Numerical calculations of the B1g Raman spectrum of the two-dimensional Heisenberg model

The B-1g Raman spectrum of the two-dimensional S = 1/2 Heisenberg model is discussed within Loudon-Fleury theory at both zero and finite temperature. The exact T = 0 spectrum for lattices with up to 6 X 6 sites is computed using Lanczos exact diagonalization. A quantum Monte Carlo (QMC) method is used to calculate the corresponding imaginary-time correlation function and its first two derivatives for lattices with up to 16 X 16 spins. The imaginary-time data are continued to real frequency using the maximum-entropy method, as well as a fit based on spin-wave theory. The numerical results are compared with spin-wave calculations for finite lattices. There is a surprisingly large change in the exact spectrum going from 4 X 4 to 6 X 6 sites. In the former case then is a single dominant two-magnon peak at omega/J approximate to 3.0, whereas in the latter case there are two approximately equal-sized peaks at omega/J approximate to 2.7 and 3.9. This is in good qualitative agreement with the spin-wave calculations including two-magnon processes on the same lattices. The spin-wave results for larger lattices show how additional peaks emerge with increasing lattice size, and eventually develop into the well known two-magnon profile peaked at omega/J approximate to 3.2 and with weight extending up to omega/J approximate to 4.6. Both the Lanczos and the QMC results indicate that the actual two-magnon profile is broader than the narrow peak obtained in spin-wave theory, but the positions of the maxima agree to within a few percent. The higher-order contributions present in the numerical results are merged with the two-magnon profile and extend up to frequencies omega/J approximate to 7. The first three frequency cumulants of the spectrum are in excellent agreement with results previously obtained from a series expansion around the Ising limit. Typical experimental B-1g spectra for La2CuO4 are only slightly broader than what we obtain here. The exchange constant extracted from the peak position is J approximate to 1400 K, in good agreement with values obtained from neutron scattering and NMR experiments. We discuss the implications of our present results for more sophisticated theories of Raman scattering suggested recently.