四重各向异性有限2DXY模型的重整化群分析：铁磁薄膜磁化率的应用
Renormalization group analysis of the finite 2DXY model with fourfold anisotropy: Application to the magnetic susceptibility of a ferromagnetic ultrathin film
摘要：
本文分两步求解了描述有限的四重各向异性2DXY系统的重整化群方程组，以研究非磁性薄膜的顺磁性转变。首先，方程在临界耦合$K=J/k_BT=2/π$附近线性化。线性化方程组的精确解证实，有限大小的Kosterlitz无（KT）过渡发生在弱四重各向异性的存在下，和伊辛过渡发生强各向异性。给定系统的行为由各向异性参数和系统尺寸的对数的乘积确定。线性化RG方程在有限尺寸转变的整个温度范围内是不定量可靠的，但它们定义了第二种方法的参数空间，其中四重各向异性被视为扰动，是有效的。这种微扰处理提供了一个定量测定的重整化交换耦合，涡旋密度，和各向异性在整个过渡。特别是，耦合具有一个普遍的拐点，其中涡旋-反涡旋对解束缚（与“普遍跳跃”相反），并且在顺磁状态下渐近地变为零，正如有限系统所预期的那样。这些结果被用来计算磁化率的系统从一个占主导地位的自旋波的一个占主导地位的自由涡旋气体。各向异性的存在使得有必要包括由于磁化强度的波动而引起的磁化率$χ_{||}$和由于磁化强度绕四重易轴的角度波动而引起的$χ_\perp$。与最近的测量结果的比较，铁磁/W（001）薄膜的磁化率是非常令人鼓舞的。
英文摘要：
The renormalization group equations describing a finite 2DXY system with fourfold anisotropy are solved in two steps, in order to study the magnetic transition to paramagnetism in an ultrathin film. First, the equations are linearized near the critical coupling $K=J/k_BT=2/π$. An exact solution of the linearized equations confirms that a finite-size Kosterlitz-Thouless (KT) transition occurs in the presence of weak fourfold anisotropy, and that an Ising transition occurs for strong anisotropy. The behaviour of a given system is determined by the product of an anisotropy parameter and the logarithm of the system size. The linearized RG equations are not quantitatively reliable across the full temperature range of the finite-size transition, but they define the parameter space where a second approach, where the fourfold anisotropy is treated as a perturbation, is valid. This perturbative treatment provides a quantitative determination of the renormalized exchange coupling, vortex density, and anisotropy throughout the transition. In particular, the coupling has a universal point of inflection where vortex-antivortex pairs unbind (as opposed to a "universal jump"), and goes to zero asymptotically in the paramagnetic state, as is expected for a finite system. These results are used to calculate the magnetic susceptibility as the system moves from one dominated by spin waves to one dominated by a free vortex gas. The presence of anisotropy makes it necessary to include both the susceptibility $χ_{||}$ due to fluctuations of the magnitude of the magnetization, and $χ_\perp$ due to angular fluctuations of the magnetization about a fourfold easy axis. A comparison to recent measurements of the magnetic susceptibility of ultrathin Fe/W(001) films is very encouraging.
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