## Monday, November 11, 2013

### The Orthogonality Catastrophe (!)

Physicists sometimes like to use dramatic terminology to describe phenomena or ideas.  For example, the Ultraviolet Catastrophe was the phrase describing the failure of classical statistical physics to predict the correct form of thermal ("black body") radiation in the high frequency ("UV" at the time) limit.  The obvious disagreement between what seemed like a solid calculation (based on equipartition) and observation (that is, you are not bathed by gamma radiation from every thermal emitter around you) was a herald of the edge of validity of classical physics.

In condensed matter physics, there is another evocative phrase, Anderson's Orthogonality Catastrophe.  (Original paper)   Here's the scenario.  Suppose we have a crystal lattice, within which are the electrons of an ordinary metal.  In regular solid state physics/quantum mechanics, the idea is that the lattice provides a periodic potential energy for the electrons.  We can solve the problem of a single electron in that periodic potential, and we find that the single-particle states (which look a lot like plane waves) form bands.  The many-electron ground state is built up from products of those single-electron states (glossing over irrelevant details).  The important thing to realize is that those single-particle states form a complete basis set - any arrangement of the electrons can be written as some linear combination of those states.  (For students/nonexperts:  This is analogous to Fourier series, where any reasonable function can be written as a linear combination of sines and cosines.  Check out this great post about that.)

Now, imagine reaching in and replacing one of the atoms in the lattice with an impurity, an atom of a different type.  What happens?  Well, intuitively it seems like not much should happen; if there were 1022 atoms in the lattice, it's hard to see how changing one of them could do much of anything.  Indeed, if you compared the solutions to the single-particle problem before and after the change, you would find that the single-particle states are almost identical.  However, "almost" means "not quite".  Suppose the overlap between the new and old single particle states was 0.9999999, where 1 = no change.  For $N$ electrons, that means that the new many-particle ground state's overlap with the old many-particle ground state is goes something like $0.9999999^{N}$.  For a thermodynamically large $N$, that's basically zero.  The new many-particle ground state is therefore orthogonal to the old many-particle ground state.  In other words, in terms of the old basis, it seems like adding one impurity (!) produces an infinite number of electron-hole excitations (!!) (since it takes an infinite number of terms to write the new many-particle ground state in terms of the old).

So, where does this fall apart?  It doesn't!  It's basically correct.  The experimental signature of this ends up being apparent in x-ray absorption spectra, in a variety of meso/nano experiments (pdf), and in cold atoms (pdf).

Any other good physics examples of overly dramatic language?

Austen said...

The curious thing about the orthogonality catastrophe is that it turns out to be much less important in many-body physics than the generality of the above argument suggests.

For example, the quasiparticle weight Z (i.e. the residue of the single particle Green's function) is a measure of the overlap of the true ground state with the product state of a free particle and the ground state with one fewer particle. The orthogonality argument would suggest that Z is generically zero, while in fact finite values are the norm. 1D (i.e. in a Luttinger liquid) and the case of adding an infinite mass particle (heavy hole, in the original papers) are two of the situations where you really do get zero. PWA made numerous efforts to argue that the same happens in 2D to substantiate exotic theories of high Tc, to no avail.

Douglas Natelson said...

Hi Austen - Thanks for the insightful comment. This is something that I'd like to understand better. Is there an intuitive way to understand the cases you mention?

Austen said...

Basically it is the physics of recoil. Through interaction with the Fermi sea, the added particle can excite particle hole pairs. In general, these may be have arbitrarily low energies (i.e. hole just below the Fermi surface and particle just above) but finite momentum (because the momentum of particle and hole can point in different directions).

By momentum conservation, the balance of momentum must be taken up by the injected particle, and if its mass is finite, the resulting kinetic energy sets an energy scale p_F^2/2M. Below this scale particle hole pairs can't be excited -- or rather the phase space volume for doing so is dramatically reduced because then the directions of the momentum of particle and hole must almost coincide.

If you consider the heavy limit M to infinity, you get a true IR catastrophe. Likewise in 1D, because the momenta are pointing in the same direction anyway.

A nice discussion of the effects of recoil in different dimension is Nozieres, J. Phys. I France 4 (1994) 1275-1280.

The survival of orthogonality catastrophe for massive impurities in 1D is the main piece of physics behind the "beyond Luttinger liquid" theories reviewed in Adilet et al. Rev. Mod. Phys. 84, 1253–1306 (2012)

Anonymous said...

In the end it is an avoided event (that being a deliberate choice of words).
One could even say it's "just" an electronic reconstruction.

Another example is not from the negative dramatic side, but from the positive side: the "robustness" of surface states on topological insulators against disorder.
THey are robust against backscattering (because of time reversal symmetry), and there will be a conductive state because of the change in topological invariant. But robust against disorder is overdone - see what you see (ARPES E(k)) in a surface exposed to air. The state definitely changes width, and thus lifetime. So robust is too much.

Anyway, these are two of my pet-peeves :-)

matthew foster said...
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matthew foster said...

I prefer overly understated language myself. E.g., "Absence of Diffusion in Certain Random Lattices," Anderson's 1958 breakthrough paper on localization. Or, "Infinite conformal symmetry in two-dimensional quantum field theory," (BPZ) as opposed to, say, "Exact solution to (almost all) 2D critical phenomena," or "One (Virasoro) algebra to rule them all."

As for topological surface states, I'm happy with robust to disorder OR (sufficiently weak) interactions. The precise statement with respect to disorder is not the the prediction of vanishing linewidth, but the absence of localization in the presence of disorder. A finite lifetime just means inelastic scattering. For interactions, the robustness is the absence of an interaction-induced (i.e. Mott) gap. However, the combination of disorder and interactions can be deadly...