Friday, September 2, 2016

The mysterious origin of resistivity in Fermi liquids

It is hard to believe that we really don't understand the basic issues that I am going to discuss.

Resitivity occurs in a metal because scattering causes decay of charge currents. This means that the total momentum of the electrons in the presence of an electric field decays.
However, in a Fermi liquid metal with strong electron-electron interactions the main scattering of electrons is due to electron-electron scattering. But, in such collisions the total momentum of the two electrons is the same before and after the collision.
One can calculate the life time of the quasi-particles and it is inversely proportional to the temperature squared. The quasi-particle scattering rate ~ T^2.
Suppose one makes the relaxation time approximation in the Boltzmann equation or equivalently, neglects vertex corrections in the corresponding current-current correlation function associated with the Kubo formula for the conductivity. Then the resistivity is proportional to the quasi-particle scattering rate and one has resistivity ~ T^2. We say the transport lifetime is the same as the quasi-particle lifetime.
However, these are approximations, and strictly speaking there is no decay of the total electron momentum (or current) by electron-electron scattering and so the resistivity should be zero!
One way to save the situation is when there is Umklapp scattering. However, this requires a special relation between the shape of the Fermi surface and the Brillouin zone, as illustrated below.

These issues and puzzles are highlighted in a beautiful paper

Scalable T^2 resistivity in a small single-component Fermi surface 
Xiao Lin, Benoît Fauqué, Kamran Behnia

By chemical doping they tune the charge density and Fermi energy by several orders of magnitude, with the size of the Fermi surface increasing from some very small fraction of the Brilloiun zone.
In all cases the resistivity equals A T^2, characteristic of electron-electron scattering.
The figure below shows how the proportionality factor A scales with the density.
They also find A scales with the inverse of the effective mass squared as one expects from the Kadowaki-Woods ratio.

Yet for small densities (and Fermi surfaces) it is just not clear how one can have electron-electron scattering since Umklapp scattering is not relevant.

This major puzzle awaits an explanation.

I thank David Cavanagh, Jure Kokalj, Jernez Mravlje, and Peter Prevlosek for stimulating discussions about this topic.

Note added. The theoretical issues are nicely reviewed in
Resistivity of non-Galilean-invariant Fermi- and non-Fermi liquids 
 H. K. Pal, V. I. Yudson, D. L. Maslov


  1. Resistivity occurs as a result of irreversible processes, which can not be captured by pure dynamics obeying time reversal symmetry. Some irreversible element must be added. There is a discussion on this here:

    1. Thanks for the comment.

      Unfortunately, the link you give makes incorrect claims. Specifically, one does not have to add irreversibility, by hand. You can start with a microscopic Hamiltonian that is time reversal invariant and can calculate transport coefficients for irreversible phenomena such as resistivity. Boltzmann showed this.

    2. I think when Boltzmann used a probabilistic description of for the collision integrals, some irreversibility element has already been implemented. Taking Kubo-Greenwood: in this method, the irreversibility element is introduced by an infinitesimal imaginary part of the frequency. On the other hand, it is indeed possible to have irreversibility of a subsystem from a totally microscopic time-reversal Hamiltonian but of a bigger system. This is done in e.g. quantum open systems, where a heat bath is mimicked by e.g. a pool of bosons or fermions. A good amount of such discussions can be found in

      (1) L. A. Banyai, Lectures on Non-Equilibrium theory of condensed matter.
      (2) Hansen, Basic concepts for simple and complex liquids.

  2. Nice post. One small technical remark on your comment regarding the relaxation-time approximation and vertex corrections. As far as I recall, the "1 - cos(theta)" factor in the relaxation time can actually be shown to originate from vertex corrections in the current-current correlation function. This was discussed by Baym in "CONSERVATION LAWS AND THE QUANTUM THEORY OF TRANSPORT: THE EARLY DAYS". Correct me if I am wrong.

    1. Yes. You are correct. This is also in Mahan and mentioned in the blog post I linked to.

    2. No, you misunderstood me. Making the relaxation-time approximation is not equivalent to neglecting vertex corrections. The "1 - cos(theta)" factor originates from vertex corrections.

  3. The review you added is a very nice one. However, it seems even if one combines electron-electron and electron-impurity scattering (an alternative mechanism of T^2 law studied in the review), the data of the cited work cannot be explained. One thing that puzzles me is as follows. If impurity plays a role, the T^2 coefficent A show be a (smooth?) function of disorder strength, which to some extend is described by \rho_0. From the senond figure that you show, \rho_0 is sort of a ramdom function on n. Then, how can A be a smooth function of n? Maybe impurity is also ruled out ...

  4. I found this paper when trying to understand phonon-phonon scattering processes, which argues that the distinction between Umklapp and Normal processes is artificial.

    The crystallographic momentum is not a true momentum: it is defined modulus G, and depends on the choice of unit cell. It is only conserved modulus G. The group velocity is uniquely defined, doesn't depend on the choice of unit cell, and conserved. So the distinction between Umklapp and Normal processes is only in terms of conservation of quasimomentum, true momentum is always conserved.

    Maznev, Wright (2014). "Demystifying umklapp vs normal scattering in lattice thermal conductivity." Am.J.Phys. 82 (11), November 2014

    So could a similar process be going on here? Of course, the Fermi surface is a real thing, and is located in the first Brillouin zone (whereas with phonons you are always integrating through the zone), but is there such a strict definition between N and U processes in electron scattering?