The interest in quantum spin liquids seems to be growing in physics. Part of this is due to a flood of experiments on new materials. Many of these draw their spins from a different part of the periodic table than did the previously studied spin liquids: **Ru**, **Ir**, **Yb**, **Pr** instead of **Cu**. In these heavier elements, spin-orbit coupling (SOC) is much stronger, and may even play a dominant role in the local physics of the magnetic ion. Some hotly studied materials like this are α-RuCl_{3}, Na_{2}IrO_{3}, YbMgGaO_{4}, Yb_{2}Ti_{2}O_{7}, Pr_{2}Zr_{2}O_{7}. The upshot of the SOC is that the spins in these materials have very anisotropic interactions, that generally depend on the direction of the bond. Theory of quantum spin liquids, however, has been largely restricted to Heisenberg models with isotropic SU(2) spin-rotation symmetry.

A notable exception is Kitaev’s honeycomb model, which is extremely anisotropic, but as he showed, exactly soluble. Unfortunately, exact solutions are rare, and more general approaches are needed. Even for extensions of Kitaev’s model, which have been heavily studied in the context of materials like α-RuCl_{3} and Na_{2}IrO_{3}, the vast majority of theoretical studies have employed methods that cannot actually diagnose spin liquids, like classical energy minimization, or exact diagonalization on very small systems.

We decided to try to apply a relatively well-established approach to SU(2) invariant quantum spin liquids, the Gutzwiller construction, to a general anisotropic problem. The Gutzwiller construction works by starting with a ground state wave function of a “fake” system of free spin-1/2 fermions. A quadratic Hamiltonian for these fermions determines this wave function, generally constructed by filling momentum states. Then one “Gutzwiller” projects this fermion wavefunction onto the physical spin subspace in which there is exactly one fermion per site. Since the fermions were placed into extended momentum states, the result is highly entangled in real space. It is a pretty established method, and the Gutzwiller projected states are amenable to computationally tractable Variational Monte Carlo (VMC) simulations. They give some interpretation and classification of spin liquids, and there is a physical interpretation of the Gutzwiller fermions as “spinons”: exotic neutral excitations of the quantum spin liquid.

We took as our test problem a microscopic model of strongly anisotropic spins on the triangular lattice, which may be relevant to YbMgGaO_{4}, and is also relatively convenient as the triangular lattice has only one spin per unit cell. Still, you pay a price for giving up SU(2) symmetry: even the nearest-neighbor model has 4 independent exchange parameters, so there is a large phase space to study. Jason Iaconis, the student who spearheaded this project, also added 2nd and 3rd neighbor (isotropic) exchange which makes the phase space even bigger. To compensate, we opted to restrict the consideration of spin liquid states to just one variety, known as U(1) spin liquids, a choice motivated by some interpretations of YbMgGaO_{4}. With this assumption, there are 6 possible spin liquid states, and we tried to judge the competitiveness in terms of energy, amongst each other, as well as with magnetically ordered states. This was still a pretty ambitious problem, and Jason did a remarkable job, helped by another student, Chunxiao Liu, and a postdoc, Gábor Halász, who have become experts in projective symmetry groups, a tool important for understanding these wave functions.

I won’t try to summarize the outcome very much, except to say that a number of issues arose in the process. We found that for some purposes the Gutzwiller projected states are inadequate, and needed to be improved. Jason came up with a novel way to do that. While we hoped to find a state with a fermi sea of spinons, which had been suggested in experiment, this seemed hard to stabilize. Maybe it could be stabilized beyond Gutzwiller, but it is not clear. We did argue that this could be tested directly in experiment by measuring the thermal Hall conductivity, which we showed should be substantial in such a spinon fermi sea.

If you are interested, please have a look at the paper.