**Quasiperiodic systems**

Many systems of current interest — such as quasicrystals, twisted bilayer graphene, and cold atoms in bichromatic laser potentials — are inhomogeneous, but with quasiperiodic rather than random modulation of the couplings. Quasiperiodic patterns are deterministic and have long-range spatial correlations: thus the central limiting arguments that describe random critical points fail in the quasiperiodic case. The study of such quasiperiodic systems is in its infancy: how is quantum criticality modified in the presence of quasiperiodic modulations? What is the nature of the many-body localization transition in quasiperiodic potentials? Can quasiperiodic systems host new topological phenomena and lead to novel dynamical phases of matter? I have been working recently on these questions, as well as on the non-equilibrium dynamics of quasi-periodically driven quantum systems.

**Localization and quantum criticality in quasiperiodic potentials**

I worked on the localization properties of a quantum particle subject to Ohmic dissipation, moving in a bichromatic quasiperiodic potential. In a periodic potential the particle undergoes a zero-temperature localization-delocalization transition as dissipation strength is decreased. We showed that the delocalized phase is absent in the quasiperiodic case, even when the deviation from periodicity is infinitesimal. More recently, we studied the quantum critical behavior of quasiperiodic spin chains. We found that that for a wide class of spin chains, generic quasiperiodic modulations flow to discrete sequences under a real-space renormalization group transformation. This observation allowed for an asymptotically exact treatment of the quasiperiodic quantum critical points.

*Quantum Brownian motion in a quasiperiodic potential, A.J. Friedman, R. Vasseur, A. Lamacraft and S.A. Parameswaran, Phys. Rev. B***100**, 060301(R) (2019).- Universality and quantum criticality in quasiperiodic spin chains, U. Agrawal, S. Gopalakrishnan and R. Vasseur, Nature Communications
**11,**2225 (2020).

**Quasi-periodically**** driven systems and time quasicrystals **

We studied the dynamics of disordered interacting spin chains subject to a quasiperiodic time-dependent drive, corresponding to a stroboscopic Fibonacci sequence of two distinct Hamiltonians. We found that after an initial transient, the system exhibits a long-lived glassy regime characterized by a logarithmically slow growth of entanglement and decay of correlations. Ultimately, at long time scales, which diverge exponentially for weak or rapid drives, the system thermalizes to infinite temperature. The slow relaxation enables metastable dynamical phases, exemplified by a “time quasicrystal” in which spins exhibit persistent oscillations with a distinct quasiperiodic pattern from that of the drive.