One-dimensional projection of two-dimensional systems using spiral boundary conditions

TL;DR

This paper introduces spiral boundary conditions (SBCs) that enable the exact projection of 2D lattice models onto 1D chains, allowing the use of 1D techniques like DMRG for 2D systems.

Contribution

The authors develop a novel projection scheme using SBCs to map 2D lattice models onto 1D chains, facilitating the application of 1D computational methods to 2D problems.

Findings

Successfully projected 2D models onto 1D chains using SBCs.

Demonstrated the approach with the 2D square and honeycomb lattices.

Estimated staggered magnetization in the 2D XXZ Heisenberg model.

Abstract

We introduce spiral boundary conditions (SBCs) as a useful tool for handling the shape of finite-size periodic clusters. Using SBCs, a lattice model for more than two dimensions can be exactly projected onto a one-dimensional (1D) periodic chain with translational invariance. Hence, the existing 1D techniques such as density-matrix renormalization group (DMRG), bosonization, Jordan-Wigner transformation, etc., can be effectively applied to the projected 1D model. First, we describe the 1D projection scheme for the two-dimensional (2D) square- and honeycomb-lattice tight-binding models in real and momentum space. Next, we discuss how the density of states and the ground-state energy approach their thermodynamic limits. Finally, to demonstrate the utility of SBCs in DMRG simulations, we estimate the magnitude of staggered magnetization of the 2D XXZ Heisenberg model as a function of XXZ anisotropy.