Unconventional Phases in Doped or Frustrated Quantum Antiferromagnets: a Systematic QMC Study

We investigate the ground-state properties of two strongly-correlated systems: the two-dimensional $t-J$ model that can be used to study the high-temperature superconducting phase in the doped antiferromagnet, and the frustrated N\'eel antiferromagnet described by the $J_{1}-J_{2}$ Heisenberg model on the square lattice, which is widely considered as the prototype model for spin frustration. In this thesis, we apply to these two systems state-of-the-art quantum Monte Carlo techniques, including the variational and the Green's function Monte Carlo with the fixed-node approximation. Few Lanczos steps are used to systematically improve the accuracy of the trial wave functions. By introducing a suitable regularization scheme for the variational calculations with few Lanczos steps, stable and controllable simulations can be performed up to very large cluster sizes with very good accuracy. In the two-dimensional $t{-}J$ model at $J/t=0.4$, we show that the accuracy of the Gutzwiller-projected variational state (containing $d_{x^2-y^2}$ pairing) can be improved much by few Lanczos steps; in addition, the fixed-node Monte Carlo with these systematically improvable trial wave functions gives results that are comparable with the best accurate DMRG ones. Our main outcome is that the ground state is homogeneous and no evidence of stripes is detected around the doping $\delta=1/8$. Indeed, our best approximation to the ground state does not show any tendency towards charge inhomogeneity. Furthermore, our results show that a uniform state containing superconductivity and antiferromagnetism is stabilized at low hole doping, i.e., $\delta \lesssim 0.1$. In the $J_{1}-J_{2}$ Heisenberg model on the square lattice, we use the projected mean-field state that is built from Abrikosov fermions having a $Z_2$ gauge structure and four Dirac points in the spinon spectrum. No spin or dimer order is found in the strongly frustrated regime and our calculations imply that a spin liquid phase may faithfully represent the exact ground state around $J_{2}/J_{1}=0.5$. The few Lanczos step technique is used to systematically improve the accuracy of the variational states both for the ground state and for few relevant low-energy excitations. This procedure allows us to estimate, in a valuable and systemic way, the spin gaps within thermodynamical limit and to show a solid evidence of an unconventional gapless excitation spectrum in the strongly frustrated regime, i.e., $J_{2}/J_{1}\simeq 0.5$. In particular, we found gapless triplet excitations at momenta $(\pi,0)$ and $(0,\pi)$, which are compatible with the presence of four Dirac points at momenta $(\pm \frac{\pi}{2},\pm \frac{\pi}{2})$ in the spinon spectrum.