In the latest issue of Nature, Leon Balents has a nice comprehensive review article, Spin liquids in Frustrated Magnets.

In a Box at the side Leon considers the problem of quantifying the amount of frustration in an antiferromagnetic material (or model) and considers a measure (emphasized by Ramirez in a classic review) f=T_CW/T_N, the ratio of the Curie-Weiss temperature to the Neel temperature, at which three-di

mensional ordering occurs.

Although, indicative of frustration I think we should come up with a more precise measure. One limitation of this measure is it does not separate out the effects of fluctuations (both quantum and thermal), dimensionality, and frustration. For strictly one or two dimensional systems, T_N is zero. For quasi-two-dimensional systems the interlayer coupling determines T_N. Thus, f would be larger for a set of weakly coupled unfrustrated chains than for a layered triangular lattice in which the layers are moderately coupled together.

In 2005, Weihong Zheng, Rajiv Singh, Radu Coldea, and I published a paper, Temperature dependence of the magnetic susceptibility for triangular-lattice antiferromagnets with spatially anisotropic exchange constants.

In Section II we discuss in some detail two different measures of frustration for model Hamiltonians:

1. the number of degenerate ground states

2. the ratio of the ground state energy to the

base energy defined by Lacorre.

The base energy is the sum of all bond energies if they are independently fully satisfied.

One of the main results of our paper is the Figure below

All of the quantities plotted are meaurable in an actual material.

In some sense then the temperature Tp at which the susceptibility has a maximum and the magnitude of that susceptibility is a measure of the amount of frustration. This is consistent with our intuitive notion (prejudice, bias?) that the largest frustration occurs for the isotropic triangular lattice (J1=J2).

These measures of frustration are not dependent on dimensionality and so do not have the same problems discussed above that the ratio f does.

But, it is not clear to me that these measures distinguish quantum and classical frustration. Surely there is a difference? Could entanglement measures help?

Of course I agree - I think I even wrote it in the review - that Ramirez' f is more of a "fluctuation" parameter than a "frustration" one.

ReplyDeleteBut I think for many purposes this is the useful thing. For the search for spin liquids, the most interesting thing is to see a suppression of ordering. Even a true QSL will show a T>0 peak in the susceptibility, and will not look very "frustrated" by a T_p/\Theta_{CW} measure. That measure is really a high energy property, I would say. It is a good one for comparing to series expansions or similar methods, and for determining some exchange parameters if the basic model is pretty well established. But it has not very much to do with low energy physics. And I think its interpretation becomes rather more theory dependent. By contrast, when f is large, it has low energy implications.

As far as the particular concern about dimensionality affecting f, the point is also well made. But I think in most practical situations, it is not so important. An exception is very strongly quasi-1d systems (Cs2CuCl4 is not nearly quasi-1d enough for this). You seemed particularly concerned about 2d systems. While it's true that an ideal 2d Heisenberg system usually has no phase transitions (it can have them, but only if there are non-magnetic order parameters), that is rather fragile. If the ground state is ordered, then the correlation length grows exponentially with reducing temperature, and any very weak 3d coupling induces a rather large (logarithmically) Tc.

Just a little magnetic anisotropy can also yield Kosterlitz-Thouless

or other transitions, even without 3d coupling, with similarly extreme

sensitivity. I think in practice this is not a common cause (in

quasi-2d systems) of large f.

Anyway, I have no special love for this particular measure, but it is convenient, and does not require much theoretical modeling to interpret - at least in terms of fluctuations! On the general question of whether there is something to distinguish quantum from classical frustration, I personally don't think quantum frustration is necessarily a very useful concept. Level repulsion means that it is rather non-generic for quantum systems to have significant ground state degeneracies. The classical concept is probably useful because it tends to give a guide for how stable the (semi)classical description is. Spin wave theory works unreasonably well for many small S systems, and so frustration can at least help give a way out of this unfortunate situation.

The difference between quantum and classical frustration here sounds very suspiciously to the often discussed difference between "dynamical and non-dynamical correlation" in quantum chemistry. In the latter case, there can be no such distinction. Although the concept of a difference has its uses, it actually becomes an impediment once people begin to think its anything more than a fuzzy guide. That there can't be a real difference between the two is clear because of the statistical complementarity principle in quantum mechanics: you can't tell if an observed mixing of the density matrix on some sub-level is due to higher-level quantum correlations or just ordinary uncertainty.

ReplyDeleteAlso, its interesting to note the close correspondence between the frustration measures you suggest and quantitative descriptors invoked by Wolynes and others to describe the speed of folding in proteins (e.g. Bryngelson and Wolynes. Spin glasses and the statistical mechanics of protein folding. Proceedings of the National Academy of Sciences (1987) vol. 84 (21) pp. 7524-7528).

ReplyDeleteWhat I would like to know is what purpose a measure of frustration serves. In the early HFM days every presentation seemingly showcased the frustration index of the material but I never understood why, other than the kudos of finding the most frustrated systems, this arbitrary measure was of any interest. Likewise with any modification of this measure.

ReplyDeleteThanks for Leon for the clarification. I agree that the Ramirez measure is a simple and useful empirical measure for the search for spin liquids.

ReplyDeleteI also agree that the measure I propose is sensitive to high energy properties rather than low energy properties.

My overall goal is to move towards decoding the mantra "spin liquids are most likely to occur in materials with large fluctuations and large frustration." What is cause and effect?

I consider that frustration IS a property of the underlying "high-energy" Hamiltonian, i.e., short-range interactions. Can we quantify it?

We say that frustration increases in going from the square to the triangle to Kagome. But, what do we mean by that, particularly for the spin-1/2 Hamiltonian?

Hopefully, the above also answers Bob a's question.

While I agree with Leon's point that the uniform susceptibility based measures

ReplyDeletesay more about the short distance physics than the long distance physics,

it may be worth exploring these measures for a few different models to see how

frustrated they are by this definition.

I am pretty sure that if the calculations were done for the Triangular-Kagome Lattice,

which interpolates between Triangular and Kagome lattice, one would find Kagome

most frustrated. Basically, we are still not 100 % sure at how low a temperature the Kagome

susceptibility peaks and it clearly becomes quite large by then. So, Kagome is more

frustrated than triangular.

It would be interesting if one were to calculate the measures for a kagome-pyrochlore

structure to see how frustrated pyrochlore is relative to kagome and what is the

most frustrated point by this definition. Similarly, one should be able to come up

with an interpolation between kagome and hyper-kagome.

One place where these definitions run into problems is in models that have any

ferromagnetism or ferrimagnetism. One might need some kind of q-integrated

generalization there. Although that will mean no longer having a simple way

to measure it experimentally.

Hi Ross, maybe the quest is over, or at least we have finally something well defined for all systems and in all dimensions, in principle computable via well defined and unanmbiguous procedures (it is essentially an overlap), observable (either indirectly via stati structure factors or directly by interferometric visibility), and with a clear meaning in terms of the incompatibility between global and local orders.

ReplyDeleteSee here for a thorough application to a simple 1-D playground (and references therein for the earlier more abstract theory works that introduced the measure):

http://arxiv.org/abs/1501.04642