Energy Level Distribution of Perturbed Conformal Field Theories

TL;DR

The paper investigates energy level statistics of perturbed conformal minimal models in finite volume using a renormalization group–improved truncated conformal spectrum approach (TCSA), enabling access to thousands of low-lying levels. It demonstrates that integrable perturbations produce Poissonian level spacings while non-integrable perturbations exhibit GOE-like statistics, with crossovers between these regimes achievable by tuning the cylinder circumference $R$ and the truncation cutoff $E_c$. The study validates Berry-type expectations in a quantum field theory context and showcases a powerful numerical framework—combining TCSA with an RG-inspired sweeping scheme—to study spectral properties in perturbed CFTs. The results have implications for understanding quantum chaos, thermalization, and non-equilibrium dynamics in integrable and non-integrable quantum field theories.

Abstract

We study the energy level spacing of perturbed conformal minimal models in finite volume, considering perturbations of such models that are massive but not necessarily integrable. We compute their spectrum using a renormalization group improved truncated conformal spectrum approach. With this method we are able to study systems where more than 40000 states are kept and where we determine the energies of the lowest several thousand eigenstates with high accuracy. We find, as expected, that the level spacing statistics of integrable perturbed minimal models are Poissonian while the statistics of non-integrable perturbations are GOE-like. However by varying the system size (and so controlling the positioning of the theory between its IR and UV limits) one can induce crossovers between the two statistical distributions.

Figures (9)

• Figure 2.1: A set of lines with random slopes and intercepts. The spacing distribution is evaluated for a specific value of $l$, indicated by the red line in the figure.
• Figure 2.2: Spacing distribution for 100 000 random lines on a plane. The continuous line is the Poissonian distribution.
• Figure 2.3: Level spacing distribution for Gaussian correlated random variables with 100 000 samples taken. (a) r=0.1; (b) r=0.5; (c) r=0.9. For comparison we present the Poissonian distribution as a continuous line while the GOE is given by a dashed line.
• Figure 2.4: Level spacing distribution for non-Gaussian correlated random variables with 100 000 samples taken and r=0.5. Continuous line: Poissonian distribution; dashed line: GOE.
• Figure 4.1: Energy levels(multiplied by R) vs cylinder size. (a) Integrable case, showing several level crossings. (b) Non-integrable case, showing instead avoided crossings.