Vol.34 (Feb) 2024 | Article no.2 2024
Flat bands arise in periodic media when symmetries or fine-tuning result in perfect wavepacket localisation. Flat band localisation is fragile and exhibits remarkably sharp sensitivity to perturbations including interactions and disorder, leading to a variety of interesting quantum and classical phenomena. Originally a theoretical curiosity, advances in fabrication methods have allowed flat band physics to be observed down to the nanoscale. This article briefly reviews progress in the study of flat bands and disorder over the past decade and provides an outlook on where this exciting field is headed.
In conventional disorder-free periodic wave media, a localised excitation will spread due to diffraction, at a rate determined by the group velocity of the excited Bloch band. In certain lattices with special symmetries or subject to fine tuning, the wave group velocity can vanish throughout an entire Bloch band, resulting in a flat band; diffraction becomes completely suppressed due to destructive interference between different propagation channels. However, this interference-induced localisation is not robust but sensitive to additional weak perturbations in the form of interaction or disorder.
The peculiar properties of flat band lattices have been studied by theoretical condensed matter physicists since the 1980s. Experimental studies remained challenging for a long time, due to a lack of suitably accessible materials hosting flat bands. This has all changed in the past decade, with the emergence of precision nanofabrication technologies for photonics, the ability to create electronic flat bands by manipulating individual atoms on substrates, and the rise of fine-tuned moiré materials. We now have the tools to reliably make flat bands and exploit their exotic properties, making this an exciting time for researchers working in the field.
There are already many excellent reviews covering different aspects of flat band physics including topological phenomena [1, 2], photonic flat bands [3, 4], and moiré materials and strong correlation effects [5,6,7]. What good is one more, so soon? This is not intended to be a comprehensive review in the style of Refs. [8, 9], rather the focus here is on developments in understanding the interplay between disorder and flat dispersion over the past decade.
The outline of this article is as follows: Section 2 provides a brief review of waves in periodic media and photonic lattices, motivated by interest in photonic systems analogous to graphene. Next, Section 3 discusses the generalisation of graphene to the Lieb lattice. Efforts to understand the effect of disorder led to the study of its one-dimensional cousin, the diamond chain (Section 4), followed by a variety of other one-dimensional examples (Section 5). The article concludes with Section 6, which summarises some lessons learned and speculates on future research directions.
The 2010 Nobel Prize in Physics awarded for experiments with graphene [10] led to rapidly growing interest in the topic of wave transport in two-dimensional Dirac materials. Reviews such as Ref. [11] had already attracted thousands of citations (now tens of thousands). Graphene-like systems became a hot topic in photonics.
The two-dimensional honeycomb lattice of carbon atoms forming graphene exhibits peculiar crossing points or intersections of its energy bands, as shown in Fig. 1. The band crossing points are known as Dirac points because a long wavelength expansion of their effective Hamiltonian at the crossing point yields a two-dimensional Dirac equation. Waves propagating in the vicinity of a Dirac point thus mimic various relativistic effects [12]. Conical intersections can occur either due to some symmetry protection (as is the case in graphene), or they can also be generated by fine tuning of lattice parameters [13]. One can also find lattices hosting conical intersections involving more than two bands.
The graphene honeycomb lattice and its band structure, exhibiting conical intersections at the corners of the Brillouin zone
Many remarkable properties of graphene do not require electronic materials but already appear at the level of single (non-interacting) particle band structures, described by the Schrödinger equation
where \(\psi\) is the wavefunction, t is the evolution time, \(\boldsymbol{r} = (x,y)\) are in-plane coordinates, m is the particle mass, \(\nabla ^2 = \partial _x^2 + \partial _y^2\) is the Laplacian, and \(V(\boldsymbol{r})\) is a spatially periodic potential. Often, it is convenient to discretise the right hand side of Eq. (1) using a tight binding model that accounts only for hopping between neighbouring sites of the honeycomb lattice, but this is not essential [14]. Regardless of whether a continuum or discrete model is used, the energy eigenstates will take the form discrete bands of Bloch waves parameterised by a continuous crystal momentum \(\boldsymbol{k}=(k_x,k_y)\) restricted to the first Brillouin zone.
The paraxial wave equation describing the propagation of the envelope of a slowly varying monochromatic laser beam \(A(\boldsymbol{r},z)\) along the z axis takes a similar form to Eq. (1),
where \(\delta n(\boldsymbol{r})\) is the modulation of the refractive index with respect to a uniform background index \(n_0\) and \(k_0 = 2 \pi n_0 / \lambda\) is the wavenumber. The only (minor) difference is the opposite sign of the potential function; electrons localise to potential wells, whereas light prefers to localise to regions of higher refractive index. Thus, by studying light propagation in periodic optical media, also known as photonic lattices, it is possible to emulate certain properties of electronic condensed matter systems such as graphene.
The analogy between honeycomb photonic lattices and graphene was demonstrated experimentally in 2007 [15]. The conical dispersion of the energy bands resulted in a conical diffraction of wavepackets, which can be understood by noting that all wavevectors in the vicinity of a conical intersection propagate with the same speed, just with different (radial) directions. The resulting ring-like intensity pattern resembles the conical diffraction occurring in biaxial crystals, first observed by Lloyd in 1837 [16]. The key difference, however, is that conventional conical diffraction arises due to optical anisotropy (i.e. a polarisation-dependent refractive index), whereas Eq. (2) is a scalar wave equation with no polarisation degree of freedom. Instead, the (spatial) sublattice degree of freedom plays the role of the wave spin or polarisation, acting as a pseudospin.
In this context, a natural question is whether there are novel phenomena at conical dispersions that are hard to probe in electronic systems but might be easier to observe in photonics. The relation between pseudospin and angular momentum was naturally an interesting question to explore in photonics due to long-running interest in the angular momentum of light [17]. In 2011, Mecklenberg and Regan [18] argued that pseudospin was not just a mathematical convenience but actually was a real angular momentum. How universal was this result? Was it limited to the tight binding approximation, or did it hold more generally? What about in other lattices with different kinds of conical intersections? Could this pseudospin angular momentum be measured in optical experiments? These were questions that were hotly pursued over the following years [19].
Conical dispersion relations are not limited to graphene and can be found in a variety of other lattices [12]. Since graphene’s honeycomb lattice is obtained by removing sites from a triangular lattice, it was natural to consider applying a similar procedure to other Bravais lattices. Doing so for the square lattice leads to the face-centred square lattice, shown in Fig. 2, which is now more commonly known as the Lieb lattice [20]. The name derives from Elliot Lieb, who in 1989 studied interacting spins on this lattice as a simple model for ferrimagnetism. The Lieb lattice has three sub-lattices, so there are three energy bands. The upper and lower bands exhibit a conical intersection at the Brillouin zone corner. In addition, there is a perfectly flat band sandwiched between them. The wave energy in the middle flat band is independent of the momentum; thus, the wave group velocity vanishes. The middle band can therefore be expected to support perfect wave localisation, not due to some external potential or disorder but rather due to wave interference.
Face-centred square (Lieb) lattice, its band structure, and conical intersection including a middle flat band
As is often the case in research, multiple groups became interested in these ideas independently around the same time, revealing many peculiar properties of the Lieb lattice in the context of cold atoms trapped in optical lattices [21,22,23] and electronic condensed matter physics [13, 24,25,26]. In the context of photonic lattices, the behaviour of conical diffraction and pseudospin was of interest given the prior experiments on conical diffraction in photonic graphene [15].
In contrast to the half-integer pseudospin in graphene, the Lieb lattice presents an example of integer pseudospin. Numerical and semi-analytical calculations predicted that wavepackets that partially excited the middle flat band would lead to a peculiar conical diffraction accompanied by a non-diffracting central spot and that nonlinearity could be used to redistribute the energy of a propagating wavepacket between the different bands [27]. A few years later, this pseudospin-dependent conical diffraction was observed in laser-written waveguide arrays [28]. However, the experiments were challenging, requiring heroic efforts to minimise experimental imperfections and disorder, and these difficulties motivated more detailed studies on the effects of disorder at conical intersections.
To get non-specialist readers up to speed, disorder can have quite a profound effect on wave propagation in lattices because it can lead to phenomenon of Anderson localisation, which was first predicted in the 1950s and continues to attract interest [29]. Disorder in one-dimensional lattices, no matter how weak, results in a complete suppression of wave transport. In the presence of disorder, the lattice eigenmodes no longer take a declocalised Bloch wave form but instead become exponentially localised. This can already be captured by a simple tight binding eigenvalue problem of the form
describing a one-dimensional lattice with hopping C between neighbouring sites, with a random on-site potential drawn uniformly from the interval \([-W/2,W/2]\). In one dimension there is always localisation, with the \(\nu\)th eigenstate taking the form \(\phi _n^{(\nu )} \sim \exp (-|n - n_{\nu }| / \xi )\), where \(\xi \approx 100 C^2/W^2\) at the band centre (\(E = 0\)) [30]. In three-dimensional lattices, there is a transition from extended to localised modes at a critical disorder strength \(W_c\).
The two-dimensional case is more complicated. For spinless particles, there is localisation for arbitrarily weak disorder strengths, but for certain systems with time-reversal symmetry, there is a subtle destructive interference between different backscattering amplitudes, termed weak antilocalisation [31], which was predicted to occur in graphene in the presence of sufficiently long-ranged disorder potentials that did not induce scattering between the different Dirac cones [32]. However, weak antilocalisation does not occur in the Lieb lattice due to the lack of a \(\pi\) Berry phase around its conical intersection. The pseudospin in the Lieb lattice’s conical intersection merely made the scattering due to long-ranged disorder anisotropic, without any cancellation of backscattering.
Figure 3 shows numerical simulations of the impact of disorder in the Lieb lattice tight binding model, specifically considering the propagation of wavepackets at the conical intersection as a function of the disorder strength. There is a gradual washing out of the ideal conical diffraction pattern as the disorder becomes stronger, replaced by localisation. However, at least for the broad beam sizes required to observe conical diffraction, this cross-over requires relatively strong disorder which made any analytical perturbative explanation difficult, particularly because methods developed for studying disordered graphene are complicated by the presence of the highly degenerate flat band.
Conical diffraction in the Lieb lattice for different disorder strengths
Rather than looking at Anderson localisation in the two-dimensional Lieb lattice, a simpler alternative was to consider a one-dimensional analogue for which it was possible to obtain analytical results. Specifically, the diamond chain lattice shown in Fig. 4 has a similar band structure with two dispersive bands intersecting with a middle flat band at the Brillouin zone edge. Moreover, rather than using qualitative beam propagation simulations, by studying the energy-dependent properties of the eigenstates one could more easily distinguish the properties of the conical bands from those related to the flat band.
The quasi-one-dimensional diamond lattice, its band structure, and Anderson localisation length for different disorder strengths
Figure 4 shows the eigenstate localisation length as a function of energy, with different disorder strengths plotted in different colours. For weak disorder, there is a pronounced dip in the localisation length in the vicinity of the flat band zero at energy, indicating that the flat band is much more strongly affected by the disorder and much more strongly localised compared to the the modes belonging to the other bands. Moreover the shape of the dip, its width, and its contrast change with the disorder strength. This suggests different scaling properties of the modes at zero and non-zero energies.
Indeed, calculating the localisation length of the eigenstates as a function of the disorder strength W yields different power law behaviours [33]. Away from the flat band energy, there is the scaling exponent \(\nu = 2\), which is the well-known scaling of the conventional Anderson model, indicating that the dispersive bands behave very similar to conventional one-dimensional lattices. On the other hand, the scaling of the flat band modes at energy zero was much weaker, \(\nu \approx 1.33\), indicating qualitatively different behaviour compared to the regular Anderson model.
Attempts to explain the peculiar \(\nu \approx 1.33\) scaling by reducing the eigenvalue problem for the diamond chain to a more conventional Anderson model of localisation were unfortunately not fruitful. While one can eliminate some of the site amplitudes from the eigenvalue problem to obtain something that looks like the one-dimensional lattice Eq. (3), there are important differences which complicate further analysis: the effective disorder become not only energy-dependent, but also correlated between neighbouring sites.
Attempts to better understand the peculiar \(\nu \approx 1.33\) scaling with disorder strength led to a study of Anderson localisation and other observables in several other quasi-one-dimensional lattices hosting flat bands including the one-dimensional pyrochlore, another one-dimensional variant of the Lieb lattice, the sawtooth, and the stub [34]. In some cases, the flat band was separated from dispersive bands by a gap; in other cases, the flat band touched a band edge; and in others, it was embedded within a dispersive band. All these different scenarios could be captured by a single model with a tunable flat band energy, the cross stitch lattice (also known as the two leg ladder), which had a single flat band and a single dispersive band. The scaling of the numerically computed localisation length was universal, in that it did not depend on the choice of lattice model but only by the detuning of the flat band with respect to the nearest dispersive band. Figure 5 shows the different scaling behaviours obtained, dependent on whether the flat band was embedded in a dispersive band, located at the edge of a dispersive band, or in a band gap.
The scaling of the Anderson localisation length with the disorder strength in a flat band (FB) depends on the detuning of the flat band with respect to the nearest dispersive band (DB)
These numerical results can be understood qualitatively in terms of Fano resonances, which were another fashionable topic at the time [35]. Fano resonance refers to an interaction between a single bound mode and a continuum of delocalised modes which gives rise to a strongly asymmetric and distinct scattering spectrum. In the case of flat band lattices, the strongly localised flat band mode can be taken to be the bound state and the dispersive states which are weakly localised can form a continuum. The unconventional scaling can thus be understood in terms of a self-consistent interaction between strongly bound and weakly bound modes; disorder induces scattering and some interaction between a discrete state and the continuum.
Using this intuition and first eliminating the flat band states from the eigenvalue problem, one can obtain a regular Anderson model on a simple one dimensional lattice with a relatively simple energy-dependent disorder potential,
Here, the effective disorder potential involves two parts. The first part is related to the symmetric part of the disorder distribution and corresponds to a bounded disorder term, similar to that in the conventional Anderson model. The second term, arising due to the antisymmetric part of the local disorder potential, describes the disorder-induced coupling between a flat band state and the dispersive band states. This coupling involves an energy term on the denominator, meaning there is a resonant enhancement of the effective disorder strength for energies close to the flat band energy. This term is unbounded and the probability distribution function of the effective disorder potential exhibits fat or heavy tails obeying a Lorentz distribution.
The one-dimensional Anderson model with Lorentz distributed disorder is called the Lloyd model [36]. It had already been solved in the 1970s by Thouless [37] and Ishii [38], who in separate papers had shown how one could obtain all spectral properties of the model exactly. So, by simply mapping the flat band eigenvalue problem onto this problem that had already been solved, the unconventional scaling of the localisation length can be reproduced analytically [34].
When the flat band is spectrally isolated from other dispersive bands by a gap, there is no resonant interaction between the flat and dispersive band modes. In this case, the effective potential is bounded, and for weak disorder, the mode localisation is dictated purely by the localisation of evanescent states derived from the dispersive band. This corresponds to scaling with an exponent \(\nu = 0\), appearing as the lowest curve in Fig. 5. When the flat band is embedded in or touching a dispersive band, the resonant interaction between the two different types of modes leads to different scaling laws. For a flat band located precisely at the edge of a dispersive band, the predicted exponent is \(\nu = 1/2\), while a flat band embedded within the dispersive band has \(\nu = 1\). These scaling laws lead to much stronger localisation compared to the regular Anderson model scaling of \(\nu = 1\) and \(\nu = 2\) at the band edge and within the band, respectively. In all these cases, there is very nice agreement between the analytical prediction and the numerical results. The diamond chain with its \(\nu \approx 1.33\) remains the only anomaly.
This study served as a neat example of not only how flat bands lead to an enhanced sensitivity to disorder but also how this enhancement can be captured using simple analytical techniques. Usually, large degeneracies lead to perturbative approaches failing. This is what makes flat band systems such a valuable setting for observing and understanding a variety of strong interaction phenomena.
This article has reviewed some of the early works in the current phase of interest in flat band lattices, which began around 2010, motivated by interest in graphene and graphene-like systems exhibiting dispersion relations with a vanishing effective mass and insensitivity to disorder. Attempts to generalise graphene’s conical intersection to more intersecting bands led to peculiar dispersion relations involving flat bands with an infinite effective mass and enhanced sensitivity to perturbations including disorder.
An important take-home message for graduate students, supervisors, and funding agencies will be that research is often a highly nonlinear process. Research rarely goes all according to plan. Not all projects need to have a clear objective, obvious applications, or experimental feasibility when first embarked on; these often emerge further down the track. Today’s curiosity-driven research will lay the foundations for tomorrow’s unanticipated scientific breakthroughs. We often repeat these maxims. Then, when it comes to writing up and publishing our results, we tend to distill a sleek story from a complicated project. We pretend the many false starts and dead ends never happened.
Where to next? Funding agencies especially would like to see practical applications derived from research into flat bands. An important step in this direction is to translate simplified tight binding models of flat bands to nanophotonic devices with enhanced light-matter interactions including free electron light sources [39] and nanolasers [40]. In finite photonic systems, in contrast to electronic ones, states (almost always) have nonzero radiative losses and long range couplings. These effects are usually neglected in studies of flat bands - most models considered in the literature rely on models with nearest neighbour couplings. Nevertheless, parameter fine-tuning can still lead to collective resonances with an enhanced density of states resembling the resonant enhancement provided by flat bands [41]. Insights in this area may also be relevant to flatbands in moiré materials [6, 7].
Another important avenue is the intersection of flat band physics with machine learning and artificial intelligence, which promise new ways to do science. A few recent studies have used high-throughput numerical screening and neural networks to search for new examples of flat bands arising in crystalline materials [42,43,44,45]. These techniques will hopefully broaden the scope for potential technological applications and identify new classes of models for theorists to explore.
In summary, flatband lattices have proven quite a fun and interesting playground for exploring non-perturbative wave phenomena. We have a variety of tricks up our sleeve which allow us to obtain an analytical understanding of these complex systems. Flatband systems can now be implemented experimentally in various platforms, including photonics, cold atoms, and electronic materials. Especially, recent observations of flatbands in a variety of twisted two-dimensional materials have sparked renewed interest in this area. It will be exciting to see what surprises emerge next!
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The author wishes to express his gratitude to all of his collaborators on flat bands over the years.
The author is supported by the National Research Foundation, Singapore and A*STAR under its CQT Bridging Grant.
The author wrote, read, and approved the final manuscript.
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The author declares no competing interests.
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