Onset of Floquet Thermalisation
Asmi Haldar Roderich Moessner Arnab Das







































    ^1Department of Theoretical physics, Indian Association for the Cultivation of Science, 
    
    2A & 2B Raja S. C. Mullick Road, Kolkata 700032, India^2Max Planck Institute for the Physics of Complex Systems, 
    
    	Nöthnitzer Straße 38,
    01187 Dresden, Germany 
    
    In presence of  interactions,  a closed, homogeneous (disorder-free)
    many-body system is believed to generically heat up to an `infinite temperature' ensemble 
    when subjected to a periodic drive: in the spirit of the ergodicity hypothesis
    underpinning statistical mechanics, this happens as no energy or other conservation
    law prevents this. Here we present an interacting Ising chain driven by a  field of time-dependent strength, 
    where 
    such  heating onsets only below a threshold 
    value of the drive amplitude, above which the system exhibits non-ergodic behaviour.
    The onset appears at  strong, but not fast driving. This in particular puts
     it beyond the scope of high-frequency
    expansions. The onset
    location shifts, but it is robustly present, across wide variations of the model Hamiltonian
    such as driving frequency and protocol, as well as the initial state.  The portion of nonergodic
    states in the Floquet spectrum, while thermodynamically subdominant, has a finite entropy. We find that 
    the magnetisation as an  emergent conserved quantity underpinning the freezing; indeed
    the freezing effect is readily observed, as initially magnetised states remain partially
    frozen  up to infinite time.
    This result, which bears a family resemblance to the Kolmogorov-Arnold-Moser theorem for classical dynamical systems,
    could be a valuable ingredient for extending Floquet engineering to the interacting realm.
    
    Onset of Floquet thermalisation
        Asmi Haldar^1, Roderich Moessner^2 and Arnab Das^1
        December 30, 2023
    ======================================================
    
    
    
    § INTRODUCTION
    
    Interacting many-body systems, by the ergodic hypothesis, generically thermalise, placing them 
    in the purview of statistical mechanics and equilibrium thermodynamics <cit.>. 
    Our understanding of the corresponding situation for non-equilibrium systems is still in flux. For perhaps 
    the simplest class
    of non-equilibrium systems, periodically driven (Floquet) systems, thermalisation physics 
    at first pass looks maximally simple: removing time translation invariance destroys energy conservation, 
    and hence the concept of temperature–which means  thermalisation is to a featureless `infinite-temperature' 
    state <cit.>. 
    
    Such Floquet systems have been predicted to be  capable of sustaining new forms of spatiotemporal ordering 
    when many-body localised as a result of strong quenched disorder <cit.>. The experimental search for 
    such so-called discrete time crystals has been qualitatively more successful <cit.> 
    than may have been anticipated: the collection of systems appearing to exhibit such order now even includes 
    a dense periodic array of nuclear spins initialised in a thermal 
    state <cit.>. 
    
    All of this focuses the question on settings which permit long-lived correlations and order to persist despite 
    the presence of periodic driving even in the absence of quenched disoroder. 
    In periodically driven  non-interacting systems, quantum 
    heating can be suppressed <cit.> and an
    extensive number of periodically conserved quantities identified <cit.>. 
    In turn, a pre-thermalisation regime has been
    identified <cit.> which resembles a frozen non-thermal state
     <cit.> which can be described by a 
    periodic (generalized) Gibbs' ensemble <cit.>.  Tuning the drive parameters, and weakening the interactions, 
    can substantially enhance the prethermalization period, still expected to remain 
    finite <cit.>. In fact, 
    for disorder-free systems, a transient but exponentially long-lived
    regime exhibiting discrete time-crystalline phenomenology
    has already been identified <cit.>.
    These constitute lower bounds on the thermalisation timescales. 
    For finite-size systems, an emergent integrability structure
    for strong drives has also been proposed as a way to avoid thermalisation <cit.>.
    There is further evidence indicating  absence of heating at high drive frequencies in a 
    variety of other settings <cit.> and in specially designed models <cit.>.
    
    Here, we address the question whether there is an identifiable threshold for the ratio of 
    driving and interaction strength,
     below which the system approaches a 
    non-trivial steady state that depends on the drive and the initial state. We consider a  spin chain subject to  strong, but not fast
    driving, and use remanent infinite-time magnetisation of a  initial magnetised state as measure of failure to Floquet-thermalise. 
    As the driving is increased from low strength, where standard Floquet thermalisation is observed, we find a remarkably
    well-defined second regime, in which remanent magnetisation is present 
    even in the infinite time limit. Its value is given by the Floquet diagonal ensemble average implied by the initial state. 
    The location of this threshold moves, but its existence is stable to variations in state initialisation, 
    driving strength, driving protocol, and driving frequency. 
    
    In all cases, however, we are able to identify an emergent  approximately conserved quantity – in the case we discuss at length, 
    the magnetisation itself – which
    becomes exactly conserved if the static part of the Hamiltonian is ignored. Thus, rather than an extensive set of integrals of motion,
    as is present in the case of the periodic Gibbs ensemble<cit.> and the Floquet many-body localised 
    cases<cit.>, all that appears to be needed
    to stop the system from heating up indefinitely is a single, approximately conserved quantity. 
    
    While our numerical investigation on systems up to L=14 spins naturally limits our capacity to extrapolate these results to the `thermodynamic' limit,
    there are indications that this is not only a finite-size effect. Firstly, in plots of remanent magnetisation versus driving strength, 
    we identify a crossing point for curves for different L separating the ergodic and the non-thermal regimes. 
    Second, the set of Floquet eigenstates exhibiting memory, while accounting only for
     a vanishing fraction of the total Hilbert space, extrapolates to have 
     a finite entropy in the thermodynamic limit. This means 
     that such states can still be straightforwardly selected by an initial condition, not
    unlike initialising a static system in a low-temperature configuration. 
    
    In the following, we first we set the notation and provide a brief introduction to the 
    Floquet concepts we have used. We then define our model and  drive protocol. 
    We characterise the ergodic and the non-thermal phases and the threshold
    between them using various measures, and demonstrate robustness to variations of drive patterns and system parameters. 
    We close with an outlook and suggestions for further investigations. In particular, 
    origin and nature of the sharp features in the memory as a function of driving strength merit further study. 
    
    
    
    § MODEL
    
    In this section, we introduce notation, model Hamiltonian, drive protocol, and observables to be studied. 
    
    
    
    
    
     §.§ Floquet basics
    	
    Let us decompose the time-dependent Hamiltonian H(t) into 
     a static interacting Hamiltonian H_0 and a time-periodic drive
    H_D(t) with [H_0,H_D]  0: 
    
        H(t) = H_0 + H_D(t),
    
    The time evolution operator evolving a state through a period from
    t=ϵ to t=ϵ + T (0  ≤ϵ < T) is U (ϵ). 
    Since U(ϵ) is unitary, it can always be
    expressed in terms of a hermitian operator, the `Floquet Hamiltonian' H_eff as
    
        U(ϵ) = e^-iH_eff(ϵ)T.
    
    Formally, 
    
        exp(-iH_eff(ϵ)T)
        = 𝒯exp(-i∫_ϵ^ϵ+Tdt  H(t)),
     where 𝒯 denotes time-ordering. 
    Let |μ_i⟩ denote the i-th `Floquet eigenstate' of 
    H_eff corresponding to the `Floquet eigenvalue'  (also known as quasienergy) μ_i.
    
    
    A sequence of stroboscopic observations at instants 
    t = ϵ, ϵ + T, …, ϵ + nT (integer n) is 
    identical to that produced by the dynamics under the time-independent Hamiltonian 
    H_eff. 
    This applies for every ϵ,  
    hence we get continuous family of stroboscopic series. 
    
    In the following,
    we are interested in long-time asymptotic behaviour, so that temporal variations within 
    a driving period are of secondary importance.  Hence, we arbitrarily pick ϵ = 0.
    
     §.§ Infinite time limit: diagonal ensemble average
    
    
    The nature of the asymptotic state under the drive can be understood as follows. 
    Consider an initial state
    
    
        |ψ(0)⟩ = ∑_ic_i|μ_i⟩
    
    
    and the stroboscopic time series for an observable 
    
    
        Ô 
        = ∑_i,j O_ij|μ_i⟩⟨μ_j|.
    
    
    
    
    
        ⟨ψ(nT+ϵ) |Ô|ψ(nT+ϵ) ⟩ 
        	= ∑_i,jc_ic_j^∗ O_ije^-i(μ_i-μ_j)(nT+ϵ).
    
    
    Like in the case of static Hamiltonians, 
    under quite general and experimentally relevant conditions (see, e.g., Ref. <cit.>), 
    
    at long times (n→∞) the off-diagonal (i j) 
    terms `average to zero' and 
    the state of the system can hence be described by an effective “diagonal ensemble" (in the absence of synchronisation, e.g.
    for discrete time crystals, this is replaced by a block diagonal ensemble<cit.>). 
    This is captured by the mixed density matrix <cit.>ρ̂__DE = ∑_i|c_i|^2 |μ_i⟩⟨μ_i|.
    
    Thus, the asymptotic properties of a periodically driven system are 
    effectively given by a classical average (known as diagonal ensemble average or DEA) over the expectation 
    values of the eigenstates of H_eff,
    
        ⟨Ô⟩ (DEA) = ∑_i|c_i|^2 ⟨μ_i|Ô|μ_i⟩ .
    
    Hence it is sufficient to study the nature of the eigenstates and eigenvalues of H_eff, or equivalently of U(ϵ), 
    in order to obtain the long-time behaviour. 
    
    
    
    
     §.§ Driving protocol
    
    We consider  L spins on a chain. We chose a binary drive protocol, which switches periodically between a pair of rectangular pulses. 
    The time dependent Hamiltonian is 
    
        H(t) = H_0 + sgn(cosω t)  H_D,
    
    with the two components
    
        H_0   =    -J∑_iσ_i^xσ_i+1^x +κ∑_iσ_i^xσ_i+2^x -h_0^x∑_i^Lσ_i^x
              - h^z∑_i^Lσ_i^z; 
        
        H_D   =    - h_D^x∑_i^Lσ_i^x .
    
    
        H(t) = H_0 + sgn(cosω t)  H_D,    where
         
        H_0 = -J∑_iσ_i^xσ_i+1^x +κ∑_iσ_i^xσ_i+2^x -h_0^x∑_i^Lσ_i^x
         
        - h^z∑_i^Lσ_i^z; 
        
        H_D = - h_D^x∑_i^Lσ_i^x .
    
    The σ^α are Pauli matrices. We use periodic boundary condition, but tamper the system  
    slightly by putting J_L,1 = 1.2J and κ_L-1,1=1.2κ to break translational invariance (and 
    hence remove any remaining block-diagonal structure of the Hamiltonian). Here since we keep the interaction strengths
    constant during the drive, we use the drive amplitude itself 
    as the tuning parameter.
    
    In presence of the transverse field, the Hamiltonian H_0 is known to be ergodic due to the four-fermionic
    interaction terms arising from the next-nearest neighbour interactions under the spin to fermion mapping, 
    and also due to the longitudinal field. We  have explicitly verified that H_0 is ergodic for our case, see Suppl. Mat.). 
    
    
    We initialise the simulation in the time domain with different initial states. Unless otherwise stated, we use the default choice
    of the ground state of H(t=0). 
    
    
    
    § NUMERICAL RESULTS
    
    The central quantity is the longitudinal magnetization
    
        m^x(t) = 1/L∑_i^L⟨ψ(t)|σ^x_i|ψ(t)⟩ .
    
    We monitor its real-time dynamics
    in a stroboscopic time series. We diagnose non-thermalisation/freezing via 
    its long-time asymptotic behaviour, the remnant magnetisation, which we study 
    as a function of various model parameters.
    
    
    
    
     §.§ Onset of Floquet thermalisation
    
    In the following, we provide numerical evidence that for a strong (but not fast) drive, the system fails to Floquet thermalise, instead retaining
    memory of its initially magnetised state. We then show that the onset of Floquet thermalisation occurs at a fairly well-defined threshold
    driving strength. For the results in the main text, we have chosen J=1,κ = 0.7,h^z=1.2, and 
    h_0^x=0.02.
    
    
    
    The stroboscopic time series for the magnetisation m^x is shown in Fig. <ref>, left frame. Already at short times,
    three representative trajectories for different driving strengths show strikingly different behaviour. While for weak driving fields, the 
    magnetisation disappears almost immediately, for stronger ones, the decay slows down. Finally, for h^x_D beyond a threshold
    value, the decay is arrested: even at the longest times, a remnant magnetisation persists. 
    
    This remnant magnetisation  agrees with  the DEA of the magnetisation evaluated for the same system (see inset). Note that the
    nonvanishing DEA is already in itself a signature of the lack of Floquet thermalisation – in general, Floquet thermalised eigenstates  
    individually show no non-trivial correlations. 
    	
    In order to locate the onset, the DEA of m^x as a function of the drive 
    amplitude h_D^x  is plotted in Fig. <ref>, middle frame. 
    	A  threshold for nonzero remnant magnetisation is observed, separating 
    	 the ergodic (m^x_DE≈ 0) from the nonergodic regime. 
    	 	
    The lower inset shows freezing for an initial state with a reduced polarization in the x-direction. 
    	The black dotted
    	line shows the initial value of m^x for the state, and the curve shows that for 
    	high enough h_D^x, the DEA of m^x almost coincides with it. 
    	In detail, this initial state is given as  |ψ_0⟩ = ∑_i=1^2^L c_i |i_x⟩, 
    	where |i_x⟩ is the i-th eigenstate of the 
    	longitudinal field part (computational basis states in x-direction, or x-basis states), 
    	by choosing Re[c_i] and Im[c_i] from a 
    	uniform distribution between -1 and +1, 
    	multiplying them by e^β m_i^x, where β > 0 and m_x^i is the 
    	longitudinal magnetization 
    	of |i_x⟩, and finally normalizing the state. 
    	This gives a `generic' state with a bias towards positive longitudinal magnetization. For the plot in
    	Fig. <ref> (middle frame), we have chosen a random instance corresponding to 
    	β = 1.75.
    	The right frame of Fig. <ref> shows DEA of 1-m^x on a 
    	doubly logarithmic log-log plot zoomed
    	in around the threshold for 
    	better visibility. 
    	 
    
    
     §.§ Floquet eigenstates and an emergent conservation law
    
    
      §.§.§ Localization and magnetization
    
    We now turn to the properties of the Floquet eigenstates obtained 
    by numerically diagonalizing the time evolution operator U(0), Eq. <ref>.
    We consider first their `localisation' in Hilbert space, followed by their magnetisation content.
    
    
    In order to investigate the  localization properties of the Floquet states in the x-basis
    {|i_x⟩} we calculate the inverse participation ratio (IPR) in said basis defined as
    IPR(|μ_j⟩) = ∑_i=1^2^L |⟨ i_x | μ_j⟩|^4.
    The left frame of Fig. <ref> shows the IPR thus obtained, arranged in decreasing order. 
    Indefinite heating corresponds to the states being
    delocalized in the eigenbasis of any local operator, which implies a uniformly 
    small IPR given by the inverse dimension of Hilbert space, 1/D_H. This is indeed what is observed for small drive 
    fields. By contrast, for large drive fields, states appear which have an IPR close to 1, which indicates the presence of well-localised states,
    and hence the absence of Floquet thermalisation for the corresponding part of the spectrum. 
    
    
    
    Complementary information can be gleaned by considering the correlations encoded in the non-ergodic states. 
    The middle frame of Fig. <ref> shows the magnetisation of different Floquet eigenstates, m_i^x,  
    ordered according to their size. In the ergodic regime, these curves are featureless and 
    m_i^x is uniformly tiny, showing a tendency to increase with increasing drive strength. 
    
    Deep into the nonergodic regime, large values of m_i^x
    appear, which together form plateaux. For the largest drives h_D^x, the plateaux
    correspond to essentially an integer number of spin flips, which indicates that 
    the new basis is close to the computational basis in the x-direction mentioned above. As the drive is decreased,
    the plateaux give way to a smooth curve, which however still makes large excursion toward m^x=±1 before 
    assuming the featureless shape of the ergodic regime. 
    
    While the fraction of Floquet states with a magnetisation above a certain value is 
    thermodynamically vanishing, their entropy is
    nonetheless finite, see Fig. <ref>, middle and right frames. 
    This is analogous to the case of a finite-temperature ensemble of
    a magnet in a field, where a nonzero magnetisation arises as a thermodynamically vanishing fraction of 
    magnetised states is preferentially populated, with their energy gain compensating for 
    the entropy loss involved in 
    concentrating the probability density on them. Here, the selection of the magnetised Floquet states arises 
    via the  state initialisation. It is interesting to note that in this 1D system there would be no magnetization at 
    any finite temperature: the observation of a finite magnetization at finite energy density is purely a 
    non-equilibrium effect.
    
    
    
    
      §.§.§ Emergence of m^x as a local quasi-conserved quantity
    We next address what we believe is the central feature underpinning the non-thermalisation, namely the 
    existence of a conserved quantity in the drive Hamiltonian in isolation. In our example, this is the magnetisation in the
    x-direction, m^x, which persists as a quasi-conserved quantity even when the ratio of drive to static 
    components of the Hamiltonian is finite.The middle frame of  Fig. <ref> shows the value of m^x for the different Floquet eigenstates arranged by
    their size. For the strongest drives, the steps in this quantity are identical to the ones of the computational basis states in
    the x-basis, i.e. the steps simply reflect the number of spins flipped.The static part of the Hamiltonian then mixes the states with the same value of m^x, which is reflected in the 
    non-trivial distribution of the IPR of the Floquet states (left frame of Fig. <ref>). The growth of the size of each 
    m^x sector (except for the fully polarised one) 
    is in turn reflected in a decrease of the IPR.For lower driving strengths, h_D^x=18, the steps get washed out, but the range of m^x continues to span practically
    the full range in the interval between -1 and 1. This feature disappears below the threshold, h_D^x=10, where the curve
    flattens substantially.While the fraction of Floquet states with a non-zero magnetisation density vanishes with system size, these states nonetheless
    have nonzero entropy, Fig. <ref> right panel, as is the case for magnetised states of a paramagnet generally.The emergent quasi-conserved nature of m^x, along with the straightforward possibility of initialising the system 
    in a magnetised state, acount for the main features of the results discussed in this work.
    
    
     §.§ Robustness against variation of model and protocol parameters
    
    We first address the existence of the onset for variants of the above model. We note that so far,
    no fine-tuning was necessary. The central demand was for the drive amplitude h_D^x to be the largest scale,
    while the other parameters of the Hamiltonian were chosen all to be in the same ballpark. 
    
    
    
      §.§.§ Role of non-commuting term
    
    First, the location of the thermalisation threshold can  be moved by varying the strength of the term in the 
    static Hamiltonian H_0 which does not commute with the driving Hamiltonian H_D. Indeed, the top  left frame
    of Fig. <ref>  shows that the threshold driving field is approximately proportional to the static 
    transverse field strength h^z.
    
    
    
      §.§.§ Drive shape and initial state
    Also, we ask whether the `symmetry' of having a vanishing 
    mean drive of zero for symmetric pulse shapes about zero is an important ingredient. 
    Fig <ref>, top right frame, shows that the freezing is quite robust to addition of a  dc field of strength h^x_0.
    Indeed,  the freezing actually grows with h^x_0.Next, we consider a deviation of the drive protocol away from a time-symmetric switch in the sign of the driving term to one
    where more time is spent for one sign than the other (Fig. <ref> bottom left frame). While the latter case has
    considerably more structure at high drives, in particular an apparently regular suppression of the remnant magnetisation
    even above the onset threshold, the former curve basically acts as a high-magnetisation envelope of the latter. 
    
    Further, we consider an initial state prepared as the ground-state of a many-body problem (rather than a more simply
    prepared polarised state). This displays (Fig. <ref> bottom right frame) 
    all the salient features observed with the simply polarized ground state in
    Fig. <ref>, right frame.
    
      §.§.§ Drive frequency
    
    
    What is particularly worth emphasizing is that the non-ergodic behaviour is  not a high-frequency phenomenon. 
    While such freezing also exists in the limit of a driving frequency in excess of the many-body bandwidth of the 
    finite-size system, it is not even the case that the nonergodicity necessarily grows with frequency. This is 
    illustrated in Fig. <ref>, where the remnant magnetisation is, if anything, more 
    robust at small driving frequencies. 
    
    This is intriguing since at  lower drive frequencies, Magnus-type 
    high frequency expansions are divergent. Hence, this is an example of the breakdown of a Magnus expansion
    which is not associated with unbounded heating.
    
     §.§ Finite-size behaviour
    
    	Our results  indicate that absence of thermalization in this 
    	driven interacting system might persist even in the infinite-size limit. While there are some 
    	dips of the freezing strength in the nonergodic regime complicating a sharp identification of a threshold, the 
    	onset nonetheless appears to sharpen with increasing system-size. A closer view of the nonergodic 
    	regime, Fig. <ref>, middle frame top inset, shows smooth behaviour of the 
    	remnant magnetisation for the largest fields; this in fact  grows with increasing  system size. By contrast, for 
    	weak drives, the remnant magnetisation tends to decrease with system size. This results in a
    	crossing point
             as the curves for different system sizes of the deviation of the remnant magnetisation 
    	from its initial value, Fig. <ref>, right frame, thus approximately cross at
    	the threshold point. While it is hard entirely to rule out a slow drift to higher fields of the threshold with
    	increasing system size, 
    	these observations suggest the possibility of a sharp transition at a finite threshold field in 
    	the thermodynamic limit. 
    
    	Next, and most importantly, the step-structures in the m^x of the Floquet states are almost
    	indistinguishable from that of the x-basis states for  all system-sizes we investigated
    	(Fig. <ref>, middle frame). This absence of system-size dependence indicates 
    	that at large values of h_D^x, the drive does not mix the x-basis states of different 
    	m^x values. A decrease in the fraction of Floquet states with m^x > m_c with system-size is not 
    	because in larger systems the Floquet states are more delocalized between different magnetization 
    	sectors, but merely because the number of x-basis states in a given magnetization 
    	sector changes with the system-size. Delocalization between different magnetization sectors is 
    	suppressed strongly for all system-sizes at hand for h_D^x above the threshold. 
    	This is in keeping with the observations that on different types of initial states, 
    	Fig <ref> and Fig.<ref>, the freezing at the highest frequencies 
    	does not decrease with system size. and gives a further indication that our results are not merely 
    	finite-size effects.  
    
    	
    	 	
    
    
    § DISCUSSION
    
    
    
    
    
    We have studied the onset of Floquet thermalisation in a driven interacting spin chain. We have found a
    fairly sharp threshold for the drive strength, above which Floquet thermalisation does not take place.
    The threshold value varies in different manners with parameters
    like pulse shape, drive frequency, or the (non-commuting) transverse field strength, but the freezing
    persists robustly under all these variations. The question of the existence of such a threshold is of 
    fundamental importance, with a related issue appearing for classical dynamical systems, where the 
    Kolmogorov-Arnold-Moser theorem deals with the onset of chaotic behaviour upon breaking of integrability.
    
           An open question is the origin, and in particular the L-dependence, of the dips in the frozen component
            even beyond the threshold in the m^x vs h_D^x plots: 
            the dips touching the x-axis 
    	correspond to points of thermalization. While their occurrence for certain discrete 
    	values of h_D^x has no significant consequence, if their number diverges with
    	 L, this may lead to a destruction of the frozen regime. For drives with pulse durations
    	evenly placed about T/2, the dips disappear rapidly with increasing h_D^x. Such dips are, 
    	however, observed to persist even for very strong amplitudes for the case of drive with uneven 
    	division of the drive period (Fig. <ref>, bottom left). In this case, the total drive period 
    	is divided in two parts, T/GR and T(1-1/GR), where GR is the Golden ratio. While the depth
    	of the dips seems to increase with L, their number and locations remain surprisingly independent
    	of L, which points against their proliferation. Regarding an extrapolation to the thermodynamic limit,
    	we refer to our discussion at the end of the previous section.
    
    Comparison of the magnetization and IPR of the Floquet states in the frozen regime allows one to conclude 
    that the magnetization itself plays the role of a quasi-conserved quantity, which becomes exactly conserved 
    in the limit of infinitely strong driving. However, the emergence of only a single conserved quantity does 
    not rule out non-trivial steady states, as can be gleaned from the structure of Floquet eigenstates in the 
    frozen regime: these states have definite m^x values yet they are not fully localised in the x-basis. 
    It is also interesting to note a single local conserved quantity like m^x does not preclude a non-local 
    H_eff, yet is sufficient to result in a non-thermal Floquet spectrum.
    
    While our driving term in isolation is integrable, it appears that the existence of a conserved quantity is all that is
    required for the existence of the frozen regime. A study of a non-integrable drive with an emergent conservation law is
    therefore an obvious item for future work.  
    
    This non-ergodicity is  not a high-frequency phenomenon. Instead, it is particularly well-developed at 
    lower driving frequencies, which a priori renders attempts to construct a Magnus-type high frequency expansion 
    problematic. Instead, non-ergodicity is primarily associated  with strong driving. Note that for the driving term in isolation,
    the instantaneous eigenvectors of the Hamiltonian are time-independent, while the instantaneous eigenvalues
    change; this suggests the development of a perturbation theory controlled by the instantaneous gap, rather than a high
    frequency. It would also be interesting to investigate the connections of this problem to the case of weakly driven 
    interacting systems with approximate conservation laws<cit.>.
    
    The role of emergent conservation laws may in particular 
    be important for experimental studies of driven many-body systems. Indeed, a first sighting of the 
    physics we have analysed here has occurred in the context of an experiment of Floquet many-body localisation 
    <cit.>, where the possibility of a finite
    threshold for delocalisation was also noted for the low-disorder limit. 
    The main ingredient we have identified, an emergent conservation law,  
    turns out to also have been present in that situation. Analogously, for the searches of time crystals taking place at 
    present, it will be interesting to investigate if emergent conservation laws do, or can, play a role there as well. 
    
    Finally, while periodic driving is expected to heat a system and hence delocalize it, drive-induced
    destructive quantum interference can produce just the opposite effect. Competition of these might 
    result in unexpected freezing behaviour, as has been observed in quantum counterpart of classically chaotic systems,
    namely, in the kicked rotators (see, e.g.,  <cit.>). Such a suppression of heating <cit.> 
    might not be impossible in a quantum many-body system 
    where interactions lead to ergodicity. An absence of unbounded heating under periodic driving could be a step 
    in that direction, and the availability of emergent approximate conservation laws may turn out to be a 
    useful ingredient for many-body Floquet engineering. 
    
    
    
    § ACKNOWLEDGEMENTS
    
    The authors acknowledge  P. Bordia, A. Eckardt, V. Khemani, M. Knap and A. Polkovnikov  for discussions. 
    AD and AH acknowledge the partner group program “Spin liquids: correlations, dynamics
    and disorder" between IACS and MPI-PKS, and the visitor's program of MPI-PKS for supporting visits 
    to PKS during the collaboration.  RM is grateful to IACS for hospitality during the conclusion of this work.
    
    
    
    
    § SUPPLEMENTAL MATERIAL
    
    
    
    
    
    
    
    
    
    
     §.§ Ergodicity of H_0
    
    We first demonstrate the ergodicity of the undriven Hamiltonian H_0 for the parameter values we have used in the main text.
    This is displayed in Fig. <ref>.
    
    
    
     §.§ Freezing and Thermalization via Level Statistics
    
    Here we demonstrate the quasi-energy level repulsion and its absence for drive strengths above and below the threshold respectively.  
    We plot the quasi-energy (folded to the first Brillouin zone) gap ratio defined by
    
        r = min{δ_n,δ_n-1}/max{δ_n,δ_n-1},
     where,
    δ_n = μ_n+1-μ_n,μ_n being the n-th eigenvalue of H_eff, after folding them into the first Brillouin zone [-π,π]. 
    
    
     §.§ Quasi-energy Gap-ratio Statistics below and above the threshold:
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
     §.§ Threshold Phenomenon with Random Polarized Initial States
    
    Fig. <ref> shows the remanent magnetisation for a class of random polarised initial states. 
    
    
    
    § L-DEPENDENCE OF THE FLOQUET STATE AVERAGE MAGNETIZATION
    
    Fig. 2 (middle panel) of the main text shows that the step-structures in the m^x of the Floquet states are almost
    	indistinguishable from that of the x-basis states for all L we investigated. 
    	Here we consider the following average quantity to show on a finer scale, that this difference
    	systematically decreases with L. We order both the x-basis states and the Floquet states in order 
    	(decreasing, say) of their m^x values (eigenvalues and expectation values in respectively). Let 
    	m^x_h_D^x(i) and m^x_∞(i) denotes the m^x values of the i-th state thus ordered in respective basis. 
    	Now we compute the difference |m_∞ - m_h_D^x|, where 
    	m_∞ = 1/2^L∑_im^x_∞(i) and m_h_D^x = 
    	1/2^L∑_im^x_h_D^x(i). This provides a measure of the accuracy with which m^x is 
    	conserved (this difference vanishes if m^x is exactly conserved, since in that case each Floquet state
    	corresponds to an exact eigenstate of m^x). In Fig. <ref> we show that this difference is
    	tiny, and seems, if anything, to decrease with increasing L.
