Flat bands in condensed-matter systems -- perspective for magnetism and superconductivity

TL;DR

The article introduces nontrivial flatbands as dispersionless bands with finite hopping and non-orthogonal Wannier states arising from quantum interference and geometry. It surveys mechanisms and models (Lieb, Mielke, Tasaki) that realize such bands, explains how flatbands promote itinerant ferromagnetism and can enhance superconductivity via incipient or multi-band configurations, and connects these phenomena to topology and quantum metric. It also discusses candidate materials, topological aspects, and non-equilibrium Floquet engineering as a route to new superconducting and topological phases, including Floquet topological superconductivity. The work highlights the broad potential of flatband physics for designing correlated states and exploring quantum geometry, both in equilibrium and under laser illumination. Overall, it outlines a rich, interconnected framework linking lattice geometry, electron correlations, topology, and dynamical control toward emergent magnetism and superconductivity.

Abstract

There is a recent upsurge of interests in flat bands in condensed-matter systems and the consequences for magnetism and superconductivity. This article highlights the physics, where peculiar quantum-mechanical mechanisms for the physical properties such as flatband ferromagnetism and flatband superconductivity that arise when the band is not trivially flat but has a strange Hilbert space with non-orthogonalisable Wannier states, which goes far beyond just the diverging density of states. Peculiar wavefunctions come from a quantum-mechanical interference and entanglement. Interesting phenomena become even remarkable when many-body interactions are introduced, culminating in flatband superconductivity as well as flatband ferromagnetism. Flatband physics harbours a very wide range physics indeed, extending to non-equilibrium physics in laser illumination, where Floquet states for topologcial superconductivity is promoted in flatbands. While these are theoretically curious, possible candidates for the flatband materials are beginning to emerge, which is also described. These provide a wide and promising outlook.

Figures (37)

• Figure 1: The topics covered in the present article for various phenomena with various settings.
• Figure 2: Lieb (a), Mielke (b) and Tasaki (c) flatband models. In each panel, the lattice structure is displayed on the left with red enclosures representing (overlapping) Wannier functions, and the band dispersion on the right. Dashed lines enclose unit cells. In (b), a tetragonal (upper) and kagome (lower; line-graph constructed from honeycomb) realisations are displayed, where the latter is simpler in that no two bonds cross each other in a top view. Two interfering paths as an electron hops from a site to the next are shown in blue and green arrows. Right bottom figure is from JPS Hot Topics 2, 030 (2022).
• Figure 3: Hofstadter butterflies (energy spectra in an external magnetic field $B$) for Lieb (a), Mielke (b) and Tasaki (c) models [H. Aoki et al, Phys. Rev. B 54, R17296 (1996)]. $\tilde{B}$ is the magnetic flux penetrating a unit cell normalised by the flux quantum $\Phi_0 \equiv h/e$. Arrows mark the positions of the flat bands (in red in left insets) at $B=0$.
• Figure 4: (a) Spin configurations in usual paramagnetic and ferromagnetic metals. Arrows represent electron spins. (b) Flatband ferromagnetism. (c) Generalised Hund's coupling in $k$-space, here exemplified for an open-shell Fermi surface in the Hubbard model on a finite square lattice, for which the ground-state total spin $S$ is plotted against the number of electrons $N_e$ [After K. Kusakabe and H. Aoki, J. Phys. Soc. Jpn 61, 1165 (1992)].
• Figure 5: Phase diagram for the triangle chain against the Hubbard repulsion $U$ and the level offset $\varepsilon_0$ of the apex site for 1/4 filling (i.e., when the flatband, which is realised at $\varepsilon_0=-1$, is half-filled) [after K. Penc et al, Phys. Rev. B 54, 4056 (1996)]. Insets represent a hopping of a hole for $U\rightarrow \infty$ (top), how a cyclic permutation of electrons on a unit results in a ferromagnetic interaction (middle), and the triangle chain with overlapping Wannier orbits (bottom).