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Ralph M. Kaufmann

Publications and source records attributed to Ralph M. Kaufmann.

At least 19 recordsLinked to original sources
arXiv2026

Band Unfolding via the Quadratic Pseudospectrum

Band theory provides the foundation for understanding electronic structure in crystalline materials, but its reliance on exact translational symmetry limits its applicability to systems with defects, disorder, incommensurate modulations, or large unit cells. Here, we introduce a band unfolding framework that directly generalizes traditional band theory to systems where exact periodicity is absent, and which remains well-defined for both aperiodic and finite systems. To do so, we employ a pseudospectral approach to identify approximate joint eigenvectors of a system's Hamiltonian and translation operators, thereby yielding an unfolded band structure whose features are directly connected to the manifestation of approximate extended states simultaneously localized in energy and crystalline momentum. To reveal bulk-only spectral phenomena in finite systems, we further show that this pseudospectral framework naturally accommodates additional operators that suppress contributions from boundary-localized states, enabling the systematic isolation of intrinsic bulk behavior. We benchmark the scheme on several representative systems in one and two dimensions, including a Fibonacci chain, where our approach is able to both reveal a dispersive envelope while preserving the underlying hierarchy of spectral gaps. Looking forward, this pseudospectral approach may yield a broad framework for predicting momentum-resolved material responses in aperiodic, disordered, and finite systems where conventional band-theoretic methods are not applicable.

cond-mat.mtrl-sci↗
arXiv2026

Nonabelian elastic metamaterials using holonomies acquired by crossing degeneracies

Embedding nonabelian features into elastic metamaterials promises remarkable opportunities for wave control in many practical applications such as surface acoustic wave devices, mode multiplexers, and on-material computation. Nevertheless, current realizations are limited to arrangements of coupled resonators with fine-tuned interactions, limiting their applicability to continuous media. This theoretical and numerical study introduces a design principle for continuous nonabelian elastic metamaterial waveguides. The basic configuration consists of a composite waveguide made of multiple cylindrical waveguides coupled by spatially varying elements. These elements are engineered to follow geometrically-controlled parameter variations that cross selected degeneracies and produce a targeted nonabelian holonomy. The strategy based on crossing degeneracies fundamentally differs from abelian geometric phases, where parameters avoid and encircle degeneracies, or nonabelian Wilczek-Zee phases, where parameters are fine-tuned to maintain degeneracies throughout their cycle. The resulting holonomy transfers an input longitudinal excitation in one rod to an output response in another rod. When two such waveguides are concatenated, their ordering dictates the output response, thereby revealing the emergence of nonabelian dynamics. The nonabelian behavior persists across a broad range of frequencies and under perturbations to the geometry of coupling elements or cylinder diameters. These results establish a robust, effective, and practical route to leverage nonabelian physics in elastic metamaterials.

physics.app-ph↗
arXiv2025

Quantum Phase diagrams and transitions for Chern topological insulators

Topological invariants such as Chern classes are by now a standard way to classify topological phases. Introducing and varying parameters in such systems leads to phase diagrams, where the Chern classes may jump when crossing a critical locus. These systems appear naturally when considering slicing of higher dimensional systems or when considering systems with parameters. As the Chern classes are topological invariants, they can only change if the "topology breaks down". We give a precise mathematical formulation of this phenomenon and show that synthetically any phase diagram of Chern topological phases can be designed and realized by a physical system, using covering, aka. winding maps. Here we provide explicit families realizing arbitrary Chern jumps. The critical locus of these maps is described by the classical rose curves. These realize the lower bound on the number of Dirac points necessary obtained from viewing them as local charges. We treat several concrete models and show that they have the predicted generic behavior. In particular, we focus on different types of lattices and tight-binding models, and show that effective winding maps, and thus higher Chern numbers, can be achieved using k-th nearest neighbors. We give explicit formulas for a family of 2D lattices using imaginary quadratic field extensions and their norms. Our study includes the square, triangular, honeycomb and Kagome lattices.

math-ph↗
arXiv2025

Plethysm Products, Element and Plus Constructions

Motivated by viewing categories as bimodule monoids over their isomorphism groupoids, we construct monoidal structures called plethysm products on three levels: that is for bimodules, relative bimodules and factorizable bimodules. For the bimodules, we work in the general setting of actions by categories. We give a comprehensive theory linking these levels to each other as well as to Grothendieck element constructions, indexed enrichments, decorations and algebras. Specializing to groupoid actions leads to applications including the plus construction. In this setting, the third level encompasses the known constructions of Baez-Dolan and its generalizations, as we prove. One new result is that that the plus construction can also be realized an element construction compatible with monoidal structures that we define. This allows us to prove a commutativity between element and plus constructions a special case of which was announced earlier. Specializing the results on the third level yield a criterion, when a definition of operad-like structure as a plethysm monoid -- as exemplified by operads -- is possible.

math.AT↗
arXiv2025

Plus constructions, plethysm, and unique factorization categories with applications to graphs and operad-like theories

Baez-Dolan type plus constructions serve three main purposes: They (1) corepresent categorical bimodules that are monoids with respect to a plethysm product, (2) allow to define functors as bimodule monoids, and thereby algebras over functors, (3) provide a theory of twists of monads. Unital (monoidal) bimodule monoids yield (monoidal) categories and the corepresentation is for indexed enrichments of categories. The original Baez--Dolan construction constructed algebras over operads. We define several of these constructions in the general context of categories and (symmetric) monoidal categories, show that they are functorial, and prove their corepresentation properties. One application is that the structures corepresented by an FC, like operads, props, etc. can be defined as plethysm monoids if and only if the corepresenting FC is a plus construction. In one direction, we prove that such a plus construction is based on the new notion of {Unique Factorization Category} (UFC). We also prove that the resulting FC has special properties, like being cubical. This explains why there is no monoid formulation for cyclic or modular operads or props, but there is for operads and properads. Using the bimodule monoids point of view, we prove that as monoidal bimodule monoids FCs are characterized by the fact that the functor constructing free algebras preserves the property of being strongly monoidal. We give a local presentation, as well as a global description, and a graphical version using decorated groupoid colored graphs. The global presentation utilizes pasting diagrams from 2-categories or equivalently double categories. In the special case of a UFC, we also present a graphical formalism with groupoid colored graphs. This allows us to identify our plus constructions as the step-by-step generalization of the Baez-Dolan plus constructions.

math.CT↗
arXiv2023

Koszul Feynman Categories

A cubical Feynman category, introduced by the authors in previous work, is a category whose functors to a base category $\mathcal{C}$ behave like operads in $\mathcal{C}$. In this note we show that every cubical Feynman category is Koszul. The upshot is an explicit, minimal cofibrant resolution of any cubical Feynman category, which can be used to model $\infty$ versions of generalizations of operads for both graph based and non-graph based examples.

math.AT↗
arXiv2022

A combinatorial ${E_\infty}$-algebra structure on cubical cochains and the Cartan-Serre map

Cubical cochains are equipped with an associative product, dual to the Serre diagonal, lifting the graded commutative structure in cohomology. In this work we introduce through explicit combinatorial methods an extension of this product to a full ${E_\infty}$-structure. As an application we prove that the Cartan-Serre map, which relates the cubical and simplicial singular cochains of spaces, is a quasi-isomorphism of ${E_\infty}$-algebras.

math.AT↗
arXiv2022

Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics

We develop an algebraic theory of colored, semigrouplike-flavored and pathlike co-, bi- and Hopf algebras. This is the right framework in which to discuss antipodes for bialgebras naturally appearing in combinatorics, topology, number theory and physics. In particular, we can precisely give conditions for the invertibility of characters that is needed for renormalization in the formulation of Connes and Kreimer. These are met in the relevant examples. In order to construct antipodes, we discuss formal localization constructions and quantum deformations. These allow to define and explain the appearance of Brown style coactions. Using previous results, we can interpret all the relevant coalgebras as stemming from a categorical construction, tie the bialgebra structures to Feynman categories, and apply the developed theory in this setting.

math.QA↗
arXiv2022

Trees, graphs and aggregates: a categorical perspective on combinatorial surface topology, geometry, and algebra

Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via topology, geometry and algebra. In particular, the inclusion of trees into graphs and the dissection of graphs into aggregates yield a concise formalism for cyclic and modular operads as well as their polycyclic and surface type generalizations. The latter occur prominently in two-dimensional topological field theory and in string topology. The categorical viewpoint allows us to use left Kan extensions of Feynman operations as an efficient computational tool. The computations involve the study of certain categories of structured graphs which are expected to be of independent interest.

math.AT↗
arXiv2021

A detailed look on actions on Hochschild complexes especially the degree $1$ co-product and actions on loop spaces

We explain our previous results about Hochschild actions [Kau07a, Kau08a] pertaining in particular to the coproduct which appeared in a different form in [GH09] and provide a fresh look at the results. We recall the general action, specialize to the aforementioned coproduct and prove that the assumption of commutativity, made for convenience in [Kau08a], is not needed. We give detailed background material on loop spaces, Hochschild complexes and dualizations, and discuss details and extensions of these techniques which work for all operations of [Kau07a, Kau08a]. With respect to loop spaces, we show that the co--product is well defined modulo constant loops and going one step further that in the case of a graded Gorenstein Frobenius algebra, the co--product is well defined on the reduced normalized Hochschild complex. We discuss several other aspects such as ``time reversal'' duality and several homotopies of operations induced by it. This provides a cohomology operation which is a homotopy of the anti--symmetrization of the coproduct. The obstruction again vanishes on the reduced normalized Hochschild complex if the Frobenius algebra is graded Gorenstein. Further structures such as ``animation'' and the BV structure and a coloring for operations on chains and cochains and a Gerstenhaber double bracket are briefly treated.

math.AT↗
arXiv2021

Cochain level May-Steenrod operations

Steenrod defined in 1947 the Steenrod squares on the mod 2 cohomology of spaces using explicit cochain formulae for the cup-$i$ products; a family of coherent homotopies derived from the broken symmetry of Alexander--Whitney's chain approximation to the diagonal. He later defined his homonymous operations for all primes using the homology of symmetric groups. This approach enhanced the conceptual understanding of the operations and allowed for many advances, but lacked the concreteness of their definition at the even prime. In recent years, thanks to the development of new applications of cohomology, the need to have an effectively computable definition of Steenrod operations has become a key issue. Using the operadic viewpoint of May, this article provides such definitions at all primes introducing multioperations that generalize the Steenrod cup-$i$ products on the simplicial and cubical cochains of spaces.

math.AT↗
arXiv2020

Feynman categories and Representation Theory

We give a presentation of Feynman categories from a representation--theoretical viewpoint. Feynman categories are a special type of monoidal categories and their representations are monoidal functors. They can be viewed as a far reaching generalization of groups, algebras and modules. Taking a new algebraic approach, we provide more examples and more details for several key constructions. This leads to new applications and results. The text is intended to be a self--contained basis for a crossover of more elevated constructions and results in the fields of representation theory and Feynman categories, whose applications so far include number theory, geometry, topology and physics.

math.RT↗
arXiv2020

Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework, which is presented step-by-step with examples throughout. In this first part of two papers, we concentrate on the simplicial and operadic aspects.

math.AT↗
arXiv2020

Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework which is presented step-by-step with examples throughout. In this second part of two papers, we give the general categorical formulation.

math.AT↗
arXiv2018

Topological insulators and K-theory

We analyze the topological $\mathbb{Z}_2$ invariant, which characterizes time reversal invariant topological insulators, in the framework of index theory and K-theory. The topological $\mathbb{Z}_2$ invariant counts the parity of generalized Majorana zero modes, which can be interpreted as an analytical index. As we show, it fits perfectly into a mod 2 index theorem, and the topological index provides an efficient way to compute the topological $\mathbb{Z}_2$ invariant. Finally, we give a new version of the bulk-boundary correspondence which yields an alternative explanation of the index theorem and the topological $\mathbb{Z}_2$ invariant. Here the boundary is not the geometric boundary of a probe, but an effective boundary in the momentum space.

math-ph↗
arXiv2018

Local models and global constraints for degeneracies and band crossings

We study topological properties of families of Hamiltonians which may contain degenerate energy levels aka. band crossings. The primary tool are Chern classes, Berry phases and slicing by surfaces. To analyse the degenerate locus, we study local models. These give information about the Chern classes and Berry phases. We then give global constraints for the topological invariants. This is an hitherto relatively unexplored subject. The global constraints are more strict when incorporating symmetries such as time reversal symmetries. The results can also be used in the study of deformations. We furthermore use these constraints to analyse examples which include the Gyroid geometry, which exhibits Weyl points and triple crossings and the honeycomb geometry with its two Dirac points.

math-ph↗
arXiv2018

Comprehensive factorisation systems

We establish a correspondence between consistent comprehension schemes and complete orthogonal factorisation systems. The comprehensive factorisation of a functor between small categories arises in this way. Similar factorisation systems exist for the categories of topological spaces, simplicial sets, small multicategories and Feynman categories. In each case comprehensive factorisation induces a natural notion of universal covering, leading to a Galois-type definition of fundamental group for based objects of the category.

math.CT↗