Composite fermion

The effective magnetic field for composite fermions vanishes for ${\displaystyle B=2p\rho \phi _{0}}$, where the filling factor for electrons is ${\displaystyle \nu =1/2p}$. Here, composite fermions make a Fermi sea.[6] This Fermi sea has been observed at half filled Landau level in a number of experiments, which also measure the Fermi wave vector.[7][8][9][10]

As the magnetic field is moved slightly away from ${\displaystyle B^{*}=0}$, composite fermions execute semiclassical cyclotron orbits. These have been observed by coupling to surface acoustic waves,[7] resonance peaks in antidot superlattice,[8] and magnetic focusing.[9][10][11] The radius of the cyclotron orbits is consistent with the effective magnetic field ${\displaystyle B^{*}=0}$ and is sometimes an order of magnitude or more larger than the radius of the cyclotron orbit of an electron at the externally applied magnetic field ${\displaystyle B}$. Also, the observed direction of trajectory is opposite to that of electrons when ${\displaystyle B^{*}}$ is anti-parallel to ${\displaystyle B}$.

In addition to the cyclotron orbits, cyclotron resonance of composite fermions has also been observed by photoluminescence.[12]

As the magnetic field is moved further away from ${\displaystyle B^{*}=0}$, quantum oscillations are observed that are periodic in ${\displaystyle 1/B^{*}.}$ These are Shubnikov–de Haas oscillations of composite fermions.[13][14] These oscillations arise from the quantization of the semiclassical cyclotron orbits of composite fermions into composite fermion Landau levels. From the analysis of the Shubnikov–de Haas experiments, one can deduce the effective mass and the quantum lifetime of composite fermions.

With further increase in ${\displaystyle |B^{*}|}$ or decrease in temperature and disorder, composite fermions exhibit integer quantum Hall effect.[5] The integer fillings of composite fermions, ${\displaystyle \nu ^{*}=n}$, correspond to the electrons fillings

${\displaystyle \nu ={\frac {n}{2pn\pm 1}}.}$

Combined with

${\displaystyle \nu =1-{\frac {n}{2pn\pm 1}},}$

which are obtained by attaching vortices to holes in the lowest Landau level, these constitute the prominently observed sequences of fractions. Examples are

${\displaystyle {n \over 2n+1}={1 \over 3},\,{2 \over 5},\,{3 \over 7},\,{4 \over 9},\,{5 \over 11},\cdots }$

${\displaystyle {n \over 2n-1}={2 \over 3},\,{3 \over 5},\,{4 \over 7},\,{5 \over 9},\,{6 \over 11},\cdots }$

${\displaystyle {n \over 4n+1}={1 \over 5},\,{2 \over 9},\,{3 \over 13},\,{4 \over 17},\cdots }$

The fractional quantum Hall effect of electrons is thus explained as the integer quantum Hall effect of composite fermions.[5] It results in fractionally quantized Hall plateaus at

${\displaystyle R_{H}={h \over \nu e^{2}},}$

with ${\displaystyle \nu }$ given by above quantized values. These sequences terminate at the composite fermion Fermi sea. Note that the fractions have odd denominators, which follows from the even vorticity of composite fermions.

The above sequences account for most, but not all, observed fractions. Other fractions have been observed, which arise from a weak residual interaction between composite fermions, and are thus more delicate.[15] A number of these are understood as fractional quantum Hall effect of composite fermions. For example, the fractional quantum Hall effect of composite fermions at ${\displaystyle \nu ^{*}=4/3}$ produces the fraction 4/11, which does not belong to the primary sequences.[16]

An even denominator fraction, ${\displaystyle \nu =5/2,}$ has been observed.[17] Here the second Landau level is half full, but the state cannot be a Fermi sea of composite fermions, because the Fermi sea is gapless and does not show quantum Hall effect. This state is viewed as a "superconductor" of composite fermion,[18][19] arising from a weak attractive interaction between composite fermions at this filling factor. The pairing of composite fermions opens a gap and produces a fractional quantum Hall effect.

The neutral excitations of various fractional quantum Hall states are excitons of composite fermions, that is, particle hole pairs of composite fermions.[20] The energy dispersion of these excitons has been measured by light scattering[21][22] and phonon scattering.[23]

At high magnetic fields the spin of composite fermions is frozen, but it is observable at relatively low magnetic fields. The fan diagram of the composite fermion Landau levels has been determined by transport, and shows both spin-up and spin-down composite fermion Landau levels.[24] The fractional quantum Hall states as well as composite fermion Fermi sea are also partially spin polarized for relatively low magnetic fields.[24][25][26]

The effective magnetic field of composite fermions has been confirmed by the similarity of the fractional and the integer quantum Hall effects, observation of Fermi sea at half filled Landau level, and measurements of the cyclotron radius.

The mass of composite fermions has been determined from the measurements of: the effective cyclotron energy of composite fermions;[27][28] the temperature dependence of Shubnikov–de Haas oscillations;[13][14] energy of the cyclotron resonance;[12] spin polarization of the Fermi sea;[26] and quantum phase transitions between states with different spin polarizations.[24][25] Its typical value in GaAs systems is on the order of the electron mass in vacuum. (It is unrelated to the electron band mass in GaAs, which is 0.07 of the electron mass in vacuum.)

The following trial wave functions[5] embody the composite fermion physics:

${\displaystyle \Psi _{\nu }^{\rm {FQHE}}=P\;\;\Psi _{\nu ^{*}}^{\rm {IQHE}}\prod _{1\leq j<k\leq N}(z_{j}-z_{k})^{2p}}$

Here ${\displaystyle \Psi _{\nu }^{\rm {FQHE}}}$ is the wave function of interacting electrons at filling factor ${\displaystyle \nu }$; ${\displaystyle \Psi _{\nu ^{*}}^{\rm {IQHE}}}$ is the wave function for weakly interacting electrons at ${\displaystyle \nu ^{*}}$; ${\displaystyle N}$ is the number of electrons or composite fermions; ${\displaystyle z_{j}=x_{j}+iy_{j}}$ is the coordinate of the ${\displaystyle j}$th particle; and ${\displaystyle P}$ is an operator that projects the wave function into the lowest Landau level. This provides an explicit mapping between the integer and the fractional quantum Hall effects. Multiplication by ${\displaystyle \prod _{j<k=1}^{N}(z_{j}-z_{k})^{2p}}$ attaches ${\displaystyle 2p}$ vortices to each electron to convert it into a composite fermion. The right hand side is thus interpreted as describing composite fermions at filling factor ${\displaystyle \nu ^{*}}$. The above mapping gives wave functions for both the ground and excited states of the fractional quantum Hall states in terms of the corresponding known wave functions for the integral quantum Hall states. The latter do not contain any adjustable parameters for ${\displaystyle \nu ^{*}=n}$, so the FQHE wave functions do not contain any adjustable parameters at ${\displaystyle \nu =n/(2pn\pm 1)}$.

Comparisons with exact results show that these wave functions are quantitatively accurate. They can be used to compute a number of measurable quantities, such as the excitation gaps and exciton dispersions, the phase diagram of composite fermions with spin, the composite fermion mass, etc. For ${\displaystyle \nu ^{*}=1}$ they reduce to the Laughlin wavefunction [29] at fillings ${\displaystyle \nu =1/(2p+1)}$.

Another formulation of the composite fermion physics is through a Chern–Simons field theory, wherein flux quanta are attached to electrons by a singular gauge transformation.[6][30] At the mean field approximation the physics of free fermions in an effective field is recovered. Perturbation theory at the level of the random phase approximation captures many of the properties of composite fermions.[31]