Dynamical Freezing and Scar Points in Strongly Driven Floquet Matter: Resonance vs Emergent Conservation Laws
SScars in strongly driven Floquet matter: resonance vs emergent conservation laws
Asmi Haldar , Diptiman Sen , Roderich Moessner , and Arnab Das Indian Association for the Cultivation of Science,2A & 2B Raja S. C. Mullick Road, Kolkata 700032, India Centre for High Energy Physics and Department of Physics,Indian Institute of Science, Bengaluru 560012, India Max Planck Institute for the Physics of Complex Systems, Dresden, Germany (Dated: September 11, 2019)We consider a clean quantum system subject to strong periodic driving. The existence of adominant energy scale, h xD , can generate considerable structure in an effective description of a systemwhich, in the absence of the drive, is non-integrable, interacting, and does not host localization. Inparticular, we uncover points of freezing in the space of drive parameters (frequency and amplitude).At those points, the dynamics is severely constrained due to the emergence of a local conservedquantity, which prevents the system from heating up ergodically, starting from any generic state,even though it delocalizes over an appropriate subspace. At large drive frequencies, where a na¨ıveMagnus expansion would predict a vanishing effective (average) drive, we devise instead a strong-drive Magnus expansion in a moving frame. There, the emergent conservation law is reflected in theappearance of an ‘integrability’ of a vanishing effective Hamiltonian. These results hold for a widevariety of Hamiltonians, including the Ising model in a transverse field in any dimension and for anyform of Ising interactions . Further, we construct a real-time perturbation theory which capturesresonance phenomena where the conservation breaks down giving way to unbounded heating. Thisopens a window on the low-frequency regime where the Magnus expansion fails. I. INTRODUCTION
For closed systems with time-independent Hamiltoni-ans, the notion of ergodicity has been formulated at thelevel of eigenstates in the eigenstate thermalization hy-pothesis (ETH) . According to ETH, the expectationvalue of a local observable in a single energy eigenstateof a complex (disorder-free) many-body quantum systemis equal to the thermal expectation value of the observ-able at a temperature corresponding to the energy den-sity of that eigenstate. The implication of the ergodicityhypothesis in the context of time-dependent (‘driven’)closed quantum systems is an open question of funda-mental importance.Relatively recent progress in this line has occurred forsystems subjected to a periodic drive (namely, Floquetsystems) , which are perhaps conceptually closest to astatic system. These studies indicate that a quantumsystem that satisfies ETH, when subjected to a peri-odic drive, approaches a state which locally looks like an‘infinite-temperature’ state. This is in accordance withthe ergodicity hypothesis – in systems which satisfy ETH(we will call them generic), energy is the only local con-served quantity, and any time-dependence breaks thisconservation, allowing the system to explore the entireHilbert space.The breaking down of ETH in interacting systems dueto the presence of localized states – either due to disorder(many-body localization) or other mechanisms (likemany-body Wannier-Stark localization) is well-knownwithin the equilibrium set-up, and their persistence un-der periodic perturbations has also been observed ,but the common intuition is that a translationally in-variant, interacting, non-integrable many-body system will be ergodic. However, this intuition has encountereda number of remarkable counterexamples recently withinthe static setting. It has been shown that in such systemsthere can be highly excited energy eigenstates, dubbed asscars, which do not satisfy ETH . Most of these ex-amples (see, however, ) indicate the non-trivial (weak)breaking of ergodicity by certain eigenstates.On the non-equilibrium side, stable Floquet statesare seen in finite-size closed interacting Floquet systemswhich are not localized in the absence of a drive . Inparticular, it has recently been shown that ergodicity isbroken in disorder-free generic systems under a periodicdrive if the drive strength is greater than a thresholdvalue (compared with the interaction strength) – a KAMlike scenario .The emergence of constraints on dynamics understrong periodic driving is known for non-interacting sys-tems – for strongly driven free fermions, there exist spe-cial points in the space of the drive parameters, wherefreezing for all time are observed for any arbitrary initialstate, for any (including infinite) system-size . Thisis surprising since the appropriate description for such asystem is a periodic generalized Gibbs’ ensemble . Suchan ensemble, though much less ergodic than a thermalone due to presence of an extensive number of (period-ically) conserved quantities, leaves ample space for sub-stantial dynamics of the response in general. Hence, inaddition to the integrability, other constraints emerge atthose special freezing points. Those freezing points canbe thought of as “scars” in the space of drive Hamilto-nians. Here we uncover and similar scar phenomenologyin interacting Floquet systems, and provide an analyticalunderstanding of the phenomenon.Here we demonstrate that a generic, interacting, trans-lationally invariant Ising system can exhibit non-ergodic a r X i v : . [ c ond - m a t . o t h e r] S e p behavior under a strong periodic drive. For certain iso-lated sets of values of the drive parameters – the scarson the drive parameter space – a local quasi-conservedquantity (that exhibits only small fluctuations about itsinitial value with time) emerges. The Floquet Hamilto-nian is then no longer ergodic, i.e., its eigenstates (Flo-quet states) do not look like the otherwise expected in-finite temperature states , but instead are character-ized by eigenvalues of the quasi-conserved quantity. Thisis because the dynamics does not mix different eigen-states with different eigenvalues of the quasi-conservedquantity. This however, does not mean that there isno dynamics. Indeed, even at the scar points, we seepronounced dynamics evidenced by substantial growthin sub-system entanglement entropy as delocalizationwithin a sector takes place. A finite size analysis of thenumerical results indicates the stability of the scars underan increase in the system size.At high driving frequencies, the conventional Magnusexpansion – controlled in the driving frequency as thelargest energy scale – fails, as the average Hamiltoniangenerally does not exhibit the conservation law in ques-tion. To remedy this, we present a strong-drive Magnusexpansion, constructed in a ‘moving’ frame incorporat-ing the strong driving term. Here, the conservation lawis manifest at low order in the expansion. For a gen-eral class of Hamiltonians, including the Ising model ina transverse field in any dimension and any form of theIsing interactions, we find that the effective Hamiltonianvanishes exactly up to two leading orders for our exam-ple, capturing the freezing (observed from exact numer-ics) to a good approximation away from the resonances.This suggests that the expansion is either convergent orasymptotic, in this setting.For lower drive frequencies, no controlled approxima-tion scheme for Floquet systems is available. Here, weformulate a novel perturbation theory, Floquet-Dysonperturbation theory (FDPT), which again uses the factthat the drive amplitude is large. This we find works bestin the low-frequency regime, where we benchmark it forsimple systems against an exact solution, and againstexact numerics. This enables us to account for isolatedfirst-order resonances, which are of particular interest astheir sparseness implies stable non-thermal states to firstorder. The stability is maintained in the thermodynamiclimit if our perturbation expansion is an asymptotic one,which is indicated by the finite-size analysis of our nu-merical results – the freezing is insensitive to increase insystem-size (see the finite-size result App. A). In partic-ular, the FDPT is remarkably accurate in predicting theresonances (obtained from exact numerics) close to inte-grability, and hence at the scars. This opens up a recipeto construct stable Floquet state with desired propertiesby choosing suitable drive terms.We organize this paper as follows. After a brief intro-duction to Floquet and our notation, we first present thephenomenology of scarring. We then present the high-field Magnus expansion, and finally the FDPT. We con- clude with a summary and outlook. II. FLOQUET IN A NUTSHELL
The Floquet states | µ n (cid:105) are elements of a completeorthonormal set of eigenstates of the time-evolution op-erator U ( T,
0) for time evolution from t = 0 to t = T, fora system governed by a time-periodic Hamiltonian with aperiod T = 2 π/ω. The Floquet formalism is particularlyuseful for following the dynamics stroboscopically at dis-crete time instants t = nT. From the above definition itfollows that U ( T, | µ n (cid:105) = e − iµ n | µ n (cid:105) , (1)where the µ n ’s are real. It is customary to define aneffective Floquet Hamiltonian H eff as U ( T,
0) = e − iH eff T . (2)(We will set (cid:126) = 1 in this paper). When observed strobo-scopically at times t = nT, the dynamics can be thoughtof as being governed by the time-independent Hamilto-nian H eff , which has eigenvalues µ n /T (modulo integermultiples of 2 π/T ) and eigenvectors | µ n (cid:105) . In the infinitetime limit, the expectation values of a local operator O can be written in terms of the expectation values in theFloquet eigenstates aslim N →∞ (cid:104) ψ ( N T ) |O| ψ ( N T ) (cid:105) = (cid:88) n | c n | (cid:104) µ n |O| µ n (cid:105) = O DE , (3)where | ψ (0) (cid:105) = (cid:80) n c n | µ n (cid:105) , and the subscript “ DE ” de-notes the Diagonal Ensemble average defined as above.This is equivalent to taking a “classical” average over theproperties of the Floquet eigenstates {| µ n (cid:105)} . The diago-nal ensemble average of m x . given by m x DE = (cid:88) n | c n | (cid:104) µ n | m x | µ n (cid:105) . (4)The absence of interference between the Floquet statesin a DE average ensures that it is sufficient to study theproperties of individual Floquet states (and their spec-trum average) in order to characterize the gross behaviorof the driven system in the infinite time limit. In thefollowing we will therefore mostly concentrate on DE av-erages and the properties of the Floquet states. III. THE SCAR PHENOMENOLOGYA. Freezing and Quasi-Conservation
This section discusses the scar phenomenology for a pe-riodically driven, interacting, non-integrable Ising chain ω − . . . . . . . . − m x D E / m x β = ∞ β = 10 − − h xD − h xD /ω = 4 . . . . . . i/D H − . − . . . . m x x − basisω = 10 . ω = 12 . ω = 20 . ω = 30 . ω = 40 . FIG. 1. Scars, resonances and emergent conservation law. (a): m xDE /m x , the ratio of magnetizations after infinite (diagonalensemble average) and 0 (initial state) cycles versus drive frequency ω . Freezing, reflected in a large value of this ratio, occursover a broad range of ω , and is strongest at particular ‘scar’ points (marked with arrows) h xD = kω, where k is an integer(for h xD = −
40 here, the ten arrows marks ω = 40 /k ; k = 1 , , .., H (0) (which gives an initial magnetization m x (0) ≈ β = 10 − ( m x (0) ≈ .
05) for H I of the form H (0), but with h xD = 5 , and all other parameters the same as thedriven Hamiltonian, namely, J = 1 , κ = 0 . π/ , h x = e/ , h xD = 40 , L = 14. The sharp dips in the green lines representresonances, discussed in detail in the main text on Floquet-Dyson perturbation theory. Parameters are chosen to avoid theseresonances shown to be absent for large enough h xD (inset). (b): (cid:104) m x (cid:105) of the Floquet states plotted against the serial number(normalized by the Hilbert space dimension D H ) of the Floquet states arranged in decreasing order of (cid:104) m x (cid:105) . At the scar points( ω = 10 , , and 40) the (cid:104) m x (cid:105) values form steps coinciding with the eigenvalues of m x arranged and plotted in the same order: m x emerges as a quasi-conserved quantity, hence the freezing of m x for any generic initial state. described by H ( t ) = H ( t ) + V, where H ( t ) = H x + sgn(sin( ωt )) H D , with H x = − L (cid:88) n =1 Jσ xn σ xn +1 + L (cid:88) n =1 κσ xn σ xn +2 − h x L (cid:88) n =1 σ xn ,H D = − h xD L (cid:88) n =1 σ xn ,V = − h z L (cid:88) n =1 σ zn , (5)where σ x/y/zn are the Pauli matrices.The main result is that at large drive amplitude h xD , the longitudinal magnetization m x = 1 L L (cid:88) i σ xi (6)emerges as a quasi-conserved quantity under the drivecondition (‘scar points’ in the drive parameter space)given by h xD = kω, (7) where k are integers. Fig. 1 (a), main frame, shows thatat those scar points (marked with arrows), the diago-nal ensemble average m xDE (Eq. (3)) for m x is equal toits initial value m x (0) , to very high accuracy, indicatingthat m x remains frozen at its initial value for arbitrarilylong times. As seen from the figure, this happens for avery broad range of ω. The scar/freezing appears abovea sharp threshold value of h xD (Fig. 1 (a), inset). Thephenomenon is reminiscent of the non-monotonic peak-valley structure of freezing observed in integrable Floquetsystems in the thermodynamic limit .The figure shows that freezing happens for two verydifferent kind of initial states, namely, the highly po-larized initial ground state of H (0) as well as a high-temperature thermal state. The initial thermal densitymatrix we chose is of the form ρ Th ( t = 0) = L (cid:88) j e − βε j Z | ε j (cid:105)(cid:104) ε j | , (8)where | ε j (cid:105) is the j -th eigenstate of an initial Hamilto-nian H I , with ε j . We have chosen H I = H ( t = 0 , h xD =5 . , h x = 0 . , J = 1 , κ = 0 . , with H ( t ) from (Eq. (5)),and Z = (cid:80) j e − βε j is the partition function. This isa mixture of eigenstates | ε (cid:105) . Hence we obtain the finaldiagonal ensemble density matrix by taking the diago-nal ensemble density matrix for each | ε (cid:105) , weighted by itsBoltzmann weight in ρ Th (0) , i.e., ρ DE ( t → ∞ ) = (cid:88) j e − βε j Z (cid:32)(cid:88) k |(cid:104) ε j | µ k (cid:105)| | µ k (cid:105)(cid:104) µ k | (cid:33) = (cid:88) k (cid:88) j e − βε j Z |(cid:104) ε j | µ k (cid:105)| | µ k (cid:105)(cid:104) µ k | . (9)The quasi-conservation of m x for a generic thermalstate suggests that all the Floquet states must be orga-nized according to the emergent conservation law. Thisis shown to be true in Fig. 1 (b), which shows the expec-tation value (cid:104) m x (cid:105) in the Floquet eigenstates (correspond-ing to the drive in Fig. 1 (b)), plotted against their serialnumber (normalized by the dimension D H of the Hilbertspace), arranged in decreasing order of their (cid:104) m x (cid:105) values.For the scar points given by h xD = 40 and ω = 10 , , , the values of (cid:104) m x (cid:105) of the Floquet states coincide withthe eigenvalues of m x , indicating that all the eigenstatesof m x which participate in constituting a given Floquetstate have the same m x eigenvalues. This explains con-servation/freezing of m x for dynamics starting with anygeneric initial state. As we will see later, the condition ofthe scar (Eq. (7)) can be deduced both from the FDPTand a Magnus expansion in a time-dependent frame, andthe latter confirms the effect over the entire spectrumand explains the steps to the leading orders. B. Dynamics of the Unentangled Eigenstates of m x : Growth of Entanglement Entropy We define an unentangled, complete, orthonormal setof eigenstates of m x , which we will call the x − basis. Eachelement of the x − basis is a simultaneous eigenstate ofall the σ xi operators. The non-triviality of the dynam-ics at the scar points and the consequence of the quasi-conservation is manifested in the growth of the half-chainentanglement entropy E at the scar points, especiallywith different x -basis eigenstates of m x as initial states.We study the half-chain entanglement entropy E = − T r [ ρ log ρ ] , (10)where ρ is the density matrix of one half of the chain,obtained by tracing out the other half.The results are shown in Fig. 2. It highlights that,even though m x is conserved for large enough h xD at thescar points, there is substantial dynamics even at thosepoints. For large enough h xD , Fig. 2(d-i), we see thatdifferent eigenstates of m x evolve quite differently evenat the scar points, at which m x is conserved to a verygood approximation for all initial states. For example,for the fully polarized initial state entanglement does notgrow even after 10 drive cycles, but for the N´eel andthe L/ m x subspaces with maximaland zero magnetization. The growth of E also reflects the role of interactionsin the dynamics even at the scar points, without which wewould not see such a substantial growth of entanglement.In App. B, we show that the suppression of entangle-ment growth is robust in that it is observed for otherpatterns of drive field, as long as the concomitant emer-gent conservation law gives rise to well-defined sectorswhich contain only a small number of states. IV. STRONG-DRIVE MAGNUS EXPANSION
We next provide a modified Magnus expansion whichincorporates the large size of the drive from the start,using the (inverse of the) driving field as a small param-eter. This makes the emergence of a conserved quantitymanifest, for a wide range of Hamiltonians – the terms inthe time-independent part of the Hamiltonian that com-mutes with the time-dependent part of the Hamiltonian( H x here) can have any form. This is because the factorpre-multiplying the terms involving it, vanishes to secondorder regardless of its form. For example, it applies totransverse field Ising models in any dimension, with any Ising interaction. From this, one can immediately readoff the scars found above.The conventional Magnus expansion uses the inverseof a large frequency as a small parameter (see, e.g., )for obtaining the Floquet Hamiltonian H eff (Eq. (2)) asgiven below. H eff = ∞ (cid:88) n =0 H ( n ) F , where H (0) F = 1 T (cid:90) T dt H ( t ) ,H (1) F = 12!( i ) T (cid:90) T dt (cid:90) t dt [ H ( t ) , H ( t )] , (11)and so on. In our case, we have h xD > ω, making the se-ries non-convergent even when ω is greater than all othercouplings in the Hamiltonian, so that it is qualitativelywrong even at leading order: the first-order term H (0) is the time average over one period of H ( t ) (Eq. 5), aninteracting generic Hamiltonian which does not conserve m x . Hence we would have no hint of the scars from eventhe first-order term.This problem can be remedied when the strong driveis constituted of modulating the strength of a fixedfield/potential (the most natural way of employing a pe-riodic drive). The largest coupling ( h xD here) can beeliminated from the Hamiltonian by switching to a time-dependent frame as follows . We introduce a unitarytransformation | ψ mov ( t ) (cid:105) = W ( t ) † | ψ ( t ) (cid:105) , ˆ O mov = W ( t ) † ˆ O W ( t ) , (12)where | ψ ( t ) (cid:105) is the wave function and ˆ O is any predefined ... x ... ... x ... ... x ... (a) (b) (c) (d) (e) (g) (f) (h) (i) FIG. 2. (De)localization of the wave function over the x -basis (simultaneous eigenstates of all of the σ xi s) as evidencedby the half-chain entanglement entropy ( E ) vs system size L, for different driving strengths h xD (rows) and initial states(columns: left maximally m x polarized; middle: L/ m x ; right: N´eel state). Top row(small h xD = 5): E entropy grows linearly with system size for all initial states, signaling ergodicity. For stronger drives( h xD = 20 ,
40 in middle, bottom row, respectively), scars appear, and E depends strongly on the initial states, reflecting thesize of the emergent magnetization sectors: for the fully polarized initial states (left column), E does not grow at all for thefreezing/scar points (marked as (F) in the figure legends and represented by almost indistinguishably coincidental black andviolet triangles), while for the N´eel and the L/ E even at thescar points, reflecting (at least partial) delocalization over the large concomitant magnetization sectors. The results are for J = 1 , κ = 0 . , h x = e/ , L = 14, averaged over 10 cycles after driving for 10 cycles. operator (the subscript mov marks the quantities in themoving frame).The crux of the expansion is then apparent for a W ( t )of the following form, W ( t ) = exp (cid:20) − i (cid:90) t dt (cid:48) H D × r ( t (cid:48) ) (cid:21) , (13)where r ( t ) is T -periodic parameter. If the total Hamil-tonian were constant up to the time-dependent prefactor r ( t ), i.e. H ( t ) = r ( t ) × H (0), the above would justgive the solution of the static Schr¨odinger equation,but with a rate of phase accumulation for each (time-independent) eigenstate given by the integrand of the variable prefactor. In particular, any conservation lawof H (0) would be bequeathed to the time-dependentproblem. Now, if the drive is not the only, but stillthe dominant part, of the Hamiltonian, there will becorrections to this picture, but it suggests the eigenbasisof the drive and its conservation law(s) should remainperturbatively useful starting points.Given the form of H D ( t ) in Eq. 5 – the transformedHamiltonian reads H mov = W ( t ) † H ( t ) W ( t ) − iW ( t ) † ∂ t W , (14)where the second term exactly cancels the part from thefirst term which has h xD as its coupling, and hence H mov is free from any coupling of order h xD (see App. C fordetails). A. Scars in the Driven Interacting Ising Chain
In the case of Eq. (5), we have H ( t ) = H x + V − sgn(sin ( ωt )) h xD (cid:88) i σ xi . (15)Switching to the moving frame employing the transfor-mation in Eq. (13) gives H mov = H x − h z (cid:88) i [cos (2 θ ) σ zi + sin (2 θ ) σ yi ] , where θ ( t ) = − h xD (cid:90) t dt (cid:48) sgn(sin ωt (cid:48) ) . (16) After some algebra, we find the Magnus expansion of H mov to have the following leading terms: H (0) F = H x − h z h xD T (cid:34) sin (2 h xD T ) (cid:88) i σ zi + (1 + cos (2 h xD T ) − h xD T )) (cid:88) i σ yi (cid:35) . (17)Note that this is useful for h xD (cid:29) /T , the regimewe are interested in. The next-order term is yet morecomplex (see App. IV for a derivation of each term): H (1) F = 1 i T (Σ + Σ + Σ ) , (18)where the Σ i ’s are obtained by integrating the K i ’s, andare given byΣ = − h z [ H x , S z ] (cid:26)
14 ( h xD ) (2 cos (2 h xD T ) − cos ( h xD T ) − T h xD sin ( h xD T ) + T h xD sin ( h xD T ) (cid:27) . Σ = − h z [ H x , S y ] (cid:26)(cid:20)
12 ( h xD ) + T (cid:21) sin ( h xD T ) + T h xD [1 − cos h xD T ] (cid:27) . Σ = 0 . (19)The end result – a homogeneous expansion in the smallparameters 1 /h xD and 1 /T – given in Eqs. (17), (18) and(19) is quite remarkable.First, for h xD T = 2 kπ, H (0) F = H x , and H (1) F vanishes( k ∈ Z ), which is precisely the condition for scars ob-served numerically (Eq. (7)) and also from the FDPT(see Eq. (42)). Clearly, to this approximation, H eff doesnot only have a conservation law, but is also integrable,indeed classical, with all terms commuting. Numericalresults suggest that the above expansion (unlike the Mag-nus expansion in the static frame) is an asymptotic one,at least in the neighborhood of the scar points, since theleading order terms represent the exact numerical resultsaccurately.Second, it is clear from the forms of H (0) F and H (1) F that the results hold independently of the form of H x ;this could be in any spatial dimension, and can incor-porate any form of Ising interactions! This wide gen-erality implies that stable quasi-conservation laws andconstraints (in keeping with the possible asymptotic na-ture of the expansion) may emerge in generic interactingFloquet systems in the thermodynamic limit. V. FLOQUET-DYSON PERTURBATIONTHEORY
Here we develop a theory which opens up a windowon the otherwise difficult-to-access low-frequency regime.We first test it for an exactly soluble problem, and thenapply it to the Ising chain studied in the previous section.We find the theory provide valuable insights for bothsystems. In particular, it identifies a resonance conditioncorresponding to the dips, as well as a freezing condi-tion corresponding to the maxima in the response plot-ted in Figs. 4 and 1 respectively. A coincidence of thetwo accounts for the varying dip depths in that figure.While a comprehensive treatment of the general many-body problem is not yet possible, we believe that theseitems capture ingredients central for its understanding.We first present the general formulation of the FDPT.The goal is to construct the Floquet states | µ n (cid:105) . Thecentral ingredient is that the driven Hamiltonian H ( t ) = H ( t ) + V. (20)consists of a large time dependent term with a time-independent set of eigenvalues . This is appropriate forthe case of a strong driving field. The theory then treatsa small perturbation V which is time-independent .We thus work in the basis of eigenstates of H ( t ), de-noted as | n (cid:105) , so that H ( t ) | n (cid:105) = E n ( t ) | n (cid:105) , (21)and (cid:104) m | n (cid:105) = δ mn .Next, we assume without loss of generality that V iscompletely off-diagonal in this basis, namely, (cid:104) n | V | n (cid:105) = 0 (22)for all n . We will now find solutions of the time-dependent Schr¨odinger equation i ∂ | ψ n (cid:105) ∂t = H ( t ) | ψ n ( t ) (cid:105) (23)which satisfy | ψ n ( T ) (cid:105) = e − iµ n | ψ n (0) (cid:105) . (24)For V = 0, each eigenstate | n (cid:105) of H ( t ) is a Flo-quet state, with Floquet quasienergy µ (0) n = (cid:82) T dtE n ( t )(unique up to the addition of 2 pπ, where p is an integer).For V non-zero but small, we develop a Dyson seriesfor the wave function to first order in V . The drive ampli-tude h xD is the largest scale in H ( t ), and hence when wesay V is small, we mean | V /h xD | (cid:28) . V can otherwise becomparable to the other couplings of the undriven Hamil-tonian. In our ansatz, the n -th eigenstate is written as | ψ n ( t ) (cid:105) = (cid:88) m c m ( t ) e − i (cid:82) t dt (cid:48) E m ( t (cid:48) ) | m (cid:105) , (25)where c n ( t ) (cid:39) t while c m ( t ) is of order V (andtherefore small) for all m (cid:54) = n and all t .We find (for details of the algebra, see App. C: c m (0) = − i (cid:104) m | V | n (cid:105) (cid:82) T dt e i (cid:82) t dt (cid:48) [ E m ( t (cid:48) ) − E n ( t (cid:48) )] e i (cid:82) T dt [ E m ( t ) − E n ( t )] − . (26)We see that c m ( t ) is indeed of order V provided that thedenominator on the right hand side of Eq. (26) does notvanish; we will call this case non-degenerate. If e i (cid:82) T dt [ E m ( t ) − E n ( t )] = 1 , (27)we have a resonance between states | m (cid:105) and | n (cid:105) , and theabove analysis breaks down. Now, if there are severalstates which are connected to | n (cid:105) by the perturbation V , Eq. (26) describes the amplitude to go to each ofthem from | n (cid:105) . Up to order V , the total probabilityof excitation away from | n (cid:105) is given by (cid:80) m (cid:54) = n | c m (0) | attime t = 0. A. Single Large Spin: An Exactly Soluble Test-bed
As an illustration of the FDPT, we discuss a sys-tem with a single spin governed by a time-dependent Hamiltonian. We will briefly discuss some resultsobtained from the FDPT (which give the conditions forperfect freezing and resonances), numerical results, andexact results for the Floquet operator. The details arepresented in App. D.
Model:
We consider a single spin (cid:126)S , with (cid:126)S = S ( S +1),which is governed by a Hamiltonian of the form H ( t ) = − h x S x − h z S z − h xD sgn(sin( ωt )) S x . (28)The time period is T = 2 π/ω . Since sin( ωt ) is positivefor 0 < t < T / T / < t < T , theFloquet operator is given by U = e ( iT/
2) [( h x − h xD ) S x + h z S z ] × e ( iT/
2) [( h x + h xD ) S x + h z S z ] . (29)It is clear from the group properties of matrices of theform e i(cid:126)a · (cid:126)S , that U in Eq. (29) must be of the same formand can be written as U = e iγ ˆ k · (cid:126)S , where ˆ k = (cos θ, sin θ cos φ, sin θ sin φ ) . (30)We will work in the basis in which S x is diagonal. Sincethe eigenstates of U in Eq. (30) are the same as theeigenstates of the matrix M = ˆ k · (cid:126)S, the expectationvalues of S x in the different eigenstates take the valuescos θ times S, S − , · · · , − S . The maximum expectationvalue is given by m xmax = S cos θ . Analytical results from FDPT:
We can use the FDPTto derive the correction to m xmax to first order in the smallparameter h z /h xD . Namely, we find how the state givenby | (cid:105) ≡ | S x = S (cid:105) mixes with the state | (cid:105) ≡ | S x = S − (cid:105) .We discover that c (0) = √ S h z h xD e ih x T/ [ e ih xD T/ − cos( h x T / e ih x T − , (31)Three possibilities arise at this stage.(i) The denominator of Eq. (31) is not zero. Then theexpectation value of S x in this state will be close to S since h z /h xD is small. In addition, if the numerator ofEq. (31) vanishes, we get perfect freezing, namely, (cid:104) S x (cid:105) = S .(ii) The denominator of Eq. (31) vanishes, i.e., h x is aninteger multiple of 2 π/T , but the numerator does notvanish. This is called the resonance condition. Clearly,the perturbative result for c (0) breaks down in this case,and we have to either develop a degenerate perturbationtheory or do an exact calculation.(iii) Both the numerator and the denominator of Eq. (31)vanish. Once again the perturbative result breaks downand we have to do a more careful calculation.We would like to make a comment on the dependenceof the result in Eq. (31) on the value of S . At t = 0, theprobability of state | (cid:105) is | c (0) | and the probability ofstate | (cid:105) is 1 − | c (0) | . Hence the expectation value of S x /S is given by m xmax S = 1 S (cid:104) S (1 − | c (0) | ) + ( S − | c (0) | (cid:105) = 1 − (cid:18) h z h xD (cid:19) × ( h x T / − h x T /
2) cos ( h xD T / ( h x T / . (32)We expect Eq. (31) to break down at a sufficiently largevalue of S since it was derived using first-order per-turbation theory which is accurate only if | c (0) | (cid:28) m xmax /S inEq. (32) is independent of S . We therefore have thestriking result in this model that we can use first-orderperturbation theory for values of S which are not largeto derive an expression like Eq. (32) which is then foundto hold for arbitrarily large values of S . Numerical results:
Given the values of the parameters
S, T, h x , h z and h xD , we can numerically compute U andits eigenstates. From the eigenstates, we can calculate m xmax which is the maximum value of the expectationvalue of (cid:104) S x (cid:105) . In Fig. 3, we plot m xmax versus h x , for S = 20 , T = 10 , h z = 1, and (a) h xD = 40 and (b) h xD = 12 . π (cid:39) . h x equal to all integer multiples of 2 π/T . In Fig. 3 (b),we see large dips for h x equal to odd integer multiples of2 π/T , but the dips are much smaller for h x equal to even integer multiples of 2 π/T .We can understand these results using the FDPT. InFig. 3 (a), we have h xD = 40; hence cos( h xD T / (cid:54) = ± h x equal to all integermultiples of 2 π/T where the denominator of Eq. (31)vanishes (case (ii)). However, in Fig. 3 (b), h xD = 12 . π so that cos( h xD T /
2) = 1. Hence both the numerator anddenominator of Eq. (31) vanish when h x is equal to eveninteger multiples of 2 π/T (case (iii)). This explains whythe dips in m xmax are much smaller for h x equal to eveninteger multiples of 2 π/T , but they continue to be largefor h x equal to odd integer multiples of 2 π/T . Form of the Flouqet operator in different cases:
Wenow present expressions for the Floquet operator U inEq. (30) based on the exact results derived in App. D 1 a.The purpose of this exercise is to show that the form of U is quite different in cases (i-iii).Assuming that h xD is positive and much larger than | h x | and | h z | , we find, to zero-th order in h z /h xD , thatcos (cid:16) γ (cid:17) = cos (cid:18) h x T (cid:19) , and ˆ k = ˆ x, (33)provided that e ih x T (cid:54) = 1 (case (i)). Eq. (33) implies thatthe Floquet operator corresponds to a rotation about theˆ x axis by an angle γ . If e ih x T = 1, i.e., cos( h x T /
2) = ±
1, but cos( h x T / (cid:54) = e ih xD T/ , the denominator of Eq. (31) vanishes but thenumerator does not (case (ii), called the resonance con-dition). It turns out that we then have to expand up tosecond order in h z /h xD . This givesˆ k = cos (cid:18) h xD T (cid:19) ˆ z − sin (cid:18) h xD T (cid:19) ˆ y if cos (cid:18) h x T (cid:19) = 1 , = sin (cid:18) h xD T (cid:19) ˆ z + cos (cid:18) h xD T (cid:19) ˆ y if cos (cid:18) h x T (cid:19) = − . (34)This implies that the Floquet operator corresponds to arotation about an axis lying in the y − z plane. Thisimplies that the expectation value of S x will be zero inall the eigenstates of the Floquet operator.Finally, if e ih x T = 1 and cos( h x T /
2) = e ih xD T/ , boththe numerator and denominator of Eq. (31) vanish (case(iii)). We then discover thatˆ k = h x ˆ x − h z ˆ z (cid:112) ( h z ) + ( h x ) . (35)Hence, the Floquet operator corresponds to a rotationabout an axis lying in the x − z plane.To summarize, assuming that h z /h xD is small, we ob-tain quite different results depending on which of thethree cases (i-iii) arise. We see these differences both inthe numerical results for m xmax shown in Fig. 3 and inthe forms of the Floquet operator in Eqs. (33-35) whichare obtained by an exact calculation. mx max h x ( a ) mx max h x ( b ) FIG. 3. Plots of the maximum expectation value of S x versus h x , for S = 20 , T = 10 , h z = 1, and (a) h xD = 40 and (b) h xD = 12 . π (cid:39) . h x equal to all integer multiples of 2 π/T , while in figure (b)we see pronounced dips only when h x is equal to odd integermultiples of 2 π/T , as predicted by the FDPT result, Eq. (31). B. FDPT for the Interacting Ising Chain
Now we apply FDPT to our interacting Ising chain(Eq. (5)) studied numerically above. We set h z (cid:28) h xD , (a) (b) (c) FIG. 4. Freezing and resonances in the magnetization ratio m xDE /m x versus drive strength h xD . Observable, initial states atzero (panels a, b) and high temperature (inverse temperature β = 10 − ) (panel c) and drive parameters as described in Fig. 1a). Results shown for slow (a: ω = 0 .
4) and very slow (b, c: ω = 0 .
04) drives (green line-points). The resonances obtained fromfirst-order FDPT, Eq. (40), (purple vertical lines) show a remarkable match with the numerical values of dips in m x . (Somehigher order resonances are also visible at ω = 0 . and treat V as the perturbation. We use periodic bound-ary conditions.The eigenstates | n (cid:105) of H ( t ) are diagonal in the basisof the operators σ xn . In particular, the state in whichall spins σ xn = +1, will be denoted as | (cid:105) , and we startby calculating the Floquet state | m xmax (cid:105) (maximally po-larized Floquet state) obtained by perturbing this stateto first-order in h z /h xD . While calculating m x from theperturbation theory we use this Floquet state.The rationale for this is as follows. Firstly, if we startwith a fully polarized state in the + x direction (as isdone, for example, in the experiments by Monroe ), or,with the ground state of H (0) , with h xD (cid:29) h z , κ , then theinitial state is expected to have a strong overlap with thisparticular Floquet state; hence at very long times, theexpectation values of the observables in the wave functionwill be well approximated by the expectation value overthis Floquet state.Secondly, in this setting, the insights from the single-spin problem studied above are most directly transfer-able – in particular, we again encounter the ideas of res-onances and scars. With these in hand, we can then iden-tify a number of features present in the data more gen-erally, in particular for high-temperature states (whichare in turn of interest in the context of the NMR ex-periments by Rovny ). We find that the perturbationtheory works best in the vicinity of the scars with theiremergent integrability (see below), and present a limitedexploration of the performance of FDPT away from thesein App. D.For the expansion of the Floquet state to leading order,the computation proceeds entirely along the lines of thatpresented for the single spin. We denote the state inwhich all spins σ xn = +1 except for the site m where σ xm = − | m (cid:105) . In the limit in which h xD is much largerthan J, κ and h x , we find that, to leading order in h z /h xD , Eq. (D24) takes the form c m (0) (cid:39) h z h xD e iAT/ [ e ih xD T − cos( AT / e iAT − ,A = 4( J − κ ) + 2 h x . (36)The magnetization of this maximally polarized Flo-quet state is given as follows. The expectation value of (cid:80) Ln =1 σ xn in each of the m states is L − | (cid:105) is L . This gives m x = 1 − L L (cid:88) m =1 | c m (0) | . (37)
1. Resonances and stability of the scars
The resonance condition, Eq. (27),(36), e iAT = 1 where A = 4( J − κ ) + 2 h x , (38)signals the singularities of our expansion, where c m (0)naively diverges. For our Hamiltonian this occurs for h x = − J + 2 κ + pω , (39)where p is an integer.This suggests considering all possible first-order reso-nances based on Eq. (27), by considering the resonancecondition more generally: evaluating the change E m − E n due to the flip of only a single spin, σ ), with n -th near-est neighbor spins on the right/left denoted by σ ± n yieldsthe first-order resonance condition h x σ + Jσ ( σ − + σ ) − κσ ( σ − + σ ) = pω . (40)Here p ∈ Z denotes the number of photons involved inthe resonance. Of course, individual resonances may beabsent if there are no (net) matrix elements between thestates in question.0This approach can be rather successful at identifyingthe location of the numerically observed isolated reso-nances, as displayed in Fig. 4. There, the strength of thefreezing is displayed as a function of driving strength, forboth slow, and very slow, drives, ω = 0 . , .
04, respec-tively.The right panel of Fig. 4 emphasizes the generalityof this result: the considerations of the first-order res-onances obtained above yield the response even for theinitially weakly-polarized ( m x = 0 .
05) high-temperatureinitial state.Considering the expression for the magnetization,obtained from substituting the expression for c m (0)(Eq. (36)) into the expression of m x (Eq. (37)),1 − m x = 2 (cid:18) h z h xD (cid:19) × ( AT / − AT /
2) cos( h xD T )4 sin ( AT / , (41)we would like to make the following observations.Firstly, Eq. (41) indicates that m x should keep oscil-lating with h xD with a period ω (except when cos( AT / ω = 2 π/T is large, we can approximatecos( AT / (cid:39) − ( AT ) / AT / (cid:39) AT / − m x = 2 (cid:18) h z h xD (cid:19) − A T /
8) sin ( h xD T / A T . (42)This shows that freezing becomes weaker with increasing ω . An exception to this occurs when the numerator inEq. (42) vanishes, namely, when ω = h xD /k , where k is an integer. At these points, we have m x /m x (0) = 1,i.e., perfect freezing. Those are precisely the ‘scar’ pointsgiven by Eq. (7), where the peaks of freezing are obtainednumerically (Fig. 1).As encountered in the single spin example, there is aninteresting interplay between the scars – where no dy-namics takes place – and the resonances, where heatingis hugely amplified. When the two coincide, this can de-stroy the inertness of the scar point. This is manifestedas sharp dips in m xDE in the numerical results discussedabove, and for intermediate values of h xD in the inset ofFig. 1 (a). The FDPT predicts isolated resonances inparameter space and provides a guideline for choosingthe Hamiltonian parameters to avoid resonances and ob-serve stable scars. Our choice of parameters for Fig. 1 isguided by the theory (Eq. (40)), and we indeed observeresonance-free strong freezing at the scar points.It would clearly be desirable to embark on a more de-tailed study, both with respect to the role of higher-orderresonances (visible in the left panel of Fig. 4), and withregard to the statistics of the resonances as the systemsize increases. VI. CONCLUSIONS AND OUTLOOK
In conclusion, we have demonstrated that generic in-teracting Floquet systems subjected to a strong periodicdrive can exhibit scars, i.e., points in the drive parame-ter space at which the system becomes non-ergodic dueto the emergence of constraints in the form of a quasi-conservation law. This is captured by our strong-fieldMagnus expansion in a time-dependent frame. For lowdrive frequencies, we formulate a novel perturbation the-ory (Floquet-Dyson perturbation theory) which works,even at first order, very accurately at or near integrabil-ity of the scar points. In particular, the resonances pre-dicted by the theory accurately coincide with the sharpdips in the quasi-conserved quantity. At the resonances,the system absorbs energy without bound from the drive,and hence a scar ‘competes’ with the resonance. The res-onances predicted by the theory appear to be isolated inparameter space, and thus, the theory provides a guide-line for choosing parameters for observing resonance-freestable scars, as we demonstrate here. These results holdin particular for Ising systems in any dimension and withany form of the Ising interactions.Our work also touches on various Floquet experiments.In the original experimental work on Floquet many-bodylocalization, the interest of a large drive was alreadynoted. In the context of the studies of Floquet timecrystals, the two kinds of states studied above have alsoplayed a central role: the trapped ion experiment useda fully polarized starting state, while the NMR experi-ment employed a high temperature state.Our work points towards the important role in non-equilibrium settings played by the generation of emergentconservation laws and constraints, in contrast to onlyfocusing on those existing in the static (undriven) system,and their demise under an external drive. Our work alsoopens a door for stable Floquet engineering in interactingsystems, and indicates a recipe for tailoring interestingstates and structured Hilbert spaces by choosing suitabledrive Hamiltonians. Acknowledgments
AD thanks Subinay Dasgupta and Sirshendu Bhat-tacharyya for collaborating on a non-interacting versionof the phenomenon studied here. AD and AH acknowl-edge the partner group program “Spin liquids: correla-tions, dynamics and disorder” between IACS and MPI-PKS, and the visitor’s program of MPI-PKS for support-ing visits to PKS during the collaboration. This researchwas in part developed with funding from the DefenseAdvanced Research Projects Agency (DARPA) via theDRINQS program. The views, opinions and/or findingsexpressed are those of the authors and should not be in-terpreted as representing the official views or policies ofthe Department of Defense or the U.S. Government. RMis grateful to Vedika Khemani, David Luitz and Shivaji1 h x D mx DE
5 C y c l e s ( a ) - 8 - 7 - 6 W Fourier Weight W = 1 5 7 . 0 8 mDEx h x D - 3 - 2 - 1 Fourier Weight W W = 1 5 7 . 0 8 ( b ) FIG. 5. Periodicity in drive strength, h xD , of magnetization response (diagonal ensemble average m xDE , Eq. (3)). Top row showsperiodicity for both off-resonance (left, h x = − .
2) and on-resonance (right, h x = − .
21) drives. Other parameters, and initiallow-temperature state, as in Fig. 2b. In both cases the leading frequency of oscillations is Ω ≈ . ≈ π/ω , visible in thebottom panel, as predicted by Eq. (41). Sondhi for collaboration on related work . DS thanksDST, India for Project No. SR/S2/JCB-44/2010 for fi-nancial support. Appendix A: Finite-Size Analysis L . . . . . m x D E ω = 0 . ω = 10 . FIG. 6. The plot shows that m xDE is showing no percepti-ble L − dependence for two very different values of ω whenthe drive amplitude is large. The plot corresponds to driveHamiltonian 5 in the main text, with parameters J = 1 , κ =0 . π/ , h x = e/ , h xD = 40 , L = 14 . Here, we show that the numerical results exhibit nodiscernible finite size effect in m x (Fig. 6). Appendix B: Robustness of emergent conservationlaw with respect to variation of drive field
From Fig. 2 we note that the fully-polarized state isquite special – at the scar points not only its magnetiza-tion remains strongly frozen closed to unity, its entangle-ment entropy also does not grow. This is in stark contrastwith other x -basis states for which, though m x remainsconserved, entanglement entropy experiences substantialgrowth. This can be understood from the step-like struc-ture (Fig. 1) appearing at the scar points. We expect ω E UniformFieldDriveHalf − chain − up Half − chain − down Field Drive
10 1510 − − FIG. 7. Freezing the entanglement growth of the L/ x -basis state under half-up half-down field drive (Eq. B1).Fate of a fully polarized state under the same drive is alsoshown for comparison. The main frame shows E , while theinset shows m x , averaged over 10 cycles, after driving for 10 cycles. The results are for J = 1 , κ = 0 . , h x = e/ , h xD =40 , L = 14. this phenomenology to be present for other strong driveswhich divide up Hilbert space into sectors which are atmost weakly mixed as long as these sectors are separatedby finite gaps.We illustrate this by arresting the entangle dynamics ofthe L/ E under the drive with uniform longitudinal field(Fig. 2, middle column). Instead of a uniform field, wechoose the following drive Hamiltonian H D = − h xD L/ (cid:88) i =1 σ xi + h xD L (cid:88) i = L/ σ xi , (B1)keeping the rest of the set-up same as given by Eq. (5).For H D of above form, L/ L/ ω = 8 ,
10 and 13 . · · · which are thescar points corresponding to the applied drive amplitude h xD = 40 , while substantial growth of entanglement isobserved for the fully polarized initial state. This is instark contrast with the results for the uniform drive (leftand middle columns of Fig. 2). Appendix C: Strong-field Floquet expansion
Here, we provide the details of the derivation of theeffective Hamiltonian, Eqs. (17 - 19). Carrying out thePauli algebra gives H mov = H x − h z (cid:88) i [cos (2 θ ) σ zi + sin (2 θ ) σ yi ] , where θ ( t ) = − h xD (cid:90) t dt (cid:48) sgn(sin ωt (cid:48) ) . (C1)We note that the frame change does not affect m x , since itcommutes with W ( t ) . Now we do the Magnus expansionof H mov . We then find thatNow we consider the second-order term H (1) F = 12!( i ) T (cid:90) T dt (cid:90) t dt [ H ( t ) , H ( t )] . (C2)Arranging the terms in the commutator we get,[ H ( t ) , H ( t )] = K + K + K , where K = − h z { cos ( θ ( t )) − cos ( θ ( t )) } [ H x , S z ] ,K = − h z { sin ( θ ( t )) − sin ( θ ( t )) } [ H x , S y ] ,K = ( h z ) sin [ θ ( t ) − θ ( t )][ S z , S y ] , (C3)where S x/y/z = (cid:80) Li σ x/y/zi . Next we note that the integral in Eq. (C2) can be bro-ken in the following way. I [ f ( θ ( t ) , θ ( t ))] = (cid:90) T dt (cid:90) t dt [ f ( θ ( t ) , θ ( t ))]= I [ f ( θ ( t ) , θ ( t ))] + I [ f ( θ ( t ) , θ ( t ))] , + I [ f ( θ ( t ) , θ ( t ))] , where I [ f ( θ ( t ) , θ ( t ))] = (cid:90) T/ dt (cid:90) t dt [ f ( θ ( t ) , θ ( t ))] ,I [ f ( θ ( t ) , θ ( t ))] = (cid:90) TT/ dt (cid:90) T/ dt [ f ( θ ( t ) , θ ( t ))] ,I [ f ( θ ( t ) , θ ( t ))] = (cid:90) TT/ dt (cid:90) t T/ dt [ f ( θ ( t ) , θ ( t ))] . (C4)Finally, we note thatFor I ,θ ( t ) = − h xD t , θ ( t ) = − h xD t , For I ,θ ( t ) = − h xD ( t − T ) , θ ( t ) = − h xD t , For I ,θ ( t ) = h xD t , θ ( t ) = h xD t . (C5) Using Eqs. (C2), (C3), (C4) and (C5)) and evaluatingthe integrals, we finally get the results. Appendix D: Floquet-Dyson Perturbation Theory
We start from Eq. (23), which implies that i (cid:88) m ˙ c m ( t ) e − i (cid:82) t dt (cid:48) E m ( t (cid:48) ) | m (cid:105) = V (cid:88) m c m ( t ) e − i (cid:82) t dt (cid:48) E m ( t (cid:48) ) | m (cid:105) , (D1)where the dot over c m denotes d/dt . Taking the innerproduct of Eq. (D1) with (cid:104) n | and using Eq. (22), we find,to first order in V , that ˙ c n = 0 . (D2)We can therefore choose c n ( t ) = 1 (D3)for all t . We thus have | ψ n ( t ) (cid:105) = e − i (cid:82) t dt (cid:48) E n ( t (cid:48) ) | n (cid:105) + (cid:88) m (cid:54) = n c m ( t ) e − i (cid:82) t dt (cid:48) E m ( t (cid:48) ) | m (cid:105) . (D4)Hence Eq. (24) implies that the Floquet eigenvalue is stillgiven by µ (0) n = (cid:82) T dtE n ( t ) up to first order in V .Next, taking the inner product of Eq. (D1) with (cid:104) m | ,where m (cid:54) = n , we find, to first order in V , that˙ c m = − i (cid:104) m | V | n (cid:105) e i (cid:82) t dt (cid:48) [ E m ( t (cid:48) ) − E n ( t (cid:48) )] , (D5)so that c m ( T ) = c m (0) − i (cid:104) m | V | n (cid:105)× (cid:90) T dt e i (cid:82) t dt (cid:48) [ E m ( t (cid:48) ) − E n ( t (cid:48) )] . (D6)We now impose the condition on | ψ n ( T ) (cid:105) of Eq. (25)such that | ψ n (0) (cid:105) turns out to be a Floquet state, i.e.,(from Eq. (D4)) we must have ψ n ( T ) = e − i (cid:82) T dtE n ( t ) ψ n (0) , (D7)namely, we must have c m ( T ) = e i (cid:82) T dt [ E m ( t ) − E n ( t )] c m (0) (D8)for all m (cid:54) = n. Clearly, | ψ n (0) (cid:105) satisfying this conditioncan be identified as the Floquet state | µ n (cid:105) .3
1. Single spin model a. Model
We consider a single spin- S object which evolves ac-cording to the time-dependent Hamiltonian H ( t ) = − h x S x − h z S z − h xD sgn(sin( ωt )) S x . (D9)Since sin( ωt ) is positive for 0 < t < T / T / < t < T , where T = 2 π/ω , the Floquet operator isgiven by U = e ( iT/
2) [( h x − h xD ) S x + h z S z ] × e ( iT/
2) [( h x + h xD ) S x + h z S z ] . (D10)The group properties of matrices of the form e i(cid:126)a · (cid:126)S implythat U in Eq. (D10) must be of the same form and cantherefore be written as U = e iγ ˆ k · (cid:126)S , where ˆ k = (cos θ, sin θ cos φ, sin θ sin φ ) (D11)is a unit vector. We will work in the basis in which S x is diagonal; hence we choose the polar angles in such away that the x -component of ˆ k is equal to cos θ . Theeigenstates of U in Eq. (D11) are the same as the eigen-states of the matrix M = ˆ k · (cid:126)S . It is then clear that theexpectation values of S x in the different eigenstates takethe values cos θ times S, S − , · · · , − S . The maximumexpectation value is given by s max = S cos θ .An important point to note is that if the parameters h x , h z , h xD and T are fixed and only the spin S is varied,the values of γ and ˆ k in Eqs. (D11) do not change. Thismeans that if we can calculate these quantities for oneparticular value of S , the results will hold for all S . Inparticular, m xmax ≡ s max /S = cos θ will not depend on S . We have confirmed this numerically for a variety ofparameter values. b. Results from FDPT Next, we apply the perturbation theory developed inSec. V. Writing the Hamiltonian as H = H ( t )+ V , where H ( t ) = − h x S x − h xD sgn( sin ( ωt )) S x ,V = − h z S z , (D12)we can do perturbation theory to study how the stategiven by | (cid:105) ≡ | S x = S (cid:105) mixes with the state | (cid:105) ≡ | S x = S − (cid:105) . Following the steps leading up to Eq. (26), andusing the fact that (cid:104) | S z | (cid:105) = (cid:112) S/
2, we find that c (0) = √ S h z h xD e ih x T/ [ e ih xD T/ − cos( h x T / e ih x T − , (D13) c. Exact results It is instructive to look at the form of the Floquetoperator U in different cases. We first derive an exactexpression for U using the identity that if e iα ˆ m · (cid:126)S e iχ ˆ n · (cid:126)S = e iγ ˆ k · (cid:126)S , (D14)then cos (cid:16) γ (cid:17) = cos (cid:16) α (cid:17) cos (cid:16) χ (cid:17) − ˆ m · ˆ n sin (cid:16) α (cid:17) sin (cid:16) χ (cid:17) , ˆ k = 1sin ( γ/ (cid:104) ˆ m sin (cid:16) α (cid:17) cos (cid:16) χ (cid:17) + ˆ n sin (cid:16) χ (cid:17) cos (cid:16) α (cid:17) − ˆ m × ˆ n sin (cid:16) α (cid:17) sin (cid:16) χ (cid:17) (cid:105) . (D15)We can derive Eq. (D15) from Eq. (D14) for the case S = 1 / (cid:126)S = (cid:126)σ/
2. Eq. (D15) then follows for anyvalue of S due to the group properties of the matricesgiven in Eq. (D14).We will now use Eqs. (D14-D15) along with Eq. (D11)which can be written in the form α = T (cid:113) ( h xD − h x ) + ( h z ) , ˆ m = − ( h xD − h x ) ˆ x − h z ˆ z (cid:112) ( h xD − h x ) + ( h z ) ,χ = T (cid:113) ( h xD + h x ) + ( h z ) , ˆ n = ( h xD + h x ) ˆ x + h z ˆ z (cid:112) ( h xD − h x ) + ( h z ) , (D16)where we have assumed that h xD is positive and muchlarger than | h x | and | h z | .If e ih x T (cid:54) = 1, we can write the expressions in Eqs. (D16)to zero-th order in the small parameter h z /h xD to obtain α = T h xD − h x ) , ˆ m = − ˆ x,χ = T h xD + h x ) , ˆ n = ˆ x. (D17)Eqs. (D14-D15) then imply thatcos (cid:16) γ (cid:17) = cos (cid:18) h x T (cid:19) , and ˆ k = ˆ x. (D18)We thus find that the Floquet operator for the time pe-riod T corresponds to a rotation about the ˆ x axis.If e ih x T = 1, i.e., cos( h x T /
2) = ±
1, the denominatorof Eq. (D13) vanishes. If e ih xD T/ (cid:54) = cos( h x T / h z /h xD to find thatˆ k = cos (cid:18) h xD T (cid:19) ˆ z − sin (cid:18) h xD T (cid:19) ˆ y if cos (cid:18) h x T (cid:19) = 1 , = sin (cid:18) h xD T (cid:19) ˆ z + cos (cid:18) h xD T (cid:19) ˆ y if cos (cid:18) h x T (cid:19) = − . (D19)Hence the Floquet operator corresponds to a rotationabout an axis lying in the y − z plane. This impliesthat the expectation value of S x will be zero in all theeigenstates of the Floquet operator.If e ih x T = 1 and e ih xD T/ = cos( h x T /
2) = ±
1, both thenumerator and denominator of Eq. (D13) vanish. Wethen discover thatˆ k = h x ˆ x − h z ˆ z (cid:112) ( h z ) + ( h x ) . (D20)In this case, the Floquet operator corresponds to a rota-tion about an axis lying in the x − z plane.
2. FDPT for the Ising chain E m ( t ) − E ( t ) = 4( J − κ )+2 h x +2 h xD sgn(sin( ωt )) . (D21)We now use the notations and results from Sec. V to con-struct the Floquet state | ψ (0) (cid:105) obtained by perturbingthe unperturbed (Floquet) eigenstate | (cid:105) to first order in V given by ψ (0) = c | (cid:105) + L (cid:88) m (cid:54) =0 c m (0) | m (cid:105) = c | (cid:105) + √ L c m (0) | L − (cid:105) , (D22)where | L − (cid:105) ≡ √ L L (cid:88) m =1 | m (cid:105) (D23) is a translation invariant and normalized state in which (cid:80) m σ xm = L −
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