Conventional wisdom has it that a small subsystem of a larger system thermalizes, even under unitary dynamics --- that is, its reduced density matrix acquires a universal form in the form of a (mixed) Gibbs ensemble.

Recently, a new kind of universality was proposed where instead of completely discarding the effect of the complementary subsystem (the "bath"), one retains information about its state upon measurement. Specifically, one can look at the conditional wavefunctions of the subsystem upon projective measurements of the bath in a fixed local basis. Surprisingly, the distribution of these (pure) quantum states can be universal -- it can be maximally entropic! In quantum information theory parlance, the ensemble is said to form a "quantum state-design", which replicates the moments of a uniformly (Haar) distributed ensemble.

We proved that this occurs rigorously in the setting of dual unitary circuits [1,2]; and we further elucidated a hierarchy of times taken to form good designs for high moments upon deviating from such conditions [2]. This emergence of quantum state designs represents a deeper form of quantum thermalization and is a novel nonequilibrium phenomenon deserving to be explored further.

A central goal of condensed matter physics is the understanding of the universal, collective behavior that emerges out of many microscopic interacting constituents, leading to complex phenomena and phases of matter. In the context of fundamentally out of equilibrium settings, such as in driven systems, what new phases can there be?

The Discrete Time-Crystal (DTC) in Floquet systems is one such celebrated example, where the phase is characterized by the spontaneous breaking of a time-translational symmetry. Physical observables exhibit robust, universal, long-time dynamics responses at frequencies which are subharmonics of the original drive frequencies.

Recently, we showed in [1] how to realize a huge family of novel phases, not achievable in static or Floquet settings, via Quasiperiodic-driving. This is the idea of driving a system with two or more incommensurate frequencies, which then endows the system with the concept of "multiple time-translation symmetries". We showed how these symmetries can be spontaneously broken (in a multitude of ways) to realize Discrete Time-Quasicrystals, as well as used to protect symmetry-protected topological (SPT) states in a long-lived preheating regime, thus greatly enriching the landscape of nonequilibrium phases of matter!

What is the long-time fate of an isolated quantum many-body system? Conventional wisdom has it that it either thermalizes or many-body localizes (MBL).

Can MBL survive in the presence of SU(2) symmetry? We show in [1] that while it cannot (in accordance with general representation theory arguments), the proliferation of resonances depends weakly on system size, indicating that possible non-trivial glassy behavior in dynamics remains for systems of small to moderate sizes.

Furthermore, are there possible exceptions to the aforementioned paradigms? Quantum many-body scarring -- atypical, nonergodic many-body eigenstates in a sea of otherwise thermal states, is a new, posited weak ergodicity breaking phenomenon to explain recent experimental observations in a Rydberg quantum simulator of surprising atypical nonthermalizing dynamical behavior of long-lived coherent periodic revivals of certain initial states but not others. In our work [2], we formulate an effective "semiclassical" description of quantum dynamics using the TDVP and a suitably chosen MPS manifold, which allows us to find that periodic orbits underlie these dynamics and thus establish a strong connection to the existing theory of quantum scars in chaotic single-particle systems.

Also, recently, we investigated the algebraic structure in the Rydberg atom Hamiltonian and showed that the long-lived periodic dynamics arises due to an emergent SU(2) dynamics, even though the system does not have rotational symmetry, emergent or otherwise.

With impressive advances in controlling and designing synthetic quantum systems, periodic driving has emerged as a powerful tool to engineer effective interactions and new Hamiltonians.

We show, in one such example, that driving can create higher-body interactions in Fractional Quantum Hall (FQH) systems that may stabilize exotic, non-Abelian topological phases, by exploiting the non-commutativity of the GMP algebra [1].

Periodic driving may even create novel non-equilibrium phases of matter with no analog in static systems (such as Discrete Time Crystals [DTC], signatures of which have been experimentally realized). In our work [2], we analyze the stability of DTC order in driven dipolar systems (where d=alpha=3), finding that while not truly stable in the thermodynamic limit, dynamics can be critically slow, leading to a parameterically stable version of the time crystal called Critical Time Crystals.

Since the stability of DTC order is so sensitive to thermalization channels, it may also be used as a probe of thermalization dynamics in quantum many-body systems, as we investigate in [3], in an ensemble of interacting NV-center defects in diamond.

In [4], we also analyze the important question of state preparation of the ground state of an effective Hamiltonian in a driven system, via a quasi-adiabatic approach. We show that the ramp speed cannot be too fast, in order to remain adiabatic, but at the same time, cannot be too slow, in order to avoid heating in the system.

As noted in the previous section, periodic driving is a useful tool to engineer interactions and new phases of matter. However, due to the drive, one has to contend with unbounded heating to infinite temperature that may afflict such systems.

We show, in a series of rigorous, mathematical works [1,2], that heating in high frequency driven many-body systems with locally bounded Hilbert spaces is exponentially suppressed with the driving frequency. This gives rise to prethermalization to an effective local Hamiltonian in such systems for a parameterically long time, justifying and bolstering Floquet engineering approaches. Furthermore, the techniques we use have been further employed to predict intriguing new transient phases such as prethermal time crystals.

In [3], we also extend our result by developing new techniques to cover the case of systems with long-range interactions, relevant for NV-center setups, trapped ions, etc.

In a chaotic quantum many-body system, entanglement grows linearly in time, starting from a random initial product state. In [1], we relate the spreading of entanglement to the spreading of local operators in the systems, and show in a toy model of dynamics that the linear entanglement spreading can be understood as arising from the saturation of the Lieb-Robinson bound on dynamics in systems with local interactions.

In a seminal paper by Li & Haldane, it was demonstrated that the entanglement spectrum, a more refined characterization of the entanglement present in a reduced density matrix than the entanglement entropy, can be used as a probe of chiral topological order, via a so-called 'edge-entanglement-spectrum correspondence'.

In our work [1], we explore the edge-ES correspondence in a nonchiral topological phase, namely the Wen-plaquette model (Z2-topological order). We find in general that such a correspondence is absent, but can be reinstated by imposing translational invariance in the system, which amounts to enforcing an anyonic duality (e <->m) at the level of the bulk theory and Kramers-Wannier duality at the level of the effective boundary theory.

In [2] we show that information about the edge theory can actually simply be extracted by suitable deformations of the fixed-point wavefunction of a nonchiral topological phase, and by studying its entanglement spectrum. This circumvents having to actually diagonalize for the ground state of a perturbed Hamiltonian, is in line conceptually with the idea that topological order is intrinsic to the quantum state (contained in its entanglement), and is not fundamentally a property of the Hamiltonian.