Phases of a matrix model with non-pairwise index contractions

Abstract

<jats:title>Abstract</jats:title> <jats:p>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.</jats:p>

Counting Tensor Rank Decompositions

“…Contrary to the R = 1 case above, one might expect (10) does not always converge. The matrix model analysed in [14][15][16], corresponding to a choice of parameters of K = 3 and w = N K , did not converge in general. It had a critical value around R c ∼ 1 2 (N + 1)(N + 2), above which the → 0 + limit did not appear to converge anymore.…”

“…There, we derive a closed formula for Z R (∆), which is guaranteed to exist under the condition that a certain quantity G R , which is independent of ∆, exists and is finite. Another interesting connection to the CTM is found at this point, since this quantity G R is a generalisation of the partition function of the matrix model studied in [14][15][16]. In Section 4, the existence of G R is proven for R = 1, and numerical analysis is conducted for R > 1 for a specific choice of volume form dΦ to arrive at a conjecture for the maximal allowed value of R, called R c .…”

“…In Refs. [14][15][16], a matrix model was analysed that corresponds to a simplified wave function of the canonical tensor model. The matrix model under consideration had a partition function given by…”

“…Here, in the first step, we rescaled λ i → |Q|λ i , in the second step we introduced a new integration variable x ≡ γ|Q| 2 , and in the final step we took the limit inside the integral as is proven to be allowed in the appendix Lemma A2. Note the appearance ofZ R (γ; ) as defined in (16).…”

“…A noteworthy fact about the CTM is that it has several known exact solutions for the quantum constraint equations [11]. One of these has recently been extensively analysed due to the emergence of Lie group symmetries in this wave function, which potentially hints towards the emergence of macroscopic spacetimes [12][13][14][15][16][17]. This wave function, in the Q-representation, is closely related to a statistical model [17] that is mathematically equivalent to…”