Onset of Floquet Thermalisation
OOnset of Floquet thermalisation
Asmi Haldar , Roderich Moessner and Arnab Das Department of Theoretical physics, Indian Association for the Cultivation of Science,2A & 2B Raja S. C. Mullick Road, Kolkata 700032, India and Max Planck Institute for the Physics of Complex Systems,N¨othnitzer Straße 38, 01187 Dresden, Germany
In presence of interactions, a closed, homogeneous (disorder-free) many-body system is believedto generically heat up to an ‘infinite temperature’ ensemble when subjected to a periodic drive: inthe spirit of the ergodicity hypothesis underpinning statistical mechanics, this happens as no energyor other conservation law prevents this. Here we present an interacting Ising chain driven by a fieldof time-dependent strength, where such heating onsets only below a threshold value of the driveamplitude, 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. Theonset location shifts, but it is robustly present, across wide variations of the model Hamiltoniansuch as driving frequency and protocol, as well as the initial state. The portion of nonergodicstates in the Floquet spectrum, while thermodynamically subdominant, has a finite entropy. Wefind that the magnetisation as an emergent conserved quantity underpinning the freezing; indeed thefreezing effect is readily observed, as initially magnetised states remain partially frozen up to infinitetime . This result, which bears a family resemblance to the Kolmogorov-Arnold-Moser theorem forclassical dynamical systems, could be a valuable ingredient for extending Floquet engineering to theinteracting realm.
I. INTRODUCTION
Interacting many-body systems, by the ergodic hy-pothesis, generically thermalise, placing them in thepurview of statistical mechanics and equilibrium ther-modynamics . Our understanding of the correspond-ing situation for non-equilibrium systems is still in flux.For perhaps the simplest class of non-equilibrium sys-tems, periodically driven (Floquet) systems, thermalisa-tion physics at first pass looks maximally simple: remov-ing time translation invariance destroys energy conserva-tion, and hence the concept of temperature–which meansthermalisation is to a featureless ‘infinite-temperature’state .Such Floquet systems have been predicted to be capa-ble of sustaining new forms of spatiotemporal orderingwhen many-body localised as a result of strong quencheddisorder . The experimental search for such so-calleddiscrete time crystals has been qualitatively more suc-cessful than may have been anticipated: the collectionof systems appearing to exhibit such order now even in-cludes a dense periodic array of nuclear spins initialisedin a thermal state .All of this focuses the question on settings whichpermit long-lived correlations and order to persist de-spite the presence of periodic driving even in the ab-sence of quenched disoroder. In periodically driven non-interacting systems, quantum heating can be sup-pressed and an extensive number of periodicallyconserved quantities identified . In turn, a pre-thermalisation regime has been identified which resem-bles a frozen non-thermal state which can be describedby a periodic (generalized) Gibbs’ ensemble . Tuningthe drive parameters, and weakening the interactions,can substantially enhance the prethermalization period, still expected to remain finite . In fact, for disorder-freesystems, a transient but exponentially long-lived regimeexhibiting discrete time-crystalline phenomenology hasalready been identified . These constitute lower boundson the thermalisation timescales. For finite-size systems,an emergent integrability structure for strong drives hasalso been proposed as a way to avoid thermalisation .There is further evidence indicating absence of heating athigh drive frequencies in a variety of other settings and in specially designed models .Here, we address the question whether there is an iden-tifiable threshold for the ratio of driving and interac-tion strength, below which the system approaches a non-trivial steady state that depends on the drive and theinitial state. We consider a spin chain subject to strong,but not fast driving, and use remanent infinite-time mag-netisation of a initial magnetised state as measure of fail-ure to Floquet-thermalise. As the driving is increasedfrom low strength, where standard Floquet thermalisa-tion is observed, we find a remarkably well-defined sec-ond regime, in which remanent magnetisation is presenteven in the infinite time limit. Its value is given by theFloquet diagonal ensemble average implied by the initialstate. The location of this threshold moves, but its exis-tence is stable to variations in state initialisation, drivingstrength, driving protocol, and driving frequency.In all cases, however, we are able to identify an emer-gent approximately conserved quantity – in the case wediscuss at length, the magnetisation itself – which be-comes exactly conserved if the static part of the Hamil-tonian is ignored. Thus, rather than an extensive setof integrals of motion, as is present in the case of theperiodic Gibbs ensemble and the Floquet many-bodylocalised cases , all that appears to be needed to stopthe system from heating up indefinitely is a single, ap- a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r proximately conserved quantity.While our numerical investigation on systems up to L = 14 spins naturally limits our capacity to extrap-olate these results to the ‘thermodynamic’ limit, thereare indications that this is not only a finite-size effect.Firstly, in plots of remanent magnetisation versus driv-ing strength, we identify a crossing point for curves fordifferent L separating the ergodic and the non-thermalregimes. Second, the set of Floquet eigenstates exhibit-ing memory, while accounting only for a vanishing frac-tion of the total Hilbert space, extrapolates to have afinite entropy in the thermodynamic limit. This meansthat such states can still be straightforwardly selected byan initial condition, not unlike initialising a static systemin a low-temperature configuration.In the following, we first we set the notation and pro-vide a brief introduction to the Floquet concepts we haveused. We then define our model and drive protocol. Wecharacterise the ergodic and the non-thermal phases andthe threshold between them using various measures, anddemonstrate robustness to variations of drive patternsand system parameters. We close with an outlook andsuggestions for further investigations. In particular, ori-gin and nature of the sharp features in the memory as afunction of driving strength merit further study. II. MODEL
In this section, we introduce notation, model Hamilto-nian, drive protocol, and observables to be studied.
A. Floquet basics
Let us decompose the time-dependent Hamiltonian H ( t ) into a static interacting Hamiltonian H and a time-periodic drive H D ( t ) with [ H , H D ] (cid:54) = 0: H ( t ) = H + H D ( t ) , (1)The time evolution operator evolving a state through aperiod from t = (cid:15) to t = (cid:15) + T (0 ≤ (cid:15) < T ) is U ( (cid:15) ). Since U ( (cid:15) ) is unitary, it can always be expressed in terms of ahermitian operator, the ‘Floquet Hamiltonian’ H eff as U ( (cid:15) ) = e − iH eff ( (cid:15) ) T . (2)Formally,exp ( − iH eff ( (cid:15) ) T ) = T exp (cid:32) − i (cid:90) (cid:15) + T(cid:15) dt H ( t ) (cid:33) , (3)where T denotes time-ordering. Let | µ i (cid:105) denote the i -th‘Floquet eigenstate’ of H eff corresponding to the ‘Flo-quet eigenvalue’ (also known as quasienergy) µ i .A sequence of stroboscopic observations at instants t = (cid:15), (cid:15) + T, . . . , (cid:15) + nT (integer n ) is identical to that produced by the dynamics under the time-independentHamiltonian H eff . This applies for every (cid:15) , hence we getcontinuous family of stroboscopic series.In the following, we are interested in long-time asymp-totic behaviour, so that temporal variations within adriving period are of secondary importance. Hence, wearbitrarily pick (cid:15) = 0 . B. Infinite time limit: diagonal ensemble average
The nature of the asymptotic state under the drive canbe understood as follows. Consider an initial state | ψ (0) (cid:105) = (cid:88) i c i | µ i (cid:105) and the stroboscopic time series for an observableˆ O = (cid:88) i,j O ij | µ i (cid:105)(cid:104) µ j | . (cid:104) ψ ( nT + (cid:15) ) | ˆ O| ψ ( nT + (cid:15) ) (cid:105) = (cid:88) i,j c i c ∗ j O ij e − i ( µ i − µ j )( nT + (cid:15) ) . (4)Like in the case of static Hamiltonians, under quite gen-eral and experimentally relevant conditions (see, e.g.,Ref. ), at long times ( n → ∞ ) the off-diagonal ( i (cid:54) = j )terms ‘average to zero’ and the state of the system canhence be described by an effective “diagonal ensemble”(in the absence of synchronisation, e.g. for discrete timecrystals, this is replaced by a block diagonal ensemble ).This is captured by the mixed density matrix ˆ ρ DE = (cid:80) i | c i | | µ i (cid:105)(cid:104) µ i | . Thus, the asymptotic properties of a periodicallydriven system are effectively given by a classical aver-age (known as diagonal ensemble average or DEA) overthe expectation values of the eigenstates of H eff , (cid:104) ˆ O(cid:105) (DEA) = (cid:88) i | c i | (cid:104) µ i | ˆ O| µ i (cid:105) . (5)Hence it is sufficient to study the nature of the eigenstatesand eigenvalues of H eff , or equivalently of U ( (cid:15) ), in orderto obtain the long-time behaviour. C. Driving protocol
We consider L spins on a chain. We chose a binarydrive protocol, which switches periodically between a pairof rectangular pulses. The time dependent Hamiltonianis H ( t ) = H + sgn(cos ωt ) H D , (6)with the two components H = − J (cid:88) i σ xi σ xi +1 + κ (cid:88) i σ xi σ xi +2 − h x L (cid:88) i σ xi − h z L (cid:88) i σ zi ; (7) H D = − h xD L (cid:88) i σ xi . (8)The σ α are Pauli matrices. We use periodic boundarycondition, but tamper the system slightly by putting J L, = 1 . J and κ L − , = 1 . κ to break transla-tional invariance (and hence remove any remaining block-diagonal structure of the Hamiltonian). Here since wekeep 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 is known to be ergodic due to the four-fermionic in-teraction terms arising from the next-nearest neighbourinteractions under the spin to fermion mapping, and alsodue to the longitudinal field. We have explicitly verifiedthat H is ergodic for our case, see Suppl. Mat.).We initialise the simulation in the time domain withdifferent initial states. Unless otherwise stated, we usethe default choice of the ground state of H ( t = 0). III. NUMERICAL RESULTS
The central quantity is the longitudinal magnetization m x ( t ) = 1 L L (cid:88) i (cid:104) ψ ( t ) | σ xi | ψ ( t ) (cid:105) . (9)We monitor its real-time dynamics in a stroboscopic timeseries. We diagnose non-thermalisation/freezing via itslong-time asymptotic behaviour, the remnant magneti-sation, which we study as a function of various modelparameters. A. Onset of Floquet thermalisation
In the following, we provide numerical evidence thatfor a strong (but not fast) drive, the system fails to Flo-quet thermalise, instead retaining memory of its initiallymagnetised state. We then show that the onset of Flo-quet thermalisation occurs at a fairly well-defined thresh-old driving strength. For the results in the main text, wehave chosen J = 1 , κ = 0 . , h z = 1 . , and h x = 0 . . The stroboscopic time series for the magnetisation m x is shown in Fig. 1, left frame. Already at shorttimes, three representative trajectories for different driv-ing strengths show strikingly different behaviour. Whilefor weak driving fields, the magnetisation disappears al-most immediately, for stronger ones, the decay slows down. Finally, for h xD beyond a threshold value, the de-cay is arrested: even at the longest times, a remnantmagnetisation persists.This remnant magnetisation agrees with the DEA ofthe magnetisation evaluated for the same system (see in-set). Note that the nonvanishing DEA is already in it-self a signature of the lack of Floquet thermalisation –in general, Floquet thermalised eigenstates individuallyshow no non-trivial correlations.In order to locate the onset, the DEA of m x as a func-tion of the drive amplitude h xD is plotted in Fig. 1, middleframe. A threshold for nonzero remnant magnetisationis observed, separating the ergodic ( m xDE ≈
0) from thenonergodic regime.The lower inset shows freezing for an initial state witha reduced polarization in the x − direction. The black dot-ted line shows the initial value of m x for the state, andthe curve shows that for high enough h xD , the DEA of m x almost coincides with it. In detail, this initial stateis given as | ψ (cid:105) = (cid:80) L i =1 c i | i x (cid:105) , where | i x (cid:105) is the i − theigenstate of the longitudinal field part (computationalbasis states in x -direction, or x − basis states), by choos-ing Re [ c i ] and Im [ c i ] from a uniform distribution between-1 and +1, multiplying them by e βm xi , where β > m ix is the longitudinal magnetization of | i x (cid:105) , and finallynormalizing the state. This gives a ‘generic’ state witha bias towards positive longitudinal magnetization. Forthe plot in Fig. 1 (middle frame), we have chosen a ran-dom instance corresponding to β = 1 . . The right frameof Fig. 1 shows DEA of 1 − m x on a doubly logarithmic log − log plot zoomed in around the threshold for bettervisibility. B. Floquet eigenstates and an emergentconservation law
1. Localization and magnetization
We now turn to the properties of the Floquet eigen-states obtained by numerically diagonalizing the timeevolution operator U (0), Eq. 2. We consider first their‘localisation’ in Hilbert space, followed by their magneti-sation content.In order to investigate the localization properties ofthe Floquet states in the x − basis {| i x (cid:105)} we calculate theinverse participation ratio (IPR) in said basis defined as IP R ( | µ j (cid:105) ) = (cid:80) L i =1 |(cid:104) i x | µ j (cid:105)| . The left frame of Fig. 2shows the IPR thus obtained, arranged in decreasing or-der. Indefinite heating corresponds to the states beingdelocalized in the eigenbasis of any local operator, whichimplies a uniformly small IPR given by the inverse di-mension of Hilbert space, 1 /D H . This is indeed whatis observed for small drive fields. By contrast, for largedrive fields, states appear which have an IPR close to1, which indicates the presence of well-localised states,and hence the absence of Floquet thermalisation for the . . . . . . m x ( n T ) L=14 h xD = 2 . h xD = 10 . h xD = 30 . . . .
000 0 5 10 15 20 25 30 35 40 h xD . . . . . . m x ( D E A ) L = 11 L = 12 L = 13 L = 14
28 30 320 . (a) . . (b) L = 14
10 15 20 25 30 35 40 h xD . . . . - m x ( D E A ) T L = 11 L = 12 L = 13 L = 14 FIG. 1: Freezing and its onset threshold.
Left frame : Stroboscopic time series of magnetisation m x ( t ) for different drivingstrengths showing initial state memory for strong driving. The inset zooms in on the long-time behaviour; the black horizontalline denotes the DEA of the magnetisation. Middle frame : Remnant magnetisation as a function of driving strength for differentsystem sizes. The high-field regime (top inset) shows an increase of the remnant magnetisation with L . The bottom inset showsthe DEA of m x vs. drive amplitude for a ‘generic’ state (see the text for details) whose net initial magnetization is markedwith the horizontal line, which remains almost unchanged for very strong drives. Right frame : Same data as middle frame ona doubly logarithmic plot for 1 − m x (DEA) . The deviation away from almost complete thermalisation gets steeper and movestowards the right with increasing system size. The curves appear to accumulate from the left at a ‘threshold point’ ( T ) whichitself appears to move little as the system size is increased from L = 11 to L = 14. . . . . . . i/D H . . . . . . I P R L=14 h xD = 10 h xD = 18 h xD = 40 h xD = 600 . . . . . h xD = 40 L = 8 L = 10 L = 12 L = 14 − . − . . . . L=14 h xD = 40 h xD = 18 h xD = 10 h xD = ∞ . . . . . . i/D H − . − . . . . m x L=10
10 11 12 13 14 15 L . . . . . . . l n N c m c = 0 . h xD = 40 h xD = 18 h xD = ∞ fitting
10 12 143 . . . m c = 0 . FIG. 2: Emergent conservation law for strong drives, as reflected in the Floquet eigenstates | µ i (cid:105) . Left frame : Values of the IPRin the x -basis, arranged in decreasing order. Unbounded heating requires these states to be delocalized in the eigenbasis of anylocal operator. This is the case for the drive with amplitude below the threshold ( h xD = 10 , ) but not above ( h xD = 18 , , h xD = 40 due to the emergent conservation law evidenced inthe middle frame: m x for the Floquet eigenstates arranged in decreasing order, for different values of h xD . Black dotted lines( h xD = ∞ ) show the values of m x of the x − basis states (multiplied by a factor of 1.4 for visibility). For h xD = 40 , clear step-likestructures appear, indistinguishable from the steps of m x for x − basis states for both system sizes L = 10 ,
14 (see Suppl. Mat.for finer details of L dependence of this matching). For a lower drive value h xD = 18, close to the threshold, the curve smoothesout, indicating weakening of the quasi-conservation, yet highly polarized Floquet states are still substantial in number. Forstill lower values (e.g. h xD = 10), the curve finally flattens. The pronounced asymmetry in the Floquet magnetizations forlower values of h xD is due to the small asymmetry in the drive. Right frame:
The log of the number N c of Floquet eigenstateswith polarization above a given value m c is shown to grows approximately exponentially with system size, corresponding to (avanishing fraction of states but with a) finite entropy. For large h xD ( h xD = 40), the numerical data points fall almost exactlyon the analytically calculated (black dotted lines) corresponding to h xD = ∞ (see the matching of the step-like structures inthe middle frame). For a lower value h xD = 18 a linear fit is done for the numerical data points. corresponding part of the spectrum.Complementary information can be gleaned by consid-ering the correlations encoded in the non-ergodic states.The middle frame of Fig. 2 shows the magnetisation ofdifferent Floquet eigenstates, m xi , ordered according totheir size. In the ergodic regime, these curves are fea-tureless and m xi is uniformly tiny, showing a tendencyto increase with increasing drive strength. Deep intothe nonergodic regime, large values of m xi appear, which together form plateaux. For the largest drives h xD , theplateaux correspond to essentially an integer number ofspin flips, which indicates that the new basis is close tothe computational basis in the x − direction mentionedabove. As the drive is decreased, the plateaux give wayto a smooth curve, which however still makes large ex-cursion toward m x = ± h xD /h z . . . . . . m x ( D E A ) L=12 h z = 0 . h z = 0 . h z = 1 . h z = 2 . h z = 4 . h xD . . . a )0 1 2 3 4 h z h x D ∗ ( b ) h xD . . . . m x ( D E A ) L=12 h x = 0 . h x = 0 . h x = 1 . h xD . . . . . . m x ( D E A ) L = 10 L = 11 L = 12 L = 13
10 20 30 40 60 h xD . . . . . - m x ( D E A ) Threshold L = 10 L = 11 L = 12 L = 13 L = 14Initial m x
30 40 500 . . . . FIG. 3: Remnant magnetisation in various settings.
Top left:
Dependence of a DC the transverse field h z which does notcommute with the other, mutually commuting, terms of the model. h z enhances thermalization (upper inset). The responseapproximately scales with h xD /h z (main panel); in particular, the estimated threshold h xD ∗ is approximately proportional to h z (lower inset). Top right:
Robustness of freezing with respect to addition of DC field h x . Bottom left:
Freezing for unevendivision of the total drive period. For 0 ≤ t < rT, h xD = +40 , while for rT ≤ t < T, h xD = −
40, where r =1/(Golden Ratio).Deep freezing minima persist to high driving strengths but show little size dependence. Bottom right:
Behaviour for initialstate chosen as the ground state of the non-integrable undriven part H , with h z and h x chosen to created an initial state witha small positive polarization m x (0) ≈ . . For large h xD , freezing increases somewhat with L. tion above a certain value is thermodynamically vanish-ing, their entropy is nonetheless finite, see Fig. 2, mid-dle and right frames. This is analogous to the case of afinite-temperature ensemble of a magnet in a field, wherea nonzero magnetisation arises as a thermodynamicallyvanishing fraction of magnetised states is preferentiallypopulated, with their energy gain compensating for theentropy loss involved in concentrating the probabilitydensity on them. Here, the selection of the magnetisedFloquet states arises via the state initialisation. It is in-teresting to note that in this 1D system there would beno magnetization at any finite temperature: the obser-vation of a finite magnetization at finite energy densityis purely a non-equilibrium effect.
2. Emergence of m x as a local quasi-conserved quantity We next address what we believe is the central featureunderpinning the non-thermalisation, namely the exis-tence of a conserved quantity in the drive Hamiltonianin isolation. In our example, this is the magnetisation in the x -direction, m x , which persists as a quasi-conservedquantity even when the ratio of drive to static compo-nents of the Hamiltonian is finite.The middle frame of Fig. 2 shows the value of m x forthe different Floquet eigenstates arranged by their size.For the strongest drives, the steps in this quantity areidentical to the ones of the computational basis states inthe x -basis, i.e. the steps simply reflect the number ofspins flipped.The static part of the Hamiltonian then mixes thestates with the same value of m x , which is reflected in thenon-trivial distribution of the IPR of the Floquet states(left frame of Fig. 2). The growth of the size of each m x sector (except for the fully polarised one) is in turnreflected in a decrease of the IPR.For lower driving strengths, h xD = 18, the steps getwashed out, but the range of m x continues to span prac-tically the full range in the interval between − h xD = 10,where the curve flattens substantially.While the fraction of Floquet states with a non-zeromagnetisation density vanishes with system size, these h xD . . . . . . m x ( D E A ) ω = 0 . ω = 1 . ω = 10 . ω . . . m x ( D E A ) h xD = 10 h xD = 40 FIG. 4: Dependence of the remnant magnetisation on thedrive frequency ω for L = 14 initialised with the ground stateof H (0). The basic morphology (in particular, the two regimesand the threshold) remains the same over two decades in fre-quency. The freezing decreases as ω is increased, with thethreshold only varying slowly with ω . The inset shows m x vs ω for h xD = 10 (outside the frozen regime) and h xD = 40,where the weakening of freezing with increasing ω is evident. states nonetheless have nonzero entropy, Fig. 2 rightpanel, as is the case for magnetised states of a param-agnet generally.The emergent quasi-conserved nature of m x , alongwith the straightforward possibility of initialising the sys-tem in a magnetised state, acount for the main featuresof the results discussed in this work. C. Robustness against variation of model andprotocol parameters
We first address the existence of the onset for variantsof the above model. We note that so far, no fine-tuningwas necessary. The central demand was for the driveamplitude h xD to be the largest scale, while the otherparameters of the Hamiltonian were chosen all to be inthe same ballpark.
1. Role of non-commuting term
First, the location of the thermalisation threshold canbe moved by varying the strength of the term in the staticHamiltonian H which does not commute with the driv-ing Hamiltonian H D . Indeed, the top left frame of Fig. 3shows that the threshold driving field is approximatelyproportional to the static transverse field strength h z .
2. Drive shape and initial state
Also, we ask whether the ‘symmetry’ of having a van-ishing mean drive of zero for symmetric pulse shapesabout zero is an important ingredient. Fig 3, top rightframe, shows that the freezing is quite robust to additionof a dc field of strength h x . Indeed, the freezing actuallygrows with h x .Next, we consider a deviation of the drive protocolaway from a time-symmetric switch in the sign of thedriving term to one where more time is spent for onesign than the other (Fig. 3 bottom left frame). While thelatter case has considerably more structure at high drives,in particular an apparently regular suppression of theremnant magnetisation even above the onset threshold,the former curve basically acts as a high-magnetisationenvelope of the latter.Further, we consider an initial state prepared as theground-state of a many-body problem (rather than amore simply prepared polarised state). This displays(Fig. 3 bottom right frame) all the salient features ob-served with the simply polarized ground state in Fig. 1,right frame.
3. Drive frequency
What is particularly worth emphasizing is that thenon-ergodic behaviour is not a high-frequency phe-nomenon. While such freezing also exists in the limitof a driving frequency in excess of the many-body band-width of the finite-size system, it is not even the case thatthe nonergodicity necessarily grows with frequency. Thisis illustrated in Fig. 4, where the remnant magnetisationis, 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 Mag-nus expansion which is not associated with unboundedheating.
D. Finite-size behaviour
Our results indicate that absence of thermalization inthis driven interacting system might persist even in theinfinite-size limit. While there are some dips of the freez-ing strength in the nonergodic regime complicating asharp identification of a threshold, the onset nonethe-less appears to sharpen with increasing system-size. Acloser view of the nonergodic regime, Fig. 1, middle frametop inset, shows smooth behaviour of the remnant mag-netisation for the largest fields; this in fact grows withincreasing system size. By contrast, for weak drives,the remnant magnetisation tends to decrease with sys-tem size. This results in a crossing pointas the curvesfor different system sizes of the deviation of the remnantmagnetisation from its initial value, Fig. 1, right frame,thus approximately cross at the threshold point. Whileit is hard entirely to rule out a slow drift to higher fieldsof the threshold with increasing system size, these obser-vations suggest the possibility of a sharp transition at afinite threshold field in the thermodynamic limit.Next, and most importantly, the step-structures inthe m x of the Floquet states are almost indistinguish-able from that of the x − basis states for all system-sizes we investigated (Fig. 2, middle frame). This absence ofsystem-size dependence indicates that at large values of h xD , the drive does not mix the x − basis states of different m x values. A decrease in the fraction of Floquet stateswith m x > m c with system-size is not because in largersystems the Floquet states are more delocalized betweendifferent magnetization sectors, but merely because thenumber of x − basis states in a given magnetization sec-tor changes with the system-size. Delocalization betweendifferent magnetization sectors is suppressed strongly forall system-sizes at hand for h xD above the threshold. Thisis in keeping with the observations that on different typesof initial states, Fig 1 and Fig.3, the freezing at the high-est frequencies does not decrease with system size. andgives a further indication that our results are not merelyfinite-size effects. IV. DISCUSSION
We have studied the onset of Floquet thermalisation ina driven interacting spin chain. We have found a fairlysharp threshold for the drive strength, above which Flo-quet thermalisation does not take place. The thresholdvalue varies in different manners with parameters likepulse shape, drive frequency, or the (non-commuting)transverse field strength, but the freezing persists ro-bustly under all these variations. The question of theexistence of such a threshold is of fundamental impor-tance, with a related issue appearing for classical dy-namical systems, where the Kolmogorov-Arnold-Mosertheorem deals with the onset of chaotic behaviour uponbreaking of integrability.An open question is the origin, and in particular the L -dependence, of the dips in the frozen component evenbeyond the threshold in the m x vs h xD plots: the dipstouching the x − axis correspond to points of thermaliza-tion. While their occurrence for certain discrete valuesof h xD has no significant consequence, if their number di-verges with L , this may lead to a destruction of the frozenregime. For drives with pulse durations evenly placedabout T /
2, the dips disappear rapidly with increasing h xD . Such dips are, however, observed to persist even forvery strong amplitudes for the case of drive with unevendivision of the drive period (Fig. 3, bottom left). In thiscase, the total drive period is divided in two parts, T /GR and T (1 − /GR ) , where GR is the Golden ratio. Whilethe depth of the dips seems to increase with L, their num-ber and locations remain surprisingly independent of L, which points against their proliferation. Regarding an extrapolation to the thermodynamic limit, we refer toour discussion at the end of the previous section.Comparison of the magnetization and IPR of the Flo-quet states in the frozen regime allows one to concludethat the magnetization itself plays the role of a quasi-conserved quantity, which becomes exactly conserved inthe limit of infinitely strong driving. However, the emer-gence of only a single conserved quantity does not ruleout non-trivial steady states, as can be gleaned from thestructure of Floquet eigenstates in the frozen regime:these states have definite m x values yet they are notfully localised in the x − basis. It is also interesting tonote a single local conserved quantity like m x does notpreclude a non-local H eff , yet is sufficient to result in anon-thermal Floquet spectrum.While our driving term in isolation is integrable, it ap-pears that the existence of a conserved quantity is all thatis required for the existence of the frozen regime. A studyof a non-integrable drive with an emergent conservationlaw is therefore an obvious item for future work.This non-ergodicity is not a high-frequency phe-nomenon. Instead, it is particularly well-developed atlower driving frequencies, which a priori renders attemptsto construct a Magnus-type high frequency expansionproblematic. Instead, non-ergodicity is primarily asso-ciated with strong driving. Note that for the drivingterm in isolation, the instantaneous eigenvectors of theHamiltonian are time-independent, while the instanta-neous eigenvalues change; this suggests the developmentof a perturbation theory controlled by the instantaneousgap, rather than a high frequency. It would also be in-teresting to investigate the connections of this problemto the case of weakly driven interacting systems with ap-proximate conservation laws .The role of emergent conservation laws may in particu-lar be important for experimental studies of driven many-body systems. Indeed, a first sighting of the physics wehave analysed here has occurred in the context of an ex-periment of Floquet many-body localisation , where thepossibility of a finite threshold for delocalisation was alsonoted for the low-disorder limit. The main ingredient wehave identified, an emergent conservation law, turns outto also have been present in that situation. Analogously,for the searches of time crystals taking place at present, itwill be interesting to investigate if emergent conservationlaws do, or can, play a role there as well.Finally, while periodic driving is expected to heat asystem and hence delocalize it, drive-induced destructivequantum interference can produce just the opposite ef-fect. Competition of these might result in unexpectedfreezing behaviour, as has been observed in quantumcounterpart of classically chaotic systems, namely, in thekicked rotators (see, e.g., ). Such a suppression of heat-ing might not be impossible in a quantum many-bodysystem where interactions lead to ergodicity. An absenceof unbounded heating under periodic driving could be astep in that direction, and the availability of emergentapproximate conservation laws may turn out to be a use-ful 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” be-tween IACS and MPI-PKS, and the visitor’s program ofMPI-PKS for supporting visits to PKS during the collab-oration. RM is grateful to IACS for hospitality during theconclusion of this work. K. Huang,
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V. SUPPLEMENTAL MATERIALA. Ergodicity of H We first demonstrate the ergodicity of the undrivenHamiltonian H for the parameter values we have usedin the main text. This is displayed in Fig. 5. −
20 0 20Eigenvalues − . − . . . . EE V h σ xi σ xi +1 ih σ zi i −
20 0 20Eigenvalues − . − . . . . EE V h σ xi σ xi +1 ih σ zi i FIG. 5: Left frame shows expectation values of different localoperators over the eigenstates (EEV) of the integrable trans-verse field Ising chain ( κ = h x = 0) against the eigenvalues ofthose states ( L = 12). The non-smooth behaviour arises dueto the existence of different non-mixing (block-diagonal) sec-tors. Right frame shows that introduction of the interactionand longitudinal field breaks the integrability and the EEV’sbecome smooth, confirming that the undriven Hamiltonian H is quite generic and satisfies the eigenstate thermalizationhypothesis. The paremeters κ = 0 . h x = 1 correspondto those in the main text. B. Freezing and Thermalization via Level Statistics
Here we demonstrate the quasi-energy level repulsionand its absence for drive strengths above and below thethreshold respectively. We plot the quasi-energy (foldedto the first Brillouin zone) gap ratio defined by r = min { δ n , δ n − } max { δ n , δ n − } , where, δ n = µ n +1 − µ n , µ n being the n -th eigenvalueof H eff , after folding them into the first Brillouin zone[ − π, π ]. C. Quasi-energy Gap-ratio Statistics below andabove the threshold: . . . P ( r ) h xD = 10 . . . . . r P ( r ) h xD = 60 . FIG. 6: Gap-ratio statistics r for h xD = 10 (below the thresh-old) and 60 (far above the threshold) respectively, showingpresence and absence of quasi-energy level repulsion belowand above the threshold respectively. D. Threshold Phenomenon with Random PolarizedInitial States
Fig. 7 shows the remanent magnetisation for a class ofrandom polarised initial states. h xD . . . . m x ( D E A ) L = 8 L = 10 L = 12 L = 14 FIG. 7: Freezing for initially polarized but otherwise ran-domised states under longitudinal drive (for the same set ofparameter as in the main text). These initial states are givenas | ψ (0) (cid:105) = (cid:80) L i =1 c i | i x (cid:105) , where | i x (cid:105) is the i − th eigenstatein thecomputational basis states in x -direction, by choosing Re [ c i ] and Im [ c i ] from a uniform distribution between -1 and+1, multiplying them by e βm xi , where β >
0, where m ix isthe longitudinal magnetization of | i x (cid:105) , and finally normaliz-ing the state. Results are shown for different L values, forrandom instances generated with β = 2 . L .) The system freezes forlarge values of the drive field h xD for all accessed system-sizeswe could access, with the remanent magnetisation very closeto the diagonal ensemble average (DEA) of the longitudinalmagnetization m x . VI. L − DEPENDENCE OF THE FLOQUETSTATE AVERAGE MAGNETIZATION
Fig. 2 (middle panel) of the main text shows that thestep-structures in the m x of the Floquet states are al-most indistinguishable from that of the x − basis statesfor all L we investigated. Here we consider the followingaverage quantity to show on a finer scale, that this differ-ence systematically decreases with L. We order both the x − basis states and the Floquet states in order (decreas-ing, say) of their m x values (eigenvalues and expectationvalues in respectively). Let m xh xD ( i ) and m x ∞ ( i ) denotesthe m x values of the i -th state thus ordered in respec-tive basis. Now we compute the difference | m ∞ − m h xD | ,where m ∞ = L (cid:80) i m x ∞ ( i ) and m h xD = L (cid:80) i m xh xD ( i ) . This provides a measure of the accuracy with which m x is conserved (this difference vanishes if m x is exactly con-served, since in that case each Floquet state correspondsto an exact eigenstate of m x ). In Fig. 8 we show that this difference is tiny, and seems, if anything, to decreasewith increasing L. L − − − l n | m ∞ − m | NumericalLinearFit
FIG. 8: L − dependence of the log of the deviation of the av-erage of the magnetization of the Floquet states from thecorresponding x −−