Researchers Discover Chiral Roton and Nematic Modes in Spin-1/2 CSL Phase

Researchers are delving into the fascinating world of chiral spin liquids (CSLs), magnetic counterparts to the fractional quantum Hall effect, to understand their unique collective excitations. Hongyu Lu, Wei Zhu (from Westlake University), and Wang Yao (from The University of Hong Kong) et al. have identified and characterised two previously unseen spin-singlet collective modes within the SU(2) symmetric CSL phase of a spin-1/2 square-lattice model , a low-energy chiral roton and a higher-energy nematic mode. This work is significant because, unlike extensively studied fractional quantum Hall liquids, the collective excitations of CSLs have remained largely mysterious, and these newly discovered modes exhibit distinct fingerprints. Their findings not only advance our dynamical understanding of CSLs but also offer potential spectroscopic signatures for experimental verification in candidate materials.

Their findings not only advance our dynamical understanding of CSLs but also offer potential spectroscopic signatures for experimental verification in candidate materials.

Chiral Spin Liquid Reveals Novel Collective Modes

This allowed for the direct observation of the chiral roton and d-wave nematic modes, providing unprecedented insight into the dynamical properties of CSLs. This work establishes a crucial link between CSLs and FQH liquids, despite their differing physical origins, and opens new avenues for understanding the dynamical landscape of CSLs. The discovery of these unique collective modes, distinct from those in FQH systems, provides new spectroscopic signatures for future experimental investigations of CSL candidates, potentially enabling the identification and characterization of these exotic phases of matter. The research paves the way for a deeper understanding of CSLs from a dynamical perspective and offers a foundation for exploring the fundamental question of how CSLs differ from their FQH counterparts despite their topologically equivalent ground states.,.

Excitations in Chiral Spin Liquids via Numerical Methods

Energy spectra were computed using the Lanczos algorithm within the ED simulations, exploiting translation and total Sz conservation, and partially utilising the QuSpin package; system sizes ranged from Ns = 16 to 36, with the Ns = 36 Hilbert subspace containing approximately 2.52 × 108 basis states. Researchers constructed low-energy excitations within the single-mode approximation (SMA) by applying a local operator to the many-body ground state, validating the operator content by comparing it with ED calculations. To calculate these factors, the team computed time-dependent correlation functions using TDVP simulations, evolving the state ψ(t) step by step with ∆t = 0.05 for a total of Nt = 1000 steps, generating a frequency grid with spacing ∆ω = 0.02. A Lorentzian broadening of η = 0.05 was applied during the time-to-frequency Fourier transform, and the bond dimension for TDVP results reached χ = 400, ensuring well-converged results. Static structure factors were also calculated using equal-time correlators and Fourier transformations in momentum space. This methodological approach, combining multiple advanced numerical techniques, enabled the identification of these exotic modes and their distinct fingerprints compared to those found in fractional quantum Hall (FQH) liquids, modes previously unreported, to the best of the researchers’ knowledge.

Chiral Roton and Nematic Modes in CSLs

These exotic modes, to the best of the researchers’ knowledge, represent the first reported observations of their kind. Dynamical structure factors (DSFs) were probed for several local observables, defined as DSF O(q, ω) = X r,t eiωt−ηte−iqr √Nt (⟨O† r(t) O0(0)⟩−⟨O† r⟩⟨O0⟩), where q and ω represent crystal momentum and frequency, respectively. Results demonstrate that, targeting the Sz operator, the spin DSFs exhibit finite-momentum bound states with concentrated spectral weight in a relatively high-frequency regime. Tests prove a pronounced roton mode with a minimum energy at (π, π) when studying bond operators Bμ(r) = Sr · Sr+μ, remaining the lowest excitation across a large region of the CSL parameter space, significantly lower in energy than spin triplet excitations.

Further analysis of the By operator’s DSFs revealed non-vanishing signals at q →0, albeit weaker and at higher energies than the roton minimum. The team defined a d-wave operator Dx2−y2 ≡Bx −By, observing a pronounced spectral peak at q = 0, identified as a higher-energy nematic mode. Measurements confirm that the lowest excitation on a torus is a spin singlet located at (π, π), matching the quantum number of the observed roton mode. Wave function overlaps, calculated using the SMA, revealed high overlaps between the excited state and variational states composed of Bx and By, with the relation Bx| ψ0(0, 0)⟩= i By| ψ0(0, 0)⟩ holding exactly. This led to the definition of chiral p-wave operators P ± = Bx ±i By, exhibiting near-unit overlaps and locked chirality to the ground state chirality, even on non-symmetric tori.

Chiral Spin Liquids Exhibit Novel Dynamical Modes

These exotic modes, distinct from those found in fractional quantum Hall liquids, represent a novel finding in the field. This work establishes a dynamical understanding of CSLs, moving beyond static characterisations and offering new spectroscopic signatures for experimental verification. The authors acknowledge limitations inherent in their computational methods, specifically the finite system sizes used in the exact diagonalization calculations, which may influence the precision of the results. Future research could explore these CSL dynamics in larger systems and investigate the behaviour of these collective excitations near phase boundaries, potentially revealing new insights into the interplay between different magnetic phases.