We study wave propagation in 2D honeycomb structures with a non-commensurate or ``irrational'' line defect or edge. Our model is a Schrödinger operator which interpolates, across the edge, between two distinct bulk (asymptotic) Hamiltonians with a common spectral gap about the ``Dirac point'' of an unperturbed honeycomb operator. We seek edge states, eigenstates that are bounded and oscillatory parallel to the edge, and decaying in the transverse direction. For non-commensurate edges, the rigorous definition of these states is nontrivial due to the lack of translation invariance along the edge. To address this, we exploit quasiperiodicity along the edge by expressing the Hamiltonian as the restriction of a 3D (degenerate elliptic) Hamiltonian describing a 3D medium with a 2D interface within which there is periodicity. Via multiscale analysis, we construct approximate edge states in this 3D setting and obtain by restriction 2D edge states which are quasiperiodic along the irrational edge. These edge states are seeded by eigenfunctions of an effective Dirac operator, which has an infinite block-diagonal structure due to the non-commensurate geometry. A consequence is that infinitely many edge state eigenpairs arise, whose energies are dense in the perturbed bulk spectral gap. In a forthcoming paper, we rigorously construct these gap-filling edge states under a Diophantine condition. The main result here is a key tool in this construction: a resolvent expansion for the 3D Hamiltonian, whose leading term is the resolvent of the block-diagonal Dirac operator. The validity of this expansion requires an omnidirectional non-resonance (no-fold) condition on the dispersion functions of the unperturbed honeycomb Hamiltonian. This condition is satisfied in the strong binding regime. In contrast with earlier works on commensurate edges, the omnidirectional condition is independent of the edge.
We study wave propagation in 2D honeycomb structures with a non-commensurate or irrational'' line defect or edge. Our model is a Schrödinger operator which interpolates, across the edge, between two distinct bulk (asymptotic) Hamiltonians with a common spectral gap about the Dirac point'' of an unperturbed honeycomb operator. We seek edge states, eigenstates that are bounded and oscillatory parallel to the edge, and decaying in the transverse direction. For non-commensurate edges, the rigorous definition of these states is nontrivial due to the lack of translation invariance along the edge. To address this, we exploit quasiperiodicity along the edge by expressing the Hamiltonian as the restriction of a 3D (degenerate elliptic) Hamiltonian describing a 3D medium with a 2D interface within which there is periodicity. Via multiscale analysis, we construct approximate edge states in this 3D setting and obtain by restriction 2D edge states which are quasiperiodic along the irrational edge. These edge states are seeded by eigenfunctions of an effective Dirac operator, which has an infinite block-diagonal structure due to the non-commensurate geometry. A consequence is that infinitely many edge state eigenpairs arise, whose energies are dense in the perturbed bulk spectral gap. In a forthcoming paper, we rigorously construct these gap-filling edge states under a Diophantine condition. The main result here is a key tool in this construction: a resolvent expansion for the 3D Hamiltonian, whose leading term is the resolvent of the block-diagonal Dirac operator. The validity of this expansion requires an omnidirectional non-resonance (no-fold) condition on the dispersion functions of the unperturbed honeycomb Hamiltonian. This condition is satisfied in the strong binding regime. In contrast with earlier works on commensurate edges, the omnidirectional condition is independent of the edge.
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).
This thesis reports on the realization of Floquet topological systems with ultracold atoms in an optical honeycomb lattice. Using periodically driven quantum systems, topological phases of matter can be simulated by an effective, static Hamiltonian related to the time-evolution at integer multiples of the driving period, which is known as Floquet engineering. However, periodic driving can also give rise to genuinely time-dependent settings without static counterparts. One example is the anomalous Floquet phase in two dimensions, in which all bulk bands have a Chern number of zero but nevertheless robust chiral edge modes appear, which would be precluded by the bulk-edge correspondence in a static system. In an optical honeycomb lattice, anomalous Floquet systems can be created by continuous, periodic modulation of the laser intensities. This driving scheme results in different topological regimes, three of which are investigated, including the anomalous Floquet phase and a Haldane-like phase. Periodically driven systems feature periodic quasienergies and can be characterized in terms of winding numbers, which count the number of chiral edge modes in each quasienergy gap. By interferometric measurements of the quasienergy gaps between the first two Floquet bands, the topological phase transitions, emerging as gap closings, are located. For periodically driven systems, a modified bulk-edge correspondence can be formulated. In particular, the change of the winding number at a phase transition is related to the sign change of the local Berry curvature. The Berry curvature at the gap closing points is probed by Hall deflection measurements to obtain the winding numbers in each of the topological regimes, revealing the existence of chiral edge modes also in a setting with smooth boundaries as used in the experiments. The measured quasienergy gaps and transverse deflections are quantitatively well described by a numerical calculation of the Floquet bandstructure that includes coupling to higher bands during the driving period. To derive the spectrum of the modulated lattice in a geometry with edges, a tight-binding description of the system is discussed. Circular phase modulation of the honeycomb lattice can also give rise to an anomalous Floquet system when inversion symmetry is broken. The topological phase diagrams and the experimental feasibility are compared for both modulation schemes. Due to the vanishing Chern numbers, the bulk states in the anomalous Floquet regime can be fully localized by a disorder potential. This could prevent heating in interacting, periodically driven systems, resulting in a many-body localized bulk coexisting with thermalizing edge states. The experimental realization of disorder and strong interactions, as well as independent probes of the edge states are investigated. Bloch band geometry can be extended to multiband systems, using Wilson lines. The possible symmetry protection of their eigenvalues is discussed, along with measurements in the optical honeycomb lattice. Moreover, the onset of heating in weakly-interacting, periodically driven systems, triggered by parametric instabilities, is studied experimentally in a one-dimensional lattice.
Non-Hermitian lattice Hamiltonians generally exhibit strong boundary sensitivity, with periodic and open boundary conditions producing distinct density of states (DOS) in the complex-energy plane. This has led to the view that extended non-Hermitian systems lack a unique bulk DOS, with different prescriptions representing inequivalent bulk physics. Here, we show that this apparent ambiguity is largely illusory. For any finite-range tight-binding Hamiltonian, we establish a universal bulk structure: all DOS definitions arising as thermodynamic limits of finite systems share identical multipole moments and generate identical bulk dynamics at finite times and for observables measured far from boundaries. This universality is intimately tied to the thermodynamic Green's functions, which we show to be independent of the boundary condition for large enough complex frequencies. Among all equivalent descriptions, we identify the Brown measure - obtained via Hermitization and resolvent analysis - as a canonical and convenient representative of the bulk DOS, defined directly from the infinite-volume Hamiltonian. We further show that point-gap topology imposes additional universal constraints: boundary-dependent Green's functions are forced to coincide throughout topologically trivial point gaps. This, in particular, provides a systematic criterion, valid in arbitrary dimension, for determining where and how eigenvalues of different boundary truncations can accumulate in the complex plane, and precisely delineates the regime in which the DOS ambiguity retains physical significance.
Establishing the (non)existence of a spectral gap above the ground state in the thermodynamic limit is one of the fundamental steps for characterizing the topological phase of a quantum lattice model. This is particularly challenging when a model is expected to have low-lying edge excitations, but nevertheless a positive bulk gap. We review the bulk gap strategy introduced in [S. Warzel and A. Young, The spectral gap of a fractional quantum Hall system on a thin torus, J. Math. Phys. 63 (2022) 041901; S. Warzel and A. Young, A bulk spectral gap in the presence of edge states for a truncated pseudopotential, Ann. Henri Poincaré 24 (2023) 133–178], while studying truncated Haldane pseudopotentials. This approach is able to avoid low-lying edge modes by separating the ground states and edge states into different invariant subspaces before applying spectral gap bounding techniques. The approach is stated in a general context, and we reformulate specific spectral gap methods in an invariant subspace context to illustrate the necessary conditions for combining them with the bulk gap strategy. We then review its application to a truncation of the 1/3-filled Haldane pseudopotential in the cylinder geometry.
We study the topological properties of one-dimensional discrete-time quantum walks with Fibonacci quasiperiodic modulation. Spectral analysis under open boundary conditions reveals isolated edge modes that coexist at both zero and π energies within bulk gaps. Using the mean chiral displacement (MCD) as a dynamical bulk probe, we obtain a fractal, butterfly-like phase diagram, indicating a nontrivial bulk topology. However, since the MCD involves wave-packet averaging, its direct correspondence with boundary-localized states remains ambiguous. To resolve this issue, we determine the integer topological invariant by employing the Schur function formalism, a scattering-based approach defined on the unit disk. This analysis identifies a topological phase with winding number W=2 inside the ``wings'' of the butterfly diagram, demonstrating that the coexistence of zero- and π-energy edge modes is an intrinsic bulk property. Moreover, we find that the topological index crucially depends on the phason degree of freedom: the winding number can take values of W=0, 2, and even 4 depending on the local surface termination. This behavior originates from the surface impedance, which yields an effective winding number ranges from complete masking by reflective boundaries (W=0) to enhancement through surface resonances (W=4). Finally, we show that although individual surface terminations lead to anisotropic phase diagrams, their ensemble average restores the overall geometric structure observed in the MCD phase diagram. Our results establish a complete bulk-edge correspondence in quasiperiodic quantum walks and provide a guiding principle for surface topological engineering, where edge transport can be controlled purely through boundary manipulation.
Abstract We outline a theory of symmetry protected topological phases of one-dimensional quantum walks. We assume spectral gaps around the symmetry-distinguished points +1 and −1, in which only discrete eigenvalues are allowed. The phase classification by integer or binary indices extends the classification known for translation invariant systems in terms of their band structure. However, our theory requires no translation invariance whatsoever, and the indices we define in this general setting are invariant under arbitrary symmetric local perturbations, even those that cannot be continuously contracted to the identity. More precisely we define two indices for every walk, characterizing the behavior far to the right and far to the left, respectively. Their sum is a lower bound on the number of eigenstates at +1 and −1. For a translation invariant system the indices add up to zero, so one of them already characterizes the phase. By joining two bulk phases with different indices we get a walk in which the right and left indices no longer cancel, so the theory predicts bound states at +1 or −1. This is a rigorous statement of bulk-edge correspondence. The results also apply to the Hamiltonian case with a single gap at zero.