Quantum antiferromagnet on a Bethe lattice at percolation I. Low-energy states, DMRG, and diagnostics

We investigate ground and excited state properties of randomly diluted spin-1/2, exchange-coupled Heisenberg antiferromagnets on the Bethe lattice with coordination 3. In the case of square lattice percolation clusters, previous Quantum Monte Carlo (QMC) calculations [1] found that the singlet-triplet gaps scaled ``anomalously,'' being much smaller than the $1/N$ scaling expected from the tower of ``quantum rotor'' states (due to $E=M^2/2N\chi$). The low energies were attributed to the interaction of distant ``dangling spins,'' forced by the local imbalance of even and odd sites. In the present study we further study this effect on the Bethe lattice, using Exact Diagonalization and density-matrix RG. (DMRG applies naturally since the Bethe lattice lacks loops). We introduce inter-site correlation and susceptibility matrices as diagnostics to identify the spatial locations of the low-energy degrees of freedom, and to understand interactions between them. These matrices have been computed within the harmonic spin-wave theory, in order to highlight the deviations seen in the spin-1/2 system. In addition to the above, we propose a simple effective Hamiltonian which explains the magnitude of the singlet-triplet gap. \\[4pt] [1] L. Wang and A. Sandvik, Phys. Rev. B 81, 054417 (2010).