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Three M-Species Generalizations for Rotating Bose Gases in the Lowest Landau Level

Vidar Skogvoll

Published Jan 1, 2017• 1 citation
OpenAlex

Abstract

We generalize three known theoretical results from 1- and 2-species systems to an M-component rotating Bose gas in the lowest Landau level (LLL). First, we find a subspace of the Hilbert space which is closed under homogeneous interaction and diagonalize analytically within this subspace to find the eigenenergies. This is a straight-forward generalization of Ref. [1] from 2 to M species. Second, we state and prove the validity of a basis for a particular class of composite fermion (CF) trial wave functions, known as "simple states" [2], for M-component Bose gases. By doing so, we generalize the simple state basis for 2-component Bose gases found in Ref. [3]. The number of basis elements in the M -component basis is linked to a certain class of combinatorial polynomials called q-multinomials. The proof of the M-component basis has been condensed into an article [4] and is currently under consideration by Journal of Physics A. Third, we have developed numerical tools to study the properties of the CF formalism applied to M -species Bose gases in the LLL and show that the formalism is excellent in capturing the low-energy behavior of such systems for M > 2. This is a generalization of previous results [5, 6, 2], which has shown that the CF formalism is excellent when applied to 1- and 2-component Bose gases. As part of this third generalization, we show how the formalism of simple states captures a class of eigenfunctions of the homogeneous contact interaction. This can be used to effectively visualize the evolution of the ground state of N spin-1 bosons under spin-independent interaction as a function of angular momentum L (originally found in Ref. [7]) for L ≤ N .

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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.

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Rotational properties of two-component Bose gases in the lowest Landau level

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We study the rotational (yrast) spectra of dilute two-component atomic Bose gases in the low angular momentum regime, assuming equal interspecies and intraspecies interaction. Our analysis employs the composite fermion (CF) approach including a pseudospin degree of freedom. While the CF approach is not a priori expected to work well in this angular momentum regime, we show that composite fermion diagonalization gives remarkably accurate approximations to low-energy states in the spectra. For angular momenta $0<L<M$ (where $N$ and $M$ denote the numbers of particles of the two species, and $M\ensuremath{\ge}N$), we find that the CF states span the full Hilbert space and provide a convenient set of basis states which, by construction, are eigenstates of the symmetries of the Hamiltonian. Within this CF basis, we identify a subset of the basis states with the lowest $\ensuremath{\Lambda}$-level kinetic energy. Diagonalization within this significantly smaller subspace constitutes a major computational simplification and provides very close approximations to ground states and a number of low-lying states within each pseudospin and angular momentum channel.

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A Novel Projection: Towards a Deeper Understanding of Mathematical Structures in the Composite Fermion Formalism

Knut Halvor Helland

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.

•Jan 1, 2018•0 citations
We generalize three known theoretical results from 1- and 2-species systems to an M-component rotating Bose gas in the lowest Landau level (LLL). First, we find a subspace of the Hilbert space which is closed under homogeneous interaction and diagonalize analytically within this subspace to find the eigenenergies. This is a straight-forward generalization of Ref. [1] from 2 to M species. Second, we state and prove the validity of a basis for a particular class of composite fermion (CF) trial wave functions, known as "simple states" [2], for M-component Bose gases. By doing so, we generalize the simple state basis for 2-component Bose gases found in Ref. [3]. The number of basis elements in the M -component basis is linked to a certain class of combinatorial polynomials called q-multinomials. The proof of the M-component basis has been condensed into an article [4] and is currently under consideration by Journal of Physics A. Third, we have developed numerical tools to study the properties of the CF formalism applied to M -species Bose gases in the LLL and show that the formalism is excellent in capturing the low-energy behavior of such systems for M > 2. This is a generalization of previous results [5, 6, 2], which has shown that the CF formalism is excellent when applied to 1- and 2-component Bose gases. As part of this third generalization, we show how the formalism of simple states captures a class of eigenfunctions of the homogeneous contact interaction. This can be used to effectively visualize the evolution of the ground state of N spin-1 bosons under spin-independent interaction as a function of angular momentum L (originally found in Ref. [7]) for L ≤ N .
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.
We study the rotational properties of a two-component Bose gas trapped in a rotating harmonic potential, effectively in two dimensions. We first consider a completely homogeneous interaction between particles, and assume the bosons to be in the lowest Landau level. We derive analytical expressions for some wave functions and energies at low angular momenta, and use these results together with exact numerical diagonalization results to test the applicability of composite fermion (CF) trial wave functions to the system. We find that a general procedure based on CF wave functions reproduces the exact ground states and many of the excited states for sufficiently low angular momenta, and give very good approximate results at angular momenta up to the two-component analogy of a single vortex. Additionally, we produce very simple CF states that have remarkably high overlaps with the lowest-lying states. Finally, we briefly discuss the case of inhomogeneous interactions, pointing out the difficulties arising from this generalization.
We study the rotational (yrast) spectra of dilute two-component atomic Bose gases in the low angular momentum regime, assuming equal interspecies and intraspecies interaction. Our analysis employs the composite fermion (CF) approach including a pseudospin degree of freedom. While the CF approach is not a priori expected to work well in this angular momentum regime, we show that composite fermion diagonalization gives remarkably accurate approximations to low-energy states in the spectra. For angular momenta 0<L<M0<L<M0<L<M (where NNN and MMM denote the numbers of particles of the two species, and $M\ensuremath{\ge}N$), we find that the CF states span the full Hilbert space and provide a convenient set of basis states which, by construction, are eigenstates of the symmetries of the Hamiltonian. Within this CF basis, we identify a subset of the basis states with the lowest $\ensuremath{\Lambda}$-level kinetic energy. Diagonalization within this significantly smaller subspace constitutes a major computational simplification and provides very close approximations to ground states and a number of low-lying states within each pseudospin and angular momentum channel.
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 energy spectrum for an ultracold rotating Bose gas in a harmonic trap is calculated exactly for small systems, allowing the atoms to occupy several Landau levels. Two vortexlike states and two strongly correlated states (the Pfaffian and Laughlin) are considered in detail. In particular, their critical rotation frequencies and energy gaps are determined as a function of particle number, interaction strength, and the number of Landau levels occupied (up to three). For the vortexlike states, the lowest-Landau-level (LLL) approximation is justified only if the interaction strength decreases with the number of particles; nevertheless, the constant of proportionality increases rapidly with the angular momentum per particle. For the strongly correlated states, however, the interaction strength can increase with particle number without violating the LLL condition. The results suggest that, in large systems, the Pfaffian and Laughlin states might be stabilized at rotation frequencies below the centrifugal limit for sufficiently large interaction strengths, with energy gaps a significant fraction of the trap energy.
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.