Floquet quantum criticality William Berdaniera,1, Michael Kolodrubetza,b,c, S. A. Parameswarand,2, and Romain Vasseure aDepartment of Physics, University of California, Berkeley, CA 94720; bMaterials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720; cDepartment of Physics, The University of Texas at Dallas, Richardson, TX 75080; dThe Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, United Kingdom; and eDepartment of Physics, University of Massachusetts, Amherst, MA 01003 Edited by David A. Huse, Princeton University, Princeton, NJ, and approved August 3, 2018 (received for review April 4, 2018) We study transitions between distinct phases of one-dimensional Our approach relies on the presence of quenched disorder, periodically driven (Floquet) systems. We argue that these are which is required for a generic periodically driven system to generically controlled by infinite-randomness fixed points of a have Floquet phase structure rather than thermalize to a feature- strong-disorder renormalization group procedure. Working in less infinite-temperature state (37–40). We argue that transitions the fermionic representation of the prototypical Floquet Ising between distinct one-dimensional Floquet phases are then best chain, we leverage infinite randomness physics to provide a described in terms of an infinite-randomness fixed point (IRFP) simple description of Floquet (multi)criticality in terms of a dis- accessed via a strong-disorder real space renormalization group tinct type of domain wall associated with time translational (RSRG) procedure. In the nonequilibrium setting, the stabil- symmetry-breaking and the formation of “Floquet time crystals.” ity of IRFPs against thermalization via long-range resonances We validate our analysis via numerical simulations of free-fermion remains a topic of debate (41–43). However, even if unstable, models sufficient to capture the critical physics. we expect that they will control the dynamics of prethermaliza- tion relevant to all reasonably accessible experimental timescales quantum criticality j disordered systems j many-body localization j (44, 45). periodically driven systems j nonperturbative arguments The universality of our analysis turns on the fact that, in the vicinity of such infinite-randomness critical points, a typical con- figuration of the system can be viewed as being composed of he assignment of robust phase structure to periodically domains deep in one of two proximate phases (46–51). Transi- driven quantum many-body systems is among the most strik- T tions that do not involve TTSB (i.e., the SG/PM or 0πPM/πSG ing results in the study of nonequilibrium dynamics (1). There transitions) map to the static (random) Ising critical point and has been dramatic progress in understanding such “Floquet” can be understood in similar terms. In contrast, transitions that systems, ranging from proposals to engineer new states of mat- involve the onset of TTSB in the bulk (PM to πSG) or at the ter via the drive (2–12) to the classification of driven analogs boundary (SG to 0πPM) can be understood in terms of a class of symmetry-protected topological phases (“Floquet SPTs”) of domain wall (DW) special to driven systems that separate (13–20). These typically require that the system under inves- regions driven at a frequency primarily near 0 or near π—a pic- tigation possess one or more microscopic global symmetries. ture we verify numerically. When the Ising model is rewritten In addition, all Floquet systems share an invariance under as a fermion problem, this picture yields a simple description of time translations by an integer multiple of their drive period. Floquet criticality in terms of DWs that bind Majorana states at Unlike the continuous time translational symmetry characteristic quasienergy 0 or π, allowing us to further study the multicritical of undriven Hamiltonian systems (21–23), this discrete sym- point where all four phases meet. metry may be spontaneously broken, leading to a distinctive dynamical response at rational fractions of the drive period— a phenomenon dubbed “time crystallinity” (24–29). The time Significance translation symmetry breaking (TTSB) exhibited by Floquet time crystals is stable against perturbations that preserve the peri- Periodically driven “Floquet” systems are nonequilibrium sys- odicity of the drive, permitting generalizations of notions such tems whose time translation symmetry can give rise to a as broken symmetry and phase rigidity to the temporal set- rich dynamical phase structure. In the presence of quenched ting. Experiments have begun to probe these predictions in disorder, they can avoid thermalizing to a bland infinite tem- well-isolated systems such as ultracold gases, ion traps (30), perature state through a phenomenon known as many-body nitrogen-vacancy centers in diamond (31), and even spatially localization (MBL). The ability of these driven MBL phases ordered crystals (32, 33). to host phenomena forbidden in equilibrium, such as “time In light of these developments, it is desirable to construct crystallinity,” has gained widespread interest in recent years. a theory of Floquet (multi)critical points between distinct In this work, we consider the question of the criticality that Floquet phases. Ideally, this should emerge as the fixed point of a emerges at the transitions between distinct Floquet MBL coarse-graining/renormalization group (RG) procedure; enable phases. By providing a universally applicable picture and us to identify critical degrees of freedom, especially those respon- applying it to a prototypical driven system, the driven Ising sible for TTSB; and allow us to compute the critical scaling chain, we identify critical points and give an understanding of behavior. Floquet criticality in general. Here, we develop such a theory for a prototypical Floquet system: the driven random quantum Ising chain. Extensive anal- Author contributions: W.B., M.K., S.A.P., and R.V. designed research, performed research, ysis has shown that this model hosts four phases (1, 24). Two contributed new reagents/analytic tools, analyzed data, and wrote the paper. of these, the paramagnet (PM) and the spin glass (SG), are The authors declare no conflict of interest. present already in the static problem (34–36). A third, the π This article is a PNAS Direct Submission. SG/time crystal, has spatiotemporal long-range order and sub- Published under the PNAS license. harmonic bulk response at half-integer multiples of the drive 1 To whom correspondence should be addressed. Email: [email protected] frequency. This phase and its Ising dual—the 0π PM, which also 2 On leave from: Department of Physics and Astronomy, University of California, Irvine, exhibits TTSB but only at the boundaries of a finite sample— CA 92697.y are unique to the driven setting. A precise understanding of This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. PHYSICS the (multi)critical points between these distinct Floquet phases 1073/pnas.1805796115/-/DCSupplemental. accessed by tuning drive parameters is the subject of this work. Published online August 29, 2018. www.pnas.org/cgi/doi/10.1073/pnas.1805796115 PNAS j September 18, 2018 j vol. 115 j no. 38 j 9491–9496 Downloaded by guest on September 25, 2021 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 j α(t) = e jφα(t) , where jφα(t + T ) = jφα(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.