Abstract
Abstract 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, and discuss their implications for the canonical tensor model.
Highlights
Quantization of gravity is one of the most challenging fundamental problems in physics, and various approaches to this problem have been proposed so far
We have numerically studied a matrix model with non-pairwise index contractions by Monte Carlo simulations
The matrix model has an intimate connection to the canonical tensor model, a tensor model for quantum gravity in the Hamilton formalism [23, 24], and has a similar structure as a matrix model that appears in the replica trick of the spherical p-spin model (p = 3) for spin glasses [28, 29]
Summary
Quantization of gravity is one of the most challenging fundamental problems in physics, and various approaches to this problem have been proposed so far. There was an issue which affects the reliability of the Monte Carlo simulations: For some values of the parameters important to study its properties, the iterative updates in the radial direction of the matrix variable were too slow to reach thermodynamic equilibriums in a reasonable amount of time. This appears to occur quickly as k/λ becomes smaller at k/λ O(10−8) in our simulations, but a quantitative investigation shows that this is a smooth change, implying that there is no transition to another phase with slow dynamics. The last section is devoted to a summary and future prospects
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