Absence of localization in interacting spin chains with a discrete symmetry

We prove that spin chains symmetric under a combination of mirror and spin-flip symmetries and with a nondegenerate spectrum show finite spin transport at zero total magnetization and infinite temperature. We demonstrate this numerically using two prominent examples: the Stark many-body localization system and the symmetrized many-body localization system. We provide evidence of delocalization at all energy densities and show that the delocalization mechanism is robust to breaking the symmetry. We use our results to construct two localized systems which, when coupled, delocalize each other.

Introduction.-One of the basic assumptions of classical or quantum statistical mechanics is that interacting many-body systems thermalize, approaching local thermodynamic equilibrium under unitary dynamics. This assumption is not satisfied for localized systems, in which transport is arrested. Two well-known examples are strongly disordered or "many-body localized" (MBL) systems [1][2][3][4][5], and clean systems with a strong tilted potential ("Stark-MBL") [6, 7]. Significant suppression of dynamics was experimentally observed in both MBL [8,9] and Stark-MBL systems [10,11].
Intrinsic instability in localized noninteracting systems occurs due to resonances, which are distinct regions in space with close energies of the single-particle states [12]. The probability to have a resonance between two distinct regions decays exponentially with the distance between them [13,14]. Lowering the disorder strength increases the density of the resonances and eventually leads to delocalization at d ≥ 3 [12]. The resonances also give the dominant contribution to ac conductivity in localized systems [13][14][15][16][17].
For interacting systems, the resonant condition includes also the local interaction energy. Similarly to the noninteracting case [13,14] resonances can induce a nonlocal response [18]. Nevertheless, under the assumption that the levels of the many-body spectrum do not attract each other, it was rigorously shown that the many-body resonances cannot delocalize one-dimensional disordered systems [19,20]. The proof does not apply for higher dimensions, and it is currently unclear if localization is possible for two and higher dimensional interacting systems [21,22].
The discussion above refers to disordered systems without global symmetries. On the other hand, systems with symmetries can have either exact or a nearly degenerate spectrum, such that the rigorous proof of localization does not apply [19,20]. Discrete compact symmetriesĤPĤP do not seem to affect localization [23][24][25][26][27], however exact resonances can lead to delocalization in translationinvariant systems [7,28], as also systems with continuous non-Abelian symmetries [29,30]. Symmetry-assisted delocalization is however not stable to the addition of symmetry-breaking perturbations that lift many of the exact resonances [31].
A number of studies argue that MBL is unstable to the existence of delocalized inclusions, ruling out the existence of a mobility edge [32], and even the MBL transition itself [33][34][35]. This delocalization mechanism was numerically explored by embedding of thermal regions in MBL systems [22,36,37], or by coupling the system to a Markovian bath [38,39]. It is not clear if a similar mechanism is present in clean localized systems such as Stark-MBL.
In this Letter we prove that localization is absent in a large class of many-body spin systems with a nondegenerate spectrum. This class consists of all systems symmetric under the combination of spatial mirroring and spin flipping. By numerically verifying that the nondegeneracy assumption is fulfilled for interacting Stark-MBL and appropriately symmetrized disordered problems, we thus rule out localization in these systems and then explore the stability of these results to symmetry-breaking per-turbations. Finally, we utilize our result to construct two localized systems that delocalize each other.
General argument.-We consider a spin chain of length L described by a HamiltonianĤ, and assume the following: Assumption 1.Ĥ has a nondegenerate spectrum.
Assumption 3. The Hamiltonian is symmetric under a combination of a mirror symmetry and a spin-flip symmetry defined asPŜ whereŜ z i are spin operators of arbitrary spin size at site i, andŜ ± i are their corresponding raising (lowering) operators.
SinceP 2 =1 its eigenvalues are ±1. The commutator P , iŜ z i = 0 unless the total magnetization vanishes, and therefore we project the Hamiltonian onto the zero total magnetization sector, namely, we work in the zero magnetization sector.
We study spin transport by creating a spin excitation at site j on top of some equilibrium stateρ, such that ρ,Ĥ = 0, and assess its spreading using the connected spin-spin correlation function, where Ô ≡ Tr ρÔ . Taking the infinite-time average, , and using Assumption 1 we obtain, where 0 ≤ p α ≤ 1 are the eigenvalues ofρ. For systems without spin transport, the spin excitation is expected to be localized in the vicinity of site j at infinite times, where ξ is the localization length [40]. For systems that relax to equilibrium G ρ ij → 0 such that the excitation is uniformly spread over the lattice. Here the process is inherently many-body since it is not present for systems which can be mapped to noninteracting fermions [41]. To quantify the spreading of the excitation we use the mean-squared displacement (MSD), and its corresponding infinite-time average σ 2 ρ . For delocalized states, the infinite-time averaged MSD scales as σ 2 ρ ∼ L 2 , while for localized states σ 2 ρ ∼ ξ 2 . We now prove that for systems satisfying the assumptions above, σ 2 ρ ∼ L 2 , implying that at least a finite fraction of eigenstates are delocalized. For brevity, we only provide the sketch of the proof here; see Supplemental Material (SM) for details [41].
We takeρ =1/N where N = L L/2 is the Hilbert space dimension. This corresponds to setting p α = 1/N in Eq. (3), such that the infinite-time average of (4) becomes, We first note that N −1 α α|Ŝ z iŜ z j |α = 1 4(L−1) and therefore the second term in (5) is O L 2 [41]. To bound the first term we use the symmetryP and the identity, whereD = i iŜ z i is the dipole operator andĩ = L−i+1 the mirrored coordinate. Inserting this identity into the first term in (5) and using a combination of triangle and Hölder inequalities, we bound Since the second term in (5) can be exactly evaluated and scales as L 2 and it can be shown that D 2 1/2 = O L 3/2 then for all j −j < O L 1/2 the second term is dominating in the thermodynamic limit which yields σ 2 ∞ ∼ L 2 . It is important to note that this is not an upper bound on σ 2 ∞ but an asymptotic result, which implies delocalization of a finite fraction of eigenstates [42].
The proof does not rule out localization in noninteracting Stark or Anderson problems, since due to theP symmetry there are degeneracies in the many-body spectrum invalidating Assumption 1 [41].
In what follows, we numerically demonstrate that Assumption 1 is satisfied for two cornerstone models of localization in interacting systems and provide evidence of delocalization for all energy densities.
Applications.-We consider the Hamiltonian of a spin-1/2 chain of length L, whereŜ ± n ,Ŝ z n are spin-1/2 operators, J is the strength of the flip-flop term, ∆ is the strength of the Ising term and h n is an arbitrary magnetic field. For h n = −hñ the Hamiltonian clearly satisfies Assumptions 2 and 3. In what follows we numerically verify that Assumption 1 is also satisfied for our choices of h n . We consider two cases of ostensibly localized interacting systems: (a) h n = γ n − L+1 2 , such that all the single-particle states of the fermionic model are known to be localized for any γ and for sufficiently large γ the model is expected to be Stark many-body localized (Stark-MBL) [6, 7] . (b) h n = −hñ, but otherwise randomly and uniformly distributed in the interval [−W, W ]. We have verified numerically that all the single-particle states are strongly localized, and have only rare single-particle resonances, so that the model might be expected to be many-body localized (MBL) for sufficiently large W , by analogy with the standard MBL case [1]. We shall call case (b) symmetrized-MBL, as it obeys the symmetry embodied in Assumption 3.
We begin by verifying assumption 1 for both models, by numerically diagonalizing the Hamiltonian for systems sizes L = 11 − 19 setting J = 2 and ∆ = 1. We work in the zero (1/2) total magnetization sector for even (odd) system sizes. For the Stark-MBL case we take γ = 2.75 and for the symmetrized-MBL case W = 9. We use 10 000 disorder realizations for averaging. Figure 2 shows a histogram of the logarithm of the eigenvalue spacings, log 10 δE. Both models have a wide range of eigenvalue pairs that lie very close to each other compared to the average spacings, but are not degenerate. These quasi-degenerate pairs of states are found across the symmetry sectors ofP , as we show by restricting the eigenvalues to the even sector and calculating its distribution (see also [41]). The restricted distribution is centered around the average spacing and does not have a "fat" tail stretching to zero, which characterizes the unrestricted distribution. Since (8) satis- fies all the assumptions of our proof, we expect that a finite fraction of its eigenstates are delocalized. To confirm this, we calculate σ 2 ρ in Eq. (5) within the micro- and N E is the number of states in I. Figure 3 shows that σ 2 E /L 2 plotted vs rescaled energy for both models is nicely collapsed such that the states at all energies are delocalized, σ 2 E ∼ L 2 . We have established both analytically and numerically that both models have a delocalized excitation profile at all energy densities and infinite times. While this result is universal as long as Assumptions 1-3 are satisfied, the temporal and spatial dependence of the excitation profile (2) are model specific and are therefore left for the SM [41]. It is worthwhile to mention that both models exhibit subdiffusive transport which is better described by logarithmic subdiffusive transport, t ∼ ln L [43, 44], and not power-law subdiffusive transport, t ∼ L z (z > 2) [45,46]. For finite systems the spin-spin correlation function (2) decays to zero at all sites except j and its mirror j, which suggests a residual memory of initial conditions. The memory, however, "fades away" with increasing system size [41].
Symmetry breaking.-The proof of finite spin transport crucially depends on existence of the symmetryP . It is interesting to see if finite transport persists also when the symmetry is broken. We have numerically examined a number of ways to break the symmetry in models described by (8): taking a finite magnetization, using an odd system size, or breaking the symmetry of the magnetic field h n . All produce qualitatively similar behavior of dramatically suppressed dynamics (see for example Refs. [6,7]). Here we only present results for odd system sizes and total magnetization 1/2. To examine localization of the excitation profile (2), we compute a positive version of the MSD by taking G ρ ij (t) − G ρ ij (0) in Eq. (4) and taking an infinite-time average, σ 2 sgn . This is done to avoid the quasi-conservation of the MSD in Stark-MBL systems [47,48]. While it implies the absence of diffusion, it does not exclude subdiffusive transport [45,48]. Figure 4 shows that σ 2 sgn grows with system size for both models. It is hard to extract a reliable dependence on the system size from the accessible system sizes, but the growth is consistent with L 0.35 for the Stark-MBL system and L 1. 35 for the symmetrized-MBL system. If the growth persists in the thermodynamic limit it implies asymptotic delocalization. Instead of breaking the symmetry of the Hamiltonian we can use a nonequilibrium initial conditionρ which either satisfies or breaks the symmetry. For initial conditions that are odd or even with respect toP (such as the Neél state) we observe some memory of the initial state, however there is no asymptotic memory retention of initial conditions that break the symmetry [41].
Discussion.-We have proved that any spin chain with the symmetry given in Assumption 3 and a nondegenerate spectrum exhibits spin transport for a finite measure of its eigenstates. The proof does not apply to noninteracting systems that have degeneracies due to the symmetry, and thus can remain localized. We have numerically demonstrated delocalization of the asymptotic excitation profile for two cornerstone models of localization in many-body systems: the Stark-MBL model [49] and the symmetrized-MBL model. Our results suggest that for these models delocalization happens at all energy densities and spin transport is subdiffusive, and most probably logarithmic [41]. Moreover, our numerical results are consistent with asymptotic delocalization of the excitation profile also in the case of weak symmetry breaking, even though the finite-time dynamics is strongly suppressed.
Constructing a localized delocalizing bath.-A localized system is typically a closed system with no transport. Coupling a localized system to a Markovian heat bath induces slow transport for local coupling [50,51] or diffusive transport for global coupling [52][53][54][55][56]. A similar effect is expected to occur if a Markovian bath is replaced by a sufficiently large thermalizing system. But what if we couple two localized systems? Is it possible to induce transport in such a configuration? Since by definition there is transport in neither system, as a result of the coupling only resonant transfer between the two systems is possible. A possible guess could be couplinĝ H to itself, that we will call the "Ĥ toĤ" composite system. When the two systems are uncoupled all the spectrum is doubly degenerate and therefore resonant. Under such conditions any small coupling between the systems lifts the degeneracies and presumably results in weak transport. However numerical results suggest that this configuration does not result in obvious delocalization for neither interacting nor noninteracting systems (not shown). We conclude that the existence of exact resonances is not a sufficient condition of delocalization.
We now use our results to construct a localized system which, when attached to the edge of a given localized system, described by a localized HamiltonianĤ, delocalizes it. SinceĤ is localized, the unitarily transformed sys-temPĤP is also localized. For noninteracting systems the symmetryP implies that the single-particle spectrum ofPĤP is a reflection around zero of the singleparticle spectrum ofĤ, such that there are no exact single-particle resonances [41]. On the other hand, the many-body spectrum ofĤ is identical toPĤP and therefore has exact many-body resonances, similarly to theĤ toĤ system. Nevertheless, by couplingĤ andPĤP at the edge using a symmetric couplingPVP =V results in a composite Hamiltonian that is symmetric underP : H =Ĥ+PĤP +V (see Fig. 1). We will call this coupling "Ĥ toPĤP ". SinceĤ satisfies Assumptions 1-3 [57] it follows from the delocalization proof thatĤ is delocalized. The most dramatic demonstration of symmetryinduced delocalization can be obtained by coupling two Anderson insulatorsĤ andPĤP . A noninteracting coupling of the formV =Ŝ + + L/2+1 cannot lift the degeneracies and the system is localized. On the other hand, modifyingV to include an interacting term, such asŜ z L/2Ŝ z L/2+1 lifts the degeneracies and results in delocalization via the delocalization proof. Thus, theĤ toĤ coupled system has resonances but appears to be localized, while theĤ toPĤP coupling also has resonances, yet is delocalized. Studying the difference between these systems may provide insight into the role of resonances in delocalization.
Other open questions are whether the delocalization mechanism carries over to other spatial symmetries. What is the fastest possible transport between two coupled localized systems? Is it always logarithmic? It would be also interesting to see if our delocalization proof can be generalized to higher dimensions, the microcanonical ensemble, other conserved quantities, such as the energy, and unbounded local Hilbert space dimensions.
Moreover, implications on thermalization in systems respecting the symmetry should be also explored. In this section we examine the properties of noninteracting fermionic systems, which conserve the total particle number and are symmetric with respect toP (see main text). Specifically we consider,

J.C.H. acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant
whereĉ † i creates a fermion at site i,n i =ĉ † iĉ i is the fermion density operator, f (|i − j|) corresponds to the hopping rate of the fermions and v i is the external potential. To show the requirements on f (|i − j|) and v i for the Hamiltonian to be symmetric underP it is convenient to use the Jordan-Wigner transformation, The Hamiltonian written in terms of spins is, We see that for, the Hamiltonian is symmetric with respect toP . Using (S4) it is easy to check that the corresponding single-particle Hamiltonian, is anti-symmetric with respect to the unitary transformation, |i → (−1) i ĩ , which means that the single-particle spectrum is symmetric around zero. It is important to note that since the single-particle spectrum is symmetric, there are no exact resonances in the single-body problem, and therefore creating an excitation at site j does not create a resonant excitation at the mirrored sitej, as can be verified numerically. Nevertheless, the many-body spectrum is degenerate, and therefore Assumption 1 is not satisfied and the proof we present in Section III doesn't apply. Specifically, this includes the noninteracting Stark and the Anderson problem with the anti-symmetric disorder. We have verified numerically that these problems indeed remain localized.

II. QUASI-DEGENERACIES FOR THE STARK-MBL
The existence of the quasi-degeneracies can be analytically motivated for the Stark-MBL Hamiltonian, which is diagonal at the computational basis, namely the eigenbasis ofŜ z n operators. We will treat the flip-flop term as a perturbation,V Our goal is to show that at zero magnetization there are exponentially many states which are almost generate. We call a unit-dipole a configuration which looks like, d + i,i+1 = (↑, ↓) or equivalently d − i,i+1 = (↓, ↑). We construct a state |ψ 1 by adding to the lattice N + dipoles d + and N − = L/2 − N + dipoles . The total dipole moment of the state is proportional to, N + − N − , and its unperturbed energy is E (0) 1 . The number of such terms is L/2 N+ , namely it is exponential in the size of the system. These states have typically a different energy due to the interaction ∆. By applying the symmetry generatorP we can obtain a new state |ψ 2 =P |ψ 1 , which has the same unperturbed energy. The operatorP is a global, while the local perturbationV can only flip one unit-dipole at a time. Therefore an order of αL such flips are needed to have a non-vanishing coupling between |ψ 2 and |ψ 1 , This means that the degeneracy between the eigenvalues is removed only at order αL of the perturbation theory, where α is some constant, which depends on the structure of the state. The resulting splitting between the eigenvalues will be, Since the typical many-body energy spacing is δ = exp [−L ln 2], the states will appear quasi-degenerate for, (S11)

III. INFINITE TIME DELOCALIZATION
We start by calculation of the correlation function Ŝ z iŜ z j = N −1 Tr Ŝ z iŜ z j for arbitrary spin size and zero total magnetization. Summing over all i we have the sum rule Separating the sum to i = j and i = j and using the fact that the expectation value cannot depend on either i or j we get, Since the total magnetization is set to zero, the total spin i S i is rotationally invariant, such that (S14) Using this and the definition of the magnitude squared operator of the spin where s is the size of the spins, obtain Combining this with the sum-rule (S13) we obtain, The infinite time average of the MSD, which only assumes that there are no exact degeneracies is given by, Using (S17) we see that the last term contributes (S19) We now move to the first term which can be simplified, where the last equality follows since we are working a zero magnetization sector. Focusing on, L i=1 i 2 α|Ŝ z i |α α|Ŝ z j |α , and using the parity symmetry Changing the summation variables, i = L − i + 1 gives, where the second term vanishes at zero magnetization. We can now rearrange the terms to obtain the identity, i 2 α|Ŝ z i |α α|Ŝ z j |α = (L + 1) α|D |α α|Ŝ z j |α .
Inserting this identity into (S20) gives, (S24) where we have defined the reflectedj ≡ L − j + 1. We note that for zero magnetization L is even and therefore there is no j such that j =j. We will now proceed by bounding this term. Using the triangle inequality, now since, we can bound, The right-hand side is not a trace of a matrix, and depends on the basis. However for any diagonalizabe matrix we can change to a basis |n , whereD is diagonal, such that, using the triangle inequality again we obtain, which means that 1 N α α|D |α is maximized in the basis whereD is diagonal. We now use Jensen's inequality and obtain finally, The expectation D 2 can be evaluated exactly using (S17), Combining all the results gives, Comparing to the second-term of the MSD in (S18) we obtain that for j − j ≤ AL 1/2 , where A > 0 is some constant, which concludes the proof that at least a fraction of eigenstates in the system are delocalized.

IV. SPATIAL PROFILE OF THE SPIN EXCITATION
The mean-square displacement (MSD) is lacking the spatial information on the spreading of the spin excitation, which is contained in the infinite-time averaged spin-spin correlation function, For delocalized systems with no memory of the initial condition this function is expected to vanish at all sites. In the bottom row of Fig. S1 we calculate and plot G ∞ ij for the Stark-MBL and symmetrized-MBL problem for a number of even (left) and odd (right) system sizes. For even system sizes the total magnetization is zero and the system is symmetric with respect toP , which yields to the anti-symmetric shape G ∞ ij = −G ∞ ij . For odd system sizes the total magnetization is 1/2 and the symmetryP is broken, such that the shape of the excitation G ∞ ij doesn't have to be anti-symmetric. For even system sizes the correlation function is close to zero at all sites, except i = j and i =j, indicating a relaxation to equilibrium, however for odd system sizes all sites appear to be away from zero.
The nonzero value of G ∞ ij for i = j and i =j, indicates some memory of the initial condition, however as the top row of Fig. S1 shows this memory is decaying with the system size for both odd and even system sizes. We cannot reliably extract the dependence of the decay on the system size, but it is quite slow as one can learn from the qualitative power-law fits that are listed in the top row of Fig. S1. It is important to note that non-vanishing of G ∞ ij for a finite number of sites, is consistent with finite transport, since finite transport requires σ 2 ∞ ∼ L 2 which includes a contribution from an extensive number of sites. Therefore existence of finite memory is not in contradiction to the proof in Section. III. Here the memory of the excitation appears to fade away in the thermodynamic limit for both symmetry preserving and symmetry breaking systems.

V. EXCITATION SPREADING
In this section we consider the of the time-dependence of the excitation profile for both Stark-MBL and symmetrized-MBL systems for both symmetry preserving (even L) and symmetry breaking (odd L) system sizes. For this purpose we compute the positive MSD, which via triangle inequality bounds σ 2 ∞ (t) ≤ σ 2 sgn (t). We take σ 2 sgn (t), and not σ 2 ∞ (t), since for example for dipole preserving systems σ 2 ∞ (t) ≤ C uniformly in time while there is still slow subdiffusive transport [45,48]. From Fig. S2 we see that while the dynamical behavior of Stark-MBL and symmetrized-MBL are very similar the long time behavior of symmetry preserving and symmetry breaking systems is very different. Symmetry preserving systems have an intermediate plateau after which the positive MSD grows logarithmically towards its infinite-time value. The height of the intermediate plateau decreases weakly with increasing the tilted-field or disorder strengths and delays the approach to the asymptotic plateau. The occurrence of quasi-degeneracies in the spectrum of symmetry preserving systems doesn't allow us to numerically compute the correct dynamics of the system beyond t > 10 16 , we therefore don't present the dynamics beyond these time in Fig. S2.
The dynamics of the positive MSD for symmetry breaking systems follows the dynamics of symmetry preserving systems up to time t * , which increases with increasing the strength of the tilted field or the disorder, but does not depend on the system size. Interestingly, the intermediate plateau of symmetry preserving systems coincides with the asymptotic plateau of symmetry breaking systems. As explained in the main text the asymptotic value slowly increases with system size.
While here we present results of symmetry breaking by going to an even system size, we have observed very similar phenomenology if the symmetry is broken differently. For example, by considering a nonzero magnetization at even L or by adding weak disorder or curvature.

VI. SYMMETRY BREAKING OF INITIAL CONDITIONS
The sensitivity of both Stark-MBL and symmetrized-MBL systems to symmetry breaking can also be observed via breaking the symmetry in the initial state and not the Hamiltonian. In the left panel of Fig. ?? we show the infinite-time average of Ŝ z i (t) , for the Néel state |Néel = |↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓ , which is even underP . We see that S z i shows residual memory of this initial condition. We can quantify this memory using the imbalance, which slowly decays with the size of the system (see right panel of Fig. ??). Flipping the spins in half of the system, results in a state which is neither odd nor even underP . As can be seen from the left panel of Fig. ?? any memory of the initial condition for this state is absent already for L = 16.