In this paper, we do a complete classification of valence-bond crystals (VBCs) on the kagomé lattice based on general arguments of symmetry only and thus identify many new VBCs for different unit cell sizes. For the spin-1/2 Heisenberg antiferromagnet, we study the relative energetics of competing gapless spin liquids (SLs) and VBC phases within the class of Gutzwiller-projected fermionic wave functions using variational Monte Carlo techniques, hence implementing exactly the constraint of one fermion per site. By using a state-of-the-art optimization method, we conclusively show that the U(1) Dirac SL is remarkably stable towards dimerizing into all 6-, 12- and 36-site unit cell VBCs. This stability is also preserved on addition of a next-nearest-neighbor super-exchange coupling of both antiferromagnetic and ferromagnetic (FM) type. However, we find that a 36-site unit cell VBC is stabilized on addition of a very small next-nearest-neighbor FM super-exchange coupling, i.e. | J 2 | ≈ 0.045, and this VBC is the same in terms of space-group symmetry as that obtained in an effective quantum dimer model study. It breaks reflection symmetry, has a nontrivial flux pattern and is a strong dimerization of the uniform RVB SL.
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.
The singlet dynamics, which plays a major role in the physics of the spin-1/2 quantum Heisenberg antiferromagnet (QHAF) on the kagome lattice, can be approximately described by projecting onto the nearest-neighbor valence bond (NNVB) singlet subspace. We revisit here the effective quantum dimer model, which originates from the latter NNVB-projected Heisenberg model via a nonperturbative Rokhsar-Kivelson-like scheme. By using Lanczos exact diagonalization on a 108-site cluster supplemented by a careful symmetry analysis, it is shown that a previously found 36-site valence bond crystal (VBC) in fact competes with a new type of 12-site ``resonating-columnar'' VBC. Interestingly, these two VBCs ``emerge'' in different topological sectors. The exceptionally large degeneracy of the ground-state multiplets (144 on our 108-site cluster) and the proximity of a
Z2 dimer liquid have implications for the interpretation of numerical results on the QHAF, which are outlined. The possibility of a chiral VBC (i.e., spontaneously breaking time-reversal symmetry) is also discussed.
The spin-
21 Heisenberg model on the kagome lattice, which is closely realized in layered Mott insulators such as
ZnCu3(OH)6Cl2, is one of the oldest and most enigmatic spin-
21 lattice models. While the numerical evidence has accumulated in favor of a quantum spin liquid, the debate is still open as to whether it is a
Z2 spin liquid with very short-range correlations (some kind of resonating valence bond spin liquid), or an algebraic spin liquid with power-law correlations. To address this issue, we have pushed the program started by Rokhsar and Kivelson in their derivation of the effective quantum dimer model description of Heisenberg models to unprecedented accuracy for the spin-
21 kagome, by including all the most important virtual singlet contributions on top of the orthogonalization of the nearest-neighbor valence bond singlet basis. Quite remarkably, the resulting picture is a competition between a
Z2 spin liquid and a diamond valence bond crystal with a 12-site unit cell, as in the density-matrix renormalization group simulations of Yan et al. Furthermore, we found that, on cylinders of finite diameter
d, there is a transition between the
Z2 spin liquid at small
d and the diamond valence bond crystal at large
d, the prediction of the present microscopic description for the two-dimensional lattice. These results show that, if the ground state of the spin-
21 kagome antiferromagnet can be described by nearest-neighbor singlet dimers, it is a diamond valence bond crystal, and, a contrario, that, if the system is a quantum spin liquid, it has to involve long-range singlets, consistent with the algebraic spin liquid scenario.
The doping of quantum Heisenberg antiferromagnets on the kagome lattice by nonmagnetic impurities is investigated within the framework of a generalized quantum dimer model (QDM) describing (a) the valence-bond crystal (VBC), (b) the dimer liquid, and (c) the critical region on equal footing. Following the approach by Ralko et al. [Phys. Rev. Lett. 101, 117204 (2008)] for the square and triangular lattices, we introduce the (minimal) extension of the QDM on the kagome lattice to account for spontaneous creation of mobile
S=1/2 spinons at finite magnetic field. Modulations of the dimer density (at zero or finite magnetic field) and of the local field-induced magnetization in the vicinity of impurities are computed using Lanczos exact diagonalization techniques on small clusters (48 and 75 sites). The VBC is clearly revealed from its pinning by impurities, while, in the dimer liquid, crystallization around impurities involves only two neighboring dimers. We also find that a next-nearest-neighbor ferromagnetic coupling favors VBC order. Unexpectedly, a small-size spinon-impurity bound state appears in some region of the dimer-liquid phase. In contrast, in the VBC phase the spinon delocalizes within a large region around the impurity, revealing the weakness of the VBC confining potential. Lastly, we observe that an impurity concentration as small as 4% enhances dimerization substantially. These results are confronted to the valence-bond glass scenario [R. R. P. Singh, Phys. Rev. Lett. 104, 177203 (2010)] and implications to the interpretation of the nuclear magnetic resonance spectra of the Herbertsmithite compound 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.