Flat-band Systems Research Articles

Negative flat band magnetism in a spin–orbit-coupled correlated kagome magnet

It has long been speculated that electronic flat band systems can be a fertile ground for hosting novel emergent phenomena including unconventional magnetism and superconductivity. Although flat bands are known to exist in a few systems such as heavy fermion materials and twisted bilayer graphene, their microscopic roles and underlying mechanisms in generating emergent behavior remain elusive. Here we use scanning tunneling microscopy to elucidate the atomically resolved electronic states and their magnetic response in the kagome magnet Co3Sn2S2. We observe a pronounced peak at the Fermi level, which is identified to arise from the kinetically frustrated kagome flat band. Increasing magnetic field up to +-8T, this state exhibits an anomalous magnetization-polarized Zeeman shift, dominated by an orbital moment in opposite to the field direction. Such negative magnetism can be understood as spin-orbit coupling induced quantum phase effects tied to non-trivial flat band systems. We image the flat band peak, resolve the associated negative magnetism, and provide its connection to the Berry curvature field, showing that Co3Sn2S2 is a rare example of kagome magnet where the low energy physics can be dominated by the spin-orbit coupled flat band. Our methodology of probing band-resolved ordering phenomena such as spin-orbit magnetism can also be applied in future experiments to elucidate other exotic phenomena including flat band superconductivity and anomalous quantum transport.

Unconventional Flatband Line States in Photonic Lieb Lattices

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.

Pairing and superconductivity in the flat band: Creutz lattice

We use unbiased numerical methods to study the onset of pair superfluidity in a system that displays flat bands in the noninteracting regime. This is achieved by using a known example of flat band systems, namely, the Creutz lattice, where we investigate the role of local attractive interactions in the $U<0$ Hubbard model. Going beyond the standard approach used in these systems where weak interactions are considered, we map the superfluid behavior for a wide range of interaction strengths and exhibit a crossover between BCS and tightly bound bosonic fermion pairs. We further contrast these results with a standard two-leg fermionic ladder, showing that the pair correlations, although displaying algebraic decay in both cases, are longer ranged in the Creutz lattice, signifying the robustness of pairing in this system.

Pair Formation of Hard Core Bosons in Flat Band Systems

Hard core bosons in a large class of one or two dimensional flat band systems have an upper critical density, below which the ground states can be described completely. At the critical density, the ground states are Wigner crystals. If one adds a particle to the system at the critical density, the ground state and the low lying multi particle states of the system can be described as a Wigner crystal with an additional pair of particles. The energy band for the pair is separated from the rest of the multi-particle spectrum. The proofs use a Gerschgorin type of argument for block diagonally dominant matrices. In certain one-dimensional or tree-like structures one can show that the pair is localised, for example in the chequerboard chain. For this one-dimensional system with periodic boundary condition the energy band for the pair is flat, the pair is localised.

Thermodynamic properties of Ba2CoSi2O6Cl2 in a strong magnetic field: Realization of flat-band physics in a highly frustrated quantum magnet

The search for flat-band solid-state realizations is a crucial issue to verify or to challenge theoretical predictions for quantum many-body flat-band systems. For frustrated quantum magnets flat bands lead to various unconventional properties related to the existence of localized many-magnon states. The recently synthesized magnetic compound Ba$_2$CoSi$_2$O$_6$Cl$_2$ seems to be an almost perfect candidate to observe these features in experiments. We develop a theory for Ba$_2$CoSi$_2$O$_6$Cl$_2$ by adapting the localized-magnon concept to this compound. We first show that our theory describes the known experimental facts and then we propose new experimental studies to detect a field-driven phase transition related to a Wigner-crystal-like ordering of localized magnons at low temperatures.

Flat-band light dynamics in Stub photonic lattices

We experimentally study a Stub photonic lattice and excite their localized linear states originated from an isolated Flat Band at the center of the linear spectrum. By exciting these modes in different regions of the lattice, we observe that they do not diffract across the system and remain well trapped after propagating along the crystal. By using their wave nature, we are able to combine – in phase and out of phase – two neighbor states into a coherent superposition. These observations allow us to propose a novel setup for performing three different all-optical logical operations such as OR, AND, and XOR, positioning Flat Band systems as key setups to perform all-optical operations at any level of power.

Hard-core bosons in flat band systems above the critical density

We investigate the behaviour of hard-core bosons in one- and two-dimensional flat band systems, the chequerboard and the kagomé lattice and one-dimensional analogues thereof. The one-dimensional systems have an exact local reflection symmetry which allows for exact results. We show that above the critical density an additional particle forms a pair with one of the other bosons and that the pair is localised. In the two-dimensional systems exact results are not available but variational results indicate a similar physical behaviour.

Frustrated honeycomb-lattice bilayer quantum antiferromagnet in a magnetic field

Frustrated bilayer quantum magnets have attracted attention as flat-band spin systems with unconventional thermodynamic properties. We study the low-temperature properties of a frustrated honeycomb-lattice bilayer spin-12 isotropic (XXX) Heisenberg antiferromagnet in a magnetic field by means of an effective low-energy theory using exact diagonalizations and quantum Monte Carlo simulations. Our main focus is on the magnetization curve and the temperature dependence of the specific heat indicating a finite-temperature phase transition in high magnetic fields.

Diversity of quantum ground states and quantum phase transitions of a spin- 12 Heisenberg octahedral chain

The spin-1/2 Heisenberg octahedral chain with regularly alternating monomeric and square-plaquette sites is investigated using various analytical and numerical methods: variational technique, localized-magnon approach, exact diagonalization (ED) and density-matrix renormalization group (DMRG) method. The model belongs to the class of flat-band systems and it has a rich ground-state phase diagram including phases with spontaneously broken translational symmetry. Moreover, it exhibits an anomalous low-temperature thermodynamics close to continuous or discontinuous field-driven quantum phase transitions between three quantum ferrimagnetic phases, tetramer-hexamer phase, monomer-tetramer phase, localized-magnon phase and two different spin-liquid phases. If the intra-plaquette coupling is at least twice as strong as the monomer-plaquette coupling, the variational method furnishes a rigorous proof of the monomer-tetramer ground state in a low-field region and the localized-magnon approach provides an exact evidence of a single magnon trapped at each square plaquette in a high-field region. In the rest of parameter space we have numerically studied the ground-state phase diagram and magnetization process using DMRG and ED methods. It is shown that the zero-temperature magnetization curve may involve up to four intermediate plateaus at zero, one-fifth, two-fifth and three-fifth of the saturation magnetization, while the specific heat exhibits a striking low-temperature peak in a vicinity of discontinuous quantum phase transitions.

Linear and nonlinear compact modes in quasi-one-dimensional flatband systems

We examine analytically and numerically the spectral properties of three quasi-one-dimensional lattices, namely, kagome, Lieb, and stub lattices, which are characterized for having flatbands in their spectrum. It is observed that the degenerate eigenmodes modes of these flatbands form a Starklike ladder where each mode is shifted by one lattice site. Their combination can give rise to compact modes that do not diffract due to a geometrical phase cancellation. For all three cases we computed the stability of the fundamental band mode against perturbation of their amplitude and phase, the effect of possible anisotropy of the couplings, and the presence of small random perturbations of the coupling. For the Lieb and stub ribbon, the compact mode turns out to be quite robust and the flatband survives, while for the kagome ribbon, the compact mode is destroyed and the flatband is lost. When adding nonlinear effects, the compact mode turns out to be also a nonlinear eigenvector, with a power curve that is proportional to the eigenvalue and exists for any eigenvalue, in marked contrast to the usual case of discrete solitons, which can exist only outside the linear bands. These properties look promising for a future design of a robust system for long-distance propagation of information.

Interacting bosons in two-dimensional flat band systems

The Hubbard model of bosons on two dimensional lattices with a lowest flat band is discussed. In these systems there is a critical density, where the ground state is known exactly and can be represented as a charge density wave. Above this critical filling, depending on the lattice structure and the interaction strength, the additional particles are either delocalised and condensate in the ground state, or they form pairs. Pairs occur at strong interactions, e.g., on the chequerboard lattice. The general mechanism behind this phenomenon is discussed.

Strongly correlated flat-band systems: The route from Heisenberg spins to Hubbard electrons

On a large class of lattices (such as the sawtooth chain, the kagome and the pyrochlore lattices), the quantum Heisenberg and the repulsive Hubbard models may host a completely dispersionless (flat) energy band in the single-particle spectrum. The flat-band states can be viewed as completely localized within a finite volume (trap) of the lattice and allow for construction of many-particle states, roughly speaking, by occupying the traps with particles. If the flat-band happens to be the lowest-energy one, the manifold of such many-body states will often determine the ground-state and low-temperature physics of the models at hand even in the presence of strong interactions. The localized nature of these many-body states makes possible the mapping of this subset of eigenstates onto a corresponding classical hard-core system. As a result, the ground-state and low-temperature properties of the strongly correlated flat-band systems can be analyzed in detail using concepts and tools of classical statistical mechanics (e.g., classical lattice-gas approach or percolation approach), in contrast to more challenging quantum many-body techniques usually necessary to examine strongly correlated quantum systems. In this review, we recapitulate the basic features of the flat-band spin systems and briefly summarize earlier studies in the field. The main emphasis is made on recent developments which include results for both spin and electron flat-band models. In particular, for flat-band spin systems, we highlight field-driven phase transitions for frustrated quantum Heisenberg antiferromagnets at low temperatures, chiral flat-band states, as well as the effect of a slight dispersion of a previously strictly flat-band due to nonideal lattice geometry. For electronic systems, we discuss the universal low-temperature behavior of several flat-band Hubbard models, the emergence of ground-state ferromagnetism in the square-lattice Tasaki–Hubbard model and the related Pauli-correlated percolation problem, as well as the dispersion-driven ground-state ferromagnetism in flat-band Hubbard systems. Closely related studies and possible experimental realizations of the flat-band physics are also described briefly.

Superconducting transitions in flat-band systems

The physics of strongly correlated quantum particles within a flat band was originally explored as a route to itinerant ferromagnetism and, indeed, a celebrated theorem by Lieb rigorously establishes that the ground state of the repulsive Hubbard model on a bipartite lattice with unequal number of sites in each sublattice must have nonzero spin S at half-filling. Recently, there has been interest in Lieb geometries due to the possibility of novel topological insulator, nematic, and Bose-Einstein condensed (BEC) phases. In this paper, we extend the understanding of the attractive Hubbard model on the Lieb lattice by using Determinant Quantum Monte Carlo to study real space charge and pair correlation functions not addressed by the Lieb theorems.

Dispersion-driven ferromagnetism in a flat-band Hubbard system

We investigate a mechanism to establish ground-state ferromagnetism in flat-band Hubbard systems based on a kind of order-from-disorder effect driven by dispersion. As a paradigm we consider a frustrated diamond chain, where for ideal diamond geometry the lowest one-electron band is flat, but the ground state remains paramagnetic for arbitrary on-site repulsion $U$. We focus on half-filling of the flat band. By using numerical and analytical arguments we present the ground-state phase diagram for a distorted diamond chain, i.e., the former flat band becomes dispersive. Driven by the interplay of dispersion and interaction the ground state is ferromagnetic if the interaction exceeds a critical value ${U}_{c}$.

Phase diagram and pair Tomonaga-Luttinger liquid in a Bose-Hubbard model with flat bands

To explore superfluidity in flat-band systems, we consider a Bose-Hubbard model on a cross-linked ladder with $\pi$ flux, which has a flat band with a gap between the other band for noninteracting particles, where we study the effect of the on-site repulsion nonperturbatively. For low densities, we find exact degenerate ground states, each of which is a Wigner solid with nonoverlapping Wannier states on the flat band. At higher densities, the many-body system, when projected onto the lower flat band, can be mapped to a spin-chain model. This mapping enables us to reveal the existence of a Tomonaga-Luttinger liquid comprising it pairs of bosons. Interestingly, the high- and low- density regions have an overlap, where the two phases coexist.

High-temperature surface superconductivity in topological flat-band systems

We show that the topologically protected flat band emerging on a surface of a nodal fermionic system promotes the surface superconductivity due to an infinitely large density of states associated with the flat band. The critical temperature depends linearly on the pairing interaction and can be thus considerably higher than the exponentially small bulk critical temperature. We discuss an example of surface superconductivity in multilayered graphene with rhombohedral stacking.