Quantum skyrmions

A skyrmion is a topological defect in a ferromagnet.  In a two dimensional system, it consists of a configuration in which every possible orientation of the magnetization occurs, “wrapping” the full sphere once.  One can visualize it by putting the magnetization down at the origin and up at infinity, and rotating smoothly in a plane containing the radial and vertical directions, between these two orientations.  At some radius the magnetization is in the plane, and rotates by 360 degrees as one moves around the origin.

This is a classical topological defect.  What happens with quantum mechanics?  The skyrmion is finite object, so one might imagine that it could behave as a quantum particle.  If it is large, it involves many spins and will be heavy and classical.  But if it is small, what happens?  Rina Takashima, a graduate student from Kyoto University and a KITP graduate fellow, worked with me on this question and we found that indeed a skyrmion becomes a “quasiparticle”.  It has some unusual dynamics and this can lead to interesting physics, for example a sort of “Bose condensation” of skyrmions. So far our work is really focused on chiral magnets, for which the in-plane spin component is fixed by the material.  In the future we hope to look at non-chiral ferromagnets, where skyrmions can also occur but where the “chirality” of the in-plane twist is arbitrary and can be spontaneous.

You can read about this in our arXiv preprint.

What is a panoply?

My paper with Oleg Starykh, “Quantum Lifshitz field theory of a frustrated ferromagnet” was just published in Physical Review Letters.  Our original title, which is in the version we posted on the arXiv back in October 2015, was “A panoply of orders from a quantum Lifshitz field theory”.  We had quite some argument with a referee, who objected to the title, and despite my fondness for getting obscure words into physics publications, we adopted the new title.   A panoply is “a complete or impressive collection of things.”  This was inspired by the remarkably rich phase diagram of a very simple model: the spin-1/2 1d Heisenberg chain with ferromagnetic nearest-neighbor interaction and antiferromagnetic second-neighbor interaction.  In an applied field, a variety of interesting phases emerge.  Individually, these have been understood, but our goal was to synthesize all the different phases together to find a unified picture.


Schematic phase diagram, incorporating the main features predicted by the quantum Lifshitz field theory

The referee objected because we were not successful at a complete and rigorous synthesis. But I think we got pretty far.  What we uncovered was a simple and novel quantum critical theory, which captures the universal aspects of the system, and for which many (but not all) properties could be obtained exactly.  Moreover, we could show that it explained a first order metamagnetic transition, a vector chiral phase, and a spin nematic state.  Maybe not enough to be a panoply in terms of completeness, but I at least was impressed by how far this simple field theory could go.   You can read the paper and decide for yourself.

Icing model

Lucile Savary and I just posted a paper on the arXiv on “Disorder induced entanglement” in spin ice systems.  This was stimulated by experiments by Satoru Nakatsuji, Collin Broholm, and their collaborators on Pr2Zr2O7, which was supposed to be a type of “quantum spin ice”.   The idea was that the exchange couplings between Pr spins would involve a lot of spin-flip interactions, which induce quantum dynamics.  Their experiments revealed that actually disorder was more important than the quantum exchanges.

This turns out to be related to a bit of very common atomic physics.  Pr3+ is what is called a non-Kramers ion, which means that it has an even number of electrons.  It forms a two-level system which we can describe using a spin-1/2 operator like a Pauli matrix, but which is not quite a usual spin.  In particular, for such a non-Kramers ion, the spin operator is not odd under time-reversal.  Actually in this case the z component is odd, but the x and y components are even (this is equivalent to the condition T^2 = +1 that theorists like).  As a consequence, disorder in the material, for example misplaced ions in the Zr sites, or missing/extra O, exert electric fields inside the sample that induce local random “transverse fields”, i.e. terms like h_x S^x +h_y S^y, with different h_x,h_y for each spin.  This effect is electrostatic in origin, so it should be a robust and dominant one.


Schematic phase diagram of the random transverse field Icing model.  See the preprint for an explanation.

A transverse field is a textbook way to induce quantum dynamics in a classical system.  For example, the transverse field Ising ferromagnet is the paradigm for quantum criticality.  What we learned is that in non-Kramers systems, a transverse field can be induced without any real magnetic field, even without breaking time-reversal symmetry.  So we thought: let’s put this to work for us!  One should be able to controllably induce quantum dynamics by introducing disorder in an otherwise classical magnet.  The spin ice pyrochlores are a natural place to look.  The two most studied materials are Dy2Ti2O7 and Ho2Ti2O7.  Both seem to be modeled extremely well by a classical Ising model.  Dy3+ is a Kramers ion, so it does not work for us, but Ho3+ is a non-Kramers ion, so we’re in business!  A good model for disordered Ho2Ti2O7 is thus the classical spin ice Ising model plus a local random transverse field.  Inspired by the textbooks, we called this the random field Icing model.

Yes, the entire post has been a set up for the name.  I’m really proud of it.  The physics is pretty interesting too.  You can read about in the arXiv article.  That’s the original text – we’ve had to cut it since to fit journal length constraints, in our rather long and painful referee process which is still on-going (I don’t understand why so many referees are so bitter – lighten up!).  We rather like the original text which is more pedagogical.

Kitaev, perturbed

No, Alexei is not upset.  Only his model is being perturbed.  In 2006, Kitaev introduced his “honeycomb model” – an exactly soluble Hamiltonian describing spin-1/2 spins interacting via anisotropic exchange on the nearest neighbor bonds of a honeycomb lattice.  He showed that it is a beautiful example of a quantum spin liquid: a highly entangled very non-trivial zero temperature phase of matter.  Most remarkably, its elementary excitations are not spin waves as in a usual magnet, but instead they are Majorana fermions and some exotic “vortices”!  In the past few years, following a beautiful proposal by Jackeli and Khaliullin, it has been recognized that despite the apparently artificial appearance of the model, it might be a good first approximation to a number of real materials.  So one can hope maybe to find Kitaev’s quantum spin liquid in the laboratory!

This hope is reasonable because of stability: Kitaev showed that any small perturbation to his model which preserves time-reversal symmetry leaves the system in the same spin liquid phase.  So an experimental material doesn’t need to be extremely finely tuned to land in the state.  How would one look for it?  The most powerful probe in quantum magnetism is inelastic neutron scattering, which measures a scattering amplitude proportional to the number of excitations of a given momentum and energy created in response to flipping a spin.   It can be calculated exactly for Kitaev’s soluble model.

That is a pretty straightforward calculation, which yields a surprise: there are no excitations created below some minimum energy, or “gap”.  This appears to be a “spin gap”.  It is surprising because Kitaev’s solution shows that there is no true gap: the Majorana fermions actually have a massless relativistic dispersion, like light, so exist at arbitrarily small energies when the wavelength is arbitrarily long.


Here is the calculated structure factor – actually just the new gapless contribution.  The “obvious” part obtained from the Kitaev point adds to this, and contributes above the dashed line.

It’s a bit of a weird result, but it is so straightforward it can’t be wrong…or can it?  In fact, in a paper that just appeared on the arXiv, Xue-Yang Song, Yi-Zhuang You and I showed that the “spin gap” is only a feature of the exactly soluble model.  For any generic Hamiltonian in Kitaev’s spin liquid phase, there is not even an apparent spin gap.  By combining quantum mechanical perturbation theory and field-theoretic arguments, we were able to work out precisely how the gap fills in, and what the low energy structure of spin excitations looks like in a generic system.

This work was spearheaded by Xue-Yang Song, who is a third year undergraduate at Peking University!  Well done Xue-Yang!  You can read about it in the preprint.