Downloaded by guest on September 25, 2021 ila Berdanier William criticality quantum Floquet www.pnas.org/cgi/doi/10.1073/pnas.1805796115 work. this of of subject understanding the is phases precise parameters Floquet drive A distinct tuning sample— by these setting. finite accessed between a driven points of (multi)critical the boundaries the to the unique at drive only are the but dual—the of TTSB Ising exhibits multiples its and half-integer the phase sub- are This at third, and frequency. (SG), response A order glass bulk (34–36). long-range spatiotemporal harmonic problem spin Two has static 24). the crystal, (1, the and SG/time phases in (PM) four already paramagnet hosts present the model these, this that of anal- shown Extensive chain. has scaling Ising ysis quantum critical random driven the the compute system: to us allow and behavior. TTSB; respon- for those especially distinct freedom, sible of degrees between critical identify enable to a points us procedure; of point (RG) fixed (multi)critical the group as coarse-graining/renormalization emerge Floquet should this Ideally, of phases. Floquet theory a spatially (30), even traps and in ion (31), 33). (32, predictions gases, diamond crystals set- ordered these in ultracold temporal centers probe as the nitrogen-vacancy such to to systems begun rigidity well-isolated have phase such Experiments and notions peri- ting. of symmetry the generalizations broken preserve permitting as that drive, the perturbations of against odicity time stable The is time Floquet (24–29). crystals by period— exhibited crystallinity” (TTSB) distinctive drive breaking “time symmetry the a translation dubbed of to fractions phenomenon sym- leading rational a discrete broken, at response this spontaneously dynamical be (21–23), may systems period. under drive metry Hamiltonian invariance their undriven of an characteristic of symmetry multiple translational share time integer continuous the an systems symmetries. Unlike by global Floquet translations inves- microscopic all time under more addition, system or the In one SPTs”) analogs that possess driven require (“Floquet tigation of typically phases classification These the topological (13–20). to mat- symmetry-protected (2–12) of drive states of “Floquet” the new such engineer via to understanding ter There proposals in (1). from dynamics ranging progress systems, nonequilibrium dramatic of been study has the in results ing T systems driven periodically criticality quantum physics. critical translational the capture to free-fermion time of sufficient simulations models with numerical via crystals.” analysis time associated our “Floquet validate We dis- of wall a formation a the domain provide of and symmetry-breaking terms of to in type physics (multi)criticality tinct Floquet Ising randomness of Floquet infinite description prototypical simple leverage the in we of Working chain, representation a procedure. fermionic of are the group points these fixed renormalization that infinite-randomness strong-disorder argue by We controlled systems. 2018) generically 4, (Floquet) April one-dimensional review driven of for phases (received periodically 2018 distinct 3, between August transitions approved study and We NJ, Princeton, University, Princeton Huse, A. David by Edited and Kingdom; United 3PU, OX1 Oxford Oxford, 94720; a eateto hsc,Uiest fClfri,Bree,C 94720; CA Berkeley, California, of University Physics, of Department ee edvlpsc hoyfrapooyia Floquet prototypical a for theory a such develop we Here, construct to desirable is it developments, these of light In easgmn frbs hs tutr operiodically to strik- structure most the among phase is systems robust many-body quantum of driven assignment he c eateto hsc,TeUiest fTxsa als ihrsn X75080; TX Richardson, Dallas, at Texas of University The Physics, of Department | iodrdsystems disordered a,1 ihe Kolodrubetz Michael , | oprubtv arguments nonperturbative | aybd localization many-body e eateto hsc,Uiest fMsahsts mes,M 01003 MA Amherst, Massachusetts, of University Physics, of Department 0π a,b,c M hc also which PM, .A Parameswaran A. S. , | b aeil cecsDvso,Lwec eklyNtoa aoaoy ekly CA Berkeley, Laboratory, National Berkeley Lawrence Division, Sciences Materials π 1073/pnas.1805796115/-/DCSupplemental ulse nieAgs 9 2018. 29, August online Published at online information supporting contains article This 2 1 the under Published Submission. Direct PNAS a is article This interest. of conflict no paper. declare the authors wrote The and data, analyzed research, tools, performed reagents/analytic research, new designed contributed R.V. and S.A.P., M.K., W.B., contributions: Author on hr l orpae meet. phases four at all states where Majorana point bind of or that description 0 DWs rewritten simple of quasienergy is a terms yields in model picture criticality Ising this Floquet problem, the separate fermion When near a that numerically. or as 0 systems verify near driven we primarily frequency ture to a special at driven (DW) to regions wall (PM domain bulk the of in to that TTSB (SG transitions of boundary contrast, and onset In point terms. the critical similar involve Ising in (random) understood or static be SG/PM can the the of Transi- (i.e., to (46–51). composed TTSB map phases involve being transitions) not proximate as do two that viewed of tions one be in can deep system domains con- the typical a of points, figuration critical infinite-randomness such of vicinity prethermaliza- timescales of 45). experimental unstable, dynamics (44, accessible if reasonably the all even control to relevant will However, tion they (41–43). resonances that debate expect long-range of we via stabil- topic thermalization the a against remains setting, IRFPs nonequilibrium of the ity group In renormalization procedure. space real (RSRG) (IRFP) strong-disorder point a fixed via infinite-randomness best accessed an then of are terms in phases described Floquet to one-dimensional system transitions distinct that between argue driven We feature- (37–40). periodically a state to infinite-temperature generic thermalize less than rather a structure phase for Floquet have required is which nlaefo:Dprmn fPyisadAtooy nvriyo aiona Irvine, California, of University Astronomy, and Physics 92697.y of CA Department from: leave [email protected] y Email: addressed. be should correspondence whom To pligi oapooyia rvnsse,tedie Ising of general. understanding driven in and an the criticality give and Floquet picture system, points critical driven applicable MBL identify prototypical we chain, a universally Floquet to a distinct it that applying providing between criticality By transitions the of phases. the question the at consider emerges we years. “time work, recent as this phases in such interest In MBL widespread equilibrium, gained driven in has these forbidden crystallinity,” phenomena of host ability to many-body The as (MBL). known tem- phenomenon localization infinite a a bland through to a quenched state to of rise perature thermalizing presence avoid give the can In can they disorder, structure. symmetry phase dynamical translation rich time sys- whose nonequilibrium are tems systems “Floquet” driven Periodically Significance h nvraiyo u nlsstrso h atta,i the in that, fact the on turns analysis our of universality The disorder, quenched of presence the on relies approach Our d,2 d h uofPirsCnr o hoeia hsc,Uiest of University Physics, Theoretical for Centre Peierls Rudolf The n oanVasseur Romain and , PNAS NSlicense. 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PHYSICS Model Floquet systems are defined by a time-periodic Hamiltonian H (t) = H (t + T ). For reasons similar to Bloch’s theorem, eigen- −iEαt states satisfy |ψα(t) = e |φα(t) , where |φα(t + T ) =

|φα(t) , and we set ~ = 1 (52, 53). In contrast to the case of static Hamiltonians, the quasienergies Eα are only defined modulo 2π/T , voiding the notion of a “ground state.” An object of fundamental interest is the single-period evo- lution operator or Floquet operator F ≡ U (T ). If disorder is strong enough, F can have an extensive set of local con- served quantities. This implies area-law scaling of entangle- ment in Floquet eigenstates and consequently the absence of thermalization (54). Unlike generic (thermalizing) Floquet systems, such many- body localized (MBL) Floquet systems retain a notion of phase structure to infinitely long times. For concreteness, we focus on the driven quantum Ising chain, the simplest Floquet system that Fig. 1. Phase diagram deduced by fitting “effective central charge” from entanglement scaling (see Fig. 3 for details). (Insets) Sketches of infinite- hosts uniquely dynamical phases. The corresponding Floquet randomness coupling distributions along the critical lines (1–4) and at the operator is multicritical point (5).

−i T P J σz σz +U σz σz −i T P h σx +U σx σx F = e 2 i i i i+1 i i+2 e 2 i i i i i+1 , [1] Eq. 1. The transformations Jj 7→ Jj + π and hj 7→ hj + π both α separately map F onto another interacting Ising-like Floquet where σ are Pauli operators. Here Ji and hi are uncorre- i SI Appendix lated random variables, and U corresponds to small interaction operator with precisely the same eigenstates ( ), but possibly distinct quasienergies: Jj 7→ Jj + π preserves F exactly terms that respect the Z2 symmetry of the model generated Q x (up to boundary terms), while hj 7→ hj + π sends F 7→ FGIsing = by GIsing = i σi . For specificity, we draw couplings h, J ran- h,J GIsingF . Note that, despite not changing bulk properties of the domly with probability nπ from a box distribution of maxi- eigenstates, these transformations map the PM to the 0πPM and mal width about π—namely, [π/2, 3π/2]—and with probability π y h,J h,J the SG to the SG. Additionally, a global rotation about the n0 = 1 − nπ from a box distribution of maximal width about axis takes hj 7→ −hj . Below, we fix phase transition lines by com- 0—namely, [−π/2, π/2]. The reasons for this parametrization bining these Floquet symmetries with the usual Ising bond-field will become evident below. F corresponds to an interacting duality that exchanges h and J (and hence SG and PM in the transverse-field Ising (TFI) model where for U = 0 we strobo- static random case). scopically alternate between field and bond terms. It is help- ful to perform a Jordan–Wigner transformation to map bond Infinite-Randomness Structure and field terms to Majorana fermion hopping terms, yield- In analogy with the critical point between PM and SG phases in ing a p-wave free-fermion superconductor with density–density the static random Ising model (both at zero temperature interactions given by U . In the high-frequency limit T → 0, we and in highly excited states), we expect that the dynamical −iH T can rewrite F = e F by expanding and re-exponentiating Floquet transitions of Eq. 1 are controlled by an IRFP of order-by-order in T , and the Floquet Hamiltonian HF recov- an RSRG procedure. At a static IRFP, the distribution of ers a static Ising model. We work far from this limit, set- the effective couplings broadens without bound under renor- ting T = 1. malization, so the effective disorder strength diverges with the RG scale. A typical configuration of the system in the Phases and Duality vicinity of such a transition can be viewed as being com- h J 1 Observe that (nπ , nπ ) = π (hi , Ji ), where the bars denote dis- posed of puddles deep in one of the two proximate phases, order averages and hence tune between phases of model Eq. in contrast with continuous phase transitions in clean systems 1 analogously to h, J in the clean case. The four phases are (50, 51). summarized in the phase diagram in Fig. 1. The trivial Flo- To generalize this picture to the Floquet Ising setting, we must quet PM breaks no symmetries and has short-range spin–spin identify appropriate scaling variables. For Ji , hi  π, we recover correlations. The SG spontaneously breaks Ising symmetry with the criticality of the static model controlled by an IRFP if Ji and long-range spin correlations in time or equivalently localized hi are drawn from the same distribution. In this case, the relevant edge modes at 0 quasienergy in the fermion language. These operator at the critical point controls the asymmetry between the two phases are connected to the undriven PM and ferromag- Ji , hi distributions. At static IRFPs, critical couplings are power net/SG phases of the random Ising model (34–36). Unique to law-distributed near 0. The absence of energy conservation in the the Floquet system are the π-SG (πSG) and the 0π PM (0πPM). Floquet setting complicates this picture since there is no longer The πSG spontaneously breaks both Ising and time translation a clear notion of “low” energies. However, a natural resolution symmetry in the bulk. Often referred to as a “time crystal” (1, is to allow for fixed-point couplings to be symmetrically and 25, 27), it maps to a fermion phase with localized Majorana power law-distributed around both 0, π quasienergy (or more edge modes at π quasienergy (55). Finally, the 0πPM has short- generally, all quasienergies that can be mapped to 0 by apply- range bulk correlations but also boundary TTSB; its fermion dual ing Floquet symmetries of the drive). This introduces a special has both 0 and π-Majorana edge modes and is a simple exam- parameter for Floquet–Ising IRFPs—namely, the fractions n0 ple of a Floquet SPT. In the fermion language, DWs between and nπ of couplings near 0 and π, respectively. Evidently, we these different phases host either 0 or π-Majorana bound states have n0 = 1 − nπ. We will show that there is a type of IRFP (Fig. 2A) central to the infinite-randomness criticality discussed specific to the Floquet setting for nπ = 1/2, where the criti- below. cality is tuned by the asymmetry between the distributions at The absence of energy conservation in the Floquet setting 0 and π quasienergy, at fixed values of the Ji –hi distribution admits two new eigenstate-preserving changes of parameter to asymmetry.

9492 | www.pnas.org/cgi/doi/10.1073/pnas.1805796115 Berdanier et al. Downloaded by guest on September 25, 2021 Downloaded by guest on September 25, 2021 nesodi em fDsbtenS and SG such between state each DWs language, of fermion terms in understood between if DWs whereas to correspond these “0π 1/2), of class If new setting. a Floquet by distribu- captured and IRFP are 0 distinct physics near are there couplings and that for 0 fact tions between the from fluctuations follows spatial This quenched strong exhibit asymmetry the same. is the transition of the distributions the tuning tran- between a parameter drive relevant Although Majoranas the structure. DW DW the between here previous sition quasienergy, the zero recover global at to a still drive, factor the may we of at 1, centered ref. now but lowing For distribution these. IRFP of single terms a in DW understood each be language can fermionic state the Majorana a in of binds that terms show in results understood Standard be where can regions point between critical DWs the and case, static edne tal. et Berdanier distinct. toward are flow transition by the described is at it IRFP of Ising, an class random still universality is the transition although this Ising that usual emphasize the tunes We that transition. asymmetry field-bond emergent the of this pendently of the criticality between the π terms Crucially, tunneling short-ranged. the remain exponential Thus, stretched is 48). or coupling (46, 0 tunnel near typical are the criti- and couplings quantum the infinite-randomness where regions separating cal DWs still to are there bound distribution, where same configuration the a from from drawn start are we if between Even puddles. tunneling the the as suppressed MBL, exponentially are is Majoranas puddles 2B). intervening (Fig. chain the initial If the from independent are dynamics whose Since given 0π 2A). a (Fig. a modes, phases edge at adjacent topological the are of they modes edge the paring For Emergent Tunnel- quasienergy. configuration. or states multicritical hatched DW typical 0 and between A ing (B) either TTSB, order. SG bulk/boundary at bulk exhibit states have regions regions edge Blue topological (red). host quasienergy these language, fermionic 2. Fig. B A Mjrn hi stndby tuned is chain -Majorana ut ifrn hsc rs for arise physics different Quite W,laigt eodeegn aoaafrinchain fermion Majorana emergent second a to leading DWs, J 0π i γ ˜ , i π W ewe rxmt hsso h rvnIigmdl In model. Ising driven the of phases proximate between DWs (A) h Wcnol opet other to couple only can DW i tquasienergy at er0(n 0 near π h π -Criticality i useeg,adhne h pcrlpoete of properties spectral the hence, and quasienergy, ssaland small is π SG–0π π J γ i  0,π ssaland small is γ ˜ ,teIF itiuini iia othe to similar is distribution IRFP the 1), M wn oteglobal the to owing PM, ilstoidpnetcan rud0and 0 around chains independent two yields i 0 π tzr useeg,adtetransition the and quasienergy, zero at hsmyas eddcdb com- by deduced be also may This . J J i n i J ∼ π 0π and i π (with π  uhta h eeatcritical relevant the that such , WtasaMjrn bound Majorana a traps DW h rtclbhvo a be can behavior critical the , n h h h π i i i ∼ ∼ otepyisi essentially is physics the so n n hs where those and π π 1/2 π π ∼ = us rmbt terms both from pulse π Mjrnsbudto bound -Majoranas n cnitn with (consistent Ds nqet the to unique -DWs” GadP regions, PM and SG π e π 2 1 hr h couplings the where −` Mjrn trapped -Majorana ∼ tciiaiy,inde- criticality), at ihtesize the with , π eaanhave again we 1, π π oee,fol- However, . IAppendix ), (SI 0π ple Again, -pulse. π π M nthe In PM. -Majoranas -Majoranas J J ∼ i and  e n − π ` h √ π π of ∼ i h π π ` - . . a ecntutdepiil as explicitly additional constructed the be where can point, emergent multicritical the an at predicts try also picture Our chains. epredict we as size charge system central effective with scale should S particular entropy Assuming size entanglement in 62). system 61, and of (56, 48) system entropy 46, a entanglement 35, eigenstate This the (27, 48). eigen- in (or functions (46, dynamical correlation respectively in signatures state) quantities, universal typical have and should or lines scaling average critical for the from 1 deviation Ising as the infinite-randomness diverges characterizes length show con- correlation transitions The we the scaling: results, of IRFP all that standard clude with reasoning this Combining (Multi)Criticality Floquet static essentially the of symmetry-based all analysis both when extends used are four- (multi)criticality reasoning distributions The Floquet of four class. Ising ture all universality the which in Ising symmetric—is point—at random multicritical the phase in Ashkin–Teller always random are the critical of infinite-randomness class new universality a (58). model the to in system point the interactions intermediate drive that possibility thermaliza- might the drive multicritical open to leave the 57– likely we are at (47, tion, interactions irrelevant weak strong very relatively are and are interactions point, interactions weak as While can so long 59). and as irrelevant, also ignored are be terms such point, multicritical the Floquet-umklapp permit also interactions irrel- terms therefore While are 47). and (35, couplings other evant the than faster (U much RG and interactions 0 of the presence effective within the This Interactions in chains. persists renormaliza- Majorana also effective have decoupling two toward now the or of can decoupling 0 random we static toward crucially, the tion except for those model, self- match Ising equations which treatment. RG IRFP perturbative resulting local an disorder The a strong to in the leads justifies coupling process the consistently strongest this the putting Iterating in ones, eigenstate. involved We weaker Majoranas 56). of before 36, pair (35, couplings tech- Hamiltonians stronger RG MBL standard decimate static using for static analyzed designed effectively be niques can an chains by Majorana described two is and 0 Hamiltonian near of only criticality the couplings Appendix) instead (SI asymme- considering explicitly confirmed the by be can by picture This driven couplings. is chain 0 one the between chains, of try Majorana criticality static the effectively While ( two 0 quasienergy of under- near be terms can transitions in Floquet stood dynamical the crit- at the behavior that ical suggests hypothesis infinite-randomness above The Treatment identify RG we Therefore, system. TTSB. open in an changes of ends the the at SG–0π TTSB the Similarly, tal. the since TTSB, rtclfor critical L hrfr,frsfcetywa neatos h rtcllines critical the interactions, weak sufficiently for Therefore, PM–π the that Observe ∼ 0π (˜ c γ ˜ W stedgeso reo htaersosbefor responsible are that freedom of degrees the as DWs )ln /6) i 0 γ ˜ j 0 c ˜ γ n ˜ k π n2 ln = π J pt ouiesladtv otiuin,with contributions, additive nonuniversal to up L, F γ ˜ = PNAS and l π 2 π 2 1 e = htwudcul h rtcl0and 0 critical the couple would that Gi h rttpcleapeo iecrys- time a of example prototypical the is SG hr hr sasmer ewe and 0 between symmetry a is there where , L u oteciiaiyo h and 0 the of criticality the to due h −i γ ˜ | n pnbudr odtos h half- the conditions, boundary open and i si h sa sn hi,the chain, Ising usual the in as 0 2H etme 8 2018 18, September n h te erquasienergy near other the and ) c ˜ F n2 ln = h Mtasto novsteostof onset the involves transition PM h yaia rpriso these of properties dynamical The . π i Gtasto novsteostof onset the involves transition SG × opig r near are couplings π useege in quasienergies sn nvraiycas hspic- This class. universality Ising /2 J sn hisflwtwr under 0 toward flow chains Ising D i , 6) ttemlirtclpoint, multicritical the At (61). h = i F  √ 1 | F F 6)case. (60) o.115 vol. 2 2 hc hudhave should which , † ξ |∆ ∼ 2,2,2) o a For 27). 26, (24, F Z π 2 | eetn the reflecting | × Z −ν 2)adthe and (27) o 38 no. π 2 Z π -Majorana where , π symmetry 2 hisat chains ν hi is chain symme- π 2 = | ( 6 =0). γ ˜ 9493 i π or ∆ ). π

PHYSICS multicritical configuration with couplings near 0 or π, we find Experimental Consequences that D is distinct from the original Ising symmetry of the Let us now turn to some experimental consequences of the above model and coincides with the fermion parity of the emergent predictions. Recent advances in the control of ultracold atomic π-Majorana chain, arrays have brought models such as Eq. 1 into the realm of exper- imental realizability (65–67). The model hosts a time-crystal Y π D = γ˜j . [2] phase (the πSG), the phenomenology of which has recently been j ∈{0π DWs} directly observed (30, 31). Even though, as mentioned earlier, these critical lines may eventually thermalize due to long-range Numerics resonances (41–43), the dynamics of the Ising universality class As stressed above, our picture of these transitions relies on should persist through a prethermalization regime relevant to all the infinite-randomness assumption. To justify this and to con- reasonably accessible experimental timescales (44, 45). Thus, the firm our analytical predictions, we have performed extensive dynamical signatures of the transitions we have identified should numerical simulations on the noninteracting model, leveraging be readily experimentally observable. its free-fermion representation to access the full single-particle One prominent experimental signature of these physics is the spectrum and to calculate the entanglement entropy of arbitrary scaling behavior of the dynamical spin–spin autocorrelation func- R ∞ −iωτ z z eigenstates (SI Appendix). We average over 20,000 disorder real- tion in Fourier space C (ω, t) ≡ 0 dτe hσi (t + τ)σi (τ)i, izations (with open boundary conditions), randomly choosing a with the overline representing a disorder average (27). In Floquet eigenstate in each. accordance with the random Ising universality class, the spin– z z Given our parametrization of disorder, the combination spin autocorrelation function will scale as hσi (t)σi (0)i∼ 1 (nh − nJ ) J 1/log2−φ t 2 π π provides a measure of the asymmetry between (35), with√ the overline representing a disorder 1 h J φ = (1 + 5)/2 and h couplings, while 2 (nπ + nπ ) measures the average prob- average and the golden ratio. Performing the ability of a π coupling. Therefore, from our reasoning above Fourier transform, our analysis then predicts that along the π π and using the usual Ising duality, we expect a critical line for nh = nJ critical line of the model, the Fourier peak at 0 π π 2−φ π nJ = nh . Combining the Ising duality with Floquet symmetries quasienergy will decay as C (0, t) ∼ 1/ log t; along the nh = π π π leads to another critical line nJ + nh = 1, where we expect 0π 1 − nJ critical line, the peak at π quasienergy will decay the infinite-randomness behavior. Note that the bare disorder distri- same way as C (ω/2, t) ∼ 1/ log2−φ t, and at the multicriti- butions are far from the IRFP expected to emerge at criticality. cal point, both peaks will decay in this way simultaneously, Nonetheless, as shown in Fig. 3, we observe clear logarith- giving π π mic scaling of entanglement along the self-dual lines nJ = nh π π 1 and nJ + nh = 1 of Eq. 1. We find c˜ ≈ ln 2/2, consistent with C (0, t) ∼ C (ω/2, t) ∼ 2−φ . [3] the prediction that the lines are in the random Ising univer- log t sality class. Deep in the phases, we find c˜ ≈ 0 consistent with This slow, logarithmic decay, independently for the decoupled the area-law scaling expected for Floquet MBL phases (63, chains at 0 and ω/2, serves as an unambiguous signature of the h J 64). At the multicritical point nπ = nπ = 1/2, we find c˜ ≈ ln 2, universal multicritical physics we describe. The fact that the two consistent with our expectation of two decoupled critical Ising decays are independent is highly nontrivial, since generic Z2 × chains. Although stability to quartic interchain couplings can- Z2 multicritical points would have distinct scaling from either Z2 not be addressed in this noninteracting limit, we expect it on individually. general grounds (47, 57, 58)—modulo usual caveats on thermal- ization. Fig. 1, which shows the entanglement scaling across the Discussion entire phase diagram, summarizes these results. Finally, we have We have presented a generic picture of the transitions between also numerically calculated the relative number of single-particle MBL Floquet phases and applied it to study the criticality of quasienergies near 0 and near π, finding good agreement with a the periodically driven interacting random Ising chain. Our work simple prediction from the infinite-randomness DW picture (SI can be generalized to more intricate Floquet systems, under π π Appendix). Moreover, Fig. 1 clearly shows that changing nJ ± nh the (reasonable) assumption that they flow to infinite random- tunes across the critical lines, confirming that these parame- ness under coarse-graining. The resulting IRFP is enriched in ters control distribution asymmetries as in the IRFP picture the Floquet setting: Each distinct invariant Floquet quasienergy (Fig. 1, Insets). hosts an independent set of fixed-point coupling distributions. (For instance, the Zn model has n such invariant quasiener- gies, 2πk/n, with k = 1, ... , n.) Systems at conventional IRFPs are tuned across criticality by adjusting the imbalance between distributions of distinct couplings at the same quasienergy. At Floquet IRFPs, we may hold such single-quasienergy imbalances fixed and instead tune the imbalance between the distributions of couplings at distinct quasienergies. Transitions driven by such cross-quasienergy imbalances will usually involve an onset or change of TTSB in the bulk or at the boundary and, in this sense, describe “time crystallization.” In some cases, it may be possible to leverage a Jordan–Wigner mapping in conjunc- tion with these infinite-randomness arguments to arrive at a DW description of the critical/multicritical physics. We antici- pate that a variety of Floquet symmetry-breaking/SPTs will be amenable to similar analysis, but we defer an exhaustive study to future work. Fig. 3. Scaling with system size of disorder- and eigenstate-averaged entanglement entropy S for a cut at L/2. Dashed lines show predicted slopes Materials and Methods for strong-disorder Ising criticality along the transition lines (blue, black); Numerical simulations were performed on the TFI chain, where we extract this doubles at the multicritical point (red). the entanglement entropy across a cut of length l = L/2 from the boundary

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)γ hsrttst h ai fsnl-atceFloquet single-particle of basis the to rotates This . A oe hteeg rmorRR itr show picture RSRG our from emerge that modes = = 2 For . 2l x obigi nobokdaoa om ie ya by given form, block-diagonal into it bring to Thus, α. j L−2 j L−1 j H 0 =0 =0 C forHmloinwr o led in already not were Hamiltonian our If 2α. sin hr h oiiesg cuistemd and mode the occupies sign positive the where , +1  e o Phys Mod Rev 0.Defining (0). 2 i Q 0 γ e F 116:250401. =  2j −iT J h sin(2 cos(2 C +1 γ j A j γ 0 diag(±σ γ 2j Q 1 2j 2j and γ +1 H hsRvB Rev Phys +1 F T C 2j γ 1 e . k ij +2 2j αAB , k = γ 49:013001. t H t = C +1 σ cos(2 ) B 2j ) nteMjrn agae our language, Majorana the In . 1 (t Q j z hsRvX Rev Phys +2 2δ gr C(06 lqe niern of engineering Floquet (2016) HC agerl 89:011004. Ae ¨ and dsic)Mjrn prtr,we operators, Majorana (distinct) x = ) 2 T − Γ(t ) C Γ = D i L ij =1 −αAB ij  sin(2 {γ 2 − 82:235114. = T 0 (T Q H ) ecnte oaebc to back rotate then can We . 1 (t ≡ i ≤ C 2 2 ,

l ≤

2l T Q 1 T C o 2α cos h aoaaanti- Majorana The . T 1 γ 2δ D T k 2L ij eseta the that see we )Q, 1 2l =1 HQ 1 ) = + ij T +1 (T , ocntutthe construct To . δ T 1  = γ ij )Q 2 − i γ 2 + = P 2j = 1 B . a hr the where , T hsRvLett Rev Phys i and 2α, sin 1, ij  where , hsRvX Rev Phys i γ γ 2i 0 i † γ = σ 2i 0 Adv j x +1 [4] [6] [5] γ = i a . , 9 o ,Sejt J(06 iodrdCenisltrwt w-tpFoutdrive. Floquet two-step a with insulator Chern Disordered (2016) GJ Sreejith S, Roy one in phases 19. topological symmetry-protected Floquet Abelian (2016) F Harper R, topological Roy interacting driven of 18. Classification periodically (2016) A in Vishwanath phases T, Morimoto topological AC, of Potter Classification 17. (2016) C Nayak DV, Else interacting one-dimensional 16. of structure Phase (2016) SL Sondhi CW, Keyserlingk von interact- 15. one-dimensional of structure Phase (2016) SL Sondhi CW, Keyserlingk von topological 14. symmetry-protected Stroboscopic (2015) C Chamon LH, Santos T, Iadecola 13. graphene. in effect hall Photovoltaic G (2009) H 12. Aoki T, Oka of classification general 11. the and singularities Topological (2015) MS Rudner F, Nathan 10. is of orthogonality the From ecnwieti orlto ucina rsn rmasnl atceden- particle single a from arising matrix as sity function correlation this write can We is function two-point h suoievle r ragdsc that such define arranged Now are pseudoeigenvalues the σ n h nageetetoy rt h est arxas matrix density the write entropy, entanglement the find of part antisymmetric function the correlation diagonalize the Then, cut. entanglement the outside hs h w-on ucinis function −i two-point the Thus, via so operators Majorana from operators hn h nageetetoyi nabtayFoutegntt is eigenstate Floquet arbitrary an in −Trρ entropy entanglement the Then, rie(oSAP) n nvriyo ascuet tr-pfns(oR.V.). (to and funds W.B. start-up Massachusetts (to of California, Barbara University of and Santa University S.A.P.); the (to California, at Irvine DMR-1455366 of Grant NSF University President’s from NanoSys- support the S.A.P.); via California at the (CAIQUE) and Institute Emulation CA-15-327861 tems (PRCA) Quantum from Award Materials support for Catalyst for travel Research Institute Institute M.K.); (to Theory California initiative (BES) the (TIMES) Sciences Berkeley Spectroscopies Energy Energy US Basic and Lawrence the Depart- Energy the of of and from Science, DEAC02-05CH11231, ment of Contract under Grad- funding Office Energy Director, of Hellman the Department (LDRD) Directed by a provided Laboratory Development Laboratory, from through National support and Foundation acknowledge Hellman also Research We the (NDSEG) Fellowship. Fellowship from uate Graduate Engineering and through & (DoD) Program Defense Science of Defense Department National US the W.B. the ACI-1053575. from and Grant (70), support Foundation (XSEDE) Huse Science acknowledges Environment National used David by Discovery work supported to Engineering is This which and manuscript. grateful the Science are on Extreme comments and especially the detailed and discussions for Huse, Khemani useful David Vedika for Moore, Joel Roy Nahum, Sthitadhi Adam Nayak, Chetan Else, ACKNOWLEDGMENTS. = q γ e hsRvB Rev Phys dimension. dimension. one in phases Floquet systems. interacting phases. Symmetry-broken ii. systems. Floquet phases. topological symmetry-protected Abelian 93:245145. i. systems. Floquet ing phases. system. many-body quantum driven systems. Floquet–Bloch e D 2k   2k 0 diag r ,e l 21)Ehneetadsg hneo antccreain na in correlations magnetic of change sign and Enhancement (2018) al. et F, org ¨ γ k k 0 log −1 −1,i −e +e 2k 0 0 − − γ −1 ρ  ρ q hsRvB Rev Phys 2k 0 = −λ 0 2k = k k γ ρ = γ 94:214203. 2k ,j Y 0 hsRvB Rev Phys = = − λ k 0 i −i E λ = k −i 0 P Z 1 00 i [p =  (2c qγ Q q PNAS i l k k tanh =1 q 2k |0 92:125107. [ k k † 2k Then, . hsRvB Rev Phys p 00 hscnb civdb cu eopsto,where decomposition, Schur a by achieved be can This . c k e

0 k k ih0 −1,i 94:125105. i γ −1,i  e Phys J New log  − etakVdk hmn,SiaiSnh,Dominic Sondhi, Shivaji Khemani, Vedika thank We 2k k 0 k | γ k a o define Now . −1 ) hsgvstedniymti as matrix density the gives This 1). q 2k q 0 | = p etme 8 2018 18, September 2k 2k q, + ρ −1 γ k 00 ,j q = 2k 0 + (1 93:201103. q (δ γ T ,j

αi 2k Y σ 0 ij (1 − = hsRvX Rev Phys k = 17:125014. where q, q + o ecncntutcmlxfermion complex construct can we Now . Nature − λ p β − e e λ k k i p  )|1  δ = k

k k k kk 00 D (2c γ µ log(1 ) δ k + γ (q 0 2k 0 k αβ ih1 553:481–485. δ k † 2k 0 c e 2k = k 6:041001. γ c hsRvB Rev Phys k k −1 0 − q 2k oteol ovnsigterm nonvanishing only the so , 0 00 = k k −1) |λ 00 |], −1 −1,i sotooa and orthogonal is γ k − γ hs h nynonvanishing only the Thus, . k 2k 0 | . giving |, 2k 0

p E q −1 = k p o.115 vol. 2k σ )] = k 2 i λ 00 93:245146. +i . ,i = −i +1 k ,j hsRvB Rev Phys γ . − e 2k 0 D =   q k k , 2c e λ 2k = | + c − k † i k † 00 c tanh e o 38 no. = and ,i k k − q 79:081406. − 2k γ σ k 2k 0 1 −1 00 hsRvB Rev Phys σ . a form has E −1 ,j T )). (µ = | 2 = −i k λ 9495 .To ). −σ γ S k 2k 0 [7] [8] [9] = = , .

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