Using the fractional interaction law to model the impact dynamics in arbitrary form of multiparticle collisions

Using the molecular dynamics method, we examine a discrete deterministic model for the motion of spherical particles in three-dimensional space. The model takes into account multiparticle collisions in arbitrary forms. Using fractional calculus we proposed an expression for the repulsive force, which is the so called fractional interaction law. We then illustrate and discuss how to control (correlate) the energy dissipation and the collisional time for an individual article within multiparticle collisions. In the multiparticle collisions we included the friction mechanism needed for the transition from coupled torsion-sliding friction through rolling friction to static friction. Analysing simple simulations we found that in the strong repulsive state binary collisions dominate. However, within multiparticle collisions weak repulsion is observed to be much stronger. The presented numerical results can be used to realistically model the impact dynamics of an individual particle in a group of colliding particles.Comment: 17 pages, 8 figures, 1 table; In review process of Physical Review

Multi-Particle Pseudopotentials for Multi-Component Quantum Hall Systems

The Haldane pseudopotential construction has been an extremely powerful concept in quantum Hall physics --- it not only gives a minimal description of the space of Hamiltonians but also suggests special model Hamiltonians (those where certain pseudopotential are set to zero) that may have exactly solvable ground states with interesting properties. The purpose of this paper is to generalize the pseudopotential construction to situations where interactions are N-body and where the particles may have internal degrees of freedom such as spin or valley index. Assuming a rotationally invariant Hamiltonian, the essence of the problem is to obtain a full basis of wavefunctions for N particles with fixed relative angular momentum L. This basis decomposes into representations of SU(n) with n the number of internal degrees of freedom. We give special attention to the case where the internal degree of freedom has n=2 states, which encompasses the important cases of spin-1/2 particles and quantum Hall bilayers. We also discuss in some detail the cases of spin-1 particles (n=3) and graphene (n=4, including two spin and two valley degrees of freedom).Comment: 46 pages ; 9 tables ; no figures. (The revision fixes a number of typos and updates the formatting

Non-Abelian Anyons and Topological Quantum Computation

Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as {\it Non-Abelian anyons}, meaning that they obey {\it non-Abelian braiding statistics}. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations which are necessary for quantum computation are carried out by braiding quasiparticles, and then measuring the multi-quasiparticle states. The fault-tolerance of a topological quantum computer arises from the non-local encoding of the states of the quasiparticles, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the \nu=5/2 state, although several other prospective candidates have been proposed in systems as disparate as ultra-cold atoms in optical lattices and thin film superconductors. In this review article, we describe current research in this field, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. We address both the mathematical underpinnings of topological quantum computation and the physics of the subject using the \nu=5/2 fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.Comment: Final Accepted form for RM

Quantum Hall Physics - hierarchies and CFT techniques

The fractional quantum Hall effect, being one of the most studied phenomena in condensed matter physics during the past thirty years, has generated many groundbreaking new ideas and concepts. Very early on it was realized that the zoo of emerging states of matter would need to be understood in a systematic manner. The first attempts to do this, by Haldane and Halperin, set an agenda for further work which has continued to this day. Since that time the idea of hierarchies of quasiparticles condensing to form new states has been a pillar of our understanding of fractional quantum Hall physics. In the thirty years that have passed since then, a number of new directions of thought have advanced our understanding of fractional quantum Hall states, and have extended it in new and unexpected ways. Among these directions is the extensive use of topological quantum field theories and conformal field theories, the application of the ideas of composite bosons and fermions, and the study of nonabelian quantum Hall liquids. This article aims to present a comprehensive overview of this field, including the most recent developments.Comment: added section on experimental status, 59 pages+references, 3 figure

Charge and heat transport in topological systems

In this thesis, I address the intriguing and appealing topic of charge and heat transport in quantum Hall systems, which are among the most famous example of topological phases of matter, in presence of external time-dependent voltages. The interest of condensed matter community towards topological systems has been considerably raised in recent years. For instance, it is worth to recall the Nobel prize for physics 2016 awarded to Professors Thouless, Kosterlitz and Haldane for their contribution to the study of topological states of matter. These states are exotic phases of matter, whose properties are described in terms of quantities that do not depend on the details of a system, are very robust against defects and perturbations. The research field of topological systems takes place due to the interplay between condensed matter physics and mathematics. As a matter of fact, many concepts have been borrowed from the mathematical branch of topology in order to classify these novel states of matter. Quantum Hall effect was discovered almost forty years ago and still attracts a lot of attention from the theoretical and experimental point of view. This remarkable physical phenomenon occurs in two-dimensional electron systems in the limit of strong perpendicular magnetic fields. In quantum Hall systems, the transverse resistance, which is commonly defined Hall resistance, is very precisely quantized in terms of the resistance quantum. When this quantization occurs for integer values, this phenomenology is termed integer quantum Hall effect. It can be understood in a satysfying way by resorting to a non-interacting quantum mechanical description. The hallmark of quantum Hall systems is the emergence of one-dimensional metallic edge states on the boundaries of the system. Along these edge states particles propagate with a definite direction. As a result, they are topologically protected against backscattering. The coherence length ensured by topological protection guarantees to access the wave-like nature of electrons. Intriguingly, this investigation can be pushed to its fundamental limit by exploring quantum transport at the single-electron level. This idea embodies the core of a new field of research, known as electron quantum optics Single-electron source can be realized by applying to a quantum Hall system a periodic train of Lorentzian-shaped pulses, carrying an integer number of particles per period, thus emitting into the edge states minimal single-electron excitations, then termed levitons. Plateaus of the Hall resistance appear also at fractional values of the resistance quantum. Contrarily to the integer case, the physical explanation of fractional quantum Hall effect cannot neglect the correlation between electrons and this phase of matter is inherently strongly-correlated. Intriguingly, elementary excitations of fractional quantum Hall systems are quasi-particle with fractional charge and statistics. Remarkably, one-dimensional conducting edge states arise also in the fractional quantum Hall effect and their excitations inherit the charge and statistical properties of the one in the bulk. By considering the application of a periodic train of Lorentzian pulses to a quantum Hall system, I focus on the transport properties of levitons propagating along integer and fractional edge states. I investigate the charge density of a state composed by many levitons in the fractional quantum Hall regime, thus finding that it is re-arranged into a regular pattern of peaks and valleys, reminiscent of Wigner crystallization in strongly-interacting electronic systems. Then, I analyze heat transport properties of levitons in quantum Hall systems, which represent a new point of view on electron quantum optics, extending and generalizing the results obtained in the charge domain