The Lieb lattice, characterized by its distinctive Dirac cone and flat-band electronic structures, hosts a variety of exotic physical phenomena. However, its realization remains largely confined to artificial lattices...The Lieb lattice, characterized by its distinctive Dirac cone and flat-band electronic structures, hosts a variety of exotic physical phenomena. However, its realization remains largely confined to artificial lattices. In this work, we propose the concept of a Lieb electride, where the non-bound electrons gather at the middle edges,behaving as the quasi-atoms of a Lieb lattice, enabling the emergence of flat bands. Using crystal structure prediction method MAGUS and first-principles calculations, we predict a stable candidate, Ca_(2)I, at ambient pressure. Distinct from conventional electrides with localized electrons at cavity centers, Ca_(2)I features interstitial electrons situated at cavity edges. The resultant flat bands lie close to the Fermi level, giving rise to a pronounced peak in the density of states and leading to Stoner-type ferromagnetism. With increasing pressures, we observe quantum phase transitions from ferromagnetic to non-magnetic and finally to antiferromagnetic orders in Ca_(2)I.Intriguingly, superconductivity emerges in the antiferromagnetic region, suggesting potential competition between these correlated states. Our study not only extends the concepts of electrides but also provides a novel strategy for realizing Lieb lattices through non-bound electrons. This work establishes Ca_(2)I as a promising platform for exploring flat-band physics and correlated electronic states, opening avenues for novel quantum phenomena in electride-based materials.展开更多
For a two-dimensional Lieb lattice,that is,a line-centered square lattice,the inclusion of the intrinsic spin–orbit（ISO）coupling opens a topologically nontrivial gap,and gives rise to the quantum spin Hall（QSH） e...For a two-dimensional Lieb lattice,that is,a line-centered square lattice,the inclusion of the intrinsic spin–orbit（ISO）coupling opens a topologically nontrivial gap,and gives rise to the quantum spin Hall（QSH） effect characterized by two pairs of gapless helical edge states within the bulk gap.Generally,due to the finite size effect in QSH systems,the edge states on the two sides of a strip of finite width can couple together to open a gap in the spectrum.In this paper,we investigate the finite size effect of helical edge states on the Lieb lattice with ISO coupling under three different kinds of boundary conditions,i.e.,the straight,bearded and asymmetry edges.The spectrum and wave function of edge modes are derived analytically for a tight-binding model on the Lieb lattice.For a strip Lieb lattice with two straight edges,the ISO coupling induces the Dirac-like bulk states to localize at the edges to become the helical edge states with the same Dirac-like spectrum.Moreover,it is found that in the case with two straight edges the gapless Dirac-like spectrum remains unchanged with decreasing the width of the strip Lieb lattice,and no gap is opened in the edge band.It is concluded that the finite size effect of QSH states is absent in the case with the straight edges.However,in the other two cases with the bearded and asymmetry edges,the energy gap induced by the finite size effect is still opened with decreasing the width of the strip.It is also proposed that the edge band dispersion can be controlled by applying an on-site potential energy on the outermost atoms.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.12125404,T2495231,123B2049,and 12204138)the National Key R&D Program of China(Grant No.2022YFA1403201)+7 种基金the Advanced MaterialsNational Science and Technology Major Project (Grant No.2024ZD0607000)the Basic Research Program of Jiangsu (Grant Nos.BK20233001 and BK20241253)the Jiangsu Funding Program for Excellent Postdoctoral Talent (Grant Nos.2024ZB002,2024ZB075,2025ZB440 and2025ZB852)the China Postdoctoral Science Foundation (Grant No.2025M773331)the Postdoctoral Fellowship Program of CPSF (Grant No.GZC20240695 and GZC20252202)the AI&AI for Science Program of Nanjing UniversityArtificial Intelligence and Quantum physics (AIQ) program of Nanjing Universitythe Fundamental Research Funds for the Central Universities。
文摘The Lieb lattice, characterized by its distinctive Dirac cone and flat-band electronic structures, hosts a variety of exotic physical phenomena. However, its realization remains largely confined to artificial lattices. In this work, we propose the concept of a Lieb electride, where the non-bound electrons gather at the middle edges,behaving as the quasi-atoms of a Lieb lattice, enabling the emergence of flat bands. Using crystal structure prediction method MAGUS and first-principles calculations, we predict a stable candidate, Ca_(2)I, at ambient pressure. Distinct from conventional electrides with localized electrons at cavity centers, Ca_(2)I features interstitial electrons situated at cavity edges. The resultant flat bands lie close to the Fermi level, giving rise to a pronounced peak in the density of states and leading to Stoner-type ferromagnetism. With increasing pressures, we observe quantum phase transitions from ferromagnetic to non-magnetic and finally to antiferromagnetic orders in Ca_(2)I.Intriguingly, superconductivity emerges in the antiferromagnetic region, suggesting potential competition between these correlated states. Our study not only extends the concepts of electrides but also provides a novel strategy for realizing Lieb lattices through non-bound electrons. This work establishes Ca_(2)I as a promising platform for exploring flat-band physics and correlated electronic states, opening avenues for novel quantum phenomena in electride-based materials.
基金Project supported by the National Natural Science Foundation of China(Grant No.11274102)the Program for New Century Excellent Talents in University of the Ministry of Education of China(Grant No.NCET-11-0960)the Specialized Research Fund for the Doctoral Program of the Higher Education of China(Grant No.20134208110001)
文摘For a two-dimensional Lieb lattice,that is,a line-centered square lattice,the inclusion of the intrinsic spin–orbit（ISO）coupling opens a topologically nontrivial gap,and gives rise to the quantum spin Hall（QSH） effect characterized by two pairs of gapless helical edge states within the bulk gap.Generally,due to the finite size effect in QSH systems,the edge states on the two sides of a strip of finite width can couple together to open a gap in the spectrum.In this paper,we investigate the finite size effect of helical edge states on the Lieb lattice with ISO coupling under three different kinds of boundary conditions,i.e.,the straight,bearded and asymmetry edges.The spectrum and wave function of edge modes are derived analytically for a tight-binding model on the Lieb lattice.For a strip Lieb lattice with two straight edges,the ISO coupling induces the Dirac-like bulk states to localize at the edges to become the helical edge states with the same Dirac-like spectrum.Moreover,it is found that in the case with two straight edges the gapless Dirac-like spectrum remains unchanged with decreasing the width of the strip Lieb lattice,and no gap is opened in the edge band.It is concluded that the finite size effect of QSH states is absent in the case with the straight edges.However,in the other two cases with the bearded and asymmetry edges,the energy gap induced by the finite size effect is still opened with decreasing the width of the strip.It is also proposed that the edge band dispersion can be controlled by applying an on-site potential energy on the outermost atoms.
