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.
We revisit the challenging problem of identifying the quantum spin liquid candidate in the spin-1/2
J1-
J2 Heisenberg antiferromagnet on the square lattice. By integrating the Gutzwiller-guided density matrix renormalization group method with analytical analyses, we present clear evidence that the ground state is a Z
2 Dirac spin liquid. This state can be efficiently described by a Gutzwiller-projected parton theory characterized by its projective symmetry group. To distinguish the difference between the projected Z
2 and U(1) parton state, we investigate the chiral spin liquid ground states as topological orders by incorporating a
Jχ term into the
J1-
J2 model and observe a transition from a Z
2 chiral spin liquid to a U(1)
2 chiral spin liquid as
Jχ increases.
We explore the Dirac spin liquid (DSL) as a candidate for the ground state of the Mott insulating phase of fermions with six flavors on the kagome lattice, particularly focusing on realizations using
173Yb atoms in optical lattices. Using mean-field theory and variational Monte Carlo simulations, we demonstrate that the DSL has the lowest variational energy among SU(6) symmetry-preserving trial wave functions with a periodicity of a 12-site unit cell, as well as uniform chiral states with larger unit cells. It remains a local minimum even when small second-nearest neighbor and ring-exchange interactions are introduced. To characterize the DSL, we calculate the static and dynamic structure factor of the Gutzwiller projected wave function and compare it with mean-field calculations. The static structure factor shows triangular-shaped plateaus around the
K points in the extended Brillouin zone, with small peaks at the corners of these plateaus. The dynamical structure factor consists of a gapless continuum of fractionalized excitations. Our study also presents several complementary results, including bounds for the ground-state energy, methods for calculating three-site ring-exchange expectations in the projective mean field, the boundary of ferromagnetic states, and the nontopological nature of flat bands in the DSL band structure.
We implement the universal wave-function overlap (UWFO) method to extract modular
S and
T matrices for topological orders in Gutzwiller-projected parton wave functions (GPWFs). The modular
S and
T matrices generate a projective representation of
SL(2,Z) on the degenerate-ground-state Hilbert space on a torus and may fully characterize the 2+1D topological orders, i.e., the quasiparticle statistics and chiral central charge (up to
E8 bosonic quantum Hall states). We use the variational Monte Carlo method to computed the
S and
T matrices of the chiral spin liquid (CSL) constructed by the GPWF on the square lattice, and we confirm that the CSL carries the same topological order as the $\ensuremath{\nu}=\frac{1}{2}$ bosonic Laughlin state. We find that the nonuniversal exponents in the UWFO can be small, and direct numerical computation can be applied on relatively large systems. The UWFO may be a powerful method to calculate the topological order in GPWFs.
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 ${J}_{2}/{J}_{1}=\ensuremath{-}0.065(5)$.