Investigations of symmetry breakings by multiple condensates on exotic chiral condensed phases

TL;DR

This work tackles the problem of identifying Nambu-Goldstone (NG) modes when continuous symmetries are broken by multiple condensates in exotic chiral phases. It combines two approaches—the meson two-point-vertex method and an algebraic commutation-relations framework—to derive NG spectra in scenarios such as a tensor-type condensate coexisting with a chiral condensate and inhomogeneous chiral condensates (dual chiral density waves) within the NJL model. The authors derive concrete NG spectra, uncover mode mixing between meson channels, and formulate general rules for counting and constructing NG and Higgs modes when several condensates break one or more symmetries, including cases with spatial translations and rotations. The resulting framework clarifies when internal and spatial symmetries mix or remain independent and provides a systematic method to obtain independent NG generators and their associated fields in complex broken-symmetry phases, with potential applications to QCD-like phase structure and condensed-matter analogs.

Abstract

The Nambu-Goldstone modes on the exiotic chiral condensed phase with chiral and tensor-type quark-antiquark condensates are investigated by using of the two-point vertex functions. It is shown that one of the Nambu-Goldstone modes appears as a result of meson-mixing. As is well known, another method to find the Nambu-Goldstone modes is given by the use of the algebraic commutation relations between broken generators and massless modes obtained through the spontaneous symmetry breaking. This method is adopted to the cases of the chiral symmetry breakings due to the tensor-type condensate and the inhomogeneous chiral condensate. The result obtained by the use of the meson two-point vertex functions is obviously reproduced in the case of the tensor-type condensate. Furthermore, we investigate the general rules for determining the broken symmetries and the Nambu-Goldstone modes algebraically. As examples, the symmetry breaking pattern and the Nambu-Goldstone modes due to the tensor-type condensate or the inhomogeneous chiral condensate are shown by adopting the general rules developed in this paper in the algebraic method.