[Paper Review] Cluster update for tensor network states

Ling Wang, Frank Verstraete|arXiv (Cornell University)|Oct 19, 2011

Physics of Superconductivity and Magnetism21 citations

TL;DR

This paper introduces a cluster update method for tensor network states that improves the accuracy of infinite projected entangled pair state (iPEPS) simulations by evolving tensors in imaginary time on finite clusters, enabling better capture of long-range entanglement and critical phenomena. The method achieves high-accuracy results for the spin-1/2 dimerized antiferromagnetic Heisenberg model on the square lattice with bond dimension D=5, yielding critical point J′_c=2.501(1) and critical exponent β=0.37(1), outperforming the simple update scheme.

ABSTRACT

We propose a novel recursive way of updating the tensors in projected entangled pair states by evolving the tensor in imaginary time evolution on clusters of different sizes. This generalizes the so- called simple update method of Jiang et al. [Phys. Rev. Lett. 101, 090603 (2008)] and the updating schemes in the single layer picture of Pižorn et al. [Phys. Rev. A 83, 052321 (2011)]. A finite-size scaling of the observables as a function of the cluster size provides a remarkable improvement in the accuracy as compared to the simple update scheme. We benchmark our results on the hand of the spin 1/2 staggered dimerized antiferromagnetic model on the square lattice, and accurate results for the magnetization and the critical exponents are determined.

Motivation & Objective

• To address the limitations of the simple update method in capturing long-range entanglement and critical correlations in 2D quantum systems.
• To develop a scalable and efficient tensor network algorithm that retains the computational advantages of simple update while improving accuracy near quantum critical points.
• To enable accurate determination of critical exponents and phase transition points in strongly correlated systems using modest bond dimensions.
• To merge the efficiency of simple update with the improved environment representation of cluster-based methods, reducing computational overhead.

Proposed method

• The method generalizes the simple update by evolving tensors on clusters of size $l_x \times l_y$ instead of single sites, incorporating long-range correlations through larger clusters.
• It converts the infinite iPEPS into an open-boundary PEPS on the cluster using entanglement spectra $\Lambda$ from previous iterations to represent the environment.
• Imaginary time evolution is applied to central sites of the cluster, and the resulting wavefunction is contracted efficiently using a modified MPS-based method that avoids full Hilbert space growth.
• Updated tensors are obtained by projecting the evolved cluster state back to bond dimension $D$ using projectors $\mathbf{P}_A, \mathbf{P}_B$, replacing the original tensors in the iPEPS.
• The algorithm uses finite-size scaling with cluster size $l_x$ as a variational parameter, enabling extrapolation to the thermodynamic limit.
• Expectation values such as energy and magnetization are computed using Monte Carlo sampling on large periodic systems after tensor updates.

Experimental results

Research questions

• RQ1 Can a cluster-based update scheme improve the accuracy of iPEPS simulations near second-order quantum phase transitions compared to the simple update?
• RQ2 To what extent does increasing cluster size $l_x$ reduce errors in energy and magnetization for a given bond dimension $D$?
• RQ3 Can finite-size scaling with cluster size provide reliable extrapolation to the thermodynamic limit for critical exponents and critical points?
• RQ4 How does the cluster update method compare in accuracy and efficiency to the complete contraction algorithm and other iPEPS variants?

Key findings

• The cluster update method achieves significantly improved accuracy in energy and magnetization compared to the simple update, with energy errors reduced by orders of magnitude at $D=5$.
• For the dimerized antiferromagnetic Heisenberg model, the method yields a critical point $J'_c = 2.501(1)$, in close agreement with state-of-the-art SSE results ($J'_c = 2.5198(3)$).
• The critical exponent $\beta = 0.37(1)$ is extracted with high precision, matching the SSE result $\beta = 0.376(5)$ within error bars.
• Finite-size scaling with cluster size $l_x$ enables reliable extrapolation of sublattice magnetization, showing excellent agreement with SSE simulations.
• The method scales as $D^5$ for $2\times2$ clusters and $D^7$ for $4\times4$ clusters, remaining efficient even for moderate $D$, and avoids the complexity of full environment contraction.
• The approach successfully captures the critical behavior of the model with $D=5$, demonstrating that intermediate bond dimensions can yield high-accuracy results when combined with cluster-based updates.