Closely competing valence bond crystal orders in the ground state of the spin- 1 2 antiferromagnetic Heisenberg model on the pyrochlore lattice: A large scale unrestricted variational study
Closely competing valence bond crystal orders in the ground state of the spin-21 antiferromagnetic Heisenberg model on the pyrochlore lattice: a large scale unrestricted variational study
The spin-21 antiferromagnetic Heisenberg model on the pyrochlore lattice(PAFH) is arguably the most well known strongly frustrated quantum magnet in three spatial dimension. However, due to the rapid scaling of Hilbert space with the linear size of such a three dimensional system, the nature of its ground state in the thermodynamic limit remains elusive after about 30 years' intense debate. Here we apply a recently developed powerful algorithm to perform large scale unrestricted variational optimization of the ground state of the spin-21 PAFH from the resonating valence bond(RVB) theory perspective. We find a highly competitive candidate ground state of the system. This novel state features a maximally resonating valence bond crystal(VBC) pattern with 2a1×2a2×2a3 periodicity. There are at least four levels of hierarchical structure in such a VBC state, with the first and the second level of hierarchy related to the breaking of the inversion and the translational symmetry. Intriguingly, we find that within the RVB framework such a maximally resonating VBC state is almost degenerate with a recently proposed VBC state that is obtained from dressing hard hexagon covering of the pyrochlore lattice, although they have very different structures. We also find that further symmetry breaking will occur in the dressed hard hexagon VBC state under unrestricted optimization, which results in strong disparity in ⟨S^u2⟩ for up and down tetrahedrons as we observe in the maximally resonating VBC state. We show that the maximally resonating VBC state found here will be favored by a tiny next-neighboring exchange coupling over the dressed hard hexagon covering VBC state.
We investigate the ground state phase diagram of the spin-1/2 antiferromagnetic Heisenberg model on the star lattice using infinite projected entangled pair states (iPEPS) and high-order series expansions. The model includes two distinct couplings: Jd on the dimer bonds and Jt on the trimer bonds. While it is established that the system hosts a valence bond solid (VBS) phase for ${J}_{d}\ensuremath{\ge}{J}_{t}$, the ground state phase diagram for Jd<Jt has remained unsettled. Our iPEPS simulations uncover a first-order phase transition at ${J}_{d}/{J}_{t}\ensuremath{\approx}0.18$, significantly lower than previously reported estimates. Beyond this transition, we identify a close competition between two valence bond crystal (VBC) states; a columnar VBC and a $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC, with the latter consistently exhibiting lower energy across all finite bond dimensions. The high-order series expansion supports this by finding that the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC state indeed becomes energetically favorable, but only at sixth order in perturbation theory, revealing the subtle nature of the competition between candidate states.
Variational study of the J1−J2 Antiferromagnetic Heisenberg model on square lattice using correlated valence bond state
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.