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arXiv:1708.08816 · Physical review. B./Physical review. B

Composite fermions on a torus

Songyang Pu, Ying-Hai Wu, Jainendra K. Jain

Published Nov 7, 2017• 29 citations
arXivHTMLPDF

Abstract

We achieve an explicit construction of the lowest Landau level (LLL) projected wave functions for composite fermions in the periodic (torus) geometry. To this end, we first demonstrate how the vortex attachment of the composite fermion (CF) theory can be accomplished in the torus geometry to produce the ``unprojected'' wave functions satisfying the correct (quasi)periodic boundary conditions. We then consider two methods for projecting these wave functions into the LLL. The direct projection produces valid wave functions but can be implemented only for very small systems. The more powerful and more useful projection method of Jain and Kamilla fails in the torus geometry because it does not preserve the periodic boundary conditions and thus takes us out of the original Hilbert space. We have succeeded in constructing a modified projection method that is consistent with both the periodic boundary conditions and the general structure of the CF theory. This method is valid for a large class of states of composite fermions, called ``proper states,'' which includes the incompressible ground states at electron filling factors $\ensuremath{\nu}=\frac{n}{2pn+1}$, their charged and neutral excitations, and also the quasidegenerate ground states at arbitrary filling factors of the form $\ensuremath{\nu}=\frac{{\ensuremath{\nu}}^{*}}{2p{\ensuremath{\nu}}^{*}+1}$, where $n$ and $p$ are integers and ${\ensuremath{\nu}}^{*}$ is the CF filling factor. Comparison with exact results known for small systems for the ground and excited states at filling factors $\ensuremath{\nu}=1/3$, 2/5, and 3/7 demonstrates our LLL-projected wave functions to be extremely accurate representations of the actual Coulomb eigenstates. Our construction enables the study of large systems of composite fermions on the torus, thereby opening the possibility of investigating numerous interesting questions and phenomena.

DOI
10.1103/physrevb.96.195302

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We achieve an explicit construction of the lowest Landau level (LLL) projected wave functions for composite fermions in the periodic (torus) geometry. To this end, we first demonstrate how the vortex attachment of the composite fermion (CF) theory can be accomplished in the torus geometry to produce the unprojected'' wave functions satisfying the correct (quasi)periodic boundary conditions. We then consider two methods for projecting these wave functions into the LLL. The direct projection produces valid wave functions but can be implemented only for very small systems. The more powerful and more useful projection method of Jain and Kamilla fails in the torus geometry because it does not preserve the periodic boundary conditions and thus takes us out of the original Hilbert space. We have succeeded in constructing a modified projection method that is consistent with both the periodic boundary conditions and the general structure of the CF theory. This method is valid for a large class of states of composite fermions, called proper states,'' which includes the incompressible ground states at electron filling factors \ensuremathν=n2pn+1\ensuremath{\nu}=\frac{n}{2pn+1}\ensuremathν=2pn+1n​, their charged and neutral excitations, and also the quasidegenerate ground states at arbitrary filling factors of the form \ensuremathν=\ensuremathν∗2p\ensuremathν∗+1\ensuremath{\nu}=\frac{{\ensuremath{\nu}}^{*}}{2p{\ensuremath{\nu}}^{*}+1}\ensuremathν=2p\ensuremathν∗+1\ensuremathν∗​, where nnn and ppp are integers and \ensuremathν∗{\ensuremath{\nu}}^{*}\ensuremathν∗ is the CF filling factor. Comparison with exact results known for small systems for the ground and excited states at filling factors \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3, 2/5, and 3/7 demonstrates our LLL-projected wave functions to be extremely accurate representations of the actual Coulomb eigenstates. Our construction enables the study of large systems of composite fermions on the torus, thereby opening the possibility of investigating numerous interesting questions and phenomena.
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.
An adiabatic approach put forward by Greiter and Wilczek interpolates between the integer quantum Hall effects of electrons and composite fermions by varying the statistical flux bound to electrons continuously from zero to an even integer number of flux quanta, such that the intermediate states represent anyons in an external magnetic field with the same ``effective'' integer filling factor. We consider such anyons on a torus, and we construct representative wave functions for their ground as well as excited states. These wave functions involve higher Landau levels in general, but can be explicitly projected into the lowest Landau level for many parameters. We calculate the variational energy gap between the first excited state and the ground state, and we find that it remains open as the statistical phase is varied. Finally, we obtain from these wave functions, both analytically and numerically, various topological quantities, such as the ground-state degeneracy, the Chern number, and the Hall viscosity.
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.
We construct explicit lowest-Landau-level wave functions for the composite-fermion Fermi sea and its low-energy excitations following a recently developed approach [Pu, Wu, and Jain, Phys. Rev. B 96, 195302 (2017)] and demonstrate them to be very accurate representations of the Coulomb eigenstates. We further ask how the Berry phase associated with a closed loop around the Fermi circle, predicted to be $\ensuremath{\pi}$ in a Dirac composite fermion theory satisfying particle-hole symmetry [D. T. Son, Phys. Rev. X 5, 031027 (2015)], is affected by Landau-level mixing. For this purpose, we consider a simple model wherein we determine the variational ground state as a function of Landau-level mixing within the space spanned by two basis functions: the lowest-Landau-level projected and the unprojected composite-fermion Fermi sea wave functions. We evaluate Berry phase for a path around the Fermi circle within this model following a recent prescription, and find that it rotates rapidly as a function of Landau-level mixing. We also consider the effect of a particle-hole symmetry-breaking three-body interaction on the Berry phase while confining the Hilbert space to the lowest Landau level. Our study deepens the connection between the $\ensuremath{\pi}$ Berry phase and the exact particle-hole symmetry in the lowest Landau level.
Field theories of the composite-fermion (CF) metal model it as a Fermi sea of composite fermions coupled to an emergent gauge field. Within a random phase approximation, these theories predict that the Landau damping of the gauge field resulting from its coupling to the low-energy, long-wavelength CF particle-hole excitations modifies the electrons' density-density correlation function related to the static structure factor S(q) at wave vector q. This produces a nonanalytic correction ∝q^{3}lnq to S(q) (with the magnetic length ℓ_{B}=1). Thanks to the recently developed quaternion formulation for Jain-Kamilla projection of CF wave functions, the evaluation of S(q) from the accurate microscopic theory of composite fermions has now become possible for systems containing as many as N=900 CFs, which enables a reliable determination of the small-q behavior of S(q). We study CF metals corresponding to electrons at Landau level filling factors ν=1/2 and 1/4, and for completeness, also of bosons at ν=1 and 1/3. In the q→0 limit, our microscopic calculation reveals a q^{3} term in S(q) of the CF metals rather than q^{3}lnq. This behavior is well predicted by a model of a noninteracting Fermi sea of dipolar CFs, which also obtains its coefficient accurately.