Entanglement Entropy of Two-Dimensional Quantum States with Isometric Tensor Networks

Entanglement entropy is an important quantity in diagnosing properties of quantum phases of matter, encoding information about ground-state correlations, symmetries, criticality, localization, and topological order, among others [1]. In the absence of analytic solutions, numerical techniques for computing entanglement entropy, including quantum Monte Carlo (QMC) estimators [2] and neural-network quantum states (NQS) [3], are promising, yet their applicability for two-dimensional (2D) systems remains limited due to the QMC sign problem and NQS phase-learning and estimator-variance issues. Tensor networks are another major platform for computing entanglement entropy, but their use has been mainly confined to one-dimensional states due to the contraction cost of 2D projected entangled pair states. In this work, we make use of isometric tensor networks [4], natural extensions of canonical form in 2D, to efficiently compute subsystem Rényi-n entanglement entropy of 2D quantum many-body phases for large system sizes beyond the accessibility of exact diagonalization. We benchmark our platform across a range of interacting and noninteracting bosonic, spin, and fermionic systems by probing the ground state scaling, pinpointing quantum phase transitions, following entanglement growth under unitary dynamics, and characterizing topological order. While QMC or NQS methods are limited to measuring integer Rényi orders, our algorithm computes n < 1 cases that provide complementary constraints on the entanglement spectrum and expressive power of the tensor network ansatz. Our results enable accurate, scalable entanglement entropy evaluation in 2D quantum materials, providing a practical tool for probing low-temperature physics and nonequilibrium phenomena. 

[1] J. Eisert, M. Cramer, and M. B. Plenio. Rev. Mod. Phys. 82, 277 (2010). 

[2] M. B. Hastings, I. Gonzalez, A. B. Kallin, and R. G. Melko. Phys. Rev. Lett. 104, 157201 (2010).

[3] Z. Wang and E. J. Davis. Phys. Rev. A 102, 062413 (2020).

[4] M. P. Zaletel and F. Pollmann. Phys. Rev. Lett. 124, 037201 (2020).