Spectral intensity at the Fermi energy for a domain wall in a model of Mn3Sn.   Fermi arcs arc visible connecting projections of Weyl points (marked as dots).

Spontaneous symmetry breaking is one of the fundamental ideas in physics.  In condensed matter, the paradigm is the Ising model of magnetism.  In the Ising model, spins spontaneously orient below the critical temperature, in one of two possible time-reversed orientations, say “up” or “down”.  Without special preparation, when this happens, a system will form domains or both orientations, separated by domain boundaries.  These boundaries are a simple example of a topological defect.  Other examples are vortices in superfluids, and dislocations  and disclinations in crystals.  In general, domain walls form whenever there is a discrete broken symmetry, and so are not limited to Ising models.

These defects, which live in real space, are “old” examples of topology in physics.  The more modern versions are defects in momentum space.  Notable are Weyl points, which are linear crossings of a pair of non-degenerate bands, that can be viewed as monopoles of the Berry curvature.  They are related to a variety of cool phenomena, like chiral edge states and Hall effects, and “teleportation” of electrons from one Weyl point to another.

What happens when we bring the two types of topology together?  The magnetic order which forms a spontaneously broken symmetry state directly influences the formation and locations of Weyl points.  This is because magnetic order locally acts like a Zeeman field on electrons, and modifies their band structure.  So if Weyl points are present in a magnetic system, they generally re-orient in momentum space as one crosses a magnetic domain boundary.  We can expect intimate coupling of the extended electronic states and domain walls.  For example, chiral edge states may live on the latter, and the walls themselves may have distinct transport properties.  Conversely, when one drives a current in the system, carried by the conduction electrons, it may “push” the domain boundaries in unusual ways.

Dr. Jianpeng Liu, a postdoc here, recently studied this with me in the context of Mn3Sn, an exciting material which shows a classic indication of band topology, the anomalous Hall effect, up to room temperature.  It has interesting magnetic order, whose domains and domain boundaries we described theoretically, including their coupling to Weyl fermions.  This material is being studied by a number of experimental groups – we learned about it thanks to Professors Satoru Nakatsuji and Yoshi-Chika Otani at the ISSP in Japan.  We expect there will be a lot more to see experimentally in this interesting material!  You can read our paper in PRL.

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-RuCl₃, Na₂IrO₃, YbMgGaO₄, Yb₂Ti₂O₇, Pr₂Zr₂O₇.  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-RuCl₃ and Na₂IrO₃, 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 YbMgGaO₄, 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 YbMgGaO₄.  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.