discrete time-crystalline response stabilized by domain
TRANSCRIPT
Discrete time-crystalline response stabilized by domain-wall confinement
Mario Collura,1 Andrea De Luca,2, ∗ Davide Rossini,3, † and Alessio Lerose4, ‡
1SISSA and INFN, via Bonomea 265, I-34136 Trieste, Italy2Laboratoire de Physique Theorique et Modelisation (CNRS UMR 8089),
Universite de Cergy-Pontoise, F-95302 Cergy-Pontoise, France3Dipartimento di Fisica dell’Universita di Pisa and INFN, Largo Pontecorvo 3, I-56127 Pisa, Italy
4Department of Theoretical Physics, University of Geneva,Quai Ernest-Ansermet 30, 1205 Geneva, Switzerland
(Dated: October 29, 2021)
Discrete time crystals have been surmised as genuine nonequilibrium phases of matter, featuring anemergent spatiotemporal long-range order that spontaneously breaks time-translational symmetry.Protecting time-crystalline order necessitates a mechanism that hinders the spreading of defects ofthe ordered background, such as localization of domain walls in disordered quantum spin chains.In this work, we establish the effectiveness of a different mechanism arising in clean spin chains:the confinement of domain walls into “mesonic” bound states. We consider translationally invariantquantum Ising chains periodically kicked at arbitrary frequency, and discuss two possible routesto domain-wall confinement: longitudinal fields and interactions beyond nearest-neighbors. Ourcentral tools are the construction of domain-wall conserving effective Hamiltonians and the analysisof the resulting dynamics of domain walls. We show that the symmetry-breaking-induced confiningpotential gets effectively averaged out by the drive, leading to deconfined dynamics and hence only toa parametric decrease of the order-parameter decay rate. On the other hand, we rigorously prove thatincreasing the range R of spin-spin interactions Ji,j beyond nearest-neighbors enhances the orderparameter lifetime exponentially in R, despite the non-existence of a long-range-ordered prethermalstate. We numerically confirm our theory predictions and their robustness using a combinationof matrix-product-state simulations for infinite chains and exact diagonalization for finite chains,and suggest their experimental relevance for Rydberg-dressed spin chains. Contrary to conventionalprethermal time crystals, the long-lived stability of spatiotemporal order identified in this workrelies on the nonperturbative origin of vacuum-decay processes, consisting in the fractionalizationof critical-size excited bubbles into pairs of deconfined topological excitations.
I. INTRODUCTION AND OVERVIEW
Because of the subtle role played by the temporaldimension, spontaneous breaking of time-translationalsymmetry has long escaped conclusive theoretical formu-lations [1–3]. A meaningful characterization of an ex-tended many-body system as a time crystal requires ro-bust stationary macroscopic oscillations, without a netexchange of energy with external devices [2]. A leapforward has been taken with the realization that cer-tain nonequilibrium setups [4–7] allow to circumvent theobstacles posed by thermal equilibrium [8, 9]. Discretetime crystals (DTCs) formed by interacting periodicallydriven quantum spin systems currently represent the the-oretical paradigm of large-scale spatiotemporal order-ing [2, 10]. Signatures of their stable and robust subhar-monic response to the drive have been experimentallyobserved with several state-of-the-art quantum simula-tors [11–17].
A major challenge to realizing a DTC is the fact thatthe external drive tends to repeatedly inject excitationenergy into the system, and the resulting heating gen-erally deteriorates large-scale spatiotemporal ordering.
∗ [email protected]† [email protected]‡ [email protected]
Protecting order against melting necessitates a mecha-nism to keep the impact of dynamically generated exci-tations under control and thus prevent indefinite entropygrowth. To date, many-body localization (MBL) [18–21] represents the single fully robust mechanism to stabi-lize a persistent subharmonic DTC response: The strongquenched disorder of a MBL system freezes the mo-tion of local excitations, thereby stabilizing long-rangeorder throughout the many-body spectrum of the sys-tem [22, 23]. On the other hand, the quest for disorder-free DTCs calls for alternative mechanisms to evade ther-malization. The crucial observation that energy absorp-tion from the drive is asymptotically suppressed for largedriving frequencies [24, 25] allowed Else et al. [6] to for-mulate the notion of a prethermal DTC as an extremelylong-lived (as opposed to permanently stable) dynamicalphase exhibiting a broken time-translational symmetry.This phase relies on the existence of an effective Hamil-tonian governing the transient many-body dynamics in asuitable rotating frame, possessing an emergent Abeliansymmetry determined by the driving protocol, which canget spontaneously broken at low enough effective temper-ature. As a consequence, it is guaranteed that a relevantset of initial states will display long-lived quasistationaryoscillations of the order parameter in the original frame.
Necessary ingredients for prethermal DTC behavior in-clude a high-frequency drive and an interaction struc-ture that supports a thermal phase transition; in one
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Sec. IV : γ ∼ aR ϵ2R+1
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FIG. 1. An illustrative summary of the theory predictions of this paper for the decay of the stroboscopic order parameterm(n) = 〈Zj(n)〉 ∼ e−γn at integer times tn = n in generalized kicked Ising chains. (a) In the standard transverse-field kickedIsing chain, the decay is driven by the fractionalization of isolated spin flips, generated by small imperfections of magnitude εof the kicking protocol, into pairs of unbound traveling domain walls. The exact decay rate provided in Eq. (7) is interpretedas γ ∼ ρv, with ρ ∼ ε2 the density of spin flips and v ∼ ε the spreading velocity of domain walls. The panel below reports arepresentative example [J = 0.685, ε = 0.2 in Eq. (6)]. (b) A tilt of the kick axis (green arrow in bottom left sketches) givesrise to domain-wall confinement, hindering the spreading of reversed bubbles as long as heating is suppressed. However, theperiodic flips average out the symmetry-breaking component of the kick, dynamically restoring the symmetry. As a result,the decay rate is only parametrically modified by the tilt, as shown in the panel below for increasing value of the longitudinalkick component [h = 0.2, 0.4, 0.6 in Eq. (25)]. (c) Couplings beyond nearest neighbors give rise to a form of domain-wallconfinement, while being completely insensitive to periodic flips. The order-parameter decay is only triggered by higher-orderprocesses in the kick perturbation ε. We rigorously establish that, for a generic choice of the couplings, the decay rate changesqualitatively to γ ∼ ε2R+1, and can be ascribed to the fractionalization of rare reversed bubbles as large as the interaction rangeR into pairs of travelling domain walls. This long-term stability of long-range order is due to the intrinsically slow approach tothe (non-long-range-ordered) Floquet-prethermal state, thus evading any obvious pseudo-equilibrium description. The panelbelow shows the strong enhancement of the order-parameter lifetime when weak additional couplings extend to a distance Rincreased to 2, 3, 4 [J2 = 0.144, J3 = 0.058, J4 = 0.03 in Eq. (33); note that ε� J2,3,4].
dimension, this requires fat-tailed long-range interac-tions [6]. A natural question is whether one can real-ize a robust long-lived DTC response exploiting differentmechanisms of thermalization breakdown, such as quan-tum many-body scarring [26–28], Hilbert-space fragmen-tation [29, 30], Stark [31, 32] and other kinds of disorder-free quasi-MBL [33–42]. A possibility that recently at-tracted considerable interest is provided by the confine-ment of excitations, an archetypal phenomenon of parti-cle physics [43] which also exists in low-dimensional con-densed matter models [44–48]. By now, there is mountingevidence and convincing understanding of robust non-thermal behavior in quantum spin chains with confinedexcitations [49–58], intimately related to characteristicphenomena of (lattice) gauge theories [52, 59].
In this paper, we explore the efficacy of confinement ofexcitations to stabilize a long-lived DTC response. Moti-vated by the main limitations of the theory of prethermalDTC, we focus our analysis on arbitrary driving frequen-cies and do not require fat-tailed long-range interactions.Our main result is that a version of prethermal time crys-tal can arise due to confinement, crucially not relying onFloquet prethermalization to an effective long-range or-dered Gibbs ensemble. This occurrence is made possi-ble by the intrinsic slowness of the dynamics generatedby the effective Hamiltonian, such that processes lead-ing to order-parameter meltdown only take place overextremely long timescales. In the following, we providea more detailed account of our findings, which are sum-marized and illustrated in Fig. 1.
3
We start (Sec. II) by exactly solving for the order-parameter dynamics of the integrable periodically-kickedtransverse-field Ising chain. We establish a possibly longbut perturbative lifetime γ ∼ ε3 of DTC response, whereε represents the deviation from a kicking protocol im-plementing perfect spin flips. Furthermore, we identifythe physical mechanism leading to order-parameter melt-down as the spreading of a small density ρ ∼ ε2 of dy-namically generated reversed spins, as the domain wallsdelimiting them freely move at velocity v ∼ ε [Fig. 1(a)].In passing, we note that this result resolves previous con-troversies on finite-size scaling of the DTC signal in thismodel [60, 61].
Building on this physical intuition, we introducedomain-wall confinement via symmetry-breaking longi-tudinal fields (Sec. III). Similarly to what happens inMBL and high-frequency driven spin chains [6, 7], weshow that symmetry-breaking terms are averaged out bythe drive, generating a deconfined effective Hamiltoniandespite domain walls being instantaneously confined atall times. As a result, the order-parameter lifetime isonly parametrically increased, and its perturbative na-ture γ ∼ ε3 remains unaffected [Fig. 1(b)].
We finally consider increasing the range of spin-spininteractions, Ji,j 6= 0 for |i − j| ≤ R, as an alternativeroute to domain-wall binding, which does not suffer fromincompatibility with the (explicit or emergent) Z2 sym-metry (Sec. IV). The crucial feature that arises in thesesystems is the coexistence of topological domain-wall-likeexcitations above non-topological confined bound statesin the spectrum of excitations. While the former solelyare responsible for the order-parameter decay, their dy-namical generation is heavily suppressed by their nonlo-cal nature. In fact, we can rigorously prove a qualitativeenhancement of the order-parameter lifetime γ ∼ ε2R+1
under mild genericity assumptions on the couplings Ji,j[Fig. 1(c)]. In other words, the fastest process leading toorder melting occurs at a perturbative order that growswith the interaction range R. Leveraging this result, wefinally conjecture that for algebraically decaying interac-tions Ji,j = J/|i − j|α the decay rate is asymptoticallysuppressed faster than any power of the perturbation,γ ∼ εAε
−1/(α−2) , for all α. (As α approaches 2 fromabove, however, the lifetime 1/γ is eventually supersededby the heating timescale.) This phenomenon places DTCbehavior stabilized by confinement on a similar footingas conventional prethermal DTCs, where the milder re-quirements to observe it come at the price of reducingthe set of initial states exhibiting DTC response. As abyproduct, our result may underlie the resolution of long-standing issues on the anomalous persistence of the or-der parameter in the quench dynamics of undriven quan-tum Ising chains with algebraically decaying interactionsJi,j = J/|i − j|α, α > 2 previously observed in severalnumerical studies [56, 62, 63].
II. ORDER PARAMETER DECAY IN THEKICKED TRANSVERSE-FIELD ISING CHAIN
Our starting point is the standard Ising Hamiltonianfor L spins on a ring, namely
HJ = −JL∑j=1
ZjZj+1, (1)
where J > 0 is the ferromagnetic coupling strength,Xj , Yj , Zj are the local spin-1/2 Pauli matrices for thejth spin, and periodic boundary conditions (j + L ≡ j)are assumed. The Floquet dynamics is obtained by theevolution governed by HJ , intertwined by sudden single-spin kicks K =
∏j Kj at integer times tn = n = 1, 2, . . . .
The resulting single-cycle time-evolution (Floquet) oper-ator reads
U = K VJ , VJ = eiJ∑
jZjZj+1 . (2)
To observe the simplest realization of time-crystallinespatiotemporal order, the system is initially preparedin the fully polarized state with positive magnetizationalong the z direction, namely, one of the two degener-ate ground states of HJ . This has a product-state form|+〉 ≡ | · · · ↑↑↑ · · · 〉, where |↑〉 (|↓〉) denotes the eigenvec-tor of the Pauli matrix Z with eigenvalue +1 (−1). Thekick K is taken to rotate each spin by an angle π arounda transverse axis: [64]
K −→ Kπ/2 = eiπ2
∑jXj . (3)
In this case, the time evolved state after n kicks |Φ(n)〉 =Un|+〉 exhibits a sequence of perfect jumps between|+〉 and the other ground state of HJ , namely |−〉 ≡|· · · ↓↓↓ · · ·〉. The persistent nonvanishing value of theorder parameter
m(n) = 〈Φ(n)|Zj |Φ(n)〉 (4)
in both space and time, being equal to (−1)n, gives riseto a trivial example of a quantum many-body time crys-tal. This behavior, however, relies on a fine-tuning ofthe kick strength, g = π/2. The existence of a nonequi-librium phase of matter exhibiting time-crystal behaviorrevolves around the stability of this spatiotemporal orderto arbitrary (sufficiently weak) Floquet perturbations inthe thermodynamic limit L→∞.
A. Exact decay rate in the thermodynamic limitand finite-size effects
The simplest perturbation to the above Floquet pro-tocol consists in performing imperfect spin flips, i.e.,
K −→ Kπ/2+ε = ei(π2 +ε)
∑jXj , (5)
4
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0.8
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ϵ = 0.2
0 50 100 150 2000.01
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ϵ = 0.4
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(−1)n
m(n)
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-0.5
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m(n)
ϵ = 0 . 1, ϵ = 0 . 2, ϵ = 0 . 4
FIG. 2. (a) Evolution of the order parameter under the Floquet Ising dynamics in Eq. (6), with J = 1 and different values ofthe kick strength ε. Data have been obtained by analytically solving the dynamics of the model, as explained in Appendix A.(b-d) Absolute value of the order parameter in logarithmic scale: thick lines are the same data as in panel (a) and are forL =∞. Shaded lines are the results of ED simulations for finite systems with different sizes L = 10, 15, 20, 25, 30, from lighterto darker colors. Dashed black lines denote the asymptotic exponential decay e−γn, with γ predicted by Eq. (7).
with ε 6= 0. Since the perfect kick Kπ/2 = iLP canbe factored out of K and is proportional to the globalZ2-spin flip operator P =
∏j Xj , the expectation value
of the local order parameter (4) over n periods can beexpressed as
m(n) = (−1)n〈+|[KεVJ ]−nZj [KεVJ ]n|+〉, (6)
where we used the properties PZjP = −Zj and [VJ , P ] =0, while Kε = exp(iε
∑j Xj). Equation (6) expresses the
fact that the absolute value of the magnetization evolvesas if it were governed by the Floquet operator Kε VJ withkick strength equal to ε, where the perfect kick has beencompletely gauged away by switching to a toggling frameof reference, leading to the multiplicative factor with al-ternating sign (−1)n.
The persistence of time-crystalline order is related tothe preservation of a finite absolute value of the local or-der parameter |m(n)| for large times n→∞. This ques-tion has been addressed in previous works investigatingfinite-size chains. In particular, the analysis of Ref. [60]led to a positive answer based on finite-size scaling ofthe order parameter obtained by exact diagonalization(ED) of short chains L . 20. This finding contradictsthe generic expectations of the absence of long-range or-der in excited states of one-dimensional clean short-rangeinteracting systems.
Here, by computing the exact dynamics of the mag-netization for an infinite chain using the integrabilityof the model, as detailed in Appendix A, we establishthat the order parameter decays exponentially in time as|m(n)| ∼ e−γn. The rate γ is found to be
γ = −∫ π
0
dp
π∂pφp ln | cos(∆p)|, (7)
where the quasiparticle spectrum φp and the Bogoliu-bov angle ∆p resulting from the diagonalization of thequadratic Floquet Hamiltonian are defined by the equa-tions cos(φp) = cos(2J) cos(2ε) + sin(2J) sin(2ε) cos(p)and cos(∆p) = cos(θp) cos(p) + sin(θp) cos(2ε) sin(p), re-spectively. The explicit expression of θp is given in Ap-pendix A. By Taylor expanding the exact result (7) forsmall perturbations ε, we find that the rate γ scales as
γ = 163π sin2(2J)
|ε|3 +O(|ε|5). (8)
Figure 2 reports the exact evolution of m(n) for in-creasing values of ε, for J = 1 (colored data sets). Inall cases, the resulting decay rate excellently reproducesthe analytical result (7) (dashed black lines). As it is ev-ident in Fig. 2(a), the decay remains quite slow even formoderate values of ε. Furthermore, in Fig. 2(b)-(d), wecompare the thermodynamic-limit evolution with that offinite chains of length L = 10 ÷ 30. This illustrates thecrucial role of finite-size effects, which determine an ap-parent persistence of a stable order parameter, detectedin previous works [60, 61]. Similarly to the case of staticquantum Ising chains [65, 66], these strong finite-size ef-fects can be attributed to a large overlap of the initialstate with the magnetized ground state of the integrableFloquet Hamiltonian.
B. Physical interpretation of the order-parameterdecay as domain-wall spreading
The scaling in Eq. (8) with ε can be understood inintuitive terms, considering that the system has exact
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quasiparticles which behave as non-interacting fermions.The dynamics in Eq. (6) is equivalent to a quench fromthe ground state of the classical Ising Hamiltonian (1),evolving with a kicked Ising chain deep in the ferromag-netic phase |ε| � J . In this case, the free fermionscan be interpreted as topologically protected excitations,i.e., domain walls (kinks and antikinks), interpolating be-tween the two degenerate magnetized ground states. Tothe lowest order in the kick strength ε, the quench createsa small density ρ = O(ε2) of spin flips, whose constituentpair of domain walls freely spread along the chain withmaximum velocity v = O(|ε|). The domain of reversedspins extending between a kink-antikink pair grows lin-early in time until one of them meets another domainwall initially located far away. The decay rate of theorder parameter is thus
γ ∼ ρv = O(|ε|3). (9)
Indeed, the exact result in Eq. (7) obtained from theasymptotic expansion of the determinant of a largeblock Toeplitz matrix [67, 68], precisely takes the formof a product between the quasiparticle group velocity|∂pφp| at momentum |p| and (for small ε) the numbersin2(∆p) ' − ln | cos2(∆p)| of excited quasiparticle pairswith momenta (p,−p) in the initial state, averaged overall momenta. This substantiates the intuitive interpreta-tion above.
The exact result thus quantitatively confirms the in-tuitive model of order parameter meltdown by spread-ing domain-wall pairs, illustrated in Fig. 1(a). Moreimportantly, this picture highlights what makes time-crystalline order doomed to melt in one-dimensional sys-tems: Preventing domain-wall pairs from unbounded sep-aration requires certain microscopic mechanisms, such asAnderson (many-body) localization induced by quencheddisorder. In the rest of this article, we will investigateunder which circumstances domain-wall confinement canprovide a robust stabilization mechanism.
III. DOMAIN-WALL CONFINEMENT ANDDECONFINEMENT IN THE KICKED
MIXED-FIELD ISING CHAIN
The previous section unambiguously illustrates howdomain-wall spreading underlies the time-crystal melt-ing in clean, locally interacting, spin chains. A cele-brated proposal to overcome this occurrence and protectlong-range order out of equilibrium hinges upon disorder-induced localization: in such case, domain walls behavelike particles moving in a random background, and spa-tial localization can arise from destructive interference,as first foreseen by Anderson [18], even in the presence ofmany-body interactions [19–21]. This basic mechanismof localization-protected order [22, 23] has been proposedto stabilize time-crystalline behavior for arbitrarily longtimes [4, 5]. Here we explore a different mechanism to
prevent domain-wall spreading, namely domain-wall con-finement.
In this section we briefly review the current under-standing of this multifaceted phenomenon (Sec. III A)and extend it to driven systems, exemplified by the spin-chain dynamics (2) subject to weak kicks K ' 1 aboutan arbitrary tilted axis. Specifically, we use the tools ofmany-body perturbation theory to reformulate the orderparameter-dynamics in terms of the motion of effectivedomain walls (Secs. III B) and demonstrate domain-wallconfinement (Sec. III C). Finally, in Sec. III D, we comeback to our main discrete time-crystal problem with kicksclose to perfect spin flips K ' iLP , and understand theorder-parameter melting, illustrated in Fig. 1(b).
A. Confinement in quantum spin chains
For the benefit of readers who may not be familiar withdomain-wall confinement in quantum spin chains, in thissubsection we provide a brief overview.
Particle confinement is a non-perturbative phe-nomenon arising in certain gauge theories, which con-sists in the absence of colored asymptotic states: all sta-ble excitations of the theory above the ground state arecolorless bound states of elementary particles [43]. An in-tuitive picture of this phenomenon is given by the forma-tion of a gauge-field string connecting a quark-antiquarkpair, the energy cost of which provides an effective con-fining potential that grows linearly with the spatial sep-aration between the two particles. As a result, the quarkand antiquark bind together into composite neutral par-ticles called mesons. When a large physical separationbetween them is enforced, the potential energy storedin the string becomes sufficient to produce another pairof particles out of the vacuum, which bind with the oldparticles to form two mesons, making the observation ofisolated quarks impossible.
An analogous confinement phenomenon naturallyarises for domain walls in quantum spin chains. Its mech-anism was proposed by McCoy and Wu in 1978 [44] andlater studied in a variety of theoretical [45–48, 69, 70]and experimental [71–73] works. The core ingredienthere is a first-order quantum phase transition, i.e., theexplicit lifting of a spontaneously broken discrete sym-metry. In the ferromagnetic quantum Ising chain (H =−J
∑j ZjZj+1−g
∑j Xj , with |g| < J), this can be sim-
ply realized by introducing a longitudinal field −h∑j Zj ,
which generates an energy penalty for the reversed mag-netic domain separating a pair of domain walls, analo-gous to a string tension. The energetic cost for separat-ing the pair thus grows proportionally to the distance,giving rise to a linear confining potential that fully sup-presses the spreading at arbitrarily high energies, bindingthe pair of “topologically charged” particles (kink and an-tikink) into “topologically neutral” bound states, referredto as mesons by analogy with particle physics. In certainquasi-one-dimensional magnetic insulators, similar effec-
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tive longitudinal fields are provided at a mean-field levelby inter-chain interactions; the resulting tower of mesonicexcitations (spinon bound states) has been spectacularlyobserved with inelastic neutron scattering [71–73].
Recently, it has been realized that confinement inquantum spin chains and in (1 + 1)-dimensional lat-tice gauge theories can be generically mapped onto eachother [52] via elimination/introduction of matter degreesof freedom exploiting the local constraints posed by gaugeinvariance [74, 75]. This substantiates intuitive picturesof the nonequilibrium dynamics of spin chains in terms ofprototypical phenomena in gauge theories, such as vac-uum decay [52, 59, 76, 77], string dynamics [50–53], andstring inversions [74].
Dynamical signatures of domain-wall confinement havebeen recently attracting a growing interest, starting fromRef. [49]. The suppression of domain-wall spreading sta-bilizes the order parameter even out of equilibrium. Thisstabilization also applies to the dynamics starting fromthe “false vacuum” magnetized against the longitudinalfield, which is a very atypical highly excited state withoutdomain walls [51, 52]. The basic explanation of this phe-nomenon is that domain-wall pairs excited on top of thefalse vacuum are also confined into “anti-mesons”. Fur-thermore, domain-wall pair production out of the falsevacuum is strongly suppressed, despite being energeti-cally allowed and entropically favorable, as it requires alocally created virtual pair to tunnel across a high-energybarrier. This effect, akin to the Schwinger mechanismin quantum electrodynamics [78], results in an exponen-tially long lifetime of the order parameter [52, 59].
B. Effective domain-wall dynamics for weak kicks
Here we show that analogous considerations ondomain-wall confinement carry over to driven quantumIsing chains, under the assumption that the periodic kicksare weak. We consider the spin chain evolving withthe Floquet operator in Eq. (2), with the simplest Z2-symmetry-breaking periodic kicks, i.e., a tilted rotationaxis away from the x− y plane:
K −→ Kε,h = ei∑
j(εXj+hZj) . (10)
We define ε = max(ε, h) to conveniently keep track ofthe global magnitude of the kick. For clarity, we stressthat here, we are considering a weak kick Kε,h, withε�1, in contrast with the almost perfect spin flip Kπ/2+ε,(compare with Eq. (5)) which is relevant for the stabilityof DTC.
The first problem that we encounter is related to thebreaking of integrability for h 6= 0: domain walls cease toexist as exact quasiparticles throughout the many-bodyFloquet spectrum, seemingly obscuring the physical in-terpretation of the magnetization dynamics. However,we can still make sense of an effective picture of domain-wall dynamics for weak perturbations, using the rigor-ous theory of prethermalization of Ref. [79]. Namely,
one can construct a sequence of time-periodic, quasi-localunitary transformations {eiεmSm}m=1,...,p, designed to re-move from the time-dependent Hamiltonian all the termsthat change the total number of domain walls
D1 ≡∑j
Dj,j+1 , Dj,j+1 ≡1− ZjZj+1
2 . (11)
The resulting transformed Floquet operator
Up = e−iεpSp · · · e−iεS1 Kε,hVJ e
iεS1 · · · eiεpSp (12)
conserves D1 up to terms of order p+ 1, i.e.,[Up, D1
]=
O(εp+1). The perturbative series generated by thisconstruction represents a non-trivial generalization ofthe well-known Schrieffer-Wolff transformation for staticHamiltonians [80–83]. The generators {Sm} take theform of sums of local operators, whose number and sup-port size grow proportionally to the perturbative orderm.
The construction of the transformation eiS≤p ≡eiεS1 · · · eiεpSp for arbitrarily large p aims at asymptot-ically producing an exactly domain-wall-conserving Flo-quet operator. However, the resulting perturbative se-ries is expected to have divergent (asymptotic) characterin general, similarly to the static case [83], suggesting alate-time breakdown of the conservation of D1 and aneventual heating to infinite-temperature. Nevertheless,the transformed picture still contains extremely usefulinformation on the transient dynamics. The analysis ofRef. [79] shows that, when J is strongly incommensuratewith the driving frequency 2π, [84] the breakdown of D1conservation —and hence heating— must be extremelyslow. In fact, the optimal (with respect to suitable oper-ator norms) truncation order depends on the magnitudeof the perturbation as p∗ ∼ C/ε3+δ, where δ > 0 is ar-bitrary and C = C(δ) is a constant. The very fact thatp∗ scales up with the smallness of ε itself yields a non-perturbatively small truncation error. In turn, this trans-lates into a quasi-exponentially long time window [79][85]
Tpreth ≥1ε
exp(
c
ε1/(3+δ)
)(13)
within which —for the purpose of computing the dynam-ics of local observables— the non-equilibrium evolutionof an initial state |ψ0〉 can be approximated as
Un |ψ0〉 ' eiS≤p∗ (U ′p∗)ne−iS≤p∗ |ψ0〉 , (14)
where U ′p∗ is the approximate Floquet operator obtainedfrom Up∗ by truncating terms beyond the order εp∗ . Byconstruction, U ′p∗ exactly conserves the number of do-main walls,
[U ′p∗ , D1
]= 0.
The physical consequence of this analysis is that the“bare” (i.e., unperturbed) domain-wall occupation num-ber Dj,j+1 on the bond (j, j + 1) acquires a perturbativequasi-local “dressing” e−iS≤p∗Dj,j+1e
iS≤p∗ for small per-turbations. In the integrable limit h = 0, the underlying
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algebraic structure of the model produces cancellationsto all orders which make this perturbative series conver-gent, leading to exact dressed domain-wall quasiparticles.As soon as h 6= 0, instead, these emergent domain wallsare expected to only be extremely long-lived, rather thaninfinitely stable (see also Refs. [50, 86]). Thus, as long aswe deal with dynamical phenomena occurring in the longFloquet-prethermal time window 0 ≤ t ≤ Tpreth, we canswitch to the effective picture where states and observ-ables are transformed via the unitary operator eiS≤p∗ ,and work therein with the effective domain-wall conserv-ing Floquet operator U ′p∗ , as expressed by Eq. (14).
As it results from the above discussion, we can analyzethe evolution of the order parameter by switching to thetransformed picture,
m(n) = 〈ψ0|U−nZjUn|ψ0〉 ' 〈ψ′0|(U ′p∗)−nZ ′j(U ′p∗)n|ψ′0〉 ,(15)
where |ψ′0〉 = e−iS≤p∗ |ψ0〉, Z ′j = e−iS≤p∗ZjeiS≤p∗ , and
the approximation, due to truncating Up∗ to U ′p∗ , holdsup to the long timescale in Eq. (13). In this transformedpicture, the number D1 of domain walls is an exact quan-tum number, and the Hilbert space fractures into sepa-rate blocks labelled by D1.
The perturbative construction introduced in Ref. [79]to prove the theorem leading to Eq. (13) is hardly man-ageable for explicit low-order computations. In prac-tice, we have found it more convenient to resort to acombination of the replica resummation of the Baker-Campbell-Hausdorff (BCH) expansion of Ref. [87] anda standard static Schrieffer-Wolff transformation (see,e.g., Ref. [83]). In both approaches, the existence ofa well-defined expansion requires incommensurability ofthe coupling J with 2π. This condition is necessary toensure that the domain-wall number uniquely labels theunperturbed sectors of the Floquet operator, and thusremains a good quantum number throughout the pertur-bative construction. We remark that while the schemeof Ref. [79] generally produces a time-dependent effec-tive Hamiltonian, the combined BCH resummation andSchrieffer-Wolff transformation produce a static effectiveHamiltonian, with presumably similar convergence prop-erties.
In Appendix B 1, we use the latter approach to derivethe expression of S and U ′ to lowest order p = 1, reportedhere:
U ′1 = e+i∑
jJ ZjZj+1+ε (P↑
j−1XjP↓j+1+P↓
j−1XjP↑j+1)+hZj ,
(16)
S≤1 = − ε2∑j
P ↑j−1[Xj − cot(2J)Yj ]P ↑j+1+
+ P ↓j−1[Xj + cot(2J)Yj ]P ↓j+1 . (17)
In Eqs. (16), (17), the operators P ↑,↓j represent local pro-jection operators onto the “up” and “down” states alongz of spin j. The off-diagonal processes in U ′1 are given
by the terms proportional to ε, which describe nearest-neighbor hopping of domain walls to the left or to theright. Processes which create or annihilate pairs of do-main walls have been removed from the evolution opera-tor U1 through the unitary transformation eiS≤1 ; in thisformulation, these processes only show up in the expres-sion of the transformed initial state |ψ′0〉 = e−iS≤1 |ψ0〉.
The general structure of the expansion and further de-tails are discussed in Appendix B 1. Higher-order termsgenerate further small corrections in the effective Flo-quet Hamiltonian. In particular, at order p, one hasterms in U ′p that correspond to at most p displacementsof one domain wall or more adjacent domain walls toa neighboring bond. For example, at second order onehas diagonal terms, plus nearest-neighbor hopping termsanalogous to those in U ′1, plus new terms proportionalto P ↑j−1(S+
j S+j+1 + S−j S
−j+1)P ↓j+1 (next-nearest-neighbor
hopping), P ↓j−1(S+j S−j+1 + S−j S
+j+1)P ↓j+1 (pair nearest-
neighbor hopping), and analogous flipped combinations.These additional terms do not modify the conclusions be-low. Likewise, the higher-order Schrieffer-Wolff generatorS≤p flips at most p neighboring spins.
C. Domain-wall confinement and order-parameterdynamics
Armed with the rigorously established picture of effec-tive domain-wall dynamics for weak kicks, we now discussdomain-wall confinement and its implications for the evo-lution of the order parameter. Here, as in the previoussubsection, we discuss the Floquet dynamics generatedby weak kicks U = Kε,hVJ . In the next subsection wewill go back to studying robustness of the DTC signalwith U = Kπ/2Kε,hVJ .
Equation (15) describes the evolution of the magne-tization in the transformed domain-wall picture. Thetransformed initial state |+′〉 consists of a low density oforder ε2 of flipped spins,
〈+|eiS≤1 Dj,j+1 e−iS≤1 |+〉 = ε2
2 sin2 2J+O(ε3) , (18)
as, to lowest order, this state is obtained by rotating thespins in |+〉 by an angle ε/ sin(2J) around a transverseaxis (note that each spin flip carries two domain walls).Furthermore, since Z ′j = Zj + O(ε), for the purpose ofunderstanding the nature of the evolution of m(n) (i.e.,persistent or decaying), we can drop the correction to Zj .
Since the initial state is composed of dilute tight pairsof domain walls, we can enlighten the resulting order-parameter dynamics by studying the two-body problem.The intuitive picture of the evolution of m(n) in terms ofthe motion of domain-wall pairs becomes asymptoticallyexact in such a low-density limit. The nature of this evo-lution (persistent or decaying order) depends on the effec-tive Floquet operator U ′ governing the motion of domainwalls. Equation (16) describes domain walls of “mass”
8
Hcm = 2hQ − 4ϵ cos P
⋮
⋮ ,
energy ΔE = 2J∑j
Dj, j+1 + 2h∑j
P↓j
⋯⋯
⋯⋯
D1 = 2
D1 = 4
(a) (b)
ℓ=�
ℓ=�
ℓ=�
ℓ=�
ℓ=��
0 10 20 30 400.0
0.2
0.4
0.6
0.8
1.0
r
|ψℓ(r)2
FIG. 3. (a) Sketch of the low-energy spectrum of the Floquet operator in Eq. (16) for ε → 0. The vertical axis representsthe relative eigenphase ∆E with respect to the polarized state |+〉. (b) A selection of mesonic eigenfunctions of the two-bodyproblem, cf. Eq. (21), for ε/h = 1.5.
2J , hopping to neighboring lattice bonds with amplitudeε, and experiencing a confining potential V (r) = 2hrwhich ties them to a neighbor at a distance r.
The two-body problem is obtained by projecting theeffective Hamiltonian H1, defined by the exponent of U ′1in Eq. (16), onto the sector spanned by two-particle basisstates |j1, j2〉, where the integers j1 < j2 label the posi-tions of two domain walls along the chain. The resultingtwo-body Hamiltonian is
H2-body =∑j1<j2
V (j2 − j1) |j1, j2〉 〈j1, j2|
− ε∑j1<j2
(|j1 + 1, j2〉+ |j1, j2 + 1〉
)〈j1, j2|+ H.c.
(19)
where the hard-core constraint |j1 = j2〉 ≡ 0 is under-stood. The second term represents nearest-neighbor hopsof the domain walls, whereas the first one acts as a lin-ear confining potential V (r) = 2hr as a function of thedistance r = j2 − j1 > 0. While for h = 0 one has acontinuum of unbound traveling domain walls, as soonas h 6= 0 the spectrum changes nonperturbatively to aninfinite discrete tower of bound states.
Due to translational invariance of the initial state|+′〉, domain-wall pairs are only generated with vanishingcenter-of-mass momentum K = 0. The relative coordi-nate wavefunction ψ(r) satisfies a Wannier-Stark equa-tion with a hard-wall boundary condition ψ(0) = 0, yield-ing [52, 88] the exact mesonic masses
E` = 2h ν`(2ε/h) ≡ −2h× {`-th zero of x 7→ Jx(2ε/h)}(20)
` = 1, 2, . . . , and wavefunctions
ψ`(r) = Jr−ν`(2ε/h), (21)
where J is the standard Bessel function. [89] For ε→ 0,one finds the energy levels E` = 2h`, corresponding to a
domain of ` reversed spins, ψ`(r) = δ`,r. Figure 3(a) re-ports a sketch of the low-energy spectrum of the FloquetHamiltonian in this limit. For finite ε/h, the eigenfunc-tions can still be adiabatically labelled by the integer `.Panel (b) reports a selection of mesonic eigenfunctionsψ`(r) in the center-of-mass frame, for ε/h = 1.5. Bound-ary effects are visible for small ` . 2ε/h, whereas forlarger ` the wavefunctions become essentially Wannier-Stark localized orbitals, i.e., rigidly shifted copies of eachother.
We can formulate a more intuitive analysis of the two-body dynamics, which will turn out to be fruitful laterto analyze the time-crystalline behavior. To this aim,we introduce the canonically conjugated operators Q,Pdefined by
Q =∑r
r |r〉 〈r| , eiP =∑r
|r + 1〉 〈r| , (22)
which correspond to the position and the momentum inthe center-of-mass frame, i.e., the distance between thetwo domain walls and their relative momentum; one ver-ifies [Q,P ] = i. [90] Using these variables, the center-of-mass frame Hamiltonian becomes
Hcm = 2hQ− 4ε cosP, (23)
where the domain is Q > 0 and a hard-wall boundarycondition at Q = 0 is understood. Classical trajectoriesare bounded in the Q direction and are translationallyinvariant away from the boundary Q = 0. Indeed, theHeisenberg evolution equations can be integrated exactlyin the bulk Q� 2ε/h (i.e., neglecting the boundary con-dition), which givesQ(t) =Q(0)
+ 2εh
[sin 2ht sinP (0) + (1− cos 2ht) cosP (0)
],
P (t) =P (0) + 2ht .(24)
9
|+⟩
|−⟩
h = 0
h = 0.1
h = 0.2
0 20 40 60 80 1000.0
0.2
0.4
0.6
0.8
1.0
m(n)
h = 0
h = 0.1
h = 0.2
0 20 40 60 80 100-1.0
-0.8
-0.6
-0.4
-0.2
0.0
n
m(n)
(b)
(a)
FIG. 4. Floquet dynamics of the order parameter induced bythe kick Kε,h, with ε = 0.2 and different values of h, in an infi-nite chain L =∞, obtained through iTEBD simulations. Thesystem is initially prepared either (a) in the “vacuum” |+〉,or (b) in the “false vacuum” |−〉. The order-parameter melt-down for h = 0 (dashed black lines) gets strongly suppressedby the presence of a small h 6= 0 (full lines).
Equation (24) indicates that the relative momentum Prevolves freely around the Brillouin zone, whereas therelative coordinate Q performs stable Bloch oscillations,remaining localized near the corresponding initial condi-tion. In physical terms, the mutual confining potentialcreates an effective Wannier-Stark ladder which pins thedistance between the two domain walls.
As anticipated above, the solution of the two-bodyproblem sheds light on the many-body dynamics. Eachof the dilute spin flips in the initial state |+′〉 overlaps sig-nificantly with the lightest mesonic wavefunctions. [91]The total magnetization thus exhibits a persistent oscil-latory behavior, characterized by multiple frequencies as-sociated with the “masses” of the mesonic bound states.As long as the typical separation between distinct ini-tial domain-wall pairs —the inverse ' ε−2 of the den-sity in Eq. (18)— exceeds by far the size ' 1 + ε/h ofthe excited bound states, the evolution of the magneti-zation m(n) can be described as resulting from the inco-herent superposition of independent small-amplitude me-son oscillations, similarly to the undriven case discussedin Ref. [49]. We note that inelastic meson scattering isexpected to trigger asymptotic thermalization; however,the inelastic cross section drops rapidly for small ε [92],making the transient out-of-equilibrium state extremelylong lived [93, 94].
Analogous considerations can be made for the dynam-
ics starting from the “false vacuum” state |−〉 (or, equiv-alently, taking h 7→ −h). Two consecutive domain wallsare subject to an anticonfining potential, decreasing lin-early with their separation. Due to lattice effects, how-ever, the domain walls cannot accelerate arbitrarily to-wards large distances, as their hopping kinetic energy isbounded. Thus, they form a tower of bound states, for-mally analogous to the true ground-state excitations (21)discussed above. The dynamics of the order parameterthus follows a similar pattern, exhibiting a stable anti-magnetization with small-amplitude multiple-frequencyoscillations superimposed due to antimesons.
The ultimate decay of the antimagnetization due toresonant domain-wall pair production is expected to oc-cur at very long times, as the high energy cost 4J ofcreating two domain walls has to be compensated bya gain 2hr associated with a large ground-state bubbleof size r extending between them. To realize this, thetwo locally created virtual domain walls have to tunnelthrough a distance r∗ ' 2J/h, which suggests a non-perturbative lifetime. The more rigorous estimation ofRef. [79] leads to the quasi-exponentially long time inEq. (13). This phenomenon is closely related to theSchwinger mechanism in quantum electrodynamics, asexplained in Ref. [52].
The confinement scenario for the periodically kickedmixed-field Ising chain has been verified numerically, sim-ulating the Floquet dynamics induced by Z2-symmetry-breaking periodic kicks Kε,h by means of the infinitetime-evolving block decimation (iTEBD) algorithm [95].Figure 4 reports the behavior of the order parameter asa function of the number of n kicks. In fact, either start-ing from the vacuum |+〉 [panel (a)], or from the false-vacuum |−〉 [panel (b)], a small value of the longitudinalcomponent h is sufficient to induce a non-perturbativechange in the order parameter dynamics: domain wallsget confined into (anti)mesons, thus hindering the melt-ing of the order parameter, which remains finite for ex-ponentially long times (continuous colored lines).
D. Deconfinement by driving and enhancement ofDTC response lifetime
In the last section we established that domain-wall con-finement induced by a Z2-symmetry-breaking kick com-ponent stabilizes both the magnetization when quench-ing from the “true vacuum” and the antimagnetizationwhen quenching from the “false vacuum” (cf. Fig. 4).We are now ready to come back to our main problem oftime-crystalline order, and discuss how generic (non-Z2-symmetric) kick imperfections impact the order parame-ter lifetime determined in Sec. II.
We consider kicks K in Eq. (2) of the form of imperfectspin flips:
K −→ Kπ/2Kε,h = iLPei∑
j(εXj+hZj). (25)
To make progress, generalizing the approach of Sec. II,
10
we switch to the toggling frame, i.e., rewrite the two-cycleFloquet operator reabsorbing the perfect kick:
U2 = (Kπ/2Kε,hVJ) (Kπ/2Kε,hVJ)≡ (−)L (Kε,−hVJ) (Kε,hVJ) (26)
where, as in Eq. (6), we have exploited the fact that(−i)LKπ/2 = P flips the Z axis, leaving the Z2-symmetric interactions VJ invariant. Here, however, thetoggling frame makes the symmetry-breaking longitudi-nal component h of the kick periodically flip in sign. Thedynamics can thus be seen as generated by strong in-teractions and weak kicks only (without perfect flips),alternating the sign of the longitudinal component of thekick at each period.
The theory of Ref. [79] discussed above for U , straight-forwardly applies to U2 as well: it guarantees the exis-tence of a close-to-identity time-periodic unitary trans-formation eiS≤p such that, in the transformed frame, thetwo-cycle Floquet dynamics described by Eq. (26) ap-proximately conserves the number of domain walls over along prethermal timescale analogous to Eq. (13). In theprevious section we have discussed the transformationeiS≤p for U = Kε,hVJ [see Eqs. (12) and (14)]. Unfortu-nately, there is generally no simple relation between eiS≤pand eiS≤p , because the generators S≤p and the effectiveFloquet operator U ′p depend on both ε and h. In particu-lar, h and −h produce generally different operators S≤p.This fact prevents us from straightforwardly combiningthe two transformations for Kε,hVJ and Kε,−hVJ into asingle one for U2.
However, the lowest-order result in Eq. (17) shows thatS≤1 is actually independent of h. This simplification al-lows us to directly combine the two transformations intoa single one for U2 right away: substituting into Eq. (26),we find that the resulting lowest-order transformationeiS≤1 coincides with eiS≤1 in Eq. (17):
U2 ' (−)LeiS≤1U ′1(−h)U ′1(h)e−iS≤1
≡ (−)LeiS≤1(U2)′
1e−iS≤1 (27)
where U ′1(h) is expressed in Eq. (16). In contrast withU ′1(±h),
(U2)′
1 is not expressed as the exponential of atime-independent local Hamiltonian, but it results fromtwo time steps where the longitudinal field switches be-tween h and −h. On the other hand, the occurrencethat the lowest-order transformation S≤1 = S≤1 is time-independent here relies on the special form (25) cho-sen for the kick perturbation, which makes the presentderivation especially simple.
Thus our problem amounts to study the driven dynam-ics of a dilute gas of domain-wall pairs. To the level ofapproximation considered above, domain walls are sub-ject to the unitary dynamics expressed by Eq. (16) where,crucially, the confining string tension h regularly flips insign at integer times, as dictated by Eq. (27). General-izing the analysis of Sec. III C, the nature of the evolu-tion of the order parameter —and hence the fate of time-crystal behavior in the presence of confinement— will be
0 50 100 150 2000.6
0.7
0.8
0.9
1.0
|m(n)|
h = 0.2
0 50 100 150 2000.6
0.7
0.8
0.9
1.0h = 0.4
0 50 100 150 2000.6
0.7
0.8
0.9
1.0
n
|m(n)|
h = 0.6
0 50 100 150 2000.6
0.7
0.8
0.9
1.0
n
h = 0.8
(b)(a)
(d)(c)
(−1)n
m(n)
(−1)n
m(n)
FIG. 5. Log-linear plots of the order parameter time-evolutionunder the Floquet dynamics induced by Kπ/2Kε,hVJ for ε =0.1 and different values of h. Thick lines are the iTEBD data,which are compared with ED results for finite system withdifferent sizes L = 10, 15, 20, 25, 30 (shaded lines, from lighterto darker). Dashed black lines are the results for h = 0 andare plotted for comparison.
essentially determined by the solution of the two-bodyproblem.
Solving the two-body dynamics amounts to composingthe evolution map in Eq. (24) with +h and −h. Eventhough domain walls are completely bound into mesonsor antimesons within each individual period, we demon-strate that the periodic switching between the two leadsto deconfinement, and thus meltdown of the system mag-netization. The exact two-cycle Floquet map restrictedto the two-body space in the “bulk” (i.e., for Q� 2ε/h)is equivalent to the composition of two maps given byEq. (24) for t = 1, with +h and −h, respectively. Theresult of this composition isQ(2t+ 2) = Q(2t) + 4 ε
h
[cosP (2t)− cos(P (2t) + h)
],
P (2t+ 2) = P (2t) .(28)
This two-cycle map is equivalent to one generated by aneffective static Hamiltonian,{
Q(2t+ 2) = e+i2Hcm Q(2t) e−i2Hcm ,
P (2t+ 2) = e+i2Hcm P (2t) e−i2Hcm ,(29)
which reads
Hcm = 2εh
[sinP − sin(P + h)
], (30)
11
ϵ = 0.1
ϵ = 0.2
0.0 0.2 0.4 0.6 0.8 1.00.5
0.6
0.7
0.8
0.9
1.0
h
γ/γ 0
ϵ = 0.1
ϵ = 0.2
0.1 0.2 0.5 1
0.0050.010
0.0500.100
0.5001
h
1-γ/γ 0
ϵ = 0.1
ϵ = 0.2
0.0 0.2 0.4 0.6 0.8 1.00.5
0.6
0.7
0.8
0.9
1.0
h
γ/γ 0
0.1 0.2 0.5 1
0.0050.010
0.0500.100
0.5001
1-γ/γ0
FIG. 6. Scaling of the decay rate γ = γ(ε, h) of the orderparameter (cf. Fig. 5) as a function of the longitudinal kickcomponent h for different fixed values of ε. Rates have beenrescaled by γ0 = γ(ε, h = 0). The dashed line is the resultof the fit γ/γ0 = 1 − c h2 for the relative correction, wherec ' 1/2. The inset shows the same data in log-log scale.
as can be readily verified.Remarkably, Hcm is a pure-hopping Hamiltonian,
without interaction potentials. Its eigenstates are nolonger bound states localized around a finite value of Q,but deconfined plane waves with a definite momentumP . The periodic switching between +h and −h averagesout the Z2-breaking confining potential, and effectivelyrestores the symmetry, similarly to what happens in high-frequency-driven discrete time crystals [6].
A semiclassical description of the effective domain-walldynamics is portrayed in Fig. 1(b), where it is highlightedhow the periodic switching of the sign of the confiningpotential h leads to an effective ballistic spreading of thereversed bubble delimited by two domain walls, with arenormalized maximum effective velocity veff = veff(ε, h).From Eq. (30), we find the approximate dispersion rela-tion v(k) = 2ε
h [sin k − sin(k + h)], whence
veff(ε, h)veff(ε, 0) = 1−O(h2). (31)
As we neglected all terms in H2-body beyond the first or-der in ε and h [cf. Eq. (19)], we cannot expect the correc-tion to be quantitatively accurate. However, numericalsimulations shown in Fig. 5 and 6 clearly indicate a rela-tive reduction of the decay rate γ of the order parameterby a factor
γ(ε, h)γ(ε, 0) ' 1− h2
2 , (32)
which is compatible with a relative decrease of order h2
for both the density ρ of spin flips in the initial state andthe spreading velocity, according to the formula γ ∼ ρveff.
The bottom line of this analysis is that the symmetry-breaking perturbation, leading to confinement of excita-
tions in the static case, does enhance the lifetime of time-crystalline behavior, although only by parametrically de-creasing the prefactor a = a(J, h) of the perturbativedecay law γ ∼ aε3. The above arguments can be eas-ily modified to account for generic symmetry-breakingperturbations of the perfect kick. Such perturbations,thus, do not qualitatively modify the picture of the or-der parameter meltdown found in the integrable case inEq. (8). Although we derived this result in the lowestperturbative order, we expect that the Z2 symmetry isrestored to all orders, similarly to what happens in MBLand high-frequency driven prethermal DTCs [6, 7]. Oursimulations of Fig. 5 show no signature of slowdown ofthe computed exponential decay, clearly confirming theexpectation.
We finally remark that, similarly to the integrable case,the order-parameter dynamics is strongly affected by thefinite size of the chain: our ED data for L = 10 ÷ 30,shown in Fig. 5 (shaded lines), display a deceptive per-sistence of the order parameter, as previously observedin different contexts [96], whereas it eventually decaysto zero for systems in the thermodynamic limit (thickstraight lines). Note that both operators K and V canbe exactly applied to the many-body wavefunction, whichallows us to efficiently simulate dynamics of 30 spins witha reasonable amount of resources.
IV. STABILIZATION OF DTC RESPONSE BYINTERACTIONS BEYOND NEAREST
NEIGHBORS
A version of domain-wall confinement also arises in theordered phase of spin chains with an interaction rangeextended over multiple sites, even in the absence of ex-plicit symmetry-breaking fields. The basic mechanismwas identified in Ref. [56]: the separation of two domainwalls involves an increase of the configurational energy,due to the increase in the number of frustrated bondsbetween pairs of spins beyond the nearest neighbors. Asdiscussed in Refs. [56, 57] and experimentally verified inRef. [58], this gives rise to an effective attractive potentialvα(r) between two domain walls. The resulting physicsis thus reminiscent of that generated by a longitudinalfield. Here, however, the interaction tail tunes the shapeand depth of the effective potential well.
Below we show that this general domain-wall bindingmechanism also arises in the Floquet context. Crucially,as it does not rely on explicitly breaking the symmetry, itdoes not suffer from time-averaging effects demonstratedin Sec. III, opening the door to a true enhancement ofthe time-crystal order lifetime, as pictorially illustratedin Fig. 1(c). As a core result of this section, we establishthat an increase of the interaction range R (i.e., Ji,j 6= 0only if |i− j| ≤ R) leads to an exponential enhancementof the order-parameter lifetime, γ ∼ a ε2R+1. We fur-ther discuss the practical conditions for observing thisconfinement-stabilized DTC behavior, and the implica-
12
tions for experiments with Rydberg-dressed interactingatomic chains, recently realized in the lab [97, 98]. Basedon this result, we will finally argue that long-range alge-braically decaying interactions with a generic exponentα (not necessarily smaller than 2) stabilize the order pa-rameter over timescales longer than any inverse power ofthe kick perturbation ε.
This section is organized as follows. In Sec. IV A,we generalize the derivation of an effective domain-wallconserving Floquet Hamiltonian of Sec. III B to a chainwith arbitrary Ising couplings Ji,j beyond nearest neigh-bors. Hence, in Sec. IV B we study the two-body prob-lem, which is richer than the corresponding one stud-ied in Sec. III C due to the coexistence of bound statesand deconfined continuum, and determine the conditionsfor domain-wall binding and their implications for theevolution of the order parameter. Building on the in-tuition from the two-body problem, in Sec. IV C weswitch to a more scrupulous analysis and prove that,in the asymptotic regime of weak kick perturbation ε,the order-parameter decay rate gets heavily suppressedas γ ∼ a ε2R+1. This and related theory predictions arenumerically verified in Sec. IV D. Finally, in Sec. IV Ewe discuss the limit of long-range interactions and arguethat the decay becomes non-perturbatively small.
A. Effective domain-wall dynamics for weak kicks
We generalize Eq. (2) by considering a kicked Isingchain with arbitrary longer-range couplings, defined bythe Floquet operator
U = KVJ, VJ = ei∑L
j=1
∑R
r=1JrZjZj+r (33)
with periodic boundary conditions. We have denoted byJ ≡ (J1, J2, . . . , JR) the array of coupling strengths atincreasing distances, with π/2 > J1 ≥ J2 ≥ · · · ≥ JR >0, and we implicitly assumed R < L/2. We consider therange R fixed and independent of the system size L (thelong-range limit R ∝ L will be discussed in Sec. IV E).We take the kick K as in Eq. (5), i.e.,
K −→ Kπ/2+ε = ei(π/2+ε)
∑jXj = iLPe
iε∑
jXj . (34)
We discard explicit symmetry-breaking components ofthe kick: as shown in the previous Sec. III, the lat-ter are not expected to qualitatively enhance the time-crystal lifetime. Note that the couplings beyond nearest-neighbors break integrability. As in Eq. (6), transformingto the toggling frame gives
m(n) = (−1)n〈+|[KεVJ]−nZj [KεVJ]n|+〉, (35)
so we focus on the effect of weak kicks Kε close to theidentity.
Along the lines of the first part of this paper, we wantto build intuition on the evolution of the order param-eter in terms of the dynamics of domain walls. The
construction of the effective domain-wall conserving Flo-quet Hamiltonian of Sec. III B can be straightforwardlygeneralized to the present case of the Floquet operatorU = KεVJ with interactions beyond nearest neighbors.To this aim, we first work in the regime of “weak con-finement” J2, . . . , JR � J1 As the dominant scale J1 inthe problem couples to the number of domain walls D1[defined in Eq. (11)], we can set up a perturbation theorysimilar to that of Sec. III B, where we had h� J .
The derivation closely parallels that of Sec. III B, basedon the general theory of Ref. [79], where in this case theperturbative parameter is ε = max(ε, J2, . . . , JR). Therigorous bounds of Ref. [79] ensure that the perturba-tively dressed domain walls remain accurately conservedover the very long prethermal timescale in Eq. (13),where the numerical constant c is here adjusted to ac-count for the longer range R of the perturbation opera-tor.
Similarly to the case discussed in Sec. III B, for low-order explicit computations it is more practical to followa different scheme from Ref. [79] and aim for a static ef-fective Floquet Hamiltonian HF by combining the twoexponentials of the product KεVJ using the replica re-summation of the BCH expansion [87], order by orderin the kick imperfection ε, and hence perform a conven-tional static Schrieffer-Wolff transformation on HF . Thepresence of arbitrary Ising couplings here requires a non-trivial generalization of the calculation in Ref. [87]. Thestructure of the resulting expansion is worked out in Ap-pendix B 1.
The lowest-order result reported here is a simple gen-eralization of Eqs. (16), (17):
HF1,1 = −
∑j,r≥2
Jr ZjZj+r−ε (P ↑j−1XjP↓j+1+P ↓j−1XjP
↑j+1),
(36)
S≤1 = − ε2∑j
P ↑j−1[Xj − cot(2J)Yj ]P ↑j+1+
+ P ↓j−1[Xj + cot(2J)Yj ]P ↓j+1 . (37)
Due to the arbitrary range of the perturbation, the ap-pearance of higher order terms in the effective Hamilto-nian and in the Schrieffer-Wolff generator is more cum-bersome than that discussed in Sec. III B. Appendix B 1presents the general hierarchical structure of the expan-sion and explicitly reports the second-order result, forillustration. Terms of order p in HF contain at most pspin-flip operators S±j ≡ 1
2 (Xj ± iYj) separated by a dis-tance at most R, i.e., they contain strings of the formSµ1j1· · ·Sµqjq with µ1, . . . , µq = ±, q ≤ p, and j1 < · · · < jq
with jn+1 − jn ≤ R for all n. Such a product of spin-flipoperators (which we occasionally refer to as a quasilocal“cluster”) is multiplied by complicated products of diag-onal Zj operators - with coefficients depending on thecouplings {Jr} - located not farther away than R sitesfrom the cluster. Finally, projectors P ↑j , P
↓j applied to
spins adjacent to the position of the spin-flip operators
13
ensure that the spin flips only move domain walls withoutcreating or annihilating them. Likewise, the Schrieffer-Wolff generator at order p contains clusters of at most pspin-flip operators, with the same locality properties asdiscussed above.
B. Domain-wall binding
Similarly to Sec. III C, we now analyze the dynamicsof the order parameter in terms of the motion of con-served domain walls in the Schrieffer-Wolff-transformedpicture. The initial state in the transformed picture|+′〉 = e−iS≤p∗ |+〉 can be viewed as a dilute gas ofquasilocal clusters of p = 1, 2, . . . , p∗ flipped spins, therespective density being suppressed as ε2p. To lowest or-der p = 1, these are isolated spin flips, i.e., adjacent pairsof domain walls. Thus, to this order of approximation,the evolution of the order parameter can be understoodstarting from the two-body problem.
Let us begin with qualitative considerations. Inter-actions beyond the nearest neighbors lead to the pres-ence of a discrete set of bound states, coexisting with acontinuum of unbound domain walls for larger energy.Furthermore, interactions between distant spins also fa-vor the formation of more structured “molecular” boundstates out of larger clusters of domain walls. The richnonequilibrium dynamics of the system, including theanomalously slow decay of the order parameter, resultsfrom the coexistence of topological and nontopologicalexcitations in the spectrum (i.e., unbound domain wallsand bound pairs) which we now turn to quantitativelyanalyze.
The effective domain-wall conserving Floquet Hamilto-nian (36) projected onto the two-particle sector gives thefollowing first-quantized two-body problem, analogous toEq. (19):
H2-body =∑j1<j2
vJ(j2 − j1) |j1, j2〉 〈j1, j2|
− ε∑j1<j2
(|j1 + 1, j2〉+ |j1, j2 + 1〉
)〈j1, j2|+ H.c.,
(38)
where
vJ(r) =
4
r∑d=1
dJd + 4rR∑
d=r+1Jd, if r < R,
4R∑d=1
dJd ≡ vJ(∞), if r ≥ R.
(39)
The two-body potential vJ(r) grows as a function of thedistance up to r = R, then flattens out. Thus, the po-tential well hosts a finite number of bound states, whichgrows to R−1 upon decreasing ε→ 0. In field-theoreticallanguage, this discrete set of energy levels forms the mass
spectrum of nontopological particles. Above this, a con-tinuum of scattering states appears, built out of two un-bound domain walls; in field-theoretical language, thespectrum contains stable topologically charged particles,i.e., kinks and antikinks. The topological nature of theseexcitations stems from the fact that they can be locallycreated or destroyed in globally neutral pairs only, notindividually.
The bound (nontopological) and unbound (topologi-cal) excitations can be distinguished by being labelled bya real-space or momentum-space quantum number. Tounderstand this, let us transform to the center-of-massframe Ψ(j1, j2) = eiK(j1+j2)ψ(j2− j1) and set K = 0 dueto translational invariance of the nonequilibrium initialstate (cf. Sec. III C). The reduced wavefunction ψ satis-fies the Schrodinger equation
vJ(r)ψ(r)− 2ε[ψ(r + 1) + ψ(r − 1)] = Eψ(r) (40)
in the domain r > 0, subject to the boundary con-dition ψ(0) ≡ 0. This equation defines the center-of-mass frame Hamiltonian Hcm. For ε → 0, the eigen-functions in the center-of-mass frame ψ`(r) = δ`,r cor-respond to contiguous reversed domains of ` spins, witheigenvalues E` = vJ(`). In this limit, the discrete label` = 1, . . . , R−1 thus has the physical meaning of distancebetween the two domain walls. Due to the discretenessof the spectrum, this labelling can be adiabatically con-tinued to finite ε, where eigenfunctions feature quantumfluctuations of the physical distance. On the other hand,the degenerate levels ER = ER+1 = · · · = vJ(∞) splitinto a continuous band Ek = vJ(∞)−4ε cos k, with eigen-functions labelled by the relative momentum k of the twodomain walls. The binding potential vJ only affects theseeigenfunctions via the relative scattering phase eiδk be-tween incoming and outgoing waves. We can estimatethe stability range of the most excited [(` = R − 1)-th]bound state by the condition that the continuum bandEk does not overlap the discrete energy level ER−1. Thisyields the range ε . JR; for larger ε, the bound state hy-bridizes with the unbound continuum. Energy levels andeigenfunctions are further renormalized by higher-orderprocesses of order ε2, ε3,. . . to be added to H2-body, notincluded in Eq. (38). These additional terms consist ofdiagonal terms and longer hops of domain walls by atmost 2, 3, . . . lattice sites, respectively. For small enoughε, the resulting quantitative corrections do not alter thequalitative structure of the spectrum discussed here.
Figure 7(a) reports a sketch of the eigenstates of theFloquet Hamiltonian for ε → 0. Unlike in Fig. 3, bothdiscrete “mesonic” bound states and unbound “domain-wall-like” states appear. The latter are highlighted byblue lines in the pictorial sketch, and their energy levelsare marked by thick blue bars, indicating that they formcontinuous bands of width ∝ ε. Panel (b) reports a se-lection of eigenfunctions of the two-body problem (40) inthe center-of-mass frame, for algebraically decaying cou-pling Jr = 1/rα truncated to the range r ≤ R = 10,α = 3, and ε = 0.1139. In this case, due to small cou-
14
D1 = 2
Hcm = vJ(Q) − 4ϵ cos P
⋮
⋮
energy ΔE = 2J1 ∑j
Dj, j+1 + … + 2JR ∑j
Dj, j+R
⋯
⋯⋮ ⋮
⋯ ⋯ ⋯
D1 = 4
(a) (b)
ℓ=�
ℓ=�
ℓ=�
���������
0 10 20 30 40 500.0
0.2
0.4
0.6
0.8
1.0
r
|ψℓ(r)2
FIG. 7. (a) Sketch of the eigenstates of the Floquet Hamiltonian HF for ε → 0. The vertical axis represents the excitationenergy ∆E above the Floquet ground state. (b) A selection of eigenstates of the two-body problem (40) in the center-of-massframe, for Jr = 1/rα for r ≤ R = 10, α = 3, and ε = 0.1139.
plings beyond the nearest neighbors and comparativelylarge hopping ε, the number of bound states (red curves)is only 3. As ε→ 0, it grows to R−1 = 9. Higher excitedeigenstates are unbound plane waves (blue curves).
The two-body problem already gives us importanthints on the nonequilibrium evolution of the order pa-rameter. As a matter of fact, confined domain wallsonly produce a weak oscillatory behavior of m(n) withfrequencies related to the excitation energies (masses)of the bound states. Within the prethermal time win-dow 0 ≤ t ≤ Tpreth in Eq. (13), the order parameterdecay is only ascribed to the dynamical production of un-bound domain walls. As discussed above, domain-wall-like excitations exist as higher energy excitations. Whilein thermal equilibrium such domain-wall excitations arefinitely populated, imperfect kicks Kε will only excite thedomain-wall continuum very weakly, precisely with am-plitude εR, as generating unbound domain walls requiresto flip a critical contiguous domain of at least R spins.This key insight suggests that the destruction of long-range order in the Floquet-prethermal Gibbs ensemblee−βH
F
/Z is a very slow process, leaving large room fornonequilibrium time-crystalline behavior.
C. Exponential suppression of the order-parameterdecay rate
Generalizing the reasoning of Sec. II B, the solutionof the two-body problem suggests that the decay rate isseverely suppressed: The density of critical domains inthe initial state is of order ρ ∼ (εR)2, and the spreadingvelocity of their constituents domain walls is v ∼ |ε|,
leading to the estimate
γ ∼ ρv = O(ε2R+1). (41)
For R = 1, domain walls are the only stable excitationsin the spectrum, and we recover the exact result γ ∼ ε3
of Sec. II A. For R > 1, the appearance of bound states isexpected to significantly slow down the order parameterdecay.
The argument based on the two-body problem is,however, too naive, as it completely neglects all multi-body processes and interactions between confined domainwalls. In fact, not only adjacent domain walls attracteach other via the two-body potential vJ(r), but alsomesonic bound states themselves are subject to an effec-tive attraction when their distance is less than R. [99]Such attractive forces can be thought of as residual in-teractions between bound states of elementary particles,physically analogous to nuclear forces that keep togetherprotons and neutrons (i.e., bound states of quarks), orto molecular forces that keep together atoms (i.e., boundstates of electrons and nuclei). The meson-meson attrac-tive potential can be defined as
wJ,`1,`2(r) = EJ(`1, r, `2)− vJ(`1)− vJ(`2) < 0, (42)
where EJ(`1, r, `2) is the configurational energy of tworeversed domains of length `1 and `2, separated by r spinsin between. One finds
wJ,`1,`2(r) = −4∑
1≤i≤r1
∑`1+r+1≤j≤`1+r+`2
J|i−j|
= −4min(`1+`2+r−1,R)∑
s=r+1nsJs, (43)
with ns ≡ min(s− r, `1, `2, r + `1 + `2 − s). Consideringfixed `1 and `2, this two-body potential wJ,`1,`2(r) grows
15
isolated flipped spinsp < R
⋯1 2 p
⋯ ⋯bubble of size R
1 2 p−1
⋯
FIG. 8. A quasilocal cluster of few (p < R) spin flips (topconfiguration) could agglomerate tightly and convert the ac-cumulated extra interaction energy into the formation of areversed bubble of size R (bottom configuration), thus releas-ing a free domain wall, while conserving the total number ofdomain walls. While energetically allowed, such a process canbe shown to be necessarily off-resonant under generic incom-mensurability assumptions on the couplings {Jr} (see maintext).
monotonically as a function of the distance r, from w(r =1) < 0 to w(r ≥ R) ≡ 0. For instance, two isolated spinflips (`1 = `2 = 1) experience an attractive potentialproportional to the bare coupling, wJ,1,1(r) = −4Jr.
It is easy to imagine a process where a cluster of fewflipped spins or small domains, initially distant from eachother, agglomerate more tightly, making the gained re-ciprocal meson-meson interaction energy available to pro-gressively enlarge a domain and finally release a domain-wall from the attraction of the rest of the cluster at dis-tance > R, such that it can freely travel away and meltthe order parameter. In Fig. 8 we sketch such an ex-ample, where a finite (R-independent) cluster of initialspin flips possesses sufficient energy to release a travel-ing domain wall. It is a-priori conceivable that suchprocesses could trigger a comparatively fast decay of theorder parameter at low perturbative order. Remarkably,however, it is possible to rigorously exclude such a sce-nario, and prove that the fastest decay process occurs atorder R.
To establish this result, we need to modify the pertur-bative Schrieffer-Wolff transformation of Sec. III B to ex-plicitly account for the fact that all couplings J1, . . . , JRare large compared to ε when the asymptotic regimeε → 0 is considered. The unperturbed Floquet operatorVJ = e
i∑
j,rJrZjZj+r defines highly degenerate sectors of
the many-body Hilbert space, identified by the energylevels
E(n1, . . . , nR) = EGS + 2∑r
nrJr, (44)
where EGS = −L∑r Jr is the unperturbed ground-state
energy of the fully polarized state |+〉, and the non-negative integer nr ∈ N has the meaning of total numberof frustrated bonds at distance r. Under the assumption
of strong incommensurability of the couplings {Jr} inEq. (B14) —necessary to derive a well defined static Flo-quet Hamiltonian, as shown in Appendix B 1— each de-generate sector is in one-to-one correspondence with theset {nr}. For ε→ 0, transitions between such sectors areenergetically suppressed and can be adiabatically elimi-nated. In other words, we can dress the effective Hamil-tonian within each sector, order by order in ε, to accountfor all resonant processes occurring via virtual transitionsbetween different sectors. The strong incommensurabil-ity condition on the couplings guarantees that each suchtransition is accompanied by a finite energy denomina-tor. The construction can thus be formally carried outto all orders in ε [79].
Let us illustrate how this procedure works withinour time-independent approach. Starting from the Flo-quet operator U = KεVJ, we combine the two expo-nentials into a Floquet Hamiltonian HF
p by generaliz-ing the replica calculation of Ref. [87] as detailed inAppendix B 1. Hence, we seek a modified Schrieffer-Wolff unitary transformation eiεmSm , iteratively for m =1, . . . , p, which eliminates from HF
p all terms of order εmthat violate the conservation of the any of the operators
Dr =∑j
1− ZjZj+r2 , (45)
r = 1, . . . , R, respectively coupled to Jr in the unper-turbed Floquet Hamiltonian. These operators are thusapproximate conservation laws, meaning that their eigen-values {nr} are good quantum numbers to label theeigenstates of the resulting truncated Schrieffer-Wolff-transformed Floquet Hamiltonian HF
R,p:
HFp = eiS≤pHF
R,pe−iS≤p +O(εp+1), (46)
[HFR,p, Dr] = 0 for all r = 1, . . . , R. (47)
where eiS≤p ≡ eiεS1 · · · eiεpSp . We remark that thisscheme should be distinguished from the perturbationtheory of Sec. IV A, aimed at the conservation of thequantity D1 only for the effective Floquet HamiltonianHF
1,p; on the contrary, the effective Floquet Hamilto-nian HF
R,p obtained here conserves all quantities Dr,r = 1, . . . , R. To emphasize this distinction, here we usea calligraphic notation for the generator S of the pertur-bative scheme in powers of the kick ε, to avoid confusionwith the previous generator S of the perturbative schemein powers of ε.
As usual in the many-body context (cf. Sec. III B), theproliferation of possible processes as well as the decreaseof energy denominators at high perturbative orders is ex-pected to lead to a divergent (asymptotic) character ofthe series, corresponding to a mutual hybridization ofthe many-body energy bands arising from the splittingof the highly-degenerate unperturbed levels. However,the perturbative series provides valuable information on
16
the slowness of the dynamical delocalization in Hilbertspace. The timescale Tpreth over which the approximateconserved quantities {D′r ≡ e−iS≤p∗Dre
iS≤p∗}Rr=1 appre-ciably depart from their initial value can be estimated byfinding the optimal truncation order p∗ of the series. [100]Within the time-dependent construction of Ref. [79], p∗is found to depend on the magnitude of the perturbationas Cε−1/(2R+1+δ) where δ > 0 is arbitrary and C = C(δ)is a constant. This result leads to a stretched exponentiallower bound on the prethermal timescale, which general-izes Eq. (13):
Tpreth ≥1ε
exp(
c
ε1/(2R+1+δ)
). (48)
Within the long Floquet-prethermal window 0 ≤ t ≤Tpreth, the dynamics is guaranteed to take place withinthe Hilbert space sectors defined by the set of eigenval-ues nr of the dressed operators D′r, i.e., the (dressed)total numbers of frustrated bonds at distance r. Sincethe time-independent approach followed here (replica re-summation + standard static Schrieffer-Wolff) basicallyproduces the same set of conserved quantities as the time-dependent scheme of Ref. [79], it is natural to assume thatthis alternative scheme yields similar nonperturbativelylong heating timescales (see also the relative discussionin Refs. [79, 87]).
Since HFR,p∗ conserves all operators {Dr}, an initial
configuration can only evolve within the correspondingresonant subspace with fixed number of frustrated bondsnr at distance r. The effective Hamiltonian HF
R,p∗ is thusmuch more constrained than HF
1,p∗ in Eq. (36). Thetransformed-picture initial state e−iS≤p∗ |+〉 is a low-density superposition of isolated quasilocal clusters offlipped spins. At pth order in ε, these clusters may com-prise at most p flipped spins.
We are now in a position to clearly formulate ourquestion on the order parameter evolution: Can aninitial cluster of p < R flipped spins evolving viaHFR,p∗ resonantly excite unbound domain walls? Under
the single assumption that the array (J1, . . . , JR, 2π) isstrongly incommensurate [as specified by Eq. (B14) inAppendix B 1], we prove that such a configuration withp < R is never resonant in energy with order-meltingconfigurations, i.e., configurations possessing a contigu-ous domain of R reversed spins.
Referring to the illustration in Fig. 9, the claim is thatthe top and bottom configurations are necessarily sep-arated by an energy mismatch. By the incommensura-bility condition, the existence of this energy mismatch isequivalent to the occurrence that nr 6= nLr +nRr for some1 ≤ r ≤ R (note that the process in Fig. 8 is a particularcase of Fig. 9).
The proof follows from theLemma. A domain-wall-like configurationhas nr ≥ r.
To show this, we consider a domain-wall-like configura-tion, with all spins pointing up (down) for j ≤ jL and
⋯
⋯bubble of size R
flipped spinsp < R
RL
FIG. 9. A candidate low-order process leading to the melt-down of the order parameter. In the main text, we prove thatenergy resonance between the two configurations is not possi-ble under a generic assumption of strong incommensurabilityof the couplings.
down (up) for j > jR (where jR ≥ jL). We focus onthe r sublattices {i+ jr|j ∈ Z}, labelled by i = 1, . . . , R.Each sublattice exhibits a domain-wall-like configuration,and hence, it has at least one frustrated nearest-neighborbond. Each such bond maps to a bond at distance r inthe original lattice. Thus, the original configuration hasat least r frustrated bonds at distance r, QED.
Using this lemma, it is easy to prove the main claim:considering again Fig. 9, the bottom configuration isenergetically equivalent to two isolated domain-wall-likeconfigurations, defined by the content of the regions de-noted L and R, as the reversed bubble in between canbe made infinitely large without frustrating any furtherbonds. By the lemma, nLr ≥ r and nRr ≥ r. In particular,nLR+nRR ≥ 2R. On the other hand, the top configurationin Fig. 9 has p flipped spins, each of which can frustrateat most two bonds at distance r, for all r. In partic-ular, nR ≤ 2p. Hence, clearly, the resonance conditionnR = nLR+nRR is impossible to satisfy when p < R, whichconcludes the proof, QED.
The result proved here implies that processes involvingat least R spin flips are responsible for the order param-eter decay. In particular, we already identified above thefastest such process, arising from terms
∝ εR∑j
P ↑−R+1 · · ·P↑j S−j+1 · · ·S
−j+R P
↑j+R+1 · · ·P
↑j+2R
(49)in S≤p∗ [recall S±j ≡ 1
2 (Xj ± iYj)]. These terms giverise to isolated domains of R contiguous reversed spins,with density ∝ ε2R, appearing in the transformed-pictureinitial state |+′〉 = eiS≤p∗ |+〉 at order R. The two do-main walls separated by R sites are free to hop away fromeach other with amplitude ε, thus spreading the reverseddomain and melting the system magnetization at a rateγ ∼ ε2R+1, cf. Eq. (41).
We finally remark that tunable-range Ising interactionsof the form
Jr = J
1 + (r/rc)6 (50)
17
ϵ = 0.051
ϵ = 0.062
ϵ = 0.076
ϵ = 0.093
ϵ = 0.110 50 100 150 200
0.80
0.85
0.90
0.95
1.00
n
|m(n)|
J1 = 0.831907, J2 = 0.103988
(b)
(a)
(c)
(a)
(b)
~ ϵ5
Eq.(7)
0.01 0.02 0.05 0.1010-8
10-7
10-6
10-5
10-4
0.001
ϵ
γ0 50 100 150 2000.90
0.92
0.94
0.96
0.98
1.00|m(n)|
J1 = 0.684832, J2 = 0.143968(−
1)nm
(n)(−
1)nm
(n)J1 ≃ 0.685, J2 ≃ 0.144
J1 ≃ 0.832, J2 ≃ 0.104
FIG. 10. (a-b) Log-linear plot of the order-parameter timeevolution under the Floquet dynamics in the quantum Isingchain with next-to-nearest-neighbor interactions, obtained bymeans of iTEBD simulations. We set J1 = 1/ζ(α) and J2 =(1/2)α/ζ(α), with α = 2.25 (a) and 3 (b). Dashed black linesare exponential fits. (c) Scaling of the decay rate γ as functionof the kick strength ε, where symbols correspond to the dataextracted from panels (a) and (b). The agreement with thepredicted ε5 law is perfect.
can be realized in Rydberg-dressed atomic spin lattices,as demonstrated in Ref. [97]. These interactions have analmost flat core for r . rc, and cross over to a quick de-cay in the intermediate range r ≈ rc. The value of theeffective range rc can be efficiently tuned in the experi-ment, making this setup an ideal platform to observe theconfinement-stabilized DTC response identified in thiswork.
D. Numerical simulations for R > 1
The above theoretical analysis is strongly supportedby extensive numerical simulations that we performed forR = 2 (namely, for the Ising chain with next-to-nearest-neighbor interactions). In our numerics, the couplingshave been fixed as J1 = 1/ζ(α) and J2 = (1/2)α/ζ(α),where ζ(x) =
∑∞r=1 1/rx denotes the Riemann zeta func-
tion. We fix either α = 2.25 or 3 (thus larger than 2), tobe consistent with the next section concerning long-rangesystems. This choice is largely arbitrary at this level; weanticipate that it allows a direct comparison with thedata presented in the next section. Note that the incom-mensurability of the couplings {Jr} would be guaranteedfor irrational α’s, but we take rational values to furthertest of the robustness of our analytical predictions.
L = 10
L = 15
L = 20
L = 25
L = 30
L = ∞0 50 100 150 200
0.90
0.92
0.94
0.96
0.98
1.00
n
|m(n)|
R = 2, ϵ ≃ 0.1139
(−1)n
m(n)
R = 2, ϵ ≃ 0.114
FIG. 11. Same as in Fig. 10(a), for ε = 0.11390625. TheiTEBD data (thick dark line) are compared with ED for finitesystem with different sizes L = 10, 15, 20, 25, 30 (shaded redlines, from lighter to darker).
Some representative iTEBD data, for systems in thethermodynamic limit, are reported in panels (a) and (b)of Fig. 10, which highlights the exponential decay of theabsolute value of the order parameter under the Floquetdynamics induced by a non-vanishing kick strength ε. Aspredicted by our theory, the scaling of the decay-rate γundergoes a qualitative change to ∼ ε5, as compared tothe nearest-neighbor limit α → ∞: this is verified inpanel (c), which clearly shows how the numerical datapoints follow such predicted scaling (dashed black lines).The nearest-neighbor analytical result, given by Eq. (7)and valid for α → ∞ (i.e. R = 1), is also reported forcomparison (dotted black line).
In Fig. 11 we compare the thermodynamic data withthe finite-size ED results. (Note that for small ε theentanglement entropy growth is slow enough to pushiTEBD simulations to unusually large numbers of driv-ing periods [101].) Also in this case, the absolute valueof the order parameter gets stuck to a non-zero valuefor L < ∞. However, both iTEBD data and finite-sizeresults show the same high-frequency oscillations.
The frequencies of such oscillations can be extractedfrom the Fourier power spectrum of the time series for|m(n)| (Fig. 12). We observe that data in the thermody-namic limit and at finite size are consistent, and manifestthe same main peaks symmetrical around ω = π [panel(a)]. (Note that the finite-size spectrum shows some spu-rious frequencies due to time-recurrence effects.) Furtherconfirmation of the validity of our perturbative analysisis provided by the scaling of the position of the mainpeak, which is approaching the “classical” value of the
18
ϵ = 0.015
ϵ = 0.051
ϵ = 0.11
0 1 2 3 4 5 6
10-11
10-9
10-7
10-5
0.001
0.100
ω
ℱ(ω
)J1 = 0.684832, J2 = 0.143968
L = ∞
L = 28
0 0.04 0.08 0.123.10
3.15
3.20
3.25
3.30
3.35
3.40
ϵω
*
(b)(a)
J1 ≃ 0.685, J2 ≃ 0.144
FIG. 12. (a) Discrete Fourier transform of the time series (upto 800 kicks) in Fig. 10 for representative values of the kickstrength ε (full lines). Dashed lines are the analogous dataobtained via ED for L = 28 and longer time series (up to 2000kicks). (b) Position of the main peak (at ω > π) as a functionof ε, which is expected to match the “classical” single spin-flipexcitation value for ε→ 0 (see main text for details).
single spin-flip excitation 4J1 + 4J2 for ε→ 0 [panel (b)].
E. Long-range limit
The results of the previous section demonstrate thatthe order parameter decay rate is suppressed as γ ∼aRε
2R+1, as ε → 0. In this last section, we discuss thelong-range limit R → ∞. In particular, we focus onthe experimentally relevant case of algebraically decay-ing interactions Jr = J/rα. This model is relevant tothe dynamics of effective qubits in quantum simulatorsbased on trapped ions (tunable 0 < α < 3) [102, 103]and Rydberg atoms (α = 6) [104, 105]. In these setups, anontrivial quantum dynamics is generated by additionalmagnetic fields acting on the spins, whose spatiotempo-ral variations can be efficiently controlled in the exper-iment. We thus hereafter interpret J = {J1, . . . , JR} asa finite-range truncation of the parent sequence of cou-plings {Jr = J/rα}∞r=1. [Note that for a generic (ir-rational) value of α, these truncated sequences are ex-pected to satisfy the strong incommensurability condi-tion in Eq. (B14).] Within this perspective, it is interest-ing to shift our viewpoint to the functional dependencyof the rate γ on R.
Extracting the scaling of the prefactor aR would in-volve keeping track of the magnitude of the subset ofprocesses of order εR which trigger the order parameterdecay in our combined replica + Schrieffer-Wolff trans-formations. This is in principle straightforward but prac-tically unfeasible, due to the rapid growth of the com-plexity of high-order perturbative computations. How-ever, as a crude conservative estimate, we can bound aRfrom above by the total magnitude of all terms of orderεR. This type of bounds are worked out in the related
analysis of Ref. [79], as well as in many previous workson rigorous prethermalization theory [6, 24, 25, 83, 106],to obtain estimates of the thermalization timescales, likeEq. (48). The ubiquitous scenario resulting from theseworks is that the total magnitude (measured by a rel-evant operator norm) of all terms perturbatively gener-ated at order p first decreases exponentially with p, beforeplateauing at p = p∗ and finally diverging rapidly. In thecase of interest here, Ref. [79] finds p∗ ∼ ε−1/(2R+1+δ).Since any finite range R is largely superseded by p∗ forsmall enough perturbation ε, the exponential suppressionε2R+1 dominates over the prefactor aR, and the decreaseof γ upon increasing R is effectively exponential.
However, taking the limit R → ∞ is subtle, as itdoes not commute with the asymptotic perturbative limitε → 0: Setting heuristically R = ∞, the heating boundin Eq. (48) trivializes. Taken literally, this occurrencesuggests that a fast violation of the effective conserva-tion laws of {D′r} has to be expected for the long-rangeinteracting system, which could in principle lead to a fastorder parameter meltdown. In particular, this would pre-clude any meaningful extrapolation of the results of theprevious section to long-range interactions.
Numerical simulations, however, suggest the oppositebehavior: we find that increasing the truncation radiusR of long-range interactions Jr = J/rα to the maximumR = L/2, for arbitrary α, leads to a dramatic increaseof the order parameter lifetime, as clearly visible fromthe data shown in Fig. 13 for α = 2.25 [panels (a)-(b)]and α = 3 [panels (c)-(d)]. In these simulations, wehave chosen Kac rescaled interactions, i.e., J = 1/ζ(α)(cf. Fig. 10), so that results for different α can be fairlycompared. The reported data point at a robust stabi-lization of the DTC signal, well beyond our analyticaltheory of Sec. IV C: in fact, one can see that the kickstrength taken there, ε = 0.114 and 0.171, correspondto quite big rotations of the spins at each kick, by an-gles ≈ 13◦ and ≈ 20◦, respectively. These perturbationsare actually much larger than the considered couplingsbeyond nearest neighbors. In spite of this, the order pa-rameter decay is extremely suppressed upon increasingR, and hardly visible in the long-range limit; moreover,this occurrence is not a finite-size effect.
To resolve this apparent contradiction, we observe thatthe bound (48) on prethermalization is unnecessarilypretentious for our purposes: it expresses the expectedtimescale of quasiconservation of a large number of opera-tors {D′r}Rr=1, uniformly in the many-body spectrum. Aswe take R = L/2, energy levels become infinitely densein the thermodynamic limit away from the band edges,thanks to the strong incommensurability condition whichprevents them from being degenerate. In fact, the preser-vation of an extensive number of commuting operators{D′r}
L/2r=1 would make the system effectively many-body
localized, contrary to conventional delocalization scenar-ios for translationally invariant models [34, 39]. Theslow dynamics of highly excited states resulting fromthis long-range limit is thus nonstandard, and the ac-
19
R = 1
R = 2
R = 3
R = ∞
0 10 20 30 40 500.6
0.7
0.8
0.9
1.0
n
|m(n)|
α = 2.25, ϵ ≃ 0.1709
R = 1
R = 2
R = 3
R = ∞
0 10 20 30 40 500.6
0.7
0.8
0.9
1.0
n
|m(n)|
α = 3, ϵ ≃ 0.1709
R = 1
R = 2
R = 3
R = ∞
0 10 20 30 40 50 60 700.80
0.85
0.90
0.95
1.00
n
|m(n)|
α = 2.25, ϵ ≃ 0.1139
R = 1
R = 2
R = 3
R = ∞
0 10 20 30 40 50 60 700.80
0.85
0.90
0.95
1.00
n
|m(n)|
α = 3, ϵ ≃ 0.1139
(b)(a)
(d)(c)
(−1)n
m(n)
(−1)n
m(n)
α = 2.25, ϵ ≃ 0.114
α = 3, ϵ ≃ 0.114 α = 3, ϵ ≃ 0.171
α = 2.25, ϵ ≃ 0.171
FIG. 13. Order parameter decay for increasing interaction range R of the Ising couplings, from nearest-neighbor (R = 1) tolong-range (R =∞) interactions, as indicated in the legends. Data come from ED simulations for chains of length L = 20, 25, 30(shaded lines from lighter to darker). The various panels represent different values of the decay exponent α and of the kickstrength ε.
tual heating timescales (or thermalization timescales fortime-independent systems) are presently unclear, evenfor static (undriven) systems [56, 57, 62, 63].
On the other hand, the evolution of the order param-eter relevant to this work takes place in a particular cor-ner of the many-body Hilbert space, corresponding tothe low-energy sector of an approximate Floquet Hamil-tonian. While the perturbative series might be severelydivergent at low orders in the long-range limit when mea-sured by uniform operator norms, the same bounds arefar too loose when the construction is restricted to a low-energy sector with dilute excitations, relevant for the pur-pose of this work. [107] In other words, while low-orderperturbative transitions from highly excited configura-tions with extensively many frustrated bonds are verylikely to hit resonances, this becomes extremely unlikelywhen the initial state is the polarized state |+〉 consideredin this work, since quasilocal clusters of flipped spins gen-erated by the weak kicks are far away from each other andhence can hardly cooperate to produce resonant transi-tions. This argument suggests that the lower bound onthe timescale Tpreth in Eq. (48) is far too conservativefor the dynamics originating from the state |+〉, for arbi-
trary R. A tighter bound would be needed to correctlyaccount for the decrease of the density of states at lowenergy. In the same regime, a long-lasting suppressionof heating has to be expected in the limit of long-rangeinteractions R→∞, as well.
Setting up a generalized Schrieffer-Wolff perturbativescheme aimed at estimating the timescales involved inthe intricate slow dynamics of the long-range interactingchain appears as a formidable problem, which we leaveto future investigations. Here, building on the insight ofSec. IV C and above, we formulate the conjecture thatthe timescale of the fastest process leading to the orderparameter decay can be estimated in terms of the initialdensity of bubbles of reversed spins, whose walls are freeto spread away from each other. Remarkably, such plau-sible scenario leads to a decay rate γ beyond perturbationtheory, i.e., smaller than any power of ε.
As suggested in the discussion above, even though theelimination of domain-wall non-conserving processes isformally valid throughout the entire many-body spec-trum only for α � 1, in the low-energy sector one cannaively perform several perturbative steps to preserve D1(and a few other operators Dr, as well) for arbitrary α.
20
Thus, we reconsider the two-body problem of Eq. (38),and set Jr = J/rα and R =∞. We obtain
H2-body =∑j1<j2
vα(j2 − j1) |j1, j2〉 〈j1, j2|
+ ε∑j1<j2
(|j1 + 1, j2〉+ |j1, j2 + 1〉
)〈j1, j2|+ H.c., (51)
where
vα(r) = 4r∑d=1
dJ/dα + 4r∞∑
d=r+1J/dα . (52)
The potential grows from vα(r = 1) = 4Jζ(α) to vα(r =∞) = 4Jζ(α − 1) (for α > 2) or ∞ (for α ≤ 2). Theasymptotic behavior at large r is
vα(r) ∼r→∞
4J×
r2−α
(2− α)(α− 1) for α < 2,
log r for α = 2,
ζ(α− 1)− r−(α−2)
(α− 2)(α− 1) for α > 2.
(53)Let us now discuss the result in Eq. (53). For α ≤ 2,
the binding potential is confining at large distances.Hence, the hierarchy of domain-wall bound states ex-hausts the excitation spectrum [56]. The infinite en-ergetic cost of isolated domain walls for α ≤ 2 under-lies the persistence of long-range order at finite temper-ature [108, 109], circumventing the standard Landau ar-gument against the existence of thermal phase transi-tions in one dimension. Indeed, at a prethermal level,the effective Floquet Hamiltonian resulting from the re-summation of the BCH expansion supports long-rangeorder at finite temperature for small but finite ε. On theother hand, the two-body potential is bounded at largedistances for α > 2. In this case, freely travelling domain-wall states appear, similarly to finite-range systems. Dueto their nonlocal nature, isolated domain-wall-like exci-tations cannot be locally created or destroyed; thus, theygive rise to topologically protected quasiparticles, whichform a continuum in the excitation spectrum above adiscrete sequence of nontopological bound states. Un-like the case R < ∞, however, the potential only flat-tens out asymptotically for r → ∞. Consequently, thenumber Nα of such bound states critically depends onthe hopping amplitude ε. The finite statistical weight ofdomain-wall-like quasiparticles in thermal equilibrium iswhat prevents long-range order at finite temperature forα > 2.
Figure 14 reports an illustration of the two-body spec-trum, obtained within a semiclassical approximation(which becomes quantitatively accurate in the contin-uum limit, i.e., for highly excited states). Here we setα = 3 and ε = 0.1139. Quantized trajectories undergoa transition between spatially localized (red) and delo-calized (blue), representing a discrete sequence of non-topological confined bound states below a continuum of
0 5 10 15 20 250
π
2 π
Q
P
FIG. 14. Semiclassical energy eigenstates of the two-bodyproblem in Eq. (51) with α = 3 and ε = 0.1139, rep-resented by classical trajectories in phase space (Q,P ) ∈[0,∞)× [0, 2π) governed by the Hamiltonian H2-body(Q,P ) =vα(Q) − 4ε cosP , encircling an area equal to a multiple ofPlanck’s constant h = 2π~ (Bohr-Sommerfeld quantizationrule). Here we take ~ = 1, so that h ≡ 2π. The continu-ous function vα(Q) has been taken as in Eq. (53). For thischoice of parameters, the center-of-mass potential well hostsfour bound states, marked by red trajectories, bounded in Q.All higher excited eigenstates (blue trajectories) are unboundplane waves, and form a continuum labelled by the asymptoticmomentum P for Q→∞.
topological unbound domain walls. For this choice ofparameters, Nα(ε) = 4; however, this number grows asε→ 0. A clear signature of these bound states is furthergiven by the presence of pronounced peaks in the powerspectrum of the absolute value of the order parametertime series. In Figure 15 we show the Fourier transformof finite-size data for α = 2.25. In the long-range limit(R = ∞) the two main peaks correspond to the lower-energy bound states, namely the single and the doublespin-flip excitation. In the classical limit (ε → 0) theirenergies are given respectively by ω1 = 4
∑L/2r=1 r
−α/ζ(α)and ω2 = ω1 + 4
∑L/2r=2 r
−α/ζ(α). On the contrary, themain signatures in the order parameter evolution of thepresence of unbound domain walls in the spectrum is the(very slow) overall decay of the signal in Fig. 13.
Generalizing our argument of Sec. IV B, the numberof bound states Nα can be estimated starting from theobservation that isolated domain walls can freely hop toneighboring sites with amplitude ε, which gives the dis-persion law Ek = 1
2vα(∞) − 2ε cos k. The unperturbedbound-state wavefunctions for ε→ 0, ψ`(r) = δ`,r, ` ∈ N,are precluded from hybridizing with the domain-wall con-tinuum when their unperturbed energy E` = vα(`) is be-low the “ionization threshold” vα(∞)−4ε. The equation
vα(`) = vα(∞)− 4ε (54)
thus identifies the highest stable bound state ` ≡ Nα.Using the asymptotic expansion in Eq. (53), we obtain
Nα ∼ (cαε/J)−1/(α−2) (55)
where cα = 4(α− 2)(α− 1).This result expresses how the number of bound states
diverges as ε→ 0 for all α > 2. Accordingly, the physicalsize of a critical reversed bubble triggering the order pa-rameter meltdown grows unbounded in the asymptotic
21
(b)(a)R = 2
R = 3
R = ∞
0 1 2 3 4 5 610-14
10-12
10-10
10-8
10-6
10-4
ω
ℱ(ω
)α = 2.25, ϵ = 0.015
ω1 ω2
ω1ω2
0 0.04 0.08 0.12
4.0
4.5
5.0
ϵ
ω*
R = ∞
FIG. 15. (a) Discrete Fourier transform of the order parame-ter time series for ε = 0.015, α = 2.25 and different interactionranges R. Data have been obtained via ED with L = 28 andtime series up to 2000 kicks. (b) The position of the first twopeaks (at ω > π) as a function of ε in the long-range caseR =∞, which are expected to match the “classical” single ordouble spin-flip excitation value, for ε = 0 (see main text fordetails).
regime of weak perturbation. In other words, our conjec-ture that the formation of a critical bubble is the fastestprocess leading to the decay of the order parameter, sug-gests that this decay has a non-perturbative origin in thelong-range interacting spin chain,
γ ∼ ε2Nα+1 ∼ εAε−1/(α−2)
, (56)
where we have defined A ≡ 2(cα/J)−1/(α−2). We notethat as α approaches 2 from above, the lifetime γ−1 fromEq. (56) diverges. The lack of an appropriate heatingbound in this regime (as discussed above) prevents usfrom estimating the location of a presumable crossoverregion α ≈ α∗ ≥ 2 between a vacuum-decay driven(α & α∗) and a heating-driven (α . α∗) order-parameterdecay. In any case, we reiterate that the order-parameterlifetime is expected to be non-perturbatively long for allα’s.
To summarize, while the Floquet-prethermalized statefeatures a nonvanishing density of travelling domain wallsprecluding large-scale spatiotemporal order, the dynami-cal production of these excitations by kick imperfectionsis extremely slow, delaying the prethermalization itselfand the eventual heating, ultimately paving the way for along window of nonequilibrium time-crystalline behavior,compatible with the numerical simulations. Our conjec-ture that the fastest order-melting process is the fraction-alization of a critical-size excited bubble into a pair of de-confined kink and antikink leads to the non-perturbativelifetime in Eq. (56).
V. CONCLUSIONS
In this paper, we have established a framework to un-derstand and compute the order-parameter evolution inperiodically driven quantum spin chains, hinging uponthe effective dynamics of emergent domain-wall excita-tions. Within this framework, we have analyzed the im-pact of domain-wall confinement on the order-parameterdecay, and established that a slight increase of the in-teraction range can result in a dramatic extension ofthe lifetime of DTC response, despite the absence of along-range ordered Floquet-prethermal states. The re-sults of this paper delimit and characterize the theoryof time crystals for disorder-free, finite-range interact-ing quantum spin chains driven at arbitrary frequencies,extending the current state of the art. Furthermore,these theoretical results have direct interesting applica-tions for observing confinement-stabilized DTC responsein Rydberg-dressed spin chains [97, 98].
This work corroborates the idea that long-range inter-actions are, in a broad sense, a robust source of non-thermal behavior in quantum many-body systems [56,57, 110–115], and help realizing exotic nonequilibriumstates [116, 117].
An important remark is that the set of initial stateswhich give rise to confinement-stabilized DTC responseis more limited than for the prethermal DTC. Roughlyspeaking, the affordable effective temperature scale is setby the perturbation ε. This is, in a sense, reminiscentof DTC behavior associated with quantum many-bodyscars in systems of periodically driven Rydberg atom ar-rays [28].
As a byproduct, the results reported here resolve anumber of previous controversies on the role of the sys-tem size in clean short-range interacting “time crystals”,consistent with the numerical analysis of Ref. [118], andclarify the origin of the anomalous persistence of the or-der parameter after a quench observed in several numeri-cal investigations of long-range interacting quantum Isingchains [56, 62, 63].
On a technical level, a few points remain open and areleft to future work, including a more rigorous estimate ofthe order-parameter lifetime for algebraically decayinginteractions with α > 2, and a better understanding ofthe role of resonances in the unperturbed spectrum.
ACKNOWLEDGMENTS
We thank D. A. Abanin and W. De Roeck for discus-sions at the early stages of this project, and M. Heyl andR. Moessner for interesting comments on the results. Wegratefully acknowledge M. Heyl for drawing our attentionon the implementation of Rydberg-dressed spin latticesin Ref. [97]. A.L. acknowledges support by the SwissNational Science Foundation.
22
Appendix A: Exact kicked dynamics in theintegrable Ising chain
We start by analyzing the high-frequency regime J, ε�1, which allows us to approximate
KεVJ ' eiJ∑
j[ZjZj+1+(ε/J)Xj ] , (A1)
using the BCH formula truncated to the lowest order.The resulting stroboscopic dynamics is equivalent tothat of a static transverse-field Ising Hamiltonian aftera quench of the field from 0 to ε/J . Exact calculationsof the large-distance behavior of the two-point functionin the thermodynamic limit have shown that the orderparameter does relax to zero exponentially at late time(here |ε/J | < 1) [68],
|m(n)| '[
1 +√
1− (ε/J)2
2
]1/2
e−n/τI , (A2)
with τ−1I = J
[ 43π |ε/J |
3 +O(|ε/J |5)], confirming the fact
that the relaxation time can be very long, depending onthe specific driving parameters. Even though the for-mula (A2) is not expected to be accurate away from thehigh-frequency regime [i.e., for arbitrary J = O(1)], westill do expect the real relaxation time to scale as ∼ |ε|−3
for |ε| � 1.Next, we proceed by analytically solving the dynamics
induced by the Floquet operator KεVJ . To this aim, weexploit the fact that both Kε and VJ are Gaussian op-erators in terms of the spinless fermions c(†)j , introducedby the following Jordan-Wigner transformation
Zj =j−1∏i=1
(1− 2c†i ci)(cj + c†j), Xj = 1− 2c†jcj , (A3)
where {ci, c†j} = δij and {ci, cj} = 0. In terms of suchfermions, the evolution operators read
Kε = eiε∑
j(cjc†j−c
†jcj), VJ = e
iJ∑
j(c†j−cj)(c†j+1+cj+1)
,(A4)
whose product can be diagonalized by first going to theFourier space cj = 1√
L
∑p e−ipjηp, where the sum runs
over momenta that are quantized depending on the spe-cific sector, namely p = 2πn/L + (π/L)(1 + P )/2 withn = −L/2, . . . , L/2 − 1. In terms of positive momenta,both operators can be recast in the following forms
Kε =∏p>0
eη†pKηp , VJ =∏p>0
eη†pVηp , (A5)
where we defined the row vector η†p ≡ (η†p, η−p), and the2× 2 matrices
K = −2iε Z, V = 2iJ [cos(p)Z − sin(p)Y ]. (A6)
For different momenta, the operators entering Eq. (A5)commute between each other, therefore we only need to
combine, for each p > 0, two local Gaussian operators byexploiting the identity [119]
eη†pKηp eη†pVηp = eη†pHηp , eH = eKeV. (A7)
The product of the two matrix exponents can beparametrized as an SU(2) rotation, such that one has
H = iφp rp · σ, (A8)
where σ = {X,Y, Z} is the vector of the Pauli matri-ces, and φp and rp are defined below [cf. Eqs. (A15)and (A17)]. The matrix H is then diagonalized by thefollowing unitary transformation
Up = e−iεZe−iθpX/2, (A9)
with θp defined below in Eqs. (A19). The orthonormaleigenvectors in Up can be used to construct the followingfermionic operators
ξ†p ≡ (ξ†p, ξ−p) = η†pUp, (A10)
in terms of which the Floquet operator is diagonal andreads
KεVJ = ei∑
p>0φpξ†pZξp . (A11)
The dynamics induced by the Floquet operator inEq. (A11) can be understood as a quench originated fromthe vacuum state in the P = +1 sector. In practice, wemay solve the dynamics, by connecting the pre- and post-quench diagonal fermions. The initial diagonal fermionsξ0p are those which diagonalize the Ising Hamiltonian in
Eq. (1), and they are related to the model-independentmomentum-space fermions ηp via the Bogoliubov trans-formation ξ0
p = eipX/2ηp. The pre- and post-quench di-agonal fermions are thus related by the unitary transfor-mation ξp = eiθpX/2eiεZe−ipX/2ξ0
p.To use the asymptotic prediction of the order-
parameter decay in Ref. [68], we need to reshapethe aforementioned unitary transformation connect-ing pre- and post-quench diagonal fermions intoe−iαpZei∆pX/2eiα
0pZ , so that we can absorb the irrele-
vant phases into a redefinition of new diagonal fermionsξp ≡ eiαpZξp and ξ0
p ≡ eiα0pZξ0
p. These finally identify asingle Bogoliubov rotation with angle ∆p, such that
cos(∆p) = cos(θp) cos(p) + sin(θp) cos(2ε) sin(p). (A12)
The Floquet dynamics therefore induces an unavoidableexponential decay of the magnetization, with exact decayrate given by
γ = −∫ π
0
dp
π∂pφp ln | cos(∆p)|. (A13)
A Taylor expansion of Eq. (A13) for small values of εproduces the asymptotics in Eq. (8).
23
1. The exact Floquet operator
The Floquet operator for the integrable kicked Isingmodel has been constructed from the matrices K and V
defined in Eq. (A6) so that the product of the two matrixexponent reads
eKeV =[e−2iε[cos(2J) + i sin(2J) cos(p)] −e−2iε sin(2J) sin(p)
e2iε sin(2J) sin(p) e2iε[cos(2J)− i sin(2J) cos(p)]
]. (A14)
This is a unitary matrix with det[eKeV] = 1 and eigen-values {eiφp , e−iφp} such that tr[eKeV] = 2 cos(φp) with
cos(φp) = cos(2J) cos(2ε) + sin(2J) sin(2ε) cos(p).(A15)
The matrix in Eq. (A14) can be thus parametrized as anSU(2) rotation
eKeV = eiφp rp·σ = cos(φp) + i sin(φp)rp · σ, (A16)where we introduced the unit vector rp =tr[eKeUσ]/[2i sin(φp)], whose components are
rp = 1sin(φp)
sin(2J) sin(2ε) sin(p)− sin(2J) cos(2ε) sin(p)
sin(2J) cos(2ε) cos(p)− cos(2J) sin(2ε)
.(A17)
We can parametrize rp with the angles {θp, ξp} suchthat (notice that is not the standard polar-azimuthparametrization)
rp =
sin(θp) sin(ξp)− sin(θp) cos(ξp)
cos(θp)
. (A18)
Comparing the definition in Eq. (A18) with the resultsin Eq. (A17), we may easily identify ξp = 2ε (for −π/2 <2ε < π/2) independent of the momentum, and
sin(θp) = sin(2J) sin(p)sin(φp)
, (A19a)
cos(θp) = sin(2J) cos(2ε) cos(p)−cos(2J) sin(2ε)sin(φp)
,(A19b)
so that the effective Hamiltonian H = iφp rp ·σ takes theform
H = iφp
[cos(θp) i sin(θp)e−2iε
−i sin(θp)e2iε − cos(θp)
], (A20)
and is diagonalized by the unitary transformation
Up =[
cos(θp/2)e−iε −i sin(θp/2)e−iε−i sin(θp/2)eiε cos(θp/2)eiε
], (A21)
which can be rewritten as the product of two SU(2) ro-tations
Up = e−iεZe−iθpX/2. (A22)Let us notice that, for ε = 0, the transformation inEq. (A22) reduces to a single rotation around x with an-gle θp = p, inducing the usual Bogoliubov transformationdiagonalizing the Ising Hamiltonian in Eq. (1).
2. The order-parameter dynamics
The free-fermion techniques outlined above allowthe computation of the discrete time evolution of thetwo-point function 〈Z0(n)Z`(n)〉, starting from a Z2-symmetric state. Here we assume to work in the P = +1sector, thus preparing the initial state to be |0+〉 ≡(|+〉 + |−〉)/
√2, namely the vacuum of the ξ0
p fermions.From the large-distance behavior of the two-point func-tion we have |m(n)|2 = lim`→∞〈Z0(n)Z`(n)〉.
The order-parameter correlation function can be com-puted using the Majorana fermions
a2j = i(c†j − cj), a2j−1 = c†j + cj , (A23)
which are Hermitian operators satisfying {ai, aj} = 2δij .Indeed, the expectation value of the two-point func-tion involves the evaluation of the Pfaffian of a skew-symmetric real matrix, namely 〈Z0(n)Z`(n)〉 = pf[M(n)],where Mij(n) = −i〈ai(n)aj(n)〉 + iδij with {i, j} ∈{0, 1, . . . , 2`− 1}. Actually, since we only need the mod-ulus of the two-point function, we may just compute thedeterminant, since det[M(n)] = pf[M(n)]2.
Therefore, the building block is the fermionic two-pointfunction, whose time evolution can be computed by re-writing the Majorana operators in Eq. (A23) in termsof the new diagonal operators in Eq. (A10), whose time-evolution is a trivial phase factor. In particular, by usingthe fact that, in the thermodynamic limit,
aj =∫ π
−π
dp
2π e−ipjAηp, A =
[1 1−i i
], (A24)
where a†j ≡ (a2j−1, a2j), we easily obtain after n periods
〈aj(n)a†l (n)〉 =∫ π
−π
dp
2π e−ip(j−l) Wp(n) 〈ηpη†p〉W†p(n),
(A25)where Wp(n) = A Up einφp Z U†p, and the initial cor-relation function is easily computed as 〈ηpη†p〉 =e−ipX/2〈ξ0
pξ0†p 〉eipX/2, with 〈ξ0
pξ0†p 〉 = (1 + Z)/2.
Appendix B: Derivation of the effective FloquetHamiltonian
In this section, we present the explicit construction ofthe Floquet Hamiltonian which conserves at any order
24
the number of domain walls D1 in (11). As explainedin the main text, we follow a different procedure fromRef. [79], which allows us to obtain explicit low-orderformulas more easily. For the sake of generality, we con-sider a kicked Ising chain with variable-range interactionsas in Eq. (33) and mixed-field kick as in Eq. (10). Forconvenience, we parametrize ε = η cos θ, h = η sin θ inEq. (B1). It is useful here to introduce the associatedHamiltonians
HK = −∑j
(cos θXj+sin θZj) , H0 = −∑j
R∑r=1
JrZjZj+r .
(B1)The procedure is based on two steps:
1. using the replica resummation of Ref. [87], we com-bine the kick generator K = e−iHK with the inter-actions VJ = e−iH0 into an approximate FloquetHamiltonian,
K VJ ' e−iHF
; (B2)
2. we apply a static Schrieffer-Wolff transformation toHF to cancel order-by-order all the terms which donot commute with D1.
The two procedures are detailed in the next two subsec-tions, respectively.
1. Replica resummation for the kickedvariable-range and mixed-field Ising chain
To keep the calculation as general as possible, we ac-tually consider a generic 2-body interactions in the z-direction
HK =∑i<j
Ji,jZiZj . (B3)
The translationally invariant case in Eq. (B2) can bereadily recovered at the end setting Ji,j = Jr=j−i.
The approach of Ref. [87] allows to obtain HF definedin Eq. (B2) perturbatively in small η.
HF = H0 + ηH1 + η2H2 + . . . . (B4)
While this series is expected to be generically divergent,signalling the non-existence of a physically meaningfulconserved energy, truncations of the series do have a cru-cial physical meaning as approximate conserved energyfor long times (see Ref. [87] and the discussion in themain text).
The p-th coefficient Hp of the replica expansion, p ≥ 1,is obtained by writing the logarithm
HF = i log e−iηHKe−iH0 (B5)
as a limit log x = limn→01n (xn−1), hence taking the p-th
derivative with respect to η at η = 0 before analyticallycontinuing the result for non-integer n and taking thereplica limit n→ 0,
Hp= i limn→0
1n
1p!∂p
∂ηp
∣∣∣∣η=0
(e−iηHKe−iH0)· · ·(e−iηHKe−iH0)︸ ︷︷ ︸n times
.
(B6)The building block for explicit computations is the op-
erator
Hm = eimH0HKe−imH0 . (B7)
In the model under consideration, one obtains
Hm = − cos θ∑j
[cos(
2mζj)Xj
− sin(
2mζj)Yj
]− sin θ
∑j
Zj . (B8)
where ζj =∑i( 6=j) Ji,jZi is the effective field acting on
spin j. The first term of the replica expansion is thencomputed as
H1 = limn→0
1n
n−1∑m=0
Hm
= −cos θ∑j
(ζj cot ζj Xj + ζj Yj
)−sin θ
∑j
Zj , (B9)
We see that first-order terms contain at most one lo-cal spin-flip operator (Xj or Yj). Compared to the barekick generator, such spin-flip terms are “decorated” bydiagonal operators. The range of these decorations is in-herited from that of the interaction couplings Ji,j in H0.In particular, the amplitude of a local spin-flip processis only influenced by the configuration of the spins in aneighborhood of radius R. In the case of the conven-tional quantum Ising chain (R = 1), Eq. (B9) reduces tothe expression first obtained in Ref. [87]. Equation (B9)constitutes a generalization to Ising chains with arbitrarycouplings. In the high-frequency limit Ji,j → 0, we cor-rectly retrieve the standard BCH expression at lowestorder, H1 ≡ HK . Importantly, divergences appear in thereplica coefficients of particular off-diagonal (“resonant”)transitions. This brings in the need for an incommensu-rability assumption on the nonvanishing interaction cou-plings, as discussed below.
The complexity of the computation increases rapidlywith the perturbative order. To illuminate the generalstructure of the replica series expansion, it is instructiveto report here the result for p = 2. The second term iscomputed as [87]
H2 = i
2 limn→0
1n
∑0≤m1<m2<n
[Hm2 , Hm1
]. (B10)
Calculation gives
25
H2 = + cos2 θ∑j
λjZj
+ cos2 θ∑j1,j2
(µXXj1,j2
Xj1Xj2 + µY Yj1,j2Yj1Yj2 + µXYj1,j2
Xj1Yj2 + µY Xj1,j2Yj1Xj2
)+ cos θ sin θ
∑j
(νXj Xj + νYj Yj
).
(B11)
The coefficients λ, µ, ν are diagonal operators obtained via the following analytical replica limits:
λj = limn→0
1n
∑0≤m1<m2<n
−12 sin[2(m1 −m2)ζj ]
µXXj1,j2= limn→0
1n
∑0≤m1<m2<n
sin(2m2ζj2) sin(2m1ζ′j1,j2
) sin(2m1Jj1,j2)− [m1 ↔ m2]
µY Yj1,j2= limn→0
1n
∑0≤m1<m2<n
cos(2m2ζj2) cos(2m1ζ′j1,j2
) sin(2m1Jj1,j2)− [m1 ↔ m2]
µXYj1,j2= limn→0
1n
∑0≤m1<m2<n
cos(2m2ζj2) sin(2m1ζ′j1,j2
) sin(2m1Jj1,j2)− [m1 ↔ m2]
µY Xj1,j2= limn→0
1n
∑0≤m1<m2<n
sin(2m2ζj2) cos(2m1ζ′j1,j2
) sin(2m1Jj1,j2)− [m1 ↔ m2]
νXj = limn→0
1n
∑0≤m1<m2<n
sin(2m1ζj)− sin(2m2ζj)
νYj = limn→0
1n
∑0≤m1<m2<n
cos(2m1ζj)− cos(2m2ζj)
(B12)
In these equations, ζ ′j1,j2≡∑i(6=j1,j2) Ji,j1Zi = ζj1 −
Jj1,j2Zj2 . The evaluation of the replica limits inEq. (B12) can be easily performed with the help of alge-braic manipulators such as Wolfram Mathematica. Inparticular, finite replica limits exist in all cases, andµY Xj1,j2
≡ 0. A crucial property is that
µαβj1,j2= O(Jj1,j2) as Jj1,j2 → 0 (B13)
for all α, β = X, Y . This implies that for interactionswith finite range R, pairs of spin flips in H2 can onlyoccur at maximum distance R. Furthermore, for long-range interactions, large-distance pairs are suppressed asrapidly as the interaction tail. We note that, comparedto the first order term, a larger (finite) set of resonancesappear at second-order.
The second-order processes described by Eq. (B11) in-clude diagonal terms (∝ ε2, first line), single local spin-flips (∝ hε, third line), and pairs of spin-flips (∝ ε2, sec-ond line). As in the first order term (B9), spin-flip op-erators (Xj , Yj , Xj1Xj2 , Xj1Yj2 , Yj1Yj2) are “decorated”by diagonal operators determined by the configuration ofspins in a neighborhood of radius R from the location ofthe flips. Furthermore, double spin-flips come in quasilo-cal pairs, i.e., spins are flipped at most R sites away fromeach other.
Higher-order terms in the replica expansion have in-creasingly complex coefficients, but crucially exhibit a
hierarchical structure in terms of the maximum numberand the quasilocality of off-diagonal spin-flip processes.This property follows from expressing the p-th order termas p nested commutators of the building block Hm be-fore taking the replica limit [87]. In particular, terms oforder εp in the Floquet Hamiltonian feature a quasilocalproduct of at most p spin-flip local operators (X or Y ),each one located at most R sites away from the next,“dressed” by diagonal coefficients involving operators Zand couplings Jr.
As already noted above, these coefficients may havepoles when particular integer combinations of the cou-plings
∑r drJr, dr ∈ Z, equal a multiple of 2π. Such
divergences stem from corresponding sequence of per-turbative transitions generated by the kick Kε hittinga resonance in the unperturbed spectrum of VJ. Theseresonances make the perturbative series ill-defined. Toresolve this issue, we must assume a condition of strongincommensurability of the couplings, such that integercombinations of {Jr} and 2π remain sufficiently removedfrom zero unless correspondingly large integers are cho-sen. More precisely, we assume the following Diophantinecondition: there exist x, τ > 0, such that for all nonzerointeger arrays n ≡ (n0, n1, . . . , nR) ∈ ZR+1,∣∣∣∣ R∑
r=1nrJr − 2πn0
∣∣∣∣ > x
||n||τ , (B14)
26
where ||n|| ≡ max(n0, n1, . . . , nR). It is not hard toshow that almost all choices of the couplings Jr satisfythis strong incommensurability condition for some x > 0when τ > R (see, e.g., the aforementioned Ref. [79]).This condition guarantees that the resummation of theBCH series is formally well defined to all orders in ε,and that the coefficients do not grow too wildly with theperturbative order.
2. Derivation of the domain-wall conservingeffective Hamiltonian
In this subsection we apply the Schrieffer-Wolf trans-formation to the Floquet Hamiltonian in Eq. (B2), toobtain a new operator which conserves the number ofdomain walls D1. For simplicity, we focus on the shortrange case R = 1 and we restrict to the first order. In thiscase, the effective fields reduce to ζj = J(Zj+1 + Zj−1)and the Floquet Hamiltonian up to first order takes theform
H≤1 = J∑j
ZjZj+1 + ηH1 (B15)
where
H1 = − cos θ[(J cot 2J + 1
2
)∑j
Xj
+(J cot 2J−1
2
)∑j
Zj−1XjZj+1+J∑j
(Zj−1+Zj+1)Yj]
− sin θ∑j
Zj . (B16)
Note that for J � 1 one has J cot 2J ' 1/2, hence weretrieve the Trotter limit of the BCH formula, with HF ∼H0 +HK .
Hence, we split the first-order term H1 into two or-thogonal components, a domain-wall conserving and adomain-wall non-conserving part:
H1 = D + V, (B17)
with[D,∑j ZjZj+1
]= 0 and V purely off-diagonal be-
tween sectors with different number of domain walls. Ex-plicitly we have
D =− cos θ∑j
P ↑j−1XjP↓j+1 + P ↓j−1XjP
↑j+1+
− sin θ∑j
Zj , (B18)
V = (B19)
− cos θ(
2J cot 2J∑j
P ↑j−1XjP↑j+1 + P ↓j−1XjP
↓j+1+
+ 2J∑j
P ↑j−1YjP↑j+1 − P
↓j−1YjP
↓j+1
), (B20)
Now we fix S1 in such a way to exactly cancel V in thetransformed Floquet operator, i.e.,
H ′≤1 = e−iηS1H≤1eiηS1 = J
∑j
ZjZj+1 + ηD+
+ η
!= 0︷ ︸︸ ︷(V − iJ
[S1,∑j
ZjZj+1
])+O(η2). (B21)
This condition for S1 is solved by
S1 = cos θ2∑j
P ↑j−1[Xj − cot(2J)Yj ]P ↑j+1+
+ P ↓j−1[Xj + cot(2J)Yj ]P ↓j+1 (B22)
which corresponds to Eq. (17) in the main text. Withsuch a choice, one recovers the Floquet operator U ′1 =e−iH
′≤1 in Eq. (16).
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