We consider an exchange model describing two isotropic spin-1/2 Heisenberg antiferromagnets coupled by a quartic term on the square lattice. The model is relevant for systems with orbital degeneracy and strong electron-vibron coupling in the large Hubbard repulsion limit, and is known to show a spin-Peierls-like dimerization in one dimension. In two dimensions we calculate energy gaps, susceptibilities, and correlation functions with a Green's function Monte Carlo method. We find a finite triplet and singlet gap, with no evidence of order of any kind. We conclude that the ground state is, most likely, a spin liquid of resonating valence bonds.
We consider the exchanged Hamiltonian HST=−J∑〈rr′〉(2Sr⋅Sr′−12)(2Tr⋅Tr′−12), describing two isotropic spin-1/2 Heisenberg antiferromagnets coupled by a quartic term on equivalent bonds. The model is relevant for systems with orbital degeneracy and strong electron-vibron coupling in the large Hubbard repulsion limit. To investigate the ground state properties we use a Green’s Function Monte Carlo, calculating energy gaps and correlation functions, the latter through the forward walking technique. In one dimension we find that the ground state is a “crystal” of valence bond dimers. In two dimensions, the spin gap appears to remain finite in the thermodynamic limit, and, consistently, the staggered magnetization—signal of Néel long range order—seems to vanish. From the analysis of dimer–dimer correlation functions, however, we find no sign of a valance bond crystal. A spin liquid appears as a plausible scenario compatible with our findings.
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 perform highly accurate density matrix renormalization group (DMRG) simulations to investigate the ground-state properties of the spin-
1/2 antiferromagnetic square lattice Heisenberg
J1-
J2 model. Based on studies of numerous long cylinders with circumferences of up to 14 lattice spacings, we obtain strong evidence for a topological quantum spin liquid state in the region $0.41\ensuremath{\le}{J}_{2}/{J}_{1}\ensuremath{\le}0.62$, separating conventional N\'eel and striped antiferromagnetic states for smaller and larger
J2/J1, respectively. The quantum spin liquid is characterized numerically by the absence of magnetic or valence bond solid order, and nonzero singlet and triplet energy gaps. Furthermore, we positively identify its topological nature by measuring a nonzero topological entanglement entropy $\ensuremath{\gamma}=0.70\ifmmode\pm\else\textpm\fi{}0.02$, extremely close to $\ensuremath{\gamma}=\mathrm{ln}(2)\ensuremath{\approx}0.69$ (expected for a
Z2 quantum spin liquid) and a nontrivial finite size dimerization effect depending upon the parity of the circumference of the cylinder. We also point out that a valence bond solid, and indeed any discrete symmetry breaking state, would be expected to show a constant correction to the entanglement entropy of opposite sign to the topological entanglement entropy.
The large-N limit of a nearest-neighbor SU(N) antiferromagnet on a bipartite lattice exhibits in dimensions d\ensuremath{\ge}2 a zero-temperature phase transition between a N\'eel ordered state and a resonating-valence-bond state. Here it is shown in d=1,2 that topological effects produce spin-Peierls or valence-bond-solid order in the non-N\'eel phase with a ground-state degeneracy which varies periodically with ``spin'' for fixed N with periodicity given by the coordination number of the lattice. Thus a non-N\'eel phase of the spin-(1/2 Heisenberg model on a square lattice would be a spin-Peierls state with a fourfold degeneracy due to broken lattice rotational symmetry.
We consider a resonating-valence-bond state as a trial wave function for the ground state of the antiferromagnetic Heisenberg model on the square lattice. The energy and the staggered spin-spin correlation functions are calculated on a small lattice and are compared with the results of the exact diagonalization of Oitmaa and Betts. The trial energy is only about 4.7% different from the exact ground-state energy on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. The importance of the singlet pairs between the next-nearest antiferromagnetic sites is inferred. An argument against the previous interpretation of the numerical data as being evidence of long-range order is presented.
We study the zero-temperature quantum phase diagram of a spin-1 Heisenberg antiferromagnet on the honeycomb lattice with both nearest-neighbor exchange coupling
J1>0 and frustrating next-nearest-neighbor coupling ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, using the coupled cluster method implemented to high orders of approximation, and based on model states with different forms of classical magnetic order. For each we calculate directly in the bulk thermodynamic limit both ground-state low-energy parameters (including the energy per spin, magnetic order parameter, spin stiffness coefficient, and zero-field uniform transverse magnetic susceptibility) and their generalized susceptibilities to various forms of valence-bond crystalline (VBC) order, as well as the energy gap to the lowest-lying spin-triplet excitation. In the range $0<\ensuremath{\kappa}<1$ we find evidence for four distinct phases. Two of these are quasiclassical phases with antiferromagnetic long-range order, one with two-sublattice N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}=0.250(5)$, and another with four-sublattice N\'eel-II order for $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}=0.340(5)$. Two different paramagnetic phases are found to exist in the intermediate region. Over the range ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}^{i}=0.305(5)$ we find a gapless phase with no discernible magnetic order, which is a strong candidate for being a quantum spin liquid, while over the range ${\ensuremath{\kappa}}_{c}^{i}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$ we find a gapped phase, which is most likely a lattice nematic with staggered dimer VBC order that breaks the lattice rotational symmetry.