# Classic Rock Revival

I’ve written a commentary for June’s version of the online journal club for condensed matter physics.  Get it here in advance!

Paper 1 (arXiv:1705.00570) : High pressure floating-zone growth of perovskite nickelate LaNiO3 single crystals, by Junjie Zhang, Hong Zheng, Yang Renb and J. F. Mitchell.

Paper 2 (arXiv:1705.02589) : LaNiO3 – a highly metallic and antiferromagnetic strongly correlated transition metal oxide, by Z. W. Li, H. Guo, Z. Hu, L. Zhao, C.-Y. Kuo, W. Schmidt, A. Piovano, T. W. Pi, D. I. Khomskii, L. H. Tjeng and A. C. Komarek.

The rare earth nickelates, with the chemical formula RNiO3 and the perovskite structure based on the cubic lattice, are one of the paradigmatic families of materials undergoing metal-insulator transitions (MITs). They are prominently featured in the authoritative reivew from 1998 on Mott MITs by Imada, Fujimori, and Tokura. With increasing ionic radius of the rare earth element R, the nickelates become increasingly metallic, and in the standard phase diagram, a line of finite temperature MITs terminates at a zero temperature MIT lying between R=Pr, with an antiferromagnetic insulating ground state and the end member, R=La, with a paramagnetic metallic ground state. This phase diagram has probably been reproduced hundreds of times in the literature. In the intervening decades the nickelates have re-risen periodically as prominent research subjects for diverse reasons.

It might be surprising to learn that, despite the fact that the nickelates were an established topic already in the 1990s, this classic rock (mineral) LaNiO3 (LNO) did not actually exist, at least in single crystal form. All the work on LNO was carried out on polycrystalline powders and thin films. Single crystals are not necessarily better than the former, but they are often different, and usually studying them is clarifying. Rather remarkably, as explained in the two featured papers above, after all this time, this year two groups have managed to produce single crystals of LNO, using the floating zone technique with high oxygen pressure.

Both group’s crystals are rhombohedral, with R-3c symmetry, as expected from polycrystal studies. They display a broad maximum of magnetic susceptibility somewhat above 200K, and show metallic behavior down to the lowest measured temperatures, both features consistent with the standard phase diagram. However, a major surprise occurred in the study by Li et al (paper 2) who observed an antiferromagnetic ordering transition at 135K. Neutron diffraction showed clear peaks at the wavector (1/4, 1/4, 1/4) in pseudocubic notation, which is the same type of antiferromagnetic order found in the low temperature phase of all the other nickelates (R=Pr,Nd etc.). Thus according to this study the ground state of LNO is not a paramagnetic but an antiferromagnetic metal! A transition is not observed in paper 1 above, but I would note that the residual resistivity (zero temperature limit of the resistivity) in paper 1 is about 14μΩ-cm, about 36 times larger than the 0.38μΩ-cm found in paper 2, which may indicate that extreme purity is required to realize the antiferromagnetic state.

Added comment 6/26:  My colleague Susanne Stemmer pointed out to me that paper 2’s resistivity may be too extremely low.  The value they show is as low as that of copper, even up to room temperature – this seems particularly surprising and perhaps indicates some issue with the resistivity measurement.  It may be that the difference between the two crystals is something other than purity…Anyway, everyone should be aware that any time two experiments on the same system disagree, there is uncertainty and further work is needed to resolve what is different, and why.  So…please read on, but keep this in mind: be careful, and remember that what follows discusses the implications assuming the other results are solid (the resistivity is not crucial for the conclusions about magnetism).

Assuming the result holds up, what do we learn from this revision of the nickelate phase diagram? It has bearing on a long-standing debate in the literature on the driving force behind the MIT and antiferromagnetism in these compounds. For the more insulating nickelates, R=Sm and smaller rare earths, two transitions occur on lowering temperature. The first transition coming from high temperature is the MIT one, and occurs without magnetic ordering. It is characterized by a symmetry lowering from orthorhombic to monoclinic, driven by an alternating contraction and expansion of NiO6 octahedra around every other Ni site. At lower temperature, antiferromagnetic order at the (1/4, 1/4, 1/4)  wavevector develops. Partly based on these observations, a view that the non-magnetic structural changes drive the MIT and the antiferromagnetism was proposed. Initially thought of as “charge ordering” of differently charged nickel ions, this has also been deemed “charge disproportionation” (I will use this name for concreteness) and most recently a “site selective Mott transition”, viewed through the lens of Dynamical Mean Field Theory (DMFT). In the Nd and Pr compounds, the structural and magnetic transitions occur together, leaving open the chicken and egg question of whether charge disproportionation or magnetism are the driving force here. DMFT studies seem to support the continuing importance of charge disproportionation, while other work advocated for a magnetic mechanism.

At least for LNO, the new work points to a clear answer, since antiferromagnetism occurs in the metallic state. A preliminary search by the authors did not reveal any structural or electronic changes associated with charge disproportionation, though it remains possible that the effect is simply very small. Antiferromagnetism seems to be in the drivers seat in LNO. It would be natural to think then that in the nearby materials PrNiO3 and NdNiO3, antiferromagnetism might again also be a driving factor. There one does observe charge disproportionation, but in fact it follows on symmetry grounds: in an underlying orthorhombic crystal (which these are), charge disproportionation is always induced by (1/4, 1/4, 1/4) antiferromagnetism as a secondary order parameter. This is not true for rhombohedral symmetry, consistent with the absence of charge disproportionation in antiferromagnetic LNO.

Many prior experimental and theoretical results should be revisited in light of these findings. Very narrow atomic scale superlattices of LNO, found to display an antiferromagnetic metallic state, might be stabilizing the inherent three-dimensional order of the crystals. A pseudogap observed in tunneling measurements of LNO films might be indicative of local formation of the antiferromagnetic state. More generally it is inspiring to see previously “impossible” crystals grown using modern machines, and we may hope to enjoy the fruits of these labors more broadly across interesting electronic materials.

# SYKness makes a metal incoherent

Sometimes ideas come from surprising places.  Recently Alexei Kitaev made a splash by pointing out that a problem studied decades ago by Subir Sachdev and Jinwu Ye, who were studying quantum spin glasses, has bearing on the problem of quantum gravity.  What is now called the Sachdev-Ye-Kitaev (SYK) model became the subject of intense study as a toy model in which gravity and black hole physics emerges from totally different microphysics: quantum mechanics of many fermions.  It has a very simple Hamiltonian:

$H_{SYK} = \sum_{ijkl} U_{ijkl} c_i^\dagger c_j^\dagger c_k^{\vphantom\dagger} c_l^{\vphantom\dagger}$ ,

where the coefficients $U_{ijkl}$ are totally random.  Here I’ve written what is called a “complex” SYK model, slightly different from what Kitaev studied, which was written for Majorana fermions.  The difference is not crucial for the gravity connection, but it is helpful for us.  The key feature is that according to this Hamiltonian electrons move collectively, transitioning from two “orbitals” (kl) to another two (ij) in pairs, never alone.  You could visualize it like this:

This is a “toy model” because what emerges (in the limit of a large number of orbitals N) is not the gravity of our world, but a slightly different, and lower-dimensional version.  But still it is remarkable and beautiful.

The SYK model also describes the physics of strongly interacting electrons, forming an unconventional metallic state.  This metal is what condensed matter physicists refer to as a “non-Fermi liquid”, which behaves differently from typical metals like copper, in which electrons act almost independently.  In the SYK metal, you cannot separate motions of single electrons, but instead only see a collective dance of the whole electron fluid. The problem of understanding non-Fermi liquids is a forefront one in the field of quantum materials, but there are very few solvable theoretical models for them.  As such, the SYK model is quite valuable.

The drawback of the SYK model is that it is “zero dimensional” — every electron in it interacts with every other one, and so there is no built-in locality to the problem.  Because of this we cannot imagine driving a flow from one side of the system to another, and hence cannot discuss conductivity.  We should view the SYK model as describing a strongly-interacting “quantum dot”: a little chunk of non-Fermi liquid metal.

A natural thing to do is to try to string together these quantum dots to build up a higher dimensional system.  Already a couple of very nice papers along these lines have appeared: this paper, and this one which is a generalization.  In both these works, coupling between the SYK dots is chosen to preserve the scaling of the SYK model: electrons must hop in pairs between the dots.  Consequently the whole system remains a non-Fermi liquid and no new energy scale is introduced by the coupling.

We just posted a paper in which we make a different choice — just ordinary one-electron hopping between the dots:

$H = \sum_x H_{SYK}(x) - \sum_{\langle x,x'\rangle} t_{ij}(x,x') c_i^\dagger(x) c_j^{\vphantom\dagger}(x') + {\rm h.c.}$ .

In this model, there is a competition between the SYK interaction U and the hopping t.  For t<<U, we have a strongly correlated system.  The header image is suppose to show this: electrons zipping between the correlated fluid dots.  Fortunately, it remains soluble in the large N limit and one can obtain a rich structure.  Briefly, at low temperatures and low energy, the system is a Fermi liquid.  We can extract the parameters of the Fermi liquid theory: effective mass, Fermi liquid interactions, quasiparticle residue, etc.  However, at higher temperatures it is an incoherent metal, and displays properties that are a mixture of those of the pure SYK model and a regular metal.  Together, it is striking how many of the properties are similar to those of experimental correlated metals:

• Small coherence scale Ec = t2/U
• Large effective mass (Sommerfeld coefficient) γ ~ m*/m ~ U/t
• Small quasiparticle weight Z ~ t/U
• T2 low-temperature resistivity with Kadowaki-Woods ratio A/γ2 = constant
• linear in temperature resistivity at high temperature
• linear in temperature thermal “resistivity” T/κ at high temperature
• Fermi liquid Lorenz number L = κ/(T σ) = π2/3 at low T
• non-Fermi liquid Lorenz number L = π2/8 at high T

Anyway, I’m pretty pleased with how it all works out.  Maybe the most remarkable part of this story is that the person who really spearheaded all the calculations, Xue-Yang Song, is still an undergraduate.  Pretty impressive.  You can read the preprint here.