Phononic enhancement and detection of hidden spin-nematicity and dynamics in quantum magnets

Abstract

The spin nematic phase, characterized by long-range order of spin quadrupole moments in the absence of dipolar magnetism, presents a significant challenge for conventional experimental detection. We propose a novel method to detect this elusive order in quantum magnets with an illustration in the spin-1 triangular lattice Mott insulator. By integrating out the phonon degrees of freedom, we obtain a phase diagram with substantially enlarged regions for the spin-nematic and spin-nematic-supersolid phases. We then demonstrate that through the spin-lattice coupling, the emergence of spin nematic order imprints a distinctive signature onto the phonon spectra, providing a clear spectroscopic signature for the quadrupolar order accessible via Raman or inelastic X-ray scattering. Our formalism offers a direct and powerful method to uncover the hidden spin nematicity, opening a new pathway for diagnosing multipolar orders in quantum magnets.

Figures (5)

• Figure 1: (a) Triangular lattice structure with ordering wavevector at $K$ where a unit cell (dashed lines) contains three sublattices as colored in red (A), blue (B) and green (C). The black solid (dashed) arrows are the lattice vectors for the supercell (original triangular lattice). (b) Phase diagram of $H_{s}^{\rm eff}$ in \ref{['eq:H_eff_SW']} with the averaged quadrupolar order $|Q|\equiv \sum_{\alpha\in\{A,B,C\}}{[\braket{Q^{x^2\!-\!y^2}_{\alpha}}^2+\braket{Q^{xy}_{\alpha}}^2]^{1/2}/3}$. Here and in the following, $D=4J$ and $\Delta=1.2$ are used in the calculation.
• Figure 2: Phonon–magnon band spectra in SN phase under different magnetic fields in the absence [(a), (c), and (e)] and presence [(b), (d), and (f)] of magnon–phonon coupling. The color represents the normalized relative weight of the phonon-like and magnon-like components of each band. The intrinsic frequency $\hbar \omega_0/ J=0.41$.
• Figure 3: Magnitude of the avoided crossing gap $\Delta_{a}$ as a function of magnetic field in the SN phase. The black arrow indicates the critical magnetic field $B/J\approx 7.72$ where $\Delta_{a}$ vanishes at the $K$ point.
• Figure 4: Phonon–magnon band spectra in SNS phase with magnetic field $B/J=0.06$ [first row, (a-c)] and $B/J=0.12$ [second row, (d-f)]. In the first column [(a) and (d)], the magnon-phonon coupling is turned off ($\eta=0$). In the second column [(b) and (e)], the magnon-phonon coupling is turned on ($\eta\neq 0$). (c) and (f) are the same as (b) and (e) but with the fictitious optical modes projected out. The intrinsic frequency $\omega_0$ takes the same value as in Fig. \ref{['fig:SN-bands']}.
• Figure S1: (a) The original BZ (black solid lines) of the triangular lattice and the reduced BZ (gray solid line) corresponding to the enlarged unit cell in the SNS phase. (b) Free phonon band structure along the green dashed path. (c) Free phonon band structure along the blue dashed path. (d) Same as (c), but with the fictitious optical modes removed by projection $P^{ac}$.