Study of Exactly Solvable Model Systems in Statistical Mechanics

拟议的研究的主要目标是开发新的数学和新的数值计算程序，以获得精确的表达相关函数的统计力学模型系统与程序，以评估他们的高精度。这是当前的兴趣，因为精确可解（或可积）的多体系统出现在物理和数学的许多领域。人们现在可以构建各种各样的实验可达的低维系统。从这些模型的精确理论研究中获得的见解是确定的。新的精确结果增强了对这些实验系统的理解。对于可积模型完全可解通过解决方案的杨？巴克斯特方程，人们通常可以找到自由能和序参量。各种所谓的伊辛型模型在杨？巴克斯特第一类现在有一个相当完整的了解对相关性，部分是由于最近的工作的PI。也有最近的进展，在计算相关函数的其他一些模型的杨？巴克斯特级。但这些结果仍然是相当不完整的，到目前为止所得到的表达式往往是笨重的。寻找更容易获得的结果是拟议研究的一个主要目标。许多潜在的应用依赖于对这些相关性的良好理解。在拟议的研究中，PI将特别关注手性Potts模型。它的可积子情形仍然是一个巨大的数学挑战。 更好地理解这个模型可能会导致物理和数学几个领域的理论突破。最近巴克斯特成功地证明了十六岁的猜想的序参数。将他的技术与PI开发的其他方法相结合，将研究剩余的许多具有挑战性的问题中的一些，例如自旋-自旋和能量-能量相关性。更广泛的影响：拟议的研究涉及凝聚态物理学、量子计算、弦理论和高能物理学、纽结理论、量子群以及各种其他数学领域。待研究的模型应导致更好地理解凝聚态物理学中研究的各种现象。 这些模型也与量子计算机理论中的纠缠研究、弦论中的Bethe-Anterior计算、结不变量、算子代数、特殊函数有关，并可能导致生物膜和DNA科学的新见解。一些调查结果可能会被纳入明天？的软件包和研究生和高级本科教材。这些结果可能会产生新的材料和设备。纳米技术中的某些设备、表面涂层的进步、高温超导性、细胞膜以及量子计算中的潜在应用只是许多例子中的几个。任何进展都可能产生新的重要影响。所要求的支持包括培训研究生的资金，一些资金让本科生参与兼职，旅行和咨询资金，以继续和扩大与美国和国际研究合作伙伴的内部和跨学科互动，以及更换过时的计算机和软件的资金。

The main objective of the proposed study is to develop new mathematics and new numerical procedures to obtain exact expressions for correlation functions in statistical mechanical model systems together with procedures to evaluate them to high precision. This is of current interest as exactly solvable (or integrable) many-body systems appear in many areas of physics and mathematics. One can now construct a wide variety of experimentally accessible low-dimensional systems. The insight obtained from exact theoretical studies of such models is definitive. New exact results enhance the understanding of these experimental systems. For integrable models exactly solvable through solutions of the Yang?Baxter equations, one can usually find the free energy and order parameters. For various so-called Ising-type models within the Yang?Baxter class one now has a rather complete understanding of the pair correlations, in part due to recent work of the PIs. There has also been recent progress in calculating correlation functions for some of the other models of the Yang?Baxter class. But these results are still rather incomplete and the expressions derived so far are often unwieldy. The search for more accessible results is a major goal of the proposed research. Many potential applications depend on a good understanding of these correlations. In the proposed research the PIs will pay special attention to the chiral Potts model. Its integrable subcase remains a great mathematical challenge. Better understanding of this model could lead to theoretical breakthroughs in several areas of physics and mathematics. Recently Baxter succeeded in giving a proof of the sixteen year old conjecture of the order parameter. Combining his techniques with other methods developed by the PIs, some of the many challenging problems remaining will be investigated, such as the spin-spin and energy-energy correlations.Broader Impact: The proposed research relates to condensed matter physics, quantum computing, string theory and high-energy physics, knot theory, quantum groups, and various other areas of mathematics. The models to be studied should lead to a better understanding of a variety of phenomena studied in condensed matter physics. The models are also relevant for the study of entanglement in the theory of quantum computers, Bethe-Ansatz calculations in string theory, knot invariants, operator algebras, special functions, and could lead to new insights in biomembranes and DNA science. Some of the findings may be incorporated in tomorrow?s software packages and graduate and advanced undergraduate textbooks. The results may enable new materials and devices. Certain devices in nanotechnology, advances in coating of surfaces, high-temperature superconductivity, cell membranes, and potential applications in quantum computing are only a few of the many examples. Any progress made could have new important implications. The requested support includes funds for training a graduate student, some funds to get an undergraduate student involved part-time, funds for travel and consulting in order to continue and expand the intra- and interdisciplinary interactions with U.S. and international research partners, and funds to replace outdated computers and software.