Dice Lattice Research Articles

Topological edge states of massless fermions with nonquantized and zero Berry phases

We investigate the bulk-boundary correspondence for massless Dirac fermions in α−T3 lattice where the Berry phase can be continuously tuned from π (graphene) to 0 (T3 or dice lattice) without modifying the energy dispersion. The topological origin of edge states is revealed by the unitary transformation of the Hamiltonian, which maps an α−T3 zigzag ribbon (ZR) onto a stub Su-Schrieffer-Heeger (SSH) chain. In the Majorana representation of the eigenstates, the topological invariant is manifested by the azimuthal winding number on the Bloch sphere. Particularly, T3 ZR exemplifies a topologically nontrivial system with zero Berry phase. Contrary to conventional topological insulators, the gap closing in the corresponding stub SSH chain is not accompanied by a topological phase transition. Published by the American Physical Society 2024

Nernst effect of the dice lattice in a strong magnetic field

The dice lattice bears a similar honeycomb lattice structure to graphene but with a non-dispersive flat band intersecting the Dirac bands at the band center. In this work, we investigate Nernst effect of the dice lattice in a strong magnetic field, focusing on the role of the flat band. By using the Chebyshev polynomial Green’s function method, we show that no Nernst effect (Sxy=0) is around the Dirac point in the clean limit contrary to the graphene case because of the existence of a zero Hall conductivity platform. However, an unconventional negative Sxy of the double-peak structure emerges instead when the flat band is broadened by disorder and temperature. In addition, when a mass term of Dirac electrons is introduced in the system to open an energy gap, a negative single peak of Sxy appears at the Dirac point and this is due to the derivative quantum Hall effect of non-Dirac electrons in the flat band appearing in the energy gap.

Floquet Modification of the Bandgaps and Energy Spectrum in Flat-Band Pseudospin-1 Dirac Materials

In this paper, we investigate the so-called electronic dressed states, a unified quasiparticle resulting from the interaction between electrons in a two-dimensional material with an off-resonance optical dressing field. If the frequency of this field is much larger than all characteristic energies in the system, such as the Fermi energy or bandgap(s), the electronic band structure is affected by radiation so that some important properties of the electron dispersions could be modified in a way desirable for practical applications. For example, circularly polarized light can be used to vary the bandgap of Dirac materials: it opens a gap in graphene and other metallic and semimetallic lattices, or it modifies the magnitude of an existing gap. This will either enhance or reduce a gap, depending on its initial value as well as properties of a host material. Here, we consider gapped dice and Lieb lattices as samples, and we put forward a full theoretical model to reveal how these electronic states are deformed by elliptically-polarized irradiation with a focus on the generation and modification of a bandgap.

Twisted graphene superlattices: resonating valence bond states and magnetic properties

Resonating valence bond (RVB) states are fundamental for understanding quantum spin liquids in two-dimensional (2D) systems. The RVB state is a collective phenomenon in which spins are uncoupled. 2D lattices such as triangular, honeycomb, and dice lattices were investigated using the Hubbard model and exact diagonalization method. We analyzed the total spin, spin–spin correlation functions, local magnetic moments, and spin and charge gaps as a function of on-site Coulomb repulsion, electron concentration, and electronic hopping parameters. Phase diagrams showed that RVB states can live in half-filled and hole-doped anisotropic triangular lattices. We found two types of RVB states: one in the honeycomb sublattice and the other in the centered hexagons in the triangular lattices. Owing to the novel discovery of exotic magnetic ordering in triangular moiré patterns in twisted bilayer graphene and transition metal dichalcogenide systems, our results provide physical insights into the onset of magnetism and possible spin liquid states in these layered materials.

Klein Tunneling in β12 Borophene.

Motivated by the recent observation of Klein tunneling in 8-Pmmn borophene, we delve into the phenomenon in β12 borophene by employing tight-binding approximation theory to establish a theoretical mode. The tight-binding model is a semi-empirical method for establishing the Hamiltonian based on atomic orbitals. A single cell of β12 borophene contains five atoms and multiple central bonds, so it creates the complexity of the tight-binding model Hamiltonian of β12 borophene. We investigate transmission across one potential barrier and two potential barriers by changing the width and height of barriers and the distance between two potential barriers. Regardless of the change in the barrier heights and widths, we find the interface to be perfectly transparent for normal incidence. For other angles of incidence, perfect transmission at certain angles can also be observed. Furthermore, perfect and all-angle transmission across a potential barrier takes place when the incident energy approaches the Dirac point. This is analogous to the "super", all-angle transmission reported for the dice lattice for Klein tunneling across a potential barrier. These findings highlight the significance of our theoretical model in understanding the complex dynamics of Klein tunneling in borophene structures.

Corner transfer matrix renormalization group approach in the zoo of Archimedean lattices.

We develop a new methodology to contract tensor networks within the corner transfer matrix renormalization group approach for a wide range of two-dimensional lattice geometries. We discuss contraction algorithms on the example of triangular, kagome, honeycomb, square-octagon, star, ruby, square-hexagon-dodecahedron, and dice lattices. As benchmark tests, we apply the developed method to the classical Ising model on different lattices and observe a remarkable agreement of the results with the available from the literature. The approach also shows the necessary potential to be applied to various quantum lattice models in a combination with the wave-function variational optimization schemes.

Andreev reflection in hybrid [formula omitted] lattices junction

We explore local Andreev reflection and crossed Andreev reflection in a normal metal (N)–superconductor (S)–normal metal (N) hybrid junction based on the α−T3 lattice. Initially, both the left and right sections feature dice lattice structures (α=1). We compute probabilities for electron reflection, local Andreev reflection, electron transmission, and crossed Andreev reflection, as well as local conductance, electron transmission conductance, and crossed Andreev reflection conductance. Subsequently, we analyze structures with the left side featuring dice lattice and the right side graphene (α=0), as well as the reverse configuration. Our findings reveal that crossed Andreev reflections are more prevalent in the graphene–superconductor–dice lattice (GSD) junction, particularly with moderate superconducting region lengths, larger electron incidence angles, and lower electron incidence energy.

Valley-Controlled Anomalous Nernst Effect in an Optically Driven Biased Dice Lattice

Valley-Controlled Anomalous Nernst Effect in an Optically Driven Biased Dice Lattice

Spin and charge persistent currents in a Kane Mele α-T 3 quantum ring

We conduct a thorough study of different persistent currents in a spin-orbit coupled α-T 3 (pseudospin-1) fermionic quantum ring (QR) that smoothly interpolates between graphene (α = 0, pseudospin-1/2) and a dice lattice (α = 1, pseudospin-1) in presence of an external perpendicular magnetic field. In particular, we have considered effects of intrinsic (ISOC) and Rashba spin-orbit couplings (RSOC) that are both inherent to two dimensional quantum structures and yield interesting consequences. The energy levels of the system comprise of the conduction bands, valence bands, and flat bands which show non-monotonic dependencies on the radius, R of the QR, in the sense that, for small R, the energy levels vary as 1/R , while the variation is linear for large R. The dispersion spectra corresponding to zero magnetic field are benchmarked with those for finite fields to enumerate the role played by the spin–orbit coupling terms therein. Further, it is noted that the flat bands demonstrate dispersive behavior, and hence is able to contribute to the transport properties only for finite ISOC. Moreover, RSOC yields spin-split bands, thereby contributing to the spin-resolved currents. The charge and the spin-polarized persistent currents are hence computed in presence of these spin-orbit couplings. The persistent currents in both the charge and spin sectors oscillate as a function of the magnetic field with a period equal to the flux quantum, as they should be; although they now depend upon the spin–orbit coupling parameters. Interestingly, the ISOC distorts the current profiles, owing to the distribution of the flat band caused by it, whereas RSOC alone preserves the flat band and hence a perfect periodicity of the current characteristic is maintained. Further, we have explored the role played by the parameter α in our entire analysis to enable studies while interpolating from graphene to a dice lattice.

Size effects on atomic collapse in the dice lattice.

We study the role of size effects on atomic collapse of charged impurity in the flat band system. The tight-binding simulations are made for the dice lattice with circular quantum dot shapes. It is shown that the mixing of in-gap edge states with bound states in impurity potential leads to increasing the critical charge value. This effect, together with enhancement of gap due to spatial quantization, makes it more difficult to observe the dive-into-continuum phenomenon in small quantum dots. At the same time, we show that if in-gap states are filled, the resonant tunneling to bound state in the impurity potential might occur at much smaller charge, which demonstrates non-monotonous dependence with the size of sample lattice. In addition, we study the possibility of creating supercritical localized potential well on different sublattices, and show that it is possible only on rim sites, but not on hub site. The predicted effects are expected to naturally occur in artificial flat band lattices.

Highly efficient quantum heat engine operating at maximum power in the α-T3 lattice

The α−T3 lattice, an interpolation of graphene's honeycomb lattice and the dice lattice, has drawn a lot of attention. Herein, the thermoelectric properties of the α−T3 lattice based junction are investigated. The parameter φ and the gate voltage Vg can be utilized for adjusting thermoelectric properties. Without the mass term Δ, the efficiency at maximum power η is relatively low, similar to two-dimensional graphene. When Δ is introduced, the initially flat band warps and disperses, resulting in a band gap between the distorted flat band and the conduction (valence) band. In this case, the maximum power and η are both greatly enhanced and are adjustable by Vg and φ. With high output power, η can be higher than 0.3 Carnot efficiency. As a result, it is suggested that the α−T3 lattice will be utilized to create a highly efficient quantum heat engine.

Isolated flat bands in 2D lattices based on a novel path-exchange symmetry

The increased ability to engineer two-dimensional (2D) systems, either using materials, photonic lattices, or cold atoms, has led to the search for 2D structures with interesting properties. One such property is the presence of flat bands. Typically, the presence of these requires long-ranged hoppings, fine-tuning of nearest neighbor hoppings, or breaking time-reversal symmetry by using a staggered flux distribution in the unit cell. We provide a prescription based on carrying out projections from a parent system to generate different flat band systems. We identify the conditions for maintaining the flatness and identify a path-exchange symmetry in such systems that cause the flat band to be degenerate with the other dispersive ones. Breaking this symmetry leads to lifting the degeneracy while still preserving the flatness of the band. This technique does not require changing the topology nor breaking time-reversal symmetry as was suggested earlier in the literature. The prescription also eliminates the need for any fine-tuning. Moreover, it is shown that the subsequent projected systems inherit the precise fine-tuning conditions that were discussed in the literature for similar systems, in order to have and isolate a flat band. As examples, we demonstrate the use of our prescription to arrive at the flat band conditions for popular systems like the Kagomé, the Lieb, and the Dice lattices. Finally, we are also able to show that a flat band exists in a recently proposed chiral spin-liquid state of the Kagomé lattice only if it is associated with a gauge field that produces a flux modulation of the Chern-Simons type.

Majorana corner states on the dice lattice

Lattice geometry continues providing exotic topological phases in condensed matter physics. Exciting recent examples are the higher-order topological phases, manifesting via localized lower-dimensional boundary states. Moreover, flat electronic bands with a non-trivial topology arise in various lattices and can hold a finite superfluid density, bounded by the Chern number C. Here we consider attractive interaction in the dice lattice that hosts flat bands with C = ± 2 and show that the induced superconducting state exhibits a second-order topological phase with mixed singlet-triplet pairing. The second-order nature of the topological superconducting phase is revealed by the zero-energy Majorana bound states at the lattice corners. Hence, the topology of the normal state dictates the nature of the Majorana localization. These findings suggest that flat bands with a higher Chern number provide feasible platforms for inducing higher-order topological superconductivity.

Stackings and effective models of bilayer dice lattices

Stackings and effective models of bilayer dice lattices

Optical conductivity of bilayer dice lattices

Optical conductivity of bilayer dice lattices

High Chern numbers in a perovskite-derived dice lattice (LaXO3)3/(LaAlO3)3(111) with X = Ti, Mn and Co

The dice lattice, containing a stack of three triangular lattices, has been proposed to exhibit nontrivial flat bands with nonzero Chern numbers, but unlike the honeycomb lattice it is much less studied. By employing density-functional theory (DFT) calculations with an on-site Coulomb repulsion term, we explore systematically the electronic and topological properties of (LaXO3)3/(LaAlO3)3(111) superlattices with X = Ti, Mn and Co, where a LaAlO3 trilayer spacer confines the LaXO3 (LXO) dice lattice. In the absence of spin-orbit coupling (SOC) with symmetry constrained to P3, the ferromagnetic (FM) phase of the LXO(111) trilayers exhibits a half-metallic band structure with multiple Dirac crossings and coupled electron-hole pockets around the Fermi energy. Symmetry lowering induces a significant rearrangement of bands and triggers a metal-to-insulator transition. Inclusion of SOC leads to a substantial anomalous Hall conductivity (AHC) around the Fermi energy reaching values up to sim -3e^2/h for X = Mn and Co in P3 symmetry and both in- and out-of-plane magnetization directions in the first case and along [001] in the latter. The dice lattice emerges as a promising playground to realise nontrivial topological phases with high Chern numbers.

Optical conductivity of gapped α−T3 materials with a deformed flat band

We derive and discuss the optical conductivity of $\ensuremath{\alpha}\ensuremath{-}{\mathcal{T}}_{3}$ materials with a finite band gap in their energy dispersions. In contrast with familiar $\ensuremath{\alpha}\ensuremath{-}{\mathcal{T}}_{3}$ materials, this constitutes a nontrivial case since the flat band changes into a curved one and two other branch dispersion relations become nonlinear beyond the limiting cases for graphene and the dice lattice. Such a unique band structure appears if the $\ensuremath{\alpha}\ensuremath{-}{\mathcal{T}}_{3}$ material is irradiated by a circularly polarized light nonresonantly. For this system, we have obtained optical conductivities at either zero or finite temperatures with nonzero or near-zero doping. Additionally, we demonstrate that analytical expressions can be obtained for all types of gapped $\ensuremath{\alpha}\ensuremath{-}{\mathcal{T}}_{3}$ materials, and, meanwhile, provide closed-form analytical expressions for a gapped dice lattice. Our current paper reproduces known properties of optical conductivity in unirradiated $\ensuremath{\alpha}\ensuremath{-}{\mathcal{T}}_{3}$ materials and silicene with two nonequivalent band gaps and, furthermore, demonstrates some very interesting irradiation-enabled behaviors which are absent in any Dirac materials. The discovery of unusual properties for the optical conductivity of gapped $\ensuremath{\alpha}\ensuremath{-}{\mathcal{T}}_{3}$ in this paper is expected to have many promising device applications, such as photodetectors, optical modulators, and metasurfaces.

One-dimensional flat bands and Dirac cones in narrow zigzag dice lattice ribbons

We show that four narrow zigzag dice lattice ribbons, which have the minimal widths among their separate categories, constitute a unique collection of systems to study physics related to one-dimensional Dirac cones and flat bands. In zero magnetic field, all three combinations, including only Dirac cones, only flat bands, coexisting Dirac cones and flat bands, are realized in the low-energy band structures of one or two of the four ribbons. In particular, we identify flat bands and Dirac cones corresponding to the edge states of wide ribbons. In a perpendicular magnetic field that gives half a flux quantum per elementary rhombus, two of the four minimal ribbons have fully pinched spectrum, and dynamical evolutions from initially localized wave packets always lead to compact Aharonov-Bohm (AB) cages. The experimental realizations of these narrow zigzag dice lattice ribbons, and the opportunities of exploring novel single-body and many-body physics therein are discussed.

Chiral zero-energy modes in the disordered α−T3 lattice

The $\ensuremath{\alpha}\text{\ensuremath{-}}{T}_{3}$ lattice is an interpolate between the graphene $(\ensuremath{\alpha}=0)$ and the Dice lattices $(\ensuremath{\alpha}=1)$ and has a nondispersive flat band across the Dirac bands at the band center $(E=0)$. In this paper, we study the delocalization effect of the additional chiral zero-energy modes (CZEM) from the vacancy disorder in the $\ensuremath{\alpha}\text{\ensuremath{-}}{T}_{3}$ lattice and address the influence of the flat band on the CZEM. It is shown that the mere broadening of the flat band without the CZEM could produce a supermetallic phase around the band center $E\ensuremath{\sim}0$ as well as an adjoined narrow localization regime. When the CZEM from the vacancy disorder is turned on, the transport property resembles the graphene case $(\ensuremath{\alpha}=0)$ that electrons in the low-energy regime are fully localized except the Dirac point of $E=0$, which is a critical delocalization point because the zero-energy conductivity (ZEC) is independent of the inelastic-scattering strength $\ensuremath{\eta}$. But the ZEC itself is shown to weakly rely on both the model parameter $\ensuremath{\alpha}$ and the vacancy density ${n}_{c}$ due to the broadening effect of the flat band. The extreme vacancy imbalance among each ABC site of a primitive cell is also studied, and the electron transport around $E\ensuremath{\sim}0$ is changed to be either the pure inelastic disorder case without vacancy or the fully localized phase without the critical delocalization point of $E=0$, which depends on the central site $B$ atom replaced by vacancy or not.

Probing topological signatures in an optically driven α−T3 lattice

The $\alpha$-$T_3$ lattice, an interpolation model between the honeycomb lattice of graphene($\alpha=0$) and the dice lattice($\alpha=1$), undergoes a topological phase transition across $\alpha=1/\sqrt{2}$ when exposed to a circularly polarized off-resonant light. We study Berry phase mediated bulk magnetic and anomalous thermoelectric responses in order to capture the topological signatures of a driven $\alpha$-$T_3$ lattice. It is revealed that both the Berry curvature and the orbital magnetic moment associated with the flat band change their respective signs across $\alpha=1/\sqrt{2}$. The off-resonant light distorts the flat band near the Dirac points when $0<\alpha<1$ which eventually introduces two distinct well separated forbidden gaps of equal width in the quasienergy spectrum. The orbital magnetization varies linearly with the chemical potential, in the forbidden gaps. The slopes of the linear regions in the orbital magnetization are closely related to the respective Chern numbers on either side of $\alpha=1/\sqrt{2}$. We find that the slope for $\alpha>1/\sqrt{2}$ is approximately two times of that for $\alpha<1/\sqrt{2}$ which essentially indicates a topological phase transition across $\alpha=1/\sqrt{2}$. However, the anomalous Nernst coefficient vanishes when the chemical potential is tuned in the forbidden gaps. The anomalous Hall conductivity in the forbidden gap(s) approaches different quantized values on either side of $\alpha=1/\sqrt{2}$. All these topological signatures can be observed experimentally.