An exactly solvable lattice with periodic inverse-square singularities exhibits two physically distinct mechanisms of spectral-gap closure. The model arises from the generalized trigonometric Pöschl-Teller potential in the coupling range where both local solutions at each singularity are square-integrable. A transmitting, current-preserving self-adjoint matching condition connects neighboring cells and generates a nontrivial Bloch spectrum. For a generic lattice with two inequivalent singular sublattices, the spectrum contains two alternating families of gaps, and the band edges reproduce the scale-invariant spectra of the associated finite singular system. One gap family closes when the two sublattices become equivalent. In this symmetric limit, however, the primitive period halves, so the apparent band crossing results from Brillouin-zone folding. A different family closes at a critical but genuinely asymmetric configuration. Because the primitive period is preserved, adjacent bands then meet at the center of the original Brillouin zone in a genuine massless Dirac-like cone. Near either critical limit, a small finite gap acts as an effective Dirac mass: the dispersion remains quadratic near a band edge but crosses over to a massive Dirac-like form beyond that narrow region. The band curvatures define effective electron and hole masses. Exact Bloch eigenfunctions further reveal that isospectral lattices can display different single-particle probability densities. This model therefore provides an analytically controlled one-dimensional setting in which conventional quadratic carriers, massive Dirac-like band dispersions, and a tunable massless contact coexist within the same singular lattice.
An exactly solvable lattice with periodic inverse-square singularities exhibits two physically distinct mechanisms of spectral-gap closure. The model arises from the generalized trigonometric Pöschl-Teller potential in the coupling range where both local solutions at each singularity are square-integrable. A transmitting, current-preserving self-adjoint matching condition connects neighboring cells and generates a nontrivial Bloch spectrum. For a generic lattice with two inequivalent singular sublattices, the spectrum contains two alternating families of gaps, and the band edges reproduce the scale-invariant spectra of the associated finite singular system. One gap family closes when the two sublattices become equivalent. In this symmetric limit, however, the primitive period halves, so the apparent band crossing results from Brillouin-zone folding. A different family closes at a critical but genuinely asymmetric configuration. Because the primitive period is preserved, adjacent bands then meet at the center of the original Brillouin zone in a genuine massless Dirac-like cone. Near either critical limit, a small finite gap acts as an effective Dirac mass: the dispersion remains quadratic near a band edge but crosses over to a massive Dirac-like form beyond that narrow region. The band curvatures define effective electron and hole masses. Exact Bloch eigenfunctions further reveal that isospectral lattices can display different single-particle probability densities. This model therefore provides an analytically controlled one-dimensional setting in which conventional quadratic carriers, massive Dirac-like band dispersions, and a tunable massless contact coexist within the same singular lattice.
In this paper, we consider a broad class of continuum two-scale Hamiltonians \begin{align*}
H_\varepsilon:=T(-i\nabla_x+\mathbf A(x,\varepsilon x))+V(x,\varepsilon x),\qquad x\in \mathbb{R}^d \end{align*} where T represents either a Dirac operator or a Schrödinger operator, and x↦A(x,X) and x↦V(x,X) are L-periodic with respect to some lattice L⊂Rd. No periodicity assumption is imposed on the second variable X.
Let \begin{align*}
\mathbb{R}^d\times \mathbb{R}^d\ni (k,X) \mapsto h(k,X):=T(-i\nabla_x+k+\mathbf A(x,X))+V(x,X) \end{align*} be a family of operators acting on L2(Rd/L) with periodic boundary conditions. We assume only local spectral information near one point (k0,X0): an isolated J-dimensional Bloch bands at energy e0, possibly with internal crossings, and a Hermitian matrix-valued homogeneous function of degree m∈{1,2}. From these data we derive an effective Hamiltonian h. Then every sufficiently localized eigenpair (v,μ) of h gives an approximate eigenfunction Θε such that for ε small enough, ∥Θε∥L2(Rd)=∣Ω∣−1/2+O(ε1/2),∥(Hε−e0−εm/2μ)Θε∥L2(Rd)=O(εm/2+1/4).
Fractional calculus has become an essential framework in geophysics, optics, and biological systems to capture long-range correlations and anomalous transport. In this article, we extend this framework to explore a particle in a periodic potential, where the Schrodinger equation is extended to its fractional form. This allows us to study how the Levy index q governs the formation and inversion of energy bands, offering a pathway to engineer new physical behaviors by tuning q in periodic quantum systems. We solve the fractional Schrodinger equation (FSE) for periodic rectangular potentials of varying height V0, barrier thickness L, and well width W using an imaginary-time evolution algorithm, supplemented by Gaussian process regression. This analysis reveals a qualitative shift in the system's band structure at q=2, separating into distinct regimes of dispersion for q>2 and q<2. For q>2, the energy bands invert as symmetric minima emerge within the first Brillouin zone and shift from k=0 toward k=±π/a with increasing q. These degenerate minima define a Bloch-momentum qubit, suggesting an analog to valley degrees of freedom in valleytronics. The q at which inversion completes scales as q∝V0−0.28±0.05, q∝L−0.35±0.08, and q∝W−0.49±0.06 when varying parameters individually, indicating a tunable sensitivity to potential geometry. In contrast, for q<2, the ground band hardens around k=0, with a dispersion following C∣k∣q+E0 near k=0. This suggests an effective mass of 0 for 1<q<2 at the band's lowest energy state. These results demonstrate that the Levy index serves as a tunable degree of freedom in quantum periodic systems, capable of driving band inversion, modulating the band gap, and reshaping carrier dynamics through effective-mass control.
Dirac points lie at the heart of many fascinating phenomena in condensed matter physics, from massless electrons in graphene to the emergence of conducting edge states in topological insulators. At a Dirac point, two energy bands intersect linearly and the particles behave as relativistic Dirac fermions. In solids, the rigid structure of the material sets the mass and velocity of the particles, as well as their interactions. A different, highly flexible approach is to create model systems using fermionic atoms trapped in an optical lattice, a method which so far has only been applied to explore simple lattice structures. In my talk I will report on the creation of Dirac points with adjustable properties in a tunable honeycomb optical lattice [1]. Using momentum-resolved interband transitions, we observe a minimum band gap inside the Brillouin zone at the position of the Dirac points. We exploit the unique tunability of our lattice potential to adjust the effective mass of the Dirac fermions by breaking the inversion symmetry of the lattice. Moreover, changing the lattice anisotropy allows us to move the position of the Dirac points inside the Brillouin zone. When increasing the anisotropy beyond a critical limit, the two Dirac points merge and annihilate each other. We map out this topological transition in lattice parameter space and find excellent agreement with ab initio calculations. Our results not only pave the way to model materials where the topology of the band structure plays a crucial role, but also provide the possibility to explore many-body phases resulting from the interplay of complex lattice geometries with interactions.
Symmetry-protected ideal flat bands in one-dimensional (1D) Hermitian lattices are populated by compact localized states (CLS) - a special class of localization with wavefunctions confined within a small region. In this work, we discover that the non-Hermitian skin effect (NHSE) can appear in a flat band. Unlike conventional NHSEs for dispersive bands that are protected by nontrivial point-gap topology, the flat band remains a point on the complex-energy plane and is therefore always topologically trivial. We found that, intriguingly, the flat-band skin effect (FBSE) is associated with the non-trivial spectral topology of the dispersive bands enclosing the flat band on the complex-energy plane, so it only emerges within a finite range of non-Hermitian parameters and can counterintuitively disappear at large non-Hermiticity. Moreover, the gaps between the flat and the dispersive bands can close at higher-order exceptional points under both periodic and open boundary conditions. The flat-band wavefunctions are discontinuous in quantum distance across these exceptional points, signifying that the gap-closing is singular. The FBSE was experimentally observed in a non-Hermitian mechanical lattice. Our work reveals flat-band phenomena unique to non-Hermitian systems and highlights new possibilities in quantum geometry and localization control.
With recent developments in fractional quantum mechanics, we introduce a fractional generalization of the Bernevig-Hughes-Zhang (BHZ) model to investigate the effects of fractional dispersion on band topology and quantum geometry. In the low-energy limit, the model reduces to a fractional Dirac Hamiltonian while preserving momentum-space periodicity, thereby ensuring a compact Brillouin zone and a well-defined topological invariant. We further show that the corresponding real-space tight-binding model can be constructed directly through a Fourier-series transformation, providing a simpler and more general alternative to the methods typically employed for fractional lattice systems. This approach readily extends beyond the generalized BHZ model considered here. Using the Bloch eigenstates, we compute the quantum geometric tensor and analyze its components, namely the Berry curvature and quantum metric. We find that fractional tuning redistributes these quantities throughout the Brillouin zone as the dispersion exponent becomes fractional, leading to pronounced modifications of the local band geometry. In contrast, the Chern number remains invariant, demonstrating the robustness of the global topological phase against fractional deformation. We further argue that this invariance persists for a broad class of fractional models.
We identify the gapless-point geometry of the honeycomb lattice model, where its three bond classes carry mutually anticommuting Hermitian-unitary matrix hoppings. For the chosen 8 × 8 Jordan-Wigner realization, we demonstrate that the exact zero condition detQ(k)=0⟺∑ie2ik⋅δi=0 yields eight inequivalent gapless momenta within the first Brillouin zone. Each zero point hosts four two-component low-energy cones with projected velocity v=3TA/(2ℏ). We further construct a two-stage mass mechanism: NNN mixing achieves 2/4 pairing selection within the test supercell domain, while the on-site marker mass 21(3−χ) has its spectrum restricted to {1,1,1,2,2} in units of εE. The ratio of 20:1 originates entirely from the traceless involution χ (full-channel parity with the exact symmetry {H,χ⊗I}=0 of the hopping Hamiltonian), giving 2=(N+1)/(N−1) at channel number N=3 ([12]). The analytical results are complemented by exact diagonalization and supercell folding checks.