Composite fermions are a new class of exotic particles discovered in condensed matter physics. Composite fermions were originally predicted theoretically to explain the "fractional quantum Hall effect" (FQHE), one of the most spectacular phenomena discovered in physics during the past three decades, but are now known to describe a superstructure that encompasses phenomena beyond FQHE as well.


Fractional quantum Hall effect

When a two-dimensional electron system is exposed to a strong transverse magnetic field, it forms a new quantum liquid whose phenomenology rivals superconductivity and superfluidity in its richness and elegance. The investigation of this wonderful mystery has led to the discovery of some of the most profound and unprecedented emergent structures in modern physics.

In 1980, Klaus von Klitzing made the dramatic discovery of the "integral quantum Hall effect" (IQHE), namely extremely precise quantization of the Hall resistance at RH=h/ie2 where i is an integer, for which he received the 1985 Nobel Prize. The origin of the IQHE is conceptually straightforward. In the presence of a magnetic field, the kinetic energy levels of electrons become quantized, called Landau levels.  The IQHE follows from the fact that a unique, gapped ground state is obtained when an integral number of Landau levels are fully occupied. (The number of filled Landau levels is called the filling factor.) The IQHE is the most striking manifestation of the quantization of the electron's kinetic energy into Landau levels.

In 1982, Daniel C. Tsui, Horst L. Stormer, and Arthur C. Gossard discovered the "1/3 effect," namely a pleateau on which the Hall resistance is quantized at RH=h/(1/3)e2 (Nobel Prize, 1998). This was the beginning of the FQHE, which refers to the existence of accurately quantized Hall resistance plateaus at values given by RH=h/fe2, where f is a rational fraction. More than 70 fractions have been observed to date. They all have odd denominators, with one exception.

The FQHE occurs at high magnetic fields, most prominently when all electrons fall into their lowest Landau level. A model that neglects interactions is inadequate, and theory must deal head-on with the problem of interacting electrons confined to the lowest Landau level. The difficulty lies in the fact that the FQHE is a nonperturbative effect, as can be appreciated from the following observations:

  1. The degeneracy problem- Given that we do not even know how to solve the general problem of three interacting particles, a collection of macroscopically many interacting particles can be expected to be incredibly complex. Sometimes, a perturbative treatment of the interaction may be satisfactory, as is the case for Landau Fermi liquids. Such a treatment is not possible for the FQHE problem, because switching off the Coulomb interaction produces an astronomically large number of degenerate ground states.
  2. Absence of a "normal state"- For certain problems, such as weakly coupled superconductors, we begin with a normal state, which is the solution of the noninteracting problem, and ask what instability occurs when an appropriate interaction is turned on. As indicated in the preceding paragraph, for the FQHE problem no unique state is obtained in the absence of interaction. In other words: The FQHE has no normal state.
  3. Lack of a small parameter- We often treat interaction as a perturbation. For the FQHE problem, the interaction energy is the only energy scale, so cannot be treated as small.
  4. Nonperturbative physics-Through FQHE, nature is telling us that the repulsive interelectron interaction has a nonperturbative effect: at certain filling factors, it removes the enormous degeneracy of the noninteracting system to produce unique and robust ground states which are separated from excitations by a gap. The gaps open for arbitrarily weak interaction strengths.
 

The situation may appear hopeless. We do not even know where to begin, and there exists no small parameter to guide our thinking. It may therefore come as a pleasant surprise that a secure and well established theoretical understanding of this phenomenon has been achieved.  To summarize:

A single emergent principle, namely the formation of exotic topological particles called composite fermions, provides a unified explanation of the vast phenomenology of the fractional quantum Hall effect, makes unexpected and nontrivial predictions that have subsequently been verified, and leads to a parameter-free microscopic theory that produces accurate numbers for experimentally measurable quantities! Composite fermions have been directly observed in several experiments, their quantum numbers have been measured, and many of their states and phenomena have been confirmed. The exotic state of matter in the lowest Landau level is, in fact, a liquid of composite fermions.

Composite fermions are bound states of electrons and quantized vortices. They are the fundamental building blocks of the FQHE -- they play the same role in the FQHE as electrons do in the integral quantum Hall effect. The existence of composite fermions has numerous definite, unambiguous, unexpected, and inescapable consequences, listed below, which have been critically examined time and again through a number of experimental and numerical tests, within and beyond the FQHE.

Unlike superconductors and superfluids, the composite-fermion liquid has no underlying Bose-Einstein condensation, no spontaneously broken symmetry, and no order parameter. It represents a new paradigm for collective behavior, whose investigation has led to the development of a new conceptual framework and a new language.

Composite fermion theory

The composite fermion theory postulates that when a two-dimensional electron system is exposed to a strong transverse magnetic field, electrons minimize their interaction energy by capturing an even number of quantized vortices to create new particles called composite fermions. It further postulates that composite fermions themselves are weakly interacting (for many situations); that is, the only nonperturbative effect of interactions is to produce composite fermions. The complex, strongly correlated liquid of interacting electrons thus transforms into a weakly interacting gas of composite fermions. (An artistic depiction by Kwon Park.)  The composite fermion is sometimes also envisioned and modeled as an electron carrying an even number of magnetic flux quanta; while this definition captures the topological character of the composite fermion, it is not to be taken literally.

An important accomplishment of the composite fermion theory is the immediate explanation of the FQHE: The mysterious FQHE is a manifestation of the IQHE of composite fermions.  The unification of the FQHE and the IQHE was the original inspiration for the composite fermion theory.

The most fundamental property of composite fermions is that they experience an effective magnetic field (B*) that is reduced from the external magnetic field (B) by an amount proportional to the number of vortices bound to composite fermions. (The Berry phases due to vortices partly cancel the Aharonov Bohm phases from the external magnetic field to produce an effective magnetic field.) This single property is responsible for most of the unexpected phenomenology in the lowest Landau level. Composite fermions form Landau-like levels, called Λ levels, in the effective magnetic field.  Their filling factor is much larger than the filling factor of electrons.  The explicit expressions for the effective magnetic field and the composite-fermion filling factor can be found here. Based on this physics, parameter-free wave functions for ground and excited states of interacting electrons at B can be constructed in terms of the known wave functions for the ground and excited states of noninteracting fermions at B* (the explicit form given here), which enable an accurate microscopic description of composite fermions.

A field theoretical formulation of composite fermions proceeds by making an exact transformation (a singular Chern-Simons gauge transformation) to attach point flux quanta to electrons, followed by a mean field approximation that spreads the bound magnetic flux uniformly to produce particles in a new magnetic field.

The composite fermion state is a "hidden Fermi liquid" in the sense defined by P.W. Anderson: It is a non-Fermi liquid (not perturbatively connected to a system of noninteracting electrons), but is related to an ordinary Fermi liquid in an unphysical Hilbert space, namely a weakly interacting system of fermions in an effective magnetic field. 

The exotic character of composite fermions is illustrated by their following properties:

    1. Topological particles: Composite fermions are "topological particles" because one of their constituents, namely the quantized vortex, is a topological object. A vortex produces a phase of 2π  for a closed loop around it, independent of the size and the shape of the loop; this topological feature is what allows us to count the number of vortices. The topological character of composite fermions manifests most directly through the effective magnetic field.
    2. Topological quantum fluids: Because composite fermions are topological particles, all of their states, such as their Fermi sea or QHE, are topological quantum fluids.
    3. Nonperturbative particles: Composite fermions are nonperturbative entities. Being quantized due to their topology, the vortices cannot be bound to electrons "continuously" -- they are either bound or not. Composite fermions are thus not perturbatively connected to electrons, but are topologically distinct particles. The formation of composite fermions reveals the intrinsically nonperturbative nature of the FQHE state.
    4. Complex collective bound states: Because all electrons participate in the formation of a single vortex, each composite fermion is an incredibly complex collective entity from the vantage point of the electrons. Nonetheless, composite fermions behave as legitimate particles with mass, charge, spin, statistics, and other properties that we associate with particles. Surprisingly, composite fermions can be treated as almost free for a qualitative understanding of many (but not all) essential experimental observations.

    Successes of the composite fermion theory

"A theory is the more impressive the greater the simplicity of its premises, the more different kinds of things it relates, and the more extended its area of applicability." -Albert Einstein

Confirming the validity of a theory is perhaps more straightforward when it concerns an unconventional state with a large body of anomalous experimental facts. Some salient successes of the composite fermion theory for the explanation of the phenomenology of interacting electrons in the lowest Landau level are listed here; more details and references to the original articles (which are too numerous to list here) can be found in the books and review articles listed below.

      1. the quantum numbers of the low-energy eigenstates of the interacting system in the FQHE regime have a one to one correspondence with those of noninteracting fermions at an effective magnetic field (the latter can be determined trivially);
      2. with no adjustable parameters, the predicted energies are correct to within 0.1-0.2%, and the wave functions have a close to 100% overlap with the exact wave functions. 
      1. the occurrence of FQHE and its similarity to the IQHE;
      2. the appearance of sequences that correspond to the integer sequence of the noninteracting fermions;
      3. formation of a Fermi sea at 1/2 filled Landau level;
      4. direct measurements of the radius of the cyclotron orbit of the current carrying entities, which conforms to the reduced effective magnetic field rather than the applied magnetic field;
      5. exact diagonalization calculations that show that the low energy states of the problem of interacting electrons have a one to one correspondence with that of noninteracting fermions at a reduced effective magnetic field;
      6. a close relation between the wave function of interacting electrons in the external magnetic field and noninteracting fermions at the reduced magnetic field.
      1. their IQHE (there is no IQHE for bosons);
      2. their Fermi sea;
      3. correct counting of low energy eigenstates in terms of a model of noninteracting fermions.
      1. the existence of composite fermions;
      2. effective magnetic field;
      3. similarity between the FQHE and the IQHE;
      4. sequences of fractions;
      5. composite fermion Fermi sea;
      6. spin polarizations of FQHE states and the composite fermion Fermi sea;
      7. excitation energies;
      8. composite fermion mass;
      9. all of the hundreds of numbers computed with the composite fermion theory that have been verified by computer and / or  laboratory experiments.

    The composite fermion theory possesses many qualities we desire in a theory.
     

Open issues and challenges:



Further reading

A huge number of scientists have made significant contributions to the field of composite fermion. The interested reader may find it useful to consult the following books and review articles:

             Books

    1. "Composite Fermions," Jainendra Jain, Cambridge University Press, 2007.
    2. "Composite Fermions," O. Heinonen (editor), World Scientific, 1998.
    3. "Perspectives in Quantum Hall Effects," S. Das Sarma and A. Pinczuk (editors), Wiley, 1997.
    4. "Quantum Theory of the Electron Liquid," G.F. Giuliani and G. Vignale, Cambridge University Press, 2005.
    5. "Quantum Hall Systems: Braid Groups, Composite Fermions, and Fractional Charge," Lucjan Jacak, Piotr Sitko, Konrad Wieczorek, and Arkadiusz Wojs, Oxford University Press, 2003.

               Pedagogical articles
  1. "The fractional quantum Hall effect," Rev. Mod. Phys. 71, S298-S305 (1999), H.L. Stormer, D.C. Tsui, A.C. Gossard
  2. "Composite Fermions," H.L. Stormer and D.C. Tsui, in "Perspectives in Quantum Hall Effects," edited by S. Das Sarma and A. Pinczuk, Wiley, 1997.
  3. "The composite fermion: A quantum particle and its quantum fluids,"  Physics Today, 53(4), 39 (2000), J.K. Jain
  4. "Research Topic: Composite Fermions ,"  K. von Klitzing group web page
  5. "Composite fermions and the fractional quantum Hall effect in a two-dimensional electron system," I.V. Kukushkin, J.H. Smet, and K. von Klitzing, in "Problems of Condensed Matter Physics," edited by Alexei L. Ivanov and Sergei G. Tikhodeev.
  6. "Composite fermions and the Fermion-Chern-Simons Theory," Physica E 20, 71-78 (2003), B.I. Halperin
  7. "The role of analogy in unraveling the fractional quantum Hall effect mystery," Physica E 20, 79-88 (2003), J.K. Jain