The dynamical structure factor of the SU(3) Heisenberg chain: The variational Monte Carlo approach

We compute the dynamical spin structure factor $S(k,\omega)$ of the SU(3) Heisenberg chain variationally using a truncated Hilbert space spanned by the Gutzwiller projected particle-hole excitations of the Fermi sea, introduced in [B. Dalla Piazza et al., Nature Physics 11, 62 (2015)], with a modified importance sampling. We check the reliability of the method by comparing the $S(k,\omega)$ to exact diagonalization results for 18 sites and to the two-soliton continuum of the Bethe Ansatz for 72 sites. We get an excellent agreement in both cases. Detailed analysis of the finite-size effects shows that the method captures the critical Wess-Zumino-Witten SU(3)$_1$ behavior and reproduces the correct exponent, with the exception of the size dependence of the weight of the bottom of the conformal tower. We also calculate the single-mode approximation for the SU($N$) Heisenberg model and determine the velocity of excitations. Finally, we apply the method to the SU(3) Haldane-Shastry model and find that the variational method gives the exact wave function for the lowest excitation at $k=\pm 2\pi/3$.