We investigate the impact of spin-phonon coupling on the
S=1/2 Heisenberg model on the kagome lattice. For the pure spin model, there is increasing evidence that the low-energy properties can be correctly described by a Dirac spin liquid, in which spinons with a conical dispersion are coupled to emergent gauge fields. Within this scenario, the ground-state wave function is well approximated by a Gutzwiller-projected fermionic state [Ran, Hermele, Lee, and Wen, Phys. Rev. Lett. 98, 117205 (2007)]. However, the existence of
U(1) gauge fields may naturally lead to instabilities when small perturbations are included. Phonons are ubiquitous in real materials and may play a relevant role in the determination of the actual physical properties of the kagome antiferromagnet. Therefore, we perform a step forward in this direction, including phonon degrees of freedom (at the quantum level) and applying a variational approach based upon Gutzwiller-projected fermionic Ans\"atze. Our results suggest that the Dirac spin liquid is stable for small spin-phonon couplings, while valence-bond solids are obtained at large couplings. Even though different distortions can be induced by the spin-phonon interaction, the general aspect is that the energy is lowered by maximizing the density of perfect hexagons in the dimerization pattern.
Motivated by recent density-matrix renormalization group (DMRG) calculations [Yan, Huse, and White, Science 332, 1173 (2011)], which claimed that the ground state of the nearest-neighbor spin-
1/2 Heisenberg antiferromagnet on the kagome lattice geometry is a fully gapped spin liquid with numerical signatures of
Z2 gauge structure, and a further theoretical work [Lu, Ran, and Lee, Phys. Rev. B 83, 224413 (2011)], which gave a classification of all Schwinger-fermion mean-field fully symmetric
Z2 spin liquids on the kagome lattice, we have thoroughly studied Gutzwiller-projected fermionic wave functions by using quantum variational Monte Carlo techniques, hence implementing exactly the constraint of one fermion per site. In particular, we investigated the energetics of all
Z2 candidates (gapped and gapless) that lie in the neighborhood of the energetically competitive U(1) gapless spin liquids. By using a state-of-the-art optimization method, we were able to conclusively show that the U(1) Dirac state is remarkably stable with respect to all
Z2 spin liquids in its neighborhood, and in particular for opening a gap toward the so-called ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state, which was conjectured to describe the ground state obtained by the DMRG method. Finally, we also considered the addition of a small second nearest-neighbor exchange coupling of both antiferromagnetic and ferromagnetic type, and obtained similar results, namely, a U(1) Dirac spin-liquid ground state.
The parton approach for quantum spin liquids gives a transparent description of low-energy elementary excitations, e.g., spinons and emergent gauge-field fluctuations. The latter ones are directly coupled to the hopping/pairing of spinons. By using the fermionic representation of the
U(1) Dirac state on the kagome lattice and variational Monte Carlo techniques to include the Gutzwiller projection, we analyze the effect of modifying the gauge fields in the spinon kinematics. In particular, we construct low-energy monopole excitations, which are shown to be gapless in the thermodynamic limit. States with a finite number of monopoles or with a finite density of them are also considered, with different patterns of the gauge fluxes. We show that these chiral states are not stabilized in the Heisenberg model with nearest-neighbor superexchange couplings, and the Dirac state corresponds to the lowest-energy ansatz within this family of variational wave functions. Our results support the idea that spinons with a gapless conical spectrum coexist with gapless monopole excitations, even for the spin-1/2 case.
By using the variational Monte Carlo technique, we study the spin-
21 XXZ antiferromagnetic model (with easy-plane anisotropy) on the kagome lattice. A class of Gutzwiller projected fermionic states with a spin Jastrow factor is considered to describe either spin liquids [with
U(1) or
Z2 symmetry] or magnetically ordered phases [with
q=(0,0) or $\mathbf{q}=(4\ensuremath{\pi}/3,0)$]. We find that the magnetic states are not stable in the thermodynamic limit. Moreover, there is no energy gain to break the gauge symmetry from
U(1) to
Z2 within the spin-liquid states, as previously found in the Heisenberg model. The best variational wave function is therefore the
U(1) Dirac state, supplemented by the spin Jastrow factor. Furthermore, a vanishing
S=2 spin gap is obtained at the variational level, in the whole regime from the
XY to the Heisenberg model.
We assess the ground-state phase diagram of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the Kagome lattice by employing Gutzwiller-projected fermionic wave functions. Within this framework, different states can be represented, defined by distinct unprojected fermionic Hamiltonians that include hopping and pairing terms, as well as a coupling to local Zeeman fields to generate magnetic order. For
J2=0, the so-called U(1) Dirac state, in which only hopping is present (such as to generate a $\ensuremath{\pi}$-flux in the hexagons), has been shown to accurately describe the exact ground state [Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, Phys. Rev. B 87, 060405(R) (2013); Y.-C. He, M. P. Zaletel, M. Oshikawa, and F. Pollmann, Phys. Rev. X 7, 031020 (2017).]. Here we show that its accuracy improves in the presence of a small antiferromagnetic superexchange
J2, leading to a finite region where the gapless spin liquid is stable; then, for
J2/J1=0.11(1), a first-order transition to a magnetic phase with pitch vector
q=(0,0) is detected by allowing magnetic order within the fermionic Hamiltonian. Instead, for small ferromagnetic values of
∣J2∣/J1, the situation is more contradictory. While the U(1) Dirac state remains stable against several perturbations in the fermionic part (i.e., dimerization patterns or chiral terms), its accuracy clearly deteriorates on small systems, most notably on 36 sites where exact diagonalization is possible. Then, on increasing the ratio
∣J2∣/J1, a magnetically ordered state with $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ periodicity eventually overcomes the U(1) Dirac spin liquid. Within the ferromagnetic
J2 regime, evidence is shown in favor of a first-order transition at ${J}_{2}/{J}_{1}=\ensuremath{-}0.065(5)$.
In a recent paper [arXiv:1601.02165], Tao Li claimed that a gapped
Z2 spin liquid, obtained using projected Gutzwiller fermionic wave functions, can be stabilized in the Heisenberg model on the kagome lattice. We perform very accurate variational calculations and confirm with unprecedented accuracy the fact, that the
Z2 spin liquid is a local energy minimum that goes away with system size, thus reaffirming the scenario of a gapless
U(1) Dirac spin liquid ground state.