Institut für Theoretische Physik, Universität Frankfurt, Max-von-Laue Strasse 1, 60438 Frankfurt, Germany.
Phys Rev E. 2017 Oct;96(4-1):042107. doi: 10.1103/PhysRevE.96.042107. Epub 2017 Oct 6.
We develop a functional renormalization group (FRG) approach for the two-dimensional XY model by combining the lattice FRG proposed by Machado and Dupuis [Phys. Rev. E 82, 041128 (2010)PLEEE81539-375510.1103/PhysRevE.82.041128] with a duality transformation that explicitly introduces vortices via an integer-valued field. We show that the hierarchy of FRG flow equations for the infinite set of relevant and marginal couplings of the model can be reduced to the well-known Kosterlitz-Thouless renormalization group equations for the renormalized temperature and the vortex fugacity. Within our approach it is straightforward to include weak amplitude as well as out-of-plane fluctuations of the spins, which lead to additional interactions between the vortices that do not spoil the Berezinskii-Kosterlitz-Thouless transition. This demonstrates that previous failures to obtain a line of true fixed points within the FRG are a mathematical artifact of insufficient truncation schemes.
我们通过将 Machado 和 Dupuis 提出的格点 FRG（Phys. Rev. E 82, 041128 (2010)PLEEE81539-375510.1103/PhysRevE.82.041128）与对偶变换相结合，为二维 XY 模型开发了一种功能重整化群（FRG）方法，对偶变换通过一个整数值场显式引入涡旋。我们表明，模型中无限多相关和边缘耦合的 FRG 流方程的层次结构可以简化为众所周知的 Kosterlitz-Thouless 重整化群方程，用于重整化温度和涡旋逸度。在我们的方法中，很容易包括弱幅度和自旋的离面波动，这导致涡旋之间的额外相互作用，而不会破坏 Berezinskii-Kosterlitz-Thouless 转变。这表明，之前在 FRG 中未能获得真正的固定点线是截断方案不足的数学假象。
