Local Integrals of Motion for Topologically Ordered Many-Body Localized Systems
aa r X i v : . [ c ond - m a t . d i s - nn ] J u l Local Integrals of Motion for Topologically Ordered Many-Body Localized Systems
Thorsten B. Wahl and Benjamin B´eri
1, 2 DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom T.C.M. Group, Cavendish Laboratory, University of Cambridge,J.J. Thomson Avenue, Cambridge, CB3 0HE, United Kingdom
Many-body localized (MBL) systems are often described using their local integrals of motion,which, for spin systems, are commonly assumed to be a local unitary transform of the set of on-site spin- z operators. We show that this assumption cannot hold for topologically ordered MBLsystems. Using a suitable definition to capture such systems in any spatial dimension, we demon-strate a number of features, including that MBL topological order, if present: (i) is the same for alleigenstates; (ii) is robust in character against any perturbation preserving MBL; (iii) implies thaton topologically nontrivial manifolds a complete set of integrals of motion must include nonlocalones in the form of local-unitary-dressed noncontractible Wilson loops. Our approach is well suitedfor tensor-network methods, and is expected to allow these to resolve highly-excited finite-size-splittopological eigenspaces despite their overlap in energy. We illustrate our approach on the disorderedKitaev chain, toric code, and X-cube model. I. INTRODUCTION
Systems displaying many-body localization [1, 2](MBL) violate the eigenstate thermalization hypothe-sis [3] and therefore do not thermalize. (See Refs. 4, and5 for some recent reviews.) MBL occurs in strongly disor-dered interacting lattice systems. Recent analytical andnumerical work has put the effect on firm theoretical foot-ing in one dimension [2, 6–8], whereas the existence ofMBL in higher dimensions is still debated [9, 10]. How-ever, MBL has been observed experimentally both in one-[11] and two-dimensional systems [12]. This might be dueto extremely long relaxation times [13], which are unob-servable in the experiments. Numerical simulations areconsistent with two-dimensional MBL-like behavior [14].All eigenstates of MBL systems are area-law entan-gled [15, 16]. This makes MBL compatible with the sce-nario where all eigenstates are topologically ordered [15,17–24]. In one-dimensional systems, nontrivial topologyrequires the presence of certain symmetries. In higherdimensions, however, topological states can exist with-out symmetries and have fractionalized quasiparticles(anyons) and spectral degeneracies dependent only onthe topology of the system’s configuration space (the twofeatures are interlinked [25]). It is these states that wecall here topologically ordered, while we call topologicalstates requiring symmetries for their existence symmetry-protected topological (SPT) states [26]. Topologicallyordered MBL systems have been suggested to provideimproved protection of quantum information against per-turbations compared to their clean counterparts [15, 27].Fully MBL (FMBL) systems can be described in termsof local integrals of motion (LIOMs) , which are exponen-tially localized operators which commute with each otherand the Hamiltonian [5, 28–36]. LIOMs are commonlyassumed to form a complete set arising as a local unitarytransform of the set of on-site spin- z operators. Thistacitly assumes the absence of topological order: it im-plies that FMBL eigenstates arise from product states ofthis spin- z basis via local unitary transformation, which guarantees [37, 38] that they are topologically trivial.Topologically ordered FMBL systems therefore requirea more general notion of LIOMs. The key goals of this pa-per are to (i) develop such a “ topological LIOM (tLIOM) ”notion and thereby provide a precise definition of FMBLwith topological order; to investigate (ii) what proper-ties of topological FMBL phases follow from this defini-tion, and (iii) how tLIOMs may be used to characterizesuch phases in numerical simulations. The concept oftLIOMs can be illuminated by placing LIOMs into thebroader context of the stabilizer formalism [39]. Fromthis perspective, one sees the on-site spin- z operators asjust one choice of local stabilizers, namely those of prod-uct states in this spin- z basis. Topological LIOMs mustthen correspond to a different set, namely the local sta-bilizers in the commuting projector limits of topologicalphases. These tLIOMs are closely related to the approachof Ref. 19 to many-body localizability, where the use ofcommuting projector limits as reference points was firstsuggested and used to investigate topological MBL stateswithout fractionalized quasiparticles (i.e., “integer” topo-logical states and SPTs). However, since all non-chiralforms of Abelian topological order admit such commutingprojector limit [40], the tLIOMs capture all Abelian non-chiral FMBL eigenstate topological orders. The scopeis not restricted by dimensionality; it even includes pu-tative FMBL cousins of recently introduced fractonicphases [41–44]. Furthermore, tLIOMs lend themselvesto be combined with tensor-network methods thus farused for non-topological FMBL systems [7, 8, 14]. ThistLIOM—tensor-network combination is the perspectivethrough which we shall seek new avenues for the numer-ical characterization of topological FMBL systems. II. NON-TOPOLOGICAL FMBL
For concreteness we consider an N -site spin-1 / d -dimensional square lattice. The ex-tension to higher-spin systems and fermionic systems is FIG. 1. A one-dimensional depth-two quantum circuit withlength- ℓ unitaries (denoted as boxes). In d -dimensional sys-tems, ℓ is allowed to grow sublinearly with N /d [8]. straightforward. The FMBL phase is defined by a com-plete set of LIOMs τ zi ( i = 1 , . . . , N ) which commutewith the Hamiltonian H and each other,[ H, τ zi ] = [ τ zi , τ zj ] = 0 , (1)and are exponentially localized, i.e., their non-trivial ma-trix elements decay exponentially with distance from site i . The corresponding decay length, the localization length ξ i , must fulfill ξ i /N /d → i in the thermodynamiclimit N → ∞ . The τ zi can be constructed from a unitary U which diagonalizes the Hamiltonian as τ zi = U σ zi U † ,where σ zi is the third Pauli operator acting on site i .A key feature that makes nontopological FMBL systemsspecial, and computationally tractable [7, 8, 14], is that U is a local unitary , i.e., efficiently approximable by aconstant-depth quantum circuit of unitary gates withlength (i.e., linear size) ℓ sublinear in N /d [8]. [Theerror scales as exp( − ℓ/ξ max ) where ξ max is the largest ξ i .] Fig. 1 shows a depth-two example.The τ zi ’s are also known as l-bits (l=localized) [29];they label, due to Eq. (1), all eigenstates | ψ l ...l N i of H , τ zi | ψ l ...l N i = l i | ψ l ...l N i , l i = ± . (2)The σ zi operators are called p-bits (p=physical) [29].Eq. (2) implies | ψ l l ...l N i = U | l l . . . l N i in terms of thep-bit product states | l l . . . l N i . For d >
1, the LIOMdescription might not apply exactly due to delocalizationon extremely long time scales [9, 13]. However, on exper-imental time scales, the description in terms of LIOMsseems to be appropriate [14], which is what we restrictourselves to in the following.
III. TOPOLOGICAL FMBL PHASES
In the topological FMBL case it is conjectured that alleigenstates display topological order [15, 17, 20]. How-ever, even if only one eigenstate | ψ l l ...l N i is topologi-cally non-trivial, there exists no local unitary U such that | ψ l l ...l N i = U | l l . . . l N i [37, 38]. Hence, if a LIOM de-scription applies, it is not of the form τ zi = U σ zi U † . Wemust therefore use a more general notion of (t)LIOMsappropriate also for topological FMBL systems.We begin by explaining how p-bits and the correspond-ing product eigenstates fit in the broader context of local stabilizer codes [39] (see also Ref. 19 for a closely relatedapproach to MBL). For concreteness, we present the ideafor N -site qubit systems, however the scope of stabilizercodes, and hence our construction, is much more general.A local stabilizer code may be thought of in terms of astabilizer Hamiltonian H sc = − N X i =1 S i , (3)where the local stabilizers S i = S † i are Pauli strings withlocal support around site i , and [ S i , S j ] = 0. (Thus, up toa constant, H sc is a commuting-projector Hamiltonian.)On a topologically trivial manifold, S i are independent( Q i S x i i = 1 only if x i = 0 , ∀ i ; x i ∈ { , } ), hence theireigenvalues s i = ± S i as s-bits . The p-bits and their producteigenstates correspond to S i = σ zi and the eigenstates of H sc , respectively. The scope of stabilizer codes, however,is much wider (and not limited to qubits): they cap-ture all non-chiral topological orders [40], from the toriccode [45] to fracton models [41–44], as well as fermionicsystems such as the Kitaev chain [46] or various otherMajorana fermion codes [47]. We propose the followingdefinition of FMBL and (t)LIOMs to capture Abeliantopological order on topologically trivial manifolds: Definition 1.
Let S i , i = 1 , . . . , N be a complete setof local stabilizers on a topologically trivial manifold M .We call a local Hamiltonian H on M FMBL, if thereexists a local unitary ˜ U such that T i = ˜ U S i ˜ U † , [ H, T i ] = [ T i , T j ] = 0 , (4) and the same property holds in an ǫ > open envi-ronment around H [i.e., for local perturbations δH ofstrength k δH k / k H k < ǫ where k · k is a suitable norm],where for N sufficiently large, ǫ does not depend on N [i.e., ǫ is nonzero in the thermodynamic limit]. We callthe complete set T i tLIOMs (or topological l-bits) if S i are the stabilizers of topologically ordered states. We shall come back to discussing various aspects ofour definition, including observations for topologicallynontrivial M and non-Abelian topological order. Fornow, we note that Definition 1 implies, firstly, | ψ { s i } i =˜ U |{ s i }i where | ψ { s i } i and |{ s i }i are respective eigen-states of the FMBL Hamiltonian H and H sc . Secondly,up to an additive constant, and with c ijk... ∈ R , H = X i c i T i + X i The FMBL eigenstates have the sametopological order as those of H sc . Furthermore, since any set of the stabilizers S i can beflipped by a suitable Pauli string [39], and since any Paulistring is a local unitary, every eigenstate of H sc has thesame topological order. Hence: Statement 2. All eigenstates of topological FMBL sys-tems display the same topological order. Definition 1 requires that if the system is FMBL, itshould also be FMBL after having applied a local per-turbation (e.g., a translation-invariant nearest-neighborcoupling term) of relative strength ǫ , and this should holdfor ǫ sufficently small but nonzero even in the thermody-namic limit: otherwise we would be at a phase transitionpoint. The requirement of robustness against perturba-tions in a nonzero environment around the Hamiltonian H thus amounts to describing a localized phase .Conversely, Definition 1 excludes systems with(t)LIOMs that easily delocalize. An example of such sys-tems is H sc itself [which satisfies Eq. (4) with ˜ U = 1], asillustrated by S i = σ zi and H ( δt ) = H sc + δt P i ( σ xi σ xi +1 + σ yi σ yi +1 ) in d = 1: for any δt = 0, Jordan-Wigner trans-formation reveals the integrals of motion as plane-wave-operators, hence not related to S i by a local unitary. Fora local stabilizer code with topological order, the stabiliz-ers with s i = − T i . By Definition 1, this support does not easily delocal-ize: we find anyon localization.In Definition 1 we specialized to a topologically triv-ial M and Abelian topological order so that eigenstatesare fully characterized by the local s-bit strings { s i } . Fora topologically nontrivial M and/or non-Abelian topo-logical order, the set of local S i is not complete; { s i } label subspaces of degeneracy depending on the topol-ogy of M and/or the anyon fusion [48]. A complete setincludes nonlocal S nl i required to resolve these degenera-cies. In the non-Abelian case, due to the nonzero densityof anyons in generic eigenstates, the number of S nl i isextensive; tLIOMs give a highly incomplete characteri-zation [49]. Hence we focus on the Abelian case. There, S nl i are noncontractible Wilson loops (e.g., certain Paulistrings for qubits) on M . We can thus complete the set ofFMBL integrals of motion by T nl i = ˜ U S nl i ˜ U † . Therefore: Statement 3. On a topologically nontrivial M , the com-plete set of FMBL integrals of motion must include T nl i =˜ U S nl i ˜ U † where S nl i are noncontractible Wilson loops re-solving the eigenspace degeneracies of H sc . These T nl i give an operational definition of the fattenedWilson loops in Ref. 15. Aiming at { T i } via { S i } in a tensor-network calculation [7, 8, 14], one may in principleresolve highly-excited topological multiplets despite thehuge density of states.We next introduce an FMBL notion of topologicalequivalence: Definition 2. Two FMBL Hamiltonians H and H arein the same topological phase if and only if for all suf-ficiently large N there exists a continuous parameteriza-tion H ( λ ) , λ ∈ [0 , , with H (0) = H and H (1) = H ,such that H ( λ ) is FMBL for all λ ∈ [0 , and it convergesto a continuous parameterization as N → ∞ . In other words, one cannot connect topologically inequiv-alent FMBL Hamiltonians without delocalizing the sys-tem along the way. Furthermore, Statement 4. Two FMBL Hamiltonians H and H arein the same topological phase if and only if their eigen-state topological order is the same. We demonstrate this in the scope of Definition 1. Wefirst show that the same eigenstate topological order im-plies the same FMBL topological phase. If the eigenstatetopological order is the same, then the sets { S i, } Ni =1 and { S i, } Ni =1 corresponding to Hamiltonians H and H , re-spectively, are mapped by a local unitary U S (with de-gree of locality linked to those of the eigenstate-mappingunitaries): S i, = U S S i, U † S for all i . We consider H α (with α = 0 , 1) expanded according to Eq. (5) with T i,α = ˜ U α S i,α ˜ U † α and coefficients c ijk...,α . We define c ijk... ( λ ) = (1 − λ ) c ijk..., + λc ijk..., . We also define a lo-cal unitary ˜ U ( λ ) such that ˜ U (0) = ˜ U and ˜ U (1) = ˜ U U S and ˜ U ( λ ) a continuous function of λ [38]. H ( λ ) definedas in Eq. (5) gives a continuous path connecting H and H preserving FMBL (Definition 1) for all λ ∈ [0 , H ( λ ) ful-fills Definition 1 for all λ ∈ [0 , ǫ ( λ ) environment around H ( λ ). We nextobserve that the eigenstate topological order is the sameacross this ǫ ( λ ) environment: otherwise, since topolog-ical order cannot be changed continuously [48] (as em-bodied by the discreteness of the topological equivalenceclasses of stabilizers [40]), the environment would haveto contain points where the spectrum of the system hasdegeneracies such that the local perturbation δH in Def-inition 1 can switch the topological order. However, Def-inition 1 also requires that (t)LIOMs stay local, hencedegenerate states that can be coupled by δH must differat most in the action of a locally supported operator.(In other words, they must form dilute, well-isolated,small resonant clusters [50].) The corresponding uni-tary rotations in the degeneracy spaces amount to a lo-cal unitary update of ˜ U , hence topological order cannotbe switched. Using this constancy of the topological or-der within the ǫ ( λ ) environments, the rest of our demon-stration is straightforward: the continuity of H ( λ ), to-gether with the fact that I λ = [0 , 1] is compact and con-nected, implies that H ( I λ ) is compact and connected. Itsopen cover consisting of the ǫ ( λ ) environments thus hasa finite subcover which we can choose such that succes-sive (in λ ) environments overlap with each other. [The N -independence of ǫ ( λ ) ensures that the number of ǫ -environments in this subcover is N -independent.] Henceeigenstate topological order is the same on the entire path H ( I λ ): H and H must have the same topological order.An implication of Statement 4 is that the (t)LIOMsalong the path I λ can be written as T j ( λ ) = ˜ U ( λ ) S j ˜ U † ( λ )such that ˜ U ( λ ) remains a local unitary for any λ . Go-ing along I λ , one encounters resonances at certain valuesof λ . While, as we noted in our demonstration, FMBLrequires these to form small resonant clusters, the set ofpoints where such resonances occur becomes increasinglydense in I λ as N → ∞ due to the increasing number ofpossible spatial locations for these clusters. One mightthus wonder why the corresponding extensive number oflocal unitary updates applied successively as we go along I λ cannot result in ˜ U ( λ ) ceasing to be a local unitary.An intuitive reasoning for this is as follows: Firstly, thelocality of a resonant cluster implies that, upon crossingthe corresponding value in λ , the matrix ˜ U ( λ ) is multi-plied by is not merely a local unitary, but (to exponentialaccuracy) a locally-supported gate with size set by thatof the resonant cluster, the localization lengths ξ i of thecorresponding (t)LIOMs and the decay length of the cou-plings in Eq. (5). The locality of FMBL physics impliesthat in any fixed-size spatial region, the number of suchresonance clusters occurring as one crosses I λ becomes N -independent for large N . Moreover, due to the ran-domness inherent to FMBL systems, the correspondinglocal gates appear in random locations. These consider-ations lead to a circuit with N -independent depth andgate-length sublinear in N /d , hence the cumulative ac-tion corresponding to these resonances is a local unitary. IV. TOPOLOGICAL LIOMS: EXAMPLES Next, we illustrate our tLIOMs on three examples, in-cluding qubits and fermions in d = 1 , , 3. We point outwhich non-local integrals of motion emerge for those sys-tems and how they give rise to approximate degeneraciesarbitrarily high up in the spectrum. For the disorderedtoric code ( d = 2), we suggest a way of resolving suchalmost degenerate eigenstates despite the significantlysmaller average level spacing. A. Disordered Kitaev chain ( d = 1 ) This is the spinless-fermion Hamiltonian on an N -siteopen chain [46] H K = 14 N − X n =1 t n (cid:16) a n a n +1 + a n a † n +1 + h.c. (cid:17) , (6) where a n annihilates a fermion at site n and t n isGaussian-random with zero mean and unit variance. In-troducing the Majorana operators γ n − = 12 ( a n + a † n ) , γ n = 12 ( − ia n + ia † n ) , (7)the Hamiltonian can be rewritten as H K = i N − X n =1 t n γ n γ n +1 . (8)Although it is a (non-interacting) commuting projectorHamiltonian ( S n = iγ n γ n +1 ), it fulfills Definition 1 be-cause the localization of all eigenstates is stable as inter-actions are introduced [17]. It is also topological: Thes-bits S n ( n = 1 , . . . , N − S nl N = iγ N γ ,cannot be connected to local fermion-product-state p-bits ( − a † n a n by any fermion-parity conserving local uni-tary [51]. (The same holds for a closed chain where S N is also local.) Here, fermion-parity is a protectingsymmetry, although often taken as given in which caseEq. (8) is considered topologically ordered. For Eq. (8),the tLIOMs are T n = S n ; upon adding weak disor-der (e.g., i P Nn =1 µ n γ n − γ n with zero-mean Gaussian-random µ n of much smaller than unit variance) they be-come T n = ˜ U S n ˜ U † , where ˜ U is a parity-conserving localunitary. In addition to T n , the nonlocal T nl N = ˜ U S nl N ˜ U † also appears in the expansion (5), but with magnitudeexponentially suppressed in the linear system size L .We note that the topological phase characterized by thetLIOMs T n = ˜ U iγ n γ n +1 ˜ U † accounts for the missing Z index in the classification of fermionic one-dimensionaltopological MBL phases (with a symmetry) using onlyshort-depth quantum circuits [22]. B. Disordered Toric code ( d = 2 ) We consider qubits on the links of a square lattice ona torus and H tc = − X v J v A v − X p K p B p , (9)where A v = Q i ∈ v σ xi and B p = Q i ∈ p σ zi act on ver-tices v and plaquettes p of the lattice, respectively [45].The couplings J v and K p are again Gaussian-randomwith mean 0 and variance 1. Eq. (9) is also a com-muting projector Hamiltonian, but is expected to fallunder Definition 1 because upon adding weak disorder,such as a randomly fluctuating magnetic field P i h i σ zi ,the Hamiltonian is believed to remain FMBL [15, 20].The local s-bits are { A v } , { B p } ; they are completed by,e.g., the noncontractible Wilson loops S nl1 , = Z , = Q i ∈C , σ zi on the two generating cycles C , of thetorus. Hence, s-bit strings specify topologically degen-erate eigenspaces |{ s i } , z , z i with z i = ± Z i . Upon adding weak disorder, tLIOMs be-come { ˜ U A v ˜ U † } , { ˜ U B p ˜ U † } , where again ˜ U is a local uni-tary. These are completed by the nonlocal integrals ofmotion, e.g., ˜ U Z i ˜ U † ; as before, for finite linear systemsize L these also appear in the expansion (5), but withexponentially suppressed coefficients. The correspond-ing ∝ exp( − L ) splitting of topological degeneracies ismuch larger than the level spacing ∝ exp( − L ). Thismakes detecting topological multiplets impossible in ex-act diagonalization [15, 17, 20]. As we noted earlier, ourframework can avoid this problem: it allows us to takeadvantage of the efficient approximation of ˜ U by a quan-tum circuit U qc , and employ the methods of Refs. [8, 14]to numerically minimize P i tr (cid:0) [ H, T qc ,i ][ H, T qc ,i ] † (cid:1) with T qc ,i = U qc S i U † qc . Thus obtaining U qc gives the ap-proximation of the tLIOMs T qc ,i , of their nonlocal com-pletion T nlqc ,i = U qc Z i U † qc , and of the topological multi-plets U qc |{ s i } , z , z i . In addition to demonstrating theseFMBL topological multiplets, the presence of FMBLtopological order could be tested by comparing to min-imizing with τ qc ,i = U qc σ zi U † qc instead of T qc ,i . If T qc ,i perform significantly better, this would indicate that thesystem is in a topological FMBL phase. C. Disordered X-cube model ( d = 3 ) Fractons are emergent excitations which either cannotmove without creating additional fractons (at an energycost) or can move only along certain directions [41–44].Here, we focus on the so-called X-cube model [42] of frac-tons of the latter type. This model has qubits on the linksof a cubic lattice on the 3-torus. The Hamiltonian is H X = − X c u c A c − X v ; µ ∈{ xy,xz,yz } K µv B µv , (10)where A c = Q i ∈ c σ xi , B µv = Q i ∈ v ( µ ) σ zi with c denotinga cube and v ( µ ) the sites around vertex v lying par-allel to plane µ = xy, xz, yz . The couplings u c and K µv are Gaussian-random with mean 0 and variance 1.This, again, is a commuting projector Hamiltonian, butmay satisfy Definition 1 as fracton models may becomeFMBL [20]. The local s-bits are { A c } , { B µv } . On a 3-torus of linear size L , they are completed by 6 L − S nl i [43]. The subextensive scaling of the number of thesesuggests that a tLIOM description may be useful, whichproceeds analogously to the toric code case. V. CONCLUSION AND OUTLOOK Defining topological FMBL phases using tLIOMs pro-vides a transparent framework for establishing a num-ber of features on the character and robustness of thesephases. Firstly, it allowed us to show that (i) all eigen-states of topological MBL systems must have the same topological order (Statement 2) and that (ii) this or-der must be the same throughout the topological FMBLphase (Statement 4). The topological properties of alleigenstates are thus robust to small perturbations; chang-ing them requires delocalization (see Ref. 52 for a nu-merical study). Features (i) and (ii) are shared [21–23]with FMBL SPT systems [18, 19, 24], however unlikein those cases [which are local-unitary related to on-siteproduct states and hence allow conventional LIOMs tobe used to establish (i) and (ii) for SPTs], establishingthese results in the topologically ordered case relies on thetLIOM approach in an essential way. Furthermore, ourapproach also allowed us to capture features without SPTcounterparts. In particular, we showed (iii) how anyonlocalization (implied by Definition 1) and (iv) spectraldegeneracy on topologically nontrivial manifolds (State-ment 3) follow from our framework. These results notonly put certain thus far only heuristically establishedfindings [15, 27] on firm footing, but en route to (iv) wealso demonstrated the existence, and provided the op-erational definition, of nonlocal integrals of motion un-derlying the topological multiplets and complementingthe tLIOMs to form a complete set. Besides providing atransparent theoretical picture, such an operational def-inition has practical significance: upon combining ourtLIOMs with tensor-network approaches, it allows one,e.g., to numerically resolve topological multiplets at highenergies, an objective hitherto considered infeasible dueto the mean level spacing scaling faster to zero with sys-tem size than the multiplet splitting [15, 17, 18, 20].In closing, we mention a few examples of future di-rections where our tLIOM framework may find uses orgeneralizations. The l-bit description gave key insightsinto the phenomenology of FMBL systems, including intothe dynamics of quantum information [29]. A naturalquestion is: what new features may arise in tLIOM-Hamiltonians (5) due to topological order? The suit-ability of our framework for tensor-network methods alsoopens the door for numerically addressing questions withat most heuristic answers thus far: under what condi-tions is FMBL present e.g., in the disordered toric codeand what level of improvement may FMBL provide inprotecting the encoded quantum information? How dothe conditions for FMBL depend on the type of Abeliantopological order beyond the toric code? How do tLIOMsbehave near the topological MBL transition? FMBL isoften invoked for protecting driven (Floquet) phases fromheating [53]; our framework may thus find applicationsin novel topologically ordered driven phases of matter. Itwould also be interesting to generalize our approach tosymmetry-enriched topological phases [26] in the FMBLregime. The quantum circuit formalism of Ref. 23 mightbe of particular relevance for this endeavor. 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