Related papers: Tangle Machines II: Invariants
The topology of entanglement in multipartite states with translational invariance is discussed in this article. Two global features are foundby which one can distinguish distinct states. These are the cyclic unit and the quantised geometric…
In this work we develop a theory of Vessels. This object arises in the study of overdetermined 2D systems invariant in one of the variables, which are usually called time invariant. To each overdetermined time invariant 2D systems there is…
We introduce the notion of quantum duplicates of an (associative, unital) algebra, motivated by the problem of constructing toy-models for quantizations of certain configuration spaces in quantum mechanics. The proposed (algebraic) model…
In this note we give a simple unifying proof of the undecidability of several diagrammatic properties of term rewriting systems that include: local confluence, strong confluence, diamond property, subcommutative property, and the existence…
The aim of this work is to establish the existence of invariant manifolds in complex systems. Considering trajectory curves integral of multiple time scales dynamical systems of dimension two and three (predator-prey models, neuronal…
Algorithms which learn environments represented by automata in the past have had complexity scaling with the number of states in the automaton, which can be exponentially large even for automata recognizing regular expressions with a small…
We present mathematical details of the construction of a topological invariant for periodically driven two-dimensional lattice systems with time-reversal symmetry and quasienergy gaps, which was proposed recently by some of us. The…
Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…
The density matrix of composite spin system is discussed in relation to the adjoint representation of unitary group U(n). The entanglement structure is introduced as an additional ingredient to the description of the linear space carrying…
Recent evidence indicates that the abundance of recurring elementary interaction patterns in complex networks, often called subgraphs or motifs, carry significant information about their function and overall organization. Yet, the…
Topologically ordered systems are characterized by topological invariants that are often calculated from the momentum space integration of a certain function that represents the curvature of the many-body state. The curvature function may…
The first-principles band theory paradigm has been a key player not only in the process of discovering new classes of topologically interesting materials, but also for identifying salient characteristics of topological states, enabling…
We study the topological characterization of the energy gaps in general two-dimensional quasiperiodic systems consisting of multiple periodicities, represented by twisted two-dimensional materials. We show that every single gap is uniquely…
Much attention has been devoted to the use of machine learning to approximate physical concepts. Yet, due to challenges in interpretability of machine learning techniques, the question of what physics machine learning models are able to…
A new family of graphs, {\it entangled networks}, with optimal properties in many respects, is introduced. By definition, their topology is such that optimizes synchronizability for many dynamical processes. These networks are shown to have…
Recent work suggests that topological features of certain quantum gravity theories can be interpreted as particles, matching the known fermions and bosons of the first generation in the Standard Model. This is achieved by identifying…
Turing's theory of pattern formation is a universal model for self-organization, applicable to many systems in physics, chemistry and biology. Essential properties of a Turing system, such as the conditions for the existence of patterns and…
Based on the Kauffman bracket at $A=e^{i \pi/4}$, we defined an invariant for a special type of $n$-punctured ball tangles. The invariant $F^n$ takes values in the set $PM_{2\times2^n}(\mathbb Z)$ of $2\times 2^n$ matrices over $\mathbb Z$…
One of the apparent advantages of quantum computers over their classical counterparts is their ability to efficiently contract tensor networks. In this article, we study some implications of this fact in the case of topological tensor…
We construct an explicit categorification of the action of tangles on tensor powers of the fundamental representation of quantum sl(2).