Erratum: Asymptotic work statistics of periodically driven Ising chains ournal of Statistical Mechanics: J Theory and Experiment

. We study the work statistics of a periodically-driven integrable closed quantum system, addressing in particular the role played by the presence of a quantum critical point. Taking the example of a one-dimensional transverse Ising model in the presence of a spatially homogeneous but periodically time-varying transverse field of frequency ω 0 , we arrive at the characteristic cumulant generating function G ( u ), which is then used to calculate the work distribution function P ( W ). By applying the Floquet theory we show that, in the infinite time limit, P ( W ) converges, starting from the initial ground state, towards an asymptotic steady state value whose small- W behaviour depends only on the properties of the small-wave-vector modes and on a few important ingredients: the time-averaged value of the transverse field, h 0 , the initial transverse field, h i , and the equilibrium quantum critical point h c , which we find to generate a sequence of non-equilibrium critical points value of c | | which is entirely determined by h i . The form of that singularity — Dirac delta derivative or square root — depends on h 0 being or not at a non-equilibrium critical point h * l . On the contrary, when u decays as a power-law for large u , leading to different types of edge singularity at = W 0 th . Generalizing our calculations to the case in which we initialize the system in a finite temperature density matrix, the irreversible entropy generated by the periodic driving is also shown to reach a steady state value in the infinite time limit.

1 The demonstration in appendix C, starting from the fourth line after equation C. 5 to the end, is incorrect. This does not change the main results, but modifies some details. Here is the corrected version: We see, from equation (7), that the symmetry relation  In general the coefficients a x , a y and a z are non-vanishing; they can vanish in some cases, giving rise to interesting phenomena which we will discuss later. Whenever the resonance condition equation (C.3) is not fulfilled (hence ≠ h 0), the second-order-in-k expansion of Floquet modes (expressed in the basis which applies to the case > h 0; these two states should be exchanged if < h 0. Moving now to the initial Hamiltonian ground state ψ k gs , the diagonalization of equation (7)   . This is generally true, up to special cases where there is coherent destruction of tunnelling (CDT) [48]: here = = a a 0 x y , and we fall back to equation (B.6). In the Supplemental material of [28] we show that, in the case of a sinusoidal driving ( ) Abstract. We study the work statistics of a periodically-driven integrable closed quantum system, addressing in particular the role played by the presence of a quantum critical point. Taking the example of a one-dimensional transverse Ising model in the presence of a spatially homogeneous but periodically timevarying transverse field of frequency ω 0 , we arrive at the characteristic cumulant generating function G(u), which is then used to calculate the work distribution function P(W ). By applying the Floquet theory we show that, in the infinite time limit, P(W ) converges, starting from the initial ground state, towards an asymptotic steady state value whose small-W behaviour depends only on the properties of the small-wave-vector modes and on a few important ingredients: the time-averaged value of the transverse field, h 0 , the initial transverse field, h i , and the equilibrium quantum critical point h c , which we find to generate a sequence of non-equilibrium critical points When ≠ h h

Introduction
In recent years, there have been many theoretical studies aimed at understanding the non-equilibrium dynamics of closed quantum systems [1,2], inspired by a series of experiments on cold atomic gases, which are nearly isolated systems with long phasecoherence times, allowing for the study of a coherent quantum dynamics over long time scales [3][4][5][6][7][8]. These experiments have paved the way to addressing many fundamental questions, such as the role of integrability in thermalization following a quench [1] or the universal scaling of the defects generated when a system is driven across a quantum critical point [2,9]. Usually, a quantum system is driven out of equilibrium by a slow ramping (annealing) or by a sudden quench of a parameter of the Hamiltonian (for example, the transverse magnetic field in the quantum Ising model discussed in this paper); the subsequent dynamical response of the system is encoded in several quantities, e.g. the Loschmidt echo [10][11][12][13][14], dynamical correlation functions [15], the growth of entanglement entropy [16], time evolution of observables [17][18][19] and dynamical response functions [20] following a sudden quench, or the change in diagonal entropy [21]. In parallel, in the context of the Jarzynski equality [22] and non-equilibrium fluctuation relations [23], the question of the emergence of thermodynamical laws from a finite quantum system driven out of equilibrium and the generation of irreversible entropy have been addressed in several recent works [24,25]. One of the ways to characterize the dynamics of an out-of-equilibrium quantum system is to explore the statistics of the performed work, both at zero [26][27][28] and finite temperatures [29]. Given the non-equilibrium nature of the driving protocol, the work (W ) is a stochastic variable and hence described by a probability distribution P(W ). Recently, the W = 0 work distribution function iˆ/ 0 2 f , which can be viewed as the probability of doing no work in a time t during a doublequench process, has been shown to display non-analyiticities as a function of t which can mark the existence of a sequence of dynamical phase transitions in real time [30]. Furthermore, the knowledge of P(W ) enables us to obtain information about some universal features, by connecting it to the critical Casimir effect, for sudden quenches ending near the critical point [27]; in particular, there exists a power-law edge singularity in P(W ), for small W, which is characterized by an exponent that is independent of the choice of protocol, but rather depends just on the initial and final values of the parameter being quenched [31]. It is usually convenient to define and work with the characteristic function G(u), obtained by Fourier transforming P(W ). The characteristic function G(u) has been shown to be closely related to the Loschmidt echo of a quenched quantum system, both at zero [10,26] and finite temperature [32].
In this paper we focus on a periodically driven integrable closed quantum system, namely the transverse Ising chain [33,34], with a spatially homogeneous but timeperiodic transverse field π ω = + h t h t ( ) ( 2 / ) 0 , and calculate P(W ) stroboscopically at the end of n complete periods τ π ω = 2 / 0 . It has been shown in the literature [35,36] that the system reaches a periodic steady state in the limit → ∞ n : the residual energy (which is in fact the first moment of P(W ))-and indeed essentially any local observablereaches a stationary value, when observed at times τ = t n n , in the thermodynamic J. Stat. Mech. (2015) P08030 limit. The system does not heat-up indefinitely, in spite of the driving, because the model is integrable, as discussed in [35]. We have observed a similar relaxation to a periodic steady condition also for a genuinely quantum non-local object, the so-called dynamical fidelity [37]. In the present work we will investigate what happens to G(u), and hence to P(W ), under such periodic driving in the asymptotic limit → ∞ n , and address the question of the universal behaviour emerging in the small-W region.
Working within the framework of the Floquet theory [38,39], and assuming that the initial state is a Gibbs state at temperature T, we provide an analytical form of the characteristic function G(u) in terms of Floquet quasi-energies and corresponding overlaps between the initial state and the Floquet eigenstates. G(u) is then used to arrive at an exact expression of P(W ) at zero temperature: we demonstrate that indeed P(W ) also tends to 'synchronize' with the periodic driving in the limit → ∞ n , converging to a well-defined asymptotic work distribution function P W ( ) ∞ , whose small-W behaviour depends only on the properties of the small-wavevector modes, ultimately controlled by the time-averaged value of the transverse field The non-equilibrium phase transitions we find are reminiscent of those discussed in [40], but are different in many aspects, notably in residing at low frequency, as we will discuss.
An important aspect of these results is that the small-W properties of P W ( ) ∞ are determined (as we are going to show in detail in the paper) exclusively by the small-k modes: the aspects of the dynamics independent of the details of the driving protocol rely only on the large wavelenght modes which encode the universal properties of the ground state in the static system [41,42]. Remarkably, a similar fact can be observed in two coupled Luttinger liquids undergoing a quantum quench [43]: if the quenched operator is relevant in the renormalization group sense (and then affecting mainly the long wavelength modes) then the phase coherence evolves with a universal scaling function.
Finally we move to the finite temperature case: starting from an initial mixed state at finite temperature, we will show that the irreversible entropy generated during the periodic driving tends to 'synchronize' with the driving for → ∞ n . It saturates to a steady state value for large ω 0 , displaying a sequence of dips and peaks for small and intermediate values of ω 0 , respectively. We note in passing a recent study on the work statistics of a periodically driven system which has explored the universal properties of the rate function, that is found to satisfy a lower bound and has a zero when W matches the residual energy [44].

J. Stat. Mech. (2015) P08030
The structure of the paper is as follows. In section 2 we summarize the basic definitions and properties of P(W ) and G(u). Next, in section 3 we focus on the case of a quantum Ising chain undergoing a uniform generic periodic driving, showing in section 4 that the stroboscopic G(u) and P(W ) converge to an asymptotic value (see appendix A for mathematical details). In section 5 we discuss the behaviour of the asymptotic P(W ) in the small-W regime, showing that it is independent of the details of the driving protocol; we substantiate our analysis with numerical results obtained in the case of a sinusoidal driving. In section 6 we discuss briefly the case with finite temperature T in the context of irreversible entropy generation, and in section 7 we draw our conclusions together with perspectives of future work. Technical details of our derivations are contained in three appendices.

The work distribution P(W ) and its characteristic function G(u)
We present here, for the readers' convenience, some basic facts about the work distribution function, following [23,45,46]. Suppose that a closed quantum system undergoes a time-dependent driving such that its Hamiltonian is H t( ), while the system was at the initial time t 0 = 0 in a given (possibly mixed) state ˆ(0) ρ . If p n (0) is the probability that the system has energy E n (0) at the initial time, and P mt n ( 0) f | the conditional probability that the system is observed to have energy E t ( ) m f at some later time t f , then the work distribution function is defined as: denote the zero-work Dirac delta, while the red arrow at W th is a way of hinting to a Dirac delta derivative singularity. h c denotes the equilibrium critical point, ) denote points where the Floquet spectrum is gapless.
With simple manipulations, and introducing the unitary evolution operator U t( , 0) f for the closed quantum system, it results that: , 0 e ,0 e 0 Tr e e 0 , , 0 is the final Hamiltonian in Heisenberg representation, and we have assumed that the initial state is such that H [ˆ(0),ˆ(0)] 0 ρ = (a Gibbs or micro-canonical state would do that). We note, in passing, that the quantum Jarzynski equality [22,25] follows immediately from the previous expression by taking β = u i and assuming an initial where F 0 and F f , correspond to the free energies of the initial and final equilibrium states, respectively.

The uniform periodically driven quantum Ising chain
Let us now specialize our discussion to the quantum Ising chain in a uniform timeperiodic transverse field (although some progress might be done in the general nonhomogeneous case). The Hamiltonian we consider is [41]: The Hamiltonian can always be recast in the form of a quadratic fermionic model, thanks to the Jordan-Wigner transformation [47]. Upon Fourier transforming in space to the relevant Jordan-Wigner fermions, we get: where the × Notice that the previous Hamiltonian is really only a part of the total = + + − H H Ĥˆˆ, i.e. the part living in the subspace with even fermion-parity, for which anti-periodic boundary conditions (ABC) apply, and k n L . This is certainly enough for describing the ground state and the dynamics starting from the ground state. At finite temperature, the contributions due to the extra odd-fermion-parity term, − Ĥ , corresponding to periodic boundary conditions k-values, are automatically accounted for when we transform the sum over k into an integral over the half of the Brillouin zone π ∈ k [0, ]. We assume that the system is, at time t = 0, in the Gibbs ensemble at temperature T for the Hamiltonian Ĥ(0). Following a standard procedure [35][36][37], the initial problem is diagonalized by introducing new fermionic operators with corresponding Floquet quasi-energies µ ± k ; this must be done, in general, by a numerical integration of the Schrödinger equation, for each k, over a single period [35,37]. The outcome of that calculation provides the relevant overlaps r (0) , and, by unitarity, (In principle, one could also consider τ δ + G n t , but that requires the knowledge of the full Floquet modes i.e. the time-periodic part of the Floquet states, while for δ = t 0, it is enough to know ) With these basic ingredients, the derivation of a general expression of G u ( ) nτ for the uniformly driven Ising chain follows essentially the steps outlined in [37], generalized to an arbitrary finite temperature. The final result can be cast in the form: denotes the Fermi occupation function (observe that creating an excitation costs here an energy 2E k ) and Had we chosen to work with the Laplace transform of the P(W ), rather than with the Fourier transform G(u), we would have obtained a similar expression with the formal replacement → u s i . For T = 0 we would get which actually shows that the expression for G s (i ) is better behaved at T = 0, at the expense of having to perform an inverse Laplace transform to recover P(W ). In the thermodynamic limit, transforming the sum over k into an integral, we can finally write: This object (see also equation (15)) is the so-called cumulant generating function and coincides 6 with the expression of Smacchia et al [31]. We observe also that, in the limit → ∞ s , we recover the expression for g n , the logarithm of the dynamical fidelity n discussed in [37] (see equation (8) there). It is worth mentioning that τ n ( ) F has been found [37] to exhibit sharp non-analyticities as a function of τ when ⩽ ω π τ = J 2 / 4 0 ; whether these non-analyticities are connected to the time-periodic counterpart of the dynamical phase transitions studied in [28] is an open question at the moment.

Steady state of the work probability distribution P nτ (W ) for n → ∞
The question we are now going to address is if the probability distribution of the work P W ( ) t f tends to 'synchronize' with the periodic driving in the asymptotic limit → ∞ t f . When viewing at the system stroboscopically at the times τ = t n n (integer multiples of the period τ), what we want to understand is if τ P W ( ) n converges towards a well defined asymptotic work distribution for → ∞ n . The answer is positive, as we are now going to show. We know from previous work [35] that the quantum average of the work (i.e. the first moment of P W ( ) nτ ) indeed reaches a 'steady state' for → ∞ n , in the thermodynamic limit 7 . Moreover, as we have recently shown in [37], the δ W ( ) Dirac-delta part of the distribution P W ( ) ∞ (see, for instance, equation (22) below) alias the large-s limit -which corresponds to the dynamical fidelity-also reaches a well defined 'steady state' for → ∞ n : Our claim now is that all the cumulants of τ P W ( ) n reach such a 'steady state', and therefore so does the whole probability distribution. To see this, it is enough to show that the whole cumulant generating 6 The relevant quantity in [31] translates as follows in terms of our quantities: We also mention that the analysis of [31] identifies the large-s limit of G s L ln (i )/ n T 0 τ = with a 'surface' free-energy contribution of a 1-dimensional quantum system having a 2-dimensional classical counterpart which coincides with our g n . 7 Strictly speaking, when → ∞ L the P(W ) becomes narrower and narrower, on the scale of the average work ∼ W L, with fluctuations which scale as L. Nevertheless, when L is large but not ∞ and the approximation of having set L → ∞ in transforming the sum into an integral holds only until a certain finite time * t L ∼ , the question we are asking is meaningful, provided the 'steady state' is effectively reached before t*.
function, equation (12), reaches a steady state. With an argument which generalizes that of [37], whose details are given in appendix A, one can show that when → ∞ n this quantity tends to the stationary value where s r r ( ) 4 ( 1 e ).
The numerical results shown in figure 2 perfectly confirm this analytic prediction: there is a transient-whose details depend on the parameters, for instance on the average field h 0 -which then leads to an asymptotic result for → ∞ n given by the simple analytic expression in equation (13). Figure 2 also illustrates (bottom panel) a case in which the frequency ω 0 is such that ω ≈ J A (2 / ) 0 0 0 , where J 0 (x) is the 0th-order Bessel function, and there is coherent destruction of tunnelling [39,48]: observe that G s L ln i n ( ) τ has a very small magnitude for this value of ω 0 .
The cumulants of the asymptotic work distribution are obtained as: The first cumulant (which is the quantum average of the work performed) is given by: This coincides with the result we would obtain by evaluating gs which is quite easy to calculate directly. The second cumulant is the variance of the work distribution and is given by: We notice that the P(W ) tends to become narrower and narrower in the thermodynamic limit, as expected, because σ ∝ ∞ ∞ W L / 1/ .

Universal edge singularity at small W in P ∞ (W )
Inspired by the results of [31], we now discuss the behaviour of the asymptotic work probability distribution at small values of W, especially in connection with aspects which are independent of the details of the specific driving protocol. From a technical point of view, the small-W behaviour of ∞ P W ( ) is encoded in the large-s behaviour of ∞ G s (i ), which we can evaluate by means of equation (13). We will show that, indeed, the important ingredients are: (i) the value h i of the initial transverse field h(t = 0), and (ii) the value of the average field h 0 (and the frequency ω 0 ), determining if the Floquet spectrum shows a resonance, and is gapless, at k = 0, or not. Indeed, the value of h i determines the position W th of the singularity in P(W ) which we observe, while the form of this singularity is determined by the possibility that the Floquet spectrum has a resonance at k = 0, which is entirely determined by the time-averaged field h 0 (see equation (6) is obeyed. Although these dynamical critical points were already found, with sinusoidal driving, in a high frequency regime within the rotating wave approximation (RWA) [40], here we find them for a generic periodic driving h(t): details can be found in appendix C. The small-W universal behaviour of ∞ P W ( ) we describe below relies, in the end, only on the properties of the small-k modes, in particular on the small-k behaviour of | | + r k 2 in equation (14): we find that if the resonance condition in equation (18) is ful- otherwise The precise values of α and β depend on the specific form of driving, but the functional form of | | + r k 2 and the Floquet spectrum being gapless at k = 0 or not depend only on the fulfillment of equation (18) The inverse Laplace transform predicts that the small-W behaviour of P W ( ) ∞ is given by 9 which applies whenever W h h (2 ln 2) i c + | − |. In both expressions the constant a is given by: , exchanging the roles of r k | | + and | | − r k has no effect, as equation (13) is symmetric under such an exchange. 9 Remarkably, the form of the singularity in equation (22) for the asymptotic work-distribution function is the same found in [31] in the case of a generic quench starting and ending into the same paramagnetic or ferromagnetic case. This is exactly what we are doing in this periodic driving protocol.

J. Stat. Mech. (2015) P08030
where α 2 is such that for small k (see equation (20)). Equation (22) predicts an edge singularity in the asymptotic work distribution function at a precise value of W which is totally independent of the details of the periodic protocol (and even of the frequency) but depends only on the initial value h i of the field. The details of the protocol enter into the strength of the singularity (the coefficient a). We notice that the threshold | − | h h 2 i c is the energy which has to be provided to the system to generate an excitation in the k = 0 mode; moreover, also the form of the singularity is only determined, through the constant a, by the small-k Floquet modes, as detailed in appendix C. So, the behaviour at small W of the work distribution function is dominated by the modes of lowest energy: in retrospective, this is a very reasonable finding. We stress that the previous analytical expressions are approximations valid in a precise range of s or W. The only condition for the validity of these approximate formulas, as detailed in appendix B, is that the driving field does not start from the critical point value, i.e. ≠ h h i c and there are no resonances at k = 0 in the Floquet spectrum. We show some instances of the validity of equation (21) in the upper panel of figure 3, where we numerically evaluate as long as there is no coherent destruction of tunnelling, i.e. for a resonance of order l, we have that ω ≠ J A (2 / ) 0 l 0 (J l being the Bessel function of the first kind of order l [49]). The resulting P(W ), after inverse Laplace transform, has a very singular contribution ).
The lower panel of figure 3 shows plots of R s g s where θ is the Heaviside function. We notice that in [31] the authors find a very similar formula for P t (W ) (when the time t is finite) in an Ising chain undergoing showing the convergence towards a finite limit for large s, in agreement with equation (24). Notice that cases where there is coherent destruction of tunnelling fail to be described by such a formula.  ln which generalizes equation (16) to finite T and finite n. When → ∞ n , the rapidly oscillating term n cos (2 ) k µ τ gives a contribution that averages to zero, and we get: The statistical nature of the work for a finite system-evidenced by equation (1) . The heat transfer between the closed system and the bath being zero, the entire contribution to the entropy generation is in fact due to 〈 〉 W irr , and one can define the irreversible entropy generated as β ∆ = 〈 〉 S W irr i rr [22,24].  Since at stroboscopic times τ = t n n the Hamiltonian returns to the original value Ĥ(0), so that ∆ = F 0, one can define an irreversible entropy increase, in the limit → ∞ n , as In figure 5 we show that for a driving protocol ω = + h t t ( ) 1 cos( ) 0 , ∆S irr indeed saturates to a steady state value, like the residual energy [35], displaying a sequence of well defined dips and peaks as a function of ω 0 : in the small ω 0 regime, there are dips at certain frequencies for which J A (2 / ) 0 0 0 ω = , a consequence of the coherent destruction of tunnelling [48]. In the intermediate range, on the contrary, one finds peaks at ω = p 4/ 0 , with p integer, due to quasi-degeneracies in the Floquet spectrum [35].

Conclusion
In conclusion, we have studied a periodically driven transverse-field Ising model and analyzed the behaviour of the stroboscopic characteristic function G s (i ) nτ , and hence the stroboscopic work distribution function P W ( ) nτ , after n complete periods of driving. Our study establishes that, in the thermodynamic limit, τ P n indeed converges for → ∞ n towards a well defined steady state value ∞ P W ( ) which reproduces the exact asymptotic value of the first cumulant 〈 〉 ∞ W (i.e. the asymptotic value of the average work performed on the system) derived earlier [35]. In the limit → ∞ s , on the other hand, τ G s (i ) n reduces to the stroboscopic dynamical fidelity [37]. For large s, we are able to provide asymptotic analytical expressions for ∞ G s (i ) and, by means of inverse Laplace transforms, we can derive corresponding expressions describing the small-W behaviour of P W ( ) ∞ . The small-W properties of P W ( ) ∞ depend strongly on the fact that there is a static critical point h c , and any periodic driving induces further non-equilibrium critical points where the gap in the Floquet spectrum closes up. This finding is in line with the study reported in [40], where, however, the relevant regime was one of large-amplitude driving at large frequencies, and a rotating wave approximation was appropriate. Here, on the contrary, the exact resonances we find reside at low frequencies: for a fixed average field h 0 , at ω = − h h l 2( )/ 0 0 c . In any case, the form of the singularity in the work distribution turns out to be a useful detector of such non-equilibrium phase transitions. According to the way the external periodic driving field relates to these critical points we can observe different phenomena. The time-averaged value of the field h 0 and its initial value h i happen to be crucial. Whenever h i is different from the static critical point h c and h 0 differs from any non-equilibrium critical point (i.e. temperature case, we have shown that the irreversible entropy ∆S irr , obtained using the first cumulant of the finite temperature characteristic function, also synchronizes with the periodic driving for → ∞ n and converges to a steady state value for large ω 0 . Summarizing, we see a strong relationship between the features of P W ( ) ∞ , the existence of nonequilibrium quantum critical points and the way the driving field relates to them. This work is a first step towards the application of time-periodic probes to understand the existence of a quantum phase transition by looking at the work distribution function. In this sense, we are generalizing the very interesting works in [26][27][28]50], which refer to the case of a sudden quench. In this perspective, it is also interesting to see if it is possible to induce non-equilibrium phase transitions in systems without static transitions and how this influences the work statistics. Another possible direction is to see how the quantum driven system being regular or ergodic ( [51][52][53][54]) influences the work statistics. A lot of work still remains to be done, starting from the consideration of other tractable cases, like the Dicke model [50].
To that purpose, we first show that the second term inside the logarithm is <1. We know that ⩽ µ τ n sin ( ) 1 where we have exchanged the integral and the sum over m, due to the dominated convergence theorem. We then write a binomial expansion of the sine term in terms of exponentials: The sum over j, for each m, has a finite number of terms: there is no problem in exchanging the integral over k with this sum. We now observe that the ≠ j m terms contain rapidly oscillating factors and vanish in the limit → ∞ n , thanks to the Riemann-Lebesgue lemma and the smoothness of the factors ξ k m . Hence, in the limit → ∞ n , we retain only the j = m terms and write which can be integrated to give The expansion is questionable whenever there are k-points such that ξ = 1 k . We will see below that this is indeed the case at k = 0 whenever the field oscillates around a non-equilibrium critical value given by equation (18) and the Floquet spectrum is resonant in k = 0. Even restricting ourselves to non-resonant cases, we would possibly find k-points where ξ = 1 k 0 , but this time with k 0 > 0. This would still seem to be an issue at first glance: indeed, near these points we would expand ξ k quadratically, . This is indeed a convergent expression. More importantly, when ≠ k 0 0 the singularities needing such a special treatment are in a region where the integrand is exponentially smaller, due to the prefactor − e sE 2 k 0 , then the main contribution comes from the k = 0 region, which we are now going to analyze.
To upper integration limit to ∞ and expanding the factor multiplying − e sE 2 k to the lowest order in k. But here the k = 0 point plays a tricky role. For a generic periodic driving where h(t) oscillates around the average value h 0 one quickly realizes, by focusing on the evolution operator U ( , 0) k τ for modes with a small k (as detailed in appendix C) that there are two cases:  So, restricting our consideration to non-critical initial fields, ≠ h h i c , and non-resonant Floquet spectrum, and performing the appropriate Gaussian integral emerging from equation (B.2), we finally arrive at: So, our theory predicts a square-root edge singularity in the asymptotic work distribution at a precise value simply the initial gap of the system, a value which is totally independent of the details of the periodic protocol (and even of the frequency), which only enter into the prefactor a.
Finally, let us briefly consider the remaining cases where the previous Gaussian analysis fails. There are, essentially, two cases left: The resulting P(W ), after inverse Laplace transform, has a very singular contribution proportional to a Dirac delta derivative: where θ is the Heaviside function. But this does not exhaust all the possibilities: when h 0 = 0 we find that the leading asymptotics of G s (i ) ∞ is 1/s 3 rather than 1/s whenever the Floquet-resonant condition equation (18) is fulfilled. A thorough study of the gapless scenario is left to future studies.

Appendix C
In this appendix we prove the statements leading to the resonance condition equation (18)  In general the coefficients a x , a y and a z are non-vanishing; they can vanish in some cases, giving rise to interesting phenomena which we will discuss later in some detail. which applies to the case > h 0; these two states should be exchanged if < h 0. Notice that a x and a z do not enter at this order of approximation. Moving now to the initial Hamiltonian ground state ψ | k gs , the diagonalization of equation (7)  , which is a corollary of the Liouville's theorem [56].