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![Flat band formation. (a, top): The flat band represents a topological object in momentum space. The dispersionless Fermi band is analogous to a soliton terminated by a half-quantum vortex: the phase of the Green's function changes by π around the edge of the flat band. Twodimensional flat band appears on the surface of gapless systems with a topologically protected nodal line[1, 2]. (a, bottom): The nodal spiral in a semi-metal. This nodal line has a non-zero topological charge in momentum space, N 2 = 1, and this charge protects the surface states with zero energy (p x , p y ) = 0, in the whole region within the projection of the spiral on the surface. (b): One-dimensional topologically protected flat band emerges on the zig-zag edge of graphene. Projections of the Dirac points on the edge determine the boundaries of the flat band [16]. (c): Nodes (nodal lines) in cuprate superconductors give rise to the two-dimensional flat bands on the lateral surface [16]. Projections of the nodes on the surface determine the boundaries of the flat bands. (d): One-dimensional topologically protected flat band emerges in the core of linear topological defects (such as the vortex with winding number n = 1 in real space in the bottom right corner) in three-dimensional topological matter with Dirac points. The projections of the Dirac points on the direction of the vortex line (along the z-axis) determine the boundaries of the region where the spectrum of fermions bound to the vortex core is exactly zero, (p z ) = 0.](profile/N-Kopnin/publication/47865067/figure/fig1/AS:669963748659232@1536743213826/Flat-band-formation-a-top-The-flat-band-represents-a-topological-object-in-momentum.png)
Flat band formation. (a, top): The flat band represents a topological object in momentum space. The dispersionless Fermi band is analogous to a soliton terminated by a half-quantum vortex: the phase of the Green's function changes by π around the edge of the flat band. Twodimensional flat band appears on the surface of gapless systems with a topologically protected nodal line[1, 2]. (a, bottom): The nodal spiral in a semi-metal. This nodal line has a non-zero topological charge in momentum space, N 2 = 1, and this charge protects the surface states with zero energy (p x , p y ) = 0, in the whole region within the projection of the spiral on the surface. (b): One-dimensional topologically protected flat band emerges on the zig-zag edge of graphene. Projections of the Dirac points on the edge determine the boundaries of the flat band [16]. (c): Nodes (nodal lines) in cuprate superconductors give rise to the two-dimensional flat bands on the lateral surface [16]. Projections of the nodes on the surface determine the boundaries of the flat bands. (d): One-dimensional topologically protected flat band emerges in the core of linear topological defects (such as the vortex with winding number n = 1 in real space in the bottom right corner) in three-dimensional topological matter with Dirac points. The projections of the Dirac points on the direction of the vortex line (along the z-axis) determine the boundaries of the region where the spectrum of fermions bound to the vortex core is exactly zero, (p z ) = 0.
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Topological media are systems whose properties are protected by topology and
thus are robust to deformations of the system. In topological insulators and
superconductors the bulk-surface and bulk-vortex correspondence gives rise to
the gapless Weyl, Dirac or Majorana fermions on the surface of the system and
inside vortex cores. Here we show that i...
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Citations
... This places naturally these systems in the class of highly correlated materials and opens the access to exotic and unexpected physical phenomena and quantum phases. Undeniably, one of the most striking feature is the possibility of high critical temperature superconductivity (SC) in compounds where the Fermi velocity vanishes [9][10][11][12][13][14][15][16][17][18]. This unconventional form of SC is of inter-band nature and characterised by a geometrical quantity known as the quantum metric (QM). ...
... The partially flat bands of the zigzag graphene nanoribbon attracted tremendous attention due to their instability toward the halfmetallic ground state 65 . Another well-known example is the drum-head surface states of nodal-line semimetals [61][62][63][64] , which are expected to lead to the high-temperature superconductivity 66 . ...
The localized nature of a flat band is understood by the existence of a compact localized eigenstate. However, the localization properties of a partially flat band, ubiquitous in surface modes of topological semimetals, have been unknown. We show that the partially flat band is characterized by a non-normalizable quasi-compact localized state (Q-CLS), which is compactly localized along several directions but extended in at least one direction. The partially flat band develops at momenta where normalizable Bloch wave functions can be obtained from a linear combination of the non-normalizable Q-CLSs. Outside this momentum region, a ghost flat band, unseen from the band structure, is introduced based on a counting argument. Then, we demonstrate that the Wannier function corresponding to the partially flat band exhibits an algebraic decay behavior. Namely, one can have the Wannier obstruction in a band with a vanishing Chern number if it is partially flat. Finally, we develop the construction scheme of a tight-binding model for a topological semimetal by designing a Q-CLS.
... This interesting observation has recently been documented in a material possessing a Kagome lattice structure. [30][31][32] This observation implies that CuSe may exhibit superconducting properties. Since electrons that traverse the Fermi level are primarily responsible for defining the Fermi surface, they are considered an essential quantity for comprehending the electronic properties of metallic materials. ...
In the realm of materials science, a fascinating class of substances known as topological nodal line semimetals have garnered significant attention. These materials possess distinctive quantum states, characterized by topologically nontrivial conduction and valence bands that intricately intersect with the Fermi level. Such unique band structures give these materials a plethora of surprising properties, including the emergence of flat Landau levels and the appearance of long‐range Coulomb interactions. Notably, certain bulk materials, namely PbTaSe2, ZrSiS, PtSn4, and Cu2Si, exhibit these intriguing topological nodal lines. However, the theoretical investigation pertaining to CuSe and its Dirac nodal line properties remains relatively limited. In this work, we have employed simulations on CuSe to elucidate its distinctive physical properties such as a Dirac matter. There exist Dirac bands and band inversion in the vicinity of Fermi levels, and Fermi surface holds a Horn‐like structure. We pointed out the occurrence of mirror reflection symmetry‐protected Dirac nodal line fermions in copper selenide (CuSe) that make CuSe a candidate for Dirac nodal line fermion started by the Se 4p and Cu 3d orbitals. The nontrivial topological edge states further verify our proposal of the mirror reflection symmetry is broken by spin‐orbit coupling. These findings provide a new perspective that advances our comprehension of the remarkable properties exhibited by these materials, thereby paving the way for the development of high‐speed and low‐dissipation devices in the future.
... More recently MBL is discussed in systems which exhibit flat bands (FBs) [42][43][44][45][46][47][48][49][50][51][52]. FBs can appear in different kind of systems and can even appear as topologically protected surface states [53][54][55][56][57][58]. FBs may remain stable even for strong interactions [59]. ...
We study the broadening of initially localized wave packets in a quasi one-dimensional diamond ladder with interacting, spinless fermions. The lattice possesses a flat band causing localization. We place special focus on the transition away from the flat band many-body localized case by adding very weak dispersion. By doing so, we allow propagation of the wave packet on significantly different timescales which causes anomalous diffusion. Due to the temporal separation of dynamic processes, an interaction-induced, prethermal equilibrium becomes apparent. A physical picture of light and heavy modes for this prethermal behavior can be obtained within Born–Oppenheimer approximation via basis transformation of the original Hamiltonian. This reveals a detachment between light, symmetric and heavy, anti-symmetric particle species. We show that the prethermal state is characterized by heavy particles binding together mediated by the light particles.
... The QM has been found to play a key role in the unconventional form of superconductivity that takes place in FBs [62]. Hence, in the presence of attractive electron-electron interaction in the Stagome lattice, one may expect substantial values of the superfluid weight and thus potentially large superconducting critical temperatures [63][64][65]. We recall that, the QM is related to the real part of the quantum geometric tensor (QGT) and provides a measure of the typical spread of the FB eigenstates [66,67]. ...
We present the Stagome lattice, a variant of the Kagome lattice, where one can make any of the bands completely flat by tuning an externally controllable magnetic flux. This systematically allows the energy of the flat band (FB) to coincide with the Fermi level. We have analytically calculated the compact localized states (CLS) associated to each of these flat bands appearing at different values of the magnetic flux. We also show that, this model features nontrivial topological properties with distinct integer values of the Chern numbers as a function of the magnetic flux. We argue that this mechanism for making any of the bands exactly flat could be of interest to address the FB superconductivity in such a system. Furthermore, we believe that the phenomenon of photonic flat band localization could be studied in the Stagome lattice structure, designed for instance using femtosecond laser induced single-mode waveguide arrays.
... Even in pristine graphene, two valleys with opposite Berry phases are connected by the zero-energy flat band along zigzag edges. These well-known zigzag edge states (ZESs) in graphene have a topological origin, are quite robust against weak perturbations, and have been experimentally verified [3][4][5][6][7][8]. The magnetic structures of zigzag-edge flat bands have been widely studied based on the electron-electron interaction [9][10][11][12][13][14][15][16][17]. ...
Typically, edge states in graphene are known to exist solely along zigzag edges. However, in this paper, we present a theoretical discovery of edge states along armchair edges in graphene under shear strain. This phenomenon arises from shear strain causing a separation between two inequivalent Dirac cones in the Brillouin zone (BZ) along the zigzag direction. Consequently, these armchair edge states appear as flat bands, connecting the two Dirac points at the two edges of armchair graphene nanoribbons (AGNRs). The length of these flat bands in the BZ and the penetration depth of the edge states are directly and inversely proportional to the strain, respectively. In monolayer AGNRs, possible magnetic configurations of flat bands resulting from electron-electron interactions are investigated. The edge-to-edge antiferromagnet (AFM) ground state is found in neutral AGNRs, while the AFM to ferromagnet (FM) transition can occur and be controlled by the strain in low-doped AGNRs. In gapped bilayer AGNRs, the armchair edge states evolve into quantum valley Hall edge states (QVHESs), which significantly improves the conductivity of QVHESs at realistic imperfect sample edges. These armchair edge states present a promising and tunable platform for exploring topological edge states in graphene.
... Just beyond the few-layer graphene systems, ABC-stacked multilayer graphene (ABC-MLG) [54], a finite stacking sequence of rhombohedral graphite, has notably been known for some time to host topological flat band surface states at zero energy (or charge neutrality) [55,56], also more recently observed in experiments [57,58]. The bulk topology generates the flat bands on surfaces perpendicular to the stacking direction, given a sufficient number of layers and thereby clearly appearing in thicker ABC stack, beyond five to six layers. ...
... Thereafter follows the interlayer hopping between different sublattices of neighboring layers t ⊥ = 0.1t = 0.3 eV since each carbon atom has another atom directly above or below it. A Hamiltonian consisting only of these two hopping amplitudes is already a good approximation for the noninteracting normal state electronic structure of ABC-MLG, as it captures the bulk topology and the associated topological surface flat bands [56,74]. For N L layers, this Hamiltonian reads as [56,73,87] ...
... A Hamiltonian consisting only of these two hopping amplitudes is already a good approximation for the noninteracting normal state electronic structure of ABC-MLG, as it captures the bulk topology and the associated topological surface flat bands [56,74]. For N L layers, this Hamiltonian reads as [56,73,87] ...
ABC-stacked multilayer graphene (ABC-MLG) exhibits topological surface flat bands with a divergent density of states, leading to many-body instabilities at charge neutrality. Here, we explore electronic ordering within a mean-field approach with full generic treatment of all spin-isotropic, two-site charge density and spin interactions up to next-nearest-neighbor (NNN) sites. We find that surface superconductivity and magnetism are significantly enhanced over bulk values. We find spin-singlet s wave and unconventional NNN bond spin-triplet f wave to be the dominant superconducting pairing symmetries, both with a full energy gap. By establishing the existence of ferromagnetic intrasublattice interaction (J2<0) we conclude that the f-wave state is favored in ABC-MLG, in sharp contrast to bulk ABC-graphite where chiral d- or p-wave states, together with s-wave states, display stronger ordering tendencies albeit not achievable at charge neutrality. We trace this distinctive surface behavior to the strong sublattice polarization of the surface flat bands. We also find competing ferrimagnetic order, fully consistent with density functional theory (DFT) calculations. The magnetic order interpolates between sublattice ferromagnetism and antiferromagnetism, but only with the ratio of the sublattice magnetic moments (R) being insensitive to the DFT exchange correlation functional. We finally establish the full phase diagram by constraining the interactions to the R value identified by DFT. We find f-wave superconductivity being favored for all weak to moderately strong couplings J2 and as long as J2 is a sufficiently large part of the full interaction mix. Gating ABC-MLG away from charge neutrality further enhances the f-wave state over the ferrimagnetic state, establishing ABC-MLG as a strong candidate for f-wave superconductivity.
... The FB induced SC had been shown to be quantitatively different from that of single band BCS physics. The order parameter ∆ and the critical temperature T c scale linearly with the interaction strength |U | [19][20][21][22] while BCS theory predicts ∆ ∝ k B T c ∝ t e −1/ρ(E F )|U | [23]. These unusual features in FBs have a geometrical interpretation, and can be connected to the concept of quantum metric (QM) [24] as defined in Ref. [25]. ...
... While superconductivity in this material has not yet been replicated by other groups, a Density Functional Theory (DFT) calculation recently predicted the appearance of two nearly flat bands in this material close to the Fermi level [3]. Since flat bands are known to host strongly correlated physics [4][5][6][7], this could potentially provide a microscopic mechanism for the putative high-T c superconductivity reported in CuPb 9 (PO 4 ) 6 OH 2 . In order to study the potentially rich physics hosted by the two flat bands, it would be beneficial to have a minimal two-band model which reproduces their main qualitative features. ...
Two recent preprints gave evidence that a copper-substituted lead apatite, denoted as CuPb$_9$(PO$_4$)$_6$OH$_2$ and also known as LK99, could be a room-temperature superconductor. While other research groups have not yet replicated the superconductivity in this material, a recent Density Functional Theory (DFT) calculation indicated the presence of two nearly flat bands near the Fermi level. Such flat bands are known to exhibit strongly correlated physics, which could potentially explain the reported high-$T_c$ superconductivity. In order to facilitate the theoretical study of the intriguing physics associated with these two flat bands, we propose a minimal tight-binding model which reproduces their main features. We also discuss implications for superconductivity.
... Topologically nontrivial flat bands have proven to be promising in the search for high temperature superconductivity [1][2][3][4][5][6][7][8][9][10][11][12]. The merit of superconductivity on isolated flat bands is two-fold: (a) due to quenched kinetic energy any finite interaction is relevant, resulting in a strongly correlated system, where superconducting transport arises from the correlated hopping of the fermions; and (b) both the critical temperature (T c ) and superfluid weight (D s ) depend linearly on the attractive interaction (U ), for weak interaction, thus is exponentially enhanced compared to dispersive bands with weak attraction between electrons [7][8][9][13][14][15]. In these systems, flat band superconductivity at weak interaction was shown to be dominated by a single isolated flat band [2,[5][6][7]. ...
... The first ingredient for superconductivity is the existence of the Cooper pairs, quantified by the pairing order parameter, ∆ z /U , and the single particle correlation length, ξ, characterizing the spatial extent of the pair. For dispersive bands in 2D, the critical temperature of pairing is of the order of k B T c ∼ ∆(T = 0) ∼ e −a/U , but it has been shown that k B T c ∼ ∆(T = 0) ∼ U for flat bands [7,8,[13][14][15][16]. We show in Figs. 3 and 4 the z-band filling, ρ z , and pairing order parameter, ∆ z /U , at F = 0.5 and F = 1, with a range of attraction strengths 0.1 ≤ U ≤ 7. ...
We investigate superconducting transport in the DC field induced Wannier-Stark flat bands in the presence of interactions. Flat bands offer the possibility of unconventional high temperature superconductivity, where the superfluid weight, $D_s$, is enhanced by the density overlap of the localized states. However, construction of flat bands typically requires very precise tuning of Hamiltonian parameters. To overcome this difficulty, we propose a feasible alternative to realize flat bands by applying a DC field in a commensurate lattice direction. We systematically characterize the superconducting behavior on these flat bands by studying the effect of the DC field and attractive Hubbard interaction strengths on the wavefunction, correlation length, pairing order parameter and the superfluid weight $D_s$. Our main result is that the superfluid weight is dramatically enhanced at an optimal value of the interaction strength and weak DC fields.
