We compute the signature of the Milnor fiber of certain type of non-isolated complex surface singularities, namely, images of finitely determined holomorphic germs. An explicit formula is given in algebraic terms. As a corollary we show that the signature of the Milnor fiber is a topological invariant for these singularities. The proof combines complex analytic and smooth topological techniques. The main tools are Thom-Mather theory of map germs and the Ekholm-Szűcs-Takase-Saeki formula for immersions. We give a table with many examples for which the signature is computed using our formula.

In quantum mechanics, the Schrieffer-Wolff (SW) transformation (also called quasi-degenerate perturbation theory) is known as an approximative method to reduce the dimension of the Hamiltonian. We present a geometric interpretation of the SW transformation: We prove that it induces a local coordinate chart in the space of Hermitian matrices near a k-fold degeneracy submanifold. Inspired by this result, we establish a 'distance theorem': we show that the standard deviation of k neighboring eigenvalues of a Hamiltonian equals the distance of this Hamiltonian from the corresponding k-fold degeneracy submanifold, divided by root k. Furthermore, we investigate one-parameter perturbations of a degenerate Hamiltonian, and prove that the standard deviation and the pairwise differences of the eigenvalues lead to the same order of splitting of the energy eigenvalues, which in turn is the same as the order of distancing from the degeneracy sub-manifold. As applications, we prove the 'protection' of Weyl points using the transversality theorem, and infer geometrical properties of certain degeneracy submanifolds based on results from quantum error correction and topological order.

Parameter-dependent quantum systems often exhibit energy degeneracy points, whose comprehensive description naturally leads to the application of methods from singularity theory. A prime example is an electronic band structure where two energy levels coincide in a point of momentum space. It may happen, and this case is the focus of our work, that three or more levels coincide at a parameter point, called multifold degeneracy. Upon a generic perturbation, such a multifold degeneracy point is dissolved into a set of Weyl points, that is, generic twofold degeneracy points. In this work, we provide an upper bound to the number of Weyl points born from the multifold degeneracy point. To compute this upper bound, we describe the geometric degeneracy variety in the space of complex matrices. We compute its multiplicity at certain singular points corresponding to a multifold degeneracy, and the multiplicity of holomorphic map germs with respect to this variety. Our work covers physics and mathematics aspects in detail and attempts to bridge the two disciplines and communities. For self-containedness, we survey examples of multifold degeneracies in quantum systems and condensed-matter physics, as well as the established tools of local algebraic geometry that we use to identify the upper bound.

The image of a finitely determined holomorphic germ $\Phi$ from $\mathbb{C}^2$ to $\mathbb{C}^3$ defines a hypersurface singularity $(X,0)$, which is in general non-isolated. We show that the diffeomorphism type of the boundary of the Milnor fibre $\partial F$ of $X$ is a topological invariant of the germ $\Phi$. We establish a correspondence between the gluing coefficients (so-called vertical indices) used in the construction of $\partial F$ and a linking invariant $L$ of the associated sphere immersion introduced by T. Ekholm and A. Sz\H{u}cs. For this we provide a direct proof of the equivalence of the different definitions of $L$. Since $L$ can be expressed in terms of the cross cap number $C(\Phi)$ and the triple point number $T(\Phi)$ of a stable deformation of $\Phi$, we obtain a relation between these invariants and the vertical indices. This is illustrated on several examples.

Degeneracy points of parameter-dependent Hermitian matrices play a fundamental role in quantum physics, as illustrated by the concept of Berry phase in quantum dynamics, the Weyl semimetals in condensed-matter physics, and the robust ground-state degeneracies in topologically ordered quantum systems. Here, we construct simple ball-and-spring mechanical systems, whose eigenfrequency degeneracies mimic the behaviour of degeneracy points of electronic band structures. These classical-mechanical arrangements can be viewed as de-quantized versions of Weyl Josephson circuits, i.e., superconducting nanostructures proposed recently to mimic band structure effects of Weyl semimetals. In the mechanical setups we study, we identify degeneracy patterns beyond simple Weyl points, including the chirality flip effect and a quadratic degeneracy point. Our theoretical work is a step toward simple and illustrative table-top experiments exploring topological and differential geometrical aspects of physics.

In this work, we describe the phenomenon of Weyl-point teleportation. Weyl points usually move continuously in the configuration parameter space of a quantum system when the control parameters are varied continuously. However, there are special transition points in the control space where the continuous motion of the Weyl points is disrupted. In such transition points, an extended nodal structure (nodal line or nodal surface) emerges, serving as a wormhole for the Weyl points, allowing their teleportation in the configuration space. A characteristic side effect of the teleportation is that the motional susceptibility of the Weyl point diverges in the vicinity of the transition point, and this divergence is characterized by a universal scaling law. We exemplify these effects via a two-spin model and a Weyl Josephson circuit model. We expect that these effects generalize to many other settings including electronic band structures of topological semimetals.

Weyl points are generic and stable features in the energy spectrum of Hamiltonians that depend on a three-dimensional parameter space. Nongeneric isolated twofold degeneracy points, such as multi-Weyl points, split into Weyl points upon a generic perturbation that removes the fine tuning or protecting symmetry. The number of the resulting Weyl points is at least |Q|, where Q is the topological charge associated to the nongeneric degeneracy point. Here, we show that such a nongeneric degeneracy point also has an birth quota, i.e., a maximum number of Weyl points that can be born from it upon any perturbation. This upper bound is a local multiplicity associated to the nongeneric degeneracy point, an invariant of map germs known from singularity theory. This holds not only for the case of a three-dimensional parameter space with a Hermitian Hamiltonian, but also for the case of a two-dimensional parameter space with a chiral-symmetric Hamiltonian. We illustrate the power of this result for band structures of two- and three-dimensional crystals. Our work establishes a strong and powerful connection between singularity theory and topological band structures and, more broadly, parameter-dependent quantum systems.

Changes in the number of Weyl nodes in Weyl semimetals occur through merging processes, usually involving a pair of oppositely charged nodes. More complicated processes involving multiple Weyl nodes are also possible, but they typically require fine tuning and are thus less stable. In this Letter, we study how symmetries affect the allowed merging processes and their stability, focusing on the combination of a twofold rotation and time-reversal (C_{2}T) symmetry. We find that, counterintuitively, processes involving a merging of three nodes are more generic than processes involving only two nodes. Our Letter suggests that multi-Weyl merging may be observed in a large variety of quantum materials, and we discuss SrSi_{2} and bilayer graphene as potential candidates.

Weyl points (WP) are robust spectral degeneracies, which can not be split by small perturbations, as they are protected by their non-zero topological charge. For larger perturbations, WPs can disappear via pairwise annihilation, where two oppositely charged WPs merge, and the resulting neutral degeneracy disappears. The neutral degeneracy is unstable, meaning that it requires the fine-tuning of the perturbation. Fine-tuning of more than one parameter can lead to more exotic WP mergers. In this work, we reveal and analyze a fundamental connection of the WP mergers and singularity theory: phase boundary points of Weyl phase diagrams, i.e., control parameter values where Weyl point mergers happen, can be classified according to singularity classes of maps between manifolds of equal dimension. We demonstrate this connection on a Weyl--Josephson circuit where the merger of 4 WPs draw a swallowtail singularity, and in a random BdG Hamiltonian which reveal a rich pattern of fold lines and cusp points. Our results predict universal geometrical features of Weyl phase diagrams, and generalize naturally to creation and annihilation of Weyl points in electronic (phononic, magnonic, photonic, etc) band-structure models, where Weyl phase transitions can be triggered by control parameters such as mechanical strain.

It was recently proved that for finitely determined germs Φ : ( ℂ^2, 0) → ( ℂ^3, 0) the number C(Φ ) of Whitney umbrella points and the number T(Φ ) of triple values of a stable deformation are topological invariants. The proof uses the fact that the combination C(Φ )-3T(Φ ) is topological since it equals the linking invariant of the associated immersion S^3 ↬ S^5 introduced by Ekholm and Szűcs. We provide a new, direct proof for this equality. We also clarify the relation between various definitions of the linking invariant.

Weyl points are generic and stable features in the energy spectrum of Hamiltonians that depend on a three-dimensional parameter space. Non-generic isolated two-fold degeneracy points, such as multi-Weyl points, split into Weyl points upon a generic perturbation that removes the fine-tuning or protecting symmetry. The number of the resulting Weyl points is at least $|Q|$, where $Q$ is the topological charge associated to the non-generic degeneracy point. Here, we show that such a non-generic degeneracy point also has a birth quota, i.e., a maximum number of Weyl points that can be born from it upon any perturbation. The birth quota is a local multiplicity associated to the non-generic degeneracy point, an invariant of map germs known from singularity theory. This holds not only for the case of a three-dimensional parameter space with a Hermitian Hamiltonian, but also for the case of a two-dimensional parameter space with a chiral-symmetric Hamiltonian. We illustrate the power of this result for electronic band structures of two- and three-dimensional crystals. Our work establishes a strong connection between singularity theory and topological band structures, and more broadly, parameter-dependent quantum systems.

Quantum systems are often described by parameter-dependent Hamiltonians. Points in parameter space where two levels are degenerate can carry a topological charge. Here we theoretically study an interacting two-spin system where the degeneracy points form a nodal loop or a nodal surface in the magnetic parameter space, similarly to such structures discovered in the band structure of topological semimetals. Key results of our work are that (1) we determine the topological charge distribution along these degeneracy geometries and (2) we show that these nonpointlike degeneracy patterns can be obtained not only by fine-tuning but they can be stabilized by spatial symmetries. Since simple spin systems such as the one studied here are ubiquitous in condensed-matter setups, we expect that our findings, and the physical consequences of these nontrivial degeneracy geometries, are testable in experiments with quantum dots, molecular magnets, and adatoms on metallic surfaces.

The thesis deals with holomorphic germs Φ: (ℂ^2, 0) → (ℂ^3,0) singular only at the origin, with a special emphasis on the distinguished class of finitely determined germs. The results are published in two articles (arXiv:1404.2853 and arXiv:1902.01229), joint with András Némethi. In Chapter 3 of the thesis we study the associated immersion S^3 ↬ S^5, while Chapter 5 contains an algorithm providing the Milnor fibre boundary of the non-isolated hypersurface singularity determined by the image of Φ. These results create bridges between different areas of complex singularity theory and immersion theory. The background of these topics is summerized in Chapter 1, 2 and 4.

Let $ \Phi: ({\mathbb C}^2, 0) \to ( {\mathbb C}^3, 0) $ be a finitely determined complex analytic germ and let $(\{f=0\},0)$ be the reduced equation of its image, a non-isolated hypersurface singularity. We provide the plumbing graph of the boundary of the Milnor fibre of $f$ from the double-point-geometry of $\Phi$.

A holomorphic germ \Phi: (C^2, 0) \to (C^3, 0), singular only at the origin, induces at the links level an immersion of S^3 into S^5. The regular homotopy type of such immersions are determined by their Smale invariant, defined up to a sign ambiguity. In this paper we fix a sign of the Smale invariant and we show that for immersions induced by holomorphic gems the sign-refined Smale invariant is the negative of the number of cross caps appearing in a generic perturbation of \Phi. Using the algebraic method we calculate it for some families of singularities, among others the A-D-E quotient singularities. As a corollary, we obtain that the regular homotopy classes which admit holomorphic representatives are exactly those, which have non-positive sign-refined Smale invariant. This answers a question of Mumford regarding exactly this correspondence. We also determine the sign ambiguity in the topological formulae of Hughes-Melvin and Ekholm-Szucs connecting the Smale invariant with (singular) Seifert surfaces. In the case of holomorphic realizations of Seifert surfaces, we also determine their involved invariants in terms of holomorhic geometry.