Composite Fermions

Though mathematically elegant, the hierarchy approach of Haldane and Halperin appears to incorrectly predict the relative strengths of the fractional quantum Hall states [58]. For example, ν = 3/7 and ν = 5/13 are both direct “daughter” states of

ν = 2/5, but 3/7 is far more prevalent than 5/13 in real samples. As an alternative to the hierarchy approach, Jain reformulated the problem of interacting electrons moving in a magnetic field by describing it as a system of composite particles known as composite fermions [57]. A composite fermion is an electron bound to an even number of quantized vortices in the multiparticle wave function. Each vortex takes the form of (zj −zk) and thus the wave function Ψν of the electrons at filling factor

ν can be related to the composite fermion wave function Φν∗ in the following way:

Ψν = Y j<k

(zj−zk)2pΦν∗ (1.19)

Like in the Laughlin states, the Q

j<k(zj − zk)2p will cause the electrons to avoid

each other and make the formation of composite fermions energetically favorable. By minimizing the Coulomb repulsion, it is typical to assert that Φν∗ represents a wave

function for a system of weakly interacting composite fermions. We will show that because the binding of vortices can be related to the binding of fictitious magnetic flux, the composite fermions subsequently move about in a reduced magnetic fieldB∗

and fill up an integer number of fictitious Landau levels generated byB∗.

We start by considering a system of noninteracting electrons at integer filling fac- tor ν∗ =n=NSφ0/|B∗|. Here, NS is the number density of electrons and φ0 =hc/e

This can be accomplished formally through a Chern-Simons gauge transformation, which associates with each electrons the requisite amount of flux [137, 77, 50, 97]. The flux attachment does not change any observables because they make zero net Aharonov-Bohm phase contribution to any Feynman path-integrals. The new system of composite fermions will be incompressible because the original system of noninter- acting electrons was also incompressible. We note that these fictitious magnetic flux quanta are identical to the vortices (zj −zk)2p because they both cause the phase of

the multiparticle wave function to increase by the same amount when one moves one particle in a complete circle around another.

Next, we adiabatically spread each attached flux until it merges with the external magnetic field. This adiabatic evolution is permitted due to the finite energy gap. So long as this gap does not close (which we assume that it does not), we will avoid passing through a phase transition into a completely different state. Thus, we map the problem of noninteracting composite fermions with integer filling factor ν∗ into a

problem of electrons moving in an external magnetic field B =B∗+ 2pNSφ0. Using |B∗| =NSφ0/n and taking B to be positive, the electrons will have fractional filling

factor,

ν= n

2pn±1. (1.20) Therefore, the fractional quantum Hall effect for electrons can ultimately be explained in terms of an integer quantum Hall effect for composite fermions. A simple example of this is the case of n = 1 and p= 1. This gives the ν = 1/3 Laughlin state, which can be rewritten as Ψ1/3 = Y j<k (zj −zk)3exp " −1 4 X i |zi|2 # =Y j<k (zj −zk)2Φ1. (1.21)

Here, the term Q

j<k(zj −zk)

2 _{represents the binding of two flux quanta to each}

electron and Φ1 = Q j<k(zj −zk) exp −1 4 P i|zi| 2

29

for a completely filled Landau level, corresponding to a ν∗ = 1 state of composite fermions.

The treatment of composite fermions presented here is an oversimplification of the complete theory. It provides an intuitive explanation for an energy gap for a system of interacting electrons at fractional filling factor without accounting for the exact evolution of the energy levels during the attaching and spreading of fictitious magnetic flux. For example, one might incorrectly posit that the energy gap should be given by the cyclotron energy for electrons at filling factor ν∗, which would be dependent on the effective mass of the electrons. This clearly cannot be the case because the projection of the system into the lowest Landau level will quench the kinetic energy of the electrons and cause the spectrum to only depend on the Coulomb repulsion term. Thus, the true energy gap for a system of composite fermions at filling factor

ν∗ should somehow scale with the Coulomb energy EC. For a more complete review

of composite fermions, see reference [58].

The theory of composite fermions makes a number of testable predictions about the energy spectrum of fractional quantum Hall states. For example, one would ex- pect that the energy gap of a particular FQHE would be grow along with the effective magnetic fieldB∗, which governs the cyclotron splitting of the fictitious Landau levels of the composite fermions. This is borne out in experiments [15, 74, 16] that observe Shubnikov–de Haas oscillations in the vicinity of ν = 1/2, whose amplitude grows as one moves away from ν = 1/2. Crucially, the theory of composite fermions also sug- gests that at half-filling factor the composite fermions should feel no effective magnetic field other than a Zeeman field leading to partial spin polarization. Consequently, the composite fermions form a compressible Fermi sea. This is quite remarkable because the system would have an effective mass that is determined by the Coulomb energy rather than the effective mass of the underlying electrons. Experimentally, no quan- tum Hall plateau is visible atν = 1/2 in single-layer 2DESs, which is consistent with a compressible system. However, the composite fermion system has no energy gap at ν = 1/2, so it is unclear if it can survive gauge fluctuations during the adiabatic creation of the composite fermions. Nonetheless, Halperin, Lee, and Read [50] have

fermion Fermi sea at ν= 1/2.