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arXiv:1303.2809 · Physical Review B

Landau level mixing in the perturbative limit

Steven H. Simon, Edward H. Rezayi

Published Apr 22, 2013• 76 citations
arXivHTMLPDF

Abstract

We study the effects of Landau level mixing in the limit of weak electron interaction. We use a numerical method to obtain the two- and three-body corrections to quantum Hall pseudopotentials, which are exact to lowest order in the Landau level mixing parameter. Our results are in general agreement with certain analytic results (some derived here; some derived by other authors) in the thermodynamic limit. We find that the convergence to this thermodynamic limit can be slow. This suggests that errors could occur if one tries to use pseudopotentials derived in a thermodynamic limit for numerical work on finite systems.

DOI
10.1103/physrevb.87.155426

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We study the effects of Landau level mixing in the limit of weak electron interaction. We use a numerical method to obtain the two- and three-body corrections to quantum Hall pseudopotentials, which are exact to lowest order in the Landau level mixing parameter. Our results are in general agreement with certain analytic results (some derived here; some derived by other authors) in the thermodynamic limit. We find that the convergence to this thermodynamic limit can be slow. This suggests that errors could occur if one tries to use pseudopotentials derived in a thermodynamic limit for numerical work on finite systems.
The discovery of the quantum Hall effect in 1980 opened to physics one of the simplest systems for studying many-body correlations. Numerical techniques and trial wavefunctions have proven useful for describing the novel collective behavior of the electrons, but have not fully explained all features of the fractional quantum Hall effect. For example, it is predicted that Landau level mixing should have a moderate effect on the system for all but the very strongest magnetic fields, but this effect has not been extensively studied. Among the tools most useful to modeling and describing the quantum Hall system is the Haldane pseudopotential, which is the energy of electron pairs as a function of their coupled total angular momentum. Because the pseudopotential uniquely determines the correlative behaviors in the quantum Hall system, the study of the pseudopotential is particularly important. In this dissertation I will first derive novel analytic expressions for the pair interactions and the pseudopotential using the techniques of angular momentum algebra. Then, I will use these results to numerically determine the effect of Landau level mixing on the pair pseudopotentials.
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We derive full analytic expressions of three-body interactions from Landau-level (LL) mixing in fractional quantum Hall systems with Schrieffer-Wolff transformation. The formalism can be applied to any LL, and to very general systems without rotational or Galilean invariance. We illustrate how three-body pseudopotentials can be readily computed from the analytical expressions for a wide variety of different systems, and show that for realistic systems, softening the bare Coulomb interactions (e.g., finite thickness or screening) can significantly suppress three-body interactions. More interestingly, for experimental systems without Galilean invariance (which is common for real materials), there is strong evidence that higher orders in band dispersion can drive the Moore-Read state from anti-Pfaffian to Pfaffian phase. Our analysis points to the importance of the realistic band structure details to the non-Abelian topological phases, and the analytical expressions we derived can also be very useful for high fidelity numerical computations.
A major motivation for building a quantum computer is that it provides a tool to efficiently simulate strongly correlated quantum systems. In this paper, we present a detailed roadmap on how to simulate a two-dimensional electron gas---cooled to absolute zero and pierced by a strong transversal magnetic field---on a quantum computer. This system describes the setting of the fractional quantum Hall effect, one of the pillars of modern condensed-matter theory. We give analytical expressions for the two-body integrals that allow for mixing between NNN Landau levels at a cutoff MMM in angular momentum and give gate-count estimates for the efficient simulation of the energy spectrum of the Hamiltonian on an error-corrected quantum computer. We then focus on studying efficiently preparable initial states and their overlap with the exact ground state for noisy as well as error-corrected quantum computers. By performing an imaginary time evolution of the covariance matrix, we find the generalized Hartree-Fock solution to the many-body problem and study how a multireference state expansion affects the state overlap. We perform small-system numerical simulations to study the quality of the two initial state Ans\"atze in the lowest Landau level approximation.
Exact diagonalization of a two-dimensional electron gas in a strong magnetic field in the disk geometry shows that there exists a filling-factor range in the second Landau level where the states significantly differ from those in the lowest Landau level. We show that the difference arises because the interaction between composite fermions is not negligible in higher Landau levels, as indicated by a substantial mixing between composite-fermion Landau-like levels, or $\ensuremath{\Lambda}$ levels. We find that the exact ground state is well reproduced by composite-fermion theory with $\ensuremath{\Lambda}$-level mixing incorporated at the lowest level of approximation. Using the same variational approach in the spherical geometry we estimate the excitation gap at filling 1∕31∕31∕3 in the second Landau level (which corresponds to 7∕37∕37∕3 of experiment).