Quench spectroscopy of amplitude modes in a one-dimensional critical phase

Abstract

We investigate the emergence of an amplitude (Higgs-like) mode in the gapless phase of the $(1+1)$D XXZ spin chain. Unlike conventional settings where amplitude modes arise from spontaneous symmetry breaking, here, we identify a symmetry-preserving underdamped excitation on top of a Luttinger-liquid ground state. Using nonequilibrium quench spectroscopy, we demonstrate that this mode manifests as oscillations of U(1)-symmetric observables following a sudden quench. By combining numerical simulations with Bethe-ansatz analyses, we trace its microscopic origin to specific families of string excitations. We further discuss experimental pathways to detect this mode in easy-plane quantum magnets and programmable quantum simulators. Our results showcase the utility of quantum quenches as a powerful tool to probe collective excitations, beyond the scope of linear response.

Figures (3)

• Figure 1: (a) Quench protocol: the system is initialized in the ground state of the XXZ Hamiltonian ${H}(\Delta_i)$ and subsequently evolved under ${H}_f$$=$${H}(\Delta_f)$. (b) Dynamics of the U(1)-symmetric probe $\mathcal{O}^{zz}$ after a quench from $\Delta_i = 0.1,\ldots, 1.0$ (in steps of 0.1) to the free-fermion point $\Delta_f=0$. Similar underdamped oscillations are found for other $|\Delta_f|\le 1$ as well. (c) Dynamics after quenches from $\Delta_i = 0$ to $\Delta_f = 0.1,\ldots, 1.0$. The oscillation frequency is determined by $\Delta_f$, while its amplitude is set by $\lvert \Delta_f-\Delta_i \rvert$.
• Figure 2: (a) Parametrization of the four crystal momenta $k_{p+}$, $k_{p-}$, $k_{h+}$, and $k_{h-}$ in terms of three parameters $Q$, $p$, and $q$, in consistency with momentum conservation. (b) Density of states $g(\omega)$ and Fourier component $\langle \mathcal{O}\rangle^{(1)}(\omega)$ as functions of $\omega/J$. (c--g) Surface morphology in the parameter space $(Q,p,q)$, where the energy difference $\varepsilon$ matches the frequency. While the figure sketches $\mathcal{D}$, our calculations are restricted to $\mathcal{D}_{>0}$ to prevent double counting from the indistinguishability of $p\leftrightarrow-p$ and $q\leftrightarrow-q$. Shown are selected values of $\omega/J$: (c) $0.5$, (d) $2$, (e) $2.3$, (f) $3\sqrt{3}/2\approx2.6$, and (g) $3.5$. The DOS is nondifferentiable at $\omega/J=2$ (gray dotted line) and exhibits a cusp at $\omega/J=3\sqrt{3}/2$ (red dotted line), corresponding to an extremal point where punctures begin to appear on the surface as it attaches to the boundary of the domain $\mathcal{D}$ for $\omega/J < 3\sqrt{3}/2$.
• Figure 3: (a--c) Schematic illustrations of the three excitation types (I--III) discussed in the main text. Open circles denote real-axis holes, while filled circles represent 2-strings or length-1 complex rapidities. (d--f) Density of states $g(\omega)$ for $\Delta \in \{0.1, 0.4, 0.7\}$ for each excitation family. For small $\Delta$ (e.g., $\Delta=0.1$), $g(\omega$) for the type-I family smoothly connects to the free-fermion DOS: the peak coincides with the free-fermion value, and the spectrum extends continuously down to $\omega=0$.