One-dimensional systems---ranging from travelling light to circuit cables and from DNA to superstrings---are ubiquitous and critically important to the human knowledge of the universe. However, our engagement with one-dimensional systems in the research and education of spontaneous phase transitions, the phenomena wherein materials can change rapidly between different phases (e.g., gas, liquid, solid, etc.) on their own, has not been largely exercised, since it was proven that one-dimensional systems do not contain phase transitions in the textbook Ising model almost 100 years ago [1] and its quantum counterpart, the Heisenberg model, over 50 years ago [2]. Recently, a spontaneous marginal phase transition (MPT) was discovered in a one-dimensional Ising model containing strong geometrical frustration [3]. Here, by exploring the symmetry of the new mathematical structure underlying the MPT, I report the finding and classification of an infinite number of MPT cases---with highly tunable intriguing behaviors like phase reentrance, the dome shape of transition temperature, pairing, and gauge freedom. These discoveries reveal the possibility of building the MPT-based one-dimensional Ising Machine that can be used to simulate the complex phenomena of phase competition in strongly correlated systems and provide insights with its unambiguous exact solutions. They also form a rich playground for exploring unconventional phase transitions in one-dimensional Heisenberg models.
我们计算与TAP自由能的最小值和指数一鞍点相关的复杂性（TAP状态数的对数）。高指数鞍具有较小的复杂性。两个前导复杂度相等，与关于转向点总数的Morse定理一致，并且具有[A]中给出的值。J. Bray和M. A. Moore，J.Phys.C13，L469（1980）]。在热力学极限下，所有自由能的TAP状态变得边缘稳定。
Surface tension and interfacial fluctuations in d-dimensional Ising model
根据已有的结果和本文的一些原始计算，讨论了二维和三维伊辛模型中共存相间粗糙界面的表面张力。结果总结在一个公式中，它允许插值校正有限尺寸标度之间的二维和三维。讨论了空间维数d的非整数解析延拓的物理意义。具有适当定义的分形维数的晶格和界面应该满足某些要求，才可能具有d维超立方体解析延拓的性质。在这里，2表示d的边际值，低于这个值，（d-1）维界面就会分裂成不连续的部分。一些唯象的论点提出来描述这样的接口。结果表明，d = 2时界面涨落的性质<2 is not the same as provided by a formal analytic continuation from d-dimensional hypercubes with d >与实验值吻合较好。对于相关的临界指数，这可能也是正确的。
Critical behavior of the two-dimensional N-component Landau-Ginzburg Hamiltonian with cubic anisotropy
本文研究了具有立方各向异性的二维N分量Landau-Ginzburg哈密顿量。计算并分析了重整化群函数到四个环的定维微扰展开式。讨论了这些模型与N色Ashkin-Teller模型、离散立方模型、四阶各向异性平面模型以及吸附单分子膜结构相变的关系。对于N=2（XY模型，具有立方各向异性），我们的结果与存在一条连接Ising和O（2）不动点的不动点线是相容的.沿着这条线，指数$η$具有常数值1/4，而指数$ν$以连续和单调的方式从1到$\infty$（从Ising到O（2））。对于\geq 3，我们发现一个三次不动点的区域$u, v \geq 0$，这是边际稳定或不稳定，根据扰动的符号。对于N=3的物理相关的情况下，我们发现指数$η=0.17(8)$和$ν=1.3(3)$在立方过渡。
