Ostilli M, Rocha G W C, Bezerra C G, Viswanathan G M
Institute of Physics, Federal University of Bahia, Salvador, Bahia, 40210-340, Brazil.
Physics Department, Federal University of Rio Grande do Norte, Natal, Rio Grande do Norte, 59072-970, Brazil.
Sci Rep. 2024 Sep 27;14(1):22375. doi: 10.1038/s41598-024-71809-1.
The generating function for polygons on the square lattice has been known for many decades and is closely related to the path integral formulation of a free fermion model. On the cubic and hypercubic lattices the generating function is still unknown and the problem remains open. It has been conjectured that the three-dimensional (3D) and higher dimensional problems are not solvable-or, to be more precise, that there are no differentiably finite (D-finite) solutions. In this context, very recently a Berezin integral of an exponentiated Grassmann action was found for the polygon generating function on the cubic lattice, making explicit the connection between 3D polygons and a model of interacting fermions. Here we address the problem of how to generalize the 3D result to higher dimensions. We derive a Grassmann representation in terms of a Berezin integral for the generating function of polygons on d-dimensional hypercubic lattices. On the one hand, this new result admittedly brings us no closer to the problem of finding an explicit analytic expression for the desired generating function for polygons. On the other hand, however, the significant advance reported here precisely quantifies the remarkable mathematical difficulty of finding the explicit generating function. Indeed, the non-quadratic functional form of the Grassmann action that we derive here provides a very clear picture of the formidable mathematical obstruction that would need to be overcome. Specifically, in d dimensions, the Grassmann action contains terms of degree , so the model describes interacting rather than free fermions. It is an open problem whether or not these models of interacting fermions can in principle be free fermionized through some still undiscovered algebraic method, but it is widely believed that this goal is mathematically impossible.
正方形晶格上多边形的生成函数已经被知晓数十年了，并且与自由费米子模型的路径积分表述密切相关。在立方晶格和超立方晶格上，生成函数仍然未知，问题仍然悬而未决。据推测，三维（3D）及更高维的问题是不可解的——或者更确切地说，不存在可微有限（D - 有限）解。在这种背景下，最近发现了立方晶格上多边形生成函数的指数化格拉斯曼作用量的贝雷辛积分，明确了三维多边形与相互作用费米子模型之间的联系。在此，我们探讨如何将三维结果推广到更高维度的问题。我们针对d维超立方晶格上多边形的生成函数，根据贝雷辛积分推导出一种格拉斯曼表示。一方面，诚然这个新结果并没有让我们更接近找到多边形所需生成函数的显式解析表达式的问题。然而，另一方面，这里所报道的重大进展恰恰量化了找到显式生成函数的显著数学难度。实际上，我们在此推导出的格拉斯曼作用量的非二次泛函形式非常清晰地展现了需要克服的巨大数学障碍。具体而言，在d维中，格拉斯曼作用量包含次数为的项，所以该模型描述的是相互作用的费米子而非自由费米子。这些相互作用费米子模型原则上是否能通过某种尚未被发现的代数方法转化为自由费米子模型是一个未解决的问题，但人们普遍认为这个目标在数学上是不可能的。
