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The notions of Bloch wave, crystal momentum, and energy bands are commonly regarded as unique features of crystalline materials with commutative translation symmetries. Motivated by the recent realization of hyperbolic lattices in circuit quantum electrodynamics, we exploit ideas from algebraic geometry to construct the first hyperbolic generalization of Bloch theory, despite the absence of commutative translation symmetries. For a quantum particle propagating in a hyperbolic lattice potential, we construct a continuous family of eigenstates that acquire Bloch-like phase factors under a discrete but noncommutative group of hyperbolic translations, the Fuchsian group of the lattice. A hyperbolic analog of crystal momentum arises as the set of Aharonov-Bohm phases threading the cycles of a higher-genus Riemann surface associated with this group. This crystal momentum lives in a higher-dimensional Brillouin zone torus, the Jacobian of the Riemann surface, over which a discrete set of continuous energy bands can be computed.
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Hyperbolic lattices are a new form of synthetic quantum matter in which particles effectively hop on a discrete tessellation of two-dimensional hyperbolic space, a non-Euclidean space of uniform negative curvature. To describe the single-particle eigenstates and eigenenergies for hopping on such a lattice, a hyperbolic generalization of band theory was previously constructed, based on ideas from algebraic geometry. In this hyperbolic band theory, eigenstates are automorphic functions, and the Brillouin zone is a higher-dimensional torus, the Jacobian of the compactified unit cell understood as a higher-genus Riemann surface. Three important questions were left unanswered: (1) whether a band theory can be expected to hold for a non-Euclidean lattice, where translations do not generally commute; (2) whether a formal Bloch theorem can be rigorously established; and (3) whether hyperbolic band theory can describe finite lattices realized in experiment. In the present work, we address all three questions simultaneously. By formulating periodic boundary conditions for finite but arbitrarily large lattices, we show that a generalized Bloch theorem can be rigorously proved, but may or may not involve higher-dimensional irreducible representations (irreps) of the nonabelian translation group, depending on the lattice geometry. Higher-dimensional irreps corrrespond to points in a moduli space of higher-rank stable holomorphic vector bundles, which further generalizes the notion of Brillouin zone beyond the Jacobian. For a large class of finite lattices, only one-dimensional irreps appear, and the hyperbolic band theory previously developed becomes exact.
We present a method for efficiently enumerating all allowed, topologically distinct, electronic band structures within a given crystal structure. The algorithm applies to crystals with broken time-reversal, particle-hole, and chiral symmetries in any dimension. The presented results match the mathematical structure underlying the topological classification of these crystals in terms of K-theory, and therefore elucidate this abstract mathematical framework from a simple combinatorial perspective. Using a straightforward counting procedure, we classify the allowed topological phases in any possible two-dimensional crystal in class A. We also show how the same procedure can be used to classify the allowed phases for any three-dimensional space group. Employing these classifications, we study transitions between topological phases within class A that are driven by band
We identify and investigate two classes of non-Hermitian systems, i.e., one resulting from Lorentz-symmetry violation (LSV) and the other from a complex mass (CM) with Lorentz invariance, from the perspective of quantum field theory. The mechanisms to break, and approaches to restore, the bulk-boundary correspondence in these two types of non-Hermitian systems are clarified. The non-Hermitian system with LSV shows a non-Hermitian skin effect, and its topological phase can be characterized by mapping it to the Hermitian system via a non-compact $U(1)$ gauge transformation. In contrast, there exists no non-Hermitian skin effect for the non-Hermitian system with CM. Moreover, the conventional bulk-boundary correspondence holds in this (CM) system. We also consider a general non-Hermitian system in the presence of both LSV and CM, and we generalize its bulk-boundary correspondence.
2+1 dimensional topological insulator described by the Kane-Mele model in the presence of Rashba spin-orbit interaction is considered. The effective action of the external fields coupled to electromagnetic and spin degrees of freedom is accomplished within this model. The Hamiltonian methods are adopted to provide the coefficients appearing in the action. It is demonstrated straightforwardly that the coefficients of the Chern-Simons terms are given by the first Chern number attained through the related non-Abelian Berry gauge field. The effective theory which we obtain is in accord with the existence of the spin Hall phase where the value of the spin Hall conductivity is very close to the quantized one.
Among non-Hermitian systems, pseudo-Hermitian phases represent a special class of physical models characterized by real energy spectra and by the absence of non-Hermitian skin effects. Here, we show that several pseudo-Hermitian phases in two and three dimensions can be built by employing $q$-deformed matrices, which are related to the representation of deformed algebras. Through this algebraic approach we present and study the pseudo-Hermitian version of well known Hermitian topological phases, raging from two-dimensional Chern insulators and time-reversal-invariant topological insulators to three-dimensional Weyl semimetals and chiral topological insulators. We analyze their topological bulk states through non-Hermitian generalizations of Abelian and non-Abelian tensor Berry connections and quantum metric. Although our pseudo-Hermitian models and their Hermitian counterparts share the same topological invariants, their band geometries are different. We indeed show that some of our pseudo-Hermitian phases naturally support nearly-flat topological bands, opening the route to the study of pseudo-Hermitian strongly-interacting systems. Finally, we provide an experimental protocol to realize our models and measure the full non-Hermitian quantum geometric tensor in synthetic matter.
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