## About

64

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Introduction

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Education

September 2013 - August 2017

September 2010 - June 2013

September 2007 - June 2010

## Publications

Publications (64)

We establish a connection between the electromagnetic Hall response and band topological invariants in hyperbolic Chern insulators by deriving a hyperbolic analog of the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula. By generalizing the Kubo formula to hyperbolic lattices, we show that the Hall conductivity is quantized to -e^2C_{ij}/h − e 2...

Multifractal analysis is a powerful tool for characterizing the localization properties of wave functions. Despite its utility, this tool has been predominantly applied to disordered Hermitian systems. Multifractal statistics associated with the non-Hermitian skin effect remain largely unexplored. Here, we demonstrate that the tree geometry induces...

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 establish a connection between the electromagnetic Hall response and band topological invariants in hyperbolic Chern insulators by deriving a hyperbolic analog of the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula. By generalizing the Kubo formula to hyperbolic lattices, we show that the Hall conductivity is quantized to $-e^2C_{ij}/h$, wh...

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...

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...

Curved spaces play a fundamental role in many areas of modern physics, from cosmological length scales to subatomic structures related to quantum information and quantum gravity. In tabletop experiments, negatively curved spaces can be simulated with hyperbolic lattices. Here we introduce and experimentally realize hyperbolic matter as a paradigm f...

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...

Motivated by the recent experimental realizations of hyperbolic lattices in circuit quantum electrodynamics and in classical electric-circuit networks, we study flat bands and band-touching phenomena in such lattices. We analyze noninteracting nearest-neighbor hopping models on hyperbolic analogs of the kagome and dice lattices with heptagonal and...

We study the Hatano-Nelson model, i.e., a one-dimensional non-Hermitian chain of spinless fermions with nearest-neighbor nonreciprocal hopping, in the presence of repulsive nearest-neighbor interactions. At half filling, we find two PT transitions, as the interaction strength increases. The first transition is marked by an exceptional point between...

We study the Hatano-Nelson model, i.e., a one-dimensional non-Hermitian chain of spinless fermions with nearest-neighbor nonreciprocal hopping, in the presence of repulsive nearest-neighbor interactions. At half filling, we find two PT transitions, as the interaction strength increases. The first transition is marked by an exceptional point between...

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...

Pontrjagin's seminal topological classification of two-band Hamiltonians in three momentum dimensions is hereby enriched with the inclusion of crystallographic rotational symmetry. The enrichment is attributed to a new topological invariant which quantifies a 2π-quantized change in the Berry-Zak phase between a pair of rotation-invariant lines in t...

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...

Motivated by the recent experimental realizations of hyperbolic lattices in circuit quantum electrodynamics and in classical electric-circuit networks, we study flat bands and band-touching phenomena in such lattices. We analyze noninteracting nearest-neighbor hopping models on hyperbolic analogs of the kagome and dice lattices with heptagonal and...

We introduce and experimentally realize hyperbolic matter as a novel paradigm for topological states, made of particles moving in the hyperbolic plane with negative curvature. Curvature of space is emulated through a hyperbolic lattice using topolectrical circuit networks relying on a newly developed complex-phase circuit element. This original met...

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 study the Hatano-Nelson model, i.e., a one-dimensional non-Hermitian chain of spinless fermions with nearest-neighbor nonreciprocal hopping, in the presence of repulsive nearest-neighbor interactions. At half-filling, we find two $\mathcal{PT}$ transitions, as the interaction strength increases. The first transition is marked by an exceptional p...

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...

Pontrjagin's seminal topological classification of two-band Hamiltonians in three momentum dimensions is hereby enriched with the inclusion of a crystallographic rotational symmetry. The enrichment is attributed to a new topological invariant which quantifies a $2\pi$-quantized change in the Berry-Zak phase between a pair of rotation-invariant line...

We introduce the exceptional topological insulator (ETI), a non-Hermitian topological state of matter that features exotic non-Hermitian surface states which can only exist within the three-dimensional topological bulk embedding. We show how this phase can evolve from a Weyl semimetal or Hermitian three-dimensional topological insulator close to cr...

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...

Being Wannierizable is not the end of the story for topological insulators. We introduce a family of topological insulators that would be considered trivial in the paradigm set by the tenfold way, topological quantum chemistry, and the method of symmetry-based indicators. Despite having a symmetric, exponentially localized Wannier representation, e...

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...

The code contains a Mathematica implementation of an algorithm to compute the second Chern number of a four-dimensional topological-insulator Hamiltonian as described in the following work:
M. Mochol-Grzelak, A. Dauphin, A. Celi, and M. Lewenstein, Quantum Sci. Technol. 4 014009 (2019)
DOI: https://doi.org/10.1088/2058-9565/aae93b

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...

Weyl semimetals in three-dimensional crystals provide the paradigm example of topologically protected band nodes. It is usually taken for granted that a pair of colliding Weyl points annihilate whenever they carry opposite chiral charge. In stark contrast, here we report that Weyl points in systems that are symmetric under the composition of time r...

Being Wannierizable is not the end of the story for topological insulators. We introduce a family of topological insulators that would be considered trivial in the paradigm set by the tenfold way, topological quantum chemistry, and the method of symmetry-based indicators. Despite having a symmetric, exponentially-localized Wannier representation, e...

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 introduce the exceptional topological insulator (ETI), a non-Hermitian topological state of matter that features exotic non-Hermitian surface states which can only exist within the three-dimensional topological bulk embedding. We show how this phase can evolve from a Weyl semimetal or Hermitian three-dimensional topological insulator close to cr...

Implementation of the numerical algorithm to compute patch Euler class as outlined in work "Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe", Nature Physics 16, 1137--1143 (2020), and as discussed in detail in its supplemental material. The algorithm is implemented in Wolfram Mathematica.
Given a real-symmetric Bloch...

We present a framework to systematically address topological phases when finer partitionings of bands are taken into account, rather than only considering the two subspaces spanned by valence and conduction bands. Focusing on C2T-symmetric systems that have gained recent attention, for example in the context of layered van-der-Waals graphene hetero...

An Alice string is a topological defect with a very peculiar feature. When a defect with a monopole charge encircles an Alice string, the monopole charge changes sign. In this paper, we generalize this notion to the momentum space of periodic media with loss and gain. In particular, we find that the generic band-structure node for a three-dimension...

Mathematica code that generates the illustrations of nodal-line compositions in work Physical Review B 101, 195130 (2020). The work was supported by the Ambizione Program of the Swiss National Science Foundation, Grant No. 185806

Nodal lines inside the momentum space of three-dimensional crystalline solids are topologically stabilized by a π flux of Berry phase. Nodal-line rings in PT-symmetric systems with negligible spin-orbit coupling (here described as “nodal class AI”) can carry an additional “monopole charge,” which further enhances their stability. Here, we relate tw...

We revisit the problem of classifying topological band structures in non-Hermitian systems. Recently, a solution has been proposed, which is based on redefining the notion of energy band gap in two different ways, leading to the so-called “point-gap” and “line-gap” schemes. However, simple Hamiltonians without band degeneracies can be constructed w...

We present a framework to systematically address topological phases when finer partitionings of bands are taken into account, rather than only considering the two subspaces spanned by valence and conduction bands. Focusing on C2T -symmetric systems that have gained recent attention, for example in the context of layered van-der-Waals graphene heter...

We revisit the problem of classifying topological band structures in non-Hermitian systems. Recently, a solution has been proposed, which is based on redefining the notion of energy band gap in two different ways, leading to the so-called "point-gap" and "line-gap" schemes. However, simple Hamiltonians without band degeneracies can be constructed w...

Electron energy bands of crystalline solids generically exhibit degeneracies called band-structure nodes. Here, we introduce non-Abelian topological charges that characterize line nodes inside the momentum space of
𝓟𝓣 -symmetric crystalline metals with weak spin-orbit coupling. We show that these are quaternion charges, similar to those describing...

We illustrate a procedure that transforms non-Abelian charges of Weyl
nodes via braid phase factors, which arise upon exchange inside the reciprocal momentum space. This phenomenon derives from intrinsic symmetry properties of topological materials, which are increasingly becoming available due to recent cataloguing insights. Specifically, we show...

An Alice string is a topological defect with a very peculiar feature. When a defect with a monopole charge encircles an Alice string, the monopole charge changes sign. In this paper, we generalize this notion to the momentum space of periodic media with loss and gain. In particular, we find that the generic band-structure node for a three-dimension...

Nodal lines inside the momentum space of three-dimensional crystalline solids are topologically stabilized by a π flux of Berry phase. Nodal-line rings in PT-symmetric systems with negligible spin-orbit coupling (here described as “nodal class AI”) can carry an additional “monopole charge,” which further enhances their stability. Here, we relate tw...

Nodal lines inside the momentum space of three-dimensional crystalline solids are topologically stabilized by a π-flux of Berry phase. Nodal-line rings in PT-symmetric systems with negligible spin-orbit coupling (here described as "nodal class AI") can carry an additional "monopole charge", which further enhances their stability. Here, we relate tw...

We show that the Nielsen-Ninomiya no-go theorem still holds on a Floquet lattice: there is an equal number of right-handed and left-handed Weyl points in a three-dimensional Floquet lattice. However, in the adiabatic limit, where the time evolution of the low-energy subspace is decoupled from the high-energy subspace, we show that the bulk dynamics...

We introduce non-Abelian topological charges for nodal-line band degeneracies in momentum space of PT-symmetric crystalline metals with weak spin-orbit coupling. We show that these are quaternion charges, similar to those describing vortices in biaxial nematics. Starting from two-band considerations, we develop the complete many-band description of...

We show that the Nielsen-Ninomiya no-go theorem still holds on Floquet lattice: there is an equal number of right-handed and left-handed Weyl points in 3D Floquet lattice. However, in the adiabatic limit, where the time evolution of low-energy subspace is decoupled from the high-energy subspace, we show that the bulk dynamics in the low-energy subs...

According to a widely-held paradigm, a pair of Weyl points with opposite chirality mutually annihilate when brought together. In contrast, we show that such a process is strictly forbidden for Weyl points related by a mirror symmetry, provided that an effective two-band description exists in terms of orbitals with opposite mirror eigenvalue. Instea...

The subject matter of this thesis is the appearance of various nodal semimetals in crystalline solids. The valence and the conduction bands of such materials touch at points or along lines in momentum space, and their chemical potential is adjusted to the energy of this touching. As a consequence, these materials have a vanishing or a very small de...

Weyl points in three spatial dimensions are characterized by a Z-valued charge { the Chern number
{ which makes them stable against a wide range of perturbations. A set of Weyl points can mutually
annihilate only if their net charge vanishes, a property we refer to as robustness. While nodal loops
are usually not robust in this sense, it has recent...

The band theory of solids is arguably the most successful theory of condensed-matter physics, providing a description of the electronic energy levels in various materials. Electronic wavefunctions obtained from the band theory enable a topological characterization of metals for which the electronic spectrum may host robust, topologically protected,...

We study the electronic properties of strongly spin-orbit coupled electrons
on the elastic pyrochlore lattice. Akin to the Peierls transition in
one-dimensional systems, the coupling of the lattice to the electronic degrees
of freedom can stabilize a spontaneous deformation of the crystal. This
deformation corresponds to a breathing mode, which bre...

Both the pairing and the pair-breaking modes lead to similar kinks of the
electron dispersion curves in superconductors, and therefore the photoemission
spectroscopy can not be straightforwardly applied in search for their pairing
glue. If the momentum-dependence of the normal and anomalous self-energy can be
neglected, manipulation with the data d...

(BSc thesis in Slovak langauge)
In this thesis, we study superconductivity in the disordered Hubbard model. We focus on the 3-dimensional cubic lattice and the 2-dimensional graphene lattice. In the first chapter we discuss notions concerning electron states in ideal and disordered lattices, in the second chapter those concerning the phenomenon of...

(book in Slovak language)
Collection of advanced and tricky high-school physics problems together with their model solutions. Created while training students for the International Physics Olympiad and similar physics competitions.
Published by Trojsten (2009).
Online access: http://old.fks.sk/fx/zbierka/knizka.pdf