Schwinger-Keldysh effective field theory of type-B Goldstone: near-diagonal geometry and Berry term

Abstract

In this work, we formulate a finite temperature Schwinger-Keldysh effective field theory for type-B Goldstone modes. In particular, we organize the required two time contour structure by a near-diagonal geometry in which the r-type field is interpreted as the physical Goldstone configuration and the a-type field is identified as corresponding tangent displacement. Within this geometric viewpoint, the Berry term essential for type-B Goldstones is obtained directly through the transgression of the Berry curvature. Moreover, we show that this exact Berry transgression is compatible with dynamical KMS condition. In order to construct the conservative, dissipative and noise sectors, we classify possible tensors globally defined on the coset manifold. Based on this framework, we discuss in detail two concrete model examples. The dispersion relations of the Goldstone modes and the associated two-point correlation functions are calculated in the presence of dissipation.