Abstract
Abstract We discuss spontaneous symmetry breaking of open classical and quantum systems. When a continuous symmetry is spontaneously broken in an open system, a gapless excitation mode appears corresponding to the Nambu–Goldstone mode. Unlike isolated systems, the gapless mode is not always a propagation mode, but it is a diffusion one. Using the Ward–Takahashi identity and the effective action formalism, we establish the Nambu–Goldstone theorem in open systems, and derive the low-energy coefficients that determine the dispersion relation of Nambu–Goldstone modes. Using these coefficients, we classify the Nambu–Goldstone modes into four types: type-A propagation, type-A diffusion, type-B propagation, and type-B diffusion modes.
Highlights
Spontaneous symmetry breaking is one of the most important notions in modern physics
The relation between them was first shown for relativistic systems [1,2,3], where the number of NG modes is equal to the number of broken symmetries or generators
It was extended to isolated systems without Lorentz symmetry [4,5,6,7,8], where the number of NG modes does not coincide with the number of broken symmetries [9,10,11]
Summary
Spontaneous symmetry breaking is one of the most important notions in modern physics. U (1) phase symmetry is spontaneously broken, and the diffusion mode appears as the NG mode, which has different dispersion from that in isolated systems. Another example is ultracold atoms in an optical cavity [16]. We generalize it to analysis with finite momentum, and derive the low-energy coefficients for the inverse of the Green functions in the NG mode channel Using these coefficients, we classify the NG modes, and discuss the relation between these modes and the broken generators.
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