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 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.
Quantum spin liquids are exotic many-body states featured with long-range entanglement and fractional anyon quasiparticles. Quantum phase transitions of spin liquids are particularly interesting problems related with novel phenomena of anyon condensation and anyon confinement. Here we study a quantum dimer model, which implements a transition between a Z 2 spin liquid ( Z 2 SL ) and a valence bond solid (VBS) on the kagome lattice. The transition is driven by the condensation of vison excitation of the Z 2 spin liquid, which impacts on other anyon excitations especially leading to the confinement of spinon excitations. By numerical exact diagonalization of the dimer model, we directly measure the vison condensation using vison string operators, and explicitly check a confining potential acting on spinon excitations in the VBS state. It is observed that topological degeneracy of the spin-liquid state is lifted concomitantly with the vison condensation. The dimer ordering pattern of the VBS state is identified by investigating dimer structure factor. Furthermore, we find an interesting state that exhibits features of spin liquid and VBS simultaneously. We discuss the origin of the mixed behaviors and possible scenarios expected in thermodynamic limit. This work complements the previous analytical studies on the dimer model [ and ] by providing numerical evidences on the vison condensation and the spinon confinement in the Z 2 SL -to-VBS transition. Published by the American Physical Society 2024
We present numerical evidence for the existence of an extended valence bond solid (VBS) phase at
T=0 in the kagome
S=1/2 Heisenberg antiferromagnet with ferromagnetic further-neighbor interactions. The VBS is located at the boundary between two magnetically ordered regions and extends close to the nearest-neighbor Heisenberg point. It exhibits a diamondlike singlet covering pattern with a 12-site unit cell. Our results suggest the possibility of a direct, possibly continuous, quantum phase transition from the neighboring
q=0 coplanar magnetically ordered phase into the VBS phase. Moreover, a second phase which breaks lattice symmetries, and is of likely spin-nematic type, is found close to the transition to the ferromagnetic phase. The results have been obtained using large-scale numerical exact diagonalization. We discuss implications of our results on the nature of the nearest-neighbor Heisenberg antiferromagnet.
The spin-1/2 quantum antiferromagnet on the kagome lattice provides a quintessential example in strongly correlated electron physics in which the effects of both geometric frustration and quantum fluctuation are pushed to their limit. Among possible nonmagnetic ground states, the valence bond solid (VBS) with a 36-site unit cell is one of the most promising candidates. A natural theoretical framework for the analysis of such VBS order is to consider quantum states on a bond connecting the nearest-neighboring sites as fundamental quantum modes of the system and treat them as effectively independent ``bond particles.'' While correctly describing the VBS order in the ground state, this approach, known as the bond operator theory, significantly overestimates the lowest spin excitation energy. To overcome this problem, we take a next logical step in this paper to improve the bond operator theory and consider extended spin clusters as fundamental building blocks of the system. Depending on two possible configurations of the VBS order, various spin clusters are considered: (i) in the VBS order with staggered hexagonal resonance, we consider one spin cluster for a star of David shape and two spin clusters with each composed of a perfect hexagon and three attached dimers, and (ii) in the VBS order with uniform hexagonal resonance, we consider one spin cluster composed of a star of David and three attached dimers. It is shown that the majority of low-energy spin excitations are nearly or perfectly flat in energy. With most of its weight coming from the star of David, the lowest spin excitation has a gap much lower than the previous value obtained by the bond operator theory, narrowing the difference against exact diagonalization results. Despite the better agreement, however, the fact that the lowest spin excitation mostly comes from the star-of-David structure is at odds with the bond operator theory as well as previous series expansion studies. Expecting that the true low-energy spin excitations have contributions from both the star of David and its surroundings, we propose using two superclusters of spin in future work, treating spin excitations from both contributions on a more equal footing.