L-7

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

Short recap: wavefunctions and eigenstates

Before discussing localization we briefly recall a few basic notions of quantum mechanics.

A quantum particle in one dimension is described by a wavefunction $\psi (x,t)$. The quantity

$|\psi (x,t){|}^2$

is the probability density of finding the particle at position $x$ at time $t$. The wavefunction therefore satisfies the normalization condition

$$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dx|\psi (x,t){|}^2=1.$$

The time evolution of the wavefunction is governed by the Schrödinger equation

$$i\hslash {\partial }_t\psi (x,t)=H\psi (x,t),$$

where $H$ is the Hamiltonian of the system.

Eigenstates

A particularly important class of solutions are the eigenstates of the Hamiltonian

$$H{\psi }_n(x)=E_n{\psi }_n(x).$$

If the particle is in an eigenstate the full solution reads

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

The probability density $|{\psi }_n(x,t){|}^2$ is therefore independent of time: eigenstates are stationary states.

Discrete and continuous spectra

Two situations may occur.

• Discrete spectrum

The energies take isolated values $E_n$. This happens when the particle is confined in a finite region (for instance in a potential well). The eigenstates are normalizable and labeled by an integer $n$.

• Continuous spectrum

The energy can take any value in a continuous interval. This happens for instance for a free particle. The eigenstates are plane waves

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

These states are not normalizable in the usual sense. They are instead normalized using Dirac delta functions and serve as a basis for constructing physical wave packets.

Probability current

Besides the probability density one can define a probability current

$$J(x,t)=\frac{\hslash }{2mi}({\psi }^{*}\frac{d\psi }{dx}-\psi \frac{d{\psi }^{*}}{dx}).$$

This quantity measures the flow of probability across a point in space.

Its expression follows from combining the Schrödinger equation with the conservation of probability. Indeed the probability density

$\rho (x,t)=|\psi (x,t){|}^2$

satisfies a continuity equation

$${\partial }_t\rho (x,t)+{\partial }_{x}J(x,t)=0,$$

which has the same structure as a conservation law in hydrodynamics. The current originates from the kinetic term of the Hamiltonian, which contains spatial derivatives.

For a plane wave

$$\psi (x)=e^{ikx}$$

one finds

$$J=\frac{\hslash k}{m}.$$

Thus plane waves describe particles propagating through space and carrying a non–zero current.

By contrast, for bound states belonging to a discrete spectrum the current is zero. In this case the eigenfunctions can be chosen real and the two terms in the expression of $J$ cancel. Physically this corresponds to a standing wave rather than a propagating wave.

Scattering states

Transport problems often involve a localized potential (for instance a disordered region or a sample) surrounded by free space. Outside this region the solutions of the Schrödinger equation are plane waves.

When a particle interacts with the sample, the wavefunction generally contains three contributions:

• an incoming wave,
• a reflected wave,
• a transmitted wave.

The corresponding solutions are called scattering states.

For example, a particle incoming from the left is described asymptotically by

$${\psi }_{k,L}(x)=\{\begin{array}{ll}{e}^{ikx}+r{e}^{-ikx}& x\to -\mathrm{\infty }\\ t{e}^{ikx}& x\to +\mathrm{\infty }\end{array}$$

The first term represents the incoming wave, the second the reflected wave, while the transmitted wave propagates to the right with amplitude $t$.

Using the expression of the probability current, one finds that a plane wave $e^{ikx}$ carries a current

$$J=\frac{\hslash k}{m}.$$

Thus the incoming, reflected and transmitted waves correspond to incoming, reflected and transmitted probability currents. Since probability is conserved, the current is the same on both sides of the sample. This implies the relation

$$R+T=1,$$

where

$$R=|r{|}^2,T=|t{|}^2$$

are the reflection and transmission probabilities.

These quantities will play a central role in transport problems, where the electrical current is determined by the transmission probability of the sample.

Superposition principle

The Schrödinger equation is linear. Any linear combination of eigenstates is therefore again a solution.

For a continuous spectrum one writes

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

By choosing the coefficients $c(k)$ appropriately one can construct a localized wave packet describing a particle initially confined in space.

In the next section we study the time evolution of such a packet.

Free particles and ballistic behaviour

Consider now the case of a free particle. The Hamiltonian is

$$H=-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}.$$

The stationary solutions are plane waves

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

These states are completely delocalized and cannot represent a physical particle. Physical states are therefore constructed as superpositions of plane waves.

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

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

$\sqrt{⟨x^2⟩}\sim \sqrt{t}$

• Localized regime

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

Understanding how disorder modifies ballistic motion and eventually suppresses transport is the main goal of the following sections.

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, known as 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 expands 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

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

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

Because the eigenstates are localized, the wave packet is built from states localized near its initial position. Eigenstates far from the packet contribute exponentially little, and states composing the packet decay exponentially away from their center.

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

Conductance and diffusive transport

In most materials weak disorder leads to diffusion rather than localization.

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

In this regime Ohm's laws hold.

• First law

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

• 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.

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 system size

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

The “Gang of Four” scaling theory

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

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 asymptotic behaviours are

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

If the beta function is monotonic the scaling theory predicts

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