The one-band model is the preferred model in this field, because of its conceptual simplicity. (a) Nearest-neighbour hoping of the hole in the antiferromagnetic lattice creates a string of wrongly oriented spins (shown in red); (b) Spin-fluctuations allow pairs of wrongly oriented spins to un-flip themselves. Combined together, the processes in (a) and (b) allow the hole to move efficiently to any of its second or third nearest-neighbour sites, without disturbing the magnetic order. © 2016 IOP Publishing Ltd.

Mona Berciu from the University of British Columbia describes the differences between the insulating limit quasiparticles of one-band and three-band cuprate models based on her recently published JPCM paper.

Dear reader, we bet you have heard of the following story at least once in a talk about cuprates. A hole (the story goes) cannot move in an antiferromagnet in the absence of spin fluctuations. This is because as it hops it disrupts the magnetic order, creating a string of wrongly oriented spins whose energy increases linearly with its length. This would confine the hole near its original location, if it wasn’t for spin fluctuations that allow pairs of wrongly oriented spins to un-flip themselves, thus removing the string and releasing the hole.

There is nothing very wrong with this story as such (although Trugman loops allow the hole to move, albeit extremely slowly, even without spin fluctuations). The problem is that the resulting dispersion — the dependence of the hole’s energy on its momentum — cannot be like that measured in insulating cuprates. As we show in our work, destructive interference between second and third nearest-neighbour effective hoppings arising from processes, as in the figure above, guarantees a nearly flat dispersion for k_x=π-k_y, whereas experiments find considerable dispersion.

A CuO2 layer is a charge transfer insulator. Upon doping, the hole is located on O (black squares), not on the Cu (blue circles) ; the latter host the spin degrees of freedom. This problem is described by the three-band model. Our work shows that the resulting hole dynamics is very different from that described by the much-simpler one-band model. © 2016 IOP Publishing Ltd.

For agreement with experiments, one must also include second (t’) and third (t”) nearest-neighbour bare hopping of the hole. This is not surprising, because the one-band model aims to describe the actual motion of the doped hole, which is known to be located on the oxygen (O) atoms, through the background of spins hosted at copper (Cu) sites. Clearly, O-O hoping t_pp allows the hole to move without distorting the spin background, and that is what t’ and t” do in the one-band model.

So, do these models describe the same physics?
The answer is: No!

To get the correct dispersion in the one-model model, t’ and t” must be comparable to the antiferromagnetic exchange J (for reasons explained in our paper). In other words, here both spin-fluctuations and longer-range hopping are equally important for the hole dynamics.

In the three-band model, however, t_pp is about 4 times larger than J (and is supplemented by a “spin-swap” O-O hopping that is nearly as big as t_pp). As a result, here the background spin fluctuations have very little effect on the dispersion: the hole moves much faster than the characteristic time-scale on which spin fluctuations act, and the latter cannot influence its dynamics.

The popular story of the importance of spin fluctuations to the dynamics of the hole in an antiferromagnet, while true for one-band models, is not relevant to hole-doped cuprates. Of course, spin-fluctuations may be important in mediating an effective attraction between holes — a possibility we are currently investigating.

About the authors

Hadi Ebrahimnejad received his PhD from University of British Columbia in 2014. He is currently a data scientist at RichRelevance, where he works on developing machine learning models for recommender systems.

George Sawatzky received his PhD from University of Manitoba in 1969. After postdoctoral work at University of Groningen, The Netherlands, he was appointed as professor at the same institution in 1971. He moved back to Canada in 2001 with a Tier 1 Canada Research Chair in Physics and Chemistry at the University of British Columbia.

Mona Berciu received her PhD from University of Toronto in 1999. After postdoctoral work at Princeton University, she joined University of British Columbia in 2002, where she is now a professor in the Department of Physics and Astronomy.

This work is licensed under a Creative Commons Attribution 3.0 Unported License. Figures adapted from H Ebrahimnejad et al 2016 J. Phys.: Condens. Matter 28 105603. © 2016 IOP Publishing Ltd.

Categories: Journal of Physics: Condensed Matter

Tags: Antiferromagnetics, Condensed matter physics, Cuprates, Guest blog, Nearest-neighbour