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# Floquet quantum criticality

Edited by David A. Huse, Princeton University, Princeton, NJ, and approved August 3, 2018 (received for review April 4, 2018)

## Significance

Periodically driven “Floquet” systems are nonequilibrium systems whose time translation symmetry can give rise to a rich dynamical phase structure. In the presence of quenched disorder, they can avoid thermalizing to a bland infinite temperature state through a phenomenon known as many-body localization (MBL). The ability of these driven MBL phases to host phenomena forbidden in equilibrium, such as “time crystallinity,” has gained widespread interest in recent years. In this work, we consider the question of the criticality that emerges at the transitions between distinct Floquet MBL phases. By providing a universally applicable picture and applying it to a prototypical driven system, the driven Ising chain, we identify critical points and give an understanding of Floquet criticality in general.

## Abstract

We study transitions between distinct phases of one-dimensional periodically driven (Floquet) systems. We argue that these are generically controlled by infinite-randomness fixed points of a strong-disorder renormalization group procedure. Working in the fermionic representation of the prototypical Floquet Ising chain, we leverage infinite randomness physics to provide a simple description of Floquet (multi)criticality in terms of a distinct type of domain wall associated with time translational symmetry-breaking and the formation of “Floquet time crystals.” We validate our analysis via numerical simulations of free-fermion models sufficient to capture the critical physics.

- quantum criticality
- disordered systems
- many-body localization
- periodically driven systems
- nonperturbative arguments

The assignment of robust phase structure to periodically driven quantum many-body systems is among the most striking results in the study of nonequilibrium dynamics (1). There has been dramatic progress in understanding such “Floquet” systems, ranging from proposals to engineer new states of matter via the drive (2⇓⇓⇓⇓⇓⇓⇓⇓⇓–12) to the classification of driven analogs of symmetry-protected topological phases (“Floquet SPTs”) (13⇓⇓⇓⇓⇓⇓–20). These typically require that the system under investigation possess one or more microscopic global symmetries. In addition, all Floquet systems share an invariance under time translations by an integer multiple of their drive period. Unlike the continuous time translational symmetry characteristic of undriven Hamiltonian systems (21⇓–23), this discrete symmetry 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 translation symmetry breaking (TTSB) exhibited by Floquet time crystals is stable against perturbations that preserve the periodicity of the drive, permitting generalizations of notions such as broken symmetry and phase rigidity to the temporal setting. Experiments have begun to probe these predictions in well-isolated systems such as ultracold gases, ion traps (30), nitrogen-vacancy centers in diamond (31), and even spatially ordered crystals (32, 33).

In light of these developments, it is desirable to construct a theory of Floquet (multi)critical points between distinct Floquet phases. Ideally, this should emerge as the fixed point of a coarse-graining/renormalization group (RG) procedure; enable us to identify critical degrees of freedom, especially those responsible for TTSB; and allow us to compute the critical scaling behavior.

Here, we develop such a theory for a prototypical Floquet system: the driven random quantum Ising chain. Extensive analysis has shown that this model hosts four phases (1, 24). Two of these, the paramagnet (PM) and the spin glass (SG), are present already in the static problem (34⇓–36). A third, the π SG/time crystal, has spatiotemporal long-range order and subharmonic bulk response at half-integer multiples of the drive frequency. This phase and its Ising dual—the

Our approach relies on the presence of quenched disorder, which is required for a generic periodically driven system to have Floquet phase structure rather than thermalize to a featureless infinite-temperature state (37⇓⇓–40). We argue that transitions between distinct one-dimensional Floquet phases are then best described in terms of an infinite-randomness fixed point (IRFP) accessed via a strong-disorder real space renormalization group (RSRG) procedure. In the nonequilibrium setting, the stability of IRFPs against thermalization via long-range resonances remains a topic of debate (41⇓–43). However, even if unstable, we expect that they will control the dynamics of prethermalization relevant to all reasonably accessible experimental timescales (44, 45).

The universality of our analysis turns on the fact that, in the vicinity of such infinite-randomness critical points, a typical configuration of the system can be viewed as being composed of domains deep in one of two proximate phases (46⇓⇓⇓⇓–51). Transitions that do not involve TTSB (i.e., the SG/PM or

## Model

Floquet systems are defined by a time-periodic Hamiltonian

An object of fundamental interest is the single-period evolution operator or Floquet operator

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 hosts uniquely dynamical phases. The corresponding Floquet operator is

## Phases and Duality

Observe that **1** analogously to *A*) central to the infinite-randomness criticality discussed below.

The absence of energy conservation in the Floquet setting admits two new eigenstate-preserving changes of parameter to Eq. **1**. The transformations *SI Appendix*), but possibly distinct quasienergies:

## Infinite-Randomness Structure

In analogy with the critical point between PM and SG phases in the static random Ising model (both at zero temperature and in highly excited states), we expect that the dynamical Floquet transitions of Eq. **1** are controlled by an IRFP of an RSRG procedure. At a static IRFP, the distribution of the effective couplings broadens without bound under renormalization, so the effective disorder strength diverges with the RG scale. A typical configuration of the system in the vicinity of such a transition can be viewed as being composed of puddles deep in one of the two proximate phases, in contrast with continuous phase transitions in clean systems (50, 51).

To generalize this picture to the Floquet Ising setting, we must identify appropriate scaling variables. For

## Emergent π-Criticality

For

Quite different physics arise for *A*). Since they are topological edge modes, a given π-Majorana trapped at a *B*). If the intervening puddles are MBL, the tunneling between π-Majoranas is exponentially suppressed as *SI Appendix*), and the typical tunnel coupling is stretched exponential

Observe that the PM–πSG transition involves the onset of TTSB, since the πSG is the prototypical example of a time crystal. Similarly, the SG–

## RG Treatment

The above infinite-randomness hypothesis suggests that the critical behavior at the dynamical Floquet transitions can be understood in terms of two effectively static Majorana chains, one near quasienergy 0 (*SI Appendix*) by considering instead the criticality of

Therefore, for sufficiently weak interactions, the critical lines are always in the random Ising universality class. The four-phase multicritical point—at which all four distributions are symmetric—is in the Ising × Ising universality class. This picture of Floquet (multi)criticality extends both symmetry-based reasoning used when all

## Floquet (Multi)Criticality

Combining this reasoning with standard IRFP results, we conclude that all of the transitions show infinite-randomness Ising scaling: The correlation length diverges as

## Numerics

As stressed above, our picture of these transitions relies on the infinite-randomness assumption. To justify this and to confirm our analytical predictions, we have performed extensive numerical simulations on the noninteracting model, leveraging its free-fermion representation to access the full single-particle spectrum and to calculate the entanglement entropy of arbitrary eigenstates (*SI Appendix*). We average over 20,000 disorder realizations (with open boundary conditions), randomly choosing a Floquet eigenstate in each.

Given our parametrization of disorder, the combination **1**. We find *SI Appendix*). Moreover, Fig. 1 clearly shows that changing *Insets*).

## Experimental Consequences

Let us now turn to some experimental consequences of the above predictions. Recent advances in the control of ultracold atomic arrays have brought models such as Eq. **1** into the realm of experimental realizability (65⇓–67). The model hosts a time-crystal phase (the πSG), the phenomenology of which has recently been directly observed (30, 31). Even though, as mentioned earlier, these critical lines may eventually thermalize due to long-range resonances (41⇓–43), the dynamics of the Ising universality class should persist through a prethermalization regime relevant to all reasonably accessible experimental timescales (44, 45). Thus, the dynamical signatures of the transitions we have identified should be readily experimentally observable.

One prominent experimental signature of these physics is the scaling behavior of the dynamical spin–spin autocorrelation function in Fourier space

## Discussion

We have presented a generic picture of the transitions between MBL Floquet phases and applied it to study the criticality of the periodically driven interacting random Ising chain. Our work can be generalized to more intricate Floquet systems, under the (reasonable) assumption that they flow to infinite randomness under coarse-graining. The resulting IRFP is enriched in the Floquet setting: Each distinct invariant Floquet quasienergy hosts an independent set of fixed-point coupling distributions. (For instance, the

## Materials and Methods

Numerical simulations were performed on the TFI chain, where we extract the entanglement entropy across a cut of length *l* = L/2 from the boundary in an arbitrary eigenstate. We use the fact that the noninteracting TFI chain can be efficiently described as a system of free Majorana fermions (68, 69), details of which follow.

First, let us apply a Jordan–Wigner transformation to the TFI chain

Let us first construct the Floquet evolution operator. To see how a Hamiltonian evolves the correlation function, first note that in the Heisenberg picture,

The initial correlation function is simple in this basis: It will also be block diagonal with blocks of

Note that the 0 and π modes that emerge from our RSRG picture show up numerically as nearly degenerate states whose quasienergies must be resolved. In practice, this required implementing high-precision numerics—beyond conventional machine precision—which is described in more detail in *SI Appendix*.

The above steps allow us to access the correlation function in any Floquet eigenstate. Let us now show how to calculate the entanglement entropy from that C matrix. First, trace out the degrees of freedom outside the entanglement cut. Then, diagonalize the antisymmetric part of the correlation function

## Acknowledgments

We thank Vedika Khemani, Shivaji Sondhi, Dominic Else, Chetan Nayak, Adam Nahum, Joel Moore, David Huse, and especially Sthitadhi Roy for useful discussions and are grateful to David Huse and Vedika Khemani for detailed comments on the manuscript. This work used the Extreme Science and Engineering Discovery Environment (XSEDE) (70), which is supported by National Science Foundation Grant ACI-1053575. W.B. acknowledges support from the US Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program and from the Hellman Foundation through a Hellman Graduate Fellowship. We also acknowledge support from Laboratory Directed Research and Development (LDRD) funding from Lawrence Berkeley National Laboratory, provided by the Director, Office of Science, of the US Department of Energy under Contract DEAC02-05CH11231, and the Department of Energy Basic Energy Sciences (BES) Theory Institute for Materials and Energy Spectroscopies (TIMES) initiative (to M.K.); travel support from the California Institute for Quantum Emulation (CAIQUE) via President’s Research Catalyst Award (PRCA) CA-15-327861 and the California NanoSystems Institute at the University of California, Santa Barbara (to W.B. and S.A.P.); support from NSF Grant DMR-1455366 at the University of California, Irvine (to S.A.P.); and University of Massachusetts start-up funds (to R.V.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: wberdanier{at}berkeley.edu. ↵

^{2}On leave from: Department of Physics and Astronomy, University of California, Irvine, CA 92697.

Author contributions: W.B., M.K., S.A.P., and R.V. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1805796115/-/DCSupplemental.

Published under the PNAS license.

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