Fate of Algebraic Many-Body Localization under driving

In this work we investigate the stability of an algebraically localized phase subject to periodic driving. First, we focus on a non-interacting model exhibiting algebraically localized single-particle modes. For this model we find numerically that the algebraically localized phase is stable under driving, meaning that the system remains localized at arbitrary frequencies. We support this result with analytical considerations using simple renormalization group arguments. Second, we inspect the case in which short-range interactions are added. By studying both, the eigenstates properties of the Floquet Hamiltonian and the out-of-equilibrium dynamics in the interacting model, we provide evidence that ergodicity is restored at any driving frequencies. In particular, we observe that for the accessible system sizes localization sets in at driving frequency that are comparable with the many-body bandwidth and thus it might be only transient, suggesting that the system might thermalize in the thermodynamic limit.


I. INTRODUCTION
Understanding the breakdown of ergodicity in the quantum realm is an active and fast growing front of research, motivated by recent developments of controllable quantum simulations which allows one to access out-ofequilibrium dynamics [1,2]. In particular, many-body localization (MBL) describes the paradigm of ergodicity breaking in quantum phases of matter, generalizing the concept of Anderson localization to the interacting case [3][4][5][6].
An MBL phase is described by an emergent form of integrability, i.e., the existence of a complete set of quasilocal integrals of motion (LIOMs) [7][8][9][10]. These LIOMs are adiabatically connected to the integrals of motion of the non-interacting model and thus are typically exponentially localized. As a consequence, transport is absent, some memory of the local structure of its initial state is retained during the quantum evolution and the interactions between the LIOMs allow a slow information propagation through the system [11][12][13]. The MBL phase should be distinguished from an ergodic/thermal one, in which eigenstates are chaotic and are expected to obey the eigenstate thermalization hypothesis (ETH) [14][15][16][17].
Several works have studied the stability of the MBL phase in the case of interacting one-dimensional shortrange systems subject to strong disorder [12,[18][19][20][21][22]. Recently, the possible existence of MBL has been shown in systems in which its single-particles are power-law localized, dubbed algebraic MBL [23,24]. Algebraic MBL generalizes the paradigm of power-law localization to the interacting case and therefore its LIOMs develop algebraically decaying tails. An immediate consequence of having power-law localized LIOMs is that the bipartite entanglement entropy after a quantum quench has an unbounded algebraic growth in time [23,25,26], unlike the case with exponentially localized single-particles for which the entanglement propagates logarithmically slowly [11,13]. Furthermore, an increasing amount of In this work we investigate the e↵ects of configurational diso properties of a tight-binding model on a quasi-one-dimensional decorated with linear o↵shoots of randomly distributed leng exponentially localized along the backbone of the comb. Moreo extensive number of compact localized states with precisely z analytical understanding of these states and show that they su interactions along the backbone of the system where, for su they form many-body scar states. Finally, we discuss the im states on the dynamical properties of systems with configura appearance of long-lived transient behaviour in the time ev states. The random quantum comb: from compact localized states to many-bod In this work we investigate the e↵ects of configurational disorder on the eigenstates and dynamic properties of a tight-binding model on a quasi-one-dimensional comb lattice, consisting of a backbo decorated with linear o↵shoots of randomly distributed lengths. We show that all eigenstates a exponentially localized along the backbone of the comb. Moreover, we demonstrate the presence of extensive number of compact localized states with precisely zero localization length. We provide analytical understanding of these states and show that they survive in the presence of density-dens interactions along the backbone of the system where, for su ciently low but finite particle densiti they form many-body scar states. Finally, we discuss the implications of these compact localiz states on the dynamical properties of systems with configurational disorder, and the correspondi appearance of long-lived transient behaviour in the time evolution of physically relevant produ states.
The random quantum comb: from compact loc In this work we investigate the e↵ects of configurational dis properties of a tight-binding model on a quasi-one-dimensiona decorated with linear o↵shoots of randomly distributed len exponentially localized along the backbone of the comb. More extensive number of compact localized states with precisely analytical understanding of these states and show that they s interactions along the backbone of the system where, for su they form many-body scar states. Finally, we discuss the i states on the dynamical properties of systems with configura appearance of long-lived transient behaviour in the time ev states.
The random quantum comb: from compact localized states to many-body scars In this work we investigate the e↵ects of configurational disorder on the eigenstates and dynamical properties of a tight-binding model on a quasi-one-dimensional comb lattice, consisting of a backbone decorated with linear o↵shoots of randomly distributed lengths. We show that all eigenstates are exponentially localized along the backbone of the comb. Moreover, we demonstrate the presence of an extensive number of compact localized states with precisely zero localization length. We provide an analytical understanding of these states and show that they survive in the presence of density-density interactions along the backbone of the system where, for su ciently low but finite particle densities, they form many-body scar states. Finally, we discuss the implications of these compact localized states on the dynamical properties of systems with configurational disorder, and the corresponding appearance of long-lived transient behaviour in the time evolution of physically relevant product states.
The random quantum comb: from compact localize In this work we investigate the e↵ects of configurational disorder properties of a tight-binding model on a quasi-one-dimensional com decorated with linear o↵shoots of randomly distributed lengths. exponentially localized along the backbone of the comb. Moreover, extensive number of compact localized states with precisely zero l analytical understanding of these states and show that they survive interactions along the backbone of the system where, for su cient they form many-body scar states. Finally, we discuss the implic states on the dynamical properties of systems with configurationa appearance of long-lived transient behaviour in the time evoluti states.
The random quantum comb: from compact localized states to many-body sc In this work we investigate the e↵ects of configurational disorder on the eigenstates and dynamical properties of a tight-binding model on a quasi-one-dimensional comb lattice, consisting of a backbone decorated with linear o↵shoots of randomly distributed lengths. We show that all eigenstates are exponentially localized along the backbone of the comb. Moreover, we demonstrate the presence of an extensive number of compact localized states with precisely zero localization length. We provide an analytical understanding of these states and show that they survive in the presence of density-density interactions along the backbone of the system where, for su ciently low but finite particle densities, they form many-body scar states. Finally, we discuss the implications of these compact localized states on the dynamical properties of systems with configurational disorder, and the corresponding appearance of long-lived transient behaviour in the time evolution of physically relevant product states.
FIG. 1: Schematic phase diagram of an algebraically localized phase under driving. α is the rate of decay of the hopping amplitudes and ω is the frequency of the driving. The non-interacting localized phase is robust under driving (left). Instead ergodicity is restored at any driving frequency for the interacting case (right). In the limiting case α → ∞ (short-range), the system undergoes an MBL transition at a finite driving frequency.
efforts has been devoted to study the fate of MBL under periodic driving [27][28][29][30][31][32][33][34][35]. In Refs. 27 and 28 it was shown that at high frequency an MBL phase evades the fate of thermalization remaining localized, while at sufficiently low frequency the system is delocalized. As a consequence of the robustness of an MBL phase under driving, it has been possible to explore novel non-equilibrium quantum phases of matter, i.e., discrete time-crystals which are characterized by a spontaneous breaking of discrete time translation symmetry [29][30][31][32][33]. The aim of this work is to investigate the stability of an algebraic MBL under driving. In particular, we periodically drive (Floquet) a system of spinless fermions with long-range hopping known to have algebraically localized single-particle wavefunctions (|ψ(x)| ∼ 1/|x| α ) [36][37][38][39][40][41].
First, we study the stability of a non-interacting alge-arXiv:2108.02215v1 [cond-mat.dis-nn] 4 Aug 2021 braically localized phase by numerically inspecting several probes to distinguish a localized from an extended phase. We conclude that for the non-interacting case the localized phase does not break down under periodically driving. We support this claim by analytical considerations based on strong-disorder renormalization group techniques. Next, we focus on the interacting case. By studying both the eigenstates properties of the Floquet Hamiltonian and the out-of-equilibrium dynamics, we provide numerical evidence that the algebraic MBL phase breaks down under driving. We surrogate this conjecture with simple analytical considerations. Thus, ergodicity is restored at any finite drive's frequency in the thermodynamic limit. This result might imply the nonexistence of time-crystals with algebraically localized LI-OMs. Figure 1 summarizes schematically the phase diagram of an (non-)interacting algebraically localized phase under driving. The rest of the work is organized as follows. In Sec. II we introduce both, the non-interacting and the interacting model and we discuss the different probes that have been used to detect localization/delocalization. Section III A is devoted to probe the stability of a noninteracting case. We support these findings using a renormalization group approach, which is explained in Sec. III B. Finally, in Sec. III C we study numerically the stability of an algebraic MBL phase subject to driving.

II. MODEL & METHODS
We consider a periodically driven system of spinless fermions on an one-dimensional lattice with long-range hopping where J ij = J ji = µ ij / 1 + |i − j| 2α describes a disordered long-range hopping involving all pairs of sites, with an algebraically decaying amplitude w.r.t. their distance according to an exponent α > 0 [36][37][38][39][40][41][42][43]. Quenched hopping disorder is given by {µ ij }, which is uniformly distributed between [−1, 1], and on-site disorder is added by {h i } which are random fields uniformly distributed between [−W, W ]. V is the interaction strength and L is the number of sites and we indicate with N = L/2 (half-filling) the number of particles. We choose a binary driving protocol of period T , i.e., δ(t) = −δ for 0 < t < T /2 and δ(t) = +δ for T /2 < t < T and δ(t + nT ) = δ(t), n ∈ Z. As an energy reference we define the hopping unit J =

2L
i =j J 2 ij , where · indicates the average over disorder and by ω D = 2π/T we denote the driving frequency in the following.
We now briefly summarize the dynamical phase diagram of the non-driven Hamiltonian H(δ = 0) in Eq. (1).
The non-interacting case without driving (V, δ = 0), referred as power-law banded random matrix (PLBRM), is known to have a metal-insulator transition at α = 1 from an extended (α < 1) to an algebraically localized phase (α > 1) [44][45][46]. Thus, for α > 1 the single-particle orbitals are algebraically localized meaning that they possess power-law decaying tails around their centers of lo- In this work, we are interested in the robustness of this localized phase under driving, therefore we mostly consider α > 1.
As interactions are turned on (V > 0), the non-driven system goes through an MBL transition as a function of the disorder strength W [23,47]. For V = 1 and α = 3, the critical point has been estimated to be W c ≈ 3, ergodic for W < W c and localized for W > W c . The MBL phase of H in Eq. (1) is described by a complete set of LIOMs It is important to point out, that the question whether MBL is stable in higher dimensions or for long-range Hamiltonians, is still under debate [49][50][51][52][53][54][55][56][57][58][59][60][61]. Recent experiments in fairly large system sizes and for relevant time-scales, have shown evidence of an MBL phase for both, two-dimensional systems and long-range Hamiltonians [62][63][64][65]. However, in quantum-avalanche theory the proposed mechanism for delocalization invokes the existence of rare entropic thermal bubbles, which will have effect at very long time scales and large system-sizes [49-51, 66, 67].
We will numerically show that the algebraic MBL phase is vastly unstable against external driving. This fragility appears to be independent of the driving frequency, although partially accompanied with finite-size effects depending on the domain of frequency. As a result, we observe a qualitative difference with respect to the behavior from short-ranged, or a "conventional" MBL phase under driving, in which the LIOMs are exponentially localized. For the latter, it is believed that a system-size independent critical driving frequency ω c exists, above which the system remains localized [27,28,34].
Throughout this manuscript, numerical results were obtained using free-fermions techniques for the noninteracting case and exact diagonalization for the interacting one. In order to establish the existence of a localized phase, we use a variety of probes, each giving insights to the dynamical phase structure from different facets. In particular, for each quantity we average over at least 10 2 random configurations and over energy spectrum.
In the remainder of this section we describe all utilized probes in detail.

A. Level spacing statistics
The distribution of level spacing has been found to be an useful probe to distinguish an ergodic phase from an MBL one [17,68,69]. In an ergodic phase at infinite temperature, the level spacing is expected to be distributed as the one of a random matrix belonging to the same universality class of the Hamiltonian, namely the Gaussian Orthogonal Ensemble (GOE). However, in an MBL phase, due to the existence of LIOMs, energy levels are weakly correlated and the level spacings are Poisson distributed.
Specifically, the degree of proximity to a random matrix ensemble can be extracted from the distribution of level spacing through the so called r-value, or gap-ratio parameter, which is defined as where δ n are level spacing of two consecutive level energies and · refers to the disorder and spectrum average [68,69]. The r-value in the ergodic phase is For the periodically driven system in Eq. (1), we consider the quasi-energies of the Floquet Hamiltonian H F defined by which serves as generator for the discrete translation in time of period T [70].

B. Energy propagation
In driven systems the energy is not conserved during time evolution and at any discrete time step an extensive amount of energy is pumped up in the system. Thus, in an ergodic phase the long-time asymptotic steady state is expected to be a structureless infinite temperature state. However, in an MBL phase the expected heating is effectively prevented at high driving frequencies. This can be understood from the emergence of an extensive number of locally conserved quantities, which effectively hinder the system to exchange energy beyond a characteristic localization length [27,28,34].
In order to study the energy absorbed through the quantum time evolution, we define the dimensionless parameter where τ = t/T and |ψ(τ ) is the time-evolved state, starting from the ground-state of H(τ = 0). Here, E min and E max refer to the lowest and highest eigenvalues of H(τ = 0), respectively. In an ergodic phase in the thermodynamic limit the asymptotic long-time value of (τ → ∞) = 1/2 (middle of the spectrum), while in an MBL phase (τ → ∞) < 1/2.

C. Entanglement entropy
The MBL transition can be understood in terms of a rearrangement of the local structure of entanglement in the system [18,19,21]. In particular, the half-chain bipartite entanglement entropy (EE) measures the amount of entanglement through the cut and can be used, both to detect the MBL transition and to characterize the two phases. The EE (S) is defined through the half-partition reduced density matrix ρ L/2 , where the sites belonging to one half of the system have been traced out In an ergodic phase at infinite temperature, the eigenstates behave like random vectors and therefore the EE reaches the so called Page value S Page = L/2 log 2 − 1/2 [71], [72].
For the out-of-equilibrium dynamics, we compute the EE following a quantum quench starting from the Néel state |ψ(0) = N i c † 2i |0 . In an ergodic system driven in time, S(τ ) increases up to the value of an infinite temperature thermal state, S Page = L/2 log 2 − 1/2. In an MBL phase, on the other hand, S(τ ) is known to saturate to an extensive value which is not thermal [11,13], thus we expect lim t→∞ S(τ )/S Page < 1.
Furthermore, we compute the EE for the eigenstates of the effective Hamiltonian H F , which will have a volumelaw S ∼ O(L) in the ergodic phase and a sub-volume scaling in the localized one lim L→∞ S/L → 0 [73].

D. Inverse partition ratio
Finally, for the non-interacting case (V = 0), we compute the inverse partition ratio (IP R) [74] for the singleparticle eigenfunctions of the Floquet Hamiltonian H F , which is defined as where i indicates the lattice site and φ ν (i) refers to the single-particle wavefunction with single-particle energy ν . The IP R separates a delocalized phase from a localized one from its scaling with L, IP R ∼ O(L −1 ) in a delocalized phase and IP R ∼ O(L 0 ) in a localized one. We emphasize that in driven systems the density of states (DOS) is constant, up to finite-size fluctuations, due to the lack of energy ordering, therefore we can safely average IP R( ν ) over all energies.

A. Non-interacting case
In this section, we inspect the eigenstate properties of the Floquet Hamiltonian H F for the non-interacting case. In this case, H F is still a quadratic Hamiltonian Thus, we can simply focus on the single-particle energies { ν } and eigenstates {φ ν } of H F . We start our analysis by inspecting the level statistics of the single-particle Floquet Hamiltonian h F . Fig. 2 shows the level spacings parameter r computed with { ν } as a function of the driving frequency ω D = 2π/T . As expected, at low frequencies (ω D J) the system is more delocalized than higher ones (ω D J). Indeed, at low frequencies and relatively small system sizes L, the r-value is close to the one of a random matrix (r GOE ≈ 0.52), see Fig. 2 (a). However, an overall scaling towards localization (Poissonian statistics) with increasing L is visible in Fig. 2 (a),(b). As shown in Fig. 2 (c), (d), we found the scaling law which describes the crossover from delocalization to localization with increasing L r = r(ω/J · L 2(α−1) , δ, W ).
Thus, only in the L-dependent frequency regime ω D JL −2(α−1) the single-particle wavefunction are delocalized. Importantly, the above scaling law breaks down at the metal-insulator transition (α = 1) of the PLBRM. Indeed, for α < 1 we expect level energies to show repulsion, as visible in Fig. 3 (a), where the r-value is shown as a function of α at fixed frequency for several system sizes L. Finite-size scaling analyse shows that the phase diagram of the PLBRM, delocalized for α < 1 and localized for α > 1, remains unchanged under driving, see inset in Fig. 3 (a). Thus, we conclude that the localized phase is robust under driving, meaning that for α > 1 and at any finite frequency ω D the spectrum of the single-particle quasienergy remains Poissonian in the thermodynamic limit (L → ∞). For the sake of completeness, we further confirm this result by inspecting the IP R in Eq. (6) for the single-particle wavefunctions {φ ν } of the Floquet Hamiltonian, which quantifies their degree of localisation in real space. In Fig. 4 we present the averaged IP R for two values of α and fixed ω D /J = 0.1. In agreement with the analysis of the r level statistics, the IP R converges to a finite value with L (lim L→∞ IP R(L) > 0) and therefore the system is localized (see inset in Fig. 4 (a) for α = 2.5). We complete our analysis for the non-interacting case by investigating the behavior of two dynamical indicators, the energy transport and the information propagation quantified by (τ ) in Eq. (4) and S(τ ) in Eq. (5), respectively. As explained in Sec. II, the energy density (τ ) is computed starting from the ground-state of H(τ = 0) and for the EE from the Néel state N i c † 2i |0 . As expected, in the localized phase the system does not heat up to infinite temperature, meaning that (τ ) < 1/2 and the saturation value of EE is sub-volume (lim L→∞ S(∞)/S Page → 0) [73], as one can see in Fig. 5. In particular, we study the long-time saturation value of (τ ) as function of the parameter α. As one can see in Fig. 3 (b), (∞) ≈ 1/2 for α ≤ 1, meaning that in the delocalized phase energy diffuses to its maximum value.
To summarize, by inspecting several indicators, i.e., level spacing, IP R, entanglement and transport properties, we have provided numerical evidence that the noninteracting localized phase remains localized once the system is driven in time.

B. Driven flow equations
In the previous section, we have shown numerically that the non-interacting localized phase is robust under driving. We now present an analytical argument supporting our results.
For this purpose, we closely follow a recent Renormalization Group (RG) approach [45,75] which uses flowequations [76] to diagonalize a static Hamiltonian. After briefly reviewing this RG approach, we extend it to the periodic driving case, δ(τ ) = 0, and show that the driving becomes irrelevant in the RG sense. Thus, we will conclude that external driving does not destroy the phase structure of the non-interacting case, in agreement with our numerical results.
In Ref. 45 Quito et al. have shown that the diagonalization of Eq. (1), with V, δ(τ ) = 0, can essentially be decomposed into a sequence of two-body problems, i.e., a successive decimation of bonds, whereby the levelspacing statistics of the entire Hamiltonian is maintained. This strictly holds as long as α > 1/2 and becomes asymptotically exact in the limit α → ∞ [45]. Let us consider a single RG-step, or in other words, the decimation of a bond between two lattice sites i, j. i c j − h.c. the above Heisenberg equation can be straightforwardly expressed by the following coupled differential (flow-)equations, where J → 0 for Γ → ∞ with arbitrary initial conditions. A characteristic RG time scale on which J decays, is given by τ Γ = 1/r 2 with r 2 = 4J 2 + x 2 being an integral of motion w.r.t. Γ in Eq. (8). The time scale τ Γ sets an order of the RG-steps beginning with the largest r 2 max . The fact that r 2 is conserved gives rise to an intuitive picture of the renormalization process: each RG-step moves one point (bond) within the x − J plane towards the x-axis on a circle of radius r. Thereby, x 2 Γ≈τΓ → x 2 Γ=0 + 4J 2 Γ=0 gets renormalized. As a side note it should be mentioned that a renormalization of adjacent bonds affected by the flow can be neglected here as the distribution P (J ) behaves invariant under the full RG-flow [45]. The above picture of renormalization therefore connects the initial distribution P Γ=0 (J , x) to the final one (P Γ→∞ (x)). The latter is equivalent to the level-spacing distribution P (s) in the limit x → 0. With this picture in mind, of the RG accumulating decimations on a circular shell, two phases can be immediately identified: (i) For an initially uniform distribution P Γ=0 (J , x) ∼ const, one obtains an extended phase, exhibiting level-repulsion, i.e., P Γ→∞ (x) ∼ x. This is because the number of accumulated decimations N dec within a shell [r − dr, r] is N dec ∝ 2πrdr in this case [45]. (ii) For any kind of short-ranged distribution, P Γ=0 (J , x) ∼ e −J /ξ , where ξ r max is a characteristic length scale, the RG leads to a localized phase, since P Γ→∞ (x) ∼ const (no level-repulsion).
The initial marginal distribution P (J ) in our system consists of two regimes separated by a critical J c ∼ 1 L α [45], set by the typical hopping amplitude at maximum distance L, As described above by means of the RG, these two regimes map to two different phases: For J < J c , the distribution in Eq. (9) is constant, leading to level-repulsion, while for J > J c it is concentrated near J = 0 and thus avoids level-repulsion. At the crucial scale of the mean level spacing δ ∼ 1 L (bandwidth can roughly be considered size-independent), the normalized critical j c defined by j c = J c /δ ∼ 1 L α−1 then gives a critical point at α = 1.

Dynamical extension
In this section we derive a dynamical extension of Eqs. (8) that accounts for a periodic driving term δ(t) = 0 in the Hamiltonian. As a simplification we start with a harmonic driving ansatz, where Ω denotes a driving frequency and δ a dimensionless driving strength. A time-dependent change of basis is generally given by In order to continuously dampen the off-diagonal hopping terms of H 1 , we use the following generator where ω = Ω/x represents a normalized driving frequency. For simplicity, this generator is chosen such that it does not create a time-dependence for the off-diagonal hopping terms during the flow. It can be seen that the continuous change of basis with S(τ ) can be expressed in closed form by the following, "driven" flow equations, where we omit the explicit Γ-dependencies A few considerations are in order. (i) We neglected higher-harmonic terms ∝ δ 4 , which naturally arise from a time-dependent S(τ ). (ii) The driving strength δ enters quadratically and therefore explains the need for a relatively large value of δ = 1/2 to see significant effects from driving in our numerics. (iii) In the infinite frequency limit ω → ∞ the flow equations just reduce to the static ones, consistent with the universal scaling law of Eq. (7). For extraction of the phase structure we are only interested in the behavior at large RG times τ Γ , since they correspond to small values of x of the level-spacing. This implies that large RG times also correspond to high values of ω = Ω/x. The above set of equations thus can be further simplified by considering only terms up to leading order in 1/ω, Within this approximation we find a first-order correction of the integral of motion r 2 tor 2 = 4J 2 (1 + 2δ 2 ω ) + x 2 . Hence, a typical level spacing from the static case gets scaled by 2δ 2 ω . By identifying τ Γ ≈ 1/x 2 , the driving correction becomes manifestly irrelevant in the course of the RG, 2δ 2 ω ≈ 1 √ τΓ 2δ 2 Ω . As a result, the non-interacting algebraically localized phase is robust against driving.

C. Algebraic MBL
Having established that the non-interacting algebraic phase is stable under periodic driving, we now turn to the interacting case by setting V = 1 in Eq. (1). With the aim to understand the robustness of an algebraic MBL phase, we fix the parameters α and W of H in Eq. (1) such that the non-driven system is localized [47,73,81]. The numerical results have been computed using exact diagonalization techniques, thus we will be limited by relatively small systems L ∈ [6,16]. However, by studying several probes to distinguish an ergodic phase from an MBL one, even if affected by finite-size effects, we provide numerical evidence that at any finite frequency, ergodicity is expected to be restored in the thermodynamic limit.
As well as for the non-interacting case, we start our analysis by inspecting the spectrum of the effective Hamiltonian H F in Eq. 3. Fig. 6 shows the r-value as function of the driving frequency ω D , for several L. A clear trend towards delocalization (r ≈ 0.531) with increasing L is visible in Fig. 6 (a)-(b). The r-value change their behavior at high-frequency ω D J, where a frequency-independent plateau is formed close to the Poissonian value r ≈ 0.386. This plateau belongs to the finite size perturbative regime, in which the system is close to the non-driven case U (T ) ≈ I − iT 2 (H(T /2) + H(T )), which disappears in the thermodynamic limit. Generically, a driving localized phase remains localized at enough high frequencies. In particular, this will happen if the driving frequency ω D exceed the many-body bandwidth Λ(L) of the energy spectrum of the non-driven Hamiltonian [27]. We compute the averaged density of states of the non-driven Hamiltonian and using a Gaussian fit we extrapolate its typical bandwidth Λ(L) (vertical dashed line in Fig. 6 for the largest L).
Thus, for ω D ≥ Λ our numerical result cannot be used to understand the dynamical phase diagram of H(t). By safely ignoring the high-frequency regime (ω D ≥ Λ), the r-value of H F shows a clear tendency towards delocalization (rigid spectrum), allowing us to claim that ergodicity might be restored in the thermodynamic limit. This is visible in Fig. 6 (c),(d), where r-value is showed as function of ω D /Λ(L), providing evidence that the criti- cal frequency ω c for which localization sets on, scales as ω c ∼ Λ(L) ∼ √ L. With the aim to further support the claim of delocalization of an algebraic MBL phase under driving, we study the entanglement properties of the eigenstates of the Floquet Hamiltonian (H F ). Fig. 7 shows the bipartite EE averaged over disorder and eigenstates and its fluctuations δS 2 as function of the driving frequency [19,21]. The EE has a similar behavior as the r level statistics parameter, meaning that low frequency EE reaches the ergodic values (see inset in Fig. 7 (a)) and then the curves indicate a shift to delocalization (volume law scaling) before approaching the natural cut-off Λ. Furthermore, we study fluctuations around the mean value S, which are quantified by the variance (δS 2 ) over random samples and eigenstates [19,21]. At low frequencies δS 2 decays with L in agreement with the ETH (see inset in Fig. 7  (d)). Meanwhile at moderate frequencies δS 2 develops a growing peak with L, a feature which has been used as a fingerprint for the MBL transition [19,21]. However, as one can observe, the peak shifts inexorably with system size, getting closer to Λ.
Finally, we inspect some dynamical properties of the interacting driven system, i.e., energy and entanglement propagation.
As for the non-interacting case, we start with the prop- agation of energy. Fig. 8 (a)-(b) shows the energy density (τ ) at moderate frequency ω D ≈ 4J. For both values, α = 2.5, 3 and for our available system sizes, the infinite temperature value (ε = 1/2) is not reached yet. However, in long-time limit, (τ ) tends to ε ≈ 1/2, which it also supported by the finite-size scaling (see inset in Fig. 8). The remaining two panels, Fig. 8 (c), (d), are dedicated to the asymptotic long-time saturation value of S(τ ). As a function of the driving frequency, we compute S(τ → ∞)/S Page , where S(τ → ∞) is determined by averaging over a long-time window in which S(τ ) reaches its saturation value. The numerical analysis of S(τ → ∞)/S Page in Fig. 8 (c), (d), in which is visible the tendency S(τ → ∞)/S Page → 1 as L increases. Thus, in agreements with the results of the spectrum and eigenstates of H F , the finite size analysis of S(τ → ∞) further supports the conjecture that ergodicity is restored in the thermodynamic limit.
Unlike the case in which the LIOMs have exponentially decaying tails (α → ∞), the algebraic MBL phase shows a clear tendency towards delocalization at any finite-driving frequencies. To better pin-down this fundamental difference, we study a Floquet minimal MBL model [23,81,82], which can be considered as the strong for α = 2.5. In both panels W = 6 and δ = 0.5.
disorder limit of H in Eq. (1) with where { ± l } and {I ± l } are the single-particle energies and LIOMs of the non-interacting model (V = 0) in Eq. (1) with ±δ, respectively [23,81]. The couplings B pq = i {|φ ± p (i)| 2 |φ ± q (i + 1)| 2 − φ ± p (i)φ ± p (i + 1)φ ± q (i)φ ± q (i + 1)} are obtained using first order perturbation theory in H in the limit V /W 1 [23,81]. By construction, H ± is localized meaning that it has Poissonian spectrum and integrals of motion are algebraically localized for α < ∞ and exponential as α → ∞. As a further consequence, for α < ∞ the couplings B pq decay algebraically w.r.t. the distance between the centers of localisation of φ p and φ q , while for α = ∞ the decay is exponentially fast. The spatial structure of the decay of B pq determinants the growth in time of the EE after a global quantum quench with H ± . Thus, for α < ∞ the EE has an unbounded algebraic growth, while for α = ∞ the growth is logarithmic slow.
Returning back to the driven case, in Fig. 9 (a)-(b) is shown the r-value as function of the driving frequency ω D computed with the eigenvalues of the Floquet Hamiltonian of the minimal model in Eq. (15) for α = 2.5 (algebraically LIOMs) and α → ∞ (exponentially LI-OMs). The crossing between curves for different system sizes, which may demarcate the existence of a putative MBL transition as function of ω D , is located in proximity of the many-body width cut-off for α = 2.5 and at smaller frequencies for α → ∞. Considering only values ω D ≤ Λ (see Fig. 9 (a) and its inset), and the tendency towards delocalization for the algebraic case, we may conclude that the localization will be completely destroyed in the thermodynamic limit. Indeed, the crossing between curves for different L happens for frequencies ω D ≥ Λ. However, in the limiting case of exponential LIOMs (α → ∞), we observe a much more robust and defined cross-over between the two regimes which occurs before the natural bound ω D ∼ Λ(L), and therefore it may culminate to an MBL transition at a finite frequency strength [27,28,34]. Now, we provide an analytical argument based on Ref. 34 to support the delocalization of an algebraic MBL phase under periodic drive. For the sake of simplicity, consider the case where the driving terms contain only of one harmonic and the time-dependent Hamiltonian can be separated in two parts H(t) = H −bit (W ) + V H 1 (t). The Hamiltonian H −bit (W ) is algebraic MBL and H 1 (t) ∼ cos(2πt/T ) is the time-depended part which allows jumps between the localized eigenstates of H −bit (W ). W and V are the disorder and interaction strength, respectively. We assume that the disorder is strong enough to localize the instantaneous eigenstates. If T is large enough, we can estimate the probability of mixing the instantaneous eigenstates |E α (t) , |E β (t) using the Landau-Zener formula [83][84][85] where M α,β = E α |H 1 (t c )|E β is the matrix element coupling the eigenstates, the Landau-Zener velocity v α,β = T ∂ t (E α − E β )| t=tc and t c the time at which the crossing takes place. In the adiabatic (C α,β ≈ 0) and diabatic (C α,β 1) limits the system remains localized after the crossing. In the intermediate situation eigenstates are characterized by different values of LIOMs mixing together and starting to thermalize. As argued in Ref. 34, there exist two length scales x 1 and x 2 , which determine the behavior of P α→α . Consider two eigenstates |E α (t) , |E β (t) which differs from the values of LIOMs only in a region of size x. Thus, these eigenstates are the same outside a region of size x. There are ∼ 2 x of these eigenstates which typically belong to an energy shell with bandwidth ∼ W √ x and the mean level spacing On the other hand, the typical gap at the crossing is controlled by V H 1 (t c ) In the limit δE α,β ∆ multiple crossing between eigenstates which differ only in the region of size x happen. This sets the length scale x 1 by comparing δE α,β (x 1 ) ∼ ∆(x 1 ) The length x 2 determinants the character of the transition, which is directly connected to C α,β ∝ |M α,β | 2 /v α,β . The matrix element between two algebraically localized states can be estimated by which decays as a power-law in distance x since the LIOMs are algebraically localized. Instead, the Landau-Zener velocity is Thus, the length x 2 is defined by C α,β (x 2 ) ∼ 1 which separate the adiabatic limit from the diabatic one, In the regime x 1 /x 2 1, the system undergoes multiples crossing in an intermediate regime C α,β ∼ 1 which implies the thermalization of the region of size x x 1 x 2 . The relation x 1 /x 2 1 implies that Although this simple argument (Eq. (23)) does not strictly prove the delocalization of the system at any frequency ω D = 2π T , it shows that the delocalization is parametrically favorable, meaning that it is exponentially sensitive to ω D ∼ 1/T .

IV. CONCLUSION
In this work we have investigated the stability of an algebraically localized phase subject to periodically global driving for both, the non-interacting and the interacting case. For sufficiently strong disorder, the non-interacting algebraically localized phase is robust once short-range interactions are turned on, generalizing the paradigm of algebraic localization to the interacting case (algebraic MBL). As a consequence, in an algebraic MBL phase, ergodicity is broken and the phase is fully characterized by an extensive number of integrals of motion with algebraic decaying tails, which are believed to be adiabatically connected to the algebraic decaying integrals of motion of the non-interacting case.
In particular, we studied a periodically driven (Floquet) spinless fermionic chain with long-range random hopping and density-density short-range interactions. The non-driven non-interacting limit is known as the power-law random banded matrix model and hosts an algebraically localized phase, meaning that its singleparticle eigenstates are localized with power-law tails.
First, we considered the non-interacting case. With a combination of analytical arguments and exact numerical simulations, we provided evidence that the algebraically localized phase is stable under driving. The spectrum of the signle-particle Floquet Hamiltonian shows Poissonian statistics and its eigenstates are localized. We supported this claim by using analytical arguments based on strong disorder renormalization group techniques.
Second, we have tackled the interacting case by turning on short-range density-density type interactions. At strong disorder and for sufficiently large decay rate of the hopping the non-driven case is believed to be in an algebraic MBL phase. Several MBL markers have been used to numerically inspect the stability of the algebraic MBL phase under driving. After a careful analysis of the finitesize effects, we argue that ergodicity might be restored in the thermodynamic limit for any driving frequencies.
For finite-size systems we showed that localization sets in for driving frequency that are comparable with the many-body bandwidth. This indicates that, the finitesize localized phase might disperse in the thermodynamic limit. Analytical arguments were made to support our numerical results by showing that delocalization appears parametrically favorable.
Thus, unlike the non-interacting case, we showed that the algebraic MBL phase is unstable once the system is periodically driven. This result should be compared to the case in which the LIOMs are exponentially localized. For conventional MBL phases, it is believed that a finite, critical frequency exists, which separates an ergodic phase from an MBL one. To further point out this difference, in the final part of the work we inspected a toy Floquet model, in which we can tune the type of local-ization, from algebraic to exponential. We confirmed the existence of a critical driving frequency in a conventional MBL phase and showed the breakdown of an algebraic MBL phase under driving.
Our work imposes strong constraints on the possible existence of a genuine MBL phase for long-range models under driving. This result could be relevant for trapped-ions experiments, where long-range interactions are present. It is important to point out that even though we do not expect the existence of an MBL phase, interesting non-ergodic behavior could persist for fairly long time scales and large systems. Thus, understanding the time scales for thermalization is an important question which is left for future research.