Schwinger boson mean field perspective on emergent spins in diluted Heisenberg antiferromagnets

Using an adaptation of Schwinger boson mean field theory (SBMFT) for nonuniform systems, we study the nature of low-energy spin excitations on the square and Bethe lattices at their percolation threshold. The optimal SBMFT parameters are interpreted as on-site potentials and pairing amplitudes, which enables an explanation of why emergent local moments develop in this system on dilution [L. Wang and A. W. Sandvik, Phys. Rev. Lett. 97, 117204 (2006); H. J. Changlani et al., Phys. Rev. Lett. 111, 157201 (2013)] and why the corresponding single-particle frequencies are driven to anomalously low values. We discuss how our mean field calculations suggest the strong link between the presence of sublattice imbalance and long-range antiferromagnetic order and why linear spin wave theory is inadequate for capturing this relation. Within the SBMFT framework, we also extract an energy scale for the interaction between emergent moments, which show qualitative agreement with many-body calculations.

Figure 1

(a) and (c) The optimized SBMFT parameters ${\lambda }_i$, proportional to radius of circles on sites, and $|{\tilde{Q}}_{ij}|-\text{min}\{|{\tilde{Q}}_{ij}\left|\right\}$ for nearest-neighbor bonds, proportional to the thickness of the bonds, on a diluted Bethe and square lattices, respectively. (b) and (d) The lowest single-particle eigenmode ${\psi }_{i0}^{+}$, whose amplitude is proportional to the radius of the circles and sign is denoted by the color. For more details about the interpretation of the parameters and eigenmodes refer to the text.

Figure 2

(a) Mode amplitudes ${\psi }_{\mathrm{in}}$ for the two lowest-frequency single-particle modes $n=0,1$ for the Bethe lattice percolation cluster shown in Figs. 1 and 1. Each mode (blue or red points) is plotted as a function of separation from the dangling spin at the fork tips. The fitted exponential decay is shown by solid black lines. (b) Single-particle frequencies for the same cluster obtained within SBMFT and LSWT. The two anomalously low Goldstone frequencies are indicated by an arrow and labeled as ${\omega }_{\ell \mathrm{ow}}$; their numerical values and mode numbers $n$ are indicated in the inset. The lowest nonuniform spin wave frequency is indicated by ${\mathrm{\Delta }}_{LSWT}$.

Figure 3

Disorder-averaged single-particle density of states within (a) SBMFT and (b) LSWT frameworks. An ensemble of 400 Bethe lattice percolation clusters, each consisting of 50 sites, was considered. The histograms were generated by grouping frequencies in bins of size $0.01J$. The SBMFT calculations show a set of low-lying frequencies missing in the LSWT spectrum, which have been labeled as ${\omega }_{\ell \mathrm{ow}}$.

Figure 4

Nearest-neighbor in-plane spin-spin correlations, defined to be $\langle S_i^x{S}_j^x+S_i^y{S}_j^y\rangle$, for bonds taken along the path shown in Fig. 5, from various methods: linear spin wave theory (LSWT), Hartree-Fock (HF), Schwinger boson mean field theory (SBMFT), and the density matrix renormalization group (DMRG). The HF results are used from the iteration before the convergence of the self-consistency cycle failed. LSWT does not capture the variations in the values of the correlation functions, while going to order $1/S$ ($S$ is the spin length) using HF shows tendency to capture the behavior of SBMFT and DMRG.

Figure 5

(a) Spin-spin correlations comparison between SBMFT and exact DMRG results for the Bethe cluster of Figs. 1 and 1, also shown in the inset. The correlations are between the tip spin labeled “tip” and all spins along the path shown in orange on the cluster. (b) Comparison of correlations similar to that in (a) for the square lattice cluster in Figs. 1 and 1. (c) Comparison between ground-state energies (sorted by value) from SBMFT (red) $e_{gs}^{SBMFT}$ (10) and DMRG (blue) for an ensemble of 400 size 50 clusters. The SBMFT energy is obtained by summing over nearest-neighbor spin correlations.

Figure 6

The decay of effective interactions $J_{ij}^{\mathrm{eff}}$ vs an effective distance ${\tilde{d}}_{ij}$ (see text). All data are for $N_s=50$ sized clusters.