Spontaneous symmetry breaking and Nambu–Goldstone modes in open classical and quantum systems

Abstract

We discuss spontaneous symmetry breaking of open classical and quantum systems. When a continuous symmetry is spontaneously broken in an open system, there appears a gapless excitation mode 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.

Magnetophonons & type-B Goldstones from hydrodynamics to holography

“…The last case, that of diffusive Goldstones, is more recent. It has been noticed, and then formalized using out-of-equilibrium effective field theory (EFT) methods [19][20][21], that in dissipative systems (e.g. open systems) diffusive Goldstone modes can appear:…”

Quasinormal modes of a semi-holographic black brane and thermalization

“…The transformation of massless propagating Goldstone modes of the type ω = ±k (which is exactly how the modes of χ behave in the decoupling limit) to a pair of diffusion and a quasi-hydro modes has been observed before in other models of holography [36,37] where such modes are also obtained from an explicit and soft symmetry breaking [13,36,38] 8 and have been also discussed from an effective field theory point of view [40,41]. 9 For a recent review, see [42].…”

Spontaneous Symmetry Breaking, Spectral Statistics, and the Ramp

“…We make use of the fact that a path integral in one dimension is equivalent to a quantum mechanics problem. The Hilbert space is just a suitable space of functions on the configuration space of the path integral, in this case U (1) 2 .…”

“…The phenomenon of SSB can occur whenever a system possesses a symmetry and a suitable thermodynamic limit. SSB is said to occur at a given energy density if, after first taking the thermodynamic limit, a vanishingly small symmetry breaking perturbation in the Hamiltonian leads to a non-symmetric equilibrium state (reviews include [1,2]). Because the system's symmetry constrains the structure of the energy spectrum, SSB also manifests as a certain reorganization of the energy levels as a function of energy density.…”