Higgs and Nambu-Goldstone modes in a spin-1 \textit{XY} model with long-range interactions

TL;DR

This work analyzes Higgs and Nambu-Goldstone modes in a spin-1 XY model with long-range dipolar interactions ($\alpha=3$) in $d$ dimensions, focusing on the two-dimensional case relevant to Rydberg-atom arrays. Using a mean-field background, Schwinger-boson representation, Holstein-Primakoff expansion, Bogoliubov diagonalization, and finite-temperature Green's functions, the authors derive the Higgs and NG dispersions and compute the Beliaev damping rate of the Higgs mode near the ordered-to-disordered quantum phase transition. They find that in $d=2$, $\alpha=3$ the Higgs mode has a linear dispersion in $|\mathbf k|$, while the NG mode scales as $|\mathbf k|^{1/2}$; crucially, long-range interactions strongly suppress the Beliaev damping, making the Higgs mode long-lived near the critical point at finite temperatures. The results imply that the Higgs amplitude mode could be experimentally excited and observed in Rydberg-atom platforms, and the paper outlines a protocol to create and probe the Higgs oscillations via controlled quenches of the Zeeman terms.

Abstract

We theoretically study the collective excitations in a spin-1 $XY$ model with a quadratic Zeeman term and a long-range interaction that decays algebraically with the distance. Using the quantum-field theory based on the finite-temperature Green's function formalism, we analyze properties of the Nambu-Goldstone (NG) and Higgs modes in order to analytically evaluate the damping rate of the Higgs mode in the $XY$ ferromagnetic ordered phase near the quantum phase transition to the disordered phase. When the power of the algebraic decay is 3 as in the case of dipole-dipole interactions in Rydberg-atom systems, we show that at two dimensions the excitation energy of the Higgs mode exhibits a linear dispersion whereas the dispersion of the NG mode becomes proportional to the square root of the momentum. We find that the damping of the Higgs mode is significantly suppressed by the long-range interaction. We also propose how to excite and probe the Higgs mode in Rydberg-atom experiments.

Figures (4)

• Figure 1: Ground-state phase diagram of the spin-1 $XY$ model of Eq. (\ref{['eq:Hamiltonian']}) obtained within the mean-field approximation for $\alpha>d$. The blue solid line represents the phase boundary between the XY-ferromagnetic ordered phase and the disordered phases. The red dashed line indicates the line of $\frac{p}{\Delta} = 0$ in the ordered phase region.
• Figure 2: Dispersion relations of the Higgs (blue solid line) and NG (red dashed line) modes, which are expressed in Eqs. (\ref{['eq:higgsE']}) and (\ref{['eq:ngE']}), for $d=2$, $\alpha=3$, and ${\bf k}=(k,0)$, where $u =0.9(< u_{\rm c})$ (a) and $u = u_{\rm c}$ (b).
• Figure 3: Higgs gap $\Delta_{\rm h}$ as a function of $u$, whose analytical expression is given in Eq. (\ref{['eq:HiggsGap']}).
• Figure 4: Beliaev damping rate of the Higgs mode as a function of $u$ for $d = 2$ and $\alpha = 3$, which is expressed in Eq. (\ref{['eq:DipolarDampingRate']}). (a) $\Gamma_{\mathrm{A}, \mathbf{k} = \mathbf{0}} / (J z_{\mathrm{eff}})$ and (b) $\Gamma_{\mathrm{A}, \mathbf{k} = \mathbf{0}} / \Delta_{\mathrm{h}}$.