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arXiv:2512.24557 · Physical Review A

Higgs and Nambu-Goldstone modes in a spin-1 X Y model with long-range interactions

Daiki Kawasaki, Ippei Danshita

Published May 11, 2026• 1 citation
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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.

DOI
10.1103/75k9-hx9k

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We theoretically study the collective excitations in a spin-1 XYXYXY 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 XYXYXY 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.
We study collective modes of superfluid Bose gases in optical lattices at commensurate fillings.We focus on the vicinity of the quantum phase transition to the Mott insulator, where there exists the Higgs amplitude mode in addition to the Nambu-Goldstone phase mode associated with the spontaneous U(1) symmetry breaking.We analyze finite-temperature effects on the damping of the collective modes by using an effective spin-1 model and field-theoretical methods based on the finite-temperature Green's function.We calculate the damping rates up to 1-loop order and evaluate them analytically and numerically.We show that the damping rate of the Higgs mode increases with increasing temperature but it remains underdamped up to the typical temperature achieved in experiments.Moreover, we find that the Nambu-Goldstone mode attenuates via a Landau damping process resulting from interactions with the Higgs mode and it can be overdamped at the typical temperature in a certain parameter region..............
Higgs and Goldstone modes manifest as fluctuations in the order parameter of system, offering insights into its phase transitions and symmetry properties. Exploring the dynamics of these collective excitations in a Rydberg atoms system advances various branches of condensed matter, particle physics, and cosmology. Here, we report an experimental signature of Higgs and Goldstone modes in a U(1) symmetry-broken Rydberg atomic gases. By constructing two probe fields to excite atoms, we observe the distinct phase and amplitude fluctuations of Rydberg atoms collective excitations under the particle-hole symmetry. Due to the van der Waals interactions between the Rydberg atoms, we detect a symmetric variance spectrum divided by the divergent regime and phase boundary, capturing the full dynamics of the additional Higgs and Goldstone modes. Studying the Higgs and Goldstone modes in Rydberg atoms allows us to explore fundamental aspects of quantum phase transitions and symmetry breaking phenomena, while leveraging the unique properties of these highly interacting systems to uncover new physics and potential applications in quantum simulation.
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Employing large-scale quantum Monte Carlo simulations, we systematically compute the energy spectra of the two-dimensional (2D) spin-1/2 Heisenberg model with long-range interactions. With the $1/{r}^{\ensuremath{\alpha}}$ ferromagnetic and staggered antiferromagnetic interactions, we find the explicit range in $\ensuremath{\alpha}$ for the short-range Goldstone-type (gapless), anomalous Goldstone-type (gapless), and Higgs-type (gapped) spectra. Accompanied by the spin-wave analysis, our numerical results vividly reveal how the long-range interactions alter the usual linear and quadratic magnon dispersions in 2D quantum magnets and give rise to anomalous dynamical exponents. Moreover, we find the explicit case where the gapped excitation emerges at a noninteger decay exponent $\ensuremath{\alpha}$ for the antiferromagnetic Hamiltonian. This work provides the first set of unbiased dynamical data of long-range quantum many-body systems and suggests that many universally accepted low-energy customs for short-range systems need to be substantially modified for long-range ones, which are of immediate relevance to the ongoing experimental efforts from quantum simulators to 2D quantum moir\'e materials.
Using an effective field theory we describe the low energy bosonic excitations in a three-dimensional ultracold mixture of spin-1 bosons and spin-1/2 fermions. We establish an interesting fermionic excitation induced generic damping of the usual undamped long-wavelength bosonic collective Goldstone modes. Two states with bosons forming either a ferromagnetic or polar superfluid are studied. The linear dispersion of the bosonic Bogoliubov excitations is preserved with a renormalized sound velocity. For the polar superfluid we find both gapless modes (density and spin) are damped, whereas in the ferromagnetic superfluid we find the density (spin) mode is (not) damped. We find that this holds for any mixture of bosons and fermions that are coupled through at least a density-density interaction. In addition, we predict the existence of the Kohn anomaly in the bosonic excitation spectrum of Bose-Fermi mixtures. We discuss the implications of our many-body interaction results for experiments on Bose-Fermi mixtures.