We study the ground-state phase diagram of the spin-
1/2${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice with an accurate bosonic resonating-valence-bond (RVB) wave function. In contrast to the RVB ansatz based on Schwinger fermions, the representation based on Schwinger bosons, supplemented by a variational Monte Carlo technique enforcing the exact projection onto the physical subspace, is able to describe a fully gapped spin liquid in the strongly frustrated regime. In particular, a fully symmetric
Z2 spin liquid is stable between two antiferromagnetic phases; a continuous transition at
J2=0.4J1, when the Marshall sign rule begins to be essentially violated, and a first-order transition around
J2=0.6J1 are present. Most importantly, the triplet gap is found to have a nonmonotonic behavior, reaching a maximum around
J2=0.51J1, when the lowest spinon excitation moves from the $\ensuremath{\Gamma}$ to the
M point, i.e., $k=(\ensuremath{\pi},0)$.
Using variational wave functions and Monte Carlo techniques, we study the antiferromagnetic Heisenberg model with first-neighbor
J1 and second-neighbor
J2 antiferromagnetic couplings on the honeycomb lattice. We perform a systematic comparison of magnetically ordered and nonmagnetic states (spin liquids and valence-bond solids) to obtain the ground-state phase diagram. N\'eel order is stabilized for small values of the frustrating second-neighbor coupling. Increasing the ratio
J2/J1, we find strong evidence for a continuous transition to a nonmagnetic phase at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.23$. Close to the transition point, the Gutzwiller-projected uniform resonating valence-bond state gives an excellent approximation to the exact ground-state energy. For $0.23\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.36$, a gapless
Z2 spin liquid with Dirac nodes competes with a plaquette valence-bond solid. In contrast, the gapped spin liquid considered in previous works has significantly higher variational energy. Although the plaquette valence-bond order is expected to be present as soon as the N\'eel order melts, this ordered state becomes clearly favored only for ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.3$. Finally, for $0.36\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\le}0.5$, a valence-bond solid with columnar order takes over as the ground state, being also lower in energy than the magnetic state with collinear order. We perform a detailed finite-size scaling and standard data collapse analysis, and we discuss the possibility of a deconfined quantum critical point separating the N\'eel antiferromagnet from the plaquette valence-bond solid.
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
J1−J2 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
dx2−y2 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
δ=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.,
δ≲0.1. In the
J1−J2 Heisenberg model on the square lattice, we use the projected mean-field state that is built from Abrikosov fermions having a
Z2 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
J2/J1=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.,
J2/J1≃0.5. In particular, we found gapless triplet excitations at momenta
(π,0) and
(0,π), which are compatible with the presence of four Dirac points at momenta
(±2π,±2π) in the spinon spectrum.
Twenty-five years after the first proposal, the question whether the ground state of a frustrated spin-half system is well described by a spin-liquid Resonating Valence Bond (RVB) wave function is still controversial. A physically transparent representation of a RVB state can be obtained in fermionic representation with a standard BCS-type pairing wave function, working in the subspace with fixed number of electrons and no double occupancies. In this work, we show that, using this variational wave function with a careful parameterization of the pairing function, it is possible to obtain an extremely accurate {\em ansatz} for the ground state of the Heisenberg antiferromagnet with next-nearest neighbors interactions (
J1−J2 model) in the regime of strong frustration. Indeed, in the spin-half realization of this model, it is known that the combined effect of frustration and zero-point motion interferes with the mechanism of spontaneously broken symmetry, giving rise to a non-magnetic phase of purely quantum-mechanical nature (
J2/J1≃0.5). This wave function is proposed to represent the generic spin-half RVB ground state in spin liquids.
We construct a family of simple fermionic projected entangled pair states (fPEPS) on the square lattice with bond dimension
D=3 which are exactly hole-doped resonating valence bond (RVB) wave functions with short-range singlet bonds. Under doping the insulating RVB spin liquid evolves immediately into a superconductor with mixed
d+is pairing symmetry whose pair amplitude grows as the square root of the doping. The relative weight between
s-wave and
d-wave components can be controlled by a single variational parameter
c. We optimize our ansatz with respect to
c for the frustrated $t\text{\ensuremath{-}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ model (including both nearest and next-nearest neighbor antiferromagnetic interactions
J1 and
J2, respectively) for ${J}_{2}\ensuremath{\simeq}{J}_{1}/2$ and obtain an energy very close to the infinite-PEPS state (using full update optimization and same bond dimension). The orbital symmetry of the optimized RVB superconductor has predominant
d-wave character, although we argue a residual (complex
s-wave) time reversal symmetry breaking component should always be present. Connections of the results to the physics of superconducting cuprates and pnictides are outlined.
We propose a class of variational wave functions, namely the correlated valence bond states, for the frustrated Hamiltonian in the paramagnetic phase. This class of wave functions admits negative amplitude and the same sub-lattice pairing when a bipartite lattice is considered, thus suffers from the negative sign problem. However if applied to small systems, the sign problem is manageable using the standard variational Monte Carlo method. We optimize the wave functions for the J1-J2 Anti ferromagnetic Heisenberg model on a square lattice in the coupling region J2/J1 \in [0.45 : 0.56] for system sizes L = 4, 6, 8. To calculate the correlation functions and the order parameters for larger systems, we make the extensive Monte Carlo samplings using the variational parameters optimized at system size L = 8. We find that the paramagnetic phase is a gap-less spin liquid in the entire range of J2/J1 \in [0.45 : 0.56] with a gap-less singlet excitation and a gaped triplet excitation.