We demonstrate how table-top settings combining hyperbolic lattices with nonlinear dynamics universally lead to a genuine realization of the bulk-boundary-correspondence between gravity in anti-de-Sitter (AdS) space and conformal field theory (CFT). Our concrete and broadly applicable holographic toy model simulates gravitational self-interactions in the bulk and features an emergent CFT with nontrivial correlations on the boundary. We measure the CFT data contained in the two- and three-point functions and clarify how a thermal CFT is simulated through an effective black hole geometry. As a concrete example, we propose and simulate an experimentally feasible protocol to measure the holographic CFT using electrical circuits.

Tight-binding models on periodic lattices are commonly studied using Bloch band theory, which provides an efficient description of their energy spectra and wave functions. Besides Abelian Bloch states characterized by a momentum vector, the band theory of hyperbolic lattices involves non-Abelian Bloch states that have so far remained largely inaccessible to analytical treatments. Here, we systematically construct non-Abelian Bloch states by generalizing to hyperbolic lattices the familiar solid-state-physics notions of supercells and zone folding.

One of the cornerstones of condensed matter physics, the description of wave functions on periodic lattices in terms of energy bands of Bloch states, serves as the unifying thread in this thesis. This description is often referred to as band theory. Within its context, topological states of matter and metamaterials have taken shape as key frontiers in recent years. Related to those frontiers, this thesis delves into seemingly distinct areas: multigap topology and lattices in negatively curved space, known as hyperbolic lattices. While these two themes may appear disconnected at first, they are intrinsically tied together by concepts such as symmetry, topology, metamaterials, and the ubiquitous role of band theory.

Particles hopping on a two-dimensional hyperbolic lattice feature unconventional energy spectra and wave functions that provide a largely uncharted platform for topological phases of matter beyond the Euclidean paradigm. Using real-space topological markers as well as Chern numbers defined in the higher-dimensional momentum space of hyperbolic band theory, we construct and investigate hyperbolic Haldane models, which are generalizations of Haldane’s honeycomb-lattice model to various hyperbolic lattices.

We extend the notion of topologically protected semi-metallic band crossings to hyperbolic lattices in negatively curved space. Due to their distinct translation group structure, such lattices support non-Abelian Bloch states which, unlike conventional Bloch states, acquire a matrix-valued Bloch factor under lattice translations. Combining diverse numerical and analytical approaches, we uncover a quartic scaling in the density of states at low energies, and illuminate a nodal manifold of codimension five in the reciprocal space. The nodal manifold is topologically protected by a non-zero second Chern number, reminiscent of the characterization of Weyl nodes by the first Chern number.

The Bloch band theory describes energy levels of crystalline media by a collection of bands in momentum space. These bands can be characterized by non-trivial topological invariants, which via bulk-boundary correspondence imply protected boundary states inside the bulk energy gap. Recently, hyperbolic lattices that tile the negatively curved hyperbolic plane emerged as a new paradigm of synthetic matter, and their energy levels were characterized by a band structure in a four- (or higher-)dimensional momentum space. To explore the uncharted topological aspects arising in hyperbolic band theory, we here introduce elementary models of hyperbolic topological band insulators: the hyperbolic Haldane model and the hyperbolic Kane-Mele model;

We extend the classification of three-fold momentum-space band structure degeneracies (triple points) to all (magnetic) space groups, including the non-symmorphic ones. The classification is based on the symmetry properties of the high-symmetry line in momentum space on which the degeneracy lies and predicts the nodal-line structure at and near the triple point. In particular, we predict the presence, number and behavior of nodal-line arcs connecting to the central nodal line (along the high-symmetry line) at nexus points. We illustrate the classification results on several material examples using first-principles calculations.

We show that pairs of triple points in systems with negligible spin-orbit coupling (three-fold momentum-space band structure degeneracies along high-symmetry lines) facilitate higher-topology. Our main result is a universal bulk-hinge correspondence of such pairs to fractional quantized jumps of the hinge charge as a function of the hinge-momentum. For the special case of \(\mathcal{P}\mathcal{T}\)-symmetric crystalline solids we further find that triple points are characterized by monople charges. We illustrate the derived phenomena on material examples using first-principles calculations.

Spaces with negative curvature are particularly difficult to realize experimentally. We show how to emulate a discretized version of constant negatively curved space, a so called hyperbolic lattice, using an electric circuit network. We present static and dynamical methods which allowed us to characterize and verify the negative curvature in the implemented model. Our experiments showcase a versatile platform to emulate hyperbolic lattices in tabletop experiments, which can be utilized to explore propagation dynamics as well as to realize topological hyperbolic matter.

We classify three-fold momentum-space band structure degeneracies (triple points) occurring along high-symmetry lines of \(\mathcal{P}\mathcal{T}\)-symmetric crystalline solids with negligible spin-orbit coupling. By employing the recently discovered non-Abelian band topology, we argue that a rotation-symmetry-breaking strain transforms a certain class of triple points into multiband nodal links. We propose material candidates and study the consequences of this phenomenon.

We consider a recently introduced real-space renormalization group (RG) algorithm which is based on maximizing real-space mutual information (RSMI). Based on general proofs and detailed studies of arbitrary coarse-grainings of the Ising chain, we argue that maximizing RSMI minimizes the range of interactions and, for disordered systems, correlations in the disorder distribution of the coarse-grained system.