Data and scripts from: Morphology of renormalization-group flow for the de Almeida–Thouless–Gardner universality class

A replica-symmetry-breaking phase transition is predicted in a host of disordered media. The criticality of the transition has, however, long been questioned below its upper critical dimension, six, due to the absence of a critical fixed point in the renormalization-group flows at one-loop order. A recent two-loop analysis revealed a possible strong-coupling fixed point, but given the uncontrolled nature of perturbative analysis in the strong- coupling regime, debate persists. Here we examine the nature of the transition as a function of spatial dimension and show that the strong-coupling fixed point can go through a Hopf bifurcation, resulting in a critical limit cycle and a concomitant discrete scale invariance. We further investigate a different renormalization scheme and argue that the basin of attraction of the limit cycle at the strong-coupling fixed point may stay finite for all dimensions.

复本对称破缺相变（replica-symmetry-breaking phase transition）在诸多无序介质中被预言存在。然而，由于单圈阶重整化群流（renormalization-group flows）中不存在临界不动点，该相变在上临界维度（upper critical dimension）6以下的临界性长期受到质疑。近期的两圈分析（two-loop analysis）揭示了可能存在强耦合不动点（strong-coupling fixed point），但鉴于强耦合区域内微扰分析（perturbative analysis）的不可控性，相关争论仍在持续。本文考察该相变随空间维度变化的本质，证明强耦合不动点可经历霍普夫分岔（Hopf bifurcation），进而产生临界极限环（critical limit cycle）与伴随的离散标度不变性（discrete scale invariance）。我们进一步研究了不同的重整化方案（renormalization scheme），并提出：强耦合不动点处的极限环吸引域（basin of attraction）在所有空间维度中或许均保持有限。