Abstract
We develop the effective field theory of diffusive Nambu-Goldstone (NG) modes associated with spontaneous internal symmetry breaking taking place in nonequilibrium open systems. The effective Lagrangian describing semi-classical dynamics of the NG modes is derived and matching conditions for low-energy coefficients are also investigated. Due to new terms peculiar to open systems, the associated NG modes show diffusive gapless behaviors in contrast to the propagating NG mode in closed systems. We demonstrate two typical situations relevant to the condensed matter physics and high-energy physics, where diffusive type-A or type-B NG modes appear.
Highlights
Symmetry and its realization give fundamental descriptions of physical systems from condensed matter physics to high-energy physics
Global symmetry of the system, if it exists, remains unbroken, or spontaneously broken in a given situation—e.g., in the ground state—and the resulting symmetry realization restricts a possible low-energy spectrum contained in the system: if spontaneous symmetry breaking (SSB) of continuous global symmetry takes place, it leads to the inevitable appearance of gapless excitations known as the Nambu-Goldstone (NG) modes according to the NG theorem [1,2,3] with a low-dimensional exception protected by the Mermin-Wagner theorem [4,5,6]
In this paper, taking one step further, we develop effective field theory (EFT) for the NG modes resulting from spontaneous symmetry breaking in nonequilibrium open systems where the conservation law is violated by considering e.g., the diffusive coupling between the system and environment [30,31,32,33]
Summary
Symmetry and its realization give fundamental descriptions of physical systems from condensed matter physics to high-energy physics. The notion of symmetry and its spontaneous breaking are extended to nonequilibrium closed systems [21,22,23,24,25,26,27,28,29] and open systems [30,31,32,33], where diffusive gapless modes appear. To achieve these developments, one most powerful tool, the effective field theory (EFT) [34,35,36], has been actively used. Generalizing the CallanColeman-Wess-Zumino’s (CCWZ) coset construction [34,35] to the Schwinger-Keldysh formalism, we lay out a solid basis to construct the general effective Lagrangian for the open system NG modes, and apply it to two typical situations where the type-A and type-B NG modes appear
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