Wednesday, May 14, 2014

Are there quantum limits to transport coefficients?

An important and fundamental question concerning a strongly interacting many-body system is whether there are fundamental limits (lower bounds) to transport coefficients such as the conductivity and viscosity. A related issue is whether there are upper bounds on energy, phase, and momentum relaxation rates, such as the buzz-concept of planckian dissipation.

There are two basic reasons why some believe this is true.
First, a simple argument is that it "does not make sense" to have mean free paths less than a lattice constant and the wavelength of the relevant quasi-particles. This leads to the Mott-Ioffe-Regel limit for the conductivity.
Second, in calculations based on the AdS-CFT correspondence one does find such bounds do hold.

However, I remain to be convinced that such bounds must hold. One reason is the existence of bad metals. In some strongly correlated electron materials the resistivity can increase smoothly above the Mott-Ioffe-Regel limit as the temperature increases. This is also seen at the level of a Dynamical Mean-Field Theory treatment of the Hubbard model.
A second reason is that I am skeptical that AdS-CFT actually corresponds to any (and certainly not all) physically relevant Hamiltonians.

Today I read an interesting paper
Quantum Mechanical Limitations to Spin Diffusion in the Unitary Fermi Gas
Tilman Enss and Rudolf Haussmann

The paper seems to presuppose that quantum limits do/should exist.
It is motivated by some very nice recent experiments on ultra cold atoms that measure spin diffusion coefficient and spin susceptibility as a function of temperature.

The main result is the graph below. The authors calculations are the red curve. The two important points are that it is a minimum as a function of temperature and that the value is Ds≃1.3ℏ/m, close to the "quantum limit".

A few minor comments.

1. It should be noted that the experimental data has been scaled down by a factor of 4.7 to allow for the inhomogeneity associated with the trapping potential. This is not as as unreasonable or as arbitrary as I first thought. At high temperatures one can calculate the correction for a harmonic trap and it is about a factor of 5.

2. The diffusion coefficient D is calculated from the "Einstein relation", D=conductivity/susceptibility.
No reference is given. This is a rather non-trivial relation, which is discussed and proven by Sachdev on page 171 of the first edition of his Quantum Phase Transitions book.

3. The authors point out that the agreement of D with experiment involves a fortuitous cancellation of errors. Their value for both the spin conductivity and susceptibility is off by a factor of about two, compared to experiment, as shown in Figures 3 and 4 in the paper.

4. I feel calling the calculation a "strong coupling Luttinger-Ward (LW) theory" is a bit too terse. To me LW theory is just a formalism [a self-consistent theory?] and the key point is that to use it one must make an approximation and decide to include only certain Feynman diagrams in the LW functional for the free energy.  My point here is similar to my post, Green's functions are just a technique.

5. To me, the experiments highlight the complementary strengths of cold atom and solid state systems. In the latter (but not the former) it is straightforward to measure the particle conductivity, maintain a homogeneous system, have good thermometry, and cool to temperatures orders of magnitude below the Fermi temperature. However, in contrast, measurements of the spin conductivity in solids are virtually non-existent.

The authors also calculate the frequency-dependent spin conductivity and find that it exhibits a broad Drude peak [the width is of the order of the Fermi energy], with [a very small amount of] spectral weight transferred to a universal high-frequency tail that is proportional to the Tan contact density C.

It is great to see the ultra cold atom community addressing these fundamental questions.

Postscript. Today (May 16) there is a paper in Science of a new measurement [by a spin echo technique] of the spin diffusion constant D, giving a value of about hbar/m, for a three-dimensional gas. The authors also cite a paper from last year which measured a value of D for a two-dimensional gas, that is about 150 times smaller than the "lower bound" of hbar/m.


  1. Do you have any idea in what sense (if any) hbar/mass sets a bound on diffusion, rather than simply a characteristic scale? As a rigorous bound, it is suspicious. What mass? Presumably not the bare mass.

  2. Such a bound is implied by AdS-CFT approaches. But, the relevance to real materials is debatable. That is some of the point of this post.

    The mass is the bare mass.

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