A Topological Visualization Framework for Quantum Spin

1999

In this paper we discuss several ways to visualize stationary and nonstationary quantum mechanical systems. We demonstrate an approach for the quantitative interpretation of probability density isovalues which yields a reasonable correlation between isosurfaces for different timesteps. As an intuitive quantity for visualizing the momentum of a quantum system we propose the probability flow density, which can be treated by vector field visualization techniques. Finally, we discuss the visualization of non-stationary systems by a sequence of single timestep images.

2020

It is proposed here to represent the quantum distribution of atomic orbitals in an unprecedented table where the quantum shells and subshells are drawn in the form of chevrons whose vertices are occupied by orbitals with the magnetic quantum number m = 0. This new representation visually shows, much better than a classic linear chart, the relationship between the number of quantum shells and the number of orbitals . Also, this new visual model can be easily used in the individual quantum depiction of the atoms represented alone or into molecules and can find its place in illustration of some two-dimensional space-time quantum theories. Finally, this graphic representation allows to introduce the hypothesis of the existence of stealth orbitals, quantum gates opening towards singularities.

2015

The state of quantum systems, their energetics, and their time evolution is modeled by abstract operators. How can one visualize such operators for coupled spin systems? A general approach is presented which consists of several shapes representing linear combinations of spherical harmonics. It is applicable to an arbitrary number of spins and can be interpreted as a generalization of Wigner functions. The corresponding visualization transforms naturally under non-selective spin rotations as well as spin permutations. Examples and applications are illustrated for the case of three spins 1/2.

Communications in Mathematical Physics, 1994

A number of interesting features of the ground states of quantum spin chains are analized with the help of a functional integral representation of the system's equilibrium states. Methods of general applicability are introduced in the context of the SU(2S + 1)-invariant quantum spin-S chains with the interaction −P (0) , where P (0) is the projection onto the singlet state of a pair of nearest neighbor spins. The phenomena discussed here include: the absence of Néel order, the possibility of dimerization, conditions for the existence of a spectral gap, and a dichotomy analogous to one found by Affleck and Lieb, stating that the systems exhibit either slow decay of correlations or translation symmetry breaking. Our representation elucidates the relation, evidence for which was found earlier, of the −P (0) spin-S systems with the Potts and the Fortuin-Kasteleyn random-cluster models in one more dimension. The method reveals the geometric aspects of the listed phenomena, and gives a precise sense to a picture of the ground state in which the spins are grouped into random clusters of zero total spin. E.g., within such structure the dichotomy is implied by a topological argument, and the alternatives correspond to whether, or not, the clusters are of finite mean length.

Journal of Molecular Spectroscopy, 2017

The ingredients of the quantum theory of orbital and spin angular momentum encounter continuing relevance in wide areas beyond the traditional ones (molecular, atomic and nuclear spectroscopies and dynamics). This paper offers a presentation of the connection at the most elementary of levels with the diagrammatic approaches to projective geometry. In particular here we exhibit how the Fano, Desargues and related incidence configurations emerge in the Racah and in the Biedenharn-Elliott identities, corresponding respectively to the hexagonal and pentagonal relationships that provide the basis for the construction of 3nj symbols and of spin networks. It is shown that the treatment, although mostly confined to the quadrangulation of the real projective plane, permits however the introduction of networks involving seven and ten spins, and preludes to developments towards computational and asymptotic approaches for quantum and semi-classical applications.

Classical and Quantum Gravity, 2011

The Spin Foams for People Without the 3d/ 4d Imagination could be an alternative title of our work. We derive spin foams from operator spin network diagrams we introduce. Our diagrams are the spin network analogy of the Feynman diagrams. Their framework is compatible with the framework of Loop Quantum Gravity. For every operator spin network diagram we construct a corresponding operator spin foam. Admitting all the spin networks of LQG and all possible diagrams leads to a clearly defined large class of operator spin foams. In this way our framework provides a proposal for a class of 2-cell complexes that should be used in the spin foam theories of LQG. Within this class, our diagrams are just equivalent to the spin foams. The advantage, however, in the diagram framework is, that it is self contained, all the amplitudes can be calculated directly from the diagrams without explicit visualization of the corresponding spin foams. The spin network diagram operators and amplitudes are consistently defined on their own. Each diagram encodes all the combinatorial information. We illustrate applications of our diagrams: we introduce a diagram definition of Rovelli's surface amplitudes as well as of the canonical transition amplitudes. Importantly, our operator spin network diagrams are defined in a sufficiently general way to accommodate all the versions of the EPRL or the FK model, as well as other possible models. The diagrams are also compatible with the structure of the LQG Hamiltonian operators, what is an additional advantage. Finally, a scheme for a complete definition of a spin foam theory by declaring a set of interaction vertices emerges from the examples presented at the end of the paper.

Topology - Recent Advances and Applications [Working Title]

This chapter reviews the implications of topology in the static and dynamics of magnetic systems. Our focus is twofold. In the first part, we describe how the application of topology allows an understanding of the structure and dynamics of magnetic textures that separate different magnetic domains in magnetic materials. Topological textures are rationalized in terms of elementary topological defects that determine complex magnetic orders and magnetization dynamics processes in the underlying magnetic systems. The second part studies topological phases and topological phenomena associated with the band theory of linear magnetic excitations. Topological spin waves are usually accompanied by exotic phenomena in magnetic materials such as the emergence of chiral edge states and the magnon Hall effect.

We present "Diagrams of States", a way to graphically represent and analyze how quantum information is elaborated during the execution of quantum circuits. This introductory tutorial illustrates the basics, providing useful examples of quantum computations: elementary operations in single-qubit, two-qubit and three-qubit systems, immersions of gates on higher dimensional spaces, generation of single and multi-qubit states, procedures to synthesize unitary, controlled and diagonal matrices. To perform the analysis of quantum processes, we directly derive diagrams of states from physical implementations of quantum circuits associated to the processes. Complete diagrams are then rearranged into simplified diagrams, to visualize the overall effects of computations. Conversely, diagrams of states help to conceive new quantum algorithms, by schematically describing desired manipulations of quantum information with intuitive diagrams and then by guessing the equivalent complete d...

Soft Computing, 2015

The standard Bloch sphere representation has been recently generalized to describe not only systems of arbitrary dimension, but also their measurements, in what has been called the extended Bloch representation of quantum mechanics. This model, which offers a solution to the longstanding measurement problem, is based on the hidden-measurement interpretation of quantum mechanics, according to which the Born rule results from our lack of knowledge of the measurement interaction that each time is actualized between the measuring apparatus and the measured entity. In this article we present the extended Bloch model and use it to investigate, more specifically, the nature of the quantum spin entities and of their relation to our three-dimensional Euclidean theater. Our analysis shows that spin eigenstates cannot generally be associated with directions in the Euclidean space, but only with generalized directions in the Blochean space, which apart from the special case of spin one-half entities, is a space of higher dimensionality. Accordingly, spin entities have to be considered as genuine non-spatial entities. We also show, however, that specific vectors can be identified in the Blochean theater that are isomorphic to the Euclidean space directions, and therefore representative of them, and that spin eigenstates always have a predetermined orientation with respect to them. We use the details of our results to put forward a new view of realism, that we call multiplex realism, providing a specific framework with which to interpret the human observations and understanding of the component parts of the world. Elements of reality can be represented in different theaters, one being our customary Euclidean space, and another one the quantum realm, revealed to us through our sophisticated experiments, whose elements of reality, in the quantum jargon, are the eigenvalues and eigenstates. Our understanding of the component parts of the world can then be guided by looking for the possible connections, in the form of partial morphisms, between the different representations, which is precisely what we do in this article with regard to spin entities.

Classical and Quantum Gravity, 2002

Within the context of loop quantum gravity there are several operators which measure geometry quantities. This work examines two of these operators, volume and angle, to study quantum geometry at a single spin network vertex -"an atom of geometry." Several aspects of the angle operator are examined in detail including minimum angles, level spacing, and the distribution of angles. The high spin limit of the volume operator is also studied for monochromatic vertices. The results show that demands of the correct scaling relations between area and volume and requirements of the expected behavior of angles in three dimensional flat space require high-valence vertices with total spins of approximately 10 20 .