arXiv subjects

Heinrich-Gregor Zirnstein

Publications and source records attributed to Heinrich-Gregor Zirnstein.

7 recordsLinked to original sources

arXiv2020

Bulk-boundary correspondence for non-Hermitian Hamiltonians via Green functions

Genuinely non-Hermitian topological phases can be realized in open systems with sufficiently strong gain and loss; in such phases, the Hamiltonian cannot be deformed into a gapped Hermitian Hamiltonian without energy bands touching each other. Comparing Green functions for periodic and open boundary conditions we find that, in general, there is no correspondence between topological invariants computed for periodic boundary conditions, and boundary eigenstates observed for open boundary conditions. Instead, we find that the non-Hermitian winding number in one dimension signals a topological phase transition in the bulk: It implies spatial growth of the bulk Green function.

cond-mat.mes-hall↗

arXiv2020

Topological Magnetoelectric Effect: Nonlinear Time-Reversal-Symmetric Response, Witten Effect, and Half-Integer Quantum Hall Effect

Topological insulators (TIs) in three space dimensions can be characterized by a quantized magnetoelectric coefficient. However, this coupling does not have experimentally observable consequences in the presence of time-reversal symmetry, because the contributions of both bulk and surface states cancel each other. Instead, the characteristic response of a TI is a nonlinear magnetoelectric effect. Field theoretic aspects of the nonlinear magnetoelectric response and numerical calculations for experimentally relevant geometries are discussed. Distinct from this effect, the magnetoelectric coupling would bind a charge $\pm e/2$ in response to a monopole, which is referred to as Witten effect. If time reversal is broken explicitly, for instance, by a Zeeman field acting on the surface layer, the electromagnetic response of a TI can be described by a half-integer quantum Hall effect of surface electrons.

cond-mat.mes-hall↗

arXiv2020

Exponentially growing bulk Green functions as signature of nontrivial non-Hermitian winding number in one dimension

A nonzero non-Hermitian winding number indicates that a gapped system is in a nontrivial topological class due to the non-Hermiticity of its Hamiltonian. While for Hermitian systems nontrivial topological quantum numbers are reflected by the existence of edge states, a nonzero non-Hermitian winding number impacts a system's bulk response. To establish this relation, we introduce the bulk Green function, which describes the response of a gapped system to an external perturbation on timescales where the induced excitations have not propagated to the boundary yet, and show that it will grow in space if the non-Hermitian winding number is nonzero. Such spatial growth explains why the response of non-Hermitian systems on longer timescales, where excitations have been reflected at the boundary repeatedly, may be highly sensitive to boundary conditions. This exponential sensitivity to boundary conditions explains the breakdown of the bulk-boundary correspondence in non-Hermitian systems: topological invariants computed for periodic boundary conditions no longer predict the presence or absence of boundary states for open boundary conditions.

cond-mat.mes-hall↗

arXiv2018

Evolution of the transmission phase through a Coulomb-blockaded Majorana wire

We present a study of the transmission of electrons through a semiconductor quantum wire with strong spin-orbit coupling in proximity to an s-wave superconductor, which is Coulomb-blockaded. Such a system supports Majorana zero modes in the presence of an external magnetic field. Without superconductivity, phase lapses are expected to occur in the transmission phase, and we find that they disappear when a topological superconducting phase is stabilized. We express tunneling through the nanowire with the help of effective matrix elements, which depend on both the fermion parity of the wire and the overlap with Bogoliubov-de-Gennes wave functions. Using a modified scattering matrix formalism, that allows for including electron-electron interactions, we study the transmission phase in different regimes.

cond-mat.mes-hall↗

arXiv2017

Time-reversal-symmetric topological magnetoelectric effect in three-dimensional topological insulators

One of the hallmarks of time-reversal-symmetric topological insulators in three dimensions is the topological magnetoelectric effect (TME). So far, a time-reversal breaking variant of this effect has attracted much attention, in the sense that the induced electric charge changes sign when the direction of an externally applied magnetic field is reversed. Theoretically, this effect is described by the so-called axion term. Here, we discuss a time-reversal-symmetric TME, where the electric charge depends only on the magnitude of the magnetic field but is independent of its sign. We obtain this nonperturbative result by a combination of analytic and numerical arguments, and suggest a mesoscopic setup to demonstrate it experimentally.

cond-mat.mes-hall↗

arXiv2013

Cancellation of quantum anomalies and bosonization of three-dimensional time-reversal symmetric topological insulators

The strong time-reversal symmetric (TRS) topological insulator (TI) in three space dimensions features gapless surface states in the form of massless Dirac fermions. We study these surface states with the method of bosonization, and find that the resulting bosonic theory has a topological contribution due to the parity anomaly of the surface Dirac fermions. We argue that the presence of a quantum anomaly is, in fact, the main reason for the existence of a surface state, by the principle that anomalies of a surface and bulk must cancel. Inspecting other classes of topological insulators, we argue that this principle holds in general. Moving beyond purely topological considerations, we incorporate the dynamics of the surface electron states into the bosonic theory. Additionally, we discuss the thermodynamics of the bosonic theory and propose a representation of the surface Dirac fermions in terms of the bosonic fields.

cond-mat.mes-hall↗

arXiv2012

Formulating Szemerédi's Theorem in Terms of Ultrafilters

Van der Waerden's theorem asserts that if you color the natural numbers with, say, five different colors, then you can always find arbitrarily long sequences of numbers that have the same color and that form an arithmetic progression. Szemerédi's theorem generalizes this statement and asserts that every subset of natural numbers with positive density contains arithmetic progressions of arbitrary length. Van der Waerden's theorem can be proven using elementary combinatorics, but it is also possible to give an interpretation and a short proof in terms of ultrafilters and the Stone-Čech compactification βN. This diploma thesis gives an interpretation of Szemerédi's theorem in terms of ultrafilters as well. In particular, van der Waerden's theorem is equivalent to the existence of a single ultrafilter with special properties and we will show that Szemerédi's theorem is equivalent to the fact that with respect to a counting measure on βN, almost all ultrafilters have these properties.

math.FA↗