# Optimization of infinite projected entangled pair states: The role of multiplets and their breaking

## TL;DR

This paper investigates the optimization challenges in infinite projected entangled pair states (iPEPS) for 2D quantum systems, highlighting how multiplet structures and symmetry breaking affect the accuracy of ground state approximations.

## Contribution

It reveals the impact of multiplet structures on tensor optimization in iPEPS and proposes understanding their role to improve ground state calculations.

## Key findings

- Multiplet structures cause symmetry breaking during tensor optimization.
- Choosing appropriate bond dimensions D is crucial to avoid artificial magnetization.
- Selective D values recover correct physical behavior in the Heisenberg model.

## Abstract

The infinite projected entangled pair states (iPEPS) technique [J. Jordan {\it et al.}, Phys. Rev. Lett. {\bf 101}, 250602 (2008)] has been widely used in the recent years to assess the properties of two-dimensional quantum systems, working directly in the thermodynamic limit. This formalism, which is based upon a tensor-network representation of the ground-state wave function, has several appealing features, e.g., encoding the so-called area law of entanglement entropy by construction; still, the method presents critical issues when dealing with the optimization of tensors, in order to find the best possible approximation to the exact ground state of a given Hamiltonian. Here, we discuss the obstacles that arise in the optimization by imaginary-time evolution within the so-called simple and full updates and connect them to the emergence of a sharp multiplet structure in the "virtual" indices of tensors. In this case, a generic choice of the bond dimension $D$ is not compatible with the multiplets and leads to a symmetry breaking (e.g., generating a finite magnetic order). In addition, varying the initial guess, different final states may be reached, with very large deviations in the magnetization value. In order to exemplify this behavior, we show the results of the $S=1/2$ Heisenberg model on an array of coupled ladders, for which a vanishing magnetization below the critical interladder coupling is recovered only for selected values of $D$, while a blind optimization with a generic $D$ gives rise to a finite magnetization down to the limit of decoupled ladders.