Zero-energy Flat Band Research Articles

Mixed higher-order topology, and nodal and nodeless flat band topological phases in a superconducting multiorbital model

We investigate the topological phases that appear in an orbital version of the Benalcazar-Bernevig-Hughes (BBH) model in the presence of conventional spin-singlet s-wave superconductivity and with the possibility of tuning an in-plane magnetic field. We chart out the phase diagram by considering different boundary conditions, with the topology of the individual phases further examined by considering both the Wannier and entanglement spectra, as well as the Majorana polarization. For weak to moderate values of magnetic field and superconducting pairing amplitude, we find a second-order topological superconductor phase with eight zero-energy corner modes. Further increasing field or pairing, half of the corner states can be turned into zero-energy edge-localized modes, thus forming what we name hybrid-order phase. Then, we find two different putative first-order topological phases, a nodal and a nodeless phase, both with zero-energy flat bands localized along mirror-symmetric open edges. For the nodal phase, the flat bands are, as expected, localized between the nodes in reciprocal space, while in the nodeless phase, the zero-energy boundary flat band instead spans the whole Brillouin zone and appears disjoint from the fully gapped bulk spectrum. As a consequence, this model present several unexpected phases with unusual surface states that can be tuned via an external magnetic field. Published by the American Physical Society 2024

Complete zero-energy flat bands of surface states in fully gapped chiral noncentrosymmetric superconductors

Complete zero-energy flat bands of surface states in fully gapped chiral noncentrosymmetric superconductors

Designer artificial chiral kagome lattice with tunable flat bands and topological boundary states

The kagome lattice is a well-known model system for the investigation of strong correlation and topological electronic phenomena due to the intrinsic flat band, magnetic frustration, etc. Introducing chirality into the kagome lattice would bring about new physics due to the unique symmetry, which is still yet to be fully explored. Here we report the investigation on a two-dimensional chiral kagome lattice utilizing tight binding band calculation and topological index analysis. It is found that the periodic chiral kagome lattice would bring about a robust zero-energy flat band. Furthermore, in the Su–Schrieffer–Heeger type dimer-/trimerized breathing chiral kagome lattice with particular edge terminations, topological corner states or metallic edge states would appear, implying new candidates for the second-order topological insulator. We also proposed the construction strategy for such lattices employing the scanning tunneling microscope atom manipulation technique.

Symmetry-protected third-order exceptional points in staggered flatband rhombic lattices

Higher-order exceptional points (EPs), which appear as multifold degeneracies in the spectra of non-Hermitian systems, are garnering extensive attention in various multidisciplinary fields. However, constructing higher-order EPs still remains a challenge due to the strict requirement of the system symmetries. Here we demonstrate that higher-order EPs can be judiciously fabricated in parity–time ( PT )-symmetric staggered rhombic lattices by introducing not only on-site gain/loss but also non-Hermitian couplings. Zero-energy flatbands persist and symmetry-protected third-order EPs (EP3s) arise in these systems owing to the non-Hermitian chiral/sublattice symmetry, but distinct phase transitions and propagation dynamics occur. Specifically, the EP3 arises at the Brillouin zone (BZ) boundary in the presence of on-site gain/loss. The single-site excitations display an exponential power increase in the PT -broken phase. Meanwhile, a nearly flatband sustains when a small lattice perturbation is applied. For the lattices with non-Hermitian couplings, however, the EP3 appears at the BZ center. Quite remarkably, our analysis unveils a dynamical delocalization-localization transition for the excitation of the dispersive bands and a quartic power increase beyond the EP3. Our scheme provides a new platform toward the investigation of the higher-order EPs and can be further extended to the study of topological phase transitions or nonlinear processes associated with higher-order EPs.

Generalized Lieb's theorem for noninteracting non-Hermitian n -partite tight-binding lattices

Hermitian bipartite models are characterized by the presence of chiral symmetry and by Lieb's theorem, which derives the number of zero-energy flat bands of the model from the imbalance of sites between its two sublattices. Here, we introduce a class of non-Hermitian models with an arbitrary number of sublattices connected in a unidirectional and cyclical way and show that the number of zero-energy flat bands of these models can be found from a generalized version of Lieb's theorem, in what regards its application to noninteracting tight-binding models, involving the imbalance between each sublattice and the sublattice of lowest dimension. Furthermore, these models are also shown to obey a generalized chiral symmetry, of the type found in the context of certain clock or parafermionic systems. The main results are illustrated with a simple toy model, and possible realizations in different platforms of the models introduced here are discussed.

Zigzag dice lattice ribbons: Distinct edge morphologies and structure-spectrum correspondences

Ribbons of two-dimensional lattices have properties depending sensitively on the morphology of the two edges. For regular ribbons with two parallel straight edges, the atomic chains terminating the two edges may have more than one choice for a general edge orientation. We enumerate the possible choices for zigzag dice lattice ribbons, which are regular ribbons of the dice lattice with edges parallel to a zigzag direction, and explore the relation between the edge morphologies and their electronic spectra. A formula is introduced to count the number of distinct edge termination morphologies for the regular ribbons, which gives 18 distinct edge termination morphologies for the zigzag dice lattice ribbons. For the pure dice model, because the equivalence of the two rim sublattices, the numerical spectra of the zigzag ribbons show qualitative degeneracies among the different edge termination morphologies. For the symmetrically biased dice model, we see a one-to-one correspondence between the 18 edge termination morphologies and their electronic spectra, when both the zero-energy flat bands and the dispersive or nonzero-energy in-gap states are considered. We analytically study several interesting features in the electronic spectra, including the number and wave functions of the zero-energy flat bands, and the analytical spectrum of unique in-gap states. The in-gap states of the zigzag dice lattice ribbons both exhibit interesting similarities and show salient differences when compared to the spectra of the zigzag ribbons of the honeycomb lattice.

General bounded corner states in two-dimensional off-diagonal Aubry–André–Harper model with flat bands

We investigate the general bounded corner states in a two-dimensional off-diagonal Aubry–André–Harper square lattice model supporting flat bands. We show that for certain values of the nearest-neighbor hopping amplitudes, triply degenerate zero-energy flat bands emerge in this lattice system. Moreover, the two-dimensional off-diagonal Aubry–André–Harper model splits into isolated fragments and hosts some general bounded corner states, and the absence of the energy gap results in that these general bounded corner states are susceptible to disorder. By adding intracellular next-nearest-neighbor hoppings, two flat bands with opposite energies split off from the original triply degenerate zero-energy flat bands and some robust general bounded corner states appear in real-space energy spectrum. Our work shows a way to obtain robust general bounded corner states in the two-dimensional off-diagonal Aubry–André–Harper model by the intracellular next-nearest-neighbor hoppings.

Flux crystals, Majorana metals, and flat bands in exactly solvable spin-orbital liquids

Spin-orbital liquids are quantum disordered states in systems with entangled spin and orbital degrees of freedom. We study exactly solvable spin-orbital models in two dimensions with selected Heisenberg-, Kitaev-, and $\mathrm{\ensuremath{\Gamma}}$-type interactions, as well as external magnetic fields. These models realize a variety of spin-orbital-liquid phases featuring dispersing Majorana fermions with Fermi surfaces, nodal Dirac or quadratic band touching points, or full gaps. In particular, we show that Zeeman magnetic fields can stabilize nontrivial flux patterns and induce metamagnetic transitions between states with different topological character. Solvable nearest-neighbor biquadratic spin-orbital perturbations can be tuned to stabilize zero-energy flat bands. We discuss in detail the examples of $\mathrm{SO}(2)$- and $\mathrm{SO}(3)$-symmetric spin-orbital models on the square and honeycomb lattices, and use group-theoretical arguments to generalize to $\mathrm{SO}(\ensuremath{\nu})$-symmetric models with arbitrary integer $\ensuremath{\nu}>1$. These results extend the list of exactly solvable models with spin-orbital-liquid ground states and highlight the intriguing general features of such exotic phases. Our models are thus excellent starting points for more realistic modelling of candidate materials.

WKB Estimate of Bilayer Graphene's Magic Twist Angles.

Graphene bilayers exhibit zero-energy flatbands at a discrete series of magic twist angles. In the absence of intrasublattice interlayer hopping, zero-energy states satisfy a Dirac equation with a non-Abelian SU(2) gauge potential that cannot be diagonalized globally. We develop a semiclassical WKB approximation scheme for this Dirac equation by introducing a dimensionless Planck's constant proportional to the twist angle, solving the linearized Dirac equation around AB and BA turning points, and connecting Airy function solutions via bulk WKB wave functions. We find zero-energy solutions at a discrete set of values of the dimensionless Planck's constant, which we obtain analytically. Our analytic flatband twist angles correspond closely to those determined numerically in previous work.

Localization of rung pairs in a hard-core Bose-Hubbard ladder

Quantum simulation in experiments of many-body systems may bring new phenomena which are not well studied theoretically. Motivated by a recent work of quantum simulation on a superconducting ladder circuit, we investigate the rung-pair localization of the Bose-Hubbard ladder model without quenched disorder. Our results show that, in the hard-core limit, there exists a rung-pair localization both at the edges and in the bulk. Using center-of-mass frame, the two-particle system can be mapped to an effective single-particle system with an approximate sub-lattice symmetry. Under the condition of hard-core limit, the effective system is forced to have a defect at the left edge leading to a zero-energy flat band, which is the origin of the rung-pair localization. We also study the multi-particle dynamics of the Bose-Hubbard ladder model, which is beyond the singleparticle picture. In this case, we find that the localization can still survive despite of the existence of interaction between the pairs. Moreover, the numerical results show that the entanglement entropy exhibits a long-time logarithmic growth and the saturated values satisfy a volume law. This phenomenon implies that the interaction plays an important role during the dynamics, although it cannot break the localization. Our results reveal another interesting type of disorder-free localization related to a zero-energy flat band, which is induced by on-site interaction and specific lattice symmetry.

Band structure and the exceptional ring in a two-dimensional superconducting circuit lattice

We investigate the energy band structure and exceptional ring in a two-dimensional superconducting circuit lattice. In the case of a Hermitian Hamiltonian, by modulating the on-site potential, we find that the single Dirac point splits into four degenerate points. For the non-Hermitian situation, we find that the flat band is significantly destroyed by the presence of gain and loss and the introduction of long-range coupling makes the zero-energy flat band survive because of the protection of chiral symmetry. Meanwhile, the purely real spectrum can be transformed into the purely imaginary spectrum via modulating the parameters appropriately. Furthermore, when the nonreciprocal next-nearest-neighbor couplings are continuously modulated, the two exceptional rings become one and a Dirac-like point occurs inside the exceptional ring. Our scheme opens the promising possibilities for exploring the topological property of two-dimensional superconducting circuit lattice system.

Correlation-induced valley splitting and orbital magnetism in a strain-induced zero-energy flatband in twisted bilayer graphene near the magic angle

In the vicinity of the magic angle in twisted bilayer graphene (TBG), many exotic correlated states, such as superconductivity, ferromagnetism, and topological phases, are observed because the two low-energy Van Hove singularities (VHSs) become exceedingly narrow, i.e., become flatbands. Heterostrain, strain in van der Waals structures where each layer is strained independently, can modify the single-particle band structure of the TBG and lead to various properties. Here, we show that heterostrain in a TBG near the magic angle generates a zero-energy flatband between the two VHSs. Doping the TBG to partially fill the zero-energy flatband, we observe a correlation-induced valley-polarized gap of \ensuremath{\sim}10 meV. By applying perpendicular magnetic fields, a large and linear response of the gap to magnetic fields is observed, attributing to the large orbital magnetic moments in TBG. The measured orbital magnetic moment per moir\'e unit cell is $\ensuremath{\sim}15\phantom{\rule{0.16em}{0ex}}{\ensuremath{\mu}}_{B}$ in the TBG, which is well consistent with our calculations.

Compact localized states and localization dynamics in the dice lattice

The dice lattice supports Aharonov-Bohm caging when all the energy bands are flat for the half-quantum magnetic flux enclosed in each plaquette of the lattice. We analytically investigate the eigenstates and discuss the localization dynamics. We find that arbitrary excitation is compactly confined within the excited-site-related snowflake structures of the dice lattice; as a consequence that the nonzero-energy flatband localizes in the single snowflake, whereas the zero-energy flatband localizes in three nearest snowflakes that are connected in the form of a trident star. The localization dynamics of an arbitrary excitation is grasped from two dynamical behaviors of single-site excitation. For the single-site excitation at the center of a snowflake, the excitation is localized in that snowflake; whereas for the single-site excitation at the branch site of a snowflake, the excitation is localized in the three snowflakes that the branch site belongs to. Our findings deepen the understanding of destructive interference and the dynamics of Aharonov-Bohm caging in the dice lattice.

Large Josephson current in Weyl nodal loop semimetals due to odd-frequency superconductivity

Weyl nodal loop semimetals (WNLs) host a closed nodal line loop Fermi surface in the bulk, protected zero-energy flat band, or drumhead, surface states, and strong spin-polarization. The large density of states of the drumhead states makes WNL semimetals exceedingly prone to electronic ordering. At the same time, the spin-polarization naively prevents conventional superconductivity due to its spin-singlet nature. Here we show the complete opposite: WNLs are extremely promising materials for superconducting Josephson junctions, entirely due to odd-frequency superconductivity. By sandwiching a WNL between two conventional superconductors we theoretically demonstrate the presence of very large Josephson currents, even up to orders of magnitude larger than for normal metals. The large currents are generated both by an efficient transformation of spin-singlet pairs into odd-frequency spin-triplet pairing by the Weyl dispersion and the drumhead states ensuring exceptionally proximity effect. As a result, WNL Josephson junctions offer unique possibilities for detecting and exploring odd-frequency superconductivity.

Spin liquid mediated RKKY interaction

We propose an RKKY-type interaction that is mediated by a spin liquid. If a spin liquid exists such an interaction could leave a fingerprint by ordering underlying localised moments such as nuclear spins. This interaction has a unique phenomenology that is distinct from the RKKY interaction found in fermionic systems; most notably the lack of a Fermi surface and absence of the requirement for itinerant electrons, since most spin liquids are insulators. We demonstrate that the interaction is predominately shaped by the lattice symmetries of the underlying spin liquid. As a working example we investigate the possible ordering of nuclear spins that interact through an underlying lattice of the two-dimensional spin-1/2 kagome antiferromagnet (KHAF), although the treatment remains general and can be extended to other spin liquids and dimensions. We find that several different nuclear spin orderings minimise the RKKY-type energy induced by the KHAF but are unstable due to a zero-energy flat magnon band in linear spin-wave theory. Despite this we show that a small magnetic field is able to gap out this magnon spectrum resulting in an intricate nuclear magnetism.

Supersymmetry in an interacting Majorana model on the kagome lattice

We construct a supersymmetric model of interacting Majorana fermions on the kagome lattice. In the infinite-coupling limit, the model exhibits an extensively degenerate ground state manifold separated in two topological sectors, in addition to two parity supersectors. An exact solution for thin-torus geometries allows us to analytically construct the entire zero-energy ground state manifold. Upon inclusion of hopping terms with energy scale $g$, the supersymmetry is spontaneously broken with the ground state energy density scaling as $g^4$. We also briefly discuss the non-interacting limit of the model, which exhibits a zero-energy flat band and Majorana Chern bands.

Spontaneous symmetry breaking at surfaces of d -wave superconductors: Influence of geometry and surface ruggedness

Surfaces of $d$-wave superconductors may host a substantial density of zero-energy Andreev states. The zero-energy flat band appears due to a topological constraint, but comes with a cost in free energy. We have recently found that an adjustment of the surface states can drive a phase transition into a phase with finite superflow that breaks time-reversal symmetry and translational symmetry along the surface. The associated Doppler shifts of Andreev states to finite energies lower the free energy. Direct experimental verification of such a phase is still technically difficult and controversial, however. To aid further experimental efforts, we use the quasiclassical theory of superconductivity to investigate how the realization and the observability of such a phase are influenced by sample geometry and surface ruggedness. Phase diagrams are produced for relevant geometric parameters. In particular, critical sizes and shapes are identified, providing quantitative guidelines for sample fabrication in the experimental hunt for symmetry-breaking phases.

Electrostatic formation of the Majorana quasiparticles in the quantum dot-nanoring structure

Zero-energy Majorana quasiparticles can be induced at the edges of low dimensional systems. Non-Abelian statistics of these states make them valid candidates for the realisation of topological quantum computer. From the practical point of view, it is crucial to obtain a system in which an on demand creation and manipulation of this type of bound states is feasible. In this article, we show such a possibility in a setup comprising a quantum nanoring in which we specify a quantum dot region via electrostatic means. The presence of quantum dot can lead to the emergence of Andreev and Majorana bound states in the investigated system. We study the differences between those two types of bound states and the possibility of their manipulation. Moreover, the exact calculation method for spectral function has been proposed, which can be used to study the influence of bound states on the band structure of the proposed system. Using this method, it can be shown that the Majorana bound states, induced at the edge of the system, present themselves as a dispersionless zero-energy flat-band.

Non-Hermitian nodal-line semimetals with an anomalous bulk-boundary correspondence

Recently, topological quantum states of non-Hermitian systems, exhibiting rich new exotic states, have attracted great attention in condensed-matter physics. As for the demonstration, most of non-Hermitian topological phenomena previously focused on are in one- and two-dimensional systems. Here, we investigate three-dimensional non-Hermitian nodal-line semimetals in the presence of a particle gain-and-loss perturbation. It is found that this perturbation will split the original nodal ring into two exceptional rings (ERs). The topological nature of the bulk electronic structure is characterized by two different topological invariants, namely, the vorticity and the winding number defined for a one-dimensional loop in momentum space, both of which are shown to take half-integer (integer) values when an odd (even) number of ERs thread through the loop. The conventional bulk-surface correspondence in non-Hermitian nodal-line semimetals is found to break down, where the surface zero-energy flat bands are no longer bounded by projections of bulk ERs. Alternatively, a macroscopic fraction of the bulk eigenstates can be localized near the surface, thus leading to the so-called non-Hermitian skin effect.

Nodal Topological Phases in s-wave Superfluid of Ultracold Fermionic Gases**Supported by National Natural Science Foundation of China under Grant Nos. 11547047 and 11504143

The gapless Weyl superfluid has been widely studied in the three-dimensional ultracold fermionic superfluid. In contrast to Weyl superfluid, there exists another kind of gapless superfluid with topologically protected nodal lines, which can be regarded as the superfluid counterpart of nodal line semimetal in the condensed matter physics, just as Weyl superfluid with Weyl semimetal. In this paper we study the ground states of the cold fermionic gases in cubic optical lattices with one-dimensional spin-orbit coupling and transverse Zeeman field and map out the topological phase diagram of the system. We demonstrate that in addition to a fully gapped topologically trivial phase, some different nodal line superfluid phases appear when the Zeeman field is adjusted. The presence of topologically stable nodal lines implies the dispersionless zero-energy flat band in a finite region of the surface Brillouin zone. Experimentally these nodal line superfluid states can be detected via the momentum-resolved radio-frequency spectroscopy. The nodal line topological superfluid provide fertile grounds for exploring exotic quantum matters in the context of ultracold atoms.