L-7

Goal. This lecture introduces the phenomenon of localization. Localization is a wave phenomenon induced by disorder that suppresses transport in a system.

Free particles and ballistic behaviour

The Schrödinger equation governs the evolution of the quantum state of a particle in one dimension:

$$i\hslash {\partial }_t\psi (x,t)=H\psi (x,t),H=-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+V(x).$$

Here $H$ is the Hamiltonian. For a free particle the potential vanishes, $V(x)=0$.

One first looks for separable solutions, also called eigenstates or stationary solutions. If the particle is in an eigenstate, all physical observables are time independent.

For a free particle the stationary solutions are plane waves

$${\psi }_k(x,t)=\frac{1}{\sqrt{2\pi }}{e}^{ikx}{e}^{-i{E}_{k}t/\hslash },E_k=\frac{{\hslash }^2{k}^2}{2m}.$$

Here $k$ is a real number and the spectrum is continuous. These solutions are completely delocalized over the real axis and cannot be normalized.

Physical states are obtained as superpositions of eigenstates. By the superposition principle any linear combination of stationary solutions is again a solution of the Schrödinger equation. Hence we can construct a localized wave packet:

$$\psi (x,t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dkc(k){\psi }_k(x,t),\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dk|c(k){|}^2=1.$$

Evolution of a Gaussian wave packet

• Initial state.

At time $t=0$ consider the Gaussian packet

$$\psi (x,0)=\frac{e^{-x^2/(4a^2)}}{(2\pi a^2{)}^{1/4}},|\psi (x,0){|}^2=\frac{e^{-x^2/(2a^2)}}{\sqrt{2\pi a^2}}.$$

Show that the coefficients of the plane wave decomposition are

$$c(k)={\left(\frac{2a^2}{\pi }\right)}^{1/4}{e}^{-a^2{k}^2}.$$

• Time evolution.

Define the spreading velocity

$$v_s=\frac{\hslash }{2m{a}^2}.$$

Show that the time evolution of the packet is

$$\psi (x,t)=\frac{e^{-x^2/(4a^2(1+i{v}_{s}t))}}{[2\pi a^2(1+i{v}_{s}t){]}^{1/4}}.$$

• Ballistic spreading.

The probability density becomes

$$|\psi (x,t){|}^2=\frac{e^{-x^2/(2a^2(1+v_s^2{t}^2))}}{\sqrt{2\pi a^2(1+v_s^2{t}^2)}}.$$

Hence

$$\sqrt{⟨x^2⟩}={(\int dx{x}^2|\psi (x,t){|}^2)}^{1/2}=a\sqrt{1+v_s^2{t}^2}.$$

At long times

$$\sqrt{⟨x^2⟩}\sim v_{s}t.$$

This behaviour is called ballistic spreading.

It should be contrasted with the two other possible transport regimes:

• Diffusive motion (random walk):

$\sqrt{⟨x^2⟩}\sim \sqrt{t}$

• Localized regime:

$\sqrt{⟨x^2⟩}$ saturates at long times.

Understanding how disorder changes ballistic motion into diffusion and eventually localization is the main goal of the following lectures.

Localization of the packet: general idea and experiment

BEC condensate expanding in a 1D disordered potential. Billy et al., Nature (2008).

In 1958, P. Anderson proposed that disorder can suppress transport in quantum systems. This phenomenon, called Anderson localization, has since been observed both numerically and experimentally.

In the experiment shown here, a Gaussian packet of a Bose–Einstein condensate is prepared in a harmonic trap. When the trap is removed the cloud initially starts to expand but quickly stops spreading and remains localized.

To understand this behaviour we must solve the eigenstate problem of the Hamiltonian in the presence of disorder.

Semilog plot of the particle density. Billy et al., Nature (2008).

In a disordered potential an eigenstate of energy $E_k$ has the form

$${\psi }_k(x,t)={\psi }_k(x)e^{-i{E}_{k}t/\hslash }.$$

The spatial part of the wavefunction is localized around some position $\overline{x}$ and decays exponentially:

$${\psi }_k(x)\sim e^{-|x-\overline{x}|/{\xi }_{\text{loc}}(E_k)}.$$

Here ${\xi }_{\text{loc}}$ is the localization length.

Because the eigenstates are localized, each of them has support only in a finite spatial region. The wave packet is therefore built from eigenstates localized near its initial position.

Two important observations follow:

• eigenstates localized far from the packet contribute exponentially little to the wavefunction near the packet,
• eigenstates contributing to the packet decay exponentially far from their center.

As a consequence, transport far away from the initial position of the particle is exponentially suppressed.

Localization therefore does not mean that the particle is frozen. The wave packet can still evolve in time, but it cannot propagate arbitrarily far.

Conductance and diffusive transport

In most materials the effect of weak disorder is not localization but diffusion.

In the Drude picture electrons scatter randomly on impurities. Beyond the mean free path the motion of electrons becomes diffusive.

In this regime Ohm's laws hold.

• First law

$$\frac{V}{I}=R,\frac{I}{V}=G.$$

Here $R$ is the resistance and $G$ the conductance.

• Second law

$$R=\rho \frac{L}{S}\sim \rho L^{2-d},G=\sigma \frac{S}{L}\sim \sigma L^{d-2}.$$

Here $\rho$ and $\sigma$ are the resistivity and conductivity. These are material properties independent of the geometry of the sample.

These phenomenological laws are the macroscopic manifestation of diffusive transport.

Conductance in the localized regime

When disorder is strong, diffusion is suppressed and the system becomes insulating.

In the localized phase the conductance decays exponentially with the system size:

$$G\sim e^{-2L/{\xi }_{\text{loc}}}.$$

The localization length ${\xi }_{\text{loc}}$ characterizes the spatial decay of the eigenstates.

The “Gang of Four” scaling theory

In a famous PRL (1979), Abrahams, Anderson, Licciardello and Ramakrishnan proposed a scaling theory of localization.

It is based on the idea that the relevant quantity is the dimensionless conductance

$$g=\frac{G\hslash }{e^2}.$$

The scaling equation reads

$$\frac{d\ln g}{d\ln L}=\beta (g).$$

The function $\beta (g)$ depends only on $g$ and the spatial dimension.

The asymptotic behaviours are

$$\beta (g)=\{\begin{array}{ll}d-2& g\to \mathrm{\infty }\\ \sim \ln g& g\to 0\end{array}$$

The second relation reflects the exponential suppression of conductance in the localized regime.

If the beta function is monotonic, the scaling theory predicts:

• a metal–insulator transition for $d>2$,
• complete localization for $d\le 2$.