Monday, December 9, 2013

Effect of frustration on the thermodynamics of a Mott insulator

When I recently gave a talk on bad metals in Sydney at the Gordon Godfrey Conference, Andrey Chubukov and Janez Bonca asked some nice questions that stimulated this post.

The main question that the talk is trying to address is: what is the origin of the low temperature coherence scale T_coh associated with the crossover from a bad metal to a Fermi liquid?
In particular,  T_coh is much less than the Fermi temperature of for non-interacting band structure of the relevant Hubbard model [on an anisotropic triangular lattice at half filling].

Here is the key figure from the talk [and the PRL written with Jure Kokalj].
It shows the temperature dependence of the specific heat for different values of U/t for a triangular lattice t'=t. Below T_coh, the specific heat becomes approximately linear in temperature. For U=6t, which is near the Mott insulator transition, T_coh ~t/20. Thus, we see the emergence of the low energy scale.

Note that well into the Mott phase [U=12t] there is a small peak in the specific heat versus temperature. This is also seen in the corresponding Heisenberg model and corresponds to spin-waves associated with short-range antiferromagnetic order.

So here are the further questions.
What is the effect of frustration?
How does T_coh compare to the antiferromagnetic exchange J=4t^2/U?

The answers are in the Supplementary material of PRL. [I should have had them as back-up slides for the talk]. The first figure shows the specific heat for U=10t and different values of the frustration.
t'=0 [red curve] corresponds to the square lattice [no frustration] and t'=t [green dot-dashed curve] corresponds to the isotropic triangular lattice.

The antiferromagnetic exchange constant J=4t^2/U is shown on the horizontal scale.  For the square lattice there is a very well defined peak at temperature of order J. However, as the frustration increases the magnitude of this peak decreases significantly and shifts to a much lower temperature.
This reflects that there are not well-defined spin excitations in the frustrated system.

The significant effect of frustration is also seen in the entropy versus temperature shown below. [The colour labels are the same]. At low temperatures frustration greatly increases the entropy, reflecting the existence of weakly interacting low magnetic moments.


  1. Hi, nice post.
    As a young researcher, I have a question : you use the terms "very well defined peak", "decreases significantly", "much lower temperature" or "frustration greatly increases", but that does not seem to be quantitative. For instance, in the case of the entropy, and just looking at the figure without really knowing the physics, the increase does not seem that great to me.

    How do you quantify that ? What is your criterion ? Is it because (still in the case of the entropy) you know that another unfrustrated lattice would not change the entropy with respect to the square lattice ?



  2. Hi Adam,

    thanks for the questions and for reminding us to quantify our statements more regularly and to clarify the comparisons we refer to. Some more details are in the paper, but let me try to help here a bit.

    With the "very well defined peak" we practically mean that the peak is nicely seen and can be distinguished from the background and that one could extract its width and position, but we do not refer to width/position ratio here. Also, for such systems the appearance of a peak can not be a priori anticipated.

    The "significant decrease" of coherence temperature and "greatly increased entropy" refers more to physics and to comparison with unfrustrated systems. The coherence temperature is around 0.1t, which is much smaller than the band width or U, and in particular if one increases t' from 0 to t, the band-width energy scale is increased, but the coherence temperature is actually decreased. Similarly for entropy. Introducing another unfrustrated J (due to t') would increase the energy scale (something like 'spinons' or 'magnons' bandwidth) which should result in decrease of entropy at certain fixed temperature, but in our case due to frustration the trend is opposite with increase of the entropy by a factor of more than 3 at certain temperatures.

    This is still not quantified as it could ideally be, but hopefully helps a bit with understanding.


  3. Hi Adam,

    Thanks for your questions. I like Jure's answers.

    You are raising some important issues that go beyond my post. Generally, these are qualitative statements that are hard to quantify. Furthermore, they are subjective. Different scientists may have different personal views about what is a "clear" effect and what is a "large" effect. Ultimately, it depends on the context and experience. But, we should be more careful about this. In a similar vein, an earlier post discussed the issue of where authors claim "excellent" agreement between theory and experiment.

  4. Thank you for your answers.
    I agree that it is mainly a matter of interpretation, depending strongly on the context (thanks for the link to the blog post !). The problem when you're new to a community is to understand what is convincing or not for this community

    In my community (strongly correlated cold atoms), if the result of a calculation does not go through the experimental data point, then there is not really a point to publish it (even though the error bars are usually meaningless, when they exist). That is, I guess, really different from your community, where finding a trend (a power law, scaling, etc) is already a breakthrough...