# Infinite Matrix Product States vs Infinite Projected Entangled-Pair States on the Cylinder: a comparative study

## TL;DR

This study compares the effectiveness of infinite matrix product states and infinite projected entangled-pair states in modeling two-dimensional lattice systems on cylindrical geometries, revealing specific crossover points where one method outperforms the other.

## Contribution

It provides a systematic comparison of these two tensor network methods on cylindrical geometries using benchmark models, highlighting their relative strengths and crossover points.

## Key findings

- Projected entangled-pair states outperform matrix product states at larger widths for the Heisenberg model.
- Matrix product states are more effective for narrower cylinders in the Hubbard model.
- Crossover widths are approximately 11 sites for Heisenberg and 7 for Hubbard models.

## Abstract

In spite of their intrinsic one-dimensional nature matrix product states have been systematically used to obtain remarkably accurate results for two-dimensional systems. Motivated by basic entropic arguments favoring projected entangled-pair states as the method of choice, we assess the relative performance of infinite matrix product states and infinite projected entangled-pair states on cylindrical geometries. By considering the Heisenberg and half-filled Hubbard models on the square lattice as our benchmark cases, we evaluate their variational energies as a function of both bond dimension as well as cylinder width. In both examples we find crossovers at moderate cylinder widths, i.e. for the largest bond dimensions considered we find an improvement on the variational energies for the Heisenberg model by using projected entangled-pair states at a width of about 11 sites, whereas for the half-filled Hubbard model this crossover occurs at about 7 sites.