Collective excitations in chiral spin liquid: chiral roton and long-wavelength nematic mode

TL;DR

The paper investigates the dynamical properties of the SU(2)-symmetric chiral spin liquid (CSL) realized in the spin-1/2 $J_1-J_2-J_\chi$ square-lattice model. By combining exact diagonalization, DMRG/ED analyses, and time-dependent variational principle (TDVP) simulations, it identifies two spin-singlet collective modes across the CSL: a chiral p-wave roton at $\mathbf{q}=(\pi,\pi)$ and a zero-momentum d-wave nematic mode, with the roton displaying chirality tied to the CSL ground state. The roton remains the lowest excitation and is distinct from magneto-roton behavior in FQH liquids, while the nematic mode can soften with increasing $J_2$, signaling potential instabilities toward nematic or stripe orders; triplet spinon-bound states also appear at finite momentum. These findings offer new spectroscopic signatures for CSLs and highlight qualitative dynamical differences from FQH/FCI paradigms, guiding future experimental probes such as Raman scattering and momentum-resolved spectroscopy.

Abstract

Chiral spin liquid (CSL) is a magnetic analogue of the fractional quantum Hall (FQH) liquid. Collective excitations play a vital role in shaping our understanding of these exotic quantum phases of matter and their quantum phase transitions. While the magneto-roton and long-wavelength chiral graviton modes in the FQH liquids have been extensively explored, the collective excitations of CSLs remain elusive. Here we explore the collective excitations in the SU(2) symmetric CSL phase of the spin-1/2 square-lattice $J_1-J_2-J_χ$ model, where an intriguing quantum phase diagram was recently revealed. Combining exact diagonalization and time-dependent variational principle calculations, we observe two spin-singlet collective modes: a chiral p-wave (low-energy) roton mode at finite momentum and a d-wave (higher-energy) nematic mode at zero momentum, both of which are prominent across the CSL phase. Such exotic modes exhibit fingerprints distinct from those of FQH liquids, and to the best of our knowledge, are reported for the first time. By tuning $J_2$, we find the nematic mode to be pronouncedly soft, together with the spin-triplet two-spinon bound states, potentially promoting strong nematic and spin stripe instabilities. Our work paves the way for further understanding CSL from the dynamical perspective and provides new spectroscopic signatures for future experiments of CSL candidates.

Figures (13)

• Figure 1: Lattice model and phase diagram. (a) The $J_1-J_2-J_\chi$ model on square lattice. (b) Schematic plot of the two exotic spin-singlet collective modes deep inside the CSL phase. For positive $J_\chi$ the chiral roton mode lies in the $p+\mathrm{i}p$ channel. (c) The quantum phase diagram with fixed $J_1=1,\ J_2=0$ and tuning $J_\chi$. The GS in the shaded region is still under debate and is not the focus of this work.
• Figure 2: Dynamical structure factors of CSL. Different panels show dynamical response of (a1-a3) spin $S_z$, (b1-b3) x-direction bond operator $B_x$, (c1-c3) y-direction bond operator $B_y$ and (d) d-wave bond operator $D$ (details see main text). The calculations are performed by the TDVP for $J_2=0$ and $J_\chi=2$ on $L_y\times L_x=4\times32$ cylinders. The horizontal/vertical axis is the energy/$q_x$ in each panel with different fixed $q_y$. The dashed lines in (a1-a3) correspond to the energy scale of the singlet roton mode ($\omega\sim0.28$). In (d3), the DSF along $q_y=\pi$ are multiplied by a factor of three for visibility.
• Figure 3: Spectral property of CSL. (a) Low-energy spectrum obtained on a $6\times6$ torus by setting $J_2=0$, $J_\chi=2$. A roton mode appears at $(\pi,\pi)$. The states of larger $S$ are at higher energies. The static structure factors (SSFs) of $B_x,B_y$ operator are shown in (b-d) with panels (c) and (d) sharing the same color bar.
• Figure 4: The chiral roton. DSFs of the chiral p-wave operators $\hat{P}^{\mp}$ at fixed $\omega=0.28$ (energy scale of the roton mode) for $J_2=0$ and $J_\chi=2$ are shown, respectively. The calculations are performed by the TDVP on $L_y\times L_x=4\times32$ cylinders. The black dashed lines denote $q_y=\frac{\pi}{2},\pi,\frac{3\pi}{2}$, respectively.
• Figure 5: Soft collective modes. (a) The energy spectrum of $J_2=0.6,J_\chi=0.5$ on the $6\times6$ torus. The TDVP results of different channels on the $4\times32$ cylinders are shown in panels (b-f). The horizontal/vertical axis is the energy/$q_x$ in each panel with different fixed $q_y$.