# Scaling theory of Anderson localization

Initially, Anderson studied the eigenstates of the tight-binding Hamiltonian

$$H = \sum_n \epsilon_n a_n^\dagger a_n + V \sum_{m,n} a_m^\dagger a_n .$$

His question was whether the eigenstates are localized or extended. But in the paper by the 'Gang of Four', the four introduced the dimensionless conductance

$$g(L) = \frac{2 \hbar}{e^2} G(L) .$$

And it seems that this quantity plays an central role.

But how is this quantity related to the original problem of Anderson? How is it related to the localization/delocalization of the eigenstates?

Is $g(L)$ determined by the Hamiltonian above, or some more parameters are needed, say, the temperature?

But how is this quantity related to the original problem of Anderson? How is it related to the localization/delocalization of the eigenstates ?

Let say we have a finite disorder system of size $$L$$ and dimension $$d$$ in which we put some particule. If the system is closed, in the sense that the particle cannot get out of the system, then the possible states are bound to the system, such that the energy eigenstates form a discrete spectrum. Considering all these discrete eigenstates, one can compute the mean level spacing $$\Delta E$$ separating each eigenstate from each other, which is associated to the mean density of states $$\nu$$ (by unit volume) : $$\nu=\frac{1}{\Delta E\;L^d}.\tag{1}$$ In turn, if the system is open, the particle can eventually get out of the system by reaching the boundary. The time $$\tau_D$$ it takes to the particle to diffuse to the boundary is determined such that : $$L\sim\sqrt{D\,\tau_D}\quad\text{i.e.}\quad\tau_D\sim\frac{L^2}{D},\tag{2}$$ where $$D$$ is the diffusion coefficient.

This means that, through its diffusion in the disorder, the particle can only resolve individual eigenstates with an energy resolution $$\delta E$$ such that : $$\delta E\,\tau_D\sim\hbar\quad\text{i.e.}\quad \delta E\sim\frac{\hbar D}{L^2}.\tag{3}$$ where $$\delta E$$ is oftenly called the "Thouless energy".

Then, the Thouless criterion for localization [1] tells us that the system is Anderson localized when this energy resolution $$\delta E$$ is much smaller than the mean level spacing $$\Delta E$$. Indeed, in this limiting case $$\delta E\ll\Delta E$$, the particle can resolve each one of the eigenstate and occupy one of them for ever.

Conversly, in the limit where $$\delta E\gg\Delta E$$ the different energy levels overlap on each other, meaning that the particle can be coupled and jump from one state to another, thus leading to diffusion toward the boundary of the system and delocalization.

After this discussion, it seems quite natural to define the quantity : $$g=\frac{\delta E}{\Delta E}\sim\hbar D\,\nu\,L^{d-2}\tag{4}$$ as an order parameter of the Anderson metal/insulator transition, for which $$g\gg 1$$ corresponds to the delocalized/metallic phase, and $$g\ll 1$$ to the localized/insulator phase.

At this point, one could argue that the link between this new quantity $$g$$ and the usual conductivity $$G=\sigma\,S/L\sim\sigma\,L^{d-2}$$ according to the Ohm's law is not very clear.

But since the conductivity $$\sigma$$ and the diffusion coefficient $$D$$ are linked one to each other through the Einstein relation : $$\sigma\propto e^2\,\nu\,D\quad\text{such that}\quad G\sim e^2\,\nu\,D\,L^{d-2}\tag{5}$$ it is actually easy to see by comparing the expression (4) and (5) that $$g$$ has no dimension and gives the conductance $$G\sim (e^2/h)\,g$$ in unit of conductance quantum $$e^2/h$$.

So, to the questions :

Is g(L) determined by the Hamiltonian above ?

It is determined by the spectral properties of the eigenstates of the hamiltonian such as the density of states $$\nu$$ and the mean level spacing according to the size $$L$$ of the system.

or some more parameters are needed, say, the temperature ?

Everything that has been told before was for a system at $$T=0$$. From the point of view of the phase transition between localized and diffusive states, the only relevant information is the scaling of $$g$$ according to the size $$L$$ of system, which is completely determined by the $$\beta$$-function of the very famous scaling theory of localization.

For those who are interested in getting into the field of Anderson localization, I would advice to check the lecture notes of Les Houches School of Physics on "Ultracold gases and quantum information" (2009) given by Delande and Müller. IMO it is one of the rare references that achieve to introduce localization phenomena in simple terms while kipping tracks with the experiments on the subject.

[1] : a good review about this Thouless criterion can be found in Disordered electronic systems, P. A. Lee and T. V. Ramakrishnan, Rev.Mod.Phys. 57, 287.

As you have noticed, the literature on Anderson localization uses several different definitions of localization, including (but not limited to!):

1. A transition from extended to localized eigenstates. This implies a change from finite to zero diffusion of a particle initialized in some region.

2. A transition from finite to zero conductivity.

3. A change in the statistics of the distribution of the energy eigenstates, from some non-Poissonian (often Gaussian) distribution to Poissionian distributed levels.

These observables correspond, roughly, to the easiest signature of localization to investigate for theory, experiment, and numerics. The different limitations of each of these types of investigations is why so many definitions have been used.

There is some relation between each of these notions of localization, especially in the simplest case of non-interacting particles. Localized eigenstates mean that states nearby in energy are likely to be separated spatially and nearly uncoupled, which leads to Poissonian level statistics because there isn't much avoided crossing. Localized states also mean that particles have to tunnel from spot to spot to move across a sample, with many different energy mismatches, which leads to a vanishing conductivity in the thermodynamic limit.

However, beyond general arguments of this type, it is not obvious that these definitions should precisely coincide, and arguments used to relate them are generally non-rigorous. For example, the paper given by the OP more or less asserts a connection between the sensitivity of the eigenstates of a system to boundary conditions, parameterized by the change in energy $\Delta E$ between periodic and anti-periodic boundary conditions, and the dimensionless conductance (from the Kubo formula). So the chain of logic is roughly that extended versus localized states leads to more or less sensitivity to boundary conditions, which in turn leads to a conductivity that is finite or zero in the thermodynamic limit. The relationship between these quantities, mentioned in this paper, is given in a slightly more elaborate way in the earlier paper of Thouless. For reference, that formula, as shown in the 'Gang of Four' paper, is:

$$\frac{\Delta E}{dE/dN}=\frac{2\hbar}{e^2}\sigma L^{d-2}$$

where $dE/dN$ is the mean level spacing and $L$ is the sample size. This is probably the closest thing you will find to a quantitive relation between localization in terms of eigenstates and localization in terms of conductivity. However, in the author's words, "The equivalence of the Kubo-Greenwood formula and the breadth $\nu$ of the distribution of $\Delta E$ as described in Ref. 3a is not precisely provable."

The best reference I know for comparing these localization criteria in a more general way is the review by Van Tiggelen, found here (paywalled, sorry). He concludes that the criterion in terms of localized wavefunctions is stronger than the criterion for vanishing diffusion, and also compares them to other criteria for localization that I haven't mentioned here.

Finally, you ask whether $g(L)$ is determined by the Hamiltonian alone, or also the states that are occupied (or the temperature if it is a thermal distribution). The precise value of $g$ certainly should depend on things like the temperature, but the main question that this paper investigated was whether it always went to zero in the thermodynamic limit. When it does, as they concluded is the case for $d=1$ and $2$, then all the states in the system are localized, and it doesn't matter which states are occupied. In $d=3$ (and above), there is a critical mesoscopic conductance $g_c$ above which the system becomes conducting in the thermodynamic limit, and below which it is insulating. This does depend on which states are occupied, not only the Hamiltonian. This is because the Hamiltonian has a mobility edge, where states below some critical energy are localized and states above it are extended.