##### Orthogonal Quantum Many-Body Scars

H. Zhao, A. Smith, F. Mintert, J. Knolle

Physical Review Letters 127, 150601 (2021).

Quantum many-body scars have been put forward as counterexamples to the eigenstate thermalization hypothesis. These atypical states are observed in a range of correlated models as long-lived oscillations of local observables in quench experiments starting from selected initial states. The long-time memory is a manifestation of quantum nonergodicity generally linked to a subextensive generation of entanglement entropy, the latter of which is widely used as a diagnostic for identifying quantum many-body scars numerically as low entanglement outliers. Here we show that by adding kinetic constraints to a fractionalized orthogonal metal, we can construct a minimal model with orthogonal quantum many-body scars leading to persistent oscillations with infinite lifetime coexisting with rapid volume-law entanglement generation. Our example provides new insights into the link between quantum ergodicity and many-body entanglement while opening new avenues for exotic nonequilibrium dynamics in strongly correlated multicomponent quantum systems.

##### Simple mitigation of global depolarizing errors in quantum simulations

J. Vovrosh, K.E. Khosla, S. Greenaway, C. Self, M.S. Kim, J. Knolle

Physical Review E 104, 035309 (2021).

To get the best possible results from current quantum devices error mitigation is essential. In this work we present a simple but effective error mitigation technique based on the assumption that noise in a deep quantum circuit is well described by global depolarizing error channels. By measuring the errors directly on the device, we use an error model ansatz to infer error-free results from noisy data. We highlight the effectiveness of our mitigation via two examples of recent interest in quantum many-body physics: entanglement measurements and real-time dynamics of confinement in quantum spin chains. Our technique enables us to get quantitative results from the IBM quantum computers showing signatures of confinement, i.e., we are able to extract the meson masses of the confined excitations which were previously out of reach. Additionally, we show the applicability of this mitigation protocol in a wider setting with numerical simulations of more general tasks using a realistic error model. Our protocol is device-independent, simply implementable, and leads to large improvements in results if the global errors are well described by depolarization.

##### Classical Prethermal Phases of Matter

A. Pizzi, A. Nunnenkamp, J. Knolle

Physical Review Letters 127, 140602 (2021).

Systems subject to a high-frequency drive can spend an exponentially long time in a prethermal regime, in which novel phases of matter with no equilibrium counterpart can be realized. Because of the notorious computational challenges of quantum many-body systems, numerical investigations in this direction have remained limited to one spatial dimension, in which long-range interactions have been proven a necessity. Here, we show that prethermal nonequilibrium phases of matter are not restricted to the quantum domain. Studying the Hamiltonian dynamics of a large three-dimensional lattice of classical spins, we provide the first numerical proof of prethermal phases of matter in a system with short-range interactions. Concretely, we find higher-order as well as fractional discrete time crystals breaking the time-translational symmetry of the drive with unexpectedly large integer as well as fractional periods. Our work paves the way toward the exploration of novel prethermal phenomena by means of classical Hamiltonian dynamics with virtually no limitations on the system's geometry or size, and thus with direct implications for experiments.

##### Classical approaches to prethermal discrete time crystals in one, two, and three dimensions

A. Pizzi, A. Nunnenkamp, J. Knolle

Physical Review B 104, 094308 (2021).

We provide a comprehensive account of prethermal discrete time crystals within classical Hamiltonian dynamics, complementing and extending our recent work [A. Pizzi, A. Nunnenkamp, and J. Knolle, Phys. Rev. Lett. 127, 140602 (2021)]. Considering power-law interacting spins on one-, two-, and three-dimensional hypercubic lattices, we investigate the interplay between dimensionality and interaction range in the stabilization of these nonequilibrium phases of matter that break the discrete time-translational symmetry of a periodic drive.

##### Rigorous Bounds on the Heating Rate in Thue-Morse Quasiperiodically and Randomly Driven Quantum Many-Body Systems

T. Mori, H. Zhao, F. Mintert, J. Knolle, R. Moessner

Physical Review Letters 127, 050602 (2021).

The nonequilibrium quantum dynamics of closed many-body systems is a rich yet challenging field. While recent progress for periodically driven (Floquet) systems has yielded a number of rigorous results, our understanding on quantum many-body systems driven by rapidly varying but aperiodic and quasiperiodic driving is still limited. Here, we derive rigorous, nonperturbative, bounds on the heating rate in quantum many-body systems under Thue-Morse quasiperiodic driving and under random multipolar driving, the latter being a tunably randomized variant of the former. In the process, we derive a static effective Hamiltonian that describes the transient prethermal state, including the dynamics of local observables. Our bound for Thue-Morse quasiperiodic driving suggests that the heating time scales like (omega/g)(-C) (ln()(omega/)(g)) with a positive constant C and a typical energy scale g of the Hamiltonian, in agreement with our numerical simulations.

##### Flat and correlated plasmon bands in graphenek/alpha-RuCl3 heterostructures

H.-K. Jin, J. Knolle

Physical Review B 104, 045140 (2021).

We develop a microscopic theory for plasmon excitations of graphene/alpha-RuCl3 heterostructures. Within a Kondo-Kitaev model with various interactions, a heavy Fermi liquid hosting flat bands emerges in which the itinerant electrons of graphene effectively hybridize with the fractionalized fermions of the Kitaev quantum spin liquid. We find novel correlated plasmon bands induced by the interplay of flat bands and interactions and argue that our theory is consistent with the available experimental data on graphene/alpha-RuCl3 heterostructures. We predict novel plasmon branches beyond the long-wavelength limit and discuss the implications for probing correlation phenomena in other flat band systems.

##### Variational quantum algorithm with information sharing

C.N. Self, K.E. Khosla, A.W.R. Smith, F. Sauvage, P.D. Haynes, J. Knolle, F. Mintert, M.S. Kim

Npj Quantum Information 7, 116 (2021).

We introduce an optimisation method for variational quantum algorithms and experimentally demonstrate a 100-fold improvement in efficiency compared to naive implementations. The effectiveness of our approach is shown by obtaining multi-dimensional energy surfaces for small molecules and a spin model. Our method solves related variational problems in parallel by exploiting the global nature of Bayesian optimisation and sharing information between different optimisers. Parallelisation makes our method ideally suited to the next generation of variational problems with many physical degrees of freedom. This addresses a key challenge in scaling-up quantum algorithms towards demonstrating quantum advantage for problems of real-world interest.

##### One-dimensional long-range Falikov-Kimball model: Thermal phase transition and disorder-free localization

T. Hodson, J. Willsher, J. Knolle

Physical Review B 104, 045116 (2021).

Disorder or interactions can turn metals into insulators. One of the simplest settings in which to study this physics is given by the Falikov-Kimball (FK) model, which describes itinerant fermions interacting with a classical Ising background field. Despite the translational invariance of the model, inhomogeneous configurations of the background field give rise to effective disorder physics which lead to a rich phase diagram in two (or more) dimensions with finite-temperature charge-density wave (CDW) transitions and interaction-tuned Anderson versus Mott localized phases. Here, we propose a generalized FK model in one dimension with long-range interactions which shows a similarly rich phase diagram. We use an exact Markov chain Monte Carlo method to map the phase diagram and compute the energy-resolved localization properties of the fermions. We compare the behavior of this transitionally invariant model to an Anderson model of uncorrelated binary disorder about a background CDW field which confirms that the fermionic sector only fully localizes for very large system sizes.

##### Confinement and entanglement dynamics on a digital quantum computer

J. Vovrosh, J. Knolle

Scientific Reports 11, 11577 (2021).

Confinement describes the phenomenon when the attraction between two particles grows with their distance, most prominently found in quantum chromodynamics (QCD) between quarks. In condensed matter physics, confinement can appear in quantum spin chains, for example, in the one dimensional transverse field Ising model (TFIM) with an additional longitudinal field, famously observed in the quantum material cobalt niobate or in optical lattices. Here, we establish that state-of-the-art quantum computers have reached capabilities to simulate confinement physics in spin chains. We report quantitative confinement signatures of the TFIM on an IBM quantum computer observed via two distinct velocities for information propagation from domain walls and their mesonic bound states. We also find the confinement induced slow down of entanglement spreading by implementing randomized measurement protocols for the second order Rényi entanglement entropy. Our results are a crucial step for probing non-perturbative interacting quantum phenomena on digital quantum computers beyond the capabilities of classical hardware.

##### Higher-order and fractional discrete time crystals in clean long-range interacting systems

A. Pizzi, J. Knolle, A. Nunnenkamp

Nature Communications 12, 2341 (2021).

Discrete time crystals are periodically driven systems characterized by a response with periodicity nT, with T the period of the drive and n>1. Typically, n is an integer and bounded from above by the dimension of the local (or single particle) Hilbert space, the most prominent example being spin-1/2 systems with n restricted to 2. Here, we show that a clean spin-1/2 system in the presence of long-range interactions and transverse field can sustain a huge variety of different 'higher-order' discrete time crystals with integer and, surprisingly, even fractional n>2. We characterize these (arguably prethermal) non-equilibrium phases of matter thoroughly using a combination of exact diagonalization, semiclassical methods, and spin-wave approximations, which enable us to establish their stability in the presence of competing long- and short-range interactions. Remarkably, these phases emerge in a model with continous driving and time-independent interactions, convenient for experimental implementations with ultracold atoms or trapped ions.

##### Gapless state of interacting Majorana fermions in a strain-induced Landau level

A. Agarwala, S. Bhattacharjee, J. Knolle, R. Moessner

Physical Review B 103, 134427 (2021).

Mechanical strain can generate a pseudomagnetic field, and hence Landau levels (LL), for low-energy excitations of quantum matter in two dimensions. We study the collective state of the fractionalized Majorana fermions arising from residual generic spin interactions in the central LL, where the projected Hamiltonian reflects the spin symmetries in intricate ways: emergent U(1) and particle-hole symmetries forbid any bilinear couplings, leading to an intrinsically strongly interacting system; also, they allow the definition of a filling fraction, which is fixed at 1/2. We argue that the resulting many-body state is gapless within our numerical accuracy, implying ultra-short-ranged spin correlations, while chirality correlators decay algebraically. This amounts to a Kitaev 'non-Fermi' spin liquid and shows that interacting Majorana Fermions can exhibit intricate behavior akin to fractional quantum Hall physics in an insulating magnet.

##### Butterfly effect and spatial structure of information spreading in a chaotic cellular automaton

S.-W. Liu, J. Willsher, T. Bilitewski, J.-J. Li, A. Smith, K. Christensen, R. Moessner, J. Knolle

Physical Review B 103, 094109 (2021).

We show how measuring real space properties such as the charge density in a quasiperiodic system can be used to gain insight into their topological properties. In particular, for the Fibonacci chain, we show that the total on-site charge oscillates when plotted in the appropriate coordinates, and the number of oscillations is given by the topological label of the gap in which the Fermi level lies. We show that these oscillations have two distinct interpretations, obtained by extrapolating results from the two extreme limits of the Fibonacci chain—the valence bond picture in the strong modulation limit, and perturbation around the periodic chain in the weak modulation limit. This effect is found to remain robust at moderate interactions, as well as in the presence of disorder. We conclude that experimental measurement of the real space charge distribution can yield information on topological properties in a straightforward way.

##### Anomalous Quantum Oscillations in a Heterostructure of Graphene on a Proximate Quantum Spin Liquid

V. Leeb, K. Polyudov, S. Mashhadi, S. Biswas, R. Valenti, M. Burghard, J. Knolle

Physical Review Letters 126 (9), 097201 (2021).

The quasi-two-dimensional Mott insulator alpha-RuCl3 is proximate to the sought-after Kitaev quantum spin liquid (QSL). In a layer of alpha-RuCl3 on graphene, the dominant Kitaev exchange is further enhanced by strain. Recently, quantum oscillation (QO) measurements of such alpha-RuCl3 and graphene heterostructures showed an anomalous temperature dependence beyond the standard Lifshitz-Kosevich (LK) description. Here, we develop a theory of anomalous QO in an effective Kitaev-Kondo lattice model in which the itinerant electrons of the graphene layer interact with the correlated magnetic layer via spin interactions. At low temperatures, a heavy Fermi liquid emerges such that the neutral Majorana fermion excitations of the Kitaev QSL acquire charge by hybridizing with the graphene Dirac band. Using ab initio calculations to determine the parameters of our low-energy model, we provide a microscopic theory of anomalous QOs with a non-LK temperature dependence consistent with our measurements. We show how remnants of fractionalized spin excitations can give rise to characteristic signatures in QO experiments.

##### Vacancy-Induced Low-Energy Density of States in the Kitaev Spin Liquid

W.H. Kao, J. Knolle, G.B. Halasz, R. Moessner, N.B. Perkins

Physical Review X 11 (1), 011034 (2021).

The Kitaev honeycomb model has attracted significant attention due to its exactly solvable spin-liquid ground state with fractionalized Majorana excitations and its possible materialization in magnetic Mott insulators with strong spin-orbit couplings. Recently, the 5d-electron compound H3LiIr2O6 has shown to be a strong candidate for Kitaev physics considering the absence of any signs of a long-range ordered magnetic state. In this work, we demonstrate that a finite density of random vacancies in the Kitaev model gives rise to a striking pileup of low-energy Majorana eigenmodes and reproduces the apparent power-law upturn in the specific heat measurements of H3LiIr2O6. Physically, the vacancies can originate from various sources such as missing magnetic moments or the presence of nonmagnetic impurities (true vacancies), or from local weak couplings of magnetic moments due to strong but rare bond randomness (quasivacancies). We show numerically that the vacancy effect is readily detectable even at low vacancy concentrations and that it is not very sensitive either to the nature of vacancies or to different flux backgrounds. We also study the response of the site-diluted Kitaev spin liquid to the three-spin interaction term, which breaks time-reversal symmetry and imitates an external magnetic field. We propose a field-induced flux-sector transition where the ground state becomes flux-free for larger fields, resulting in a clear suppression of the low-temperature specific heat. Finally, we discuss the effect of dangling Majorana fermions in the case of true vacancies and show that their coupling to an applied magnetic field via the Zeeman interaction can also account for the scaling behavior in the high-field limit observed in H3LiIr2O6.

##### Bistability and time crystals in long-ranged directed percolation

A. Pizzi, A. Nunnenkamp, J. Knolle

Nature Communications 12, 1061 (2021).

Stochastic processes govern the time evolution of a huge variety of realistic systems throughout the sciences. A minimal description of noisy many-particle systems within a Markovian picture and with a notion of spatial dimension is given by probabilistic cellular automata, which typically feature time-independent and short-ranged update rules. Here, we propose a simple cellular automaton with power-law interactions that gives rise to a bistable phase of long-ranged directed percolation whose long-time behaviour is not only dictated by the system dynamics, but also by the initial conditions. In the presence of a periodic modulation of the update rules, we find that the system responds with a period larger than that of the modulation for an exponentially (in system size) long time. This breaking of discrete time translation symmetry of the underlying dynamics is enabled by a self-correcting mechanism of the long-ranged interactions which compensates noise-induced imperfections. Our work thus provides a firm example of a classical discrete time crystal phase of matter and paves the way for the study of novel non-equilibrium phases in the unexplored field of driven probabilistic cellular automata. A model of a classical discrete time crystal satisfying the criteria of persistent subharmonic response robust against thermal noise and defects has been lacking. Here, the authors show that these criteria are satisfied in one-dimensional probabilistic cellular automata with long-range interactions and bistability.

##### Seasonal epidemic spreading on small-world networks: Biennial outbreaks and classical discrete time crystals

D. Malz, A. Pizzi, A. Nunnenkamp, J. Knolle

Physical Review Research 3, 013124 (2021).

We study seasonal epidemic spreading in a susceptible-infected-removed-susceptible model on small-world graphs. We derive a mean-field description that accurately captures the salient features of the model, most notably a phase transition between annual and biennial outbreaks. A numerical scaling analysis exhibits a diverging autocorrelation time in the thermodynamic limit, which confirms the presence of a classical discrete time crystalline phase. We derive the phase diagram of the model both from mean-field theory and from numerics. Our paper demonstrates that small worldness and non-Markovianity can stabilize a classical discrete time crystal, and links recent efforts to understand such dynamical phases of matter to the century-old problem of biennial epidemics.

##### Random Multipolar Driving: Tunably Slow Heating through Spectral Engineering

H.Z. Zhao, F. Mintert, R. Moessner, J. Knolle

Physical Review Letters 126 (4), 040601 (2021).

Driven quantum systems may realize novel phenomena absent in static systems, but driving-induced heating can limit the timescale on which these persist. We study heating in interacting quantum many-body systems driven by random sequences with n-multipolar correlations, corresponding to a polynomially suppressed low-frequency spectrum. For n >= 1, we find a prethermal regime, the lifetime of which grows algebraically with the driving rate, with exponent 2n + 1. A simple theory based on Fermi's golden rule accounts for this behavior. The quasiperiodic Thue-Morse sequence corresponds to the n -> infinity limit and, accordingly, exhibits an exponentially long-lived prethermal regime. Despite the absence of periodicity in the drive, and in spite of its eventual heat death, the prethermal regime can host versatile nonequilibrium phases, which we illustrate with a random multipolar discrete time crystal.

##### Time crystallinity and finite-size effects in clean Floquet systems

A. Pizzi, D. Malz, G. De Tomasi, J. Knolle, A. Nunnenkamp

Physical Review B 102 (21), 214207 (2020).

A cornerstone assumption that most literature on discrete time crystals has relied on is that homogeneous Floquet systems generally heat to a featureless infinite temperature state, an expectation that motivated researchers in the field to mostly focus on many-body localized systems. Some works have, however, shown that the standard diagnostics for time crystallinity apply equally well to clean settings without disorder. This fact raises the question whether a homogeneous discrete time crystal is possible in which the originally expected heating is evaded. Studying both a localized and an homogeneous model with short-range interactions, we clarify this issue showing explicitly the key differences between the two cases. On the one hand, our careful scaling analysis confirms that, in the thermodynamic limit and in contrast to localized discrete time crystals, homogeneous systems indeed heat. On the other hand, we show that, thanks to a mechanism reminiscent of quantum scars, finite-size homogeneous systems can still exhibit very crisp signatures of time crystallinity. A subharmonic response can in fact persist over timescales that are much larger than those set by the integrability-breaking terms, with thermalization possibly occurring only at very large system sizes (e.g., of hundreds of spins). Beyond clarifying the emergence of heating in disorder-free systems, our work casts a spotlight on finite-size homogeneous systems as prime candidates for the experimental implementation of nontrivial out-of-equilibrium physics.

##### Magneto-optical conductivity in generic Weyl semimetals

M. Stalhammar, J. Larana-Aragon, J. Knolle, E.J. Bergholtz

Physical Review B 102 (23), 235134 (2020).

Magneto-optical studies of Weyl semimetals have been proposed as a versatile tool for observing low-energy Weyl fermions in candidate materials including the chiral Landau level. However, previous theoretical results have been restricted to the linearized regime around the Weyl node and are at odds with experimental findings. Here, we derive a closed form expression for the magneto-optical conductivity of generic Weyl semimetals in the presence of an external magnetic field aligned with the tilt of the spectrum. The systems are taken to have linear dispersion in two directions, while the tilting direction can consist of any arbitrary continuously differentiable function. This general calculation is then used to analytically evaluate the magneto-optical conductivity of Weyl semimetals expanded to cubic order in momentum. In particular, systems with arbitrary tilt, as well as systems hosting trivial Fermi pockets are investigated. The higher-order terms in momentum close the Fermi pockets in the type-II regime, removing the need for unphysical cutoffs when evaluating the magneto-optical conductivity. Crucially, the ability to take into account closed over-tilted and additional trivial Fermi pockets allows us to treat model systems closer to actual materials and we propose a simple explanation why the presence of parasitic trivial Fermi pockets can mask the characteristic signature of Weyl fermions in magneto-optical conductivity measurements.

##### Local probes for charge-neutral edge states in two-dimensional quantum magnets

J. Feldmeier, W. Natori, M. Knap, J. Knolle

Physical Review B 102 (13), 134423 (2020).

The bulk-boundary correspondence is a defining feature of topological states of matter. However, for quantum magnets in two dimensions such as spin liquids or topological magnon insulators, a direct observation of topological surface states has proven challenging because of the charge-neutral character of the excitations. Here we propose spin-polarized scanning tunneling microscopy as a spin-sensitive local probe to provide direct information about charge-neutral topological edge states. We show how their signatures, imprinted in the local structure factor, can be extracted by specifically employing the strengths of existing technologies. As our main example, we determine the dynamical spin correlations of the Kitaev honeycomb model with open boundaries. We show that by contrasting conductance measurements of bulk and edge locations, one can extract direct signatures of the existence of fractionalized excitations and nontrivial topology. The broad applicability of this approach is corroborated by a second example of a kagome topological magnon insulator.

##### Disorder-free localization in a simple U (1) lattice gauge theory

I. Papaefstathiou, A. Smith, J. Knolle

Physical Review B 102 (16), 165132 (2020).

Localization due to the presence of disorder has proven crucial for our current understanding of relaxation in isolated quantum systems. The many-body localized phase constitutes a robust alternative to the thermalization of complex interacting systems, but recently the importance of disorder has been brought into question. A number of disorder-free localization mechanisms have been put forward connected to local symmetries of lattice gauge theories. Here, starting from translationally invariant (1 + 1)-dimensional quantum electrodynamics, we modify the dynamics of the gauge field which allows us to construct a lattice model with a U(1) local gauge symmetry revealing a mechanism of disorder-free localization. We consider two different discretizations of the continuum model resulting in a free-fermion soluble model in one case and an interacting model in the other. We diagnose the localization of our translationally invariant model in the far-from-equilibrium dynamics following a global quantum quench.

##### Dynamics of a Two-Dimensional Quantum Spin-Orbital Liquid: Spectroscopic Signatures of Fermionic Magnons

W.M.H. Natori, J. Knolle

Physical Review Letters 125 (6), 067201 (2020).

We provide an exact study of dynamical correlations for the quantum spin-orbital liquid phases of an SU(2)-symmetric Kitaev honeycomb lattice model. We show that the spin dynamics in this Kugel-Khomskii type model is exactly the density-density correlation function of S = 1 fermionic magnons, which could be probed in resonant inelastic x-ray scattering experiments. We predict the characteristic signatures of spin-orbital fractionalization in inelastic scattering experiments and compare them to the ones of the spin-anisotropic Kitaev honeycomb spin liquid. In particular, the resonant inelastic x-ray scattering response shows a characteristic momentum dependence directly related to the dispersion of fermionic excitations. The neutron scattering cross section displays a mixed response of fermionic magnons as well as spin-orbital excitations. The latter has a bandwidth of broad excitations and a vison gap that is three times larger than that of the spin-1 = 2 Kitaev model.

##### Quantum Many-Body Scars in Optical Lattices

H.Z. Zhao, J. Vovrosh, F. Mintert, J. Knolle

Physical Review Letters 124 (16), 160604 (2020).

The concept of quantum many-body scars has recently been put forward as a route to describe weak ergodicity breaking and violation of the eigenstate thermalization hypothesis. We propose a simple setup to generate quantum many-body scars in a doubly modulated Bose-Hubbard system which can be readily implemented in cold atomic gases. The dynamics are shown to be governed by kinetic constraints which appear via density-assisted tunneling in a high-frequency expansion. We find the optimal driving parameters for the kinetically constrained hopping which leads to small isolated subspaces of scared eigenstates. The experimental signatures and the transition to fully thermalizing behavior as a function of driving frequency are analyzed.

##### Electronic Properties of alpha-RuCl3 in Proximity to Graphene

S. Biswas, Y. Li, S. Winter, J. Knolle, R. Valentí

Physical Review Letters 123 (23), 237201 (2019).

In the pursuit of developing routes to enhance magnetic Kitaev interactions in alpha-RuCl3, as well as probing doping effects, we investigate the electronic properties of alpha-RuCl3 in proximity to graphene. We study alpha-RuCl3/graphene heterostructures via ab initio density functional theory calculations, Wannier projection, and nonperturbative exact diagonalization methods. We show that alpha-RuCl3 becomes strained when placed on graphene and charge transfer occurs between the two layers, making alpha-RuCl3 (graphene) lightly electron doped (hole doped). This gives rise to an insulator-to-metal transition in alpha-RuCl3 with the Fermi energy located close to the bottom of the upper Hubbard band of the t(2g) manifold. These results suggest the possibility of realizing metallic and even exotic superconducting states. Moreover, we show that in the strained alpha-RuCl3 monolayer the Kitaev interactions are enhanced by more than 50% compared to the unstrained bulk structure. Finally, we discuss scenarios related to transport experiments in alpha-RuCl3/graphene heterostructures.

##### Period-n Discrete Time Crystals and Quasicrystals with Ultracold Bosons

A. Pizzi, J. Knolle, A. Nunnenkamp

Physical Review Letter 123 (15), 150601 (2019).

We investigate the out-of-equilibrium properties of a system of interacting bosons in a ring lattice. We present a Floquet driving that induces clockwise (counterclockwise) circulation of the particles among the odd (even) sites of the ring which can be mapped to a fully connected model of clocks of two counterrotating species. The clocklike motion of the particles is at the core of a period-n discrete time crystal where L = 2n is the number of lattice sites. In the presence of a "staircaselike" on-site potential, we report the emergence of a second characteristic timescale in addition to the period n-tupling. This new timescale depends on the microscopic parameters of the Hamiltonian and is incommensurate with the Floquet period, underpinning a dynamical phase we call "time quasicrystal." The rich dynamical phase diagram also features a thermal phase and an oscillatory phase, all of which we investigate and characterize. Our simple, yet rich model can be realized with state-of-the-art ultracold atoms experiments.