arXiv 1408.70022014-08-29math-phmath.ATmath.MPquant-ph

Topology of graph configuration spaces and quantum statistics

Adam Sawicki

备注：: PhD thesis, Bristol 2014

英文摘要：

In this thesis we develop a full characterization of abelian quantum statistics on graphs. We explain how the number of anyon phases is related to connectivity. For 2-connected graphs the independence of quantum statistics with respect to the number of particles is proven. For non-planar 3-connected graphs we identify bosons and fermions as the only possible statistics, whereas for planar 3-connected graphs we show that one anyon phase exists. Our approach also yields an alternative proof of the structure theorem for the first homology group of n-particle graph configuration spaces. Finally, we determine the topological gauge potentials for 2-connected graphs. Moreover we present an alternative application of discrete Morse theory for two-particle graph configuration spaces. In contrast to previous constructions, which are based on discrete Morse vector fields, our approach is through Morse functions, which have a nice physical interpretation as two-body potentials constructed from one-body potentials. We also give a brief introduction to discrete Morse theory.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Computational topology for configuration spaces of hard disks

Stanford University（斯坦福大学）；The Ohio State University（俄亥俄州立大学）；Institute for Advanced Study（高等研究院）

We explore the topology of configuration spaces of hard disks experimentally, and show that several changes in the topology can already be observed with a small number of particles. The results illustrate a theorem of Baryshnikov, Bubenik, and Kahle that critical points correspond to configurations of disks with balanced mechanical stresses, and suggest conjectures about the asymptotic topology as the number of disks tends to infinity.

2011-11-13cs.ROmath-phmath.AT

Configuration Spaces and the Topology of Curves in Projective Space

We survey and expand on the work of Segal, Milgram and the author on the topology of spaces of maps of positive genus curves into $n$-th complex projective space, $n\geq 1$ (in both the holomorphic and continuous categories). Both based and unbased maps are studied and in particular we compute the fundamental groups of the spaces in question. The relevant case when $n=1$ is given by a non-trivial extension which we fully determine.

2001-06-29math-phmath.AGmath.AT

On the Topology of the Reduced Classical Configuration Space of Lattice QCD

We study the topological structure of the quotient of $SU(3)\times SU(3)$ by diagonal conjugation. This is the simplest nontrivial example for the classical reduced configuration space of chromodynamics on a spatial lattice in the Hamiltonian approach. We construct a cell complex structure of the quotient in such a way that the closures of strata are subcomplexes and we compute the homology and cohomology groups of the strata and their closures.

2008-03-31hep-thmath-phmath.MP

什么是自由度？构型空间及其拓扑

What is a degree of freedom? Configuration spaces and their topology

本文通过简单但非平凡的力学示例，以教学方法定义经典力学中的自由度和构型空间，并强调拓扑在理解这些概念中的关键作用。

2024-07-31math-phmath.MPphysics.class-ph

Topology Classes of Flat U(1) Bundles and Diffeomorphic Covariant Representations of the Heisenberg Algebra

The general construction of self-adjoint configuration space representations of the Heisenberg algebra over an arbitrary manifold is considered. All such inequivalent representations are parametrised in terms of the topology classes of flat U(1) bundles over the configuration space manifold. In the case of Riemannian manifolds, these representations are also manifestly diffeomorphic covariant. The general discussion, illustrated by some simple examples in non relativistic quantum mechanics, is of particular relevance to systems whose configuration space is parametrised by curvilinear coordinat…

1999-09-01hep-thmath-phmath.MP

Statistical Topology -- Distribution and Density Correlations of Winding Numbers in Chiral Systems

Statistical Topology emerged since topological aspects continue to gain importance in many areas of physics. It is most desirable to study topological invariants and their statistics in schematic models that facilitate the identification of universalities. Here, the statistics of winding numbers and of winding number densities are addressed. An introduction is given for readers with little background knowledge. Results that my collaborators and I obtained in two recent works on proper random matrix models are reviewed, avoiding a technically detailed discussion. A special focus is on the mappi…

2023-01-16cond-mat.dis-nnmath-phmath.MP

更多相关论文收起更多论文

Topology and Bistability in liquid crystal devices

We study nematic liquid crystal configurations in a prototype bistable device - the Post Aligned Bistable Nematic (PABN) cell. Working within the Oseen-Frank continuum model, we describe the liquid crystal configuration by a unit-vector field, in a model version of the PABN cell. Firstly, we identify four distinct topologies in this geometry. We explicitly construct trial configurations with these topologies which are used as initial conditions for a numerical solver, based on the finite-element method. The morphologies and energetics of the corresponding numerical solutions qualitatively agre…

2006-11-08math-phmath.GNmath.MP

基于网络的动力学模型：图拓扑统计描述的出现

Network-based kinetic models: Emergence of a statistical description of the graph topology

Politecnico di Torino（都灵理工大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；CENTAI Institute（CENTAI研究所）

本文提出利用动力学方程描述多智能体系统中图介导的 pairwise 交互产生的集体动态，证明在大图和特定交互类中，图拓扑的统计描述足以捕捉网络交互系统的集体趋势。

2024-01-04math-phmath.MPphysics.soc-ph

Quantum statistical mechanics in arithmetic topology

This paper provides a construction of a quantum statistical mechanical system associated to knots in the 3-sphere and cyclic branched coverings of the 3-sphere, which is an analog, in the sense of arithmetic topology, of the Bost-Connes system, with knots replacing primes, and cyclic branched coverings of the 3-sphere replacing abelian extensions of the field of rational numbers. The operator algebraic properties of this system differ significantly from the Bost-Connes case, due to the properties of the action of the semigroup of knots on a direct limit of knot groups. The resulting algebra of…

2016-02-16math-phmath.GTmath.MP

Quantum Statistics and Spacetime Topology: Quantum Surgery Formulas

To formulate the universal constraints of quantum statistics data of generic long-range entangled quantum systems, we introduce the geometric-topology surgery theory on spacetime manifolds where quantum systems reside, cutting and gluing the associated quantum amplitudes, specifically in 2+1 and 3+1 spacetime dimensions. First, we introduce the fusion data for worldline and worldsheet operators capable of creating anyonic excitations of particles and strings, well-defined in gapped states of matter with intrinsic topological orders. Second, we introduce the braiding statistics data of particle…

2019-01-31cond-mat.str-elhep-thmath-ph

Topology and Phase Transitions I. Preliminary Results

In this first paper, we demonstrate a theorem that establishes a first step toward proving a necessary topological condition for the occurrence of first or second order phase transitions: we prove that the topology of certain submanifolds of configuration space must necessarily change at the phase transition point. The theorem applies to smooth, finite-range and confining potentials V bounded below, describing systems confined in finite regions of space with continuously varying coordinates. The relevant configuration space submanifolds are both the level sets {Σ_v := V_N^{-1} (v)}_{v \in R} o…

2006-04-18cond-mat.stat-mechhep-thmath-ph

扰动涡旋与反场配置中的单连通拓扑

Simply Connected Topology in Perturbed Vortices and Field-Reversed Configurations

Plasma Science and Fusion Center, Massachusetts Institute of Technology（麻省理工学院等离子体科学与聚变中心）；Princeton Plasma Physics Laboratory, Princeton University（普林斯顿大学普林斯顿等离子体物理实验室）；Fusion Theory and Computation, Inc.（聚变理论与计算公司）

本文通过证明小扰动下涡旋和反场配置的磁流面变为单连通，修正了传统拓扑分类，揭示了其在等离子体约束和流体动力学中的新意义。

2026-03-13math-phmath.MPphysics.flu-dyn

Statistical mechanics of complex networks

Complex networks describe a wide range of systems in nature and society, much quoted examples including the cell, a network of chemicals linked by chemical reactions, or the Internet, a network of routers and computers connected by physical links. While traditionally these systems were modeled as random graphs, it is increasingly recognized that the topology and evolution of real networks is governed by robust organizing principles. Here we review the recent advances in the field of complex networks, focusing on the statistical mechanics of network topology and dynamics. After reviewing the em…

2001-06-06cond-mat.dis-nncond-mat.stat-mechcs.NI