The Fractional Quantum Hall Effect: The Paradigm for Strongly Interacting Systems

Abstract

The study of the electronic properties of quasi two dimensional systems has been a very exciting area of condensed matter physics during the last quarter of the 20th century. Among the most interesting discoveries in this area are the incompressible states showing integral and fractional quantum Hall effects. Incompressible quantum liquid states of the integral quantum Hall effect result from an energy gap in the single particle spectrum. The incompressibility of the fractional quantum Hall effect is completely the result of electron–electron interactions in a highly degenerate fractionally filled Landau level. Since the quantum Hall effect involves electrons moving on a two dimensional surface in the presence of a perpendicular magnetic field, we begin with a description of this problem.

Appendices

Problems

16.1

The many particle wavefunction is written, for $\nu =1$, by

$$ \varPsi _1(z_1, \ldots , z_N)=\mathcal {A}\{u_0(z_1)u_1(z_2)\cdots u_{N-1}(z_N)\} $$

where $\mathcal {A}$ denotes the antisymmetrizing operator. Demonstrate explicitly that $\varPsi _1(z_1, \ldots , z_N)$ can be written as follows:

$$ \varPsi _1(z_1, \ldots , z_N)\propto \left| \begin{array}{llcl} 1&{}1&{}\cdots &{}1 \\ z_1&{}z_2&{}\cdots &{}z_N \\ z_1^2&{}z_2^2&{}\cdots &{}z_N^2 \\ \vdots &{}\vdots &{}\cdots &{}~~\vdots \\ z_1^{N-1}&{}z_2^{N-1}&{}\cdots &{}z_N^{N-1} \end{array} \right| e^{-\frac{1}{4 l_0^2}\sum _{_{i=1,N}}|z_i|^2}. $$

16.2

Consider a system of N electrons confined to a Haldane surface of radius R. There is a magnetic monopole of strength $2Q\phi _0$ at the center of the sphere.

• (a) Demonstrate that, in the presence of a radial magnetic field $\mathbf B = \frac{2Q\phi _0}{4\pi R^2} \hat{R},$ the single particle Hamiltonian is given by

  $$ H_0=\frac{1}{2mR^2}\left( \mathbf {l}-\hbar Q \hat{R}\right) ^2. $$

  Here $\hat{R}$ and $\mathbf {l}$ are, respectively, a unit vector in the radial direction and the angular momentum operator.

• (b) Show that the single particle eigenvalues of $H_0$ are written as

  $$ \varepsilon (Q,l, m)=\frac{\hbar \omega _\mathrm{c}}{2Q}[l(l+1)-Q^2]. $$

16.3

Figure 16.5 displays $V_\mathrm{QE}(\mathcal {R})$ and $V_\mathrm{QH}(\mathcal {R})$ obtained from numerical diagonalization of N $(6\le N \le 11)$ electron systems appropriate to quasiparticles of the $\nu =1/3$ and $\nu =1/5$ Laughlin incompressible quantum liquid states. Demonstrate that $V_\mathrm{QP}(\mathcal {R})$ converges to a rather well defined limit by plotting $V_\mathrm{QP}(\mathcal {R})$ as a function of $N^{-1}$ at $\mathcal {R}=1$, 3, and 5.

Summary

In this chapter we introduce basic concepts commonly used to interpret experimental data on the quantum Hall effect. We begin with a description of two dimensional electrons in the presence of a perpendicular magnetic field. The occurrence of incompressible quantum fluid states of a two-dimensional system is reviewed as a result of electron–electron interactions in a highly degenerate fractionally filled Landau level. The idea of harmonic pseudopotential is introduced and residual interactions among the quasiparticles are analyzed. For electrons in the lowest Landau level the interaction energy of a pair of particles is shown to be superharmonic at every value of pair angular momenta.

The Hamiltonian of an electron (of mass $\mu$) confined to the x-y plane, in the presence of a dc magnetic field $\mathbf B = B \hat{z}$, is simply $H = (2\mu )^{-1} \left[ \mathbf p + \frac{e}{c}\mathbf A (\mathbf r )\right] ^2,$ where $\mathbf A (\mathbf r )$ is given by $\mathbf A (\mathbf r ) = \frac{1}{2}B (-y \hat{x} + x \hat{y} )$ in a symmetric gauge. The Schrödinger equation $(H - E)\varPsi (\mathbf r ) = 0$ has eigenstates described by

$$ \varPsi _{nm} (r,\phi ) = e^{im\phi } u_{nm} (r) \text{ and } E_{nm} = \frac{1}{2} \hbar \omega _c (2n +1 + m +|m|), $$

where n and m are principal and angular momentum quantum numbers, respectively, and $\omega _\mathrm{c}(=eB/{\mu c})$ is the cyclotron angular frequency. The lowest Landau level wavefunction can be written as $\varPsi _{0m} = \mathcal {N}_m z^{|m|} \mathrm e^{-|z|^2/4{l_0}^2}$ where $\mathcal {N}_m$ is the normalization constant and z stands for $z(=x-i y)=r e^{-i\phi }$. The filling factor $\nu$ of a given Landau level is defined by $N/N_\phi$, so that $\nu ^{-1}$ is simply equal to the number of flux quanta of the dc magnetic field per electron. The integral quantum Hall effect occurs when N electrons exactly fill an integral number of Landau levels resulting in an integral value of the filling factor $\nu$. The energy gap (equal to $\hbar \omega _c$) between the filled states and the empty states makes the noninteracting electron system incompressible. A many particle wavefunction of N electrons at filling factor $\nu =1$ becomes

$$ \varPsi _1(z_1, \ldots , z_N)\propto \prod _{_{N\ge i>j\ge 1}} z_{ij} \mathrm e^{-\frac{1}{4 l_0^2}\sum _{k=1,N}|z_k|^2}. $$

For filling factor $\nu = 1/n$, Laughlin ground state wavefunction is written as

$$ \varPsi _{1/n} (1, 2, \ldots , N) = \prod _{i>j} z_{ij}^n ~\mathrm e^{-\sum _l |z_l|^2/4l_0^2}, $$

where n is an odd integer.

It is convenient to introduce a Haldane sphere at the center of which is located a magnetic monopole and a small number of electrons are confined on its surface. The numerical problem is to diagonalize the interaction Hamiltonian $H_\mathrm{int} = \sum _{i<j} V(|\mathbf r_i - \mathbf r_j |).$ The calculations give the eigenenergies E as a function of the total angular momentum L.

Considering a two dimensional system of particles described by a Hamiltonian

$$ H=\frac{1}{2\mu }\sum _i \left[ \mathbf p_i+\frac{e}{c}\mathbf A(\mathbf r_i)\right] ^2 +\sum _{i>j}V(r_{ij}), $$

we can change the statistics by attaching to each particle a fictitious charge q and flux tube carrying magnetic flux $\varPhi$. The fictitious vector potential $\mathbf a(\mathbf r_i)$ at the position of the ith particle caused by flux tubes, each carrying flux of $\varPhi$, on all the other particles at $\mathbf r_j(\ne \mathbf r_i)$ is written as $\mathbf a(\mathbf r_i)=\varPhi \sum _{_{j\ne i}}\frac{\hat{z} \times \mathbf r_{ij}}{r_{ij}^2}.$ The Chern–Simons gauge field due to the gauge potential $\mathbf a(\mathbf r_i)$ becomes $\mathbf b(\mathbf r)=\varPhi \sum _i\delta (\mathbf r-\mathbf r_i)\hat{z},$ where $\mathbf r_i$ is the position of the ith particle carrying gauge potential $\mathbf a(\mathbf r_i)$. The new Hamiltonian, through Chern–Simons gauge transformation, is

$$ H_\mathrm{CS}=\frac{1}{2\mu }\int d^2 r \psi ^\dagger (\mathbf r) \left[ \mathbf p+\frac{e}{c}\mathbf A(\mathbf r)+\frac{e}{c}\mathbf a(\mathbf r)\right] ^2 \psi (\mathbf r) +\sum _{_{i>j}}V(r_{ij}). $$

The net effect of the additional Chern–Simons term is to replace the statistics parameter $\theta$ with $\theta +\pi \varPhi \frac{q}{hc}$. If $\varPhi =p \frac{hc}{e}$ when p is an integer, then $\theta \rightarrow \theta +\pi p q/e$. For the case of $q=e$ and $p=2$, the statistics would be unchanged by the Chern–Simons terms, and the gauge interactions convert the electrons system to the composite fermions which interact through the gauge field term as well as through the Coulomb interaction. In the mean field approach, the composite fermions move in an effective magnetic field $B^*$. The composite fermion filling factor $\nu ^*$ is given by ${\nu ^*}^{-1} = \nu ^{-1} - \alpha$. The mean field picture predicts not only the sequence of incompressible ground states, given by $\nu = \frac{\nu ^*}{1+2p\nu ^*}$ (with integer p), but also the correct band of low energy states for any value of the applied magnetic field. The low lying excitations can be described in terms of the number of quasiparticles $n_\mathrm{QE}$ and $n_\mathrm{QH}$.

In a state containing more than one Laughlin quasiparticles, quasiparticles interact with one another through appropriate quasiparticle-quasiparticle pseudopotentials, $V_\mathrm{QP-QP'}$. The total energy can be expressed as

$$ E=E_0+\sum _\mathrm{QP}\varepsilon _\mathrm{QP} n_\mathrm{QP}+ \frac{1}{2} \sum _\mathrm{QP, QP'} V_\mathrm{QP-QP'}(L) n_\mathrm{QP}n_\mathrm{QP'}. $$

If $V_\mathrm{QP-QP'}(L')$ is a “harmonic” pseudopotential of the form $V_\mathrm{H}(L')=A{\!} + {\!}BL'(L'{\!}+{\!}1)$ every angular momentum multiplet having the same value of the total angular momentum L has the same energy. We define $V_\mathrm{QP-QP'}(L')$ to be ‘superharmonic’ (‘subharmonic’) at $L'=2l - \mathcal {R}$ if it increases approaching this value more quickly (slowly) than the harmonic pseudopotential appropriate at $L' - 2$. For harmonic and subharmonic pseudopotentials, Laughlin correlations do not occur. Since the harmonic pseudopotential introduces no correlations, only the anharmonic part of the pseudopotential $\varDelta V(\mathcal {R}) = V(\mathcal {R})-V_H(\mathcal {R})$ lifts the degeneracy of the multiplets with a given L.