A Commuting Projector Model with a Non-zero Quantized Hall conductance
AA Commuting Projector Model with a Non-zero Quantized Hall conductance
Michael DeMarco and Xiao-Gang Wen Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA ∗ By ungauging a recently discovered lattice rotor model for Chern-Simons theory, we create anexactly soluble path integral on spacetime lattice for U κ (1) Symmetry Protected Topological (SPT)phases in 2 + 1 dimensions with a non-zero Hall conductance. We then convert the path integralon a 2 + 1d spacetime lattice into a 2d Hamiltonian lattice model, and show that the Hamiltonianconsists of mutually commuting local projectors. We confirm the non-zero Hall conductance bycalculating the Chern number of the exact ground state. It has recently been suggested that nocommuting projector model can host a nonzero Hall conductance. We evade this no-go theorem byconsidering a rotor model, with a countably infinite number of states per site. The discovery of the Quantum Hall Effect ignited arevolution in condensed matter physics [1–3]. Both theintegral and fractional [4] forms showed the limits of thesymmetry-breaking approach to phases of matter [5–7]and new research revealed states of matter hosting frac-tional particles [8, 9], protected edge modes [10], andtopological ground state degeneracy [5]. Together withresearch directed at understanding high- T c superconduc-tors [11–14], the new era of condensed matter has un-veiled physics which is stranger, richer, and more entan-gled than before. Certainly, more is indeed different.Commuting projector models have been a central toolfor understanding the new zoo of theories. Employedmost famously by Kitaev [15] to provide an exactly solu-ble model for the previously proposed 2+1d Z topolog-ical order [12, 13] with emergent fermions and anyons,they now describe models for a wide class of string-nettopological order [16], recently unleashed a flurry of re-search on fractons [17, 18], and continue to underlie ourmicroscopic understanding of exotic phases.It is quite surprising then that no commuting projectormodel has been discovered for gapped phases with non-zero Hall conductance. It was commonly believed thatnone could exist, and recently a no-go theorem has beenproposed [19], ruling out a large class of potential theorieswith a finite Hilbert space on each site.In this paper, we describe a commuting projectormodel for U κ (1) SPT phases with non-zero quantizedHall conductance [20–23], providing an exactly solvedmodel for U κ (1) SPT states. Related to the evasion ofthe no-go theorem, the physical degrees of freedom in ourmodel are U κ (1) rotors, with a countably infinite Hilbertspace on each site. There is no clear way to reduce theon-site Hilbert space to finite dimension, while retaining U κ (1) symmetry and commuting projector property, be-cause the Lagrangian or the Hamiltonian, while local, isnot a smooth function of the rotor variables,The phases we describe are short-range-entangled SPTphases, with a unique ground state on any manifold. Em-ploying the recent discovery of a rotor path integral [24]for K -matrix Chern-Simons theory, we ungauge [25] that ∗ [email protected] model to derive another rotor path integral for general U κ (1) SPT phases. In the Hamiltonian approach, thelatter rotor path integral yields an exactly soluble com-muting projector lattice model.First, we review the Chern-Simons model and its prop-erties. We then ungauge [25] it in the presence of a back-ground field and show that integrating out the matterfields leads to a Chern-Simons response term. A con-version from the Lagrangian to Hamiltonian approachesthen yields a commuting projector Hamiltonian for themost general U κ (1) SPT state. The Hamiltonian modelleads to a unique wavefunction, which we show has theexpected Chern number over the “holonomy torus.” Fi-nally, we discuss the relation of our model to the no-gotheorem and the theory of discontinuous group cocycles. Chern-Simons Lattice Model.
In Ref. 24, a localbosonic rotor model was constructed which realizes a2+1d topological order described by U κ (1) Chern-Simonstheory with an even K -matrix. The topological order hasa chiral central charge given by the signature of the K -matrix, and hosts an exact Z k × Z k × · · · k i are the diagonal entries of the Smith normalform of K .The model is formulated in terms of cochains on aspacetime simplicial complex. A spacetime complex (lat-tice) is a triangulation of the three-dimensional space-time with a branching structure , [20, 26], denoted as M .We denote vertices of the complex as i, j, · · · , links as (cid:104) ij (cid:105) , and so forth. For the Chern-Simons Lattice modelwith a κ × κ K -matrix, the physical degrees are rotorone-cochains, i.e. a R / Z I,ij on each link (cid:104) ij (cid:105) (which corre-sponds to latice gauge field), I = 1 ...κ . For the un-gauged model they will be rotor zero-cochains, i.e. a φ R / Z I,i on each site (which corresponds to latice scaler field).We work in units where flux and cycles are quantizedto unity; for instance, U κ (1) variables may be obtainedas u U κ (1) I,ij = e πia R / Z I,ij , ϕ U κ (1) I,ij = e πiφ R / Z I,ij . Accordingly, wewill require that all quantities be invariant under a gaugeredundancy: a R / Z I,ij → a R / Z I,ij + n I,ij , φ R / Z I,i → φ R / Z I,i + n I,i (1)with n I,ij , n
I,i ∈ Z , so that these variables are genuinely R / Z -valued. In this case, the lattice gauge fields a R / Z I,ij a r X i v : . [ c ond - m a t . s t r- e l ] F e b will describe a compact U κ (1) gauge theory, and the lat-tice scaler fields φ R / Z I,i will describe a bosonic model with U κ (1) symmetry.We will make use of the lattice differential d [27] whichsends m -cochains f m to m + 1 cochains d f m and satisfiesd = 0, and the lattice cup k-products, which take an m -cochain f m and a n -cochain g n and return a n + m − k cochain f m ∪ k g n . We will abbreviate by omitting zero-cup products f m ∪ g n = f m g n . For a review of thisnotation, see [27] and the supplemental material of [24].We can now write down the Cher-Simons latticemodel. Given a bosonic K -matrix K IJ with even di-agonal entries, define k IJ = K IJ for I (cid:54) = J and k IJ = K IJ for I = J . In terms of this reduced matrix k IJ ,the path integral is: Z = (cid:90) [ (cid:89) d a R / Z I ] e i2 π (cid:80) I ≤ J k IJ (cid:82) M d (cid:0) a R / Z I ( a R / Z J −(cid:98) a R / Z J (cid:101) ) (cid:1) e i2 π (cid:80) I ≤ J k IJ (cid:82) M a R / Z I (d a R / Z J −(cid:98) d a R / Z J (cid:101) ) −(cid:98) d a R / Z I (cid:101) a R / Z J (2)e − i2 π (cid:80) I ≤ J k IJ (cid:82) M a R / Z J (cid:94) d (cid:98) d a R / Z I (cid:101) e − (cid:82) M | d a R / Z I −(cid:98) d a R / Z I (cid:101)| g Here (cid:98) x (cid:101) denotes the nearest integer to x and (cid:82) M meansthe signed sum over all 3-simplices in M induced byevaluation against the fundamental class. The measure (cid:82) [ (cid:81) d a R / Z I ] ≡ (cid:81) (cid:104) ij (cid:105) (cid:81) I (cid:82) − d a R / Z I,ij gives rise to the pathintegral, where (cid:81) (cid:104) ij (cid:105) is a product over all the links. TheMaxwell term e − (cid:82) M | d a R / Z I −(cid:98) d a R / Z I (cid:101)| g is included to maked a R / Z I nearly an integer if we choose g to be small, whichrealizes the semi-classical limit.The invariance of (2) under eq. (1), its higher symme-try, and its semiclassical limit are described in Ref. [24].Here we focus on the SPT state obtained by ungauging. SPT State from Ungauging.
To ungauge the model of(2), we set [25] a R / Z I,ij = φ R / Z I,i − φ R / Z I,j , or a R / Z I = d φ R / Z I , (3)where φ R / Z I are zero-cochains defined on the vertices. Thepartition function is now given by: Z = (cid:90) [ (cid:89) d φ R / Z I ] e i2 π (cid:80) I ≤ J k IJ (cid:82) M d φ R / Z I d (cid:98) d φ R / Z J (cid:101) (4)where M may have boundaries, and the measure istaken to be integration over all sites, (cid:82) (cid:81) d φ R / Z I = (cid:81) i (cid:81) I (cid:82) − d φ R / Z I . Eq. (4) is the path integral descrip-tion of the commuting projector model. As the actionis a total derivative, the partition function is unity onany closed manifold, implying that the model describesa trivial topological order with zero central charge. Onany spatial boundary, the path integral defines a wave-function | ψ (cid:105) , where (cid:104) φ R / Z I,i | ψ (cid:105) = exp − πi (cid:88) I ≤ J k IJ (cid:90) M dφ R / Z I (cid:98) dφ R / Z J (cid:101) (5) This will be the ground state of the commuting projectormodel.The model has a U κ (1) symmetry: φ R / Z I,i → φ R / Z I,i + θ I (6)for constant θ I . On a manifold with a boundary, themodel is invariant under eq. (1) if and only if the reduced k -matrix k IJ is integral, i.e. if the original K -matrix K IJ is integral with even diagonals (which turns out todescribe a quantuized Hall conductance and a bosonicSPT order). In this case, the field variables are indeed R / Z valued. A reduction of this model to a known one for Z n SPT states [20] is given in the Supplemental material.In fact, the model realizes a U κ (1) SPT state. To seethe SPT order, we repeat the ungauging in the presenceof a weak background gauge field ¯ a R / Z I and evaluate theeffective action for ¯ a R / Z I . In the presence of background U κ (1) background gauge field ¯ a R / Z I , the ungauing is donevia a R / Z I = ¯ a R / Z I + d φ R / Z I . (7)Now the model is given by Z = e i2 π (cid:80) I ≤ J k IJ (cid:82) M ¯ a R / Z I (d¯ a R / Z J −(cid:98) d¯ a R / Z J (cid:101) ) −(cid:98) d¯ a R / Z I (cid:101) ¯ a R / Z J e − i2 π (cid:80) I ≤ J k IJ (cid:82) M ¯ a R / Z J (cid:94) d (cid:98) d¯ a R / Z I (cid:101) (8) (cid:90) [ (cid:89) d φ R / Z I ] e − i2 π (cid:80) I ≤ J k IJ (cid:82) M d φ R / Z J (cid:94) d (cid:98) d¯ a R / Z I (cid:101) e i2 π (cid:80) I ≤ J k IJ (cid:82) M d φ R / Z I (d¯ a R / Z J −(cid:98) d¯ a R / Z J (cid:101) ) −(cid:98) d¯ a R / Z I (cid:101) d φ R / Z J e i2 π (cid:80) I ≤ J k IJ (cid:82) ∂ M (¯ a R / Z I +d φ R / Z I )(¯ a R / Z J +d φ R / Z J −(cid:98) ¯ a R / Z J +d φ R / Z J (cid:101) ) If M is closed, this can be simplified to Z = e i2 π (cid:80) I ≤ J k IJ (cid:82) M ¯ a R / Z I (d¯ a R / Z J −(cid:98) d¯ a R / Z J (cid:101) ) −(cid:98) d¯ a R / Z I (cid:101) ¯ a R / Z J e − i2 π (cid:80) I ≤ J k IJ (cid:82) M ¯ a R / Z J (cid:94) d (cid:98) d¯ a R / Z I (cid:101) (9) (cid:90) [ (cid:89) d φ R / Z I ] e − i2 π (cid:80) I ≤ J k IJ (cid:82) M d φ R / Z J (cid:94) d (cid:98) d¯ a R / Z I (cid:101) e − i2 π (cid:80) I ≤ J k IJ (cid:82) M d φ R / Z I (cid:98) d¯ a R / Z J (cid:101) + (cid:98) d¯ a R / Z I (cid:101) d φ R / Z J Now we assume that the background field is “weak.” Be-cause we are working with R / Z valued fields, a weak fieldmeans that d¯ a R / Z I is nearly an integer, i.e. | d¯ a R / Z I −(cid:98) d¯ a R / Z I (cid:101)| < (cid:15) . Noting that this implies that d (cid:98) d¯ a R / Z I (cid:101) =0, the path integral becomes: Z = e i2 π (cid:80) I ≤ J k IJ (cid:82) M ¯ a R / Z I (d¯ a R / Z J −(cid:98) d¯ a R / Z J (cid:101) ) −(cid:98) d¯ a R / Z I (cid:101) ¯ a R / Z J (10)This is the Chern-Simons response on lattice. When M is closed, the action is invariant under gauge transforma-tions of the background gauge field ¯ a R / Z I → ¯ a R / Z I + d ϕ R / Z I .If M is a disk, the redundancy (1) can be used to set (cid:98) d¯ a R / Z I (cid:101) = 0, and the response becomes: Z = e i π (cid:80) I,J K IJ (cid:82) M ¯ a R / Z I d¯ a R / Z J (11)This Chern-Simons response, in terms of the unreduced K -matrix K IJ , describes the Hall conductance and is theSPT invariant for our model.Given that the bulk behavior of the path integral istrivial, we expect that we should be able to create anexactly soluble Hamiltonian model to describe the time-evolution. Furthermore, because the path integral definesa wavefunction on any spatial boundary independently of the bulk dynamics, we expect that this Hamiltonianmodel should be a commuting projector onto a groundstate. As we shall now see, both are true. Commuting Projector Model.
We can use the space-time formalism to construct a commuting projectorHamiltonian on a triangular lattice of the sort shown inFig. 1a. To do so, we consider the time evolution of asingle site, φ R / Z → φ R / Z , while preserving the orienta-tion of lattice links as shown in fig 1b. Evaluating thepath integral on the complex shown yields the matrix el-ements for the transition φ R / Z → φ R / Z as a function ofthe surrounding φ R / Z i : M φ R / Z I, → φ R / Z I, ( φ R / Z I, , ..., φ R / Z I, ) = exp (cid:110) πi (cid:88) I ≤ J k IJ (cid:104) φ R / Z I, (cid:16) (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) (cid:17) + φ R / Z I, (cid:16) (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) (cid:17) + φ R / Z I, (cid:16) (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) (cid:17) + φ R / Z I, (cid:16) (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) (cid:17) + φ R / Z I, (cid:16) (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) (cid:17)(cid:105)(cid:111) (12)We will interpret this transition amplitude as the ma-trix element for an operator ˆ M acting on site-4. How-ever, eq. (12) is somewhat daunting. Let us set con-sider ˆ M as an operator acting only on the Hilbert spaceon site-4. If k IJ = 0, then (cid:104) φ (cid:48) I, | ˆ M | φ I, (cid:105) = 1, andˆ M is simply the projector onto the state with zero an-gular momentum in each U (1). For nonzero k IJ , wemay rewrite the transition amplitude as M φ R / Z I, → φ R / Z I, =exp(2 πi ( f ( φ R / Z I, ) − f ( φ R / Z I, )) (note that this implies her-miticity) where f ( φ ) is a function defined in the appendixthat depends on φ and takes as parameters φ R / Z I, ...φ R / Z I, ,but not φ R / Z I, or φ R / Z I, . This implies that, up to an overallphase, the ˆ M i act asˆ M | φ I (cid:105) ∝ (cid:90) dφ I e πif ( φ I ) | φ I (cid:105) (13)We see then that these projectors may be thought of as‘twisting’ the zero angular momentum state by an phasefunction f ( φ I ) which depends on the surrounding valuesof φ I,i . The phase itself is determined by the cocycle ofthe action in (4).We may construct ˆ M i for an entire lattice. As shownin the supplemental material, the ˆ M i are hermitian and U κ (1) symmetric under φ R / Z I,i → φ R / Z I,i + θ I + n I,i .The ˆ M i inherit a number of remarkable properties fromthe fact that the 2+1d path integral action contains onlya surface term. First, they mutually commute: considerthe three spacetime complexes in figure 1c, which ad- dresses the only nontrivial case of two adjacent ˆ M i . Thetwo colored complexes correspond to time evolving eitherthe blue site followed by the red or the red followed bythe blue, respectively. Because the action contains only asurface term, and the surfaces are identical, it assigns thesame amplitude to both cases. Back in the Hamiltonianpicture, this implies that [ ˆ M i , ˆ M j ] = 0.For the same reason, the ˆ M i are projectors. The fun- FIG. 1. (Color Online).
We construct a commuting projectormodel for the lattice in ( a ) by evaluating the spacetime pathintegral for the complex in ( b ) and turning the amplitude intooperators. Because the path integral contains only a surfaceterm, we can show that ( c ) the matrix is hermitian, (d) theoperators commute, and ( e ) they are projectors. FIG. 2. (Color Online) . ( a ) The commuting projectors act ona hexagon to define a ground state. ( b ) To calculate the Chernnumber of the ground state wavefunction over the holonomytorus, we twist the boundary conditions by θ x n I and θ y m I inthe x and y directions, respectively. damental mechanism is illustrated in Fig. 1d, where wesee the effect of time-evolving twice. In the language ofeq. 12, this is the expression M φ R / Z I, → φ R / Z I, M φ R / Z I, → φ R / Z I, ; inthe Hamiltonian picture this is ˆ M i . However, becausethe path integral action depends only on the values of φ R / Z I on the surface, we could equally well drop site 5and its associated links; the amplitude will not change.This implies that ˆ M i = ˆ M i .The aforementioned hermiticity of the ˆ M i is also due tothis reason, as reversing the orientation of the link (cid:104) , (cid:105) in Fig. 1c changes the amplitude by complex conjugation.All of these properties of the ˆ M i are demonstrated in theSupplemental Material.Using the mutually commuting projectors ˆ M i , we candefine a Hamiltonian H = − g (cid:88) i ˆ M i (14)to obtain a system with ground state | ψ (cid:105) from eq. (5)and gap g .With a model producing the ground state eq. (5) nowin hand, we return to the Hall conductance. We havealready argued from the path integral that coupling thesystem to a background gauge field leads to the expectedChern-Simons response function, but here we appeal di-rectly to the wavefunction of the system on a torus toconfirm the hall conductance.Consider the wavefunction on the lattice shown in Fig.2. We twist the boundary conditions by θ x n I , θ y m I ,so that φ R / Z I,x =0 ,y = φ R / Z I,x = L x ,y − θ x n I and φ R / Z I,x,y =0 = φ R / Z I,x,y = L y − θ y m I , where the integral vectors n I , m I al-low us to describe both direct ( n I = m I ) and mixed hallresponses. Denote the ground state with boundary con-ditions θ x , θ y by | θ x , θ y (cid:105) . We are interested in the phaseaccumulated when we twist the boundary conditions byan integer, sending θ x → θ x + (cid:96) x , θ y → θ y + (cid:96) y with (cid:96) x , (cid:96) y ∈ Z . As shown in the Appendix, the wavefunctiontransforms as: | θ x + (cid:96) x , θ y + (cid:96) y (cid:105) = e − πi (cid:80) I ≤ J k IJ [ n I θ x m J (cid:96) y − m I θ y n J (cid:96) x ] | θ x , θ y (cid:105) (15) To make sense of this, apply the gauge transformation: | θ x , θ y (cid:105) → e − πi (cid:80) I ≤ J k IJ n J m I θ x θ y | θ x , θ y (cid:105) (16)In this gauge, one may replace the phase in eq. (15) by: e − πi (cid:80) I ≤ J k IJ ( n I m J + n J m I ) θ x (cid:96) y (17)We should recognize this as the boundary conditions fora particle on a torus with flux: (cid:88) I ≤ J k IJ m I n J + (cid:88) I ≥ J k IJ m I n J = (cid:88) I,J K IJ n I m J (18)We see then that our Hamiltonian system has the (mixed)Hall conductance n · K · m , in agreement with the Chern-Simons response function derived from the spacetimepath integral. In the case of κ = 1 with n = m = 1,this becomes a system with integer hall conductance K wich is an even integer. Discussion.
We have derived a model for the 2 + 1d U κ (1) SPT states in terms of both a spacetime latticepath integral and a commuting projector model. We haveconfirmed the Hall response in two ways: by coupling thespacetime model to a background gauge field, and by ex-amining the ground state wavefunction of the Hamilto-nian model on a torus with twisted boundary conditions.Now we must understand how this model evades the no-go theorem.The infinite dimensional on-site Hilbert space of therotors, combined with an action that is not a continuousfunction of the field variables, is what allows this model toexist. The discontinuous action is critical for commutingprojectors; if it were continuous, then one could trun-cate the Hilbert space to low-angular momentum modesand render the on-site Hilbert space finite while retainingthe full U κ (1) symmetry and commuting-projector prop-erty, hence running afoul of the no-go theorem [19]. Con-versely, the no-go theorem assumes that the ground statewavefunction is a finite Laurent polynomial in e iθ x , e iθ y ,an assumption which is violated in our model (See thedetails of the Chern number calculation in the Appendixfor an example). This commuting projector model rep-resents a fixed-point theory for U κ (1) SPT phases withnonzero Hall conductance; it may be that any such fixed-point theory requires an infinite on-site Hilbert space.There is a connection here to the theory of discontinu-ous group cocycles. SPT phases in d + 1 dimensions areclassified by H d +1 ( G, U (1)) [20], and their topologicalfield theory actions are given by representative cocycles.Eq. (4) is an example of this sort of cocycle. For reasonssimilar to the underlying argument of the no-go theorem,if we restrict to continuous cochains, then corrsponding H ( G, U (1)) is not large enough to classify thegroup extensions of G by U (1) [28]. It is only by allow-ing piecewise-continuous cochains, like the action in eq.(4), does corrsponding H ( G, U (1)) classify the groupextensions, as well as the projective representations of G and the 1+1d SPT orders [20].This research is partially supported by NSF DMR-2022428, the NSF Graduate Research Fellowship under Grant No. 1745302, and by the Simons Collaboration onUltra-Quantum Matter, which is a grant from the SimonsFoundation (651440). [1] K. von Klitzing, T. Chakraborty, P. Kim, V. Madhavan,X. Dai, J. McIver, Y. Tokura, L. Savary, D. Smirnova,A. M. Rey, C. Felser, J. Gooth, and X. Qi, Nature Re-views Physics , 397 (2020).[2] H. L. Stormer, D. C. Tsui, and A. C. Gossard, Rev.Mod. Phys. , S298 (1999).[3] A. Zee, in Field Theory, Topology and Condensed MatterPhysics , edited by H. B. Geyer (Springer Berlin Heidel-berg, Berlin, Heidelberg, 1995) pp. 99–153.[4] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys.Rev. Lett. , 1559 (1982).[5] X.-G. Wen, Phys. Rev. B , 7387 (1989).[6] X.-G. Wen, Int. J. Mod. Phys. B , 239 (1990).[7] T. Senthil, arXiv e-prints , cond-mat/0411275 (2004),arXiv:cond-mat/0411275 [cond-mat.str-el].[8] R. B. Laughlin, Phys. Rev. Lett. , 1395 (1983).[9] A. Stern, Annals of Physics , 204 (2008), january Spe-cial Issue 2008.[10] X.-G. Wen, Adv. Phys. , 405 (1995), arXiv:cond-mat/9506066.[11] X.-G. Wen, F. Wilczek, and A. Zee, Phys. Rev. B ,11413 (1989).[12] N. Read and S. Sachdev, Phys. Rev. Lett. , 1773(1991).[13] X.-G. Wen, Phys. Rev. B , 2664 (1991).[14] T. Senthil and M. P. A. Fisher, Phys. Rev. B , 7850(2000).[15] A. Y. Kitaev, Annals Phys. , 2 (2003), arXiv:quant-ph/9707021.[16] M. A. Levin and X.-G. Wen, Phys. Rev. B , 045110(2005), arXiv:cond-mat/0404617.[17] C. Chamon, Phys. Rev. Lett. , 040402 (2005),arXiv:cond-mat/0404182.[18] S. Vijay, J. Haah, and L. Fu, Phys. Rev. B , 235136(2015), arXiv:1505.02576 [cond-mat.str-el].[19] A. Kapustin and L. Fidkowski, Communications in Math-ematical Physics , 763 (2020).[20] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Phys.Rev. B , 155114 (2013), arXiv:1106.4772.[21] Y.-M. Lu and A. Vishwanath, Phys. Rev. B , 125119(2012), arXiv:1205.3156.[22] Z.-X. Liu and X.-G. Wen, Phys. Rev. Lett. , 067205(2013), arXiv:1205.7024.[23] T. Senthil and M. Levin, Phys. Rev. Lett. , 046801(2013), arXiv:1206.1604.[24] M. DeMarco and X.-G. Wen, Phys. Rev. Lett. ,021603 (2021).[25] X.-G. Wen, Phys. Rev. B , 205139 (2019),arXiv:1812.02517.[26] F. Costantino, Math. Z. , 427 (2005), math/0403014.[27] A. Hatcher, Algebraic topology (Cambridge Univ. Press,Cambridge, 2000).[28] F. Wagemann and C. Wockel, Trans. Amer. Math. Soc. , 1871 (2014), arXiv:1110.3304.[29] X.-G. Wen, Phys. Rev. B , 205142 (2017). Appendix A: Chern Number for Hall Conductance
Working with the wavefunction on the lattice shownin Fig. 2, recall that we label lattice points by ( x, y ) ∈ [0 ...L x − × [0 ...L y − θ x n I , θ y m I , so that φ R / Z I,x =0 ,y = φ R / Z I,x = L x ,y − θ x n I and φ R / Z I,x,y =0 = φ R / Z I,x,y = L y − θ y m I , with n I , m I ∈ Z κ .Consider first the plaquettes marked in red. The con-tribution of these plaquettes to the path integral is of theform:exp (cid:110) − πi (cid:88) I ≤ J k IJ (cid:104) ( φ R / Z I,x = L,y + n I θ x − φ R / Z I,x = L x − ,y ) (cid:98) φ R / Z J,x = L x ,y +1 − φ R / Z J,x = L x ,y (cid:101)− (cid:16) ( φ R / Z I,x = L x − ,y +1 − φ R / Z I,x = L x − ,y ) × (cid:98) φ R / Z J,x = L x ,y +1 + n J θ x − φ R / Z J,x = L x − ,y +1 (cid:101) (cid:17)(cid:105)(cid:111) Incrementing θ x by (cid:96) x changes the first term by an inte-ger, while the second term changes by ( φ R / Z I,x = L x − ,y +1 − φ R / Z I,x = L x − ,y ) (cid:96) x , and so the multiplicative change on thepath integral is: e πi (cid:80) I ≤ J k IJ ( φ R / Z I,x = Lx − ,y +1 − φ R / Z I,x = Lx − ,y ) n J (cid:96) x (A1)Similarly, the plaquettes marked in blue change by: e − πi (cid:80) I ≤ J k IJ ( φ R / Z I,x +1 ,y = Ly − − φ R / Z I,x,y = Ly − ) m J (cid:96) y (A2)On the purple plaquettes at the corner, the contributionto the path integral is:exp (cid:110) − πi (cid:88) I ≤ J k IJ (cid:104)(cid:16) ( φ R / Z I,x = L x ,y = L y − + n I θ x − φ R / Z I,x = L x − ,y = L y − ) × (cid:98) φ R / Z J,x = L x ,y = L y + m J θ y − φ R / Z J,x = L x ,y = L y − (cid:101) (cid:17) − (cid:16) ( φ R / Z I,x = L x − ,y = L y + m I θ y − φ R / Z I,x = L x − ,y = L y − ) × (cid:98) φ R / Z J,x = L x ,y = L y + n J θ x − φ R / Z J,x = L x − ,y = L y (cid:101) (cid:17)(cid:105)(cid:111) which will change by:exp (cid:110) − πi (cid:88) I ≤ J k IJ (cid:104) ( φ R / Z I,x = L x ,y = L y − + n I θ x − φ R / Z I,x = L x − ,y = L y − ) m J (cid:96) y − ( φ R / Z I,x = L x − ,y = L y + m I θ y − φ R / Z I,x = L x − ,y = L y − ) n J (cid:96) x (cid:105)(cid:111) (A3)Combining the change on the red, blue, and purple pla-quettes, the overall change to the wavefunction is: e − πi (cid:80) I ≤ J k IJ (cid:104) n I θ x m J (cid:96) y − m I θ y n J (cid:96) x − n J (cid:96) x (cid:82) γ dφ R / Z I + m J (cid:96) Y (cid:82) γ dφ R / Z I (cid:105) (A4)where γ , γ are the red and blue loops in Fig. 2, respec-tively. As that γ , γ are closed, the sums along thosecurves vanish, and we are left with the result in the maintext. Appendix B: Reducing to Z n gauge theory When κ = 1, the above becomes Z = (cid:90) [ (cid:89) d φ R / Z ] e i2 πk (cid:82) M d φ R / Z d (cid:98) d φ R / Z (cid:101) , (B1)which describe a U (1) SPT state with Hall conductance σ xy = k π .Let us compare the model (B1) for U (1) SPT statewith a model for Z n SPT state [29]: Z = (cid:88) φ Z n e i πkn (cid:82) M a Z n d a Z n ,a Z n = d φ Z n − n (cid:98) n d φ Z n (cid:101) , (B2)where φ Z n is a Z n -valued 0-cochain. The above can berewritten as Z = (cid:88) φ Z n e − i2 πk (cid:82) M d φ R / Z d (cid:98) d φ R / Z (cid:101) , φ R / Z = 1 n φ Z n . (B3)We see that the model for the Z n SPT state and themodel for the U (1) SPT state have very similar forms.Here, d φ R / Z d (cid:98) d φ R / Z (cid:101) with φ R / Z = n φ Z n is a cocyclein H ( Z n , R / Z ), while d φ R / Z d (cid:98) d φ R / Z (cid:101) is a cocycle in H ( U (1) , R / Z ). Appendix C: Properties of the CommutingProjectors
Recall the definition of the ˆ M i : M φ R / Z I, → φ R / Z I, ( φ R / Z I, , ..., φ R / Z I, ) = exp (cid:110) πi (cid:88) I ≤ J k IJ (cid:16) φ R / Z I, ( (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) )+ φ R / Z I, ( (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) )+ φ R / Z I, ( (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) )+ φ R / Z I, ( (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) )+ φ R / Z I, ( (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) ) (cid:17)(cid:111) (C1)Here we walk through the calculations to show analyti-cally that the ˆ M i are hermitian, U κ (1) symmetric, com-muting projectors.For hermiticity, we wish to show that M φ R / Z I, → φ R / Z I, ( φ R / Z I, , ..., φ R / Z I, ) ∗ = M φ R / Z I, → φ R / Z I, ( φ R / Z I, , ..., φ R / Z I, )(C2)To see this, first note that the terms in eq. (C1) with co-efficients of φ R / Z I, , φ R / Z I, , or φ R / Z I, are antisymmetric under φ R / Z I, ↔ φ R / Z I, . Next, consider the φ R / Z I, term: φ R / Z I, ( (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) ) (C3)under φ R / Z I, ↔ φ R / Z I, , this becomes: − φ R / Z I, ( (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) ) (C4)Which is precisely − φ R / Z I, term. Similarly,the φ R / Z I, term becomes minus the φ R / Z I, term. Taking allof these together with the minus sign from the factor of i , we see that the ˆ M i are hermitian.The symmetry φ R / Z I → φ R / Z I + θ I arises es-sentially because only dφ R / Z I appears in action2 π (cid:80) I ≤ J k IJ dφ R / Z I (cid:98) dφ R / Z J (cid:101) . To see this explicitly in eq.C1, first note that the θ I cancel in the rounded terms.What remains is: M φ R / Z I, → φ R / Z I, ( φ R / Z I, , ..., φ R / Z I, ) =exp (cid:110) πi (cid:88) I ≤ J k IJ (cid:16) ( φ R / Z I, + θ I )( (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) )+ ( φ R / Z I, + θ I )( (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) )+ ( φ R / Z I, + θ I )( (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) )+ ( φ R / Z I, + θ I )( (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) )+ ( φ R / Z I, + θ I ))( (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) ) (cid:17) = M φ R / Z I, → φ R / Z I, ( φ R / Z I, , ..., φ R / Z I, ) exp (cid:110) πi (cid:88) I ≤ J k IJ θ I (cid:16) (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ R / Z J, − φ R / Z J, (cid:101) (cid:17)(cid:111) = M φ R / Z I, → φ R / Z I, ( φ R / Z I, , ..., φ R / Z I, ) (C5)
03 214 5 67 8 9
FIG. 3.
One may check that remaining rounded terms cancelone-by-one, essentially because this is d (cid:98) d φ R / Z I (cid:101) evalu-ated over the closed surface of the complex. To see thatthe ˆ M i are symmetric under φ R / Z I,i → φ R / Z I,i + n I,i , firstnote that, because k IJ is integral, we need only to worryabout the rounded terms. However, one may check thateach the effect n I,i in each sum of rounded terms can-cels, essentially because each sum is d (cid:98) d φ R / Z I (cid:101) , and un-der φ R / Z I → φ R / Z I + n I this becomes d (cid:98) d φ R / Z I + d n I (cid:101) =d (cid:98) d φ R / Z I (cid:101) + d n I and d = 0. All told, we now see that the ˆ M i are symmetric under φ R / Z I,i → φ R / Z I,i + n I,i + θ I , i.e.the ˆ M i are U κ (1) symmetric.Next we check commutation. The only nontrivial caseoccurs when the ˆ M i are on adjacent sites. For this cal-culation, we will use a slightly different convention, in-dicating the time-evolved points with a prime as op-posed to a new number, so that amplitudes take the form M φ R / Z I, → ( φ R / Z I, ) (cid:48) ( ... ). We also drop the R / Z superscripts.Consider then the 2 d spatial complex in Fig. 3. We wishto compare:ˆ M ˆ M = M φ I, → φ (cid:48) I, ( φ I, , ..., φ (cid:48) I, , ..., φ I, ) (C6) × M φ I, → φ (cid:48) I, ( φ I, , ..., φ I, )to ˆ M ˆ M = M φ I, → φ (cid:48) I, ( φ I, , ..., φ (cid:48) I, , ..., φ I, ) (C7) × M φ I, → φ (cid:48) I, ( φ I, , ..., φ I, )Expanding eq. (C7), it becomes:ˆ M ˆ M = exp (cid:40) πi (cid:88) I ≤ J k IJ (cid:32) φ I, ( (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ (cid:48) I, ( (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ (cid:48) J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ (cid:48) J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ (cid:48) I, ( (cid:98) φ (cid:48) J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ (cid:48) J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ (cid:48) I, ( (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ (cid:48) J, − φ J, (cid:101) ) (cid:33)(cid:41) (C8)On the other hand,ˆ M ˆ M = exp (cid:40) πi (cid:88) I ≤ J k IJ (cid:32) φ I, ( (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48) J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ (cid:48) J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) )+ φ (cid:48) I, ( (cid:98) φ (cid:48) J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ (cid:48) J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ (cid:48) I, ( (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ (cid:48) J, − φ J, (cid:101) ) (cid:33)(cid:41) (C9)Proceeding term-by-term, one may verify that these areequal. (To simplify the calculation, note that the termswith coefficients φ I, , φ I, , φ I, , φ I, , and φ (cid:48) I, are identi- cal.)We also verify that the ˆ M i are projectors. Retainingthe same notation, we calculate: M φ (cid:48) I, → φ (cid:48)(cid:48) I, ( φ I, , ..., φ (cid:48) I, , ..., φ I, ) M φ I, → φ (cid:48) I, ( φ I, , ..., φ I, )= exp (cid:40) πi (cid:88) I ≤ J k IJ (cid:32) φ I, ( (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ (cid:48) I, ( (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ (cid:48) J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48)(cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48)(cid:48) J, (cid:101) + (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48)(cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48)(cid:48) J, (cid:101) + (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48)(cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48)(cid:48) J, (cid:101) + (cid:98) φ (cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48) J, (cid:101) )+ φ (cid:48) I, ( (cid:98) φ J, − φ (cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ (cid:48) J, − φ J, (cid:101) )+ φ (cid:48)(cid:48) I, ( (cid:98) φ J, − φ (cid:48)(cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ (cid:48)(cid:48) J, − φ J, (cid:101) ) (cid:33)(cid:41) = exp (cid:40) πi (cid:88) I ≤ J k IJ (cid:32) φ I, ( (cid:98) φ (cid:48)(cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48)(cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48)(cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48)(cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ (cid:48)(cid:48) J, − φ J, (cid:101) + (cid:98) φ J, − φ (cid:48)(cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ (cid:48)(cid:48) I, ( (cid:98) φ J, − φ (cid:48)(cid:48) J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ (cid:48)(cid:48) J, − φ J, (cid:101) ) (cid:33)(cid:41) = M φ I, → φ (cid:48)(cid:48) I, ( φ I, , ..., φ I, ) (C10)Whence ˆ M i =4 = ˆ M i =4 , and by translation ˆ M i = ˆ M i forthe entire lattice.Finally, we also note that the ˆ M i are mutually inde-pendent, so there is no condition which could allow extraground state degeneracy as in the toric code. Appendix D: Rewriting of the ˆ M i Defining f ( φ ∗ ) = (cid:88) I ≤ J k IJ (cid:104) φ I, ∗ ( (cid:98) φ R / Z J, − φ R / Z J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) )+ φ I, ( (cid:98) φ J, − φ J, (cid:101) + (cid:98) φ J, − φ J, (cid:101) ) (cid:105) (D1)we can see that eq. (C1) may be rewritten as: M φ R / Z I, → φ R / Z I, ( ... )( φ R / Z I, )= exp(2 πif ( φ R / Z I, )) exp( − πif ( φ R / Z I,4