Gentile statistics describes fractional statistical systems in the occupation number representation. Anyon statistics researches those systems in the winding number representation. Both of them are intermediate statistics between Bose-Einstein and Fermi-Dirac statistics. The second quantization of Gentile statistics shows a lot of advantages. According to the symmetry requirement of the wave function, we give the general construction of transformation between anyon and Gentile statistics. In other words, we introduce the second quantization form of anyons in a easy way. Basic relations of second quantization operators, the coherent state and Berry phase are also discussed.
The many-anyons wavefunction is constructed via the superposition of all the permutations on the direct product of single anyon states and its interchange properties are examined. The phase of permutation is not a representation but the word metric of the permutation group . Amazingly the interchange phase yields a finite capacity of one quantum state interpolating between Fermion and Boson and the mutual exchange phase has no explicit effect on statistics. Finite capacity of quantum state is defined as Gentile statistics and it is different from the fractional exclusion statistics. Some discu…
This paper presents a new way to construct single-valued many-body wavefunctions of identical particles with intermediate exchange phases between Fermi and Bose statistics. It is demonstrated that the exchange phase is not a representation character but the \textit{word metric} of the permutation group, beyond the anyon phase from the braiding group in two dimensions. By constructing this type of wavefunction from the direct product of single-particle states, it is shown that a finite \textit{capacity q} -- the maximally allowed particle occupation of each quantum state, naturally arises. The …
The exact solution of the interacting many-body system is important and is difficult to solve. In this paper, we introduce a group method to solve the interacting many-body problem using the relation between the permutation group and the unitary group. We prove a group theorem first, then using the theorem, we represent the Hamiltonian of the interacting many-body system by the Casimir operators of unitary group. The eigenvalues of Casimir operators could give the exact values of energy and thus solve those problems exactly. This method maps the interacting many-body system onto an intermediat…
In this paper, a method is developed to investigate the relativistic quantum information of anyons. Anyons are particles with intermediate statistics ranging between Bose-Einstein and Fermi-Dirac statistics, with a parameter $α$ ($0<α<1$) characteristic of this intermediate statistics. A density matrix is also introduced as a combination of the density matrices of bosons and fermions with a continuous parameter, $α$, that represents the behavior of anyons. This density matrix reduces to bosonic and fermionic density matrices in the limits $α\rightarrow 0$ and $α\rightarrow 1$,respectively. We …
Anyons, particles displaying a fractional exchange statistics intermediate between bosons and fermions, play a central role in the fractional quantum Hall effect and various spin lattice models, and have been proposed for topological quantum computing schemes due to their resilience to noise. Here we use parametric down-conversion in an integrated semiconductor chip to generate biphoton states simulating anyonic particle statistics, in a reconfigurable manner. Our scheme exploits the frequency entanglement of the photon pairs, which is directly controlled through the spatial shaping of the pum…
We present a theory of particles, obeying intermediate statistics ("anyons"), interpolating between Bosons and Fermions, based on the principle of Detailed Balance. It is demonstrated that the scattering probabilities of identical particles can be expressed in terms of the basic numbers, which arise naturally and logically in this theory. A transcendental equation determining the distribution function of anyons is obtained in terms of the statistics parameter, whose limiting values 0 and 1 correspond to Bosons and Fermions respectively. The distribution function is determined as a power series…
The idea that a system obeying interpolating statistics can be described by a deformed oscillator algebra has been an outstanding issue. This original concept introduced long ago by Greenberg is the motivation for this investigation. We establish that a q-deformed algebra can be used to describe the statistics of particles (anyons) interpolating continuously between Bose and Fermi statistics, i.e., fractional statistics. We show that the generalized intermediate statistics splits into the Boson-like and Fermion-like regimes, each described by a unique oscillator algebra. The B-anyon thermostat…
Universitat de València-CSIC（瓦伦西亚大学-CSIC）；IFIC, Universitat de València-CSIC（IFIC，瓦伦西亚大学-CSIC）；JILA, NIST, and Department of Physics, University of Colorado（JILA，美国国家标准与技术研究院，科罗拉多大学物理系）；University of Colorado（科罗拉多大学）；IUMPA - Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València（IUMPA-纯数学与应用数学大学研究所，巴伦西亚理工大学）；Adam Mickiewicz University（亚当·密茨凯维奇大学）；DIPC - Donostia International Physics Center（DIPC-圣塞巴斯蒂安国际物理中心）；IKERBASQUE, Basque Foundation for Science（IKERBASQUE，巴斯克科学基金会）；ICREA（加泰罗尼亚高级研究学院）；ETH Zurich（苏黎世联邦理工学院）
Generalized Effective Spin-Chain formalism for multicomponent anyons in one-dimensional optical lattices
Rice University（莱斯大学）；The University of Oklahoma（俄克拉荷马大学）；Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences（中国科学院精密测量科学与技术创新研究院）；Research School of Physics, Australian National University（澳大利亚国立大学物理研究学院）
The quantum-mechanical description of assemblies of particles whose motion is confined to two (or one) spatial dimensions offers many possibilities that are distinct from bosons and fermions. We call such particles anyons. The simplest anyons are parameterized by an angular phase parameter $θ$. $θ= 0, π$ correspond to bosons and fermions respectively; at intermediate values we say that we have fractional statistics. In two dimensions, $θ$ describes the phase acquired by the wave function as two anyons wind around one another counterclockwise. It generates a shift in the allowed values for the …
A local exclusion principle is observed for identical particles obeying intermediate/fractional exchange statistics in one and two dimensions, leading to bounds for the kinetic energy in terms of the density. This has implications for models of Lieb-Liniger and Calogero-Sutherland type, and implies a non-trivial lower bound for the energy of the anyon gas whenever the statistics parameter is an odd numerator fraction. We discuss whether this is actually a necessary requirement.
Utilizing quantum coherence monotone, we reexamine the thermal nature of the Unruh effect of an accelerating detector. We consider an UDW detector coupling to a n-dimensional conformal field in Minkowski spacetime, whose response spectrum generally exhibits an intermediate statistics of (1+1) anyon field. We find that the thermal nature of the Unruh effect guaranteed by KMS condition is characterized by a vanishing asymptotic quantum coherence. We show that the time-evolution of coherence monotone can distinguish the different thermalizing ways of the detector, which depends on the scaling dim…
This contribution, to be published in Imagine Math 8 to celebrate Michele Emmer's 75th birthday, can be seen as the second part of my previous considerations on the relationships between topology and physics (Mouchet, 2018). Nevertheless, the present work can be read independently. The following mainly focusses on the connection between topology and quantum statistics. I will try to explain to the non specialist how Feynman's interpretation of quantum processes through interference of classical paths (path integrals formulation), makes the dichotomy between bosons and fermions quite natural in…
An outstanding idea originally introduced by Greenberg is to investigate whether there is equivalence between intermediate statistics, which may be different from anyonic statistics, and q-deformed particle algebra. Also, a model to be studied for addressing such an idea could possibly provide us some new consequences about the interactions of particles as well as their internal structures. Motivated mainly by this idea, in this work, we consider a q-deformed Fermi gas model whose statistical properties enable effectively us to study interpolating statistics. Starting with a generalized Fermi-…
