Abstract The formalism of composite fermions (CFs) has been one of the most prominent and successful approaches to describing the fractional quantum Hall effect, in terms of trial many-body wave functions. Testing the accuracy of the latter typically involves rather heavy numerical comparison to exact diagonalization results. Thus, optimizing computational efficiency has been an important technical issue in this field. One generic (and not yet fully understood) property of the CF approach is that it tends to overcount the number of linearly independent candidate states for fixed sets of quantum numbers. Technically speaking, CF Slater determinants that are orthogonal before projection to the lowest Landau level, may lead to wave functions that are identical, or possess linear dependencies, after projection. This leads to unnecessary computations, and has been pointed out in the literature both for fermionic and bosonic systems. We here present a systematic approach that enables us to reveal all linear dependencies between bosonic compact states in the lowest CF ‘cyclotron energy’ sub-band, and almost all dependencies in higher sub-bands, at the level of the CF Slater determinants, i.e. before projection, which implies a major computational simplification. Our approach is introduced for so-called simple states of two-species rotating bosons, and then generalized to generic compact bosonic states, both one- and two-species. Some perspectives also apply to fermionic systems. The identities and linear dependencies we find, are analytically exact for ‘brute force’ projection in the disk geometry.
Composite fermion (CF) wave functions are used to describe both a two dimensional electron gas in a strong magnetic field (the quantum Hall effect system) and rotating two dimensional atomic gases that can be either bosons or fermions. In this thesis a new method for projecting fermionic CF wave functions (here called method 3) to the lowest Landau level is investigated. The new projection is based on attaching a Jastrow factor to a bosonic CF wave function projected in the standard way. This will be compared to two other projection methods: method 1 or Girvin-Jach projection and method 2 or Jain-Kamilla projection. I compare the minimal cyclotron energy compact CF ground state candidates for the ∇²δ interaction both to each other and to the exact ground state found by numerical diagonalization for up to 8 particles. The conclusion is that the CF candidates typically are good approximations of the exact ground state, and that method 3 is almost as good as the other two at this. An important motivation for method 3 is that it preserves linear dependencies from the bosonic states. The puzzle of linear dependencies is a part of CF theory that is not properly understood. Certain important results for low angular momentum states have been found for the bosonic case. Since method 3 works quite well in the cases tested here some of these results can be imported to the fermionic case. I have compared the linear dependencies among minimal cyclotron energy compact states for all three projection methods. If approximate linear dependencies are considered as proper linear dependencies then for all tested cases the same linear dependencies hold for all three projection methods, up to slightly different coefficients.
Despite its success, the composite fermion (CF) construction possesses some mathematical features that have not been fully understood until recently. In particular, this construction is known to produce wave functions that are not necessarily orthogonal, or even linearly independent, after projection to the lowest Landau level. While this is usually not a problem in practice in the quantum Hall regime, we have previously shown that it presents a technical challenge for rotating Bose gases with low angular momenta. These are systems where the CF approach yields surprisingly good approximations to the exact eigenstates of weak short-range interactions, and so solving the problem of linearly dependent wave functions is of interest. It can also be useful for studying higher bands of fermionic quantum Hall states. Here we present several ways of constructing a basis for the space of so-called ``simple'' CF states for two-component rotating Bose gases in the lowest Landau level, and prove that they all give sets of linearly independent wave functions that span the space. Using this basis, we study the structure of the lowest-lying state using so-called restricted wave functions. We also examine the scaling of the overlap between the exact and CF wave functions at the maximal possible angular momentum for simple states.
The composite fermion (CF) theory gives both a phenomenological description for many fractional quantum Hall (FQH) states, as well as a microscopic construction for large scale numerical calculation of these topological phases. The fundamental postulate of mapping FQH states of electrons to integer quantum Hall (IQH) states of CFs, however, was not formally established. The Landau level (LL) projection needed for the microscopic calculations is in some sense uncontrolled and unpredictable. We rigorously derive the unitary relationship between electrons and the CFs, showing the latter naturally emerge from special interactions within a single LL, without resorting to any projection by hand. In this framework, all FQH states topologically equivalent to those described by the conventional CF theory (e.g., the Jain series) have exact model Hamiltonians that can be explicitly derived, and we can easily generalize to FQH states from interacting CFs. Our derivations reveal fundamental connections between the CF theory and the pseudopotential/Jack polynomial constructions, and argue that all Abelian CF states are physically equivalent to the IQH states, while a plethora of non-Abelian CF states can be systematically constructed and classified. We also discuss about implications to experiments and effective field theory descriptions based on the descriptions with CFs as elementary particles.
Abstract The composite fermion (CF) formalism produces wave functions that are not always linearly independent. This is especially so in the low angular momentum regime in the lowest Landau level, where a subclass of CF states, known as simple states, gives a good description of the low energy spectrum. For the two-component Bose gas, explicit bases avoiding the large number of redundant states have been found. We generalize one of these bases to the M -component Bose gas and prove its validity. We also show that the numbers of linearly independent simple states for different values of angular momentum are given by coefficients of q -multinomials.
Single particle basis functions for composite fermions are obtained from which many-composite fermion states confined to the lowest electronic Landau level can be constructed in the standard manner, i.e. by building Slater determinants. This representation enables a Monte Carlo study of systems containing a large number of composite fermions, yielding new quantitative and qualitative information. The ground state energy and the gaps to charged and neutral excitations are computed for a number of fractional quantum Hall effect (FQHE) states, earlier off-limits to a quantitative investigation. The ground state energies are estimated to be accurate to ~0.1% and the gaps at the level of a few percent. It is also shown that at Landau level fillings smaller than or equal to 1/9 the FQHE is unstable to a spontaneous creation of excitons of composite fermions. In addition, this approach provides new conceptual insight into the structure of the composite fermion wave functions, resolving in the affirmative the question of whether it is possible to motivate the composite fermion theory entirely within the lowest Landau level, without appealing to higher Landau levels.
It is known that a subset of fractional quantum Hall wave functions has been expressed as conformal field theory (CFT) correlators, notably the Laughlin wave function [Phys. Rev. Lett. 50, 1395 (1983)] at filling factor $\ensuremath{\nu}=1∕m$ (
m odd) and its quasiholes, and the Pfaffian wave function at $\ensuremath{\nu}=1∕2$ and its quasiholes. We develop a general scheme for constructing composite-fermion (CF) wave functions from conformal field theory. Quasiparticles at $\ensuremath{\nu}=1∕m$ are created by inserting anyonic vertex operators,
P1∕m(z), that replace a subset of the electron operators in the correlator. The one-quasiparticle wave function is identical to the corresponding CF wave function, and the two-quasiparticle wave function has correct fractional charge and statistics and is numerically almost identical to the corresponding CF wave function. We further show how to exactly represent the CF wave functions in the Jain series $\ensuremath{\nu}=s∕(2sp+1)$ [Phys. Rev. Lett. 63, 199 (1989); Composite Fermions (Cambridge University Press, Cambridge, 2007)] as the CFT correlators of a new type of fermionic vertex operators,
Vp,n(z), constructed from
n free compactified bosons; these operators provide the CFT representation of composite fermions carrying
2p flux quanta in the
nth CF Landau level. We also construct the corresponding quasiparticle and quasihole operators and argue that they have the expected fractional charge and statistics. For filling fractions
2∕5 and
3∕7, we show that the chiral CFTs that describe the bulk wave functions are identical to those given by Wen's general classification [Int. J. Mod. Phys. B 6, 1711 (1992); Adv. Phys. 44, 405 (1995)] of quantum Hall states in terms of
K matrices and
l and
t vectors, and we propose that to be generally true. Our results suggest a general procedure for constructing quasiparticle wave functions for other fractional Hall states, as well as for constructing ground states at filling fractions not contained in the principal Jain series.