具有非成对指数压缩的矩阵模型的相位
Phases of a matrix model with non-pairwise index contractions
摘要：
最近在正则张量模型的背景下研究了具有非成对指数压缩的矩阵模型，正则张量模型是正则形式的量子引力的张量模型。当复制技巧应用于自旋玻璃理论中的球形$p$ -自旋模型（$p=3$）时，该矩阵模型也以相同的形式出现，具有不同的参数和变量范围。先前对该矩阵模型的研究表明，在$R\sim N^2/2$周围存在连续相变，其中$N$和$R$表示其矩阵大小$N\times R$。$N$和$R$之间的这种关系与$N$的首阶张量模型的一致性条件非常吻合，这表明张量模型位于连续相变点附近或之上，因此其连续极限自动取为$N\rightarrow \infty$极限。然而，在以前的工作中，由于蒙特卡罗模拟的速度减慢，相变的证据并不令人满意。在这项工作中，我们提供了一个新的设置为蒙特卡罗模拟通过整合出矩阵的径向方向。这种新的策略大大提高了效率，并使我们能够清楚地显示存在的相变。我们还提出了各种特征的阶段，如动态生成的配置，级联对称性破缺的尺寸，和一个参数零限制，讨论一些影响的正则张量模型。
英文摘要：
Recently a matrix model with non-pairwise index contractions has been studied in the context of the canonical tensor model, a tensor model for quantum gravity in the canonical formalism. This matrix model also appears in the same form with different ranges of parameters and variables, when the replica trick is applied to the spherical $p$-spin model ($p=3$) in spin glass theory. Previous studies of this matrix model suggested the presence of a continuous phase transition around $R\sim N^2/2$, where $N$ and $R$ designate its matrix size $N\times R$. This relation between $N$ and $R$ intriguingly agrees with a consistency condition of the tensor model in the leading order of $N$, suggesting that the tensor model is located near or on the continuous phase transition point and therefore its continuum limit is automatically taken in the $N\rightarrow \infty$ limit. In the previous work, however, the evidence for the phase transition was not satisfactory due to the slowdown of the Monte Carlo simulations. In this work, we provide a new setup for Monte Carlo simulations by integrating out the radial direction of the matrix. This new strategy considerably improves the efficiency, and allows us to clearly show the existence of the phase transition. We also present various characteristics of the phases, such as dynamically generated dimensions of configurations, cascade symmetry breaking, and a parameter zero limit, to discuss some implications to the canonical tensor model.
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