Patrick M. Lenggenhager

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Lenggenhager

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Patrick M.

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0000-0001-6746-1387

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Publications 1 - 10 of 10

Universal higher-order bulk-boundary correspondence of triple nodal points
Lenggenhager, Patrick M.; Liu, Xiaoxiong; Neupert, Titus; et al. (2022)
Physical Review B
Triple nodal points are degeneracies of energy bands in momentum space at which three Hamiltonian eigenstates coalesce at a single eigenenergy. For spinless particles, the stability of a triple nodal point requires two ingredients: rotational symmetry of order three, four, or six; combined with mirror or space-time-inversion symmetry. However, despite ample studies of their classification, robust boundary signatures of triple nodal points have until now remained elusive. In this work, we first show that pairs of triple nodal points in semimetals and metals can be characterized by Stiefel-Whitney and Euler monopole invariants, of which the first one is known to facilitate higher-order topology. Motivated by this observation, we then combine symmetry indicators for corner charges and for the Stiefel-Whitney invariant in two dimensions with the classification of triple nodal points for spinless systems in three dimensions. The result is a complete higher-order bulk-boundary correspondence, where pairs of triple nodal points are characterized by fractional jumps of the hinge charge. We present minimal models of the various species of triple nodal points carrying higher-order topology, and illustrate the derived correspondence on Sc3AlC which becomes a higher-order triple-point metal in applied strain. The generalization to spinful systems, in particular to the WC-type triple-point material class, is briefly outlined.
Triple nodal points characterized by their nodal-line structure in all magnetic space groups
Lenggenhager, Patrick M.; Liu, Xiaoxiong; Neupert, Titus; et al. (2022)
Physical Review B
We analyze triply degenerate nodal points [or triple points (TPs) for short] in energy bands of crystalline solids. Specifically, we focus on spinless band structures, i.e., when spin-orbit coupling is negligible, and consider TPs formed along high-symmetry lines in the momentum space by a crossing of three bands transforming according to a one-dimensional (1D) and a two-dimensional (2D) irreducible corepresentation (ICR) of the little cogroup. The result is a complete classification of such TPs in all magnetic space groups, including the nonsymmorphic ones, according to several characteristics of the nodal-line structure at and near the TP. We show that the classification of the presently studied TPs is exhausted by 13 magnetic point groups (MPGs) that can arise as the little cogroup of a high-symmetry line and which support both 1D and 2D spinless ICRs. For 10 of the identified MPGs, the TP characteristics are uniquely determined without further information; in contrast, for the 3 MPGs containing sixfold rotational symmetry, two types of TPs are possible, depending on the choice of the crossing ICRs. The classification result for each of the 13 MPGs is illustrated with first-principles calculations of a concrete material candidate.
Hyperbolic Non-Abelian Semimetal
Tummuru, Tarun; Chen, Anffany; Lenggenhager, Patrick M.; et al. (2024)
Physical Review Letters
We extend the notion of topologically protected semi-metallic band crossings to hyperbolic lattices in a negatively curved plane. Because of their distinct translation group structure, such lattices are associated with a high-dimensional reciprocal space. In addition, they 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 an unconventional 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 nonzero second Chern number, reminiscent of the characterization of Weyl nodes by the first Chern number.
Simulating Holographic Conformal Field Theories on Hyperbolic Lattices
Dey, Santanu; Chen, Anffany; Basteiro, Pablo; et al. (2024)
Physical Review Letters
We demonstrate how tabletop settings combining hyperbolic lattices with nonlinear dynamics universally encode aspects 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.
From triple-point materials to multiband nodal links
Liu, Xiaoxiong; Tsirkin, Stepan S.; Neupert, Titus; et al. (2021)
Physical Review B
We study a class of topological materials which in their momentum-space band structure exhibit threefold degeneracies known as triple points. Focusing specifically on PT-symmetric crystalline solids with negligible spin-orbit coupling, we find that such triple points can be stabilized by little groups containing a three-, four-, or sixfold rotation axis, and we develop a classification of all possible triple points as type A vs type B according to the absence vs presence of attached nodal-line arcs. Furthermore, by employing the recently discovered non-Abelian band topology, we argue that a rotation-symmetry-breaking strain transforms type-A triple points into multiband nodal links. Although multiband nodal-line compositions were previously theoretically conceived and related to topological monopole charges, a practical condensed-matter platform for their manipulation and inspection has hitherto been missing. By reviewing the known triple-point materials with weak spin-orbit coupling and by performing first-principles calculations to predict new ones, we identify suitable candidates for the realization of multiband nodal links in applied strain. In particular, we report that an ideal compound to study this phenomenon is Li2NaN, in which the conversion of triple points to multiband nodal links facilitates a largely tunable density of states and optical conductivity with doping and strain, respectively.
Simulating hyperbolic space on a circuit board
Lenggenhager, Patrick M.; Stegmaier, Alexander; Upreti, Lavi K.; et al. (2022)
Nature Communications
The Laplace operator encodes the behavior of physical systems at vastly different scales, describing heat flow, fluids, as well as electric, gravitational, and quantum fields. A key input for the Laplace equation is the curvature of space. Here we discuss and experimentally demonstrate that the spectral ordering of Laplacian eigenstates for hyperbolic (negatively curved) and flat two-dimensional spaces has a universally different structure. We use a lattice regularization of hyperbolic space in an electric-circuit network to measure the eigenstates of a ‘hyperbolic drum’, and in a time-resolved experiment we verify signal propagation along the curved geodesics. Our experiments showcase both a versatile platform to emulate hyperbolic lattices in tabletop experiments, and a set of methods to verify the effective hyperbolic metric in this and other platforms. The presented techniques can be utilized to explore novel aspects of both classical and quantum dynamics in negatively curved spaces, and to realise the emerging models of topological hyperbolic matter.
Non-Abelian Hyperbolic Band Theory from Supercells
Lenggenhager, Patrick M.; Maciejko, Joseph; Bzdušek, Tomáš (2023)
Physical Review Letters
Wave functions on periodic lattices are commonly described by Bloch band theory. Besides Abelian Bloch states labeled by a momentum vector, hyperbolic lattices support non-Abelian Bloch states that have so far eluded analytical treatments. By adapting the solid-state-physics notions of supercells and zone folding, we devise a method for the systematic construction of non-Abelian Bloch states. The method applies Abelian band theory to sequences of supercells, recursively built as symmetric aggregates of smaller cells, and enables a rapidly convergent computation of bulk spectra and eigenstates for both gapless and gapped tight-binding models. Our supercell method provides an efficient means of approximating the thermodynamic limit and marks a pivotal step toward a complete band-theoretic characterization of hyperbolic lattices.
Emerging avenues in band theory: multigap topology and hyperbolic lattices
Lenggenhager, Patrick M. (2023)
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. The first half of the thesis explores the implications of a multigap perspective on the topology of triple points, an instance of triply-degenerate nodal points. With the intention to shed light on unexplored connections between different manifestations of topology and material realizations of multigap topology, we study triple points in great detail. Employing minimal models, we derive a complete symmetry classification of triple points in spinless systems, predicting the presence and absence of specific additional degeneracies manifested as nodal lines. We further elucidate the role of multigap topology in the evolution of triple points into multiband nodal links. Furthermore, our analysis extends to the characterization of pairs of triple points formed by two triplets of bands from a total of four bands, which generically result in semimetallic band structures. We prove that such triple-point pairs generally exhibit signatures of higher-order topology, and, in the appropriate symmetry setting, are associated with nontrivial second Stiefel-Whitney and Euler monopole charges. With a careful analysis of tight-binding models and first-principle calculations on material candidates, we provide valuable insights into how these nodal structures and their topology manifest in realistic systems. Switching gears, the second half of the thesis ventures into the domain of hyperbolic lattices. This topic has gained traction with recent experimental realizations in several metamaterial platforms and several theoretical advancements. We start with an accessible introduction to the hyperbolic plane and regular tessellations on which hyperbolic lattices are based. Guided by this foundation, we demonstrate for the first time experimentally that hyperbolic lattices pave the way for emulating the hyperbolic plane in metamaterials, presenting an in-depth analysis of the observable signatures of negative curvature. In the rest of this part, we focus on the extension of band theory to negatively curved space. We develop an algebraic framework for labeling sites in hyperbolic lattices and forming periodic boundary conditions, thus facilitating the study of discrete symmetries and tight-binding models in these structures. Our key contribution to hyperbolic band theory is the supercell method. It provides a previously lacking systematic access to exotic non-Abelian Bloch states that exist due to the negative curvature, thereby advancing the understanding of hyperbolic reciprocal space. This pivotal step towards a complete band-theoretic characterization of hyperbolic lattices opens new pathways to a more refined understanding of these structures and their intriguing properties. Whether investigating topological aspects of semimetals or scrutinizing hyperbolic lattices realized in metamaterials, this thesis underscores the enduring centrality of band theory as a tool to uncover novel physical phenomena.
Symmetry and topology of hyperbolic Haldane models
Chen, Anffany; Guan, Yifei; Lenggenhager, Patrick M.; et al. (2023)
Physical Review B
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 present a general framework to characterize point-group symmetries in hyperbolic tight-binding models, and use this framework to constrain the multiple first and second Chern numbers in momentum space. We observe several topological gaps characterized by first Chern numbers of value 1 and 2. The momentum-space Chern numbers respect the predicted symmetry constraints and agree with real-space topological markers, indicating a direct connection to observables such as the number of chiral edge modes. With our large repertoire of models, we further demonstrate that the topology of hyperbolic Haldane models is trivialized for lattices with strong negative curvature.
Hyperbolic Topological Band Insulators
Urwyler, David M.; Lenggenhager, Patrick M.; Boettcher, Igor; et al. (2022)
Physical Review Letters
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; both obtained by replacing the hexagonal cells of their Euclidean counterparts by octagons. Their nontrivial topology is revealed by computing topological invariants in both position and momentum space. The bulk-boundary correspondence is evidenced by comparing bulk and boundary density of states, by modeling propagation of edge excitations, and by their robustness against disorder.

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Patrick M. Lenggenhager

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