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I should preface this by admitting that my physics background is rather weak so I beg you to keep that in mind in your responses. I work in mathematics (specifically probability theory) and a paper my adviser asked me to read dealt with a discrete approximation to the random Schrödinger operator $\Delta + V(x,\omega)$ where $V$ is some random function. The physical meaning of this object is completely beyond me at this point and I was hoping that someone here could shed some light on it.

The basic issues I am trying to work through now are what this actually models (one professor said something about 'disordered media') and whether there are physical reasons to believe that there should be universality of spectral statistics if one makes some kind of reasonable assumptions about the behavior of $V$. I know that proving the last statement rigorously is a major open question in mathematics, so I do want to emphasize that I am really just looking for a little hand-waving to help understand what is going on.

Based on a few explicitly solvable classes of discrete approximations in the limit, I want to say that universality should hold for some broad classes of $V$, depending on the rate of decay of $V$ since the limiting spectral statistics from the discrete approximations seem to have that property if everything is taken to be independent. I have been told that this is a somewhat common phenomenon in the study of Schrödinger operators, though again I don't really understand what is going on here.

Anything would help: links to good write-ups on the random (or not random) Schrödinger operator, a toy example that is illuminating, a suggestion on good papers to read--anything really.

Thanks :)

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Huh... interesting question, Chris. The operator $\Delta + V = \nabla^2 + V$ is basically the nonrelativistic Hamiltonian, but I'm not really familiar with the possibility of $V$ being a random function. Hopefully someone else will know. In any case, welcome to Physics Stack Exchange! –  David Z May 12 '12 at 23:22
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The physical phenomenon is Anderson localization, where adding impurities to a conductor can cause it to become insulating (mathematically this corresponds to the operator having a discrete spectrum in addition to the continuous one). Unfortunately, I cannot think of any literature which deals with the mathematics of it by actually doing analysis on the operator --- we usually skip it and averaging observables over the disorder, and this usually ends up in the realm of path integrals or the "replica trick", neither of which are mathematically rigorous. –  genneth May 13 '12 at 0:51
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Actually, I lie. This paper seems to be bordering on being mathematical, due to the heavy machinery it deploys. In particular, section 2.4 contains some references to purely mathematical literature which actually prove things about such random Schroedinger operators. –  genneth May 13 '12 at 0:54
    
Thanks to both of you! @genneth I'm not too concerned with finding mathematically rigorous explanations here, since I know a few good researchers on the math side of this who I can get that kind of information from. If you have any links to the non-rigorous arguments (even ones that average over the randomness) that give some kind of intuition, I would appreciate that. I'm really just trying to get a feel for what physically 'should' be true here. –  Chris Janjigian May 13 '12 at 16:16
    
In which case, I can recommend combing through the references in the paper I mentioned above. If you're used to path integrals, then I can suggest tcm.phy.cam.ac.uk/~bds10/publications/lesh.ps.gz –  genneth May 13 '12 at 22:37
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The random potential is a model for small fluctuations in the local energy of an electron, when there are defects or impurities that raise the energy of an electron at certain spots by a little bit, and lower it at other spots. This is a good model, although it doesn't look like one at first, because the quantum mechanical solution is not sensitive to the finest details of the random ptential V (in low dimensions), rather the quantum mechanical electron only notices the slowly varying average value of V from region to region.

This is a huge field, and I am only giving a very superficial overview. It is one of the most studied problems of the past half-century, and it deserves the attention it gets.

The random potential looks superficially intimidating, because the usual definition is that the potential V is gaussian random at every point, and generated by the probability distribution

$$ e^{-{1\over 2}\int V^2 d^dx} $$

in D-dimensions. This is an independent Gaussian at every point x of d-dimensional space, and the statistical correlation function of V for different picks from this probability distributions is

$$ \langle V(x)V(y)\rangle = \delta^d(x-y) $$

You should think of this as follows: the space is made a lattice of size $\epsilon$, the integral is replaced by a sum, and the delta function by a discrete bump of size $1\over \epsilon^d$. This is probably the discrete approximation you saw. Then the statement is that in the limit that $\epsilon$ goes to zero, you get a sensible theory. The theory is the eigenvalue of the Schrodinger operator

$$ (\Delta + \lambda V)\psi = E\psi $$

Where $\lambda$ is the disorder strength--- it tells you how strong the randomness is.

The point is that the potential energy in each lattice site is a Gaussian of width $\lambda\over \epsilon^{d\over 2}$. The Gaussian has a huge variance, and the potential swings from a huge positive value to a huge negative value every few neighboring cells. It doesn't look like a sensible notion of potential superficially.

Potentials that are this crazy make no sense classically, since you have turning points everywhere, and classically, you just oscillate around the deepest minimum, and you can't, because there are bumps everywhere blocking your path. For this problem, you have to stop thinking classically completely. In quantum mechanics the particle is tunneling around the minima, going through to other minima, and getting suppressed some whenever it has to go through a high potential region.

The main question here is whether the particle can wind its way to infinity, so that the wavefunction is extended, or if the wavefunction of every state decays exponentially at long distances. This is the question of whether the random material localizes the electrons, so that it is an insulator, or delocalizes the electrons, so that it is a conductor.

The classic reference for this is the Nobel Prize winning paper "Absence of diffusion in random lattices" by P.W. Anderson. Despite the title, it isn't talking about statistical diffusion, it is talking about this quantum mechanical thing.

Self-consistency in dimensions < 4

The first thing to do is to make sure that the problem has a self-consistent small-spacing limit. This is probably what you are interested in doing rigorously, but the heuristics are instructive.

Consider placing a small wavefunction bump of width a on top of the random potential. The thing to check is that you (almost surely) can't gain an infinite amount of energy by placing the bump in the right place, at a point where the average of V over the bump is going to be very negative. Dividing the region of size $a^d$ in to $\epsilon^d$ cubes, you find that there are $N=(a/\epsilon)^d$ of them, each random, and so by the square-root law, the typical negative value of V on this bump will have the size $1\over \sqrt{N}$, while the fluctuations in V on the scale $\epsilon$ blow up as $1\over \epsilon^{d\over 2}$. The $\epsilon$ parts cancel out, and you find the a scaling of the typical potential energy in a region of size a:

$$ V_\mathrm{typ} = - {\lambda\over\epsilon^{d\over 2}} ({\epsilon\over a})^{d\over 2} = {\lambda\over a^{d\over 2}}$$

As a gets bigger, you average over more independent random V's, and you get a smaller average, and it shrinks as this power law. As a gets smaller, you can make a more and more negative potential energy without bound, because the fluctuations get bigger (and you look for the most negative region). But there is a cost to making a smaller, the small width means there is a momentum uncertainty of order $1/a$, and this makes a kinetic enegy which goes as $1/a^2$. So so long as the kinetic energy beats the potential energy in how it blows up, there will be a good limit. Otherwise, the particle's states will collapse to a point. You probably are asked to prove this rigorously.

The condition is that d<4. So the problem makes sense in 1,2,3 dimensions, and maybe in 4. It also makes sense on fractal shapes of the appropriate fractal dimension strictly less than 4. The case of 4 dimensions is marginal, and I am not sure if the problem is well posed there, if it depends on the lattice details, or if there are different 4 dimensional fractals where it works and others where it doesn't. This is the boundary case.

To make this rigorous might be somewhat difficult, because you need to prove that with probability 1, one cannot gain energy by finding very special locations in a configuration of V where the potential energy is much much smaller than the typical energy flucuation size. There might not be good methods in mathematics for doing this rigorously right now, but it is completely obvious on physical grounds (the number of positions is only growing polynomially with the mesh shrinking, while the number of configurations on the shrinking mesh grows exponentially. The much greater number of possibilities for the microscopic V values makes it obvious that searching through polynomially many positions is not going to get you more than a small factor over the naive estimate above.)

1 dimension is exactly solvable

The one dimensional problem can be solved exactly, and this was done by Bert Halperin in http://prola.aps.org/abstract/PR/v139/i1A/pA104_1 . One way to treat this is to use the so-called "R-matrix", which is a 1-dimensional reflection/transmission matrix that tells you how plane-waves scatter/reflect off a bump.

The product of the R matrices in a series is the R-matrix for the random potential problem, and in this case, you can analyze the product numerically, and see that you get attenuation for all V's. This means that all wavefunctions in 1d are localized, they all fall off exponentially at long distances.

One surprising (but true) prediction is that all wires are eventually insulating. Since all materials have a little bit of random potential, all one-dimensional wires will localize eventually. The reason you don't see this is because long metal wires both have a relatively small disorder, and a large thickness, so that the localization length is much larger than you can measure.

2 dimensions is critical

The two dimensional problem is fascinating, because it is the critical point for a renormalization group analysis of the problem of localization. At two dimensions, the random potential problem is marginally localizing. This is an involved analysis, and I am not prepared to write about it right now.

One thing that this suggested is that 2d is like 1d, in that every $\lambda$ localizes. There were simulations, however, that suggested that this isn't so, that there is a localization transition in 2d. This debate went on for 20 years in condensed matter physics, and I have heard that it is considered resolved now, although which way, and with what methods (numerical or RG) I wouldn't be able to say.

3 dimensions: Anderson transition and weak localization

The three dimensional case is very interesting, since it has a second order phase transition between localizing and delocalized states as you tune the parameter $\lambda$.

At $\lambda=0$, you have complete de-localization, all the states are momentum states, they are spread out and do not decay at infinity. When $\lambda$ is weak, you can treat the random potential as a small perturbation, and calculate the corrections to material properties from a few orders of perturbation theory. This is not so great from a mathematical perspective, because the perturbation isn't localized at one spot, so the eigenfunctions are completely different to a mathematician, but for a physicist, the current transmission in the conductor is only altered by a sequence of scatterings from the randomness.

These scatterings have the property that if you scatter k-times, and you scatter exactly backwards k-times, you have the same phase for both processes. This leads the particle to want to stay put more than you expect, because of the constructive interference between the paths and their time-reverse. If you break the time-reversal symmetry, by introducing a magnetic field, you then allow the electrons to flow better, because you remove the constructive interference.

This effect is called weak localization, and it leads to materials having a resistivity peak at 0 applied magnetic field. This magneto-resistance was a very active subject in the 1980s and 1990s, as was weak-localization in general. One of the nice aspects of this is that it lets you figure out how long electrons maintain phase-coherence in a material, since the effect requires constructive interference of paths that are extended a significant way inside the metal.

But the standard story is just the localization. At extremely high $\lambda$, we know from the localization scaling that the wavefunctions will scruch up small, but we know from the dimensional scaling that they can't go down to $\epsilon$, but must occupy some intermediate scale.

Anderson suggested studying this starting from completely localized states. The

Approaches

Disorder problems are interesting because they require you to average over the disorder, but not like a dynamical variable that is thermally fluctuating, but like a static thing. Physicists call this "quenched" disorder, since it is analogous to quickly quenching a hot material like steel in water and freezing the impurities and disorder in place, without allowing them to come to thermal equilibrium and iron themselves out.

For quenched disorder, you want to compute correlation functions, and then average over the disorder,

$$ \langle \phi \phi \rangle_V = \sum_V P(v) \langle \phi\phi\rangle = \sum_V P(V) {\partial\over \partial J}{\partial\over \partial J} \log[Z(J)]|_{J=0} $$

The quantity $Z$ is also a sum over configurations, it is the quantum path-integral (or the quantum statistical path integral). The difference is that the average over $V$ has to be done after you take the log of $Z$, since you don't want $V$ to become a dynamical quantum field, or a classical statistically fluctuating field, but you just want to average the results of the calculation over all values of $V$.

Traditionally, there are two ways to do the average over $V$. The first is Parisi's replica trick: the idea here is to perform the sum over $V$ with $N$ copies of the system:

$$ \sum_{V\phi_1...\phi_N} e^{-S(\phi_1) - S(\phi_2) ... - S(\phi_N)} $$

This gives

$$ \langle Z^N\rangle $$

then you take the formal limit $N\rightarrow 0$, in which the leading scaling is $$\langle \log(Z)\rangle $$. This idea is probably the hardest thing to imagine making rigorous, but this replica idea has been immensely fruitful for physics. It bears a resemblance to the notion of Renyi Entropy in mathematics, but it is more formal, since the actual partition functions are only fully defined for integer $N$ greater than 1.

This is a vast field, and you can google "replica trick" and "replica symmetry breaking" to learn more. This method is indispensible to modern condensed matter physics.

The second method is the supersymmetry approach, and it relies on the following fact: in a SUSY system, the partition function is exactly 1! This might look surprising, but it is obvious in a stochastic system with a Nicolai map (see this answer: A certain $\cal{N}=2$ superconformal theory (or is it?)) . When you have a stochastic system, the partition function is constant--- it only depends on selecting the noise variable, not on the values of the field.

If you use this fact, together with the fact that the derivatives of $\log(Z) $ with respect to sources is the reciprocal of Z times the derivative of $Z$ with respect to the sources, you get that for a SUSY system, you can average $\log(Z)$ just by averaging $Z$. This is the other way of dealing with disorder.

The SUSY method and the replica method are complementary, and have given insight into different problems. The replica method has been more general although less rigorous, and more fraught with worries about whether it works.

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Thank you for taking the time to write this. It's going to take me a little while to read it carefully, but I really appreciate the effort! –  Chris Janjigian May 14 '12 at 11:12
    
I cleaned up the latex slightly, to try and perfect what is otherwise a brilliant response. You've got a dangling thought just before the section "approaches". –  genneth May 14 '12 at 11:57
    
@genneth: thanks, I didn't see the horrible tex errors. –  Ron Maimon May 14 '12 at 15:53
    
@genneth: I see the missing exposition: that was supposed to be an exposition of Anderson's "local-ator" (or whatever he called it, it's his version of time-dependent and time-independent perturbation theory where you make the hopping the perturbation, and the random potential the $H_0$. This is what shows you that there is always a pure localization limit at high disorder.) I'll fill it in. –  Ron Maimon May 15 '12 at 5:22
    
@Ron Maimon : Actually this is a very interesting answer. Could you provide me some references about the supersymmetric approach to obtain mean quantities? I am currently studying a random electromagnetic system, and after attacking it with the replica-trick, I'm willing to try the more rigorous SUSY formulation, but I cannot find more detalied information on it. –  Juan Sebastian Totero Dec 9 '12 at 13:57
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