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

任意子是遵循所谓的分数量子统计的(准)粒子--它们的统计性质既不是玻色子也不是费米子。众所周知，这种粒子存在于二维系统(2D晶格或薄金属条)和一维系统(量子线)中。关于任意子在由量子线形成的网络上的行为，即在量子图上的行为，我们知道的要少得多。形式上，我们将图视为一维CW-复形。我们通过考虑所有长度为n的无序元组由不同的点组成的空间来形成它的配置空间，S_n是置换群。N链上的图辫子群被定义为基本群，类似地，人们可以考虑拓扑空间的配置空间。一些重要的特例是R3--在三维空间中只有玻色子或费米子(有时是伪装的)。辫子群Brn(R3)简单地说是置换群Sn.R2-此设置会产生奇异的统计数据。R2的组态空间的非平凡拓扑支持其分数统计在物理模型中实现为平面辫群BRN(R2)的么正表示的任意子。它们出现在固体物理中的某些超导体模型中，出现在容错量子计算中，也出现在Chern-Simons理论中。Leinaas和Myheim(1977)表明，可以通过在组态空间的“空穴”中插入磁通量来研究任意子的动力学。然后通过最小耦合原理求出相应的哈密顿量，即第k粒子的动量由Pk+Ak给出，其中Ak是局域磁势。相应的哈密顿量pk2称为朗道算符。为了求解与时间无关的薛定谔方程，我们首先需要找到正确的粘合条件，以对应于i)粒子在图形的交界处和ii)两个粒子彼此接近的情况。告诉我们如何找到这种粘合条件的数学理论是对称算子的自伴扩张理论。目的是研究特定的图族，从最简单的T形连接开始，然后再到一般的星图、套索图、轮图等。该项目是开放式的。

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.