Local integrals of motion for topologically ordered many-body localized systems

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 MBL systems. Using a suitable deﬁnition to capture such systems in any spatial dimension, we demonstrate a number of features, including that MBL topological order, if present, (i) is the same for all eigenstates, (ii) is robust in character against any perturbation preserving MBL, and (iii) implies that on topologically nontrivial manifolds a complete set of integrals of motion must include nonlocal ones in the form of local-unitary-dressed noncontractible Wilson loops. Our approach is well suited for tensor-network methods and is expected to allow these to resolve highly excited ﬁnite-size-split topological eigenspaces despite their overlap in energy. We illustrate our approach on the disordered Kitaev chain, toric code, and X-cube model.

Introduction.-Systemsdisplaying many-body localization [1,2] (MBL) violate the eigenstate thermalization hypothesis [3] and therefore do not thermalize.(See Refs.4, and 5 for some recent reviews.)MBL occurs in strongly disordered interacting lattice systems.Recent analytical and numerical work has put the effect on firm theoretical footing in one dimension [2,[6][7][8], whereas the existence of MBL in higher dimensions is still debated [9,10].However, MBL has been observed experimentally both in one- [11] and two-dimensional systems [12].This might be due to extremely long relaxation time scales [13], which are unobservable in the experiment.Numerical simulations are consistent with twodimensional MBL-like behavior on short time scales [14].
All eigenstates of MBL systems are area-law entangled [15,16].This makes MBL compatible with the interesting scenario where all eigenstates are topologically ordered [15,[17][18][19].Such MBL systems have been suggested to provide improved protection of quantum information against external perturbations compared to their clean counterparts [15,20].In one-dimensional systems, nontrivial topology requires the presence of certain symmetries.In higher dimensions, however, topological states can exist without symmetries; it is these states that we call here topologically ordered [21].
Fully MBL (FMBL) systems can be described in terms of local integrals of motion (LIOMs), which are exponentially localized operators which commute with each other and the Hamiltonian [5,[22][23][24][25][26][27][28][29][30].LIOMs are commonly assumed to form a complete set arising as a local unitary transform of the set of on-site spin-z operators.This, however, tacitly also assumes the absence of topological order: it implies that FMBL eigenstates arise from local product states via local unitary transformation, which guarantees [31,32] that they are topologically trivial.This raises the question whether such systems can at all be FMBL (even when neglecting extremely slow relaxation effects in higher dimensions) and, if so, how one should think about their LIOMs.
In this Letter, we show that topological obstructions do not prevent FMBL and propose a notion for "topological LIOMs (tLIOMs)" suitable for topologically ordered FMBL phases.Our main insight is that the on-site spinz operators in conventional LIOMs are just one choice of local stabilizers [33], namely those of product states; to describe tLIOMs we must use a different set, namely the stabilizers in the commuting projector limits of topological phases.Since all non-chiral forms of topological order admit such commuting projector limit [34], our tLIOMs capture all non-chiral FMBL eigenstate topological orders.This tLIOM notion is very general; it applies in any dimension and its scope even includes putative FMBL cousins of recently introduced fractonic phases [35][36][37][38].
Defining topological FMBL phases using tLIOMs provides a transparent framework for establishing a number of features on the character and robustness of these phases, and it lends itself for adapting tensor-network simulations thus far used for capturing non-topological FMBL systems [7,8,14].We next turn to a more concrete exposition of our approach, first recapping the LIOM description of non-topological FMBL phases.
Non-topological FMBL.-For concreteness we consider an N -site spin-1/2 (i.e., qubit) system on a d-dimensional square lattice.The extension to higher spin systems and fermionic systems is straightforward.The FMBL phase is defined by a complete set of LIOMs τ z i (i = 1, . . ., N ) which commute with the Hamiltonian H and each other, and are exponentially localized, i.e., their non-trivial matrix elements decay exponentially with distance from site i.The corresponding decay length, the localization length ξ i , must fulfill ξ i /N 1/d → 0 for all i in the thermodynamic limit N → ∞.The τ z i can be constructed from a unitary U which diagonalizes the Hamiltonian as τ z i = U σ z i U † , where σ z i is the third Pauli operator acting on site i.
FIG. 1.A one-dimensional depth-two quantum circuit with length-ℓ unitaries (denoted as boxes).The gate length ℓ is allowed to grow subextensively with the system size [8].
A key feature that makes nontopological FMBL systems special, and computationally tractable [7,8,14], is that U is a local unitary, i.e., efficiently approximable by a constant-depth quantum circuit of subextensive-length unitary gates [8].[The error scales as exp(−ℓ/ξ max ) where ξ max is the largest ξ i and ℓ is the gate length.]Fig. 1 shows a depth-two example.The τ z i 's are also known as l-bits (l=localized) [23]; they label, due to Eq. ( 1), all eigenstates |ψ l1...lN of H, The σ z i operators are called p-bits (p=physical) [23].Eq. ( 2) implies |ψ l1l2...lN = U |l 1 l 2 . . .l N in terms of the p-bit product states |l 1 l 2 . . .l N .For d > 1, the LIOM description might not apply exactly due to delocalization on extremely long time scales [9,13].However, in the (experimentally relevant) short-time limit, the description in terms of LIOMs seems to be appropriate [14], which is what we restrict ourselves to in the following.
Topological FMBL phases.-In the topological FMBL case it is conjectured that all eigenstates display topological order [15,17,19].However, even if only one eigenstate |ψ l1l2...lN is topologically non-trivial, there exists no local unitary U such that |ψ l1l2...lN = U |l 1 l 2 . . .l N [31,32].Hence, if a LIOM description applies, it is not of the form τ z i = U σ z i U † .We must therefore introduce the new concept of (t)LIOMs appropriate also for topological FMBL systems.
We begin by explaining how p-bits and the corresponding product eigenstates fit in the broader context of local stabilizer codes [33].For concreteness, we present the idea for N -site qubit systems, however the scope of stabilizer codes, and hence our construction, is much more general.A local stabilizer code may be thought of in terms of a stabilizer Hamiltonian where the local stabilizers S i = S † i are Pauli strings with local support around site i, and [S i , S j ] = 0. (Thus, up to a constant, H sc is a commuting-projector Hamiltonian.)On a topologically trivial manifold, S i are independent ), hence their eigenvalues s i = ±1 label a complete orthonormal basis.We refer to S i as s-bits.The p-bits and their product eigenstates correspond to S i = σ z i and the eigenstates of H sc .The scope of stabilizer codes, however, is much wider (and not limited to qubits): they capture all nonchiral topological orders [34], from the toric code [39] to fracton models [35][36][37][38], as well as fermionic systems such as the Kitaev chain [40] or various other Majorana fermion codes [41].We propose the following definition of FMBL and (t)LIOMs to capture Abelian topological order on topologically trivial manifolds: Definition 1.Let S i , i = 1, . . ., N be a complete set of local stabilizers on a topologically trivial manifold M. We call a Hamiltonian H on M FMBL, if there exists a local unitary Ũ such that and the same property holds if infinitesimally small perturbations are added to H. We call the 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 of our definition, including observations for topologically nontrivial M and non-Abelian topological order.For now, we note that Definition 1 implies, firstly, |ψ {si} = Ũ |{s i } where |ψ {si} and |{s i } are respective eigenstates of the FMBL Hamiltonian H and H sc .Secondly, up to an additive constant, and with c ijk... ∈ R, (5) where for a local Hamiltonian and for describing the dynamics on experimental time scales, we can assume that |c ijk... | decay exponentially with the largest distance between the occurring sites i, j, k, . . . in any dimension d.
We next establish a number of features that follow from Definition 1, focusing again on qubit systems for concreteness.We first remind of the following result [31,32]: |ψ 1 and |ψ 2 have the same topological order if |ψ 2 = U |ψ 1 and U is a local unitary.Together with Definition 1, this means that many features directly carry over from the commuting projector limit H sc to the FMBL phase.For instance, due to |ψ {si} = Ũ |{s i } : Statement 1.The FMBL eigenstates have the same topological order as those of H sc .Furthermore, since any set of the stabilizers S i can be flipped by a suitable Pauli string [33], and since any Pauli string is a local unitary, every eigenstate of H sc has the same topological order.Hence: Statement 2. All eigenstates of topological FMBL systems display the same topological order.
Definition 1 requires robustness against small perturbation to describe a localized phase, i.e., it excludes systems with (t)LIOMs that easily delocalize.An example of the excluded systems is H sc itself [which satisfies Eq. ( 4) with Ũ = 1], as illustrated by S i = σ z i and H(δt) = H sc + δt i (σ x i σ x i+1 + σ y i σ y i+1 ) in d = 1: for any δt = 0, Jordan-Wigner transformation reveals the integrals of motion as plane-wave-operators, hence not related to S i by a local unitary.For a local stabilizer code with topological order, the stabilizers with s i = −1 indicate anyon locations; in the FMBL phase these translate to the support of the corresponding T i .By Definition 1, this support does not easily delocalize: we find anyon localization.
In Definition 1 we specialized to a topologically trivial M and Abelian topological order so that eigenstates are fully characterized by the local s-bit strings {s i }.For a topologically nontrivial M and/or non-Abelian topological order, the set of local S i is not complete; {s i } label subspaces of degeneracy depending on the topology of M and/or the anyon fusion [42].A complete set includes nonlocal S nl i required to resolve these degeneracies.In the non-Abelian case, due to the nonzero density of anyons in generic eigenstates, the number of S nl i is extensive; tLIOMs give a highly incomplete characterization [43].Hence we focus on the Abelian case.There, S nl i are noncontractible Wilson loops (e.g., certain Pauli strings for qubits) on M. We can thus complete the set of FMBL integrals of motion by These T nl i give an operational definition of the fattened Wilson loops in Ref. [15].Aiming at {T i } via {S i } in a tensor-network calculation [7,8], one may in principle resolve highly-excited topological multiplets despite the huge density of states.We next introduce an FMBL notion of topological equivalence: Definition 2. Two FMBL Hamiltonians H 0 and H 1 are in the same topological phase if there exists a parameterization In other words, one cannot connect topologically inequivalent FMBL Hamiltonians without delocalizing the system along the way.Furthermore, Statement 4. Two FMBL Hamiltonians H 0 and H 1 are in the same topological phase if and only if their eigenstate topological order is the same.
We show this in the scope of Definition 1.If the eigenstate topological order is the same, then the two corresponding sets {S (′) i } N i=1 are mapped by a local unitary (with degree of locality linked to those of the eigenstatemapping unitaries).The equivalence of the FMBL topological phases is shown by finding a path H(λ) which connects any two FMBL topological phases with localunitary related {S i } N i=1 .We thus consider H 0 as in Eq. ( 5) and where By assumption, there is a local unitary U S such that S ′ i = U S S i U † S for all i.We can thus define a local unitary V (λ) such that V (0) = Ũ and V (1) = Ũ ′ U S and V (λ) a continuous function of λ [32].H(λ) defined as in Eq. ( 5) with Ũ replaced by V (λ) and c ijk... by d ijk... (λ) gives a continuous path H(λ) connecting H 0 and H 1 preserving FMBL (Definition 1) for all λ ∈ [0, 1].
To show the converse, we take a path H = H(λ) in an FMBL phase (Definition 2) and show that the eigenstate topological order cannot change.We take the system size N finite but large.Take a unitary U tot (λ) which diagonalizes the Hamiltonian H(λ) = U tot (λ)E(λ)U † tot (λ); for finite N , U tot (λ) can be chosen as a continuous function of λ.For any λ, we have where U top (λ) diagonalizes the S i (λ) of H(λ) simultaneously.[For S i (λ), and hence U top (λ), encoding topological order, U top (λ) is not a local unitary.]We now analyze whether the eigenstate topological order can be different for λ = λ ′ − ǫ and λ = λ ′ + ǫ.Since U tot (λ) is continuous, we have in the limit ǫ → 0 As Ũ (λ) is a local unitary, it cannot change the eigenstate topological order.Hence, U top (λ ′ − ǫ) and U top (λ ′ + ǫ) encode the same same eigenstate topological order, which is thus the same along the entire path λ ∈ [0, 1].Next, we illustrate our tLIOMs on three examples, including qubits and fermions in d = 1, 2, 3.

Disordered Kitaev chain (d = 1
).-This is the spinlessfermion Hamiltonian on an N -site open chain [40] where a n annihilates a fermion at site n and t n is Gaussian-random with mean zero and unit variance.Introducing the Majorana operators the Hamiltonian can be rewritten as Although it is a (non-interacting) commuting projector Hamiltonian (S n = iγ 2n γ 2n+1 ), it fulfills Definition 1 because the localization of all eigenstates is stable as interactions are introduced [17].It is also topological: The s-bits S n (n = 1, . . ., N − 1), completed by S nl N = iγ 2N γ 1 , cannot be connected to local fermion-product-state pbits (−1) a † n an by any fermion-parity conserving local unitary [44].(The same holds for a closed chain where S N is also local.)Here, fermion-parity is a protecting symmetry, although often taken as given in which case Eq. ( 11) is considered topologically ordered.For Eq. ( 11), the tLIOMs are T n = S n ; upon adding weak disorder (e.g., i N n=1 µ n γ 2n−1 γ 2n with zero-mean Gaussianrandom µ n of much smaller than unit variance) they become T n = Ũ S n Ũ † , where Ũ is a parity-conserving local unitary.In addition to T n , the nonlocal T nl N = Ũ S nl N Ũ † also appears in the expansion ( 5), but with magnitude exponentially suppressed in the linear system size L.
Disordered Toric code (d = 2).-Weconsider qubits on the links of a square lattice on a torus and where A v = i∈v σ x i and B p = i∈p σ z i act on vertices v and plaquettes p of the lattice, respectively [39].The couplings J v and K p are again Gaussian-random with mean 0 and variance 1. Eq. ( 12) is also a commuting projector Hamiltonian, but is expected to fall under Definition 1 because upon adding weak disorder, such as a randomly fluctuating magnetic field i h i σ z i , the Hamiltonian is believed to remain FMBL [15,19].The local sbits are {A v }, {B p }; they are completed by, e.g., the noncontractible Wilson loops S nl 1,2 = Z 1,2 = i∈C1,2 σ z i on the two torus generating cycles C 1,2 .Hence s-bit strings specify topologically degenerate eigenspaces |{s i }, z 1 , z 2 with z i = ±1 the eigenvalue of Z i .Upon adding weak disorder, tLIOMs become , where again Ũ is a local unitary.These are completed by the nonlocal integrals of motion, e.g., Ũ Z i Ũ † ; as before, for finite linear system size L these also appear in the expansion ( 5), but with exponentially suppressed coefficient.The corresponding ∝ exp(−L) splitting of topological degeneracies is much larger than the level spacing ∝ exp(−L 2 ), i.e., hard to detect in exact diagonalization [15,17,19].As we noted earlier, our framework can avoid this problem: it allows us to take advantage of the efficient approximation of Ũ by a quantum circuit U qc , and employ the methods of Refs.[8,14] to numerically minimize qc .Thus obtaining U qc gives the approximation of the tLIOMs T qc,i , of their nonlocal completion T nl qc,i = U qc Z i U † qc , and of the topological multiplets U qc |{s i }, z 1 , z 2 .In addition to demonstrating these FMBL topological multiplets, the presence of FMBL topological order could be tested by comparing to minimizing with τ qc,i = U qc σ z i U † qc instead of T qc,i .If T qc,i perform significantly better, this would indicate that the system is in a topological FMBL phase.Disordered X-cube model (d = 3).-Fractonsare elementary excitations which either cannot move without creating additional fractons (at an energy cost) or otherwise only along certain directions [35][36][37][38].Here, we focus on the so-called X-cube model [36] of qubits on the links of a cubic lattice on the 3-torus.The Hamiltonian is where A c = i∈c σ x i , B µ v = i∈v(µ) σ z i with c denoting a cube and v(µ) the sites around vertex v lying parallel to plane µ = xy, xz, yz.The couplings u c and K µ v are Gaussian-random with mean 0 and variance 1.This, again, is a commuting projector Hamiltonian, but may satisfy Definition 1 as fracton models may become FMBL [19].The local s-bits are {A c }, {B µ v }.On a 3torus of linear size L, they are completed by 6L − 3 independent noncontractible commuting Wilson loops S nl i [37].The subextensive scaling of the number of these suggests that a tLIOM description may be useful, which proceeds analogously to the toric code case.
Conclusion.-The framework of tLIOMs and their nonlocal completions introduced here is naturally suited for describing topologically ordered FMBL systems.We used it to demonstrate a number of features, including Statements 1-4, and illustrated its concrete form on fermionic and qubit examples in d = 1, 2, 3.In closing, we mention a few examples of future directions where our framework may find uses or generalizations.The l-bit description gave crucial insights into the phenomenology of FMBL systems, including into the dynamics of quantum information [23].A natural question is: what new features may arise in tLIOM-Hamiltonians (5) due to the topological order encoded in these?The suitability of our framework for tensor-network methods also opens the door for numerically addressing questions with at most heuristic answers thus far: under what conditions is FMBL present e.g., in the disordered toric code and what level of improvement may FMBL provide in protecting the encoded quantum information?How do the conditions for FMBL depend on the type of Abelian topological order beyond the toric code?How do tLIOMs behave near the topological MBL transition?FMBL is often invoked for protecting driven (Floquet) phases from heating [45]; our framework may thus find applications in novel topologically ordered driven phases of matter.It would also be interesting to generalize our approach to symmetry-enriched topological phases [21] in the FMBL regime.The quantum circuit formalism of Ref. [46] might be of particular relevance for this endeavor.
We gratefully acknowledge support from the European Commission through the ERC Starting Grant No. 678795 TopInSy.

iŨ † . Therefore: Statement 3 .
On a topologically nontrivial M, the complete set of FMBL integrals of motions must include T nl i = Ũ S nl i Ũ † where S nl i are noncontractible Wilson loops resolving the eigenspace degeneracies of H sc .