# Patrick M. LenggenhagerMax Planck Institute for the Physics of Complex Systems | MPIPKS · Division of Condensed Matter

Patrick M. Lenggenhager

Dr. sc. ETH Zurich

## About

25

Publications

3,219

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315

Citations

Introduction

Additional affiliations

October 2023 - October 2023

November 2019 - September 2023

November 2019 - September 2023

Education

November 2019 - November 2023

September 2016 - March 2019

September 2012 - October 2015

## Publications

Publications (25)

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...

Hyperbolic lattices present a unique opportunity to venture beyond the conventional paradigm of crystalline many-body physics and explore correlated phenomena in negatively curved space. As a theoretical benchmark for such investigations, we extend Kitaev's spin-1/2 honeycomb model to hyperbolic lattices and exploit their non-Euclidean space-group...

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, ac...

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 systema...

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...

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...

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...

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 inacce...

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...

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...

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-dim...

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, desp...

DOI:https://doi.org/10.1103/PhysRevB.106.079903

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-dim...

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 hyperbo...

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 hyperb...

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 1D and...

The Laplace operator encodes the behaviour 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 demonstrate that the spectral ordering of Laplacian eigenstates for hyperbolic (negative curvature) a...

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: rotation symmetry of order three, four or six; combined with mirror or space-time-inversion symmetry. However, despite...

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...

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...

Recently, a novel real-space renormalization group (RG) algorithm was introduced. By maximizing an information-theoretic quantity, the real-space mutual information, the algorithm identifies the relevant low-energy degrees of freedom. Motivated by this insight, we investigate the information-theoretic properties of coarse-graining procedures for bo...

Recently a novel real-space RG algorithm was introduced, identifying the relevant degrees of freedom of a system by maximizing an information-theoretic quantity, the real-space mutual information (RSMI), with machine learning methods. Motivated by this, we investigate the information theoretic properties of coarse-graining procedures, for both tran...

Recently a novel real-space RG algorithm was introduced, identifying the relevant degrees of freedom of a system by maximizing an information-theoretic quantity, the real-space mutual information (RSMI), with machine learning methods. Motivated by this, we investigate the information theoretic properties of coarse-graining procedures, for both tran...