相关论文: Geometrical and Algebraic Structures in Quantum Ha…
The theory of Lie systems has recently been applied to Quantum Mechanics and additionally some integrability conditions for Lie systems of differential equations have also recently been analysed from a geometric perspective. In this paper…
The physics of the quantum Hall system becomes very simple when studied on a thin torus. Remarkably, however, the very rich structure still exists in this limit and there is a continuous route to the bulk system. Here we review recent…
The Quantum Hall Effects in all even dimensions are uniformly constructed. Contrary to some recent accounts in the literature, the existence of Quantum Hall Effects does not {\it crucially} depend on the existence of division algebras. For…
We gather material from many sources about the quantum potential and its geometric nature. The presentation is primarily expository but some new observations relating Q, V, and psi are indicated.
We review the investigations on the quantum structure of spactime, to be found at the Planck scale if one takes into account the operational limitations to localization of events which result from the concurrence of Quantum Mechanics and…
Some algebraic issues of the FQHE are presented. First, it is shown that on the space of Laughlin wavefunctions describing the $\nu =1/m$ FQHE, there is an underlying $W_{\infty}$ algebra, which plays the role of a spectrum generating…
The unsatisfactory status of the search for a consistent and predictive quantization of gravity is taken as motivation to study the question whether geometrical laws could be more fundamental than quantization procedures. In such an…
The paper presents an extension of the geometric quantization procedure to integrable, big-isotropic structures. We obtain a generalization of the cohomology integrality condition, we discuss geometric structures on the total space of the…
Fractional quantum Hall liquids can accomodate various degrees of spatial ordering. The most likely scenarios are a Hall hexatic, Hall smectic, and Hall crystal, in which respectively orientational, one--dimensional translational, and…
The earlier approach is used for description of qubits and geometric phase parameters, the things critical in the area of topological quantum computing. The used tool, Geometric (Clifford) Algebra is the most convenient formalism for that…
In this paper we introduce a geometric framework for mixed quantum states based on a K\"ahler structure. The geometric framework includes a symplectic form, an almost complex structure, and a Riemannian metric that characterize the space of…
General Relativity describes gravity in geometrical terms. This suggests that quantizing such theory is the same as quantizing geometry. The subject can therefore be called quantum geometry and one may think that mathematicians are…
We address two fundamental issues in the physics of the quantum Hall effect: a unified description of scaling behavior of conductances in the integral and fractional regimes, and a quasi-particle formulation of the chiral Luttinger Liquids…
Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic…
In this contribution, we present an introduction to the physical principles underlying the quantum Hall effect. The field theoretic approach to the integral and fractional effect is sketched, with some emphasis on the mechanism of…
In this paper, the key ideas of characterizing universality classes of dissipation-free (incompressible) quantum Hall fluids by mathematical objects called quantum Hall lattices are reviewed. Many general theorems about the classification…
These lecture notes yield an introduction to quantum Hall effects both for non-relativistic electrons in conventional 2D electron gases (such as in semiconductor heterostructures) and relativistic electrons in graphene. After a brief…
It has recently been pointed out that phases of matter with intrinsic topological order, like the fractional quantum Hall states, have an extra dynamical degree of freedom that corresponds to quantum geometry. Here we perform extensive…
The role of operational quantum mechanics, quantum axiomatics and quantum structures in general is presented as a contribution to a compendium on quantum physics, its history and philosophy.
Quantum geometry governs a wide range of transport and optical phenomena in quantum materials. Recent works have explored analogue electromagnetism and gravity in terms of the quantum geometric tensor, whose real and imaginary parts…