Boundary Effects in Critical Phenomena

该研究项目的目标是促进对边界效应及其基本数学结构的理解。该项目涉及来自统计物理学的三个模型。首先，两种群完全不对称排斥过程是两个耦合Burgers方程组的离散形式。PI建议确定该系统的一个新的子前导指数，并研究存在边界项时相图的性质。第二个目标是建立边界共形圈模型中的边界项与带边界算符的XX-和XXZ-量子自旋链之间的对应关系。第三个主题是一个马尔可夫链描述的增长和波动的界面存在的墙壁。PI将确定边界雪崩的临界指数，并研究与离散Boussinesq方程的联系。PI将采用从贝特-安纳托利到共形场论、有限尺寸标度和蒙特-卡罗模拟的各种方法。上述项目将提供重要的见解临界现象的理论存在的边界条件。该项目的重点是三个重要领域，即统计模型的相图中的边界的作用，在边界共形场论中保持共形不变性的所有边界项的分类和边界对马尔可夫过程的影响。还有证据表明，该项目的结果将对量子计算理论产生影响。更广泛的影响。PI在数学和物理学方面联合任命，正在通过与两个部门的同事讨论研究项目来促进这两个部门之间的跨学科对话。她还负责指导一名研究生与化学系进行联合研究项目。该项目扩大了妇女在科学领域的代表性，PI积极充当数学和物理领域妇女的榜样，并参与了“科学界妇女”和“物理界妇女”倡议。她通过普渡大学的科学K-12外展计划参加科学博览会。她被邀请的数学系教跨学科课程作为远程学习班。

The goal of this research project is to advance the understanding of boundary effects and their underlying mathematical structures. The project deals with three models stemming from statistical physics. First, the two-species totally asymmetric exclusion process, is a discrete version of a system of two coupled Burgers equations. The PI proposes to determine a conjectured new sub-leading exponent of this system and to investigate the properties of the phase diagram in the presence of boundary terms. The second objective is the establishment of a correspondence between boundary terms in the boundary conformal loop model and the XX- and XXZ- quantum spin chains with boundary operators. The third subject is a Markov chain describing a growing and fluctuating interface in the presence of a wall. The PI will determine the critical exponent of boundary avalanches and investigate the connection to the discrete Boussinesq equation. The PI will employ methods ranging from Bethe-Ansatz to conformal field theory, finite-size scaling and Monte-Carlo simulations. The above projects will provide important insights into the theory of critical phenomena in the presence of boundary terms. The project is focused on three important areas, namely the role of boundaries in phase diagrams of statistical models, the classification of all boundary terms preserving conformal invariance in boundary conformal field theories and the influence of boundaries on Markov processes. There is also evidence that the outcome of the project will have implications for the theory of quantum computing. Broader Impact. The PI, holding a joint appointment in mathematics and physics, is fostering the interdisciplinary dialogue between these two departments by discussing research projects with colleagues in both departments. She is also supervising a graduate student working on a joint research project with the chemistry department. The project broadens the representation of women in the sciences with the PI actively serving as a role model for women in mathematics and physics, and her involvement in "Women in Science" and "Women in Physics" initiatives. She participates in science fairs through the the Science K-12 Outreach Program at Purdue University. She was invited by the mathematics department to teach an interdisciplinary course as a distance learning class.