323. Fractional quantum Hall effect in some multicomponent systems

Zlatko Papic is a PhD student at LPS, Universite Paris-Sud, France (his advisors are Mark Goerbig and Nicolas Regnault) and SCL, Institute of Physics, Serbia (where his thesis advisor is Milica Milovanovic). His main interests include quantum Hall systems. Dmitry.

In recent papers, we have investigated the origins of fractional quantum Hall states that occur or may occur in certain multicomponent systems. Examples of such systems are wide quantum wells and, possibly, graphene.

I. The Quantum Hall Effect

Quantum Hall effect in the pioneering experimental work of 1980s was mainly concerned with thin layers of the so-called two dimensional electron gases (2DEGs), i.e. of electrons constrained to move in two spatial dimensions in specially prepared, high quality GaAs semiconductor heterostructures. Placing such a structure into the perpendicular magnetic field and driving current through it at very low temperatures , leads to a celebrated sequence of plateaus in the dependence of its transversal (Hall) resistance as a function of magnetic field. These plateaus occur at particular ratios between the number of electrons and the number of magnetic flux quanta that pierce the system in the direction perpendicular to the sample. This commensurability can be expressed as the filling factor in terms of integers (understood to have no common divisor), which is the single most important quantity that characterizes the quantum Hall state. The quantization is exact to impressive accuracy and is nowadays used as a standard for the unit of resistance. If , the effect can be explained in a straightforward way, by filling single-particle Landau levels (the integer quantum Hall effect, IQHE); if , the effect is highly non-trivial and presents a fascinating manifestation of fractional numbers in nature, hence the name fractional quantum Hall effect (FQHE). At places where transversal resistance is quantized, the longitudinal resistance (measured along the direction of current) drops to zero (in the limit of vanishing temperature).

II. Trial wave functions

FQHE had its pioneering explanation in terms of the Laughlin wave function for the case of . It was a remarkable “educated guess” of the wave function to describe the strongly correlated ground state of the electrons in the situation described in the previous seciton. Denoting the electron coordinates in the plane by complex numbers , the Laughlin wave function can be written as

up to some unimportant factors that are the consequence of the planar geometry and the Landau level quantization of single electron states. This wave function has many interesting properties. It describes an electron liquid, the ground state of which is incompressible (has an energy gap for all excitations) and the excited states are described by quasiparticles of fractional charge. Laughlin state is the first solid experimental observation of a topological phase of matter.

Subsequent generalizations of Laughlin’s theory came in terms of Jain’s composite fermions, applicable to general integers as long as is odd, and hierarchy theory of Haldane and Halperin. However, a state with an even denominator has also been observed in the experiments of Willet et al. but in the first excited Landau level at the filling factor . One cannot explain it in the usual Laughlin/composite fermion approach and the idea of pairing between electrons has commonly been invoked to explain the origin of this fraction. The simplest realization of pairing between spin polarized electrons is the so-called Pfaffian defined by the Moore-Read wave function:

As written, this wave function describes a state at the filling factor , with being even. The object renders the wave function totally antisymmetric in order to satisfy the Pauli principle. In recent years there has been growing evidence, from the experiments and powerful numerical techniques such as exact diagonalization, that indeed describes the physical system at . Because it supports excitations with non-Abelian statistics, it may have implications for the emerging field of topological quantum computation.

III. Halperin wave functions

Extra degrees of freedom, such as the SU(2) symmetry coming from the real spin of electrons (neglected in the discussion so far), relax the requirement of Pauli principle and hence give another route towards realizing even denominator fractions. The additional degree of freedom can be the ordinary spin or else a “pseudospin” in case of a wide quantum well, where the two lowest electronic subbands correspond to . If the experimental sample is etched in such a way to create a barrier in the middle, thus supressing tunneling between the two “sides”, one can think of it as a bilayer with denoting the left and right layer where electrons can be localized. Incompressible quantum Hall states for such systems have been theoretically predicted by Haldane and Rezayi, and experimentally confirmed for cases of bilayer at filling factor by Eistenstein et al. and by Suen et al. Later on, essentially the same quantum Hall state at was observed in a sample which had the geometry of a single wide well. We refer to the system with larger internal symmetry as being “multicomponent”.

In numerical studies such as exact diagonalization, it is handy to calculate quantum mechanical overlap between the exact ground state and some trial wave function that is believed to capture the underlying physics (Laughlin wave function is also essentially a trial wave function). Since the Hilbert space grows exponentially with the number of particles, high value of the overlap (close to 1) means that the trial wave function is very accurate in describing the ground state. Such calculations for the sample with bilayer geometry, including a realistic bilayer confinement potentials of the quantum well, established that the ground state at is well described by the so-called (3,3,1) Halperin wave function. This wave function generalizes Laughlin wave function to the case with more than one component. For example, when there are two components, the most general form of the Halperin wave function is given by:

Here the electrons are distributed over two components (labeled by ) and the exponents denote the “intra”-component correlations originating from the basic Laughlin-Jastrow building blocks within each component, whereas describes “inter”-component correlations. It is a matter of simple analytic manipulation to show that in the particular case of two components and , if one totally antisymmetrizes the Halperin wave function, one obtains the Pfaffian. There are other interesting connections between the multicomponent Halperin wave functions and the single-component quantum Hall wave functions that have been demonstrated by some of us.

IV. Multicomponent state at

Using exact diagonalization in the spherical geometry, we have revisited a problem of the electrons in a wide quantum well at the filling factor . Contrary to previous studies, we do not assume the effective-bilayer structure ad hoc, but use an infinite square well, retaining its two lowest subbands. Such a model is very versatile because it naturally interpolates between the single-component and two-component picture and we use it to study different phases that may occur in the experimental situation.

At the filling factor the incompressible phases are represented by the (3,3,1) Halperin state and the Pfaffian. As mentioned above, the antisymmetrization of the Halperin wave function leads to the Pfaffian. In experimental language, this can intuitively be understood as the variation of tunneling between the two layers (in a bilayer system) or the splitting between the two lowest subbands (in a wide quantum well). In the region of small tunneling, the ground state shows high overlap with the Halperin (3,3,1) state; as the tunneling is increased, the Halperin state is destroyed and the Pfaffian takes over. We have also related this crossover to the transition from the unpolarized to the fully polarized ground state. The value of the tunneling in experimental sample is such that it likely favors the (3,3,1) state, which is a known conclusion from earlier work.

V. Multicomponent state at

Motivated by the recent experimental paper by Luhman et al. reporting the observation of the quantum Hall state, we have used the same model of the wide quantum well to investigate possible phases at this novel fraction. The (incompressible) trial wave functions for this filling factor are the Halperin (5,5,3) and (7,7,1) states and the Pfaffian (with ). Using state of the art exact diagonalization, we have calculated the overlap between these wave functions and the exact ground state of the wide quantum well for and electrons (in the latter case, each point in the color plot was obtained by diagonalizing a matrix of dimension roughly 13 million!). Unfortunately, the finite-size effects at this filling factor are very strong and the results differ for the two particle-numbers. Nevertheless, a region with the multicomponent (5,5,3) state could be identified. The crossover towards the Pfaffian state, that is expected to happen with the increase in tunneling, is less clear. Additional Monte-Carlo calculations that we have also deployed, speak in favor of the Pfaffian state, so we may indeed have a one-component to two-component transition, like in the better studied case .

VI. Multicomponent states in graphene

Electrons in graphene may be viewed as a particular form of 2DEG, with the fundamental difference that, due to the particular band structure, their low-energy properties are discribed in terms of a zero-mass Dirac equation rather than the usual effective-mass Schroedinger equation. IQHE is also manifest in graphene, and its obervation is a spectacular proof of relativistic electrons (and holes) in graphene, due to an unusual quantization of the Hall conductivity, , in terms of the integer , as expected on theoretical grounds.

Experimental evidence for the FQHE, which is due to electron-electron interactions in a partially filled Landau level, is yet lacking in graphene. In the usual 2DEG in GaAs/AlGaAs heterostructures, the FQHE is, indeed, seen in samples with high mobilities yet unaccessed in graphene on a SiO₂ substrate. Higher mobilities have been achieved in current-annealed suspended graphene, but unexpectedly the IQHE happens to break down above 1T, probably due to extrinsic effects that are not related to the intrinsic electronic properties of these graphene samples.

In spite of the missing FQHE, interaction physics is likely to be at the origin of additional plateaus in the Hall conductivity at filling factors (and 0), where is the ratio between the carrier density ( for electron and for hole transport) and that, of the flux quanta threading the graphene sheet.

From a theoretical point of view, interactions in graphene are expected to be relevant. Unlike the quantum well of the previous sections, graphene has a potentially even richer structure because it clearly possesses fourfold spin-valley symmetry, that is described in the framework of the SU(4) group which covers the two copies of the SU(2) spin and the SU(2) valley isospin. Based on these considerations, graphene in a strong magnetic field may thus be viewed as a four-component quantum Hall system. An interesting theoretical expectation resulting from this feature is the formation of a quantum Hall ferromagnet at with SU(4)-skyrmion excitations, which may have peculiar magnetic properties. Also for the FQHE, the SU(4) spin-valley symmetry is expected to play a relevant role and has been considered within a composite-fermion approach as well as ours which is based on SU(4) Halperin wave functions (these wave functions are written down assuming there are 4 groups of electrons, two groups in each “layer”; the wave function is then denoted by where describes the correlation within each group, denotes the correlation between the two groups which are in the same “layer” and, finally, is the correlation between groups in different “layers”).

In our paper, we review how the four-component structure of graphene may have particular signatures in a possible FQHE. The electrons in graphene lose their relativistic character associated with the Lorentz invariance once they are restricted to a single Landau level. The main difference between the 2DEG and graphene arises from the approximate SU(4) spin-valley symmetry, which is respected in a wide energy range. Another difference arises from the spinor character of the wavefunctions, which yields a different effective electron-electron interaction in graphene as compared to the 2DEG. The graphene interaction potential in the first excited Landau level (in both the valence and the conduction band) is shown to be similar to that in the central zero-energy Landau level , yet with a slightly larger overall energy scale (roughly 10% larger).

The FQHE at is described as a Laughlin state with SU(4)-ferromagnetic spin-valley ordering, similar to the state at . In contrast to this state, the system profits from its internal degrees of freedom by choosing a state with partial and full SU(4)-isospin depolarisation at and , respectively. The [3;2,3] Halperin state at is a valley-isospin singlet, but its physical spin is ferromagnetically ordered and may eventually be oriented by the Zeeman effect. The state at is described in terms of a [3;2,2] Halperin wavefunction, which is an SU(4) singlet with necessarily zero spin and valley isospin polarisation. A possible FQHE at in graphene may therefore be sensitive to the Zeeman effect at high magnetic fields, and one may expect transitions between states with different polarisation, similar to the 2DEG at and .

Some relevant literature, textbooks

Quantum Hall effect

1. S. Das Sarma and A. Pinczuk, “Perspectives in quantum Hall effects”
2. D. Yoshioka, “The Quantum Hall Effect”

(see also my review of Yoshioka’s book. Dmitry)

3. T. Chakraborty and P. Pietilainen, “The Quantum Hall Effects: Integer and Fractional”

Composite fermions

1. J. Jain, “Composite fermions”
2. O. Heinonen, “Composite fermions: a Unified View of the Quantum Hall Regime”

a classic but sometimes one still needs to go back to it: R. Prange and S. Girvin, “The Quantum Hall Effect” (Springer).