# Combined use of translational and spin-rotational invariance for spin systems

## TL;DR

This paper explores combining translational and spin-rotational symmetries to efficiently study quantum spin systems, reducing computational effort and enabling analysis of larger systems through optimized symmetry projection techniques.

## Contribution

It introduces a method to effectively combine translational and spin-rotational symmetries, minimizing computational resources for larger quantum spin system simulations.

## Key findings

- Reduced computational effort for selected systems
- Largest numerically accessible cases demonstrated
- Efficient implementation of symmetry projection formulas

## Abstract

Exact diagonalization and other numerical studies of quantum spin systems are notoriously limited by the exponential growth of the Hilbert space dimension with system size. A common and well-known practice to reduce this increasing computational effort is to take advantage of the translational symmetry $C_N$ in periodic systems. This represents a rather simple yet elegant application of the group theoretical symmetry projection operator technique. For isotropic exchange interactions, the spin-rotational symmetry SU(2) can be used, where the Hamiltonian matrix is block-structured according to the total spin- and magnetization quantum numbers. Rewriting the Heisenberg Hamiltonian in terms of irreducible tensor operators allows for an efficient and highly parallelizable implementation to calculate its matrix elements recursively in the spin-coupling basis. When combining both $C_N$ and SU(2), mathematically, the symmetry projection technique leads to ready-to-use formulas. However, the evaluation of these formulas is very demanding in both computation time and memory consumption, problems which are said to outweigh the benefits of the symmetry reduced matrix shape. We show a way to minimize the computational effort for selected systems and present the largest numerically accessible cases.