Flow-Based Sampling for Entanglement Entropy and the Machine Learning of Defects

TL;DR

This work tackles the challenge of computing entanglement measures in lattice quantum field theories by estimating Rényi entropies via the replica trick. It introduces a defect-focused defect-coupling layer within Stochastic Normalizing Flows to sample partition-function ratios efficiently, with training grounded in KL minimization and supplemented by non-equilibrium MCMC concepts. Across $$(1+1)$$ and $$(2+1)$$ dimensional $\phi^4$ theories, the method outperforms traditional NE-MCMC baselines, demonstrates transfer learning across volumes and coupling values, and shows favorable scaling for larger lattices. The results establish flow-based sampling as a competitive, scalable approach for entanglement-related observables in lattice field theories, with potential applications to interfaces and topological sectors.

Abstract

We introduce a novel technique to numerically calculate Rényi entanglement entropies in lattice quantum field theory using generative models. We describe how flow-based approaches can be combined with the replica trick using a custom neural-network architecture around a lattice defect connecting two replicas. Numerical tests for the $φ^4$ scalar field theory in two and three dimensions demonstrate that our technique outperforms state-of-the-art Monte Carlo calculations, and exhibit a promising scaling with the defect size.

Figures (9)

• Figure 1: $(1+1)$-dimensional lattice with two replicas ($\tau$ is the Euclidean-time direction). Purple links connect different replicas; dashed lines separate $A$ and $B$. When defect coupling layers act on the configuration, the lattice is divided in three parts: the environment (black sites), which does not enter the coupling layer; frozen sites (empty cyan circles), that are the neural network input; active sites (orange diamonds), which are transformed by the layer.
• Figure 2: Top row: results in $1+1$ dimensions. Bottom row: results in $2+1$ dimensions. Left panels: quality of the sampling as the model, trained at the smaller value of the volume, is transferred to larger volumes. Central panel: transfer in the hopping parameter $\kappa$ of the model trained at $(\kappa_c, \lambda_c)$. Right panel: estimate of the critical behavior of $C_2$; in $1+1$ dimensions it is compared with the analytical solution Calabrese:2004eu. In all the plots, the quantities in the titles are fixed.
• Figure 3: Total numerical cost in wall-clock hours to compute the entropic c-function in $2+1$ dimensions for increasingly large volumes at fixed target accuracy. We observe that the larger the volume of the target system, the more evident the advantage. All models used for sampling were trained at the smallest volume, for fixed $l/a=1$ at the critical point.
• Figure 4: ESS of different models in $D=1+1$ during the training, with a comparison with the standard architecture. In the latter, differently from the defect block, the NF acts on the whole lattice and the coupling layer is made an even-odd masking and a mask on the replicas.
• Figure 5: Models trained for $l/a=1$ and transferred to other values of $l/a$ in $D=1+1$ (left panel) and $D=2+1$ (right panel). The ESS is approximately constant for all the values of the length of the cut.