Frustrated quantum antiferromagnetism is an important physical phenomena leading to many of the key properties of quantum matter, such as quantum spin liquids and topological order. Here, the authors employ novel techniques based on tensor network methods for the triangle-honeycomb lattice in order to compute ground-state properties of a frustrated quantum antiferromagnet on the star lattice, with high accuracy, in the thermodynamic limit. Importantly, their findings contradict some of the existing results in the literature on this model. In particular, they find a new resonating valence-bond solid (VBS) phase that had not been observed before, as well as a different VBS structure at the isotropic point than the one previously conjectured by other methods. Moreover, by using tensor network techniques, they also determine the existence of a continuous quantum phase transition in the system between different VBS orders.
We map out the ground state phase diagram of the spin-
21 isotropic Kekul\'e-Kitaev model on the honeycomb lattice in the presence of the Heisenberg exchange couplings. Our study relies on large-scale tensor network simulations based on a graph-based projected entangled pair state approach and simple tensor update in the thermodynamic limit. We find that, on top of the quantum spin liquid and conventional magnetically ordered phases which are typical of the Kitaev-Heisenberg model, the Kekul\'e-Heisenberg phase diagram hosts two plaquette valence bond solid (VBS) phases with vanishing magnetic order. While the VBS phases preserve the symmetries of the original Hamiltonian, they differ markedly from the Kitaev spin liquid by having decorated plaquette ordering which is distinguished by a plaquette order parameter.
The star lattice hosts the Yao-Kivelson chiral spin liquid, an exactly solvable Kitaev spin liquid that breaks time reversal spontaneously and is non-Abelian or Abelian depending on the ratio of its two couplings. A density-matrix renormalization group study found that an added Heisenberg exchange leaves it a wide stability window for ferromagnetic but only a narrow one for antiferromagnetic Kitaev coupling. Using projective symmetry groups (PSGs) of fermionic partons, we show that the chirality is not a peculiarity of the solvable point: whenever the
π spin rotations act on the partons as in the exact solution (spin-gauge locking), a time-reversal-symmetric mean-field state has no amplitude on the triangles, at any range, so a Kitaev-type spin liquid on this lattice must break time reversal. The Abelian and non-Abelian liquids of both signs share a single chiral PSG, within which their transition is a band inversion. The Klein duality, which reverses the Heisenberg coupling, permutes the PSGs and leaves exactly the locked ones unchanged; a
Z2 spin-liquid phase mapped onto itself is therefore locked and, if it spans a connected range of couplings, contains a Kitaev point. Unprojected mean-field theory in Kitaev's Majorana representation treats both signs alike. Exact second-order perturbation theory about the Kitaev points locates the asymmetry: the triangles respond identically for both signs, whereas on the bonds between triangles the two off-diagonal Heisenberg terms create the same four visons and interfere constructively for antiferromagnetic and almost destructively for ferromagnetic coupling, admixing a low-energy four-vison state
82 times more strongly on the antiferromagnetic side. The PSG results rest on a classification of
352 Z2 PSGs with spin-orbit-coupled amplitudes, which also identifies the fermionic partners of the known Schwinger-boson spin liquids.
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.
We explore the frustrated spin-
21 Heisenberg model on the star lattice with antiferromagnetic (AF) couplings inside each triangle and ferromagnetic (FM) intertriangle couplings (
Je<0), and calculate its magnetic and thermodynamic properties. We show that the FM couplings do not sabotage the magnetic disordering of the ground state due to the frustration from the AF interactions inside each triangle, but trigger a fully gapped inversion-symmetry-breaking trimerized valence bond crystal (TVBC) with emergent spin-1 degrees of freedom. We discover that with strengthening
Je, the system exhibits a universal scaling behavior either with or without a magnetic field
h: the order parameter, the five critical fields that separate the ${J}_{e}\text{\ensuremath{-}}h$ ground-state phase diagram into six phases, and the excitation gap obtained by low-temperature specific heat, all depend exponentially on
Je. Our work implies that the spin-1 VBCs can be stabilized by introducing small FM couplings in the geometrically frustrated spin-
21 systems.
We investigate various competing paramagnetic ground states of the Heisenberg antiferromagnet on the two-dimensional star lattice which exhibits geometric frustration. Using slave particle mean-field theory combined with a projective symmetry group analysis, we examine a variety of candidate spin liquid states on this lattice, including chiral spin liquids, spin liquids with Fermi surfaces of spinons, and nematic spin liquids which break lattice rotational symmetry. Motivated by connection to large-
N SU(N) theory as well as numerical exact diagonalization studies, we also examine various valence-bond solid (VBS) states on this lattice. Based on a study of energetics using Gutzwiller projected states, we find that a fully gapped spin liquid state is the lowest-energy spin liquid candidate for this model. We also find from a study of energetics using Gutzwiller projected wave functions and bond operator approaches that this spin liquid is unstable toward two different VBS states---a VBS state which respects all the Hamiltonian symmetries and a VBS state which exhibits $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ order---depending on the ratio of the Heisenberg exchange couplings on the two inequivalent bonds of the lattice. We compute the triplon dispersion in both VBS states within the bond operator approach and discuss possible implications of our work for future experiments on candidate materials.