Extracting conformal data from finite-size tensor-network flow in critical two-dimensional classical models

TL;DR

This paper introduces a tensor-network flow framework to extract conformal data from critical 2D classical models, accurately estimating key conformal parameters without prior detailed knowledge.

Contribution

It provides a universal, scheme-independent method to determine conformal data from transfer-matrix spectra in finite-size tensor-network analyses.

Findings

Robust universal behavior observed below the crossover scale.

Accurate extraction of conformal data up to high levels.

Operational definition of entanglement scaling for classical tensor networks.

Abstract

We present a general framework for extracting conformal data from critical two-dimensional classical lattice models using finite-size tensor-network flow. The central idea is to identify, from transfer-matrix spectra, a self-consistent finite-size window together with a crossover scale that separates the finite-size-scaling regime from the finite-entanglement-scaling regime induced by bond-dimension truncation. Within this window, the central charge, scaling dimensions, and conformal spins can be estimated without requiring a unique critical fixed-point tensor or detailed prior knowledge of the underlying conformal field theory. We benchmark the framework using three tensor-network renormalization schemes for the critical two-dimensional Ising and three-state clock models. Across schemes, we find robust universal behavior below the crossover scale, enabling accurate extraction of conformal data up to relatively high conformal levels. The analysis also yields a natural operational definition of entanglement scaling for classical tensor-network calculations and, in turn, a complementary estimator of the central charge.