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arXiv:1601.06736 · Physical review. B./Physical review. B

Energy projection and modified Laughlin states

Mikael Fremling, Jørgen Fulsebakke, Niall Moran, Johannes K. Slingerland

Published Jun 24, 2016• 15 citations
arXivHTMLPDF

Abstract

We develop a method to efficiently calculate trial wave functions for quantum Hall systems which involve projection onto the lowest Landau level. The method essentially replaces the lowest Landau level projection by projection onto the $M$ lowest eigenstates of a suitably chosen Hamiltonian acting within the lowest Landau level. The resulting ``energy projection'' is a controlled approximation to the exact lowest Landau level projection which improves with increasing $M$. It allows us to study the projected trial wave functions for system sizes close to the maximal sizes that can be reached by exact diagonalization and can be straightforwardly applied in any geometry. As a first application and test case, we study a class of trial wave functions first proposed by Girvin and Jach [Girvin and Jach, Phys. Rev. B 29, 5617 (1984)], which are modifications of the Laughlin states involving a single real parameter. While these modified Laughlin states probably represent the same universality class exemplified by the Laughlin wave functions, we show by extensive numerical work for systems on the sphere and torus that they provide a significant improvement of the variational energy, overlap with the exact wave function and properties of the entanglement spectrum.

DOI
10.1103/physrevb.93.235149

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We develop a method to efficiently calculate trial wave functions for quantum Hall systems which involve projection onto the lowest Landau level. The method essentially replaces the lowest Landau level projection by projection onto the MMM lowest eigenstates of a suitably chosen Hamiltonian acting within the lowest Landau level. The resulting ``energy projection'' is a controlled approximation to the exact lowest Landau level projection which improves with increasing MMM. It allows us to study the projected trial wave functions for system sizes close to the maximal sizes that can be reached by exact diagonalization and can be straightforwardly applied in any geometry. As a first application and test case, we study a class of trial wave functions first proposed by Girvin and Jach [Girvin and Jach, Phys. Rev. B 29, 5617 (1984)], which are modifications of the Laughlin states involving a single real parameter. While these modified Laughlin states probably represent the same universality class exemplified by the Laughlin wave functions, we show by extensive numerical work for systems on the sphere and torus that they provide a significant improvement of the variational energy, overlap with the exact wave function and properties of the entanglement spectrum.
Real-space entanglement spectrum (RSES) of a quantum Hall (QH) wave function gives a natural route to infer the nature of its edge excitations. Computation of RSES becomes expensive with increase in number of particles and number of included Landau levels (LL). RSES can be efficiently computed using Monte Carlo (MC) methods for trial states that can be written as products of determinants such as the composite fermion (CF) and parton states. This computational efficiency applies also to the RSES of lowest Landau level (LLL) projected CF and parton states, however, LLL projection to be used here requires approximations that generalize the Jain Kamilla (JK) projection. This work is a careful study of how this approximation should be done. We identify the approximation that is closest in spirit to JK projection, and perform tests of the approximations involved in the projection by comparing the MC results with the RSES obtained from computationally expensive but exact methods. We present the techniques and use them to calculate exact RSES of the exact LLL projected bosonic Jain 2/32/32/3 state in bipartition of systems of sizes up to N=24N=24N=24 on the sphere. For the lowest few angular momentum sectors of the RSES, we present evidence to show that MC results closely match the exact spectra. We also compare other plausible projection schemes. We also calculate the exact RSES of the unprojected fermionic Jain 2/52/52/5 state obtained from exact diagonalization of the Trugman-Kivelson Hamiltonian in the two lowest LLs on the sphere. By comparing with the RSES of the unprojected 2/52/52/5 state from Monte Carlo methods, we show that the latter is practically exact.
New trial wave functions corresponding to half filling quantum Hall states are proposed. These wave functions are constructed by first pairing up the quasielectrons of the 1/3 Laughlin quantum Hall state, with the same relative angular momentum for each pair, and then making the paired quasielectrons condense into a 1/4 Laughlin state. The quasiparticle excitations of the proposed wave functions carry ±1/4\pm1/4±1/4 of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta NϕN_ϕNϕ​ is shown to be related to the number of electrons NNN by Nϕ=2N−(5−q)N_ϕ = 2N-(5-q)Nϕ​=2N−(5−q) with qqq being the relative angular momentum between the quasielectrons in each pair which takes values of non-negative even integers. The overlaps are calculated between the proposed trial wave functions, including the ground state, quasiexciton states, and quasihole states, and the exact states of the finite size systems at Nϕ=2N−3N_ϕ = 2N-3Nϕ​=2N−3. The near unity overlaps are obtained in the lowest Landau level, while the moderate overlaps are obtained in the second Landau level. The relevance of the wave functions to the yet to be discovered fractional quantum Hall effect in the lowest Landau level, as well as the 5/2 quantum Hall effect is also discussed.
The most robust fractional quantum Hall states occur in the lowest Landau level at filling factors, 1/3 and 1/5. Such states are very well described by Laughlin's wave function. In this work, we have succeeded in calculating exactly the one-particle density function of the Laughlin states for some finite systems of particles in a disk geometry. The exact results we provide are not only important for the Laughlin states, but also for the general field of numerical calculations because they can serve as benchmarks to test the accuracy of various approaches, numerical schemes and computational methods used in studies of strongly correlated electronic systems.
The bootstrap method aims to solve problems by imposing constraints on the space of physical observables, which often follow from physical assumptions such as positivity and symmetry. Here, we employ a bootstrap approach to study interacting electrons in the lowest Landau level by minimizing the energy as a function of the static structure factor subject to a set of constraints, bypassing the need to construct the full many-body wave function. This approach rigorously lower bounds the ground state energy, making it complementary to conventional variational upper bounds. We show that the lower bound we obtain is relatively tight, within at most a few percent from the ground state energy computed with exact diagonalization at small system sizes, and generally gets tighter as we include more constraints. We also show that, by combining the bootstrap lower bounds with variational Monte Carlo calculations for various quantum Hall states, we can obtain two-sided bounds on the ground state energy that rigorously bounds the error to below a few percentage for large system sizes where exact diagonalization is not accessible. Beyond the energetics, our results reproduce the correct power law dependence of the pair correlation function at short distances and the existence of a large entanglement gap in the two-particle entanglement spectra for the Laughlin states at ν = 1 / 3 . We further identify signatures of the composite Fermi liquid state close to half filling. This shows that the bootstrap approach is capable, in principle, of describing nontrivial gapped topologically ordered, as well as gapless, phases. We further study the evolution of the ground state energy and correlation functions as we interpolate between the lowest and first Landau levels at half filling and find indications of a phase transition from a composite Fermi liquid to a non-Abelian state. We also discuss more generally how the geometry of the allowed set of structure factors can be used to diagnose phase transitions. At the end, we discuss possible extensions and current limitations of this approach and how they can be overcome in future studies. Our work establishes numerical bootstrap as a promising method to study many-body phases in topological bands, paving the way to its application in moiré platforms where the energetic competition between fractional quantum anomalous Hall, symmetry broken, and gapless states remains poorly understood.
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.
Although a parent Hamiltonian for the Laughlin 1/31/31/3 wave function has been long known in terms of the Haldane pseudopotentials, no parent Hamiltonians are known for the lowest-Landau-level projected wave functions of the composite fermion theory at n/(2n+1)n/(2n+1)n/(2n+1) with $n\ensuremath{\ge}2$. If one takes the two lowest Landau levels to be degenerate, the Trugman-Kivelson interaction produces the unprojected 2/5 wave function as the unique zero energy solution. If the lowest three Landau levels are assumed to be degenerate, the Trugman-Kivelson interaction produces a large number of zero energy states at $\ensuremath{\nu}=3/7$. We propose that adding an appropriately constructed three-body interaction yields the unprojected 3/73/73/7 wave function as the unique zero energy solution and report extensive exact diagonalization studies that provide strong support to this proposal.