Dynamical spin correlations in kagome antiferromagnets: comparison of Abrikosov fermion and Schwinger boson approaches beyond mean field

Abstract

Quantum spin liquids exhibit fractionalized spin excitations as a consequence of strong quantum many-body effects. The kagome antiferromagnetic Heisenberg model is a promising candidate for a quantum spin-liquid ground state; however, the nature of its excitation spectrum remains controversial, particularly regarding the presence of a spin gap and the gauge structure coupled to fractional quasiparticles. To address these issues, parton approaches have been extensively employed, where spin operators are represented in terms of fermionic or bosonic quasiparticles within the Abrikosov fermion and Schwinger boson frameworks. Thus far, these approaches have been pursued independently, and it has remained unclear how the results obtained from these frameworks compare, particularly with respect to the spin dynamics and gauge structure of the kagome antiferromagnet. Here, we investigate the dynamical spin structure factor of the antiferromagnetic Heisenberg model with a Dzyaloshinskii-Moriya interaction on the kagome lattice, relevant to herbertsmithite, by employing both approaches. We find that the dynamical spin structure factor obtained from the Abrikosov fermion mean-field theory exhibits dome-shaped features, and that its continuum structure significantly depends on the gauge structure of the spin-liquid ansatz. On the other hand, the Schwinger boson mean-field theory yields a concave-down structure in the low-energy region, distinct from that obtained using the Abrikosov fermion approach. Moreover, incorporating many-body effects beyond the mean-field approximation substantially reduces the low-energy gap and enhances the low-energy spectral weight, consistent with experimental observations. Our results suggest the importance of many-body effects in the Schwinger boson theory for capturing the low-energy spin dynamics of kagome antiferromagnets.

Figures (16)

• Figure 1: (a) Four-point vertex $V_{\bm{q}}^{\mu\nu\rho\lambda}$ defined in Eq. \ref{['eq:definition-vertex-function']}. (b) Bubble diagrams contributing to the bare susceptibility $\chi^{(0)}(\bm{q},\omega)$. In addition to the conventional particle-number-conserving bubble, anomalous bubbles (right) also contribute, reflecting the particle-number-nonconserving structure of the Schwinger boson mean-field theory. (c) Feynman-diagram representation of the Dyson equation for the RPA susceptibility. The shaded bubble denotes the RPA-dressed susceptibility $\chi_{\text{RPA}}(\bm{q},\omega)$, obtained by connecting the bare bubbles in (b) with the four-point vertex in (a).
• Figure 2: (a) Schematic picture of the kagome lattice on which the $S=1/2$ kagome antiferromagnetic Heisenberg model with a DM interaction is defined. The green arrows indicate the primitive translation vectors $\bm{a}_{1}$ and $\bm{a}_{2}$ of the unit cell, which are mainly used in the present calculations. The gray arrows on the bonds indicate the orientation of the DM vectors $\bm{d}_{ij}=(0,0,d_{ij}^{z})$, corresponding to the convention $d_{ij}^{z}>0$. The hexagon highlighted in blue and the parallelogram highlighted in orange indicate the closed loops on which the Wilson loops are defined in AFMFT to distinguish spin-liquid phases. (b) First and extended Brillouin zones of the kagome lattice. Filled symbols denote the high-symmetry points. The green arrows indicate the primitive reciprocal lattice vectors $\bm{b}_{1}$ and $\bm{b}_{2}$ corresponding to $\bm{a}_{1}$ and $\bm{a}_{2}$ in the real space.
• Figure 3: AFMFT mean-field ansatzes studied in this work. The figure specifies the pattern of the mean-field matrices $u_{ij}^{0}$ and $u_{ij}^{z}$ defined on solid, dashed, and open bonds. In all ansatzes the Lagrange multipliers $a_i^{\gamma}$ for the local constraints are taken to be uniform in site and sublattice. The labels below each ansatz indicate the IGG; for the $\mathrm{U}(1)$ states, $[\Phi_{\mathrm{Hex}},\Phi_{\mathrm{Para}}]$ specifies the flux sector. The green arrows indicate the primitive translation vectors. Note that ansatzes III and IV are analyzed in an enlarged kagome unit cell, doubled relative to the conventional primitive unit cell, and hence the unit-cell area is twice as large. The arrow direction on each bond indicates the orientation from site $i$ to site $j$.
• Figure 4: Spinon dispersion relations obtained from AFMFT for the ansatzes shown in Fig. \ref{['fig:mean_field_ansatz_AFMFT']}. We plot only the positive-energy branches $\epsilon_{\bm{k},n}\geq0$, which represent the physical quasiparticle excitations.
• Figure 5: Dynamical (top) and static (bottom) spin structure factors obtained within AFMFT for each mean-field ansatz shown in Fig. \ref{['fig:mean_field_ansatz_AFMFT']}. The dynamical spin structure factor $S(\bm{q},\omega)$ is evaluated along the high-symmetry path $\Gamma-\mathrm{M}-\mathrm{M}_{\mathrm{e}}-\mathrm{K}_{\mathrm{e}}-\mathrm{K}-\Gamma$. In the plots of the static spin structure factor $S(\bm{q})$, the inner and outer cyan dashed hexagons indicate the first and extended Brillouin zones, respectively, corresponding to those shown in Fig. \ref{['fig:kagome_lattice']}(b).