An emergent SU(4) symmetry discovered in the microscopic model for d1 honeycomb materials [M. G. Yamada, M. Oshikawa, and G. Jackeli, ] has enabled us to tailor exotic SU(4) models in real materials. In the honeycomb structure, the emergent SU(4) Heisenberg model would potentially have a quantum spin-orbital liquid ground state due to the , and we can also expect similar spin-orbital liquids in three-dimensional versions of the honeycomb lattice. In such quantum spin-orbital liquids, both the spin and orbital degrees of freedom become fractionalized and entangled together due to the strong frustrated interactions between them. Similarly to spinons in pure quantum spin liquids, quantum spin-orbital liquids can host not only spinon excitations, but also fermionic excitations at low temperature.
The enhancement of the spin-space symmetry from the usual
SU(2) to
SU(N) with
N>2 is promising for finding nontrivial quantum spin liquids, but the realization of
SU(N) spin systems in real materials is still challenging. Although there is a proposal in cold atomic systems, in magnetic materials with a spin-orbital degree of freedom it is difficult to achieve the
SU(N) symmetry by fine tuning. Here we propose a new mechanism by which the
SU(4) symmetry emerges in the strong spin-orbit coupling limit. In
d1 transition metal compounds with edge-sharing anion octahedra, the spin-orbit coupling gives rise to strongly bond-dependent and apparently
SU(4)-breaking hopping between the
Jeff=3/2 quartets. However, in the honeycomb structure, a gauge transformation maps the system to an
SU(4)-symmetric Hubbard model, which means that the system has a hidden symmetry in spite of its large spin-orbit coupling. In the strong repulsion limit at quarter filling, as expected in
α-ZrCl
3, the low-energy effective model is the
SU(4) Heisenberg model on the honeycomb lattice, which cannot have a trivial gapped ground state and is expected to host a gapless spin-orbital liquid. In such quantum spin-orbital liquids, both the spin and orbital degrees of freedom become fractionalized and correlated together at low temperature due to the strong frustrated interactions between them. Similarly to spinons in pure quantum spin liquids, quantum spin-orbital liquids can host not only spinon excitations, but also fermionic "orbitalon" excitations at low temperature, which we have named here in distinction from orbitons in the symmetry-broken Jahn-Teller phases.
While the enhancement of spin-space symmetry from the usual SU(2) to SU(N) is promising for finding nontrivial quantum spin liquids, its realization in magnetic materials remains challenging. Here, we propose a new mechanism by which SU(4) symmetry emerges in the strong spin-orbit coupling limit. In d^{1} transition metal compounds with edge-sharing anion octahedra, the spin-orbit coupling gives rise to strongly bond-dependent and apparently SU(4)-breaking hopping between the J_{eff}=3/2 quartets. However, in the honeycomb structure, a gauge transformation maps the system to an SU(4)-symmetric Hubbard model. In the strong repulsion limit at quarter filling, as realized in α-ZrCl_{3}, the low-energy effective model is the SU(4) Heisenberg model on the honeycomb lattice, which cannot have a trivial gapped ground state and is expected to host a gapless spin-orbital liquid. By generalizing this model to other three-dimensional lattices, we also propose crystalline spin-orbital liquids protected by this emergent SU(4) symmetry and space group symmetries.
We perform large-scale density matrix renormalization group simulations of the SU(4) Heisenberg model on the honeycomb lattice to address the long-standing question of its ground state in an unbiased and quantitatively controlled manner. We find reliable numerical evidence that the ground state is a gapped spin-orbital liquid, presumably with a Z_{4} topological order, characterized by a finite topological entanglement entropy close to ln(4), the absence of both SU(4) and lattice symmetry breaking, and a variationally optimized ground-state energy well below the previously proposed π-flux variational state. By exploiting full SU(4) symmetry and keeping up to 12 800 SU(4) multiplets, corresponding to more than 1×10^{6} U(1) states, we achieve unprecedented accuracy for two-dimensional SU(4) quantum magnets. Finite-size scaling of energies and entanglement entropies supports a robust gapped phase in the two-dimensional limit, while a gapless critical state on narrow cylinders is identified as a proximate remnant of a Dirac spin-orbital liquid. Our results find the SU(4) honeycomb Heisenberg model a realization of a gapped topological spin-orbital liquid and provide convincing numerical evidence for topological order in a highly symmetric two-dimensional quantum magnet.
Dirac spin liquids represent a class of highly-entangled quantum phases in two dimensional Mott insulators, featuring exotic properties such as critical correlation functions and absence of well-defined low energy quasi-particles. Existing numerical works suggest that the spin-orbital
SU(4) symmetric Kugel-Khomskii model of Mott insulators on the honeycomb lattice realizes a Dirac spin-orbital liquid, described at low energy by
(2+1)d quantum electrodynamics (QED
3) with
Nf=8 Dirac fermions. We generalize methods previously developed for
SU(2) spin systems to analyze the symmetry properties and stability of the Dirac spin-orbital liquid. We conclude that the standard Dirac state in the
SU(4) honeycomb system, based on a simple parton mean-field ansatz, is intrinsically unstable at low energy due to the existence of a monopole perturbation that is allowed by physical symmetries and relevant under renormalization group flow. We propose two plausible alternative scenarios compatible with existing numerics. In the first scenario, we propose an alternative
U(1) Dirac spin-orbital liquid, which is similar to the standard one except for its monopole symmetry quantum numbers. This alternative
U(1) state represents a stable gapless phase. In the second scenario, we start from the standard
U(1) Dirac liquid and Higgs the
U(1) gauge symmetry down to
Z4. The resulting
Z4 Dirac spin-orbital liquid is stable. We also discuss the continuous quantum phase transitions from the
Z4 Dirac liquids to conventional symmetry-breaking orders, described by the QED
3 theory with
Nf=8 supplemented with a critical charge-
4 Higgs field. We discuss possible ways to distinguish these scenarios in numerics. We also extend previous calculations of the quantum anomalies of QED
3 and match with generalized lattice Lieb-Schultz-Mattis constraints.
The SU(4) Heisenberg model can serve as a low-energy model of the Mott insulating state in materials where the spins and orbitals are highly symmetric or in systems of alkaline-earth atoms on an optical lattice. Recently, it has been argued that on the honeycomb lattice, the model exhibits a unique spin-orbital liquid phase with an algebraic decay of correlations [P. Corboz et al., Phys. Rev. X 2, 041013 (2012)]. Here we study the instability of the algebraic spin-orbital liquid toward the spontaneous formation of SU(4) singlet plaquettes (tetramerization). Using a variational Monte Carlo approach to evaluate the projected wave function of fermions with $\ensuremath{\pi}$-flux state, we find that the algebraic liquid is robust and that a small finite value of the next-nearest exchange is needed to induce tetramerization. We also studied the phase diagram of a model which interpolates between the nearest-neighbor Heisenberg model and a Hamiltonian for which the singlet-plaquette-product state is an exact ground state.
The
SU(4) Heisenberg model on the honeycomb lattice is expected to host a quantum spin-orbital liquid at low temperature with an astonishing candidate material,
α-ZrCl
3. We employed the canonical thermal pure quantum state method to investigate the finite-temperature phase of this model. Exploiting the full symmetry of
SU(4), the calculation up to a 24-site cluster, which is equivalent to 48 sites in the spin-1/2 language, is possible. This state-of-the-art computation with large-scale parallelization enables us to capture the thermodynamic properties of the
SU(4) Heisenberg model on the honeycomb lattice. In particular, the specific heat shows a characteristic peak-and-shoulder structure, which should be related to the nature of the low-temperature quantum spin-orbital liquid phase. We also discuss what can be concluded from the assumption that the ground state is gapped and symmetric in view of the generalized Lieb-Schultz-Mattis theorem.