Friday, April 25, 2014

Slow spin dynamics in the bad Hund's metal

I have the following picture of a bad metal. It is half way between a Fermi liquid and a Mott insulator. This means that although there is no energy gap, the electrons are almost localised. The imaginary part of their self energy is comparable to the electron bandwidth. Since they are almost localised they have slowly fluctuating local moments. Hence, the dynamical spin correlation function, chi_s(omega) should be narrow, on what energy scale I am not sure. Somehow I expect a qualitative change in chi_s(omega) as the temperature increases above the coherence temperature, as the system crosses over from the Fermi liquid to the bad metal.
But, I am not sure whether this picture is correct because as far as I am aware there are very few calculations of chi_s(omega), and particularly not its temperature dependence.

There are a few Dynamical Mean-Field Theory [DMFT] calculations at zero temperature, i.e., in the Fermi liquid regime, such as described here. 
In the Mott insulating phase there is a delta function peak, associated with non-interacting local moments, as described here.

There are a few calculations in imaginary time, but little discussion of exactly what this means in real time. I struggle to make the connection.
The figure below shows a DMFT calculation of the imaginary time spin correlation function for the triangular lattice, by Jaime Merino, and reported in this PRB.

Here, I want to highlight two DMFT calculations for multi-band Hubbard models including Hund's rule coupling.

Spin freezing transition and non-Fermi-liquid self-energy in a three-orbital model
Philipp Werner, Emmanuel Gull, Matthias Troyer, Andy Millis

Dichotomy between Large Local and Small Ordered Magnetic Moments in Iron-Based Superconductors
P. Hansmann, R. Arita, A. Toschi, S. Sakai, G. Sangiovanni, and Karsten Held

The first paper reports the phase diagram below, where n is the average number of electrons per lattice site. The solid vertical lines represent a Mott insulating phase.

The boundary determined between the Fermi liquid phase and the "frozen moment" phase is determined from the Figure below. The top set of curves are the spin-spin correlation function.

The important distinction is that in the Fermi liquid phase chi_s(tau beta=1/2) will go to zero linearly in temperature, whereas in the frozen moment regime it tends to a non-zero constant.

Similar results are obtained in the second paper, on a slightly different model, but they don't use the "frozen moment" language, and emphasise more the importance of the Hund's rule coupling.

Here, I have a basic question about the above results. The lowest temperature used in the Quantum Monte Carlo is T = t/100 [impressive], and so I wonder if this is above the Fermi liquid coherence temperature and if one could go to low enough temperatures one would recover a Fermi liquid.

It is known that the coherence temperature is often much smaller for two-particle properties, and Hund's rule can also dramatically lower it. Both points are discussed here.

Or is my question [bad metal vs. frozen moments] just a pedantic distinction? The important point is that, practically speaking, over a broad temperature range, the spins are effectively relaxing very slowly.

Anyway, I think there is some rich physics associated with spin dynamics in bad metals and hope it will be explored more in the next few years. I welcome discussion and particularly calculations!


  1. Hi, you say you find it hard to connect the imaginary time data with the real time results. Has Jaime tried doing the analytic continution of the correlators to real frequencies? I know this is generally a rather numerically unstable procedure, but I think progress has been made on this.

    1. Hi, Carlos,
      Thanks for the comment.
      In other papers Jaime has done analytic continuation using Pade approximats.
      For quantum monte carlo [used for the Hund's rule
      results that I showed] the best method is a MaxEnt method, reviewed here by Gubernatis and Jarrell

      But, generally doing the continuation is a far from straight-forward or robust procedure.

  2. Hi, Ross
    I would like to refer you to the review paper of Hund's metal from Antoine group, arXiv1207.3033, regarding your question on the Fermi liquid coherence scale. A similar calculation has been performed at even lower temperature T=1/800, which a crossover into FL regime is seem from the self energy as shown in Fig 7. Best

    1. Hi Xiaoyu Deng,

      Thanks for the helpful comment.
      I had forgotten about that Figure. Thanks for pointing it out.
      It is also a beautiful review article and is now available at