Sufficient symmetry conditions for Topological Quantum Order

Abstract

We prove sufficient conditions for Topological Quantum Order at zero and finite temperatures. The crux of the proof hinges on the existence of low-dimensional Gauge-Like Symmetries, thus providing a unifying framework based on a symmetry principle. These symmetries may be actual invariances of the system, or may emerge in the low-energy sector. Prominent examples of Topological Quantum Order display Gauge-Like Symmetries. New systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin exchange and Jahn–Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. We analyze the physical consequences of Gauge-Like Symmetries (including topological terms and charges) and show the insufficiency of the energy spectrum, topological entanglement entropy, maximal string correlators, and fractionalization in establishing Topological Quantum Order. General symmetry considerations illustrate that not withstanding spectral gaps, thermal fluctuations may impose restrictions on suggested quantum computing schemes. Our results allow us to go beyond standard topological field theories and engineer systems with Topological Quantum Order.

The role of invariance (symmetry) principles in accounting for observed regularities is well known (1). Invariances become particularly effective in quantum mechanics where the linear character of Hilbert space enables us to construct superpositions of states that transform as irreducible representations of various symmetry groups. First discovered were invariance principles of a geometric character relating to space-time displacements and uniform motion. The best known local invariance operations (i) relate different coordinate systems to one other by such geometric deformations that leave the (local) metric invariant and (ii) appear as local transformations that link different gauge theory representations. It was recently realized that probing nonlocal (topological) structures uncovers new invariance principles with physical (and experimental) consequences. Particle-wave type dualities link the seemingly different local and nonlocal structures.

Understanding the thermodynamic phases of matter via symmetry principles enables characterization by universal behaviors as in Landau's theory of phase transitions (2). In this theory, the order parameter(s) of the system relate to probing its local structure. A non-vanishing order parameter, encoding the breaking of a symmetry, defines the ordered state while the restoration of that symmetry signals the transition to a disordered state. A new paradigm, Topological Quantum Order (TQO), extends the Landau symmetry-breaking framework (3). In essence, this new order is associated with robustness against local perturbations, and hence cannot be described in principle by local order parameters. Thus, the underlying order remains hidden to ordinary local probes. Indeed, this new order exhibits nonlocal correlations that potentially lead to novel physical consequences. Interest in TQO is further enhanced by the prospect of using it to engineer fault tolerant hardware for quantum computation (4). A main objective of this article is to introduce a framework based on symmetry to study this new kind of order of matter. Key questions concern the physical organizing principles underlying such an order, how TQO does manifest, and how to mathematically characterize it. The symmetry based results apply also to systems displaying emergent symmetries (5). The latter are not actual invariances of the system but emerge as exact symmetries at low energies.

Several inter-related concepts are associated with TQO such as symmetry, degeneracy, fractionalization of quantum numbers, and maximal string/brane correlations (nonlocal order). It is important to determine what is needed for a system to display this order. While canonical examples such as the Fractional Quantum Hall liquids exist, there is no unambiguous definition of TQO. We start by defining this order following ref. 4 and show relations between these different concepts, establishing the equivalence between some and more lax relations among others. Most importantly, we suggest a symmetry principle for TQO at zero and finite temperatures. In particular, we (i) prove that systems with d-dimensional (with d = 1,2) Gauge-Like Symmetries exhibit TQO; (ii) analyze the resulting conservation laws and emerging topological terms; (iii) affirm that the energy spectrum on its own is insufficient for establishing the existence of this order (the devil is in the state itself); (iv) suggest that while fractionalization, string/brane-type correlators, and entanglement entropy are important concepts they are not always sufficient conditions for TQO; (v) sketch a general algorithm for constructing string/brane correlators. Our goal is to provide a unifying framework to identify new physical models with TQO (details in refs. 6–8).

We focus on quantum lattice systems (and their continuum extension) having N_s = L^D sites, with L the number of sites along each space direction, and D the dimensionality of the lattice Λ. Associated with each lattice site (or mode, or bond, or plaquette, etc.) i there is a Hilbert space H_i of finite dimension D. The full Hilbert space is the tensor product of the spaces, H = ⊗_iH_i, in the case of distinguishable subsystems, or a proper subspace in the case of indistinguishable ones. Statements about local order, TQO, fractionalization, entanglement, etc., are relative to the particular decomposition used to describe the physical system. The physically motivated natural local language (5) is often utilized. The definition of TQO requires a set of N orthonormal ground states {|g_α〉}_α=1,…,N with a gap to excited states. These states should be topologically distinct with no quasi-local operator V connecting any pair of such states. Equivalently, such operators cannot be used to distinguish between different ground states. Specifically, TQO exists if and only if for any quasi-local operator V,

where v is a constant and c is a correction that it is either zero or vanishes in the thermodynamic limit. That is, the ground states locally look identical but not globally. We will also examine a finite temperature (T > 0) extension for the diagonal elements of Eq. 1,

with ρ_α = exp[−H_α/(k_BT)] a density matrix corresponding to the Hamiltonian H when augmented by an infinitesimal operator α. At T = 0, a particular choice of such operators can be constructed to favor the state |g_α〉. We consider the most general case of arbitrary infinitesimal operators. Such infinitesimal operators may be nonlocal, e.g., may couple to the toric cycle operators (6, 7) in Kitaev's toric code model (4). A system displays finite-T TQO if it satisfies both Eqs. 1 and 2. Those equations describe mathematically the fact that physical states cannot be distinguished by local measurements. The only way to extract nontrivial information is to perform a nonlocal measurement. If the expectation value of any quasi-local operator V is independent of topological constraints that favor the state |g_α〉, or a collection of such states in a given topological sector, then the system exhibits TQO: the information encoded in the states is unaccessible (and protected) from any quasi-local perturbation. Such a topological sector may be defined by Hopf invariants, total domain wall parity, etc.

Note that the conditions in Eqs. 1 and 2 enable TQO only if the ground state sector contains, at least, two orthogonal states. According to these conditions, systems with nondegenerate ground states (such as Integer Quantum Hall liquids) do not support TQO. (In fact, such non-degenerate systems exhibit no spontaneous symmetry breaking and generally also do not support any Landau orders.) As we showed, a necessary (but not sufficient) criterion for robust topological quantum memories is directly related to the conditions of Eqs. 1 and 2 for TQO (8). That this criterion is necessary but insufficient is exemplified by Kitaev's toric code model (4), which as we discuss here and in far more detail elsewhere (8) exhibits a finite (system size independent) autocorrelation time. This criterion for robust quantum memories applies to systems with both degenerate and nondegenerate ground states (see SI Text). It shows that nondegenerate systems are trivial from another viewpoint as well. Quantum computing applications of TQO rely on the non-commutativity of different braiding operations. The presence of noncommuting symmetries ensures a topological degeneracy in the ground state sector.

To put these definitions in perspective, consider the typical example of local order: the spin s = 1/2 Heisenberg ferromagnet. The ferromagnet has a continuum of ground states |g_n̂〉 with a magnetization in the direction _n̂. In the thermodynamic limit, these states become orthogonal to one another.

However, in this case, e.g. the local spin operator V = S_i^z may distinguish between different ground states:

with ê_z a unit vector along the z axis. Thus, Landau type systems with local order parameters do not even satisfy the T = 0 condition of Eq. 1. Similarly, in three spatial dimensions, at finite temperatures, 〈V〉_n̂ ≠ 〈V〉_ℓ̂ A d-dimensional Gauge-Like Symmetry of a theory is a group of symmetry transformations G_d such that the minimal set of fields ϕ_i changed by the group operations is located on a d-dimensional subset C ⊂ Λ of the complete D-dimensional lattice (d ≤ D). These transformations can be expressed as (9): U_lk = ∏_{i ∈ Cl}g_ik, where C_l denotes the (external) spatial subregion l, Λ = ∪_lC_l, and k labels the (internal) group generators. [The extension of this definition to the continuum is straightforward.] Gauge (local) symmetries correspond to d = 0, while in global symmetries the region influenced by the symmetry operation is the full d = D-dimensional volume of the system. These symmetries may be Abelian or nonAbelian. An illustration is provided in Fig. 1, which describes three different D = 2 systems with symmetries of dimensions d = 0,1,2 : (a) The Ising gauge theory whose Hamiltonian is H = −k∑_pσ_ij^zσ_jk^zσ_kl^zσ_li^z. The sum is over all elementary squares (plaquettes) p, and on each bond (ij) we have an s = 1/2 spin ℏ${\overrightarrow{\sigma }}_{ij}/2$ (${\overrightarrow{\sigma }}_{ij}$ represents the triad of Pauli matrices). In this system, we have the local gauge symmetries G_i = ∏_s σ_is^x with the sites {s} being the nearest-neighbors of i. For any lattice site, [G_i,H] = 0. {G_i} involve a finite number of local operators and thus constitute d = 0 symmetries. (b) The anisotropic s = 1/2 orbital compass model (10), H = −∑_i[J_xσ_i^xσ_i+êx^x + J_zσ_i^zσ_i+êz^z], emulating the direction dependent interactions of electronic orbitals. In this model, exchange interactions involving the x component of the spin occur only along the spatial x direction of the lattice. Similar spatial direction dependent spin exchange interactions appear for the z components of the spin. Apart from a global reflection that only appears for the isotropic point J_x = J_z (and that may be broken at low T), the orbital compass model has the following symmetries O^α = ∏_j∈Cα iσ_j^α for α = x,z. C_α denotes any line orthogonal to the ê_α axis. On a torus, these operators are defined along toric cycles. As O^α involve O(L¹) sites, they constitute d = 1 symmetries. As we will elaborate on, these d = 1 symmetries cannot be broken at finite T. (See also Fig. 2.) It is worth emphasizing that global symmetries such as time reversal (6, 7) may be seen as composites of the minimal (more fundamental) d = 1 symmetries of O^α.* (c) The s = 1/2 XY ferromagnet H = −J ∑_〈ij〉 [σ_i^xσ_j^x + σ_i^zσ_j^z] has continuous d = D = 2 symmetry operators: U(θ) = ∏_j exp[−(i/2)θσ_j^y]. There are O(L²) local operators in the product, which defines the d = 2 operator U(θ).

Fig. 1.

Schematics of the interactions and symmetries involved in three D = 2 examples. See (A – C) of text. (A) an Ising gauge theory with local (d = 0) symmetries. (B) an orbital compass model with d = 1 symmetries; here the symmetry operations span lines. (C) an XY model with d = 2 symmetries; the symmetry here spans the entire D = 2 dimensional plane.

Fig. 2.

The physical engine behind our theorem are topological defects. For instance, introducing a (1D) soliton into an orbital compass model state such as depicted in Fig. 1 B leads to only a finite energy cost. This penalty is depicted here by a single energetic bond (dashed line). The energy-entropy balance associated with such d = 1 Ising type domain walls is the same as that in a D = 1 Ising system. At finite T, entropic contributions overwhelm energy pentalties and no local order is possible. Order is manifest in nonlocal quantities associated with topological defects. Similar results occur in other systems with low-dimensional Gauge-Like Symmetries.

The group symmetry operators and topological defects can be written in terms of unitary operators. In the continumm limit, the group elements of d = 1 Gauge-Like Symmetries are path-ordered (P) products

where C is a closed path in configuration space and A⃗ is the corresponding connection. On the lattice this expression is replaced by an equivalent discrete sum, and C is a closed path on the lattice. U is an Aharonov–Bohm phase (11) or a Wilson loop (12). For instance, for the orbital compass model [(b) above], the d = 1 group elements can be written as

A defect creating operator (see e.g., Fig. 2) is

or an exponentiated discretized sum. Here, C₊ is an open contour. For instance, the defect depicted in Fig. 2 is generated by $D_{+}=e^{i\frac{\pi }{2}{\sum }_{j\in C}+{\sigma }_j^x}$. We can interpret D₊ as creating of a defect-antidefect pair and displacing each member of the pair to the opposite endpoints of C₊. The d = 1 Gauge-Like Symmetry operators linking different ground states (U of Eqs. 4 and 5) correspond to the displacement of a defect-antidefect pair along a toric cycle. Formally, this is similar to the quasiparticle-quasihole pair creation operator linking different ground states in the Fractional Quantum Hall Effect (13). Topological concepts such as monodromy and homotopy can be realized by d = 1 symmetry operators. More complicated topological properties are associated with d = 2 symmetries.

What are the physical consequences for a system with a symmetry group G_d? Most importantly, these symmetries lead to an effective dimensional reduction in several characteristics.

The great insight behind the Landau theory of phase transitions lies in the intimate relation between the spontaneous breaking of a symmetry in the Hamiltonian and the appearance of a new ordered phase characterized by an order parameter. Can we spontaneously break d-dimensional Gauge-Like Symmetries? An important key is given by an inequality. This inequality states that the absolute values of quantities not invariant under G_d are bounded from above by the expectation values that they attain in a d-dimensional Hamiltonian H̄ (or corresponding action S̄) that is globally invariant under G_d and preserves the range of the interactions of the original system (9). (See the appendices of refs. 6 and 7 for a review of this inequality.) As the expectation values of local observables vanish in low-d systems, this bound strictly forbids spontaneous symmetry breaking of non-G_d invariant local quantities in systems with interactions of finite range and strength whenever d = 0 [Elitzur's theorem (14)], d = 1 for both discrete (see Fig. 2) and continuous G_d, and [as a consequence of the Mermin–Wagner–Coleman theorem (15, 16)] whenever d = 2 for continuous symmetries (9). Discrete d = 2 symmetries may be broken (e.g. the finite-T transition of the D = 2 Ising model, and the d = 2 Ising Gauge-Like Symmetries of D = 3 orbital compass systems). In the presence of a finite gap in a system with continuous d > 2 symmetries, spontaneous symmetry breaking is forbidden even at T = 0 (9). The absence of spontaneous symmetry breaking of d-dimensional Gauge-Like Symmetries is due to low-(d) dimensional topological defects: domain walls/solitons in systems with d = 1 discrete symmetries, vortices in systems with d = 2 U(1) symmetries, hedgehogs for d = 2 SU(2) symmetries. Transitions and crossovers can only be discerned by quasi-local symmetry invariant quantities (e.g., Wilson-like loops) or, by probing global topological properties [e.g., percolation in lattice gauge theories (17)]. Extending the bound of ref. 9 to T = 0, we now find that if T = 0 spontaneous symmetry breaking is precluded in systems with d-dimensional Gauge-Like Symmetries, then spontaneous symmetry breaking of quantities not invariant under exact or T = 0 emergent d-dimensional Gauge-Like Symmetries cannot occur as well. Exact symmetries refer to [U,H] = 0 ; in emergent symmetries (5) unitary operators U ∈ G_emergent are not bona fide symmetries ([U,H]≠0) yet become exact at low energies: when applied to any ground state, the resultant state must also reside in the ground state manifold, U|gα〉 = ∑_βu_αβ|gβ〉. In gapped systems, T = 0 spontaneous symmetry breaking of d > 2 continuous symmetries is prohibited. In particular, it can be seen (6, 7) that even emergent symmetries that appear only within the ground state sector in systems with a spectral gap preclude the appearance of spontaneous symmetry breaking. The proof of this statement relies on (i) extensions of the Mermin–Wagner–Coleman theorem to gapped systems with continuous symmetries at T = 0 (18–21) in dimensions smaller than two, and on (ii) an application of general bounds, derived in ref. 9, to these problems (6, 7).

Symmetries imply the existence of conservation laws and topological charges, with associated continuity equations when the group of symmetries is continuous. If the continuous symmetry forms a gauge group, then an additional local Gauss' law is satisfied by the conserved currents. We find that d-dimensional Gauge-Like Symmetries lead to conservation laws within d-dimensional regions. To illustrate, consider the rotational noninvariant Euclidean Lagrangian density of a complex field $\overrightarrow{\phi }(\overrightarrow{x})=({\phi }_1(\overrightarrow{x}),{\phi }_2(\overrightarrow{x}),{\phi }_3(\overrightarrow{x}))(\overrightarrow{x}=(x_1,x_2,x_3))$: L = $\frac{1}{2}{\sum }_{\mu }|{\partial }_{\mu }{\phi }_{\mu }{|}^2+\frac{1}{2}{\sum }_{\mu }|{\partial }_{\tau }{\phi }_{\mu }{|}^2+W({\phi }_{\mu })$, with $W({\phi }_{\mu })=u{({\sum }_{\mu }|{\phi }_{\mu }{|}^2)}^2-\frac{1}{2}{\sum }_{\mu }{m}^2(|{\phi }_{\mu }{|}^2)$ and μ,ν = 1,2,3 are the spatial directions and τ is the imaginary time. L displays the continuous d = 1 symmetries ϕ_μ → e^{iψ_μ({x_ν}_ν≠μ)}ϕ_μ. The conserved d = 1 Noether currents are j_μν = i[ϕ_μ*∂_νϕ_μ −(∂_νϕ_μ*)ϕ_μ], which satisfy d = 1 conservation laws [∂_νj_μν + ∂_τj_μτ] = 0 (with no summation over repeated indices implicit). Such d-dimensional Gauge-Like Symmetries may lead to a conservation law for each line associated with a fixed value of all coordinates x_ν≠μ relating to the topological charge $Q_{\mu }(\{x_{v\ne \mu }\})=\int d{x}_{\mu }{j}_{\mu \tau }(\overrightarrow{x})$ (no summation over the repeated index μ).

A similar consequence of d-dimensional Gauge-Like Symmetries is that topological terms appearing in d + 1-dimensional theories also appear in higher D + 1-dimensional systems (D > d). These topological terms appear in actions S̄ (see Criteria for Robust Topological Quantum Memory or ref. 8), which may be used to bound (both from above and below) expectation values of quantities that are not invariant under the d-dimensional Gauge-Like Symmetries. For instance, in the isotropic D = 2 (or 2+1) dimensional general spin t_2g Kugel–Khomskii model (6, 7, 22), of exchange constant J > 0, the corresponding continuum Euclidean action used to bound the d = 1 symmetry noninvariant quantities is of the 1+1 form $\overline{S}=\frac{1}{2g}{\displaystyle \int dxd\tau }\left[\frac{1}{vs}{({\partial }_{\tau }\widehat{m})}^2-v_s{({\partial }_x\widehat{m})}^2\right]+i\theta Q+S_{tr,}Q=\frac{1}{4\pi }{\displaystyle \int dxd\tau \left[\widehat{m}.(\partial \tau \widehat{m}\times {\partial }_x\widehat{m})\right]}$. Here, as in the nonlinear-σ model of a spin-s chain, m̂ a normalized slowly varying staggered field, g = 2/s, the spin wave velocity v_s = 2Js, θ = 2πs, and S_tr a transverse-field action term that does not act on the spin degrees of freedom along a given chain. Q is the Pontryagin index corresponding to the mapping between the two-dimensional space-time (x, τ) plane and the two-sphere on which m̂ resides. This 1+1 dimensional topological term appears in the 2+1 dimensional Kugel–Khomskii system even for arbitrary large positive coupling J. This, in turn, places bounds on the spin correlations enabling us to predict, for instance, that in D = 2 integer-spin t_2g Kugel–Khomskii systems, a finite correlation length exists.

Is there a relation between d-dimensional Gauge-Like Symmetries and the existence of fractional quantum numbers? Symmetries do not necessarily mandate the existence of degeneracies at all energies. Degeneracies, however, imply the existence of symmetries that effect general unitary transformations within the degenerate manifold and act as the identity operator outside it. The most general symmetry of any Hamiltonian is given by a direct product of the form ⊗_lSU(N_l) with N_l being the degeneracy of the lth eigenvalue of the Hamiltonian. Any projection of a symmetry operator onto the ground state sector must constitute an element of SU(N_g) with N_g the number of ground states. Any emergent symmetry also acts as an SU(N_g) operator. In this way, these unitary transformations may act as exact Gauge-Like Symmetries. The existence of such unitary symmetry operators (generally a subset of SU(N)) allows for fractional charge (such as the “triality” for the SU(3) group of quantum chromodynamics or “N-ality” of the SU(N) symmetry group). This suggests that degeneracy allows for a fractionalization defined by the center of the symmetry group [which may be SU(N) or any of its subgroups/quotient groups]. The (m-)rized Peierls chains constitute a typical example of a system with universal (m-independent) symmetry operators (6, 7), where fractional charge quantized in units of e* = e/m with e the electronic charge is known to occur (22). The bounds above can be generalized to apply to these symmetries. Different Peierls chain ground states break discrete d = 1 symmetries (violating Eq. 1 in this system with fractionalization), meaning that fractionalization may occur in systems with no TQO. The fermion number N_f in the Peierls chain and related Dirac theories is an integral over spectral functions (23); the fractional portion of N_f stems from soliton contributions invariant under local background deformations.

When the bound of (6, 9) is applied to correlators and spectral functions, it implies the absence of quasi-particle excitations in many instances (24). We elaborate on this: The bound of (6, 9) mandates that the absolute values of nonsymmetry invariant correlators |G| ≡ |∑_Ωja_Ωj〈∏_i∈Ωj ϕ_i〉| with Ω_j ⊂ C_j, and {a_Ωj} c-numbers, are bounded from above (and from below for G ≥ 0 (e.g., that corresponding to 〈|ϕ(k,ω)|²〉)) by absolute values of the same correlators |G| in a d-dimensional system defined by C_j. In particular, when k lies in a lower dimensional region where the d-dimensional Gauge-Like Symmetry is present (e.g., for k parallel to the ê_x or ê_z axis of the orbital compass model), the coefficients {a_Ωj} can be chosen to give the Fourier transformed pair-correlation functions. This provides bounds on viable quasi-particle weights and establishes the absence of quasi-particle excitations in many cases. In high-dimensional interacting systems, retarded correlators G generally exhibit a resonant (quasi-particle) contribution. G = G_res(k, ω) + G_non–res(k, ω) with G_res(k, ω) = $\frac{Z_k}{\omega -{\varepsilon }_k+i{0}^{+}}$. In formal terms, in high-dimensional systems, the interactions among otherwise bare free particles simply “renormalize” those free particles. The interactions lead to new quasi-particles that are for all purposes just bare particles with new renormalized weights and effective parameters. This quasi-particle behavior is captured by the poles of the retarded correlator. This is not so in low-dimensional systems (25). In low dimensions, particle-particle interactions can lead to a dramatic change of the system. The system no longer has quasi-particle type behavior: the quasi-particle weight Z_k → 0 and the poles of G are often replaced by weaker branch cut behavior. Fractionalization is associated with the disappearance of quasi-particles with sharp dispersion relations. If the momentum k lies in a lower d-dimensional region C_j and if no quasi-particle resonant terms appear in the corresponding lower-dimensional spectral functions in the presence of nonsymmetry breaking fields then the upper bound (9) on the correlator |G| (and on related quasi-particle weights given by Z_k = lim_ω→∈k(ω − ∈_k)G(k, ω)) of symmetry noninvariant quantities mandates the absence of normal quasi-particles. “Nonsymmetry breaking fields” refer to fields outside the region where the d-dimensional Gauge-Like symmetry is present - i.e. the fields {ϕ_i ∉ C}. Using the bound of (8) for Z_k = lim_ω→∈k(ω − ∈_k)G(k, ω) we see that if fractionalization occurs in the lower-dimensional system (Z_k = 0) then its higher-dimensional realization follows. These external fields do not (by the very presence of the d-dimensional Gauge-Like Symmetry: [H, U] = 0) break the symmetry. Fractionalized excitations propagate in d_s = D − d dimensional regions (25). As bounds concerning correlators in low-dimensional gapped systems may be extended to zero temperature, the bound of (9) implies the absence of quasi-particle excitations also in cases in which continuous d < 2 Gauge-like symmetries only emerge within the low energy sector.

Our central claim is that in all known systems with TQO, d-dimensional Gauge-Like Symmetries are present. Old examples include: Fractional Quantum Hall systems, ℤ₂ lattice gauge theories, the toric code model (4), and others. In all cases of TQO, we may express known topological symmetry operators as general low-dimensional d ≤ 2 Gauge-Like Symmetries (e.g. in the toric code model, there are d = 1 symmetry operators spanning toric cycles). These symmetries allow for freely propagating decoupled d-dimensional topological defects (or instantons in (d + 1) dimensions of Euclidean space-time) that eradicate local order. These defects enforce TQO. We now state a central result:

Theorem. Consider a system with interactions of finite range and strength that satisfies Eq.1. If all ground states may be linked by discrete d ≤ 1 or by continuous d ≤ 2-dimensional Gauge-Like Symmetries U ∈ G_d, then the system displays finite-T TQO.

Proof. We write V = V₀ + V_⊥. Here, V₀ is the part transforming as a singlet under G_d i.e., [U, V₀] = 0 . To prove the finite-temperature relation of Eq. 2, we note that two ground states may be related by a unitary transformation (U) and write the expectation values over a complete set of orthonormal energy states {|a〉},

We invoked U|a〉 ≡ |b〉, and E_a = E_b following from [U, H] = 0. The term ϕ_α^a is the expectation value of the infinitesimal operator α in the state |a〉. That is, 〈a|α|a〉 ≡ hϕ_α^a with h = |α| the operator norm of α. In the above derivation, ϕ_α^a = ϕ_Uα^Ua = ϕ_β^b with β ≡ U^†αU. As V_⊥ is not invariant under G_d ([U, V_⊥] ≠ 0) the theorem of ref. 9 implies that 〈V_⊥〉_α = 0, i.e., spontaneous symmetry breaking is precluded in systems with low-dimensional Gauge-Like Symmetries. [See Fig. 2.] Eq. 7 is valid whenever [U, V₀] = 0 for any symmetry U. However, [U, V_⊥] ≠ 0 implies 〈V_⊥〉_α = 0 only if U is a low-dimensional Gauge Like Symmetry. In systems in which not all ground state pairs can be linked by the use of low-dimensional Gauge-Like Symmetries U ∈ G_d (U|g_α〉 = |g_β〉), finite-temperature spontaneous symmetry breaking may occur. We conclude this proof with two remarks (6, 7): (i) T = 0 TQO holds whenever all ground states may be linked by (exact or emergent) continuous d < 2 Gauge-Like Symmetries in gapped systems. (ii) In many systems (whether gapped or gapless), T = 0 TQO states may be constructed by employing Wigner-Eckart type selection rules for d ≥ 1 Gauge-Like Symmetries. It is important to emphasize that the T = 0 conditions of TQO (Eq. 1) can be violated for some ground states of theories with local (d = 0) symmetries. In such theories, ground states can be linked by d = 0 operators; choosing V to be such a quasi-local (d = 0) operator the TQO requirement of Eq. 1 is violated.

What physical quantity characterizes TQO? We will see that neither the spectrum of H, nor the entanglement entropy (27, 28), nor string/brane correlations are sufficient criteria. That the spectrum, on its own, is insufficient is established by counter-examples. For example, Kitaev's (4) and Wen's models (30) have d = 1 symmetries and a spectrum identical to that of two uncoupled Ising chains (d = 1) with nearest-neighbor interactions [as we showed by a duality mapping (6, 7)]. While Kitaev's and Wen's models display TQO, the Ising chains do not. By duality, the exchange statistics of defects (the so-called magnetic and electric charges) of Kitaev's model that reflect the change incurred while moving one excitation completely around another (i.e., braiding), can be similarly cast as operator relations that may be represented on two Ising chains. The mapping of the braiding operations is, in general, path dependent. The electric and magnetic particles of Kitaev's model map onto domain walls in the two respective Ising chains. The operator products corresponding to full braiding operators of defects in Kitaev's model can be cast as string products of the spins on the Ising chains when domain walls are present. Our duality mappings to Ising chains demonstrate that despite the spectral gap in these systems, the toric operator expectation values and, notably, the long time autocorrelation functions, may vanish once thermal fluctuations are present [a concept we coined “thermal fragility” (6, 7)]. The Ising chains have a zero temperature transition. In Kitaev's toric code model (4), an explicit calculation of the temporal autocorrelation function among the d = 1 topological invariants using these mappings shows that the autocorrelation time is finite at any positive temperature (8). The vanishing expectation values of the toric code operator reflect a finite autocorrelation time τ . The latter is identical to that in Ising chains. For Glauber-type dynamics, τ = const/(1 − tanh[2/(k_BT)]).

For periodic boundary conditions and in the asymptotic low-temperature limit, this autocorrelation time is related to the spectral gap Δ between the ground and the next excited-state energies by τ →_T→0 exp(Δ/k_BT). Thus, although finite, the autocorrelation time can be made very large if temperatures far below the gap may be achieved. In particular, we discussed the detailed physical meaning of this result in ref. 8, illustrating that realizing finite-T fault tolerant quantum memories is a subtle problem. Finite-T TQO (Eq. 2) is not a sufficient condition for realizing a memory that is immune to thermal fluctuations. In the SI Text, we discuss conditions for a robust topological memory. A spectral equivalence exists between other TQO inequivalent system pairs. Our mappings illustrate that the quantum states themselves in a particular (operator language) representation encode TQO. The duality mappings, being nonlocal in the original representation, disentangle the order. Thus, we cannot, as coined by Kac, “hear the shape of a drum”. The information is in the eigenvectors.

It was suggested (27, 28) that a deviation γ > 0 from an asymptotic area law for the entanglement entropy is a measure of TQO. We find that on its own, this (defined) measure of topological entanglement entropy (TEE)—a linear combination of entanglement entropies to extract γ—is not a clear marker of TQO. Consider Klein spin models whose ground state basis is spanned by the product of singlet states (6, 7). In most of these ground states, local measurements can lead to different expectation values (and thus do not satisfy Eq. 1). Still, we find (6, 7) (i) an entanglement entropy that deviates from an area law within the set of all ground states that do not have TQO. Such an arbitrary—contour-shape dependent—finite TEE in a non-TQO system is not in accord with the conjecture of (27, 28). Conversely, (ii) a direct product of TQO states on decoupled spatial regions leads to a TQO state satisfying Eq. 1 with, however, a vanishing entanglement entropy across the boundaries between these different regions [γ = 0]. The criterion of (27, 28) deducing TQO from a finite γ may require additional improvements, such as the specification of an exact limit and/or average for large contours. We conjecture that for TQO we need, at the very least, not only to have a finite TEE but also that this entropy is independent of the contour chosen.

The insufficiencies of the spectra and entropy for assessing if TQO is present have counterparts in graph theory and in the Graph Equivalence Problem in particular. The adjacency matrix of a graph has elements C_ij = 1 if vertices i and j are linked by an edge and C_ij = 0 otherwise. Vertex relabeling i → p(i) leaves a graph invariant but changes the adjacency matrix C according to C → C′ = P^†CP with P representing the permutation: P = δ_j,p(i). The Graph Equivalence Problem is the following (31): “Given C and C′, can we decide if both correspond to the same topological graph?” The spectra of C and C′ are insufficient criteria. Entropic measures (31) are useful but also do not suffice.

In many systems (with or without TQO), there are nonlocal string correlators that display enhanced (or maximal) correlations vis a vis standard two-point correlation functions. We now outline an algorithm (6) for the construction of such nonlocal correlators. (In systems with uniform global order already present in their ground states, the algorithm leads to the usual two-point correlators.) We seek a unitary transformation U_s, which rotates the ground states into a new set of states that have greater correlations as measured by a set of local operators {V_i}. These new states may have an appropriately defined polarization (as manifest in the eigenvalues {v_i}) of either (i) more slowly decaying (algebraic or other) correlations, (ii) a uniform sign (partial polarization) or (iii) maximal expectation values v_i = v_max for all i (maximal polarization). Cases i or ii may lead to a lower dimensional gauge-like structure for the enhanced correlator. In systems with entangled ground states (such as in many (but not all) ground states with T = 0 TQO), U_s cannot be a uniform product of locals; U_s ≠ ∏_i∈Λ O_i. To provide a concise known example that clarifies these concepts, we focus on case ii within the well-studied AKLT spin 1 Hamiltonian (32, 33). Here, there is a nontrivial unitary operator U_s ≡ ∏_j<k exp[iπS_j^zS_k^x], ([U_s, H] ≠ 0), which maps the ground states {|g_α〉} into linear superpositions of states in each of which the local staggered magnetization V_i = (−1)ⁱS_i^z ≡ M_i is uniformly nonnegative (or nonpositive) at every site i. As all transformed states |p_α〉 = U_s|g_α〉 are superpositions of states with uniform sign v_i (allowing for two nonnegative or nonpositive v_i values at every site out of the three s = 1 states), ${\tilde{G}}_{ij}\equiv \langle p_{\alpha }\left|V_i{V}_j\right|p_{\alpha }\rangle >\langle g_{\alpha }\left|V_i{V}_j{g}_{\alpha }\right|\rangle \equiv G_{ij}$. When U_s is written in full, ${\tilde{G}}_{ij}$ becomes a nonlocal string correlator ${\tilde{G}}_{ij}={(-1)}^{|i-j|}\langle g_{\alpha }|S_i^z{\prod }_{i\&lt;k\&lt;j}exp[i\pi S_k^z]S_j^z|g_{\alpha }\rangle ={(2/3)}^2$ for arbitrarily large separation |i − j|. In the ⊗S_i^z eigen-basis, there are (2^N_s+1 − 1) states |ϕ〉 with nonnegative (nonpositive) v_i for all i (they form a Hilbert subspace). Such an exponential number of states appears in systems with a local ℤ₂ gauge structure. Here, the string correlators exhibit a symmetry under local (d = 0) gauge transformations while H itself is not invariant. These d = 0 ℤ₂ transformations correspond (in the ⊗S_i^z basis) to the creation (annihilation) of an S_j^z = 0 state at any site j followed by a unit displacement of S_k^z at all sites k > j. The operators linking the ground states form a discrete, global, (ℤ₂ × ℤ₂) symmetry group (32, 33) and, as such, T = 0 spontaneous symmetry breaking may occur. In the AKLT problem, the local expectation value 〈g_α|S_i=1^z|g_α〉 depends on |g_α〉 , violating Eq. 1 and suggesting that the AKLT chain is not topologically ordered (nonetheless, in the open chain there are spin-1/2 fractional excitations localized at the ends of the chain). The general polarization operators are intimately tied to selection rules that hold for all of the ground states; these selection rules (and a particular low-energy projection of truncated Hamiltonians for gapped systems) enable the construction of general string and higher dimensional brane correlators in general gapped systems. We may also generate maximal (${\tilde{G}}_{ij}$ = 1) string or brane type correlators; here, the number of states with maximal polarization is finite and a nonlocal gauge-like structure emerges. In systems with known (or engineered) ground states, we may construct polarizing transformations U_s.

To conclude, we developed a symmetry-based framework to study physical systems that display TQO. Symmetries may represent invariances of the actual system Hamiltonian or of emergent low-energy phenomena. In particular, we proved a theorem establishing how d ≤ 2-dimensional Gauge-Like Symmetries in gapped systems imply TQO (at all temperatures) via freely propagating low-dimensional topological defects (e.g. d = 1 solitons). This result extends the set of known TQO systems to include new orbital, spin, and Josephson junction array problems. All known TQO systems, e.g., the Quantum Dimer Model, display Gauge-Like Symmetries (6, 7). We examined physical consequences of those symmetries [such as conservation laws, dimensional reduction, and topological terms in high space dimension (D) theories], and low-dimensional gauge-like structures in theories with entangled ground states in nonmaximal string correlators (irrespective of TQO). We discussed an algorithm for the construction of nonlocal string/brane orders in systems where the ground states conform to certain selection rules and note that string/brane orders appear in general gapped systems in arbitrary space dimensions. The Hamiltonian spectra, fractionalization, entanglement entropy, and maximal string correlations do not imply TQO in all cases. We note that a non-zero topological Chern number [the Hall conductivity can be related to it (34)] does not imply TQO either, it only indicates that the system has a spectral gap with, e.g., a quantum state that breaks time-reversal symmetry (35–38), although the system can still host a Landau order parameter. Similarly, many instances of degeneracy in TQO systems follow from general time reversal invariance (Kramers' theorem); topological considerations need not be invoked to establish ground state degeneracy—especially so in the presence of a gap (6, 7, 39). In some examples of TQO, an exact dimensional reduction occurs. For example, the free energies of Kitaev's and Wen's D = 2 models are equivalent to those of Ising chains, and consequently no finite-temperature phase transition occurs (6, 7). We also drew analogies between TQO and the topology of graphs. When unbroken by external perturbations, the d-dimensional Gauge-Like symmetries that formed the focus of our study may constrain the system dynamics. Our work extends and complements different studies of protected nonlocal structures elsewhere (40).