Nonrelativistic Nambu-Goldstone Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

We show that a momentum operator of a translational symmetry may not commute with an internal symmetry operator in the presence of a topological soliton in nonrelativistic theories. As a striking consequence, there appears a coupled Nambu-Goldstone mode with a quadratic dispersion consisting of translational and internal zero modes in the vicinity of a domain wall in an O(3) $\sigma$ model, a magnetic domain wall in ferromagnets with an easy axis.

Figure 1

Schematic pictures of coupled ripplons and their propagating directions for solutions ($Z_1^{+}$, ${\alpha }_1^{+}$) (left) and ($Z_1^{-}$, ${\alpha }_1^{-}$) (right). The middle shaded area shows the region of the domain wall $|u|\approx 1$ ($n_z\approx 0$) and its tone shows the phase $\alpha$ of $u$ [direction of ($n_x$, $n_y$)]. The $+$ and $-$ signs show the areas for $\alpha >0$ and $\alpha <0$, respectively. The vertical axis is the $z$ axis and the horizontal axis shows the direction of the wave vector $\mathit{k}$ in the $xy$ plane. For left (right) figures, the coupled ripplon propagates in the right (left) direction. See the Supplemental Material for animations of their dynamics [11].