To heat or not to heat: time crystallinity and finite-size effects in clean Floquet systems
Andrea Pizzi, Daniel Malz, Giuseppe De Tomasi, Johannes Knolle, Andreas Nunnenkamp
TTo heat or not to heat: time crystallinity and finite-size effects in clean Floquet systems
Andrea Pizzi, Daniel Malz,
2, 3
Giuseppe De Tomasi,
1, 4
Johannes Knolle,
5, 3, 6 and Andreas Nunnenkamp Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-Str. 1, 85748 Garching, Germany Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Straße 38, 01187-Dresden, Germany Department of Physics, Technische Universit¨at M¨unchen, James-Franck-Straße 1, D-85748 Garching, Germany Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems,University of Nottingham, Nottingham, NG7 2RD, United Kingdom
A cornerstone assumption that most literature on discrete time crystals has relied on is that homogeneous Flo-quet systems generally heat to a featureless infinite temperature state, an expectation that motivated researchersin the field to mostly focus on many-body localized systems. Some works have however shown that the stan-dard diagnostics for time crystallinity apply equally well to clean settings without disorder. This fact raises thequestion whether an homogeneous discrete time crystal is possible in which the originally expected heating isevaded. Studying both a localized and an homogeneous model with short-range interactions, we clarify thisissue showing explicitly the key differences between the two cases. On the one hand, our careful scaling anal-ysis confirms that, in the thermodynamic limit and in contrast to localized discrete time crystals, homogeneoussystems indeed heat. On the other hand, we show that, thanks to a mechanism reminiscent of quantum scars,finite-size homogeneous systems can still exhibit very crisp signatures of time crystallinity. A subharmonic re-sponse can in fact persist over timescales that are much larger than those set by the integrability-breaking terms,with thermalization possibly occurring only at very large system sizes (e.g., of hundreds of spins). Beyond clar-ifying the emergence of heating in disorder-free systems, our work casts a spotlight on finite-size homogeneoussystems as prime candidates for the experimental implementation of nontrivial out-of-equilibrium physics.
INTRODUCTION
In the past few years, a terrific amount of excitement hasbeen raised around discrete time crystals (DTCs) [1–25]. Inessence, DTCs are periodically driven systems characterizedby a subharmonic response at a fraction of the drive frequency,thus breaking the discrete time-translational symmetry of theunderlying equations [2–5]. These nontrivial time phenomenaare collective (or ‘many-body’), meaning that they cruciallyrely on the presence of infinitely many interacting elementaryconstituents, in complete analogy with, e.g., real-space crys-tals. In this sense, they extend the notion of quantum phase ofmatter to the non-equilibrium realm. According to the conceptof universality [26], the qualitative behavior of DTCs shouldrely, by definition, neither on the specific adopted model norspecific initial conditions. Rather, they should be robust to(weak) perturbations. At the same time, the breaking of timesymmetry should not just be a transient phenomenon, butrather persist up to infinite time, analogous as to how orderis maintained over arbitrary distances in a space crystal. Inother words, a DTC should maintain an infinite autocorrela-tion time in spite of perturbations.Among the plethora of contexts in which time crystallinephenomena have been investigated, the original and arguablymost studied one is that of a quantum spin chain with localinteractions. The remainder of this work focuses on such asetting. In these systems, time crystallinity is typically provenby making use of exact diagonalization of finite-size systems,its main diagnostics being(i) the exponential scaling of the lifetime of the subhar- monic response with system size [4, 12, 21] and(ii) the presence in the system’s spectral response of a peakrigidly locked to a subharmonic frequency, not shiftingunder perturbations [5, 8, 9, 12, 18–20, 27].These two diagnostics are complimentary and interconnected.The fact that they make use of dynamical indicators is verynatural, since the dynamics of local observables such as themagnetization (cid:104) σ zi (cid:105) ( t ) is accessible in experiments (in con-trast, e.g., to the eigenstates).Under the unitary time evolution of a periodic (Floquet)Hamiltonian, short-range interacting systems are generallyexpected to thermalize, that is, to reach a featureless state atlong times [28]. This expectation shifted the focus to disor-dered models, which have the promise to evade the fate ofthermalization through the mechanism of many-body local-ization (MBL) [29–31]. Indeed, it was shown that, under suit-able conditions, all the eigenstates of the MBL Floquet opera-tor come in pairs with quasienergy difference π , which under-lines ergodicity breaking with a period-doubled subharmonicresponse for virtually any physically relevant initial condition:a sufficient condition for the realization of a DTC [27, 32, 33].For these reasons, MBL systems legitimately gained a priv-ileged position among the DTCs’ candidates. The devil’s ad-vocate, however, would argue that there is a priori no rea-son why the remarkable π -pairing condition on the eigenstatesshould be necessary for the observation of a DTC. This sus-picion is motivated by the fact that physically relevant initialstates such as ground states of local Hamiltonians or experi-mentally accessible states occupy only a small corner of thefull Hilbert space. In the undriven setting, the lesson from a r X i v : . [ c ond - m a t . d i s - nn ] N ov quantum scars is in fact that, even in homogeneous many-body systems in which most of the eigenstates look com-pletely ergodic, there might still be special initial conditionsfor which the dynamics resembles that of an integrable point,and thermalization is not straightforward [34]. Similarly, ithas been argued that even homogeneous Floquet systems can,against the general expectation, evade thermalization [35–38],as it was for instance shown for finite-size systems in the con-text of so-called dynamical localization [39–41]. Therefore,if it is true that a natural definition of the DTCs is obtainedthrough the dynamics of local variables, rather than at the levelof the eigenstates, then it becomes legitimate not to take forgranted that homogeneous Floquet systems necessarily ther-malize, and to wonder whether DTCs may be possible in theabsence of MBL. With this challenge in mind, the devil’s ad-vocate would start testing the diagnostics (i) and (ii) on fami-lies of homogeneous Hamiltonians and initial conditions.Indeed, it turns out, this endeavour can be successful: thediagnostics (i) and (ii) on the duration and robustness of thesubharmonic response, respectively, work equally well in thedisordered and homogeneous scenarios, a fact that resultedin some papers claiming the existence of homogeneous (or‘clean’) DTCs [42, 43]. In these papers, time crystallinity issupported by analyses in complete analogy with those of thepioneering MBL DTCs, although convincing cases on why theexpected ergodicity should be broken are lacking. Very re-cently, a more compelling effort in trying to understand homo-geneous DTCs has been made in Ref. [44], where the authorsascribe time crystallinity to a scar-like mechanism preventingthermalization. The apparent success of the diagnostics (i) and(ii) for various homogeneous models appears confusing. Thequestion whether MBL truly is necessary for a DTC is alsoposed by the observation of signatures of DTCs (using diag-nostic (ii)) in experiments in which the role played by disorderand interaction range has remained unclear [18, 20].Here, we clarify this issue studying both an homogeneousand a MBL model. For finite-size systems L < ∞ , we showthat a robust and exponentially long-lived subharmonic re-sponse can emerge both in the presence and in the absence oflocalization, although its origin is different in the two cases.In MBL systems, this behavior is due to the well-known π -pairing mechanism involving all the Floquet eigenstates,whereas in homogeneous systems it is due to the π -pairing ofjust two eigenstates. These have a large overlap with the ini-tial condition in a way that resembles quantum scars [34]. Weargue that special care is needed when adopting the above di-agnostics (i) and (ii) to identify a DTC in a strict sense. In fact,it happens for the homogeneous case that not only the subhar-monic response is exponentially long-lived (i) and robust (ii),but that its magnitude is also exponentially suppressed in sys-tem size, therefore disappearing in the thermodynamic limit L → ∞ , a fact that has been overlooked in the past.Our study leads to a twofold conclusion. On the one hand,we confirm that most likely no such a thing as a homogeneousDTC exists (in a strict sense and in a quantum short-range set-ting). On the other hand we show that crisp signatures of time crystallinity in these systems are nonetheless beyond doubt.In particular, the widespread belief that thermalization shouldoccur over the timescale set by integrability-breaking terms[45] holds rigorously only in the thermodynamic limit, andvery large finite-size systems can possibly behave nontriviallyfor much longer times. We show that under certain rather gen-eral and natural conditions thermalization only occurs at verylarge system sizes (of, e.g., many hundreds spins). We there-fore argue that, although homogeneous DTCs may not existaccording to the strict mathematical definitions, strong timecrystalline signatures in large (but finite) homogeneous sys-tems deserve much more consideration than they had in thepast, as they are prime candidates for the observation of non-trivial dynamical phenomena.We emphasize that the phenomenology described here ismarkedly distinct from that of prethermal DTCs, which arecharacterized by subharmonic responses that are exponen-tially large in the driving frequency, rather than in system size[8, 16, 21, 46–48]. Indeed, in the thermodynamic limit we ex-pect our system to behave in a thermal fashion, with heatinghappening over a timescale ∼ /V , V being the magnitude ofthe integrability breaking terms. The time crystalline signa-tures described here are due to remarkable finite-size effectsand hold for very general (non-integrable) models.The remainder of this paper is organized as follows. First,we introduce the models. Second, we investigate the diagnos-tics (i) and (ii) and the subtleties of finite-size effects by meansof careful scaling analyses. By inspecting the spectrum of theFloquet operator, we then reveal the mechanisms at the ori-gin of the subharmonic responses in the two cases, and usea scaling analysis to explain why thermalization is ultimatelyexpected in the thermodynamic limit. We conclude discussingthe results and their implications for future research. MODELS
We consider a chain of L interacting spins / driven ac-cording to a binary protocol, as standard for DTCs. This pro-tocol consists of the alternation of two Hamiltonians H and H , resulting in the Floquet unitary operator U F = e − iH e − iH , (1)one cycle of the driving having period T = 1 ( (cid:126) = 1 ). In thefollowing, we focus on two very general models, one homo-geneous and one MBL. As for the homogeneous model, weconsider H = π L (cid:88) j =1 σ xj + δH , (2) H = 2 L (cid:88) j =1 σ zj σ zj +1 + δH , (3) FIG. 1.
Subharmonicity plateaus: a careful scaling analysis . We investigate the scaling of the subharmonicity Z ( t ) with system size, forboth the MBL (a,c) and the homogeneous (b,d) models, and considering an initial rotation of the spins of angles θ = 0 (a,b) and θ = π/ .In all cases, the subharmonicity Z ( t ) exhibits a crisp signature of time crystallinity: an exponentially long (in system size) plateau. In theMBL system, the pleateau’s height does not decay with system size (insets), underpinning its stability and persistence in the thermodynamiclimit L → ∞ . Contrary, in the homogeneous setting, the value of the plateau decays exponentially (insets), as can be better appreciated forlarger θ (c,d), pointing towards the onset of thermalization in the thermodynamic limit, and to the impossibility of an homogeneous DTCin the strict sense. The value of the plateau’s heights in (c,d) are obtained averaging Z ( t ) over time. In (d), the plateau begins at a time τ ∼ /V ∼ (wiggly line), V being the magnitude of the integrability breaking terms. For the MBL setting (a,c), results are averaged over100 disorder realizations (one realization is plotted in faded colors for L = 10 ), whereas results in the homogeneous case (b,d) are averagedover a decade-long moving time window (one original time trace is plotted in faded colors for L = 16 ). where σ x , σ y , and σ z denote the spin / Pauli operators. TheHamiltonian H describes an imperfect π -rotation, whereasthe ZZ coupling in H should make the subharmonic re-sponse robust to perturbations. The terms δH , are smallintegrability breaking perturbations, and read δH , = 2 (cid:88) ν = x,z J ν , L (cid:88) j =1 σ νj σ νj +1 + 2 (cid:88) ν = x,z h ν , L (cid:88) j =1 σ νj . (4)The parameter h x = π/ describes the main imperfection inthe π -rotation around the x axis, J z = 0 without loss of gen-erality, and the other coefficients are small ( ∼ . ) and, justto ward off any fine-tuning, integrability, or hidden symme-try, are drawn at random: J z ≈ . , J x ≈ . , h z ≈ . , J x ≈ . , h z ≈ . and h x ≈ . .As for the localized model, we consider instead H = π L (cid:88) j =1 σ xj + δH , (5) H = 2 L (cid:88) j =1 (cid:0) J z ,j σ zj σ zj +1 + h z ,j σ zj (cid:1) + δH , (6)where J z ,j and h z ,j are uniform random numbers in [ , ] and [0 , , respectively, and where as before δH , are small integrability breaking perturbations δH , = 2 L (cid:88) j =1 (cid:88) ν = x,z (cid:16) J ν (1 , ,j σ νj σ νj +1 + h ν (1 , ,j σ νj (cid:17) . (7)The parameter h x ,j = π/
40 + δh x ,j describes the main im-perfection in the π -rotation, J z ,j = h z ,j = 0 without loss ofgenerality, and all the remaining coefficients are uniform ran-dom numbers in [0 , . . Periodic boundary conditions areassumed for both models.As an initial condition, we take that considered by Else andcollaborators in Ref. [4], that is | ψ (cid:105) = e i θ (cid:80) Lj =1 σ xj |⇑(cid:105) , (8)where |⇑(cid:105) is the product state with all the spins polarized along z and θ is the angle of rotation of the spins around the x axis.Varying θ , we can probe an entire family of physically rele-vant initial conditions. Note, our results are not contingent onthe choice of Eq. (8), but rather hold for various families ofexperimentally relevant initial conditions, such as the groundstates of some standard families of homogeneous Hamiltoni-ans, see the Supplementary Information.Time crystallinity is investigated by means of two mainobservables. The first is used for the diagnostics (i) on theduration of the subharmonic response, may be called sub-harmonicity [10], and is defined at stroboscopic times t = , , , . . . as Z ( t ) = ( − t L L (cid:88) j =1 (cid:104) σ zj ( t ) (cid:105) , (9)where (cid:104) . . . (cid:105) denotes quantum expectation value and, in theMBL case, average over disorder realizations as well. Inthe presence of a period-doubled subharmonic dynamics,the spins rotate by an angle ∼ π at every Floquet period, (cid:104) σ zj ( t ) (cid:105) takes values ∼ +1 , − , +1 , − , . . . at times t =0 , , , , . . . , and Z ( t ) ∼ . The parameter Z ( t ) can be usedto track the degree of subharmonicity of the response in time,and a finite and positive Z ( t ) up to t → ∞ is a signatureof time crystallinity. By contrast, the relaxation of Z ( t ) to corresponds to an ergodic behaviour.The second observable, useful for the diagnostics (ii) on therobustness of the subharmonic response, is the Fourier trans-form of the magnetization ˜ m ( f ) = 1 M M − (cid:88) t =0 e πift L L (cid:88) j =1 (cid:104) σ zj ( t ) (cid:105) , (10)where M is the number of Floquet periods over which thetransform is computed. The presence in the spectral response ˜ m of a peak at a subharmonic frequency f = 0 . is a signatureof time crystallinity, whereas, by contrast, no such a peak isfound when the system behaves ergodically. RESULTS
Here, we present results obtained solving the models inEqs. (2, 3) and Eqs. (5, 6) using exact diagonalization tech-niques.
Diagnostics for DTCs – The subharmonicity Z ( t ) inEq. (9) is plotted in Fig. 1 to investigate the dynamics fromthe perspective of diagnostics (i). To start with, we consider avanishing initial rotation θ = 0 of the spins. Both in the MBLand homogeneous models, Z ( t ) exhibits a non ergodic plateauof height ∼ , whose length grows exponentially with systemsize, fulfilling the diagnostics (i) for time crstallinity. For thisphenomenology, the standard argument would be that, in thethermodynamic limit L → ∞ , the period-doubled response isexpected to extend up to infinite time, thus realizing a persis-tent DTC. Naively, one may think that this reasoning worksequally well in the MBL as in the homogeneous case, but inthe latter it is actually undermined by a subtle observation.After a more careful analysis, it turns out that the height ofthe plateau for the homogeneous model decreases with sys-tem size, a crucial observation that, perhaps because hiddenby time fluctuations, has not been reported before (to the bestof our knowledge). In principle, the value of the plateau maytherefore vanish in the thermodynamic limit, and the subhar-monic response completely disappear.In the homogeneous setting, the decay of the plateau heightis a warning sign, pointing towards thermalization in the ther- modynamic limit. However, the range of plateaus values ob-tained for θ = 0 is very limited ( ∼ ), and an extrapolationto L → ∞ is difficult. The decay of the plateau value isbetter appreciated for substantially larger perturbations of theinitial condition, such as for θ = π/ . In this case, the rangeof values ( ∼ ) is broad enough to allow more confidentconclusions on its scaling, that appear exponential. In strik-ing contrast with the homogeneous scenario, the MBL modeldoes not exhibit such a scaling (fluctuations of the plateau val-ues are just due to noise, and are expected to disappear for alarge enough number of disorder realizations). This confirmsthat, as expected, MBL systems can realize a robust DTC.In the thermodynamic limit, we expect the plateau to disap-pear and thermalization to occur over a timescale τ [markedwith a wiggly line in Fig. 1(d)]. We have checked that thistimescale is set by the integrability breaking terms and scalesas τ ∼ /V , V being the magnitude of the intergability break-ing terms in the Hamiltonian. 𝑓 MBL | m | ∼ 𝐿 = J = 0.05, 𝐿 = 1010.80.60.40.2 Homogeneous | m | ∼ 𝐿 = J = 0.1, 𝐿 = 16 𝐿 | m ( . ) | ∼ N o de c a y D e c a y (a)(b) 𝐿 | m ( . ) | ∼ Decay ∼ 𝑒 - 𝜂𝐿 FIG. 2.
Subharmonic peak: a careful scaling analysis.
For boththe MBL (a) and the homogeneous (b) settings, we consider the di-agnostics (ii) regarding the presence of a subharmonic peak in thesystem response. We plot the Fourier transform ˜ m ( f ) of the mag-netization computed over the first Floquet periods. In light grey,we report for reference the case of a small J z , for which the sub-harmonic response disappears. The robust subharmonic responsesare highlighted by a peak locked to the frequency . , in both cases.The difference between the homogeneous and the MBL scenarios isthat the magnitude of the subharmonic peak does and does not de-cay with system size. Indeed, in the absence of MBL (b) we observean exponential decay of | ˜ m (0 . | with system size, suggesting thedisappearence of the subharmonic peak in the thermodynamic limit.Here, we considered an initial rotation of the spins θ = π/ . The system’s spectral response, that is a standard probe fortime crystallinity and the focus of diagnostics (ii), is insteadinvestigated in Fig. 2, where we plot the magnetization Fouriertransform ˜ m ( f ) in Eq. (10). For both the MBL and the cleanmodels, we verify the hallmark of time crystallinity: the pres- Density
MBL Homogeneous
Original spectrum 𝜋 -rotated spectrumOverlap withinitial condition 10 -5 -15 -10 -20 Quasienergies10 -5 -10 -15 ... 〉 + ... 〉 ≈ ( ( ... 〉 - ... 〉 ≈ ( ( (a) (b) FIG. 3.
Floquet spectra and the origin of subharmonicity.
Polar plots of the eigenstates of the Floquet operator U F for both the MBL( L = 14 , a) and the homogeneous ( L = 19 , b) settings. The quasienergy and the overlap with the initial condition (for θ = π/ ) areused as angular and radial coordinates, respectively. The density of points is imprinted in the colorcode. Furthermore, some eigenstates areduplicated, rotated by a phase π , and plotted as circles. This way, two π -paired eigenstates are visually signalled by a dot centered in a circle.For graphical clarity, this duplication is only performed for the and eigenstates with the largest overlaps with the initial condition forthe MBL and the homogeneous scenarios, respectively. (a) In the MBL case, all the eigenstates (or at least the considered outer ones) are π -paired, as expected. (b) In the homogeneous case, only the two outermost eigenstates (that is, those with largest overlap with the initialcondition, highlighted with red arrows) are π -paired, whereas all the others are not. These two special eigenstates are approximately givenby √ ( |⇑(cid:105) ± |⇓(cid:105) ) , and their overlap with the initial condition determines the magnitude of the subharmonic response (e.g., of the plateaus’height in Fig. 1, or of the subharmonic peak in Fig. 2). ence of a peak rigidly locked to the subharmonic frequency f = 0 . that does not shift in the presence of small perturba-tions. The genuine many-body nature of the phenomenon isobserved for both cases, as the peak locks to the subharmonicfrequency only for a large enough interaction. Again, sub-tle finite-size effects are appreciated by taking a close look atthe scaling of the magnitude of the subharmonic peak: in thehomogeneous (MBL) setting, the subharmonic peak decaysexponentially (does not decay) in system size, which suggestsits disappearence (persistence) in the thermodynamic limit, incomplete analogy with Fig. 1. π -Pairing and the origin of the DTC behavior – The re-sults of Figs. 1 and 2 show that the diagnostics (i) and (ii)work equally well in the MBL and in the homogeneous set-tings, but that in the latter clean case the subharmonic re-sponse is exponentially suppressed in system size. To gaina clearer intuition into this matter, we first have to better un-derstand the origin of the subharmonic response in finite-sizesystems of the two types. To do so, in Fig. 3 we inspect thespectrum of the Floquet operator U F . For a localized sys-tem, we confirm the expectation that all the eigenstates comein pairs with quasienergy difference approaching π exponen-tially in system size L . This π -pairing condition is distinctiveof MBL DTCs, and is in striking constrast with non-localizedsystems. In the homogeneous setting, in fact, only two eigen- states are π -paired. Remarkably, these two special eigenstatesare also those with the largest overlap with the considered ini-tial condition, which explains why a subharmonic response isstill observed, and why its intensity strongly depends on theinitial rotation θ . By inspection, we find that the two spe-cial eigenstates are approximately given by √ ( |⇑(cid:105) ± |⇓(cid:105) ) ,the approximation becoming an equality in the integrable limit(obtained for a drive with perfect π -flips and no perturbations, J x,z , = h x,z , = 0 ). We remark that, although the presenceof a few non-thermal states may not be surprising in genericHamiltonians at low energies, the π -pairing of two of them inthe Floquet setting is.Since in the homogeneous case all the subharmonic re-sponse is ascribed to just two special eigenstates, it becomescrucial to study how their overlap with the initial conditionscales with system size L . From the analysis in Fig. 4, we findthat such a scaling is clearly exponential e − L/λ ( θ ) , the decayoccurring on a characteristic system size scale λ ( θ ) that de-pends on the initial rotation angle θ . On the one hand, thefiniteness of λ < ∞ suggests that, in the thermodynamiclimit L → ∞ , the overlap with the two special states van-ishes, and no subharmonic response survives at all, indepen-dent of initial condition. On the other hand, for θ (cid:47) . wefind that λ takes very large values, in the order of a few hun-dreds. In this case, the onset of thermalization is appreciatedonly at remarkably large system sizes, way beyond the reachof exact and even approximate methods (such as density ma-trix renormalization group [42]). For instance, for the con-sidered parameters, λ has a maximum of ≈ at θ ≈ . ,see Fig. 4. An insight into the scaling is provided by the inte-grable limit, in which with a straightforward calculation onefinds that the overlap approximately scales as [cos( θ/ L ,that is, λ ( θ ) = [ − θ/ − (details in the Supple-mentary Information). System size 𝐿 T w o l a r ge s t o v e r l ap s 𝜃 ∼ 𝑒 - 𝐿 / 𝜆 Initial rotation 𝜃 -3 -2 -1 S ys t e m s i z e sc a l e 𝜆 𝜃 / 𝜆 (a) (b) FIG. 4.
Scaling of the two largest overlaps.
For the homogeneoussetting, we investigate the dependence of the two largest overlaps,corresponding to the π -paired eigenstates, on the system size L andinitial rotation θ . (a) For all considered values of θ , the sum of thetwo overlaps decays as e − L/λ (exponential fits as dotted lines). (b)The characteristic system size scale λ of the decay is plotted versusthe initial rotation θ . Even for small θ , we find that λ never diverges,meaning that thermalization is eventually expected in the thermody-namic limit. The scale λ can take remarkably large values, up to afew hundreds, and is maximal for θ ≈ . . The scaling in the inte-grable limit is plotted for reference as a dashed line. DISCUSSION
In our analysis above, we showed that an exponentially longsubharmonic response robust to perturbations can emerge inboth the MBL and the homogeneous settings for finite-sizesystems. In the MBL case this behavior is due to a π -pairingmechanism involving all the eigenstates, whereas in the ho-mogeneous setting it is instead due to the π -pairing of justtwo special eigenstates. This mechanism, which is genuinelymany-body in nature, is reminiscent of quantum scars [34], inwhich a few anomalous eigenstates that have a large overlapwith the initial condition are responsible for weak ergodicitybreaking after a quantum quench (that is, in the absence of adrive). There are however at least two important differenceswith respect to quantum scars in the non-driven setting. Thefirst is that quantum scars are fragile to perturbations, whereasthe π -pairing of the special eigenstates that we observe hereis robust to perturbations of both the Hamiltonian and the ini-tial condition, as long as these are homogeneous. The secondis that the weak ergodicity breaking of quantum scars con-sists of a few oscillations before the ultimate onset of ther-malization, occurring at relatively short timescales, whereas the subharmonic response here is exponentially long in sys-tem size. Furthermore, if the persistence of weak ergodicitybreaking from quantum scars in the thermodynamic limit isstill an open question [49], the evidence suggests that in thislimit the subharmonic dynamics in our model is completelysuppressed.Indeed, our analysis suggests that the subharmonic re-sponse in homogeneous systems is a finite-size effect. The‘critical system size’ at which thermalization can be consid-ered to take place was identified in the fitting parameter λ . Itis worth noticing that the precise estimation of this critical sizeis a hard task, that may be undermined by even more severefinite-size effects. For instance, the analyses above were per-formed in a range of system sizes L for which the spectrumof H artificially splits into ergodic ‘minibands’ (see Supple-mentary Section III), and one may expect that the scaling be-havior could change abruptly at the critical L c at which theminibands of H merge [50].Here highlighted in the Floquet scenario, subtle finite-sizeeffects in many-body systems are known more generally topossibly occur in those many-body systems in which somelength scales, such as the correlation and localization lengths,are larger than the system sizes amenable to exact techniques.Most prominently, in the non-driven setting, the debate aroundthe existence and nature of the MBL phase has shown howfinite-size effects can lead to controversial or misleading con-clusions [51–54]. A well-known example is that of the An-derson model on random-regular graphs [55, 56], for whichthe existence of a metal-insulator transition has been provenanalytically [57], and the value of its critical point is knownwithin a few percents [58–60]. For this model, exact diagonal-ization on small systems points to an incorrect critical point,and a naive analysis could suggest the absence of the local-ized phase [53] and the existence of a highly debated criti-cal/multifractal phase [55, 56]. Our study draws the attentionon analogous subtleties in the context of periodically-drivensystems and DTCs.Finally, we remark that the surprising finite-size effectsstudied here are likely of broad applicability. Indeed, our spinmodel in Eqs. (2, 3) is general, and the observation of similarphenomenologies in previous studies on (non-integrable) sys-tems of hard-core bosons [42] and spinless fermions [44] sug-gests that analyses similar to ours may apply to quite generichomogeneous Floquet systems. CONCLUSIONS
In conclusion, we clarified the issue whether MBL is trulyneeded to evade heating in a Floquet and short-ranged sce-nario. This issue has been ultimately raised by the fact that,as we observed, the standard diagnostics for DTCs (definedfor finite-size systems) are fulfilled both in the MBL and inthe homogeneous settings. We clarified it by observing that,on top of these diagnostics, there is the fact that in the cleanscenario the subharmonic response is a finite-size effect, andits intensity (e.g., the magnitude of the subharmonic peak) de-cays exponentially in system size. This, to the best of ourknowledge, has never been clarified before. Our results leadto the confirmation that only MBL systems can realize a DTCaccording to its strictest definition, stable to perturbations andpersistent to infinite times in the thermodynamic limit.Nonetheless, what is remarkable is that the subharmonicresponse in homogeneous systems can be observed for manydecades already for relatively small system sizes (e.g., ∼ ),whereas its weakening is possibly observed only at muchlarger sizes (e.g., ∼ or perhaps even ∼ ). Thismismatch makes clean moderate-size systems a unique op-portunity for implementation. Indeed, nowadays quantumsimulators are typically limited to a few dozen elementaryunits [61, 62], and their coherence times are way below thetimescales (e.g., of drive periods) that are considered intheoretical works such as ours. If the important and funda-mental questions regarding the stability of a DTC in the lim-its L → ∞ and t → ∞ made MBL a necessity in theoriesdealing with strict mathematical definitions, this necessity isrelaxed in most experimental scenarios, in which the remark-able exponentially long subharmonic response could be ob-served even in the absence of MBL. Moderate-size clean sys-tems open therefore new avenues for the observation of timecrystalline signatures in experiments, and for technologicalapplications in the next generation of quantum devices.Finally, as a brief outlook for future research, it would bedesirable to develop a general theory of finite-size clean Flo-quet matter able to capture the phenomenology and scalingsobserved here. As argued in the above Discussion, we ex-pect our results to be of broad applicability, and an analytictheory could hopefully assess the generality of our conclu-sions on rigorous grounds. Moreover, further investigationshould clarify how the transition between a strict MBL-DTCand finite-size DTC signatures relates to the MBL transition.This question could be addressed, for instance, by interpolat-ing between the clean and MBL models in Eqs. (2, 3) andEqs. (5, 6), respectively. Acknowledgements.
We thank D. Abanin for insight-ful comments on the manuscript. We acknowledge supportfrom the Imperial-TUM flagship partnership. A. P. acknowl-edges support from the Royal Society. D. M. acknowledgesfunding from ERC Advanced Grant QENOCOBA under theEU Horizon 2020 program (Grant Agreement No. 742102).A. N. holds a University Research Fellowship from the RoyalSociety.
Author contributions.
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Khe-mani, arXiv preprint arXiv:2007.11602 (2020). upplementary Information for“To heat or not to heat: time crystallinity and finite-size effects in clean Floquet systems” Andrea Pizzi, Daniel Malz, Giuseppe De Tomasi, Johannes Knolle, and Andreas NunnenkampThese Supplementary Information are devoted to technical details and complimentary results, with focus only on the homoge-neous setting of Eqs. (2-4). More specifically, in Section I we compute the scaling constant λ in the integrable limit, in SectionII we consider a different family of initial conditions as compared to the main text, and in Section III we investigate the levelstatistics of the Hamiltonian H . I) INTEGRABLE LIMIT
Here, we study the integrable limit of Eqs. (2-4), that is, we consider H = π L (cid:88) j =1 σ xj , H = 2 L (cid:88) j =1 σ zj σ zj +1 , (S1)with h x = π and J z = 1 . In this case, the Hamiltonian H acts performing a perfect π -flip of the spins from |↑(cid:105) to |↓(cid:105) andviceversa, whereas H has no effect beyond adding a phase. In this simple integrable limit, it is straightforward to see that theeigenstates of the Floquet operator are given by | s, ±(cid:105) = | s (cid:105) ± | ¯ s (cid:105)√ , (S2)where | s (cid:105) is a product state of spins in the eigenstates |↑(cid:105) and |↓(cid:105) of the operators σ zj , and | ¯ s (cid:105) is its complimentary, with (cid:12)(cid:12) ¯ ↑ (cid:11) = |↓(cid:105) ,and (cid:12)(cid:12) ¯ ↓ (cid:11) = |↑(cid:105) . It is simple to verfy that the states | s, + (cid:105) and | s, ±(cid:105) have quasienergy difference π .When deviating from the integrable limit, the eigenstates and their quasienergies get perturbed, and the π -pairing condition isbroken. As we have shown in the main text, there is nonetheless a pair of eigenstates whose quasienergies difference remainsexponentially close to π . These eigenstates are those originating from |⇑ , ±(cid:105) = |⇑(cid:105) ± |⇓(cid:105)√ , (S3)and it becomes therefore important to understand how these overlap with the initial condition. We compute this overlap with astraightforward calculation. We recall that the initial condition is given by | ψ (cid:105) = e i θ (cid:80) Lj =1 σ xj |⇑(cid:105) = L (cid:79) j =1 e i θ σ xj |↑(cid:105) j = L (cid:79) j =1 (cid:18) cos (cid:18) θ (cid:19) |↑(cid:105) + i sin (cid:18) θ (cid:19) |↓(cid:105) (cid:19) j . (S4)The overlap between | ψ (cid:105) and |⇑ , + (cid:105) is given by |(cid:104)⇑ , + | ψ (cid:105)| = 12 |(cid:104)⇑| ψ (cid:105) + (cid:104)⇓| ψ (cid:105)| + · · · = 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) cos (cid:18) θ (cid:19)(cid:21) L + (cid:20) i sin (cid:18) θ (cid:19)(cid:21) L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≈ (cid:20) cos (cid:18) θ (cid:19)(cid:21) L , (S5)where the last approximation holds for small enough θ . The same result is obtained for the overlap of the initial condition with |↓ , −(cid:105) , and the sum of the two overlaps therefore reads |(cid:104)⇑ , + | ψ (cid:105)| + |(cid:104)⇑ , −| ψ (cid:105)| ≈ (cid:20) cos (cid:18) θ (cid:19)(cid:21) L = e − ηL , (S6)with η = 1 λ = − (cid:18) cos (cid:18) θ (cid:19)(cid:19) (S7)which is the result that we used in the main text. II) FURTHER EVIDENCES FOR THE ROBUSTNESS TO PERTURBATIONS OF THE INITIAL CONDITIONS
In the main text, we considered as initial condition the state that is obtained from the fully z -polarized state |⇑(cid:105) by applyinga global rotation around the x -axis via the unitary exp (cid:16) i θ (cid:80) Lj =1 σ xj (cid:17) , in Eq. (8). In this sense, the parameter θ played the roleof magnitude of the perturbation of the initial condition. Here, we show that the results of our work are not contingent on thischoice of initial conditions, but rather hold for perturbations of the state |⇑(cid:105) more in general, at least as long as these are stillinvariant under translations. In particular, we now investigate another kind of initial condition, that is the ground states of theHamiltonian H = L (cid:88) j =1 (cid:18) σ zj σ zj +1 + 110 σ zj + h x, σ xj (cid:19) . (S8)For h x, = 0 , the initial state is | ψ (cid:105) = |⇑(cid:105) . Varying h x, (cid:54) = 0 , the initial state | ψ (cid:105) changes. The perturbation of the initialcondition is therefore parametrized by the transverse field h x, . In Fig. S1 we show that the very same analyses of the main texthold for this family of initial conditions. In particular, we show that the subharmonic response is exponentially long-lived, thatit however decays with system size, that the speed of this decay is larger for larger perturbations h x, , that the phenomenologyis due to the π -pairing of two special scarred eigenstates, and that the system size scale over which the subharmonic response issuppressed can be remarkably large. Time 𝑡 S ubha r m on i c i t y Z ( t ) Time 𝑡 𝐿 = 8 𝐿 = 10 𝐿 = 12 𝐿 = 14 𝐿 = 16 𝐿 = 18 (a) (b)Perturbation of | 𝜓 〉 h x,0 = 0.3 Perturbation of | 𝜓 〉 h x,0 = 1 P l a t. ' s he i gh t Decay
12 16System size 𝐿 ∼ 𝑒 - 𝜂𝐿 ∼ 𝑒 - 𝜂𝐿 Decay P l a t. ' s he i gh t 𝐿
10 14
Overlapwith IC 10 -5 -15 -10 -20 Q ua s i ene r g y Density(c) System size 𝐿 T w o l a r ge s t o v e r l ap s h x,0 ∼ 𝑒 - 𝐿 / 𝜆 -3 -2 -1 S ys t e m s i z e sc a l e 𝜆 h x,0 / 𝜆 (d) (e) Perturbation of | 𝜓 〉 h x,0 FIG. S1.
Results for a different family of initial conditions.
We repeat some of the analyses of the main text considering as initial conditionthe ground state of the Hamiltonian in Eq. (S8). The plots in (a), (b), (c), (d), and (e) are in complete analogy with Fig. 1(b), Fig. 1(d), Fig. 3(b),Fig. 4(a), and Fig. 4(b) in the main text, respectively. The only difference from the main text, to which we refer for a detailed interpretation ofthe results, is that the perturbation of the initial condition from the completely polarized state |⇑(cid:105) is here parametrized in h x, , rather than in θ . III) LEVEL STATISTICS
The role of the perturbations in Eqs. (2-4) is that to break integrability. To better understand to what extent integrability isbroken, in Fig. S2 we investigate the level statistics and the density of states of the spectrum of H for the homogeneous setting. p . d .f. D en s i t y o f s t a t e s ( c oun t s ) FIG. S2.
Spectral properties of H . For L = 21 sites and in the homogeneous settings, we investigate the spectrum of the Hamiltonian H . (a) The distribution of the consecutive level spacings ratio resembles that of a Gaussian orthogonal ensemble (GOE), highlighting thenon-integrable nature of H . (b) Although non-integrable, the spectrum of H is divided into separate bands, a qualitative feature likely dueto finite-size effects. More specifically, we are interested in the distribution of the ratio between consecutive level spacings, that is r n = E n +1 − E n E n − E n − , (S9)with E n the eigenvalues of H sorted in increasing order. The statistics of r n follows a GOE (Poisson) law with probabilitydensity function P GOE ( r ) = r + r (1+ r + r ) / ( P P ( r ) = r ) ) if the Hamiltonian is (is not) chaotic [ ? ]. On the one hand, thestatistics of r n in Fig. S2(a) looks mostly chaotic. On the other hand, the energy levels in Fig. S2(b) are nonetheless organizedin bands, whose separation may be a finite-size effect. In non-driven dynamical scenarios (e.g., quantum quenches), it has beenargued that the presence of these bands makes finite-size effects particularly subtle and misleading [50], and it might be that thesame extends to the driven setting in which H is alternated with H , as considered in the main text. Indeed, one expects thatthe width of these bands increases with system size quicker than their separation, so that the bands touch at a critical system size L c (beyond the reach of exact diagonalization techniques). This touching may favour thermalization, and it is possible that thetrends observed in the scaling analyses of the main text may change abruptly at L = L cc