Extended Bloch theorem for topological lattice models with open boundaries

Abstract

While the Bloch spectrum of translationally invariant noninteracting lattice models is trivially obtained by a Fourier transformation, diagonalizing the same problem in the presence of open boundary conditions is typically only possible numerically or in idealized limits. Here we present exact analytic solutions for the boundary states in a number of lattice models of current interest, including nodal-line semimetals on a hyperhoneycomb lattice, spin-orbit coupled graphene, and three-dimensional topological insulators on a diamond lattice, for which no previous exact finite-size solutions are available in the literature. Furthermore, we identify spectral mirror symmetry as the key criterium for analytically obtaining the entire (bulk and boundary) spectrum as well as the concomitant eigenstates, and exemplify this for Chern and ${\mathcal{Z}}_2$ insulators with open boundaries of co-dimension one. In the case of the two-dimensional Lieb lattice, we extend this further and show how to analytically obtain the entire spectrum in the presence of open boundaries in both directions, where it has a clear interpretation in terms of bulk, edge, and corner states.

References in corpus (45)

• Topological Insulators
• Topological insulators and superconductors
• Quantum Spin Hall Effect in Graphene
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Topological Insulators in Three Dimensions
• Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator
• Experimental realisation of the topological Haldane model
• Edge states and topological invariants of non-Hermitian systems
• Quantum Spin Hall Effect
• Quantized Electric Multipole Insulators
• Topological nodal semimetals
• Higher-Order Topological Insulators
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems
• Anomalous edge state in a non-Hermitian lattice
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Reflection symmetric second-order topological insulators and superconductors
• Non-Hermitian Chern bands
• Anatomy of skin modes and topology in non-Hermitian systems
• Recent developments in transport phenomena in Weyl semimetals
• Second-order topological insulators and superconductors with an order-two crystalline symmetry
• Line of Dirac Nodes in Hyper-Honeycomb Lattices
• Higher-order bulk-boundary correspondence for topological crystalline phases
• Topological states of non-Hermitian systems
• A new spin-anisotropic harmonic honeycomb iridate
• Impurity Bound States and Greens Function Zeroes as Local Signatures of Topology
• Non-Hermitian systems and topology: A transfer-matrix perspective
• Higher-order topological insulators protected by inversion and rotoinversion symmetries
• Topological Quadrupolar Semimetals
• Boundary Green functions of topological insulators and superconductors
• Lattice models with exactly solvable topological hinge and corner states
• Topological insulators in magnetic fields: Quantum Hall effect and edge channels with non-quantized θ-term
• Unified bulk-boundary correspondence for band insulators
• A generalization of Bloch's theorem for arbitrary boundary conditions: Theory
• Of Bulk and Boundaries: Generalized Transfer Matrices for Tight-Binding Models
• Loop-nodal and Point-nodal Semimetals in Three-dimensional Honeycomb Lattices
• Topological origin of edge states in two-dimensional inversion-symmetric insulators and semimetals
• Inversion-symmetry protected chiral hinge states in stacks of doped quantum Hall layers
• Excess charges as a probe of one-dimensional topological crystalline insulating phases
• Anatomy of Topological Surface States: Exact Solutions from Destructive Interference on Frustrated Lattices
• Generalization of Bloch's theorem for arbitrary boundary conditions: Interfaces and topological surface band structure
• Exact edge, bulk and bound states of finite topological systems
• Boundaries of boundaries: a systematic approach to lattice models with solvable boundary states of arbitrary codimension

Cited by in corpus (22)

• Exact solution of non-Hermitian systems with generalized boundary conditions: size-dependent boundary effect and fragility of skin effect
• Liouvillian Skin Effect in an Exactly Solvable Model
• Ultrafast and Anharmonic Rabi Oscillations between Non-Bloch-Bands
• Topological Phase Transition Independent of System Non-Hermiticity
• Phase Transitions and Generalized Biorthogonal Polarization in Non-Hermitian Systems
• Nonlinear non-Hermitian skin effect
• Inversion symmetric non-Hermitian Chern insulator
• Boundaries of boundaries: a systematic approach to lattice models with solvable boundary states of arbitrary codimension
• Analytical solution of open crystalline linear 1D tight-binding models
• Surface charge theorem and topological constraints for edge states: An analytical study of one-dimensional nearest-neighbor tight-binding models
• Dissipative Boundary State Preparation
• Non-Hermitian edge burst without skin localizations
• Non-Hermitian extended midgap states and bound states in the continuum
• Edge states of a diffusion equation in one dimension: Rapid heat conduction to the heat bath
• Theory of edge states based on the hermiticity of tight-binding Hamiltonian operators
• Bulk-edge correspondence in two-dimensional topological semimetals: A transfer matrix study of antichiral edge modes
• Eigenstate clustering around exceptional points
• Visualizing Topology of Real-Energy Gapless Phase Arising from Exceptional Point
• Quasi-stationary solutions in non-Hermitian systems
• Anatomy of Higher-Order Non-Hermitian Skin and Boundary Modes
• Exact lattice-model calculation of boundary modes for Weyl semimetals and graphene
• Analytical solutions of topological surface states in a series of lattice models