Tuesday, January 23, 2007

The most powerful idea in condensed matter physics?

To get some science discussion going, I thought I'd throw this out there. There are many candidates, but based purely on citations alone, one could make a credible argument that the most powerful idea in condensed matter physics is the (first) Hohenberg-Kohn theorem: the external potential V(r) of an electronic system can be determined exactly (to within a trivial additive constant) by the ground state electronic density rho(r). This means that, in principle anyway, if you know the ground state rho(r), you know everything - you've exactly specified the Hamiltonian, which means you've specified all the many-body wavefunctions for the ground and excited states of the system, all just by knowing the ground state density. Pretty impressive. It's the basis for all of density functional theory. The original paper's been cited 5059 times (as of this morning), and the followup paper that proposed a practical approximation method to make this useful for calculating electronic structure has been cited 11963 times (as of this morning).

On the other hand, I suspect that if you asked a modern CM theorist, they'd list other choices before getting to that one.

10 comments:

Incoherent Ponderer said...

I am not sure how many citations it would get, but Bloch's theorem, that provides the explanation of why electrons don't scatter off atomic lattices gets my vote.

Perhaps the fact that crystalline, i.e. periodic arrangment of atoms is favored for most substances over a glassy disordered state is an even deeper, more fundamental principle of condensed matter.

Finally, Fermi theory of free electrons, the idea that one can reduce the problem of 10^23 particles in a macroscopic system to behavior of weakly or non-interacting, low-energy excitations ("quasi-electrons") that adiabatically mapped onto most real electronic systems.

In retrospect, it is absolutely amazing how well this seeminly naive idea worked for most materials.

Imagine a world where solids always formed a ground state of aperiodic (glassy) atomic structure, all electrons are interacting with each other like crazy and scatter off everything. This would be very messy state of affairs, allowing for very little progress in the field of condensed matter physics.

CarlBrannen said...

I agree completely that the Hohenberg-Kohn theorem is #1. Furthermore, it is very much unappreciated in elementary particle physics (which has stolen so many other ideas from condensed).

From my point of view, the importance of the theorem is that it suggests that one can model fermions in the density matrix form. The reason this is important is that fermion wave functions are normally defined in the spinor formalism by requiring psi(r_1,r_2)= -psi(r_2,r_1).

And this reads into the question of whether the density matrix formalism can be assumed to be fundamental (without the spinor formalism).

It turns out that the density matrix formalism is far more restrictive, and therefore powerful, than the spinor formalism. The reason is that U(1) gauge freedom is eliminated. And this gives a hint on how to do it for the other gauge symmetries.

This is the central idea behind my description of the standard model from density operator theory.

Elia Diodati said...

"This means that, in principle anyway, if you know the ground state rho(r), you know everything - you've exactly specified the Hamiltonian, which means you've specified all the many-body wavefunctions for the ground and excited states of the system, all just by knowing the ground state density."

The weasel phrase here is "in principle", since it is certainly NOT true that excited state information can be extracted directly from the ground state density. You'd still have to fall back on wavefunction-based quantum mechanics to find excited-states, which really doesn't solve anything.

What would be really nice would be to bypass the stage of reconstructing the Hamiltonian and solving for excited state wavefunctions. The evidence so fabr suggests that you can't do so with only one-body information, but in principle the two-particle reduced density *matrix* \Gamma(r1, r2) does encode *everything* about the system, including excited states.

Peter Armitage said...

>Fermi theory of free electrons,
>the idea that one can reduce the
>problem of 10^23 particles in a
>macroscopic system to behavior of
>weakly or non-interacting,
>low-energy excitations >("quasi-electrons") that >adiabatically mapped onto most >real electronic systems.

I believe this idea is more due to Landau, that Fermi. I've referenced L. D. Landau, Sov. Phys. JETP 3, 920 (1956); 5, 101 (1957); 8, 70 (1959)... but I don't remember what is there.

If I had to come up with my top n- CM papers, it would be (in no particular order).

-Both Kohn papers that Doug mentions
-Bloch's paper on bloch's theorem
-Anderson's paper on localization
-Laughlin's paper on the FQH.
-Onsager on the Ising model
-Landau's papers on Fermi liquids
-And BCS's paper on BCS
-And whatever the most appropriate Bethe ansatz paper is.

What did I forget?

Anonymous said...

Thanks for a very interesting topic. I don't know enough about this subject to comment, but from the little I know I'd agree with all of the above, and perhaps add Philip Anderson's non-technical "More is Different"...

-Sujit

AKM Newaz said...

I think we should not forget to include also the seminal work by R. Landauer and M. Buttiker on the relation between conductance and the transmission probability for quantum transport in a mesoscopic device.

Peter Armitage said...

Roger that on including Landauer and Buttiker in my list, but not so much "More is Different" ... as much as I live my life by that paper.

I see it of the same ilk as philosophy of quantum mechanics papers. One doesn't need to have some opinion on what QM "means" to do calculations. In the same way, one doesn't need to recognize emergence as a fundamental principle to calculate something in condensed matter physics.

These ideas behind "More is different" make the field interesting to us and point us in the direction of what is important, but they don't help us calculate anything. My list enables calculatations or demonstrates novel states of matter. My $0.02.

As an aside, an anectdote was relayed to me some years ago about how when P. Anderson gave his "More is Different" lecture at UCSD (it was first given as an lecture) in ~72, guys in the audience like Kohn HATED it... I mean really really HATED it. They were of course in the process of trying to derive all of CM physics from first principles, so of course they didn't want to hear that it couldn't be done... even in principle.

smm said...

ok, here's something that i wouldn't call the most powerful idea in condensed matter physics, but i feel is underappreciated: linear response theory or the kubo formalism. why should this describe transport phenomena so well?

Doug Natelson said...

IP - Bloch's theorem is a great contender, and Fermi liquid theory is also impressive. The fact the the Pauli principle strongly limits what electron-electron interactions can really do is really the key to the remarkable success of FL theory.

Landauer-Buttiker is profound, but given that in its normal form it's a single-particle picture of wave scattering, I'm not sure that it's at the same level as some of the other things mentioned. It's the electron wave analog of the scattering matrix approach to treating, e.g., EM propagation through a complicated array of scatterers.

Anderson localization is a pretty neat idea - that disorder plus the wave nature of electrons (or light) leads to localized states. As much as I'm a fan of this with electrons, I think it's also pretty amazing that it works for photons. You can get things like "random lasers", where there really isn't a traditional cavity - just localization of light in a gain medium due to disorder scattering.

SMM - the thing that I find profound about linear response is really the underlying fluctuation/dissipation physics.

Trent Jordan said...

It turns out that the density matrix formalism is far more restrictive, and therefore powerful, than the spinor formalism.