Quantum dynamics of a domain wall in a quasi one-dimensional $XXZ$ ferromagnet

We derive an effective low-energy theory for a ferromagnetic $(2N+1)$-leg spin-$\frac{1}{2}$ ladder with strong $XXZ$ anisotropy $\left|J_{\parallel}^z\right|\ll \left|J_{\parallel}^{xy}\right|$, subject to a kink-like non-uniform magnetic field $B_z(X)$ which induces a domain wall (DW). Using Bosonization of the quantum spin operators, we show that the quantum dynamics is dominated by a single one-dimensional mode, and is described by a sine-Gordon model. The parameters of the effective model are explored as functions of $N$, the easy-plane anisotropy $\Delta=-J_{\parallel}^z/J_{\parallel}^{xy}$, and the strength and profile of the transverse field $B_z(X)$. We find that at sufficiently strong and asymmetric field, this mode may exhibit a quantum phase transition from a Luttinger liquid to a spin-density-wave (SDW) ordered phase. As the effective Luttinger parameter grows with the number of legs in the ladder ($N$), the SDW phase progressively shrinks in size, recovering the gapless dynamics expected in the two-dimensional limit $N\rightarrow\infty$.