AI 中文总结
该研究重构二维$ϕ^4$格点场论的2PI顶角，追踪其伊辛相变演化，揭示软 sector 多维性，得到近似局域接触的完全不可约顶角，为DΓA提供基准并发现拼花方程的排斥不动点行为。
AI 中文摘要
我们通过对二维单分量$ϕ^4$格点场论的连通双粒子关联函数进行蒙特卡洛测量，重构了双粒子不可约（2PI）顶角$\u0393(k,p;q)$，并追踪其在伊辛相变过程中的演化。我们在点群$C_{4v}$的不可约表示下解析该顶角，发现不稳定性由零动量转移下的$A_1$（铁磁）通道驱动，该通道的对称化Bethe–Salpeter核的主导本征值趋近于1。在所有系统尺寸下，显著的$B_1$（向列）和$B_2$（对角向列）贡献与$A_1$协同作用，凸显软 sector 具有多维性。实空间中，远离临界点时顶角呈短程特性，而在临界点处会发展出幂律尾。有序相中，$q=0$本征值坍缩，因为铁磁权重已凝聚为（单粒子可约的）序参量（或有限系统的集体坐标），尽管有限动量涨落仍然存在。通过剥离交叉通道 ladder，我们得到完全不可约的顶角，其近似为局域接触项。将该接触项插入拼花（parquet）方程和施温格-戴森（Schwinger–Dyson）方程中，其重现蒙特卡洛自能的精度优于千分之一，这为动力学局域顶角近似（DΓA）提供了第一性原理基准。此外，我们证明在临界区域，拼花方程的物理解表现为排斥不动点，初始由单一序参量模式驱动。
英文摘要
We reconstruct the 2PI vertex $Γ(k,p;q)$ from Monte Carlo measurements of the connected two-particle correlator for the two-dimensional single-component $ϕ^4$ lattice field theory and follow it across the Ising transition. Resolving the vertex in the irreducible representations of the point group $C_{4v}$, we find that the instability is driven by the $A_1$ (ferromagnetic) channel at zero transfer, whose leading eigenvalue of the symmetrized Bethe--Salpeter kernel approaches unity. Substantial $B_1$ (nematic) and $B_2$ (diagonal nematic) contributions cooperate with $A_1$ across all system sizes, highlighting that the soft sector is multidimensional. In real space, the vertex is short-ranged away from criticality while it develops a power-law tail at the critical point. In the ordered phase, the $q=0$ eigenvalue collapses because the ferromagnetic weight has condensed into the (one-particle-reducible) order parameter (or collective coordinate for a finite system), although finite-momentum fluctuations persist. By stripping the crossed-channel ladders, we obtain the fully irreducible vertex, which is a local contact -- to a very good approximation. Inserted into the parquet and Schwinger--Dyson equations, this contact reproduces the Monte Carlo self-energy with an accuracy better than one-tenth of a percent. This provides a first-principles benchmark of the dynamical local-vertex approximation (D$Γ$A). Additionally, we demonstrate that in the critical region, the physical solution of the parquet equations behaves as a repulsive fixed point, driven initially by a single order-parameter mode.
Comments13 pages, 9 figures
