Mathematical Sciences: Quantum Mechanical Classical Lattice Spin Systems

在这个项目中，主要的研究人员将研究自旋-1海森堡反铁磁一维量子自旋系统中Haldane相和二聚化相之间的相变。特别是，他将通过证明为这种相变提出的序参数在一个相中为零，而在另一个相中为非零，从而证明这种相变确实存在。此外，首席研究员还将继续他对伊辛型系统的多数规则重整化群变换的严格研究。最后，他将研究量子自旋系统哈密顿量无限体积极限的连续谱的存在性。物质在微观尺度上的行为往往决定了物质在日常生活的宏观或可观察尺度上的行为。例如，我们所说的液体或气体的温度，仅仅是无数原子和分子在物质微观尺度上运动的宏观或累积效应。在这个项目中，首席研究员将在微观或量子水平上研究原子行为的各种数学模型，以便在物质的微观和宏观性质之间建立联系。

In this project the principal investigator will study the phase transition between the Haldane phase and the dimerized phase in the spin-1 Heisenberg antiferromagnetic, one-dimensional quantum spin system. In particular, he will prove that this phase transition does indeed exist by showing that an order- parameter proposed for this transition is zero in one phase and nonzero in the other. In addition, the principal investigator will continue his rigorous study of the majority rule renormalization group transformation for Ising-type systems. Finally he will study the existence of continuous spectra for the infinite volume limit of the Hamiltonian of a quantum spin system. The behavior of matter at the microscopic scale is very often what determines the behavior of matter at the macroscopic or observable scales of ordinary life. For example, what we call the temperature of a liquid or gas is simply the macroscopic or cumulative effect of the motion of countless atoms and molecules on the microscopic scale of the material. In this project the principal investigator will study various mathematical models of the behavior of atoms at the microscopic or quantum level, in order to establish connections between microscopic and macroscopic properties of matter.