Breaking and restoration of rotational symmetry for irreducible tensor operators on the lattice

TL;DR

This paper investigates how rotational symmetry breaks on the lattice for tensor operators and proposes a method to mitigate this effect by averaging over orientations, demonstrating effectiveness in an alpha cluster model.

Contribution

It introduces a practical averaging technique to reduce rotational symmetry breaking effects in lattice calculations of tensor operators.

Findings

Symmetry breaking can be largely suppressed at lattice spacings of ≤1.7 fm.

Averaging over orientations accurately reproduces physical matrix elements.

Method is expected to improve lattice Monte Carlo calculations.

Abstract

We study the breaking of rotational symmetry on the lattice for irreducible tensor operators and practical methods for suppressing this breaking. We illustrate the features of the general problem using an $\alpha$ cluster model for $^{8}$Be. We focus on the lowest states with non-zero angular momentum and examine the matrix elements of multipole moment operators. We show that the physical reduced matrix element is well reproduced by averaging over all possible orientations of the quantum state, and this is expressed as a sum of matrix elements weighted by the corresponding Clebsch-Gordan coefficients. For our $\alpha$ cluster model we find that the effects of rotational symmetry breaking can be largely eliminated for lattice spacings of $a\leq 1.7$ fm, and we expect similar improvement for actual lattice Monte Carlo calculations.