Flat-band Systems Research Articles

Optical response of two-dimensional Dirac materials with a flat band

Two-dimensional Dirac materials with a flat band have been demonstrated to possess a plethora of unusual electronic properties, but the optical properties of these materials are less studied. Utilizing $\alpha$-$\mathcal{T}_3$ lattice as a prototypical system, where $0\le \alpha \le 1$ is a tunable parameter and a flat band through the conic intersection of two Dirac cones arises for $\alpha > 0$, we investigate the conductivity of flat-band Dirac material systems analytically and numerically. Motivated by the fact that the imaginary part of the optical conductivity can have significant effects on the optical response and is an important factor of consideration for developing $\alpha$-$\mathcal{T}_3$ lattice based optical devices, we are led to derive a complete conductivity formula with both the real and imaginary parts. Scrutinizing the formula, we uncover two phenomena. First, for the value of $\alpha$ in some range, two types of optical transitions coexist: one between the two Dirac cones and another from the flat band to a cone, which generate multi-frequency transverse electrical propagating waves. Second, for $\alpha=1$ so the quasiparticles become pseudospin-1, the flat-to-cone transition can result in resonant scattering. These results pave the way to exploiting $\alpha$-$T_3$ lattice for optical device applications in the terahertz frequency domain.

Designing spin-textured flat bands in twisted graphene multilayers via helimagnet encapsulation

Twisted graphene multilayers provide tunable platforms to engineer flat bands and exploit the associated strongly correlated physics. The two-dimensional nature of these systems makes them suitable for encapsulation by materials that break specific symmetries. In this context, recently discovered two-dimensional helimagnets, such as the multiferroic monolayer NiI2, are specially appealing for breaking time-reversal and inversion symmetries due to their nontrivial spin textures. Here we show that this spin texture can be imprinted on the electronic structure of twisted bilayer graphene by proximity effect. We discuss the dependence of the imprinted spin texture on the wave-vector of the helical structure, and on the strength of the effective local exchange field. Based on these results we discuss the nature of the superconducting instabilities that can take place in helimagnet encapsulated twisted bilayer graphene. Our results put forward helimagnetic encapsulation as a powerful way of designing spin-textured flat band systems, providing a starting point to engineer a new family of correlated moire states.

Pairing and superconductivity in quasi-one-dimensional flat-band systems: Creutz and sawtooth lattices

We study the pairing and superconducting properties of the attractive Hubbard model in two quasi one-dimensional topological lattices: the Creutz and sawtooth lattices. They share two peculiar properties: each of their band structures exhibits a flat band with a non-trivial winding number. The difference, however, is that only the Creutz lattice is genuinely topological, owing to a chiral (sub-lattice) symmetry, resulting in a quantized winding number and zero energy edge modes for open boundary conditions. We use mean field and exact density matrix renormalization group in our work. Our three main results are: (a) For both lattice systems, the superconducting weight, $D_s$, is linear in the coupling strength, $U$, for low values of $U$; (b) for small $U$, $D_s$ is proportional to the quantum metric for the Creutz system but not for the sawtooth system because its sublattices are not equivalent; (c) conventional BCS mean field is not appropriate for such systems with inequivalent sublattices. We show that, for a wide range of densities and coupling strengths, these systems are very well described by a full multi-band mean field method where the pairing parameters and the local particle densities on the inequivalent sublattices are variational mean field parameters.

Linearne kompaktne lokalizovane mode u dvodimenzionalnoj, fluksom tretiranoj, plus rešetki

It was studied a photonic linear two-dimensional plus lattice dressed by the artificial flux and explored the influence of the artificial gauge field on the energy band spectrum. The Aharonov-Bohm effect has caused the appearance of flat zones and for certain values of the flux, the lattice spectrum has been described by two momentum independent, fully degenerated flatbands and three dispersive bands. Three different types of fundamental non-orthogonal compactons have been obtained. The findings presented in this study could be tested in practice and they may be used for probing different artificial flatband systems, e. g. ultracold atoms in optical lattices, polariton condensates etc.

Higher-order exceptional point and Landau–Zener Bloch oscillations in driven non-Hermitian photonic Lieb lattices

We propose a scheme to realize parity-time (PT) symmetric photonic Lieb lattices of ribbon shape and complex couplings, thereby demonstrating the higher-order exceptional point (EP) and Landau–Zener Bloch (LZB) oscillations in the presence of a refractive index gradient. Quite different from non-Hermitian flatband lattices with on-site gain/loss, which undergo thresholdless PT symmetry breaking, the spectrum for such quasi-one-dimensional Lieb lattices has completely real values when the index gradient is applied perpendicular to the ribbon, and a triply degenerated higher-order EP (EP3) with coalesced eigenvalues and eigenvectors emerges only when the amplitude of the gain/loss ratio reaches a certain threshold value. When the index gradient is applied parallel to the ribbon, the LZB oscillations exhibit intriguing characteristics, including asymmetric energy transition and pseudo-Hermitian propagation, as the flatband is excited. Meanwhile, a secondary emission occurs each time when the oscillatory motion passes through the EP3, leading to distinct energy distribution in the flatband when a dispersive band is excited. Such novel phenomena may appear in other non-Hermitian flatband systems. Our work may also bring insight and suggest a photonic platform to study the symmetry and topological characterization of higher-order EP that may find unique applications in, for example, enhancing sensitivity.

Kondo lattice mediated interactions in flat-band systems

Electronic flat bands represent a paradigmatic platform to realize strongly correlated matter due to their associated divergent density of states. In common instances, including electron-electron interactions leads to magnetic instabilities for repulsive interactions and superconductivity for attractive interactions. Nevertheless, interactions of Kondo nature in flat band systems have remained relatively unexplored. Here we address the emergence of interacting states mediated by Kondo lattice coupled to a flat band system. Combining dynamical mean-field theory and tensor networks methods to solve flat band Kondo lattice models in one and two dimensions, we show the emergence of a robust underscreened regime leading to a magnetically ordered state in the flat band. Our results put forward flat band Kondo lattice models as a platform to explore the genuine interplay between flat band physics and many-body Kondo screening.

Geometric characterization of anomalous Landau levels of isolated flat bands

According to the Onsager’s semiclassical quantization rule, the Landau levels of a band are bounded by its upper and lower band edges at zero magnetic field. However, there are two notable systems where the Landau level spectra violate this expectation, including topological bands and flat bands with singular band crossings, whose wave functions possess some singularities. Here, we introduce a distinct class of flat band systems where anomalous Landau level spreading (LLS) appears outside the zero-field energy bounds, although the relevant wave function is nonsingular. The anomalous LLS of isolated flat bands are governed by the cross-gap Berry connection that measures the wave-function geometry of multi bands. We also find that symmetry puts strong constraints on the LLS of flat bands. Our work demonstrates that an isolated flat band is an ideal system for studying the fundamental role of wave-function geometry in describing magnetic responses of solids.

Localised Pair Formation in Bosonic Flat-Band Hubbard Models

Flat-band systems are ideal model systems to study strong correlations. In a large class of one or two dimensional bosonic systems with a lowest flat-band it has been shown that at a critical density the ground states are Wigner crystals. Under very special conditions it has been shown that pair formation occurs if one adds another particle to the system. The present paper extends this result to a much larger class of lattices and to a much broader region in the parameter space. Further, a lower bound for the energy gap between these pair states and the rest of the spectrum is established. The pair states are dominated by a subspace spanned by states containing a compactly localised pair. Overall, this strongly suggests localised pair formation in the ground states of the broad class of flat-band systems and rigorously proves it for some of the graphs in it, including the inhomogeneous chequerboard chain as well as two novel examples of regular two dimensional graphs. Physically, this means that the Wigner crystal remains intact if one adds a particle to it.

Light-matter coupling and quantum geometry in moiré materials

Quantum geometry has been identified as an important ingredient for the physics of quantum materials and especially of flat-band systems, such as moir\'e materials. On the other hand, the coupling between light and matter is of key importance across disciplines and especially for Floquet and cavity engineering of solids. Here we present fundamental relations between light-matter coupling and quantum geometry of Bloch wave functions, with a particular focus on flat-band and moir\'e materials, in which the quenching of the electronic kinetic energy could allow one to reach the limit of strong light-matter coupling more easily than in highly dispersive systems. We show that, despite the fact that flat bands have vanishing band velocities and curvatures, light couples to them via geometric contributions. Specifically, the intra-band quantum metric allows diamagnetic coupling inside a flat band; the inter-band Berry connection governs dipole matrix elements between flat and dispersive bands. We illustrate these effects in two representative model systems: (i) a sawtooth quantum chain with a single flat band, and (ii) a tight-binding model for twisted bilayer graphene. For (i) we highlight the importance of quantum geometry by demonstrating a nonvanishing diamagnetic light-matter coupling inside the flat band. For (ii) we explore the twist-angle dependence of various light-matter coupling matrix elements. Furthermore, at the magic angle corresponding to almost flat bands, we show a Floquet-topological gap opening under irradiation with circularly polarized light despite the nearly vanishing Fermi velocity. We discuss how these findings provide fundamental design principles and tools for light-matter-coupling-based control of emergent electronic properties in flat-band and moir\'e materials.

Flat bands with band crossings enforced by symmetry representation

Flat bands have band crossing points with other dispersive bands in many systems including the canonical flat-band models in the Lieb and kagome lattices. Here we show that some of such band degeneracy points of nondegenerate flat bands are unavoidable because of the symmetry representation (SR) of flat bands under unitary symmetry. We refer to such a band degeneracy point of flat bands as a SR-enforced band crossing. SR-enforced band crossing is distinct from the conventional band degeneracy protected by symmetry eigenvalues or topological charges in that its protection requires both specific symmetry representation and band flatness of the flat band, simultaneously. Even $n$-fold rotation ${C}_{n}$ ($n=2,3,4,6$) symmetry, which cannot protect band degeneracy without additional symmetries due to its abelian nature, can protect SR-enforced band crossings in flat-band systems. In two-dimensional flat-band systems with ${C}_{n}$ symmetry, when the degeneracy of a SR-enforced band crossing is lifted by a ${C}_{n}$ symmetry-preserving perturbation, we obtain a nearly flat Chern band. Our theory not only explains the origin of the band crossing points of FBs existing in various models, but also gives a strict no-go theorem for isolated FBs in a given lattice arising from the SR.

Flat and correlated plasmon bands in graphene/α−RuCl3 heterostructures

We develop a microscopic theory for plasmon excitations of graphene/${\alpha}$-RuCl${_3}$ 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/${\alpha}$-RuCl${_3}$ heterostructures. We predict novel plasmon branches beyond the long-wavelength limit and discuss the implications for probing correlation phenomena in other flat band systems.

Designing Three-Dimensional Flat Bands in Nodal-Line Semimetals

Electrons with large kinetic energy have a superconducting instability for infinitesimal attractive interactions. Quenching the kinetic energy and creating a flat band renders an infinitesimal repulsive interaction the relevant perturbation. Thus, flat band systems are an ideal platform to study the competition of superconductivity and magnetism and their possible coexistence. Recent advances in the field of twisted bilayer graphene highlight this in the context of two-dimensional materials. Two dimensions, however, put severe restrictions on the stability of the low-temperature phases due to enhanced fluctuations. Only three-dimensional flat bands can solve the conundrum of combining the exotic flat-band phases with stable order existing at high temperatures. Here, we present a way to generate such flat bands through strain engineering in topological nodal-line semimetals. We present analytical and numerical evidence for this scenario and study the competition of the arising superconducting and magnetic orders as a function of externally controlled parameters. We show that the order parameter is rigid because the quantum geometry of the Bloch wave functions leads to a large superfluid stiffness. Using density-functional theory and numerical tight-binding calculations we further apply our theory to strained rhombohedral graphite and CaAgP materials.

Inverse Anderson transition in photonic cages.

Transport inhibition via Anderson localization is ubiquitous in disordered periodic lattices. However, in crystals displaying only flatbands, disorder can lift macroscopic band flattening, removing geometric localization and enabling transport in certain conditions. Such a striking phenomenon, dubbed inverse Anderson transition and predicted for three-dimensional flatband systems, has thus far not been directly observed. Here we suggest a simple quasi one-dimensional photonic flatband system, namely, an Aharonov-Bohm photonic cage, in which correlated binary disorder induces an inverse Anderson transition and ballistic transport.

Valley-selective Floquet Chern flat bands in twisted multilayer graphene

We show that Floquet engineering with circularly polarized light (CPL) can selectively split the valley degeneracy of a twisted multilayer graphene (TMG), and thus generate a controlled valley-polarized Floquet Chern flat band with tunable large Chern number. It offers a feasible optical way to manipulate the valley degree of freedom in moir\'{e} flat bands, and hence opens new opportunities to study the valleytronics of mori\'{e} flat band systems. We thus expect that many of the valley-related properties of TMG, e.g. orbital ferromagnetism, can be switched by CPL with proper doping. We reveal a Chern number hierarchy rule for the Floquet flat bands in a generic (M+N)-layer TMG. We also illustrate that the CPL effects on TMG strongly rely on the stacking chirality, which is an unique feature of TMG. All these phenomena could be tested in the twisted double bilayer graphene systems, which is the simplest example of TMG and has already been realized in experiment.

Methods for constructing parameter-dependent flat-band lattices

We present two methods for constructing a flat-band (FB) system having a flat energy dispersion over the entire Brillouin zone within the tight-binding model, where the resulting Hamiltonian may not be easily obtained by existing methods based on a bipartite graph and line graph techniques. In the first method, we derive a set of conditions equivalent to the appearance of FBs for a given graph structure. This method allows parameters to be tuned so that systems with a small number of sites per unit cell have a FB. In the second method, we show that FB systems can be obtained by removing or adding sites to an existing FB system under specific rules. In particular, the site addition method enables us to construct multiple FB systems stemming from a single FB system. The FB system obtained by the second method has the characteristic that the component ratios in the FB eigenstate are partially common to the original system. We illustrate how lattices having a FB can be constructed by applying the latter method starting from an existing lattice such as a kagome lattice, demonstrating that a wide variety of lattices can possess a FB in the band structure.

Towards holographic flat bands

Motivated by the phenomenology in the condensed-matter flat-band Dirac systems, we here construct a holographic model that imprints the symmetry breaking pattern of a rather simple Dirac fermion model at zero chemical potential. In the bulk we explicitly include the backreaction to the corresponding Lifshitz geometry and compute the dynamical critical exponent. Most importantly, we find that such a geometry is unstable towards a nematic phase, exhibiting an anomalous Hall effect and featuring a Drude-like shift of its spectral weight. Our findings should motivate further studies of the quantum phases emerging from such holographic models.

Cascade of replica bands in flat-band systems: Predictions for twisted bilayer graphene

We investigate the effect of electron-phonon interactions (EPI) in systems exhibiting one or more flat electron bands close to the Fermi level and a comparatively large phonon energy scale. After solving the self-consistent full-bandwidth Eliashberg equations, we compute Angular Resolved Photo Emission Spectroscopy and Scanning Tunneling Spectroscopy/Microscopy spectra. We obtain a sequence of quasiparticle replica bands in both the normal and superconducting states that originate from frequency dependent features of the electron mass renormalization function. We show that these replica bands can be used to extract the relevant phonon energy scale from experiments. Focusing in particular on twisted bilayer graphene, we predict replica-band formation which, when observed, will shed light on the role of EPI in this archetypal flat-band system.

Revisiting Flat bands and localization

Flat bands imply lack of itinerancy due to some constraints that, in principle, results in anomalous behaviors with randomness. By a molecular orbital (MO) representation of the flat band systems, random MO models are introduced where the degeneracy due to the flat bands is preserved even with randomness. The zero modes of the chiral symmetric system with sublattice imbalance belong to the class. After explaining the generic flat band construction by MOs, several examples are discussed with numerical demonstration as sawtooth lattice in one dimension and hyper-Pyrochlore lattice in any d-dimensions that extends the Kagome (d=2) and Pyrochlore (d=3) lattices to general dimensions.

Quantum transport in flat bands and supermetallicity

Quantum physics in flat-band (FB) systems embodies a variety of exotic phenomenon and even counter intuitive features. The quantum transport in several graphene based compounds that exhibit a flat band and a tunable gap is investigated. Despite the localized nature of the FB states and a zero group velocity, a super-metallic (SM) phase at the FB energy is revealed. The SM phase is robust against the inelastic scattering strength and controlled only by the inter-band transitions between the FB and the dispersive bands. The SM phase appears insensitive and quasi independent of the gap amplitude and nature of the lattice (disordered or nano-patterned). The universal nature of the unconventional FB transport is illustrated with the case of electrons in the Lieb lattice.

Designer flat bands in quasi-one-dimensional atomic lattices

Certain lattices with specific geometries have one or more spectral bands that are strictly flat, i.e. the electron energy is independent of the momentum. This can occur robustly irrespective of the specific couplings between the lattices sites due to the lattice symmetry, or it can result from fine-tuned couplings between the lattice sites. While the theoretical picture behind flat electronic bands is well-developed, experimental realization of these lattices has proven challenging. Utilizing scanning tunnelling microscopy (STM) and spectroscopy (STS), we manipulate individual vacancies in a chlorine monolayer on Cu(100) to construct various atomically precise 1D lattices with engineered flat bands. We realize experimentally both gapped and gapless flat band systems with single or multiple flat bands. We also demonstrate tuneability of the energy of the flat bands and how they can be switched "on" and "off" by breaking and restoring the symmetry of the lattice geometry. The experimental findings are corroborated by tight-binding calculations. Our results constitute the first experimental realizations of engineered flat bands in a 1D solid-state system and pave the way towards the construction of e.g. topological flat band systems and experimental tests of flat-band-assisted superconductivity in a fully controlled system.