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.
The nature of quantum spin liquids is studied for the spin-
1/2 antiferromagnetic Heisenberg model on a square lattice containing exchange interactions between nearest-neighbor sites,
J1, and those between next-nearest-neighbor sites,
J2. We perform variational Monte Carlo simulations together with the quantum-number-projection technique and clarify the phase diagram in the ground state together with its excitation spectra. We obtain the nonmagnetic phase in the region
0.4<J2/J1≤0.6 sandwiched by the staggered and stripe antiferromagnetic (AF) phases. Our direct calculations of the spin gap support the notion that the triplet excitation from the singlet ground state is gapless in the range of
0.4<J2/J1≤0.5, while the gapped valence-bond-crystal (VBC) phase is stabilized for
0.5<J2/J1≤0.6. The VBC order is likely to have the columnar symmetry with a spontaneous symmetry breaking of the
C4v symmetry. The power-law behaviors of the spin-spin and dimer-dimer correlation functions in the gapless region are consistent with the emergence of the algebraic quantum-spin-liquid phase (critical phase). The exponent of the spin correlation
⟨S(0)S(r)⟩∝1/rz+η at a long distance
r appears to increase from
z+η∼1 at
J2/J1∼0.4 toward the continuous transition to the VBC phase at
J1/J1∼0.5. Our results, however, do not fully exclude the possibility of a direct quantum transition between the staggered AF and VBC phases with a wide critical region and deconfined criticality.
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., $k=(\ensuremath{\pi},0)$.
We study the plaquette valence-bond solid phase of the spin-
1/2J1-
J2 antiferromagnet Heisenberg model on the square lattice within the bond-operator theory. We start by considering four
S=1/2 spins on a single plaquette and determine the bond operator representation for the spin operators in terms of singlet, triplet, and quintet boson operators. The formalism is then applied to the
J1-
J2 model and an effective interacting boson model in terms of singlets and triplets is derived. The effective model is analyzed within the harmonic approximation and the previous results of Zhitomirsky and Ueda [Phys. Rev. B 54, 9007 (1996)] are recovered. By perturbatively including cubic (triplet-triplet-triplet and singlet-triplet-triplet) and quartic interactions, we find that the plaquette valence-bond solid phase is stable within the parameter region
0.34<J2/J1<0.59, which is narrower than the harmonic one. Differently from the harmonic approximation, the excitation gap vanishes at both critical couplings
J2=0.34J1 and
J2=0.59J1. Interestingly, for
J2<0.48J1, the excitation gap corresponds to a singlet-triplet excitation at the $\ensuremath{\Gamma}$ point while, for
J2>0.48J1, it is related to a singlet-singlet excitation at the $\mathbf{X}=(\ensuremath{\pi}/2,0)$ point of the tetramerized Brillouin zone.
The highly frustrated spin-
21 quantum Heisenberg model with both nearest (
J1) and next-nearest (
J2) neighbor exchange interactions is revisited by using an extended variational space of projected wave functions that are optimized with state-of-the-art methods. Competition between modulated valence-bond crystals (VBCs) proposed in the literature and the Dirac spin liquid (DSL) is investigated. We find that the addition of a small ferromagnetic next-nearest-neighbor exchange coupling
∣J2∣>0.09J1 leads to stabilization of a 36-site unit cell VBC, although the DSL remains a local minimum of the variational parameter landscape. This implies that the VBC is not trivially connected to the DSL; instead it possesses a nontrivial flux pattern and large dimerization.
The spin-
1/2 J1-
J2 Heisenberg model on the square lattice represents one of the simplest examples in which the effects of magnetic interactions may suppress magnetic order, eventually leading to a pure quantum phase with no local order parameters. This model has been extensively studied in the last three decades, with conflicting results. Here, by using Gutzwiller-projected wave functions and recently developed methods to assess the low-energy spectrum, we show the existence of a level crossing between the lowest-energy triplet and singlet excitations for ${J}_{2}/{J}_{1}\ensuremath{\approx}0.54$. This fact supports the existence of a phase transition between a gapless spin liquid (which is stable for $0.48\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.54$) and a valence-bond solid (for $0.54\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.6$), even though no clear sign of dimer order is visible in the correlations functions. These results, which confirm recent density-matrix renormalization calculations on cylindrical clusters [L. Wang and A. W. Sandvik, Phys. Rev. Lett. 121, 107202 (2018)], reconcile the contradicting results obtained within different approaches over the years.
We construct a class of projected entangled pair states which is exactly the resonating valence bond wave functions endowed with both short range and long range valence bonds. With an energetically preferred resonating valence bond pattern, the wave function is simplified to live in a one-parameter variational space. We tune this variational parameter to minimize the energy for the frustrated spin-1/2 J(1)-J(2) antiferromagnetic Heisenberg model on the square lattice. Taking a cylindrical geometry, we are able to construct four topological sectors with an even or odd number of fluxes penetrating the cylinder and an even or odd number of spinons on the boundary. The energy splitting in different topological sectors is exponentially small with the cylinder perimeter. We find a power law decay of the dimer correlation function on a torus, and a lnL correction to the entanglement entropy, indicating a gapless spin-liquid phase at the optimum parameter.