论文标题
d = 3的O（n）模型的边界关键性重新审视
Boundary criticality of the O(N) model in d = 3 critically revisited
论文作者
论文摘要
众所周知，尺寸$ d> 3 $的经典$ o（n）$模型在其批量关键点上允许三个边界普遍性类别：普通，非凡和特殊。对于普通过渡，同时散装和边界顺序；外部固定点对应于在有序边界存在下发生的整体转变，而特殊的固定点对应于普通和异常类之间的边界相变。虽然普通的固定点在$ d = 3 $中得以生存，但当$ d = 3 $和$ n \ ge 2 $时，尚不清楚外在和特殊固定点会发生什么。在这里，我们证明将$ n $正式视为连续参数，存在一个临界值$ n_c> 2 $分开两个不同的制度。对于$ n <n_c $，超凡的固定点在$ d = 3 $中幸存下来，尽管以修改形式：远程边界顺序丢失了，而是订单参数相关函数作为$ \ log r $的功率衰减。特别是，对于$ n = 2 $，以准长范围的顺序从表面相开始，并接近散装相变，表面顺序参数的刚度与对数分化。对于$ n> n_c $，没有固定点，订单参数相关的衰减比电力法慢；我们讨论了过去$ n = n_c $的相图演变的两个方案。我们的发现似乎与最近对$ n = 2 $和$ n = 3 $的古典模型的蒙特卡洛研究一致。我们还将结果与2+1D量子自旋模型中边界关键性的数值研究进行了比较。
It is known that the classical $O(N)$ model in dimension $d > 3$ at its bulk critical point admits three boundary universality classes: the ordinary, the extra-ordinary and the special. For the ordinary transition the bulk and the boundary order simultaneously; the extra-ordinary fixed point corresponds to the bulk transition occurring in the presence of an ordered boundary, while the special fixed point corresponds to a boundary phase transition between the ordinary and the extra-ordinary classes. While the ordinary fixed point survives in $d = 3$, it is less clear what happens to the extra-ordinary and special fixed points when $d = 3$ and $N \ge 2$. Here we show that formally treating $N$ as a continuous parameter, there exists a critical value $N_c > 2$ separating two distinct regimes. For $N < N_c$ the extra-ordinary fixed point survives in $d = 3$, albeit in a modified form: the long-range boundary order is lost, instead, the order parameter correlation function decays as a power of $\log r$. In particular, for $N=2$, starting in the surface phase with quasi-long-range order and approaching the bulk phase transition, the stiffness of the surface order parameter diverges logarithmically. For $N > N_c$ there is no fixed point with order parameter correlations decaying slower than power law; we discuss two scenarios for the evolution of the phase diagram past $N = N_c$. Our findings appear to be consistent with recent Monte-Carlo studies of classical models with $N = 2$ and $N = 3$. We also compare our results to numerical studies of boundary criticality in 2+1D quantum spin models.
