in the previous paragraph). In fact, a further generalization is possible.  Define γ_v,s,1(θ) and γ_v,s,2(θ) as 

    [ γ_v,s,1(θ); γ_v,s,2(θ) ] = [  cosθ/2  sinθ/2; -sinθ/2  cosθ/2 ][ γ_v,s,1; γ_v,s,2 ].

Essentially, ( γ_v,s,1(θ), γ_v,s,2(θ) ) correspond to a U(1)-twisted version of ( γ_v,s,1, γ_v,s,2).  We can construct a 𝒯-symmetric gapped edge for any θ by pairing γ_v,s,1(θ)'s as if the eliminated edge spins orient up and pairing γ_v,s,2(θ)'s as if the eliminated edge spins point down. The example at the beginning of the paragraph corresponds to θ = π.  Physically, θ corresponds to the phase of the superconducting order parameter that gaps the boundary.  The factor of 2 in the trigonometric functions in Eq. (<ref>) is important.  It follows that ( γ_v,s,1(2π), γ_v,s,2(2π) ) = ( -γ_v,s,1, -γ_v,s,2), though fermion-parity preservation ensures that the edge states corresponding to θ =0 and θ = 2π are identical.  The minus signs on the right side are in fact consistent with our identification of θ as the order-parameter phase.  Winding the phase by 2π is equivalent to dragging an h/(2e) superconducting vortex around the edge—which does not affect Cooper pairs but generates a minus sign for individual fermions.  
 
 

 
 






 
 


 §.§ Domain walls between incompatibly gapped edge phases





We now briefly review the physics of domain walls separating inequivalent gapped, broken-symmetry regions of the edge.  Perhaps most famously, the interface between a superconducting domain that breaks U(1) and a magnetic domain that breaks 𝒯 binds a single unpaired Majorana zero mode <cit.>.  Nontrivial domain walls are also possible even if either 𝒯 or U(1) is globally conserved.  Suppose that 𝒯 is globally conserved.  Here one can consider a 𝒯-invariant domain wall between two U(1)-breaking edge regions, one with a superconducting-order-parameter phase θ_sc = 0 and another with θ_sc = π. This π-junction domain wall hosts a Kramers pair of Majorana zero modes—which is intimately related to the fractional Josephson effect <cit.>.  Now imagine that U(1) is globally conserved.  Since 𝒯 is a discrete symmetry, 𝒯-breaking edge phases come in pairs that are time-reversed partners of each other.  In an experimental context these time-reversed partners correspond to helical edge modes gapped via a proximate ferromagnet that magnetizes either up or down; in our construction they simply reflect the two possible choices for the sign of the Zeeman field h in Eq. (<ref>) that polarizes the edge spins. A domain wall between oppositely magnetized regions of the edge hosts a (complex) fermion zero mode with fractional charge ± e/2 <cit.>.  Figure <ref> summarizes the structure of the three types of domain walls highlighted above.  



Within our framework, we can most readily capture the Majorana zero mode at a domain wall between superconducting and ferromagnetic edge phases.  To implement such a domain wall we break U(1) in one region of the edge and time-reversal symmetry in another, precisely as described in the preceding subsection.  For concreteness we arbitrarily polarize the spins up in the magnetized domain.  Recalling that Majorana fermions in layer 1 pair as if the eliminated boundary spins pointed up, one sees that the domain wall does not affect Majorana pairings in that layer.  On the contrary, Majorana fermions in layer 2 pair as if the eliminated spins pointed down—yielding an `effective magnetic domain wall' for layer 2 that does influence Majorana pairings, and in particular indeed traps the expected unpaired Majorana zero mode as sketched in Fig. <ref>(a). 

Consider next a 𝒯-invariant domain wall at which the superconducting-order-parameter phase jumps by π. In the previous subsection we established that the effective orientations of eliminated edge spins for layers 1 and 2 at θ = π are exactly opposite those at θ = 0.  Hence, there is now an effective magnetic domain wall for both layers. As Fig. <ref>(b) illustrates, we thereby obtain one Majorana zero mode from each layer that together form the expected Kramer's pair. 




We can also readily describe U(1)-invariant magnetic domain walls in our setup by simply polarizing the boundary spins up in one region and down in another via a position-dependent Zeeman field.  As seen in Fig. <ref>(c), our Majorana-pairing rules dictate that a magnetic domain wall binds two uncoupled Majorana zero modes γ_v,↑,1 and γ_v,↑,2—or equivalently one complex fermion zero mode f_0^† = γ_v,↑,1 + i γ_v,↑,2.  Under an element U_α of U(1), the complex fermion zero mode transforms as f_0^†→ e^iαf_0^†, signaling that f_0 carries a single unit of electric charge e.  Hence, if the magnetic domain wall binds charge q in a state with f^†_0 f_0 = 0, then the charge with f^†_0 f_0 = 1 must be q + e.  We will now argue that q = -e/2.  
 

 

 
As a primer, suppose that we completely gap out the boundary with a uniform Zeeman field (i.e., no magnetic domain walls).  To fix a notion of charge neutrality, it is useful to work with complex fermions f_v,s = γ_v,s,1 + i γ_v,s,2 that carry charge e and define number operators n_v,s = f_v,s^† f_v,s.  When Majoranas with spin s at vertex v pair with Majoranas with spin s' at vertex v', the state satisfies the constraint iγ_v,s,1γ_v,s',1 =  iγ_v,s,2γ_v,s',2 = ± 1. Translating into complex fermion language, one finds that configurations satisfying this constraint involve an equal superposition of a state with n_v,s = 1, n_v',s' = 0 and another state with n_v,s = 0, n_v',s' = 1.  This property indicates that when the edge is gapped out with a uniform Zeeman field (or when the system is defined without a boundary), the ground state of our model always resides exactly at half-filling.
   
 
 
  
Let us now examine a topological insulator defined on a manifold with a gapped boundary hosting a pair of magnetic domain walls, one hosting a complex fermion zero mode f_0, the other hosting f_0'.  The key observation is that exactly one of these zero modes must be occupied in the charge neutrality ground state.  The state with both zero modes vacant thus has charge -e relative to the ground state, while the state with both zero modes occupied has charge +e relative to the ground state.  Assuming symmetric charge assignments, we conclude that each magnetic domain wall binds the expected fractional charge of +e/2 or -e/2, i.e, q = -e/2 as claimed above.  


 


 


§ GAPLESS EDGE STATE OF THE 𝒯-INVARIANT TOPOLOGICAL INSULATOR



 In this section, we examine the dangling-bond edge-termination geometry in the limit where both time-reversal and U(1) symmetries are preserved everywhere.  The boundary is necessary gapless in this case, implying that the Hamiltonian can no longer consist solely of commuting projectors (which would imply a fully gapped spectrum).  Starting from our 2D model, we will nevertheless `peel off' a strictly 1D microscopic Hamiltonian whose low-energy physics exactly reproduces that of the familiar 2D topological insulator edge. Our effective 1D model and its physics can be viewed as a natural generalization of 1D models that recently appeared in Ref. jones2019 for symmetry-protected gapless edges of a 𝒯-symmetric topological superconductor and the Tarantino-Fidkowski model <cit.>.  However, our derivations and presentation differ from those of Ref. jones2019; we hope that these distinctions give readers complementary viewpoints on gapless edge states of fermionic symmetry-protected topological phases.  
 
 
 






 §.§ Symmetric edge termination


The degrees of freedom in the dangling-bond edge termination are identical to those of the 𝒯-breaking gapped boundary examined in Sec. <ref>: eight Majorana fermions per edge together with Ising spins for every plaquette, including incomplete ones at the edge.  Just as for the 𝒯-breaking case, vertex terms A_t require no modification.  The key addition here involves incomplete plaquette terms.  In the 𝒯-breaking construction, we polarized spins in incomplete plaquettes with Zeeman terms.  Instead, now we assemble B_p-like plaquette terms, denoted C_I for each incomplete plaquette I, that allow spins and Majorana pairings to fluctuate at the edge. 



 

 
Each incomplete-plaquette term is given by

    C_I  = ∑_u_I𝒞_u_Iσ_I^x𝒮_u_I
       𝒞_u_I = N_u_I( ∏_(v_1s_1,v_2s_2) ∈𝒫_u_I 2 P_v_1s_1,v_2s_2^TI) 
       ( ∏_(v_3s_3,v_4s_4) ∈𝒫_u_I  P_v_3s_3,v_4s_4^TI) .
 
Above, u_I refers to possible configurations of the five spins residing at incomplete plaquette I and the four surrounding plaquettes; and 𝒮_u_I projects onto spin configuration u_I; and  N_u_I is a normalization factor that we will specify shortly.  The set 𝒫_u_I consists of Majorana pairings enforced by spin configuration u_I according to the usual local rules, while 𝒫_u_I consists of pairings consistent with u_I but with σ^z_I flipped.  Hence, the product of projectors in 𝒞_u_I merely reconfigure the Majorana fermions, subject to the caveat below, such that they pair appropriately given the flipped incomplete-plaquette spin (the factor 2 is added for normalization). Figure <ref> illustrates the action of these incomplete-plaquette terms. 

The expression for C_I resembles the bulk plaquette term B_p from Eq. (<ref>). Two key differences do, however, arise:



  * As seen in Fig. <ref>, certain spin configurations before/after acting C_I keep some Majorana fermions unpaired;  the action of the usual bulk plaquette terms, by contrast, always keep all Majorana fermions paired.

  * Consequently, the normalization N_u_I must be more carefully chosen for C_I to be Hermitian, as opposed to the uniform normalization 1/2 for B_p in Eq. (<ref>). While more than one choice of N_u_I renders C_I properly Hermitian, we take

    N_u_I = 1/2   If u_I enforce four Majoranas to be unpaired
    
    1    else.





Due to the structure built into the incomplete-plaquette terms, C_I somewhat obviously commutes with all A_t vertex terms. A slightly less obvious fact is that C_I also commutes with all bulk plaquette terms B_p, which we investigate in detail in Appendix <ref>. We emphasize, however, that C_I's do not commute with themselves—precisely this non-commutativity underlies interesting gapless edge physics. 




 §.§ Derivation of the effective 1D model for the edge


 The fact that the low-energy physics is solely controlled by C_I terms that commute with all bulk terms suggests the possibility of `stripping away' the bulk degrees of freedom to obtain a 1D Hamiltonian describing the gapless boundary.  However, some difficulties arise in this procedure as we now briefly discuss.  
 
 The low-energy physics occurs in the subspace that satisfies the constraint A_t = B_p =1 for all vertices and all bulk plaquettes.  Consider states that strictly obey this constraint. The spin operator σ_I^z at an incomplete plaquette commutes with bulk terms, so we can further imagine states with a fixed boundary spin configuration. In the left panel of Fig. <ref>, two edge spins point up; there, bulk Majorana fermions in the grey area exhibit fluctuating pairings due to the B_p=1 constraint but never pair with edge Majoranas in the white area. One might therefore be tempted to derive an effective 1D edge models by simply throwing out Majoranas and spins in the grey region. However, the right panel of Fig. <ref> shows a different edge spin configuration which imposes pairing between Majorana fermions in the grey and white regions.  Thus one can not naively discard degrees of freedom in the grey area to derive a 1D edge model.  In fact, any choice of a `grey area'  specifying bulk degrees of freedom to be discarded similarly yields some states that entangle the `bulk' and `edge' degrees of freedom.  This conclusion reflects the familiar fact that the symmetric edge states of a quantum-spin-Hall insulator are anomalous and can not be sliced away from the accompanying bulk.    
 
 
 
 
 


Progress can nevertheless proceed if one sets out to construct a 1D lattice model whose physics at all scales may not be identical to the 2D Hamiltonian we consider, but whose physics in a certain low-energy sector certainly is. Here we only establish the basic ideas, relegating a more detailed discussion to Appendix <ref>. Consider the following family of Hamiltonians parametrized by x:

    H(x) = -a∑_t A_t -b∑_p( √(1-x^2)B_p + x σ_p^z) - c∑_I C_I

We keep a,b sufficiently larger than c so that the low-energy physics is controlled by the C_I terms.  The limit H(0) corresponds to our fully symmetric topological-insulator model. Increasing x from zero to one tunes the bulk to a trivial insulator at the cost of breaking 𝒯 symmetry. Note that σ_p^z also commutes with C_I, so here as well all bulk and edge terms commute. 






 One may naively expect that since H(x ≠ 0) does not possess exact time-reversal symmetry, the edge loses its nontrivial properties and becomes similar to the magnetically gapped edge discussed in Sec. <ref>. However, such a bulk transformation does not affect edge dynamics, so any H(x) exhibits identical low-energy physics to H(0). This conclusion follows from the fact that the bulk Zeeman terms commute with C_I's that give non-trivial edge dynamics. Hence, we are free to analyze H(1) instead, in which B_p's are completely replaced by σ_p^z's. In the low-energy subspace, all bulk spins freeze in one direction, and the bulk Majorana pairings become frozen as well.  One may then discard these degrees of freedom to study the edge physics. Making a further unitary transformations and discarding unimportant degrees of freedom (as also detailed in Appendix <ref>), we arrive at a simplified, translationally invariant 1D model with four Majorana fermions and one bosonic spin per unit cell. Figure <ref> pictorially summarizes this procedure. 
 
Let i denote 1D unit cells, and label the four Majorana fermions per unit cell as in Fig. <ref>.
Our 1D edge Hamiltonian can be written as 

    H^L = -∑_i A_i^L - ∑_i C_i^L.

(The meaning of the `L' superscripts will become apparent in the next subsection.)
The first terms are given by 

    A_i^L = ∑_s,t= ±𝒜_i, st^L1+sσ_i-1^z/21+tσ_i^z/2
       𝒜_i, –^L = 1+iγ_i-1,B,1^(1)γ_i,A^(1)/21+iγ_i-1,B^(2)γ_i,A^(2)/2
       𝒜_i, +-^L = 1+iγ_i-1,A^(1)γ_i-1,B^(1)/21+iγ_i-1,A^(2)γ_i-1,B^(2)/2
       𝒜_i, ++^L = 1+iγ_i-1,A^(1)γ_i-1,B^(1)/21+iγ_i-1,A^(2)γ_i-1,B^(2)/2
       𝒜_i, -+^L = 1.

The second terms read

    C_i^L = ∑_s,t,u = ±𝒞^L_i,stuσ_i^x1+sσ_i-1^z/21+tσ_i^z/21+uσ_i+1^z/2
       𝒞_i,stu^L = N_stu𝒜_i,st𝒜_i+1,tu𝒜_i,st𝒜_i+1,tu,

where N_stu takes an analogous role to N_u_I in Eq. (<ref>) and is defined as

    N_stu = 
    1    if s=+ and u=+
    
    2    else.




Similar to the vertex term of the original 2D model, A_i^L projects onto Majorana pairings consistent with bosonic spins.  The upper row of Fig. <ref> illustrates the relevant pairing rules.  Note that here we show the pairings imposed by A^L_i at a particular unit cell i corresponding to the rightmost spin in each column. 
In the space where A_i^L=1 is strictly enforced for all i's, Majorana fermions in each layer pair as in a non-trivial topological phase along spin-up domains, and pair as in a trivial phase in spin-down domains. The bottom of Figure <ref> illustrates an example of such pairings.  (In the upper row of Fig. <ref>, there are no green lines in unit cell i when its spin points up; these `missing' green lines are filled in by A^L_i+1 to produce a pattern like that in the last row of Fig. <ref>.)   
Domain walls therefore host two Majorana zero modes, reminiscent of the magnetic domain-wall structure discussed in Sec. <ref>.  The C_i term, derived from C_I of the original 2D model, flips a bosonic spin and modifies Majorana pairings by projection. Note that [A_i,A_j]= [C_i,A_j]=0, though different C_j's need not commute with each other. The low-energy physics is fully captured by the subspace in which A_i=1 is strictly observed.
 
 


 §.§ 𝒯-symmetry in the effective 1D edge model



 
Our 1D Hamiltonian H^L still exactly preserves U(1) symmetry, but is not invariant under 𝒯.  In fact, in our derivation we explicitly broke 𝒯-symmetry by polarizing bulk spins and threw out Kramers partners of Majoranas in the stripped-down 1D Hamiltonian.  A remnant of 𝒯 symmetry nevertheless persists in this model as nicely explained in Ref. jones2019; here, we recast their findings in a slightly different language.  

Define H^R, the `right-handed version' of our original `left-handed' H^L, such that 𝒜_i,ab^L is modified to 𝒜_i,ab^R, defined as follows:

    𝒜_i, –^R = 1+iγ_i-1,B,1^(1)γ_i,A^(1)/21+iγ_i-1,B^(2)γ_i,A^(2)/2
       𝒜_i, +-^R = 1
       𝒜_i, ++^R = 1+iγ_i,A^(1)γ_i,B^(1)/21+iγ_i,A^(2)γ_i,B^(2)/2
       𝒜_i, -+^R = 1+iγ_i,A^(1)γ_i,B^(1)/21+iγ_i,A^(2)γ_i,B^(2)/2.

We illustrate this modification in Fig. <ref> as well. 
Interestingly, the following series of transformations map H_L to H_R:


  * Global bosonic spin flip

  * Kramers-Wannier-like half-unit-cell Majorana translation combined with layer exchange, formally written as

    γ_i,A^(1)→ -γ_i,B^(2),   γ_i,A^(2)→ -γ_i,B^(1)
    γ_i,B^(1)→γ_i+1,A^(2),    γ_i,B^(2)→γ_i+1,A^(1)


  * Unitary transformation ∏_i U_i, where

    U_i = ∑_ab = ±𝒰_i,ab1+aσ_i-1^z/21+bσ_i^z/2
       𝒰_i,ab = 1+γ_i-1,B^(1)γ_i,B^(1)/√(2)1+γ_i-1,B^(2)γ_i,B^(2)/√(2)     a=-, b=+ 
    
    1    else

This unitary operator transforms γ_i-1,B^(1)→ -γ_i,B^(1), γ_i,B^(1)→γ_i-1,B^(1) and similarly for Majoranas on the second layer, but only when σ_i-1^z=-1 and σ_i^z =+1.

  * Complex conjugation.

One can check term-by-term to prove our claim. 
We argue that the above operations correspond to the implementation of 𝒯 symmetry in our 1D model.

In the low-energy sector in which A_i^R/L=1 is strictly observed, H^L and H^R are completely identical: they exhibit identical many-body spectra, and each energy level is described by the same wavefunction.  Thus within this subspace the 1D model preserves 𝒯.  Due to differences in details of A_i's, however, this correspondence does not hold outside of this subspace.  The situation is reminiscent of particle-hole symmetry in the lowest-Landau level of a 2D electron gas; such a symmetry clearly breaks down in the full unprojected Landau-level space. 

Although the 𝒯-symmetry implementation is highly non-local due to Majorana-translation operations, it is explicitly anti-unitary, and also possesses correct commutation relations with U(1) symmetry due to the fact that the Majorana translation additionally swaps the layer index. Hence, this nonlocal symmetry is valid to be viewed as an incarnation of physical time-reversal symmetry for the quantum-spin-Hall edge. Since our 1D model is exactly derived from a 2D quantum-spin-Hall Hamiltonian, 𝒯 is expected to be realized only in an anomalous fashion; in our case, this `anomaly' is manifest via the non-local nature of the symmetry.  Note that either 𝒯 or U(1) must be realized anomalously, but not both.  Indeed, U(1) admits a simple local implementation in our 1D model similar to the action in the full 2D Hamiltonian.  
 
 Finally, we have so far discussed 𝒯-symmetry only in the context of an infinitely long 1D chain. It is often more convenient to study a finite-size 1D model  with periodic or anti-periodic boundary conditions; more precisely, for an L-site chain, the projector enforcing pairing between γ_L,B and γ_1,A is taken to be (1± i γ_L,Bγ_1,A)/2, where the + and - signs respectively correspond to periodic and anti-periodic boundary conditions. With periodic boundary conditions, a straightforward generalization of 𝒯-symmetry we wrote down, whereby one identifies sites L+1 and 1, suffices. In the case of anti-periodic boundary conditions, however, 𝒯 should be slightly modified as follows: In the half-unit-cell Majorana translations, γ_L,B^(1) and γ_L,B^(2) at the very end of the chain respectively transform to -γ_1,A^(2) and -γ_1,A^(1), i.e., with a minus sign. Also, to account for the minus sign, at the last step, the unitary transformation U_L^†∏_i=1^L-1U_i should be applied instead—in particular, the unitary at the last site L is modified to its inverse. 
 


 






 §.§ Numerical results


We are unaware of any exact analytical solution to H^L or H^R.  Focusing on the restricted subspace that satisfies A_i = 1 for all i, we thus instead employ exact diagonalization to study our 1D model defined on an L-site chain with either periodic or anti-periodic boundary conditions.  



 

 

Figure <ref> plots the many-body energy levels versus the total momentum k for an L  = 14 system with anti-periodic (upper panel) and periodic (lower panel) boundary conditions.  
Eigenstates with global even fermion parity and odd fermion parity are shown.  

In both boundary conditions, the spectra are perfectly symmetric with respect to k=0—hence `right-movers' and `left-movers' are completely symmetric as expected from a 𝒯-invariant quantum-spin-Hall edge. Furthermore, the dispersion of the lowest-energy states in each momentum near k=0 is crudely linear, indicating the Dirac-fermion nature of the excitations and allowing one to identify these lowest-lying levels as single-particle states. Other parts of the spectra, however, cannot be explained by a free Dirac fermion, indicating more complex Luttinger-liquid physics arising from interaction. Since the low-energy conformal field theory is unlikely to be a minimal model, there can be an infinite number of primary operators; hence, it is difficult to read off the full identity of the low-energy conformal field theory just from the spectrum.
 
We also point out some interesting degeneracies that appear in the spectra. For the periodic-boundary-condition spectrum from Fig. <ref>, energy eigenstates at each momentum are two-fold degenerate.  This degeneracy can be explained by some extra symmetries, apart from U(1) and 𝒯, that our Hamiltonian possesses.  Specifically, our Hamiltonian separately preserves the fermion parities F_1 and F_2 for layers 1 and 2, defined by

    F_a=1,2 = ∏_i=1^n i γ_i,A^(a)γ_i,B^(a).

Additionally, one can define a unitary modification of 𝒯, which we denote 𝒯_m, by removing complex conjugation operations and modifying the Majorana translation/layer-interchange from item 2 in the numbered list of the previous subsection as follows:

    γ_i,A^(1)→γ_i,B^(1),   γ_i,A^(2)→γ_i,B^(2)
    γ_i,B^(1)→γ_i+1,A^(1),   γ_i,B^(2)→γ_i+1,B^(2).
 
One can explicitly show that 𝒯_m also maps H^L to H^R and hence is a valid symmetry on the restricted subspace of interest. Since 𝒯_m commutes with a unit-cell translation but does not involve complex conjugation, it does not change momentum of an eigenstate. Finally, 𝒯_m changes the sign of both F_1 and F_2 due to the half-unit-cell Majorana translation. It follows that energy eigenstates at a given momentum always come in degenerate pairs, each one having opposite signs of F_1 and F_2.  When applied to the k = 0 ground state, we see that a zero-momentum, zero-energy mode is guaranteed to exist. 
 
For a chain with anti-periodic boundary conditions, 𝒯_m should be modified in the same manner as we modified 𝒯 at the last part of the previous subsection. This modified symmetry action now preserves F_1 and F_2, and hence energy eigenstates at a given momentum are not always two-fold degenerate. However, any eigenstate with F_1≠ F_2 is still two-fold degenerate because of layer-interchange symmetry.  Such F_1≠ F_2 states [red circles in Fig. <ref>, upper panel] exhibit odd total fermion parity, carry momentum k=2π/L(n+1/2) for integer n, and are indeed doubly degenerate in our numerics.  Even-fermion-parity states, by contrast, have F_1 = F_2 and momentum k=2π n/L; layer-interchange symmetry clearly does not imply degeneracy for those states; in particular, the ground state at k=0 is unique.
This degeneracy can also be understood as one state being the `particle-hole conjugate' of another.  While odd-fermion states always transform nontrivially under the particle-hole-like transformation, states with the integer momentum k=2π n/L can be symmetric under that transformation and thus can exist without degeneracy, particularly when the states represent neutral excitations.
 


As an additional check, we extract the central charge c by computing the entanglement entropy of the ground state with anti-periodic boundary conditions at L=16. It is known that if the system's low-energy physics is described by a conformal field theory, the entanglement entropy of a subsystem of size l takes the form <cit.>

    S(l) = c/3lnL/πsin( lπ/L).

Figure <ref> shows the entanglement entropies versus x(l) = 1/3lnL/πsin( lπ/L) for l = 1,…, 8.  Performing a linear fit for the l = 5,6, 7, 8 data yields the relation S(l) ≈  1.0285 x(l) + 1.1366 (see dashed line in the figure), consistent with central charge c = 1 expected for a helical Luttinger liquid arising at a quantum-spin-Hall edge.  One can in principle perform other consistency checks such as extracting anomalous fermion scaling dimensions by computing fermion correlation function; we leave more detailed numerical studies of the 1D model for future work.






§ TOPOLOGICAL SUPERCONDUCTOR AND INSULATOR ON A KLEIN BOTTLE: CONSTRUCTION AND MANY-BODY INVARIANTS



The topological-superconductor construction presented in Ref. Wang2017, along with our topological-insulator generalization, implicitly assume that the manifold on which the models are defined is orientable.  In this section, we first discuss how to adapt these models to a Klein bottle manifold in a 𝒞𝒫-symmetric manner. 
Then, we compute many-body topological invariants for 𝒞𝒫-symmetric topological phases <cit.> by changing the boundary condition along a non-contractible cycle and show that they acquire non-trivial values. We finalize the section with comments on how many-body invariants originally devised for 𝒞𝒫-symmetric topological phases may work for models with only 𝒯 symmetry.
 


 §.§ Defining the lattice models on a Klein bottle


 Here, we illustrate some difficulties in naively defining our lattice models on a Klein bottle and how to overcome them. We focus on the topological superconductor; the generalization to the topological insulator is straightforward and will only be briefly discussed at the end. Imagine putting the honeycomb-lattice model on a rectangle as in Fig. <ref>(a).  The two vertical edges of the rectangle are identified in the standard way, as indicated by the accompanying aligned arrows.  The two horizontal edges, however, are identified with opposite arrows.  Hence we have `twisted' the rectangle as if making a Mobius strip, indicating the non-orientable nature of the Klein bottle.
 

 
It is useful to examine the `double cover' representation of spin configurations and corresponding Majorana pairings on the Klein bottle; see Fig. <ref>(b), which can be constructed by conjoining two rectangles with identical Majorana pairings and spins along horizontal edges, in a way that one rectangle is spatially reflected. By identifying the horizontal edges and vertical edges of this `double rectangle', as illustrated by the arrows in Fig. <ref>(b), one can construct a torus which is a double cover of the Klein bottle. For convenience we will retain the line between the conjoined rectangles, hereafter denoted as a branch cut. All A_t and B_p terms can be represented on this double cover as well.  Note that a single vertex or plaquette term corresponds to two different possible terms in the double cover due to its 2-to-1 nature.
  
 Consistently assigning Majorana pairings requires Kasteleyn orientations that satisfy the clockwise-odd rule.  It is relatively straightforward to choose Kasteleyn orientations within a single rectangle, utilizing the rules we discussed in Sec. <ref>. However, for edges that cross the branch cut, two problems arise:
 


  * Long edges usually connect vertices on different sublattices. However, due to the peculiar identification scheme adopted in the double cover, long edges that cross the branch cut actually connect vertices on the same sublattice. Hence, from the rule we specified before, it is fundamentally ambiguous what arrows one chooses here. These problematic edges are indicated with question marks in Fig. <ref>(b).

  * Even if one chooses the orientation along long edges in some way, it is now impossible to enforce the clockwise-odd rule consistently on each cycle. For example, consider the plaquette in Fig. <ref>(b) circled in orange.  There, due to the Klein-bottle twist, arrows on small triangles above the dotted line are clockwise-even, while arrows on small triangles below the line are clockwise-odd. Hence, plaquette terms cannot be constructed in a way that preserves fermion parity.

 
 To rectify these issues, we attack the problem in reverse: We will construct a different double-cover representation, deduce how to correctly represent each Majorana pairing consistent with a given spin configuration, and then construct the Hamiltonian. Let us now assign sublattices and Kasteleyn orientations on our double rectangle as if the system was defined on an ordinary torus; see Fig. <ref>(c).  The aforementioned pathology does not exist in this choice. However, all Kasteleyn orientations on the second rectangle are opposite of the original double cover, indicating that the second rectangle should be identified with the first in a different way. 
 
Specifically, we identify the second rectangle transformed by U_𝒞𝒫,TSC as the first, where U_𝒞𝒫,TSC is defined as:

    U_𝒞𝒫,TSC :    σ_p^z→ -σ_p^z
       γ_v_A,↑→γ_v_B, ↓
       γ_v_A,↓→ -γ_v_B, ↑
       γ_v_B,↑→ -γ_v_A, ↓
       γ_v_B,↓→γ_v_A, ↑.

Comparing with Eq. (<ref>), we see that U_𝒞𝒫,TSC can be understood as the `local' part of 𝒞𝒫 symmetry for the topological superconductor.  That is, 𝒞𝒫_TSC is simply U_𝒞𝒫,TSC followed by spatial inversion. This identification corresponds to what we will therefore call a U_𝒞𝒫,TSC-twisted double cover. The essential idea is that the above transformation explicitly maps any operators iγ_v,sγ_v',s' that represent Majorana pairings within the second rectangle to -iγ_v,sγ_v',s'; hence, undoing this transformation effectively flips the Kasteleyn orientation, thus recovering the Kasteleyn orientation choice for the original double cover.  Note that we explicitly swap the sublattice index in the transformation U_𝒞𝒫,TSC; when comparing the sublattice distributions in Fig. <ref>(b) and Fig. <ref>(c), it is obvious that one should explicitly change the sublattice index of the upper rectangle to properly `untwist' the U_𝒞𝒫,TSC-twisted double cover.

 Next, let us discuss how to represent spin configurations and Majorana pairings on the double cover. Starting from the U_𝒞𝒫,TSC-twisted double cover, we put bosonic spins on each plaquette, but considering the twist, bosonic spins on the second rectangle orient opposite those on the first. We pair Majorana fermions on the twisted double-cover following the usual vertex rule on the torus. To recover the original double cover from the twisted double cover, simply transform the upper rectangle with U_𝒞𝒫,TSC. We illustrate the double-cover representations of spin configurations and Majorana pairings in Fig. <ref>(a) for the twisted double cover and Fig. <ref>(b) for the untwisted one obtained by applying U_𝒞𝒫,TSC to the upper rectangle. Note that within each rectangle Majorana pairings are fairly featureless.  The nontrivial pairing configurations reflecting the Klein-bottle manifold can be seen in the orange box, wherein Majorana pairings cross the branch cut.
 


 



 We have thus constructed a double-cover representation with consistently assigned spin configurations and Majorana pairings that are free from the aforementioned pathologies. 
 From here, one can straightforwardly build A_t and B_p terms. The key modification of A_t and B_p comes from the following two features: First, due to the fact that untwisting the double cover flips bosonic spins, short-edge pairings no longer necessarily match with domain walls between Ising spins near the branch cut. Figure <ref>(a) illustrates some examples of anomalous vertex rules at the branch cut. Second, the untwisting procedure significantly modifies the nature of long-edge pairings across the branch cut, as is evident by explicitly constructing projectors corresponding to long-edge pairings of interest. We consider the following four projectors in the twisted double-cover representation:

    P_v_A↑,v_B'↑ = 1-iγ_v_A,↑γ_v_B',↑/2, P_v_A↓,v_B'↓ = 1-iγ_v_A,↓γ_v_B',↓/2
       P_w_B↑,w_A'↑ = 1+iγ_w_B,↑γ_w_A',↑/2, P_w_B↓,w_A'↓ = 1+iγ_w_B,↓γ_w_A',↓/2.

To ensure the above projectors are associated with long-edge pairings across the branch cut, we will assume that v_A and w_B belongs to the upper rectangle in Fig. <ref>(c); meanwhile, v_B' and w_A' belong to the lower rectangle. The untwisting procedure transforms the above four projector to

    P_v_B↓,v_B'↑ = 1-iγ_v_B,↓γ_v_B',↑/2, P_v_B↑,v_B'↓ = 1+iγ_v_B,↑γ_v_B',↓/2
       P_w_A↓,w_A'↑ = 1-iγ_w_A,↓γ_w_A',↑/2, P_w_A↑,w_A'↓ = 1+iγ_w_A,↑γ_w_A',↓/2.

We note that Majorana fermions with opposite spins pair along long edges crossing the branch cut as seen in both panels of Fig. <ref>(a). Also, as seen in the right panel, two long-edge pairings that belong to the same edge across the branch cut should have opposite pairing signs.   
As for `arrows' along long edges crossing the branch cut, two pairings distinguished only by spins have opposite directions on this special type of long edge. 
 
One can locally (but not globally) convert A_t and B_p terms across the branch cut to usual vertex and plaquette terms in the `bulk' by applying U_𝒞𝒫,TSC on one side.  Hence, this construction inherits all good local properties of the original model defined on an orientable manifold; most notably, our Hamiltonian is still a commuting-projector model, and B_p acts as a nontrivial unitary operator on the subspace in which all vertex constraints are satisfied. Also, the `boundary-condition prescription' that we employed preserves 𝒞𝒫 symmetry—which arises fairly straightforwardly by observing that the untwisting operation U_𝒞𝒫,TSC commutes with 𝒞𝒫 symmetry. (Commutation can be seen by how 𝒞𝒫 symmetry maps the projectors associated with long-edge pairings across the branch cut.) However, this prescription breaks 𝒯-symmetry since under 𝒯, P_v_B↓,v_B'↑ in Eq. (<ref>) transforms to a complex conjugate of P_v_B↑,v_B'↓, not P_v_B↑,v_B'↓ itself. The same is true for P_w_A↓,w_A'↑ as well. 
 

The above scheme can be straightforwardly extended to the topological-insulator model by adding one more layer. Recall that 𝒞𝒫 symmetry for the topological insulator involves layer interchange as well, so the preceding prescription enforces Majorana fermions to pair between different layers along the long edges crossing the branch cut; see Fig. <ref>(b), 
where the two layers are represented by different colors.
While this pairing breaks U(1) symmetry, we will observe in a later subsection that this `layer twist' is a key reason why a U(1)-twisted boundary condition can be defined naturally on a Klein bottle.



 §.§ Many-body invariants for topological superconductors



As an initial step towards defining a many-body invariant for the topological superconductor, we comment on a peculiar feature of the Klein-bottle manifold.  
Recall from Fig. <ref> that in our construction on orientable manifolds, Majoranas pair along short edges when there is an adjacent domain wall separating up and down bosonic spins. Hence, one can view our construction as essentially decorating topologically nontrivial Kitaev chains along spin domain walls.  
On closed, orientable manifolds that admit non-contractible cycles, domain walls and hence Kitaev chains can form along non-contractible cycles as well. Clearly there must always be an even number of such domain walls/Kitaev chains.  



But does the preceding statement continue to hold in the non-orientable Klein bottle manifold? To address this question we examine a Klein-bottle configuration with all bosonic spins down.  
As before, it is useful to consider the twisted double-cover representation, wherein such a bosonic spin configuration is represented by all-down spins in one rectangle and all-up spins in the other rectangle. Two non-contractible cycles thus form, one along each boundary between the two rectangles [see Fig <ref>(a) for an illustration]. Nevertheless, because the double cover is a two-to-one representation, and despite all spins pointing down within the Klein bottle, the pairing rules across the branch cut imply that a Kitaev chain wraps around a single non-contractible cycle.

Local transformations (i.e., B_p) that flip bosonic spins locally and reconfigure Majorana pairings accordingly may change the number of non-contractible cycles by an even number, but for all spin configurations and corresponding Majorana pairings the number of non-contractible Kitaev chains is always odd.
 
 The above observations allow us to define a topological superconductor many-body invariant by changing boundary conditions. Consider the operator

    F = ∏_(vs,v's') ∈𝒫_↓ iγ_v,sγ_v',s',

where 𝒫_↓ is a set of all (vs,v's') vertex/spin labels that pair when all bosonic spins point down. Up to some factors of i, F is a product of all Majorana operators that appear in the system; it thus defines the total fermion parity operator and indeed satisfies F = ± 1. Now, we imagine flipping the boundary condition by flipping Kasteleyn orientations along edges that cross the vertical dashed line in Figs. <ref>(a) and (b). If there is no domain wall along a given edge that crosses the vertical dashed line, then two Majorana pairings from that edge cross the dashed line; if there is a domain wall, due to presence of a non-trivial Kitaev chain along the domain wall, only one Majorana pairing crosses the dashed line. In the Klein bottle, there is always an odd number of domain walls that cross the vertical line and hence always an odd number of Majorana pairings across that line whose signs are explicitly inverted by the boundary condition change.  The boundary-condition change consequently changes the sign of F.  





 It is reasonable to assert that on a closed manifold with a given boundary condition, the ground-state fermion parity can not change provided the bulk gap is finite so that the system remains in the same phase.  A trivial insulator can be smoothly deformed into a product state with all fermions strictly localized to the lattice sites; the ground state and hence the ground-state fermion parity are then clearly independent of boundary conditions. The feature in which a system defined on a Klein bottle has a different ground-state fermion parity compared to the torus thus defines a many-body topological invariant. Similar features arise in Majorana dimer models <cit.> and p+ip superconductors <cit.>, wherein putting the system on torus and changing boundary condition changes the fermion parity of the ground states.
 


 §.§ Many-body invariants for topological insulators


Above we probed the nontrivial topological index of the topological superconductor by changing boundary conditions in a discrete way. In the case of the topological insulator, U(1) symmetry allows one to continuously vary the boundary condition by twisting the pairings that cross the vertical dashed line in Figs. <ref>(a) and (b) by a U(1) variable θ. Twisting is formally done as follows: For all projectors P_v_1s_1,v_2s_2^(1)P_v_1s_1,v_2s_2^(2) associated with pairings that cross the vertical dashed line, apply the transformation 

    P_v_1s_1,v_2s_2^(1) P_v_1s_1,v_2s_2^(2)→ P_v_1s_1,v_2s_2^(1)(θ)P_v_1s_1,v_2s_2^(2)(θ)
    
       = 1+ig_v_1v_2γ_v_1,s_1,1γ_v_2,s_2,1'/21+ig_v_1v_2γ_v_1,s_1,2γ_v_2,s_2,2'/2,

where

    [ γ_v_2,s_2,1'; γ_v_2,s_2,2' ] = U(θ) [ γ_v_2,s_2,1; γ_v_2,s_2,2 ],    U(θ) =[  cosθ  sinθ; -sinθ  cosθ ].

In the equations above we assume that v_1 and v_2 respectively reside on the right and left sides of the vertical dashed line.

 This boundary condition preserves 𝒞𝒫 symmetry, since although parity sends θ→ -θ, charge conjugation contributes another minus sign and returns θ to its original value. This boundary condition is `flat' in the sense that local terms A_t and B_p can be reverted to untwisted ones by local gauge transformations. For example, we see that the projector U(1) twisted in Eq. (<ref>) reverts back to its original form by the transformation

    [ γ_v_1,s_1,1; γ_v_1,s_1,2 ]→ 
    U(θ) [ γ_v_1,s_1,1; γ_v_1,s_1,2 ],

which is an immediate consequence of a corollary of the lemma introduced in Appendix <ref>. Using this transformation one can readily `flatten' A_t's.  Similarly, one can flatten B_p's that do not cross the branch cut by applying the same gauge transformation to Majoranas on one side of the vertical dashed line, e.g., in the blue area of Fig. <ref>(a). One should take a more careful approach for B_p's that cross the branch cut since, due to the Klein-bottle structure, the boundary condition is twisted in opposite ways across the branch cut. Hence, to flatten B_p terms locally, one should transform Majoranas above the branch cut [red area in Fig. <ref>(b)] with U(-θ) instead of U(θ), due to a different corollary presented in Appendix <ref>. We recall that long-edge pairings across the branch cut pair Majoranas with different layer indices. Hence, the corresponding projector is preserved under the gauge transformation with U(-θ) on one side and U(θ) on the other.  If we had not twisted Majorana layers across the branch cut when defining the system on the torus in an earlier subsection, it would have been impossible for us to flatten those B_p's. This once more emphasizes the importance of encoding layer interchange in the definition of 𝒞𝒫 symmetry. 

Because A_t and B_p can be locally flattened, all nice local properties are once more preserved in this boundary-condition twist. We emphasize, however, that no local gauge transformations map our system with specific θ into a system with a different U(1) boundary-condition twist. 



Let us now compute the Berry phase γ_GS of the ground state by adiabatically changing θ from 0 to 2π. First, let us consider the Berry phase for a single snapshot in which all spin configurations are fixed to be down, with Majoranas paired accordingly. Denote this state as |↓(θ)⟩. We will compute

    γ_↓ = i∫_0^2π dθ⟨↓(θ)|∂/∂θ|↓(θ)⟩.

As a first step we examine a 1D problem in which we compute the Berry phase for double Kitaev chains. 
One can easily compute that the Berry phase is precisely π. Since |↓(θ)⟩ contains an odd number of such chains, the Berry phase γ_↓ is π mod 2π. 
The full ground state of interest reads

    |GS(θ)⟩ = 1/2^n/2∑_n_p=0,1∏_p( B_p(θ) )^n_p|↓(θ)⟩,

where n is a total number of plaquettes. Since all terms in the summand have different bosonic spin configurations and hence are orthogonal to each other, one can write the Berry phase as

    γ_GS = i∫_0^2π dθ⟨GS(θ)|∂/∂θ|GS(θ)⟩
       = 1/2^n∑_n_p=0,1 i∫_0^2π dθ
       ⟨↓(θ)|( ∏_p( B_p(θ) )^n_p) ∂/∂θ( ∏_p( B_p(θ) )^n_p) |↓(θ)⟩.
 
The above equation implies that one can compute γ_GS by simply averaging the Berry phase for 2^n different configurations. Remarkably, we will show in Appendix <ref> that 

    ⟨↓(θ)|( ∏_p( B_p(θ) )^n_p) ∂/∂θ( ∏_p( B_p(θ) )^n_p) |↓(θ)⟩
       = ⟨↓(θ)|∂/∂θ|↓(θ)⟩
 
for any θ and any n_p—which guarantees that γ_GS = γ_↓ = π (mod  2π). This π Berry phase was computed from the boundary <cit.> and bulk in band-theory frameworks <cit.> and found to be quantized in 𝒞𝒫-symmetric topological phases. Strikingly, this Berry phase—which is generically nontrivial to compute and usually evaluated with small-system numerics—can be analytically determined for any system size in our exactly solvable model! We note that while there are supporting evidences that the Berry phase is quantized to either 0 or π, to our knowledge, a microscopic proof of robust quantization is not yet available.





 §.§ Towards bulk topological invariants of 𝒯-symmetric topological phases without 𝒞𝒫 symmetry



 Throughout the main text of this paper, we focused on honeycomb-lattice models with a specific choice of Kasteleyn orientation. We indeed eschewed fully general setups in favor of accessibility.  Appendix <ref>, however, shows how to define models on any trivalent lattice with any choice of Kasteleyn orientation. The particularly interesting setups relevant to this subsection occur when the trivalent lattice is chosen to violate parity symmetry; in these cases, there is no exact 𝒞𝒫 symmetry. It is natural to ask whether (i) it is possible to define models on non-orientable manifolds when the lattice is not parity-symmetric and (ii) if one can define the models on a Klein bottle, whether the many-body invariants we explored earlier in this section can be extended to these setups.
 
 In Appendix <ref>, we show that the answers to both questions are affirmative: The models can still be defined on a Klein bottle while maintaining commuting-projector properties, and the global-fermion-parity flip and π Berry phase still characterize our exactly solvable model. While we relegate technical details to the appendix, we will briefly discuss here the role of time-reversal symmetry in defining models on a Klein bottle.
 
 Recall that in the earlier subsection we stated that putting systems on a Klein bottle breaks 𝒯-symmetry. The same is true for Klein-bottle constructions presented in Appendix <ref> as well. Hence, readers may naturally wonder in what sense 𝒯 may protect many-body invariants of our interests. To see this within the context of the construction presented in this section, we emphasize that the unitary part of 𝒯-symmetry encoded in Eq. (<ref>) is actually identical to the local part of 𝒞𝒫 symmetry U_𝒞𝒫,TSC in Eq. (<ref>), modulo some sublattice relabeling in the latter equation. The Klein-bottle construction presented in the appendix is a natural extension of this observation and employs the unitary part of the time-reversal symmetry, without referring to any 𝒞𝒫 symmetry.
 
 While the previous paragraph gives us some confidence that many-body invariants on a Klein bottle may also characterize 𝒯-symmetric topological phases, whether these invariants remain quantized or valid outside of the context of our special exactly solvable models requires further investigation. Also, we did not give any detailed prescription on how to construct these many-body invariants outside in the context of our  exactly solvable models; developing a systematic procedure would be an interesting future direction.
 
 
 
 
 
 
 
 
 
 
 


§ CONCLUSIONS



In this paper we constructed a commuting-projector model for an interacting 2D quantum-spin-Hall insulator.  While such phases are of course amenable to band-theory treatments in the non-interacting limit, our perspective illuminates complementary insights that are far from obvious from that more traditional approach.  Most strikingly, from our commuting-projector model we derived a strictly 1D lattice Hamiltonian that, within a restricted subspace, faithfully reproduces the physics of a fully symmetric quantum-spin-Hall edge.  Additionally, we discovered that both the topological-superconductor model from Ref. Wang2017 and our topological-insulator model preserve 𝒞𝒫 symmetry when defined on parity-symmetric lattices, allowing us to explore bulk invariants by placing the systems on non-orientable manifolds. We further observed that computation of many-body invariants can be extended to models with 𝒯 symmetry but without explicit 𝒞𝒫 symmetry.  The latter result raises the question of whether topological invariants of time-reversal-invariant topological phases can be also explored by defining systems on non-orientable manifolds.


To close we highlight a number of other interesting open questions connected to our study:

In our 1D lattice Hamiltonian that describes the quantum-spin-Hall edge, U(1) symmetry is realized in the usual, local manner while 𝒯 is implemented nonlocally.  Can one derive an alternative strictly 1D model in which the roles of these symmetries are reversed, i.e., unconventional realization of U(1) but local 𝒯 implementation?  It is also worthwhile to contemplate practical utility of such 1D Hamiltonians.  They may, for instance, enable efficient numerical studies of 2D quantum-spin-Hall edges with interactions, noise, etc.  As an alternative application, can one interface magnetic degrees of freedom and electrons to design experimental 1D setups that emulate the quantum-spin-Hall edge?  Architectures of this type could furnish new platforms for Majorana zero modes, fractional charges, anomalous pumping cycles, and more.  

On the more technical end, are the nontrivial π Berry phase (for the topological-insulator model) and ground-state fermion-parity flip from changing fermionic boundary condition (for the topological-superconductor model) generic properties of time-reversal symmetric topological phases? If so, is there an analytical proof available in the context of (2+1)-dimensional lattice models? 
How is our construction related to the relevant Pin-structure? Kasteleyn orientations are related to spin structures <cit.>, and there is generalization of Kasteleyn orientations to the Pin^- structure as well <cit.>. The 𝒯^2=-1 topological-superconductor model is known to be related to the Pin^+ structure <cit.>, and indeed how the Kasteleyn orientation should be properly constructed for the model on a Klein bottle differs from the discrete Pin^- structure given in Ref. Cimasoni2009. 

Finally, are there higher-dimensional generalizations of the commuting-projector models that we studied here, e.g., for 3D topological insulators—potentially giving rise to strictly 2D lattice Hamiltonians that similarly capture the anomalous single-Dirac-cone surface states?


We thank Xie Chen, Gil Young Cho, Lukasz Fidkowski, Shinsei Ryu, Xiao-Qi Sun, and Zitao Wang for helpful discussions. This work was supported by the Army Research Office under Grant Award W911NF-17-1-0323; the NSF through grant DMR-1723367; the Caltech Institute for Quantum Information and Matter, an NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation through Grant GBMF1250; and the Walter Burke Institute for Theoretical Physics at Caltech.




§ MODELS ON GENERAL TRIVALENT LATTICES WITH GENERAL KASTELEYN ORIENTATIONS


 
 In this appendix, we briefly review the 𝒯-symmetric topological-superconductor model on any trivalent lattice defined on an orientable manifold, and for any choice of Kasteleyn orientation (as opposed to the specific choice of lattice and Kasteleyn orientation used elsewhere in this paper). Then, we show that on any parity-symmetric lattice and with any Kasteleyn orientation, the model preserves 𝒞𝒫 symmetry as well. Finally, we discuss how to define our models on a Klein bottle even when the lattice is not parity-symmetric and hence 𝒞𝒫 symmetry is absent. The latter discussion suggests that many-body invariants originally proposed for 𝒞𝒫-symmetric topological phases may also characterize 𝒯-symmetric phases. We stress that although we primarily focus on the constructions of topological-superconductor models, the generalization to topological insulators proceeds straightforwardly by adding another layer of Majorana fermions.
  


 §.§ Topological-superconductor models 



 The degrees of freedom are defined analogously to the honeycomb-lattice construction used earlier.  A single bosonic spin resides on each plaquette of the trivalent lattice, while four Majoranas appear at each edge—two at one end of the edge and two at the other. The pair of Majoranas at the same end of the edge will be distinguished by different spin indices. Alternatively, one can replace a vertex of the trivalent lattice with small triangles to make a triangle-decorated `pairing lattice' (i.e., the Fisher lattice in the honeycomb-lattice used in the main text); pairs of Majoranas with opposite spins can then be viewed as residing on each vertex of the pairing lattice.  As before, long edges correspond to edges of the pairing lattice from the original trivalent lattice and short edges to the edges arising from small-triangle decoration.
 
Given a Kasteleyn orientation on the pairing lattice (which can be chosen for any lattice on an orientable manifold <cit.>), we specify how Majoranas pair for each plaquette. From there the A_t vertex terms and the B_p plaquette terms can be constructed by using appropriate Majorana projectors dictated by the pairing rules below.



  * Determine which Majoranas should pair along each long edge first. Given a long edge e, vertices v and v' connected by e, and two neighboring plaquettes p_1 and p_2, these pairings are determined as follows: 


  * If Ising spins on p_1 and p_2 are identical, pair (γ_v,↑,γ_v',↑) and (γ_v,↓,γ_v',↓).

  * If those Ising spins are opposite, draw an arrow from the plaquette with up Ising spin to to the plaquette with down Ising spin, and cross product the Kastelyn arrow connecting v and v' with this arrow. If the cross product points out of the page, pair only (γ_v,↑,γ_v,↑); otherwise, pair only (γ_v,↓,γ_v',↓).


  * If all three plaquette spins adjacent to a vertex v are identical, the above rules pair all six Majoranas on the small triangle at vertex t. If not, exactly two among six Majoranas on the small triangle remain unpaired. Pair those two unpaired Majoranas.

As an exercise, one can check that for the honeycomb lattice the pairing rule given in Fig. <ref> in the main text follows the above prescription.

Time-reversal symmetry 𝒯_TSC of the Hamiltonian is implemented by the anti-unitary transformation

    𝒯_TSC : σ_p^z→ - σ_p^z,   [  γ_v,↑; γ_v',↑;  γ_v,↓; γ_v',↓ ]→[  0  0  1  0;  0  0  0 -1; -1  0  0  0;  0  1  0  0 ][  γ_v,↑; γ_v',↑;  γ_v,↓; γ_v',↓ ].

Here v and v' are vertices of the pairing lattice that are connected by a long edge, with the Kasteleyn arrow direction on that long edge pointing from v' to v. One can prove that among two Majoranas γ_v_1,s_1 and γ_v_2,s_2 that can be paired in some spin configuration, the above transformation endows a minus sign to only one of them. Hence, combined with complex conjugation that sends i → -i, iγ_v_1,s_1γ_v_2s_2 maps to iγ_v_1,s_1γ_v_2s_2, with the overlines denoting the opposite spin indices. As a consistency check, one can show that upon specializing to the honeycomb-lattice model with our previous Kasteleyn orientation choice, this transformation reduces to the operation specified in Eq. (<ref>).

 Finally, we discuss the relationship between two Hamiltonians with identical lattice but with different Kasteleyn orientations. We first recall that a `local transformation' at a vertex v of the pairing lattice is defined by flipping arrows on the three edges that meet at v. It is known that a series of local transformations map all Kasteleyn orientations that represent the same spin structure <cit.>. For topological-superconductor models, flipping an arrow direction on a long edge changes how Majoranas pair for each spin configuration. Hence, one can think of the following local transformation at v:

    L_v : [  γ_v,↑; γ_v',↑;  γ_v,↓; γ_v',↓ ]→[  0  0 -1  0;  0  0  0  1; -1  0  0  0;  0  1  0  0 ][  γ_v,↑; γ_v',↑;  γ_v,↓; γ_v',↓ ],

where v' is the vertex connected to v via a long edge. Essentially, the above transformation adds minus signs to Majoranas at v and flips the Majorana spins at both v and v'. A series of L_v transformations connect Hamiltonians defined with different Kasteleyn orientations in the same `topological sector'. We note that L_v corresponds to ℤ_2 gauge transformations on Majoranas combined with spin relabeling; if two Hamiltonians are related by a series of L_v's, they should be regarded as identical.



 §.§ 𝒞𝒫 symmetry beyond honeycomb-lattice models


 Here we present a proof that the topological-superconductor models defined on a torus or sphere and on parity-symmetric lattices always possess 𝒞𝒫 symmetry. While our proof is restricted to the case of two specific manifolds, it is natural to expect that a similar technique can be extended to any orientable manifold on which a reflection operation can be defined. 
 

 
 First, we show that there are `canonical Kasteleyn orientations' in which the reflection operation flips all arrows. As shown in Fig. <ref>(a), a line that is fixed under reflection [dotted vertical line in (a)] bisects the sphere into two parts marked blue and red in figure, one being a mirror image of the other. In the case of the torus, there are two lines that are fixed under reflection [three dotted vertical lines in Fig. <ref>(b); the two at the extreme ends are actually identical].  These two lines similarly divide a torus into two parts, which are also mirror images of each other.  
 
 One can imagine first assigning arrows for edges that reside only on one part (e.g., the red region) such that all cycles within that part satisfy the clockwise-odd rule. Note that we are not assigning arrow directions to edges that cross the fixed lines and hence belong to both parts. Then, one can define arrow directions on the other part (e.g., the blue region) such that they are opposite of the mirror image of the arrows in the first part. Now, all edges except for the ones that lie across the fixed lines have arrow assignments, and all cycles isolated to each part satisfy the clockwise-odd rule. 
 
 Thanks to the two parts being mirror images of one another, the arrows for the edges that cross the fixed line are fairly straightforward to determine as well. Plaquettes that extend across the fixed lines are guaranteed to be clockwise even without arrows for the edges on a fixed line; indeed, due to reflection symmetry, the already-assigned arrows on the left and right have the same number of edges that are directed against the clockwise cycle. Hence, one can choose arrows along fixed lines such that all of them uniformly point from one area to the other, just like the bright green arrows in Figs. <ref>(a) and (b). Plaquettes bisected by fixed lines then satisfy the clockwise-odd rule. 
 
 As a final step, we recall that there are four locally inequivalent choices of Kasteleyn orientations for the torus. The three other locally inequivalent canonical Kasteleyn orientations can be generated by flipping all arrows on any vertical or horizontal dotted line in Fig. <ref>, or both. This global transformation does not alter the defining property that the reflection transformation on these Kasteleyn orientations are equivalent to flipping all arrows. 
 
One can write down the 𝒯-symmetric topological-superconductor models based on these canonical Kasteleyn orientations. In addition to 𝒯_TSC, these models enjoy 𝒞𝒫_TSC symmetry defined by U_𝒯,TSC (unitary part of the time-reversal symmetry) followed by spatial reflection. As for the proof, one can manually check that our definition of 𝒞𝒫_TSC, when applied to a vertex term at the vertex v, properly gives its mirror partner—which suffices to prove 𝒞𝒫_TSC invariance of the whole Hamiltonian.
 
 When the Hamiltonian is defined with some arbitrary Kasteleyn orientation without a mirror property, as we saw earlier, a series of local gauge transformations L_v allow us to map the Hamiltonian to the one defined with canonical Kasteleyn orientations. Hence, we conclude that an emergence of 𝒞𝒫 symmetry on our exactly solvable model is a very general phenomenon whenever the underlying lattice is parity symmetric.
 


 §.§ Topological superconductor models on a Klein bottle without 𝒞𝒫 symmetry 



We conclude this appendix by giving a general prescription of how to define topological-superconductor models on a Klein bottle, even when the underlying lattice is not parity symmetric and hence 𝒞𝒫 symmetry is absent. We proceed similarly to the main text by building a twisted double-cover representation of Majorana pairings and spin configurations.
 
 The key ingredients in building a twisted double-cover representation is a choice of Kasteleyn orientation. Kasteleyn orientation on a twisted-double cover representation is essentially arrow assignments on a trivalent lattice on a torus, now with a special property: The lattice on the upper half is precisely the mirror image of the lower half, and the Kasteleyn orientation on the upper half is opposite of the mirror image of the lower half. We will show that such a Kasteleyn orientation can be generally chosen on a Klein bottle when each half contains an even number of vertices, using the result from Ref. Cimasoni2009. 
 
 Before stating our proof, we set up some notation. There are two lines (referred to as `cuts') on the double-cover torus where the lower half and the upper half meet. Following Ref. Cimasoni2009 we will denote edges crossing these lines as `0-edges', while edges that do not cross the lines are dubbed `1-edges'. Reference Cimasoni2009 shows that as long as the number of vertices on either half is even, a Kasteleyn orientation on an identical lattice, but with modified relations between arrows on the lower half and upper half, is guaranteed to exist: In the modified relation, 0-edges follow the same rule described in the previous paragraph. However, arrows on 1-edges crossing one cut are just the mirror image of those on the other cut (instead of being opposite of the mirror image). All we need to do to construct the Kasteleyn orientations we want for our purpose is flip arrows on 1-edges that cross one of the cuts, so that now all arrows on the upper half—regardless of whether they are on 1-edges or 0-edges—are opposite of the mirror image of the lower half. This operation is benign in the sense that all clockwise-odd cycles remain clockwise-odd.

We emphasize that this proof is expected to only work on a Klein bottle; in fact, based on the fact that there is no Pin^+ structure on ℝℙ^2, it is expected that there is no natural way to define our models consistently on some other non-orientable manifolds.
 
 The rest of the process is almost identical to the one presented in the main text: We build twisted double-cover representations by setting the bosonic spin configurations on the upper half as the opposite of the mirror image of the lower half, and pair Majoranas following the local rules as if the twisted double-cover torus was just an ordinary fully orientable torus. To build an ordinary double cover, we untwist the upper half with U_𝒯,TSC. One can show that this operation generates Majorana pairings and spin configurations where the upper half is precisely the mirror image of the lower half, and we indeed thus construct the `double cover representation' of Majorana/spin configurations on a Klein bottle. From there, one can straightforwardly write down the Hamiltonian on a Klein bottle.
 
This method of putting the model on a Klein bottle explicitly breaks time-reversal symmetry. However, we note that (i) this methods uses some form of 𝒯 symmetry and indeed can be understood as twisting the boundary condition by the unitary part of time reversal and (ii) many-body invariants presented in Sec. <ref> can be computed in this setup as well and can be shown to acquire the same values. 




§ PROOF OF A PROJECTOR LEMMA AND ITS COROLLARIES
 

Here, we prove the following lemma: We are given four Majorana operators γ_v,1, γ_v,2, γ_v',1, γ_v',2 and 2 × 2 O(2) matrices M and N. We also define

    γ'_v,i= M_ijγ_v,j,   γ'_v',i= N_ijγ_v',j.

Here and in other parts of the proof repeated indices are implicitly summed. Then the following holds:

    1+iγ'_v,1γ'_v',1/2   1+iγ'_v,2γ'_v',2/2
       = 1/4  + i/4γ_v,i( M^T N )_ijγ_v,j
       + 1/4( M^T N ) γ_v,1γ_v,2γ_v',1γ_v',2 .

Notice that regardless of the detailed form of M and N, as long as M^TN remains the same, the right-hand side remains the same as well. 

The proof proceeds by brute force:

    1+iγ'_v,1γ'_v',1/21+iγ'_v,2γ'_v',2/2
       = 1+i M_1iγ_v,i N_1jγ_v',j/21+i M_2kγ_v,k N_2lγ_v',l/2
       = 1/4 + i/4γ_v,i (M_1iN_1j + M_2iN_2j)γ_v,j
       + 1/4M_1iγ_v,i M_2kγ_v,k N_1jγ_v',j N_2lγ_v,l

By recognizing that M_1iN_1j + M_2iN_2j = M_ik^TN_kj, one can see that the second term on the right side of Eq. (<ref>) precisely matches the second term on the right side of Eq. (<ref>). Observe next that

    M_1iγ_v,i M_2kγ_v,k   = (M_11M_21 + M_12M_22) 
       + (M_11M_22 - M_12M_21)γ_v,1γ_v,2.

The first term above vanishes, due to the fact that the two row vectors of M are orthogonal to each other. The coefficient in the second term is simply M—hence M_1iγ_v,i M_2kγ_v,k = Mγ_v,1γ_v,2. One can see that the third term in the last line of Eq. (<ref>) is then precisely the last line of Eq. (<ref>). The lemma is therefore proven.  

Below we list several useful corollaries.


  * Multiplication of two projectors

    1+iγ_v,1γ_v',1/21+iγ_v,2γ_v',1/2
 
is invariant under the transformation

    γ_v,i→ U(θ)_ijγ_v,j,   γ_v',i→ U(θ)_ijγ_v',j
       U(θ) = [  cosθ  sinθ; -sinθ  cosθ ].

 This corollary can be straightforwardly proven by noticing that the above transformation is equivalent to setting M = N = U(θ). 

  * The projector

    1+iγ_v,1γ_v',2/21+iγ_v,2γ_v',2/2

appears in Sec. <ref>, where defining the model on a Klein bottle makes some Majoranas pair across the different layers. This projector is invariant under the U(1) transformation

    γ_v,i→ U(θ)_ijγ_v,j,   γ_v',i→ U(-θ)_ijγ_v',j.

This property can be proven as follows: The above projector can be obtained by choosing M = I, N = [ 0 1; 1 0 ], and the transformation is equivalent to choosing M' = U(θ) and N' = [ 0 1; 1 0 ] U_-θ. It is easy to show that M^TN = M'^TN'.

  * Let us consider

    P(θ) = 1+iγ_v,1γ_v',1'(θ)/21+iγ_v,2γ_v',2'(θ)/2,

where γ_v',1'(θ) and γ_v',2'(θ) are defined as

    [ γ_v',1'(θ); γ_v',2'(θ) ] = [  cosθ  sinθ; -sinθ  cosθ ][ γ_v',1; γ_v',2 ].

Then one can show:

    P(θ) ∂ P(θ)/∂θ P(θ) =0

This can be easily proven by noticing that

    ∂ P(θ)/∂θ = i/4[ γ_v,1 γ_v,2 ][ -sinθ  cosθ; -cosθ -sinθ ][ γ_v',1; γ_v',2 ]
       =i/4(γ_v,1γ_v',2'(θ) + γ_v,2γ_v',1'(θ)).

Then, one can deduce

    P(θ) ∂ P(θ)/∂θ P(θ) = P(θ) i/4(γ_v,1γ_v',2'(θ) + γ_v,2γ_v',1'(θ)) P(θ)
       =i/4(γ_v,1γ_v',2'(θ) + γ_v,2γ_v',1'(θ))
       1-iγ_v,1γ_v',1'(θ)/21-iγ_v,2γ_v',2'(θ)/2P(θ).

The product of the projectors on the third line is explicitly 0. This property will be used in Appendix <ref>.









§ MORE ON THE GAPLESS EDGE HAMILTONIAN



This appendix fills in technical gaps that we left open in Sec. <ref>.



 §.§ Strategy for proving [B_p,C_I]=0


 Here, we introduce a trick that allows C_I to be modified such that it can be treated more like an ordinary plaquette flip term B_p. This modification allows one to prove [B_p,C_I]=0 straightforwardly.  
First we summarize the key elements in the proof of [B_p,B_p']=0 (following similar techniques used, e.g., in Ref. Ware2016, jones2019, and Son2018). If p and p' are non-neighboring, this relation is somewhat obvious. The non-trivial feature of the proof comes from the fact that when B_p and B_p' are neighboring, there are Majorana projectors in the expression for B_p and B_p' that do not commute with each other and should be handled carefully. We emphasize that (i) the precise details of proof are only dependent on projectors involving Majoranas around two vertices and (ii) there is a step in the proof that relies on the fact that in B_p, projectors that project onto the Majorana pairings consistent with the original spin configuration and projectors that project onto the Majorana pairings consistent with the new spin configurations together form a loop around the plaquette p. 
 
Let us investigate how (i) and (ii) are retained or violated in C_I. Think of a bulk plaquette p neighboring to I, and two vertices at which I and p meet.
The operator C_I actually contains all Majorana projectors involving degrees freedom around those two vertices as if it was a normal `complete' plaquette. Hence, one may naively expect that projectors that should be treated carefully are actually just identical to those involved in the proof of [B_p,B_p']=0. However, C_I is explicitly incomplete and hence Majorana projectors do not form a loop anymore, violating (ii). Hence, if we devise an equivalent modification of C_I in which Majorana projectors now form a closed loop without affecting Majorana projectors around the two vertices of interest, then we can apply the same technique for proving [B_p,B_p']=0.



We divide into cases in which the two incomplete plaquette spins surrounding I are (i) the same or (ii) opposite. For case (i), assume that the two incomplete plaquette spins are up, and that the spin at plaquette I is up as well. Then, in the expression for 𝒞_u_I in Eq. (<ref>), the projectors in the first parenthesis explicitly keep four Majoranas at the edge unpaired, whereas the projectors in the second parenthesis keep these previously unpaired Majoranas paired with Majoranas in the bulk. Denote those four Majoranas as γ_e,s,1 γ_e,s,2, γ_e',s',1, and γ_e',s',2. One can insert the projectors

    P_es,e's'^TI = 1 ± iγ_e,s,1γ_e's',1/21 ± iγ_e,s,2γ_e's',2/2
 
to get an equivalent expression, where the signs should be determined by the clockwise-odd rule around the incomplete plaquette I. This modification makes 𝒞_u_I satisfy (ii) for the u_I we considered so far. To see why this works, let us label Majoranas (on the first layer) in a way that the second parenthesis of 𝒞_u_I enforces Majorana pairings between (γ_a_1,1,γ_a_2,1), (γ_a_3,1,γ_a_4,1),⋯, (γ_a_2n-1,1,γ_a_2n,1), and the first parenthesis projects onto the state with Majorana pairings (γ_a_2,1,γ_a_3,1),⋯,(γ_a_2n-2,1,γ_a_2n-1,1); we chose to combine spin and vertex indices of Majoranas into a single index a_i for simplicity. Note that γ_e,s,1 and γ_e',s',1 respectively corresponds to γ_a_1,1 and γ_a_2n,1. By definition, states that are not projected out by the second parenthesis have definite eigenvalue ± 1 of the following operator:

    ∏_i=1^n i γ_a_2i-1,1γ_a_2i,1 = -(iγ_a_2n,1γ_a_1,1)∏_i=1^n-1 i γ_a_2i,1γ_a_2i+1,1.

The product of operators on the right side is also fixed by the first parenthesis—implying that states obtained after applying the original 𝒞_u_I have definite iγ_a_2n,1γ_a_1,1 eigenvalues despite lacking projectors involving the latter Majoranas. Hence, adding the projector in Eq. (<ref>) that projects onto the certain eigenstate of iγ_a_2n,1γ_a_1,1 is harmless and does not change how 𝒞_u_I acts. The same logic applies for Majoranas in layer 2. See Fig. <ref>(a) for an illustration.

One can insert the same projector in the second parenthesis if the configuration u_I has spin down at plaquette I (which is the `Hermitian conjugate' of the first case we considered). One can also apply a similar modification when two surrounding incomplete plaquette spins point down instead of up.
 
 Next we introduce a modification when the two incomplete plaquette spins around I are opposite—fixed for concreteness to up on the left and down on the right—and with the spin at I pointing up. The spin configuration u_I then enforces two Majoranas γ_a_1,1 and γ_a_1,2 to be unpaired.  After action of 𝒞_u_I, the aforementioned Majoranas pair with γ_a_2,1 and γ_a_2,2, and a different set of Majoranas γ_a_3,1 and γ_a_3,2 will now be unpaired; see Fig. <ref>(b). We introduce the unitary transformation

    U = 1 ±γ_a_1,1γ_a_3,1/√(2)1 ±γ_a_1,2γ_a_3,2/√(2)1+σ_I^z/2 + 1-σ_I^z/2,

which exchanges γ_a_1,1 and γ_a_3,1 and γ_a_1,2 and γ_a_3,2 only when σ_I^z = +1.   (Whether we use the + or - sign above is unimportant.)  This unitary transformations commutes with any bulk term, hence proving [B_p, U^† C_I U] = 0 implies [B_p,C_I] = 0. The intuitive effect of this transformation is clear: As illustrated in Figs. <ref>(b) and (c), this transformation fixes Majoranas that remain unpaired, allowing 𝒞_u_I to be treated as closed loops for the u_I we are considering. This can be explicitly confirmed via projector algebra, which we leave as an exercise for readers who want to gauge their understanding of this paper (the proof can be done in three lines). As before, the above modification can be straightforwardly generalized to the other spin configurations falling into case (ii).  



 §.§ More detailed justification for polarizing bulk spins to derive the edge Hamiltonian


Next we provide a more detailed and technical justification for deriving the edge Hamiltonian by fixing bulk spins and stripping out degrees of freedom that are frozen due to lack of quantum fluctuations of bulk spins in the low-energy space. To justify this procedure, we investigate the low-energy physics and the spectrum of the Hamiltonian H = -∑ A_t -∑ C_I. We will work in the subspace 𝒮_A_t in which A_t =1 is enforced. Also, we define 𝒮_u_b, the subset of 𝒮_A_t in which bulk spin configurations are fixed to be u_b. By definition, ∪_u_b𝒮_u_b = 𝒮_A_t.  


 

One can construct a canonical isomorphism from S_u_b to S_u_b', the bulk spin configuration u_b' obtained by flipping spins at a set of plaquettes { p } from u_b, defined as:

    |ψ_u_b⟩→∏_{p }B_p|ψ_u_b⟩.
 
Moreover, C_I commutes with B_p. These two facts imply that the spectra of H = -∑ A_t -∑ C_I within all 𝒮_u_b sectors are identical. When the full Hilbert space is considered, the low-energy spectrum of H = -∑ A_t -∑ C_I is 2^n-fold degenerate (n is the number of bulk plaquette spins), each degenerate state in the spectrum coming from a different subspace S_u_b. See Fig. <ref>(a) for a schematic.

 Adding the term -∑_p B_p lifts this massive degeneracy, splitting the 2^n degenerate levels into different levels that can be labeled by B_p eigenvalues; this is possible because B_p commutes with C_I and A_t. Within the levels that possess the same set of B_p eigenvalues, their relative energies are purely given by C_I and hence are identical to those given by the Hamiltonian H= -∑_I C_I defined on the restricted Hilbert space 𝒮_u_b. See Fig. <ref>(b) for an illustration. The low-energy subspace of interest satisfies B_p=1 for all p, and therefore one can fix bulk spin configurations to study the low-energy physics of the Hamiltonian H = -∑ A_t -∑ C_I - ∑ B_p. The cost is loss of an explicit on-site time-reversal symmetry.  However, we saw that the effective 1D model we derived from this procedure retains some non-trivial incarnation of 𝒯-symmetry in the low-energy space.
 


 §.§ Simplifying the 1D model with unitary transformations and cutting out Majoranas




 After polarizing bulk spins and removing Majoranas that are frozen in the low-energy space, we obtain a 1D model with a bosonic spin in each plaquette (i.e., unit cell) and sixteen Majoranas per unit cell, eight per layer. We will label the latter as γ_i,a_1,1, γ_i,a_2,1, ⋯γ_i, a_8,1 in layer 1 and similarly for layer 2; here i denotes a unit cell index and a_1,⋯ ,a_8 are internal indices within a unit cell (see Fig. <ref> for the definition).  We now introduce a three-step local unitary transformation that reduces the number of Majorana degrees of freedom that fluctuate at low energies to four per unit cell, two for each layer: 



  * Whenever σ_i-1^z = -1 and σ_i^z=+1 apply the following unitary transformation 

    X_i   = 1-γ_i,a_1,1γ_i,a_3,1/√(2)1-γ_i,a_2,1γ_i,a_4,1/√(2)
       1-γ_i,a_1,2γ_i,a_3,2/√(2)1-5γ_i,a_2,2γ_i,a_4,2/√(2)

to each site.
The above transformation exchanges γ_a_1,1 and  γ_a_3,1, and γ_a_2,1 and γ_a_4,1 in layer 1, with the same exchanges occurring also in layer 2. In the original model, Majoranas with both spins can fluctuate; by applying the following changes, we will freeze Majoranas with a certain spin (in the language of this section, γ_i,a_1,1, γ_i,a_2,1, γ_i,a_1,2, and γ_i,a_2,2) and discard them.

  * Whenever σ_i^z = +1, apply the transformation

    Y_i = 1+γ_i,a_5,1γ_i,a_7,1/√(2)1+γ_i,a_5,2γ_i,a_7,2/√(2),

which changes γ_i,a_5,1 and γ_i,a_7,1 in layer 1 only when the spin at i points down (identical changes occur in layer 2). As a result of this transformation, in the low-energy subspace, γ_i,a_6,1 and γ_i,a_7,1 always remain paired and can be removed for the sake of studying the low-energy physics. The same goes for the analogous Majoranas in layer 2.

  * Whenever σ_i-1^z = +1 and σ_i^z=-1 apply the unitary transformation

    Z_i,+- = 1+γ_i,a_3,1γ_i,a_5,1/√(2)1+γ_i,a_3,2γ_i,a_5,2/√(2).

Meanwhile, when σ_i-1^z = -1 and σ_i^z=+1, apply 

    Z_i,-+ = 1+γ_i-1,a_8,1γ_i,a_3,1/√(2)1+γ_i-1,a_8,2γ_i,a_3,2/√(2).

Both transformations are applied when there are unpaired Majoranas between sites i-1 and i, and move the unpaired Majoranas (which due to the very first transformation we applied are always γ_i,a_3,1 and γ_i,a_3,2) to either a_8 or a_5. Also, this transformation causes γ_i,a_3,1 and γ_i,a_4,1 to always pair in the low-energy subspace—hence they can be discarded. Likewise, γ_i,a_3,2 and γ_i,a_4,2 can be thrown away.




Figure <ref> illustrates the effect of the above three transformations. Applying these unitary transformation and discarding Majoranas that are frozen in the low-energy space yields a 1D Hamiltonian H^M that is almost identical to H^L (up to some ℤ_2 Majorana gauge transformation), subject to the following difference: In H^M, there are two types of vertex terms, one originating from the fully trivalent `complete vertices' in Fig. <ref> and another originating from `incomplete vertices' where only two edges meet. Terms orignating from incomplete vertices are notably absent in H^L. However, removing these terms does not affect the subspace with A_t=1.  Hence, we conclude that the low-energy physics of H^L is completely identical to the 1D edge of our exactly-solvable topological insulator model. 




§ MORE ON THE BERRY-PHASE CALCULATION


 
The goal of this appendix is to prove Eq. (<ref>). We start by proving a related lemma.
 


 §.§ Lemma
 
Let |u(θ)⟩ denote a state with bosonic spin configuration u and Majorana pairings consistent u. Then, for any u and p, the following holds:

    ⟨u(θ)| B_p(θ) ∂ B_p(θ)/∂θ|u(θ)⟩ = 0.

Since B_p(θ) acts on some fixed spin configuration, we can write

    ⟨u(θ)| B_p(θ) ∂ B_p(θ)/∂θ|u(θ)⟩ = ⟨u(θ)|ℬ_u_p(θ)^†∂ℬ_u_p(θ)/∂θ|u(θ)⟩,

where ℬ_u_p(θ) only contains Majorana projectors (the spin parts of the two B_p operators must undo one another to get a nontrivial result).  Let us denote P_v_1s_1,v_2s_2^TI(θ) as any pair of projector that appears in the expression for ℬ_u_p(θ), and define ℬ_u_p,v_1s_1,v_2s_2(θ) as the version of ℬ_u_p(θ) with P_v_1s_1,v_2s_2^TI(θ) erased. We then have

    ⟨u(θ)|ℬ_u_p(θ)^†∂ℬ_u_p(θ)/∂θ|u(θ)⟩ = ⟨u(θ)|ℬ_u_p(θ)^†
       ∑_(v_1s_1,v_2s_2) ∈𝒫_u_p∂ P_v_1s_1,v_2s_2^TI(θ)/∂θℬ_u_p,v_1s_1,v_2s_2(θ) |u(θ)⟩
       + ∑_(v_1s_1,v_2s_2) ∈𝒫_u_pℬ_u_p,v_1s_1,v_2s_2(θ) ∂ P_v_1s_1,v_2s_2^TI(θ)/∂θ|u(θ)⟩.

 While this looks complicated, it is just a straightforward application of the product rule of derivatives. Recall that following the notation of Eq. (<ref>), (v_1s_1,v_2s_2) ∈𝒫_u_p corresponds to projectors that reconfigure Majorana pairings and are in the first parenthesis of Eq. (<ref>). In this case, following similar logic that local-fermion-parity conservation fixes one of the pairings without explicit projection, ℬ_u_p,v_1s_1,v_2s_2(θ) = ℬ_u_p(θ)! The case (v_1s_1,v_2s_2) ∈𝒫_u_p corresponds to projectors in the second parenthesis of Eq. (<ref>) that project onto Majorana pairings consistent with a spin configuration u_p around plaquette p. If Majorana pairings between (v_1s_1,v_2s_2) are reconfigured by the first parenthesis, similar logic as before guarantees that ℬ_u_p,v_1s_1,v_2s_2(θ) = ℬ_u_p(θ). If they are not reconfigured, we use the fact that P_v_1s_1,v_2s_2^TI(θ) commutes with all projectors ℬ_u_p(θ), which allows us to convert ℬ_u_p,v_1s_1,v_2s_2(θ) back to to ℬ_u_p(θ) as well. We thus deduce

    ⟨u(θ)|ℬ_u_p(θ)^†∂ℬ_u_p(θ)/∂θ|u(θ)⟩
       =  ∑_(v_1s_1,v_2s_2) ∈𝒫_u_p⟨u(θ)|ℬ_u_p(θ)^†∂ P_v_1s_1,v_2s_2^TI(θ)/∂θℬ_u_p(θ) |u(θ)⟩
       + ∑_(v_1s_1,v_2s_2) ∈𝒫_u_p⟨u(θ)|ℬ_u_p(θ)^†ℬ_u_p(θ) ∂ P_v_1s_1,v_2s_2^TI(θ)/∂θ|u(θ)⟩.

Finally, using the fact that on the third line P_v_1s_1,v_2s_2^TI|u(θ)⟩ = |u(θ)⟩ and that P^2 = P, one can equivalently substitute the partial derivatives in the above expression via

    ∂ P_v_1s_1,v_2s_2^TI(θ)/∂θ→ P_v_1s_1,v_2s_2^TI∂ P_v_1s_1,v_2s_2^TI(θ)/∂θ P_v_1s_1,v_2s_2^TI.

Appendix <ref> proved that the expression on the right side of Eq. (<ref>) vanishes. This proves Eq. (<ref>).



 §.§ Proof of Eq. (<ref>) using the lemma


Next, we note that

    ⟨↓(θ)|( ∏_p( B_p(θ) )^n_p) ( ∏_p( B_p(θ) )^n_p)  ∂/∂θ|↓(θ)⟩
       = ⟨↓(θ)|∂/∂θ|↓(θ)⟩,

where the definition of each symbol is identical to the ones in Eq. (<ref>). The proof thus really boils to down to showing

    ⟨↓(θ)|( ∏_p( B_p(θ) )^n_p) ( ∂/∂θ∏_p( B_p(θ) )^n_p)  |↓(θ)⟩ = 0.

In Eq. (<ref>) the derivative acts only on B_p operators in the second product.  

To proceed, let us set an ordering for plaquettes p_1, p_2, ⋯ p_n; the precise details of the ordering do not matter. Let us further order B_p(θ) terms in the second parenthesis according to the chosen ordering, but order terms in the first parenthesis with the inverse ordering (which can be understood as the `Hermitian conjugate' of ordering in the second parenthesis). Then, the product rule of partial derivatives allows one to show that 

    ⟨↓(θ)|( ∏_p( B_p(θ) )^n_p) ( ∂/∂θ∏_p( B_p(θ) )^n_p)  |↓(θ)⟩
       = ∑_j=1^n⟨↓(θ)|
       ( ∏_i=1^j-1( B_p_i(θ) )^n_p_i) (B_p_j(θ))^n_j( ∏_i=j+1^n( B_p_i(θ) )^n_p_i) 
       ( ∏_i=j+1^n( B_p_i(θ) )^n_p_i) ∂(B_p_j(θ))^n_j/∂θ( ∏_i=1^j-1( B_p_i(θ) )^n_p_i) |↓(θ)⟩ .

The identical product of projectors in the second and the third parenthesis multiply to 1. Also,

    ( ∏_i=1^j-1( B_p_i(θ) )^n_p_i) |↓(θ)⟩ = |u_j(θ)⟩,

where |u_j(θ)⟩ is a state with a spin configuration obtained by flipping spins of p_1, p_2, ⋯ p_j from the all-down spin configuration, and with Majorana pairings consistent with that new spin configuration. Hence, we can greatly simplify Eq. (<ref>) to be

    ⟨↓(θ)|( ∏_p( B_p(θ) )^n_p) ( ∂/∂θ∏_p( B_p(θ) )^n_p)  |↓(θ)⟩
       = ∑_j=1^n⟨u_j(θ)| (B_p_j(θ))^n_j∂(B_p_j(θ))^n_j/∂θ|u_j(θ)⟩ .
 
If n_j =0 then the summand trivially vanishes. If n_j =1 we can apply the lemma of the previous subsection to show that it also vanishes. Hence, we proved Eq. (<ref>), and the Berry phase of our topological-insulator model on a Klein bottle may be computed by computing the Berry phase at one spin configurations with corresponding Majorana pairings.




