The Spectrum of Light States in Large N Minimal Models

TL;DR

The work analyzes the spectrum of light states in the $W_{N,k}$ coset minimal models in the large-$N$ 't Hooft limit, focusing on their distribution and thermodynamic implications. By combining numerical sampling, analytic CFT insights in the $\lambda\approx 0$ and $\lambda\approx 1$ limits, and a free-fermion mapping, the authors establish a Gaussian-like distribution of light-state conformal weights with a peak at $h_{\rm peak}=\frac{N}{24}(1-\lambda^2)$ and a maximal scale $h_{\rm max}=\frac{N}{8}(1-\lambda)$. They show that the canonical partition function exhibits regime-dependent behaviour $\log Z_N\sim \tfrac{N}{2}\log(T/\lambda^2)$ for $\lambda\to 0$ and $\log Z_N\sim k\log(T/(1-\lambda))$ for $\lambda\to 1$, with no sign of a finite-temperature phase transition in the light sector. The analysis also clarifies the role of non-light primaries and descendants, arguing that their growth can dominate at large conformal weights and guiding implications for the dual higher-spin holographic picture. Overall, the results support the absence of a Hawking-Page-like transition at $T=O(1)$ and provide a detailed landscape of how the light and non-light parts of the spectrum contribute to the CFT thermodynamics.

Abstract

$W_{N,k}$ minimal models possess an interesting class of `light' primaries which control much of the low energy density of states in the large $N$ 't Hooft limit. In this paper we conduct a detailed exploration of their distribution using a combination of numerical and analytical techniques. We also make some observations about the density of states of the full CFT. Our results appear to support the contention that there is no finite temperature analogue of the Hawking-Page transition in these systems.

Figures (8)

• Figure 1: Numerical result (red circles) versus analytic fit \ref{['lhmax']} (blue line) for the maximum conformal dimension (normalized by $N$ to facilitate comparison) attained for given $N$ and $\lambda$. We obtained the full spectrum of light states for discrete values of $N \in \{2,3,\cdots,50 \}$ and $k \in \{1,2,\cdots, 200 \}$, restricting to situations where the total number of light states is less than $10^7$. We also note that the representation attaining $h_\text{max}$ agrees excellently with \ref{['lmaxOm']}.
• Figure 2: Distribution of the conformal dimensions for light states. The left panel shows the histogram of the distribution and the right panel shows a fit to a smooth probability distribution (see text for details). Color coding: $N=8,k=24$ (blue), $N=k=13$ (red), and $N=24,k=8$ (purple).
• Figure 3: Pictorial view of a random set of 10 Young diagrams representing states near the peak of the randomly generated distributions for $N=k=500$. The diagrams are mostly triangular in this asymptotic limit. Note that we have sampled $10^6$ random Young diagrams, which is a very small fraction of the total number of states (which is $\approx 10^{299}$) but apparently suffices to see the typical diagram.
• Figure 4: The distribution of primary scaling dimensions for the full spectrum obtained for different values of $N$ and $k$. The dots represent the bin counts of the histogram data and the solid curve is a fit to the Gaussian distribution. We present the same choices of $N$ and $k$ as in Fig. \ref{['f:lightdistributions']}.
• Figure 5: Distribution of light states in the regimes $\lambda \approx 0$ (left) and $\lambda \approx 1$ (right). We draw attention to two facts: (i) the number of states with dimension $h \ll 1$ grows as a power law ${\cal D}(h) \sim h^\alpha$ with $\alpha_{\lambda \approx 0} = \frac{N-3}{2}$ and $\alpha_{\lambda\approx 1} = k-1$ and (ii) the level-rank duality is clearly visible with $N - 1 \leftrightarrow k$ (see §\ref{['sec:lam1']}).