Fractal Nodal Band Structures and Fermi Surfaces

Non-Hermitian systems exhibit very interesting band structures, comprising exceptional points at which eigenvalues and eigenvectors coalesce. Novel topological phenomena were shown to arise from the existence of those exceptional points, with applications in lasing and sensing. One important open question is how the topology of those exceptional points would manifest at non-integer dimension. Here, we report on the appearance of fractal eigenvalue degeneracies in Hermitian and non-Hermitian topological band structures. The existence of a fractal nodal Fermi surface might have profound implications on the physics of black holes and Fermi surface instability driven phenomena, such as superconductivity and charge density waves.Comment: 4+1 pages, 5+1 figure

Biorthogonal Renormalization

The biorthogonal formalism extends conventional quantum mechanics to the non-Hermitian realm. It has, however, been pointed out that the biorthogonal inner product changes with the scaling of the eigenvectors, an ambiguity whose physical significance is still being debated. Here, we revisit this issue and argue when this choice of normalization is of physical importance. We illustrate in which settings quantities such as expectation values and transition probabilities depend on the scaling of eigenvectors, and in which settings the biorthogonal formalism remains unambiguous. To resolve the apparent scaling ambiguity, we introduce an inner product independent of the gauge choice of basis and show that its corresponding mathematical structure is consistent with quantum mechanics. Using this formalism, we identify a deeper problem relating to the physicality of Hilbert space representations, which we illustrate using the position basis. Apart from increasing the understanding of the mathematical foundations upon which many physical results rely, our findings also pave the way towards consistent comparisons between systems described by non-Hermitian Hamiltonians.Comment: 19 pages, 3 figure

Fractal Nodal Band Structures

Non-Hermitian systems exhibit interesting band structures, where novel topological phenomena arise from the existence of exceptional points at which eigenvalues and eigenvectors coalesce. One important open question is how this would manifest at non-integer dimension. Here, we report on the appearance of fractal eigenvalue degeneracies and Fermi surfaces in Hermitian and non-Hermitian topological band structures. This might have profound implications on the physics of black holes and Fermi surface instability driven phenomena, such as superconductivity and charge density waves

Six iterative reconstruction algorithms in brain CT- A phantom study on image quality at different radiation doses.

OBJECTIVE: To evaluate the image quality produced by six different iterative reconstruction (IR) algorithms in four CT systems in the setting of brain CT, using different radiation dose levels and iterative image optimisation levels. METHODS: An image quality phantom, supplied with a bone mimicking annulus, was examined using four CT systems from different vendors and four radiation dose levels. Acquisitions were reconstructed using conventional filtered back-projection (FBP), three levels of statistical IR and, when available, a model-based IR algorithm. The evaluated image quality parameters were CT numbers, uniformity, noise, noise-power spectra, low-contrast resolution and spatial resolution. RESULTS: Compared with FBP, noise reduction was achieved by all six IR algorithms at all radiation dose levels, with further improvement seen at higher IR levels. Noise-power spectra revealed changes in noise distribution relative to the FBP for most statistical IR algorithms, especially the two model-based IR algorithms. Compared with FBP, variable degrees of improvements were seen in both objective and subjective low-contrast resolutions for all IR algorithms. Spatial resolution was improved with both model-based IR algorithms and one of the statistical IR algorithms. CONCLUSION: The four statistical IR algorithms evaluated in the study all improved the general image quality compared with FBP, with improvement seen for most or all evaluated quality criteria. Further improvement was achieved with one of the model-based IR algorithms. ADVANCES IN KNOWLEDGE: The six evaluated IR algorithms all improve the image quality in brain CT but show different strengths and weaknesses

Homotopy, Symmetry, and Non-Hermitian Band Topology

Non-Hermitian matrices are ubiquitous in the description of nature ranging from classical dissipative systems, including optical, electrical, and mechanical metamaterials, to scattering of waves and open quantum many-body systems. Seminal K-theory classifications of non-Hermitian systems based on line and point gaps in the presence of symmetry have deepened the understanding of a wide range of physical phenomena. However, ample systems remain beyond this description; reference points and lines are in general unable to distinguish whether multiple non-Hermitian bands exhibit band crossings and braids. To remedy this we consider the complementary notions of non-Hermitian band gaps and separation gaps that crucially include a broad class of multi-band scenarios, enabling the description of generic band structures with symmetries. With these concepts, we provide a unified and systematic classification of both gapped and nodal non-Hermitian systems in the presence of physically relevant parity-time ($\mathcal{PT}$) and pseudo-Hermitian symmetries using homotopy theory. This uncovers new fragile phases and, remarkably, also implies new stable phenomena stemming from the topology of both eigenvalues and eigenvectors. In particular, we reveal different Abelian and non-Abelian phases in $\mathcal{PT}$-symmetric systems, described by frame and braid topology. The corresponding invariants are robust to symmetry-preserving perturbations that do not close band gaps, and they also predict the deformation rules of nodal phases. We further demonstrate that spontaneous symmetry breaking in $\mathcal{PT}$-symmetric systems is captured by a Chern-Euler description. These results open the door for theoretical and experimental exploration of a rich variety of novel topological phenomena in a wide range of physical platforms.Comment: 42 pages, 12 figure

Warped D-Brane Inflation and Toroidal Compactifications

We set out on the ambitious journey to fuse inflation and string theory. We first give a somewhat extensive, yet free from the most complicated details, review of string inflation, discussing concepts as flux compactifications, moduli stabilization, the η-problem and reheating. Then, we consider two specific configurations of type II supergravity; type IIB on T6 with D3-branes and O3-planes, and type IIA on a twisted torus with D6-branes and O6-planes. In both cases, we calculate the scalar potential from the metric ansatzes, and try to uplift it to one of de-Sitter (dS) type. In the IIA-case, we also derive the scalar potential from a super- and Kähler potential, before we search for stable dS-solutions.

Knotted Nodal Band Structures

It is well known that in conventional three dimensional (3D) Hermitian two band models, the intersections between the energy bands are generically given by points. The typical example are Weyl semimetals, where these singular points can be effectively described as Weyl fermions in the low energy regime. By explicitly imposing discrete symmetries or fine-tuning, the intersection can form higher- dimensional nodal structures, e.g. nodal lines. By instead considering dissipative contributions to such a system, the degeneracies will generically take the form of closed 1D curves, consisting of exceptional points, i.e. points where the Hamiltonian becomes defective. By constructing the Hamiltonian in a particular way, the 1D exceptional curves can host non-trivial topology, i.e. they can form links or knots in the Brillouin zone. In stark contrast to line nodes occurring in Hermitian systems, which inevitably rely on discrete symmetries or fine tuning, the exceptional knots are generically stable towards any small perturbation. In further contrast to point singularities and unknotted circles, the topology of knots cannot be characterized by usual integer valued invariants. Instead, the complexity of the knottedness is captured by polynomial type invariants, making the physical classification and interpretation of these system challenging. To this end, the study of knotted nodal band structures naturally brings two different aspects of topology together – mathematical knot theory on the one hand, and the physical theory of topological phases on the other hand. This licentiate thesis focuses on providing the necessary theoretical background to understand the two accompanying publications entitled Knotted non-Hermitian metals, written by Johan Carlström, together with the author of this thesis, Jan Carl Budich and Emil J. Bergholtz, published in Physical Review B on April 24 2019, and Hyperbolic nodal band structures and knot invariants, written by the author of this thesis, together with Lukas Rødland, Gregory Arone, Jan Carl Budich and Emil J. Bergholtz, published in SciPost Physics August 8 2019. An introduction to gapless topological phases in the Hermitian regime, focusing on Weyl semimetals, their classification and surface states, is provided. Then, the light is brought to non-Hermitian operators and the differences from their conventional Hermitian counterpart, such as the two different set of eigenvectors bi-orthogonal to each other, exceptional eigenvalue degeneracies and some of their consequences, are explained. Afterwards, these operators are applied to dissipative physical system, and some of the striking differences from the conventional Hermitian systems are highlighted, the main focus being the possibly non-trivial topology of the 1D exceptional eigenvalue degeneracies. In order to be somewhat self contained, a brief conceptual introduction to the utilized concepts of knot theory is given, and lastly, further research directions and possible experimental realization of the considered systems are discussed