经典二维海森堡模型的再认识：$SU(2)$对称张量网络研究
The classical two-dimensional Heisenberg model revisited: An $SU(2)$-symmetric tensor network study
摘要：
二维空间的经典海森堡模型是最典型的自旋模型之一，在统计物理和凝聚态物理中起着重要的作用。然而，尽管它的典型特征和广泛接受的禁止（连续）自发对称性破缺，争议仍然存在，该模型是否表现出在有限温度下的相变。重要的是，这个模型可以被解释为$O(3)$非线性σ模型在$1+1$维中的晶格离散化，这是最简单的量子场论之一，包含了著名的高维场论（如$3+1$维中的量子色动力学）的关键特征，即渐近自由现象。这也应该排除有限温度转变，但晶格效应可能在纠正主流图像中发挥重要作用。在这项工作中，我们利用国家的最先进的张量网络的方法，代表在热力学极限在很大的温度范围内的经典配分函数，全面探索吉布斯状态的相关结构。通过在我们的二维张量网络收缩方案中实现$SU(2)$对称性，我们能够处理高达$χ_E^\text{eff} \sim 1500$的环境的非常大的有效键尺寸，这是检测相变的关键特征。随着温度的降低，我们发现一个迅速发散的相关长度，其行为显然是兼容的两个主要矛盾的假设，在文献中已知的，即有限的$T$过渡和渐近自由，虽然有轻微的偏好第二。
英文摘要：
The classical Heisenberg model in two spatial dimensions constitutes one of the most paradigmatic spin models, taking an important role in statistical and condensed matter physics to understand magnetism. Still, despite its paradigmatic character and the widely accepted ban of a (continuous) spontaneous symmetry breaking, controversies remain whether the model exhibits a phase transition at finite temperature. Importantly, the model can be interpreted as a lattice discretization of the $O(3)$ non-linear sigma model in $1+1$ dimensions, one of the simplest quantum field theories encompassing crucial features of celebrated higher-dimensional ones (like quantum chromodynamics in $3+1$ dimensions), namely the phenomenon of asymptotic freedom. This should also exclude finite-temperature transitions, but lattice effects might play a significant role in correcting the mainstream picture. In this work, we make use of state-of-the-art tensor network approaches, representing the classical partition function in the thermodynamic limit over a large range of temperatures, to comprehensively explore the correlation structure for Gibbs states. By implementing an $SU(2)$ symmetry in our two-dimensional tensor network contraction scheme, we are able to handle very large effective bond dimensions of the environment up to $χ_E^\text{eff} \sim 1500$, a feature that is crucial in detecting phase transitions. With decreasing temperatures, we find a rapidly diverging correlation length, whose behaviour is apparently compatible with the two main contradictory hypotheses known in the literature, namely a finite-$T$ transition and asymptotic freedom, though with a slight preference for the second.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
