Effective hamiltonians for anyons on graphs via self-adjoint extensions of the Landau operator

通过 Landau 算子的自伴扩展，图上任意子的有效哈密顿量

• 批准号: 2765041
• 金额: --
• 依托单位: University of Bristol
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2765041
• 关键词: Effective; hamiltonians; anyons; graphs; via

Anyons are (quasi)particles that obey so-called fractional quantum statistics - their statistical properties are neither bosonic nor fermionic. Such particles are known to exist in two- dimensional systems (2D lattices or thin metallic strips) and one-dimensional systems (quantum wires). Much less is known about the behaviour of anyons on networks formed from quantum wires, i.e. on quantum graphs. Formally, we consider a graph as a one-dimensional CW-complex. We form its configuration space, by considering the space of all un-ordered tuples of length n that consist of distinct point Sn is the permutation group. Graph braid group on n strands is defined as the fundamental group.Similarly, one can consider a configuration space of a topological space. Some important special cases areR3 - in three-dimensional space there are only bosons or fermions (sometimes in disguise). The braid group Brn(R3) is simply the permutation group, Sn. R2 - this setting leads to exotic statistics. The nontrivial topology of the configuration space of R2 supports anyons whose fractional statistics is realised in physical models as unitary representations of the planar braid group, Brn(R2). They appear in solid state physics in certain models of superconductors, in fault-tolerant quantum computing and in Chern-Simons theories.Leinaas and Myrheim (1977) show that the dynamics of anyons can be studied by inserting magnetic fluxes in the "holes" of the configuration space. The corresponding hamiltonian is then found via the minimal coupling principle which says that the momentum of kth particle is given by pk + Ak, where Ak is the local magnetic potential. The corresponding hamiltonian, pk2, is called the Landau operator. To solve the time-independent Schrödinger equation, we first need to find the correct gluing conditions for corresponding to situations where i) a particle is on a junction of the graph and ii) two particles come close to each other. The mathematical theory that tells us how to find such gluing conditions is the theory of self-adjoint extensions of symmetric operators. The aim is to look at specific families of graphs, starting with the simplest T-junction and then proceeding to general star graphs, the lasso graph, wheel graphs, etc. The project is open-ended.

任意子是服从所谓分数量子统计的（准）粒子——它们的统计性质既不是玻色子的，也不是费米子的。已知这种粒子存在于二维系统（二维晶格或薄金属条）和一维系统（量子线）中。对于由量子线构成的网络，也就是量子图上的任意子的行为，我们所知甚少。形式上，我们认为图是一维的cw复合体。我们考虑由不同的点Sn构成的所有长度为n的无序元组的空间为置换群，形成了它的位形空间。定义n条链上的图辫群为基群。类似地，我们可以考虑拓扑空间的构形空间。一些重要的特殊情况是——在三维空间中只有玻色子或费米子（有时是伪装的）。编织群Brn（R3）就是简单的排列群Sn。R2——这个设置导致了外来的统计数据。R2位形空间的非平凡拓扑支持其分数统计量在物理模型中被实现为平面编织群Brn（R2）的幺正表示的任意子。它们出现在固态物理、某些超导体模型、容错量子计算和陈-西蒙斯理论中。Leinaas和Myrheim（1977）表明，可以通过在位形空间的“空穴”中插入磁通量来研究任意子的动力学。相应的哈密顿量通过最小耦合原理得到，即第k个粒子的动量由pk + Ak给出，其中Ak是局部磁势。相应的哈密顿算符pk2称为朗道算符。为了求解与时间无关的Schrödinger方程，我们首先需要找到对应于i)一个粒子在图的交界处以及ii)两个粒子彼此靠近的情况的正确粘合条件。告诉我们如何找到这种胶合条件的数学理论是对称算子的自伴随扩展理论。目的是研究特定的图族，从最简单的t形结开始，然后继续到一般的星图、套索图、轮图等。这个项目是开放式的。