Breaking and restoring symmetries within the nuclear energy density functional method

TL;DR

This paper reviews symmetry breaking and restoration in nuclear energy density functional methods, highlighting challenges and proposing routes for improved formulations, especially concerning particle-number and angular-momentum restoration.

Contribution

It identifies mathematical properties of energy density functions that can constrain functional kernels, advancing symmetry restoration techniques in nuclear EDF methods.

Findings

Identified an exact property of the energy density $E^{LM}(oldsymbol{R})$ for symmetry restoration.

Discussed difficulties in formulating symmetry restoration within energy functional frameworks.

Suggested potential routes for better symmetry restoration formulations.

Abstract

We review the notion of symmetry breaking and restoration within the frame of nuclear energy density functional methods. We focus on key differences between wave-function- and energy-functional-based methods. In particular, we point to difficulties to formulate the restoration of symmetries within the energy functional framework. The problems tackled recently in connection with particle-number restoration serve as a baseline to the present discussion. Reaching out to angular-momentum restoration, we identify an exact mathematical property of the energy density $E^{LM}(\vec{R})$ that could be used to constrain energy density functional kernels. Consequently, we suggest possible routes towards a better formulation of symmetry restorations within energy density functional methods.