Flat-band Models Research Articles

Thermodynamic Response and Neutral Excitations in Integer and Fractional Quantum Anomalous Hall States Emerging from Correlated Flat Bands.

Integer and fractional Chern insulators have been extensively explored in correlated flat band models. Recently, the prediction and experimental observation of fractional quantum anomalous Hall (FQAH) states with spontaneous time-reversal symmetry breaking have garnered attention. While the thermodynamics of integer quantum anomalous Hall (IQAH) states have been systematically studied, our theoretical knowledge on thermodynamic properties of FQAH states has been severely limited. Here, we delve into the general thermodynamic response and collective excitations of both IQAH and FQAH states within the paradigmatic flat Chern-band model with remote band considered. Our key findings include (i)in both ν=1 IQAH and ν=1/3 FQAH states, even without spin fluctuations, the charge-neutral collective excitations would lower the onset temperature of these topological states, to a value significantly smaller than the charge gap, due to band mixing and multiparticle scattering; (ii)by employing large-scale thermodynamic simulations in FQAH states in the presence of strong interband mixing between C=±1 bands, we find that the lowest collective excitations manifest as the zero-momentum excitons in the IQAH state, whereas in the FQAH state, they take the form of magnetorotons with finite momentum; (iii)the unique charge oscillations in FQAH states are exhibited with distinct experimental signatures, which we propose to detect in future experiments.

General construction scheme for geometrically nontrivial flat band models

A singular flat band (SFB), a distinct class of the flat band, has been shown to exhibit various intriguing material properties characterized by the quantum distance. We present a general construction scheme for a tight-binding model hosting an SFB, where the quantum distance profile can be controlled. We first introduce how to build a compact localized state (CLS), endowing the flat band with a band-touching point and a specific value of the maximum quantum distance. Then, we develop a scheme designing a tight-binding Hamiltonian hosting an SFB starting from the obtained CLS, with the desired hopping range and symmetries. We propose several simple SFB models on the square and kagome lattices. Finally, we establish a bulk-boundary correspondence between the maximum quantum distance and the boundary modes for the open boundary condition, which can be used to detect the quantum distance via the electronic structure of the boundary states.

Nematicity-enhanced superconductivity in systems with a non-Fermi liquid behavior

We explore the interplay between nematicity (spontaneous breaking of the sixfold rotational symmetry), superconductivity, and non-Fermi liquid behavior in partially flat-band (PFB) models on the triangular lattice. A key result is that the nematicity (Pomeranchuk instability), which is driven by many-body effect and stronger in flat-band systems, enhances superconducting transition temperature in a systematic manner on the T c dome. There, a plausible -wave symmetry, in place of the conventional -wave, governs the nematicity-enhanced pairing with a sharp rise in the T c dome on the filling axis. When the sixfold symmetry is spontaneously broken, the pairing interaction is shown to become stronger with more compact pairs in real space than when the symmetry is enforced. These are accompanied by a non-Fermi character of electrons in the PFBs with many-body interactions.

Machine Learning Study on the Flat-Band States Constructed by Molecular-Orbital Representation with Randomness

We study the characteristic probability density distribution of random flat band models by machine learning. The models considered here are constructed on the basis of the molecular-orbital representation, which guarantees the existence of the macroscopically degenerate zero-energy modes even in the presence of randomness. We find that flat band states are successfully distinguished from conventional extended and localized states, indicating the characteristic feature of the flat band states. We also find that the flat band states can be detected when the target data are defined in the different lattice from the training data, which implies the universal feature of the flat band states constructed by the molecular-orbital representation.

Chern-number matrix of the non-Abelian spin-singlet fractional quantum Hall effect

While the internal structure of Abelian topological order is well understood, how to characterize the non-Abelian topological order is an outstanding issue. We propose a distinctive scheme based on the many-body Chern number matrix to characterize non-Abelian multicomponent fractional quantum Hall states. As a concrete example, we study the many-body ground state of two-component bosons at the filling faction $\nu=4/3$ in topological flat band models. Utilizing density-matrix renormalization group and exact diagonalization calculations, we demonstrate the emergence of non-Abelian spin-singlet fractional quantum Hall effect under three-body interaction, whose topological nature is classified by six-fold degenerate ground states and a fractionally quantized Chern number matrix.

Construction of interacting flat-band models by molecular-orbital representation: Correlation functions, energy gap, and entanglement

Abstract We calculate correlation functions of exactly solvable one-dimensional flat-band models by utilizing the “molecular-orbital” representation. The models considered in this paper have a gapped ground state with the flat band being fully occupied, even in the presence of the interaction. In this class of models, the space spanned by the molecular-orbitals is the co-space of that spanned by the flat bands. Thanks to this property, the correlation functions are calculated by using the information of the molecular-orbitals rather than the explicit forms of the flat-band wave functions, which simplifies the calculations. As a demonstration, several one-dimensional models and their correlation functions are presented. We also calculate the entanglement entropy by using the correlation function.

General construction and topological classification of crystalline flat bands

Exotic phases of matter emerge from the interplay between strong electron interactions and non-trivial topology. Owing to their lack of dispersion at the single-particle level, systems harboring flat bands are excellent testbeds for strongly interacting physics, with twisted bilayer graphene serving as a prime example. On the other hand, existing theoretical models for obtaining flat bands in crystalline materials, such as the line-graph formalism, are often too restrictive for real-life material realizations. Here we present a generic technique for constructing perfectly flat bands from bipartite crystalline lattices. Our prescription encapsulates and generalizes the various flat band models in the literature, being applicable to systems with any orbital content, with or without spin-orbit coupling. Using Topological Quantum Chemistry, we build a complete topological classification in terms of symmetry eigenvalues of all the gapped and gapless flat bands, for all 1651 Magnetic Space Groups. In addition, we derive criteria for the existence of symmetry-protected band touching points between the flat and dispersive bands, and we identify the gapped flat bands as prime candidates for fragile topological phases. Finally, we show that the set of all (gapped and gapless) perfectly flat bands is finitely generated and construct the corresponding bases for all 1651 Shubnikov Space Groups.

Designing flat-band tight-binding models with tunable multifold band touching points

Being dispersionless, flat bands on periodic lattices are solely characterized by their macroscopically degenerate eigenstates: compact localized states (CLSs) in real space and Bloch states in reciprocal space. Based on this property, this work presents a straightforward method to build flat-band tight-binding models with short-range hoppings \emph{on any periodic lattice}. The method consists in starting from a CLS and engineering families of Bloch Hamiltonians as quadratic (or linear) functions of the associated Bloch state. The resulting tight-binding models not only exhibit a flat band, but also multifold quadratic (or linear) band touching points (BTPs) whose number, location, and degeneracy can be controlled to a large extent. Quadratic flat-band models are ubiquitous: they can be built from any arbitrary CLS, on any lattice, in any dimension and with any number $N\geq2$ of bands. Linear flat-band models are rarer: they require $N\geq3$ and can only be built from CLSs that fulfill certain compatibility relations with the underlying lattice. Most flat-band models from the literature can be classified according to this scheme: Mielke's and Tasaki's models belong to the quadratic class, while the Lieb, dice and breathing Kagome models belong to the linear class. Many novel flat-band models are introduced, among which an $N=4$ bilayer honeycomb model with fourfold quadratic BTPs, an $N=5$ dice model with fivefold linear BTPs, and an $N=3$ Kagome model with BTPs that can be smoothly tuned from linear to quadratic.

General construction of flat bands with and without band crossings based on wave function singularity

In this work, we develop a systematic method of constructing flat-band models with and without band crossings. Our construction scheme utilizes the symmetry and spatial shape of a compact localized state (CLS) and also the singularity of the flat-band wave function obtained by a Fourier transform of the CLS (FT-CLS). In order to construct a flat-band model systematically using these ingredients, we first choose a CLS with a specific symmetry representation in a given lattice. Then, the singularity of FT-CLS indicates whether the resulting flat band exhibits a band crossing point or not. A tight-binding Hamiltonian with the flat band corresponding to the FT-CLS is obtained by introducing a set of basis molecular orbitals, which are orthogonal to the FT-CLS. Our construction scheme can be systematically applied to any lattice so that it provides a powerful theoretical framework to study exotic properties of both gapped and gapless flat bands arising from their wave function singularities.

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-band solutions in D -dimensional decorated diamond and pyrochlore lattices: Reduction to molecular problem

Flat-band models have been of particular interest from both fundamental aspects and realization in materials. Beyond the canonical examples such as Lieb lattices and line graphs, a variety of tight-binding models are found to possess flat bands. However, analytical treatment of dispersion relations is limited, especially when there are multiple flat bands with different energies. In this paper, we present how to determine flat-band energies and wave functions in tight-binding models on decorated diamond and pyrochlore lattices in generic dimensions $D \geq 2$. For two and three dimensions, such lattice structures are relevant to various organic and inorganic materials, and thus our method will be useful to analyze the band structures of these materials.

Fractional quantum Hall effect of Bose-Fermi mixtures

Multicomponent quantum Hall effect, under the interplay between intercomponent and intracomponent correlations, leads us to new emergent topological orders. Here, we report the theoretical discovery of fractional quantum hall effect of strongly correlated Bose-Fermi mixtures classified by the $\mathbf{K}=\begin{pmatrix} m & 1\\ 1 & n\\ \end{pmatrix}$ matrix (even $m$ for boson and odd $n$ for fermion), using topological flat band models. Utilizing the state-of-the-art exact diagonalization and density-matrix renormalization group methods, we build up the topological characterization based on three inherent aspects: (i) topological $(mn-1)$-fold ground-state degeneracy equivalent to the determinant of the $\mathbf{K}$ matrix, (ii) fractionally quantized topological Chern number matrix equivalent to the inverse of the $\mathbf{K}$ matrix, and (iii) two parallel-propagating chiral edge branches with level counting $1,2,5,10$ consistent with the conformal field theory description.

Building flat-band lattice models from Gram matrices

We propose a powerful and convenient method to systematically design flat-band lattice models, which overcomes the difficulties underlying the previous method. Especially, our method requires no elaborate calculations, applies to arbitrary spatial dimensions, and guarantees to result in a completely flat ground band. We use this method to generate several classes of lattice models, including models with both short- and long-range hoppings, both topologically trivial and non-trivial flat bands. Some of these models were previously known. Our method, however, provides crucial new insights. For example, we have reproduced and generalized the Kapit-Mueller model [Kapit and Mueller, Phys. Rev. Lett. \textbf{105}, 215303 (2010)] and demonstrated a universal scaling rule between the flat band degeneracy and the magnetic flux that was not noticed in previous studies. We show that the flat band of this model results from the (over-)completeness properties of coherent states.

Type-III Dirac Cones from Degenerate Directionally Flat Bands: Viewpoint from Molecular-Orbital Representation

We present a systematic method of creating type-III Dirac cones from degenerate directionally flat bands. Here, the directionally flat band means a completely dispersionless band only in a certain direction. The key strategy is to retain one of the directionally flat bands while making the other dispersive, by adding a perturbation having a selected form of a Hamiltonian matrix. Such a form of a perturbation is found by using the "molecular-orbital representation" which we have developed to describe flat-band models. The models thus obtained host type-III Dirac cones even without fine-tuning hopping parameters. To demonstrate the ubiquity of this method, we study concrete examples of two-band models and a four-band model.

An Extension of the Cell-Construction Method for the Flat-Band Ferromagnetism

We present an extension of the cell-construction method for the flat-band ferromagnetism. In a rather general setting, we construct Hubbard models with highly degenerate single-electron ground states and obtain a formal representation of these single-electron ground states. By our version of the cell-construction method, various types of flat-band Hubbard models, including the one on line graphs, can be designed and shown to have the unique ferromagnetic ground states when the electron number is equal to the degeneracy of the single-electron ground states.

Systematic construction of topological flat-band models by molecular-orbital representation

On the basis of the "molecular-orbital" representation which describes generic flat-band models, we propose a systematic way to construct a class of flat-band models with finite-range hoppings that have topological natures. In these models, the topological natures are encoded not into the flat band itself but into the dispersive bands touching the flat band. Such a band structure may become a source of exotic phenomena arising from the combination of flat bands, topology and correlations.

Quantum Hall effects of exciton condensate in topological flat bands

Tunable exciton condensates in two dimensional electron gas systems under strong magnetic field exhibits anomalous Hall transport owing to mutual Coulomb coupling, and have attracted a lot of research activity. Here, we explore another framework using topological flat band models in the absence of Landau levels, for realizing the many-body exciton phases of two-component fermions under strong intercomponent interactions. By developing new diagnosis based on the state-of-the-art density-matrix renormalization group and exact diagonalization, we show the theoretical discovery of the emergence of Halperin (111) quantum Hall effect at a total filling factor $\nu=1$ in the lowest Chern band under strong Hubbard repulsion, which is classified by the unique ground state with bulk charge insulation and spin superfluidity, The topological nature is further characterized by one edge branch of chiral propagating Luttinger modes with level counting $1,1,2,3,5,7$ in consistent with the conformal field theory description. Moreover, with nearest-neighbor repulsions, we propose the Halperin (333) fractional quantum Hall effect at a total filling factor $\nu=1/3$ in the lowest Chern band.

Photonic zero-energy modes in a metal-based Lieb lattice

We design a photonic tight-binding system using the dispersive background, and observe the noncompact photonic zero-energy modes for both monopole and dipolar states in a finite Lieb lattice with flat truncations. In such a photonic Lieb system, the compact localization of s, p, and d flat bands is also checked. We show that this photonic zero-energy mode is provided by one dispersive band for singular touching, which has the same frequency with the flat band states. Specially, the zero-energy mode can be completely excited by merely one point source at the flat band frequency, covering all the minority sites and forming a noncompact state. This work may provide a deep understanding about the photonic zero-energy mode for higher order states in the Lieb or other flat band models.

Molecular-orbital representation of generic flat-band models

We develop a framework to describe a wide class of flat-band models, with and without a translational symmetry, by using “molecular orbitals” introduced in a prior work (Hatsugai Y. and Maruyama I., EPL, 95 (2011) 20003). Using the molecular-orbital representation, we shed new light on the band-touching problem between flat and dispersive bands. We show that the band touching occurs as a result of collapse, or the linearly dependent nature, of molecular orbitals. Conversely, we can gap out the flat bands by modulating the molecular orbitals so that they do not collapse, which provides a simple prescription to construct models having a finite energy gap between flat bands and dispersive bands.

Flat-band engineering in tight-binding models: Beyond the nearest-neighbor hopping

In typical flat-band models, defined as nearest-neighbor tight-binding models, flat bands are usually pinned to the special energies, such as top or bottom of dispersive bands, or band-crossing points. In this paper, we propose a simple method to tune the energy of flat bands without losing the exact flatness of the bands. The main idea is to add farther-neighbor hoppings to the original nearest-neighbor models, in such a way that the transfer integrals depend only on the Manhattan distance. We apply this method to several lattice models including the two-dimensional kagome lattice and the three-dimensional pyrochlore lattice, as well as their breathing lattices and non-line graphs. The proposed method will be useful for engineering flat bands to generate desirable properties, such as enhancement of ${T}_{c}$ of superconductors and nontrivial topological orders.