Certain lattice wave systems in translationally invariant settings have one or more spectral bands that are strictly flat or independent of momentum in the tight binding approximation, arising from either internal symmetries or fine-tuned coupling. These flat bands display remarkable strongly interacting phases of matter. Originally considered as a theoretical convenience useful for obtaining exact analytical solutions of ferromagnetism, flat bands have now been observed in a variety of settings, ranging from electronic systems to ultracold atomic gases and photonic devices. Here we review the design and implementation of flat bands and chart future directions of this exciting field. Graphical Abstract

Flatband systems typically host “compact localized states” (CLS) due to destructive interference and macroscopic degeneracy of Bloch wave functions associated with a dispersionless energy band. Using a photonic Lieb lattice (LL), such conventional localized flatband states are found to be inherently incomplete, with the missing modes manifested as extended line states that form noncontractible loops winding around the entire lattice. Experimentally, we develop a continuous-wave laser writing technique to establish a finite-sized photonic LL with specially tailored boundaries and, thereby, directly observe the unusually extended flatband line states. Such unconventional line states cannot be expressed as a linear combination of the previously observed boundary-independent bulk CLS but rather arise from the nontrivial real-space topology. The robustness of the line states to imperfect excitation conditions is discussed, and their potential applications are illustrated.

Flat bands – single‐particle energy bands – in tight‐binding lattices, aka networks, have attracted attention due to the presence of macroscopic degeneracies and their sensitivity to perturbations. They support compact localized eigenstates protected by destructive interference. This makes them natural candidates for emerging exotic phases and unconventional orders. In this review we consider the recently proposed systematic ways to construct flat band networks based on symmetries or fine‐tuning. We then discuss how the construction methods can be further extended, adapted or exploited in presence of perturbations, both single‐particle and many‐body. This strategy has lead to the discovery of non‐perturbative metal‐insulator transitions, fractal phases, nonlinear and quantum caging and many‐body nonergodic quantum models. We discuss what implications these results may have for the design of fine‐tuned nanophotonic systems including photonic crystals, nanocavities, and metasurfaces.

Many-body localization (MBL) is a result of the balance between interference-based Anderson localization and many-body interactions in a high-dimensional Fock space. It is expected that dissipation is blurring interference and destroying that balance so that the asymptotic state of a system with an MBL Hamiltonian does not bear localization signatures. This is evidently true in the case of local dephasing which drives any system into an infinite-temperature state. We demonstrate, by using a set of dissipative operators, where each one is acting nontrivially on a pair of neighboring sites (or spins), that an MBL system can be brought into a Hamiltonian-specific steady state. The difference between ergodic and MBL Hamiltonians can be seen in statistics of imbalance, entanglement entropy, and level spacing of the steady-state density operator. By introducing pairwise dissipative operators into an MBL system already exposed to dephasing, these localization signatures can be restored.

We study the dynamics of few interacting bosons in a one-dimensional lattice with dc bias. In the absence of interactions the system displays single-particle Bloch oscillations. For strong interaction the Bloch oscillation regime re-emerges with fractional Bloch periods which are inversely proportional to the number of bosons clustered into a bound state. The interaction strength affects the oscillation amplitude. Excellent agreement is found between numerical data and a composite particle dynamics approach. For specific values of the interaction strength, a particle will tunnel from the interacting cloud to a well-defined distant lattice location.

Linear wave equations on flat-band networks host compact localized eigenstates (CLS). Nonlinear wave equations on translationally invariant flat-band networks can host compact discrete breathers-time-periodic and spatially compact localized solutions. Such solutions can appear as one-parameter families of continued linear compact eigenstates, or as discrete sets on families of non-compact discrete breathers, or even on purely dispersive networks with fine-tuned nonlinear dispersion. In all cases, their existence relies on destructive interference. We use CLS amplitude distribution properties and orthogonality conditions to derive existence criteria and stability properties for compact discrete breathers as continued CLS.