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arXiv:0708.3211 · Europhysics Letters (EPL)

Monte Carlo study of Bose Laughlin wave function for filling factors 1/2, 1/4 and 1/6

Orion Ciftja

Published Mar 21, 2006• 47 citations
arXivHTMLPDF

Abstract

Strongly correlated two-dimensional electronic systems subject to a perpendicular magnetic field at lowest Landau level (LLL) filling factors: 1/2, 1/4 and 1/6, are believed to be composite-fermion (CF) Fermi liquid phases. Even though a Bose Laughlin wave function cannot describe these filling factors, we investigate whether such a wave function provides a lower energy bound to the true CF Fermi liquid energies. By using Monte Carlo simulations in disk geometry we compute the Bose Laughlin energies and compare them to corresponding results for the spin-polarized LLL CF Fermi liquid state and avalable data from the literature. We find the unexpected result that, for filling factors ν = 1/4 and 1/6, the Bose Laughlin ground-state energy is practically identical to the true CF liquid energy while this is not the case at ν = 1/2 where the Bose Laughlin ground-state energy is sizeably lower than the energy of the CF Fermi liquid state.

DOI
10.1209/epl/i2005-10545-8

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Strongly correlated two-dimensional electronic systems subject to a perpendicular magnetic field at lowest Landau level (LLL) filling factors: 1/2, 1/4 and 1/6, are believed to be composite-fermion (CF) Fermi liquid phases. Even though a Bose Laughlin wave function cannot describe these filling factors, we investigate whether such a wave function provides a lower energy bound to the true CF Fermi liquid energies. By using Monte Carlo simulations in disk geometry we compute the Bose Laughlin energies and compare them to corresponding results for the spin-polarized LLL CF Fermi liquid state and avalable data from the literature. We find the unexpected result that, for filling factors ν = 1/4 and 1/6, the Bose Laughlin ground-state energy is practically identical to the true CF liquid energy while this is not the case at ν = 1/2 where the Bose Laughlin ground-state energy is sizeably lower than the energy of the CF Fermi liquid state.
We propose a ground-state trial wave function for a two-dimensional Wigner crystal in a strong perpendicular magnetic field. The wave function includes Laughlin-Jastrow correlations between electron pairs, and may be interpreted as a crystal state of composite fermions or composite bosons. Treating the power mmm of the Laughlin-Jastrow factor as a variational parameter, we use quantum Monte Carlo simulations to compute the energy of these new states. We find that our wave functions have lower energy than existing crystalline wave functions in the lowest Landau level. Our results are consistent with experimental observations of the filling factor at which the transition between the fractional quantum Hall liquid and the Wigner crystal occurs for electron systems. Exchange contributions to the wave functions are estimated quantitatively and shown to be negligible for sufficiently small filling factors.
While an ordinary Fermi sea is perturbatively robust to interactions, the paradigmatic composite-fermion (CF) Fermi sea arises as a nonperturbative consequence of emergent gauge fields in a system where there was no Fermi sea to begin with. A mean-field picture suggests two Fermi seas, of composite fermions made from electrons or holes in the lowest Landau level, which occupy different areas away from half filling and thus appear to represent distinct states. Using the microscopic theory of composite fermions, which satisfies particle-hole symmetry in the lowest Landau level to an excellent approximation, we show that the Fermi wave vectors at filling factors ν and 1-ν are equal when expressed in units of the inverse magnetic length, and are generally consistent with the experimental findings of Kamburov et al. [Phys. Rev. Lett. 113, 196801 (2014)]. Our calculations suggest that the area of the CF Fermi sea may slightly violate the Luttinger area rule.
At low temperatures, two-dimensional electron systems in a perpendicular magnetic field exhibit remarkable quantum phenomena. In these extreme conditions, strongly correlated electron systems stabilize in different quantum phases as the filling factor of Landau levels is varied. In this Brief Report we present finite-size Monte Carlo simulation results for anisotropic quantum Hall liquid states observed at certain even-denominator filling factors. The anisotropic electronic liquid phases are described by a broken rotational symmetry wave function. Our investigations of systems of few electrons in disk geometry indicate that an anisotropic liquid-crystalline quantum Hall phase with broken rotational symmetry is energetically favored relative to an isotropic liquid one.
The composite fermion Fermi liquid (CFL) state at ν = 1 / 2 filling of a Landau level is a paradigmatic non-Fermi liquid borne out purely by Coulomb interactions. But in what ways is this exotic state of matter different from a Fermi liquid? The CFL entanglement entropy was indeed found to exhibit a significant enhancement compared to free electrons [J. Shao , ], which was subsequently ruled out as a finite-size effect by the study of a lattice CFL analog [R. V. Mishmash and O. I. Motrunich, ]. Moreover, the enhancement was not observed in a quasi-one-dimensional limit of the Coulomb ground state at ν = 1 / 2 [S. D. Geraedts , ]. Here, we revisit the problem of entanglement scaling in the CFL state realized in a two-dimensional electron gas. Using Monte Carlo evaluation of the second Rényi entropy S 2 for the CFL variational wave function, we show that the entanglement enhancement is present not only at ν = 1 / 2 , but also at ν = 1 / 4 , as well as in bosonic CFL states at ν = 1 and ν = 1 / 3 fillings. In all cases, we find the scaling of S 2 with subsystem size to be enhanced compared to the noninteracting case, and insensitive to the choice of geometry and projection to the lowest Landau level. We also demonstrate that for CFL states, the variance of the particle number in a subsystem obeys area-law scaling with a universal subleading corner contribution, in stark contrast with free fermions. Our results establish the enhanced entanglement scaling and suppressed charge fluctuations as fingerprints of non-Fermi-liquid correlations in CFL states.
We report finite-size studies of composite fermions (CF's) at even-denominator filling fractions. These show that, in the absence of strong Landau level (LL) mixing effects, the CF liquid cannot form for a half-filled second LL. We relate this to the absence of a hierarchy of incompressible states for filling fractions $2+1/3<\ensuremath{\nu}<2+2/3$. We find that a polarized CF liquid should exist for a quarter-filled second LL. We also show that, for CF states, the variation with system size of the ground state energy of interacting electrons follows that for noninteracting particles in zero magnetic field. We use this to estimate the CF effective masses. We suggest an explanation of these results based on the Rezayi-Read trial wave function and present estimates for the flux-binding energy.
We compare quantum Hall systems at filling factors $\ensuremath{\nu}=2$ to $\ensuremath{\nu}=\frac{2}{3}$ and 25\frac{2}{5}52​, corresponding to the exact filling of two lowest electron or composite fermion (CF) Landau levels. The two fractional states are examples of CF liquids with spin dynamics. There is a close analogy between the ferromagnetic (spin polarization P=1P=1P=1) and paramagnetic (P=0)(P=0)(P=0) incompressible ground states that occur in all three systems in the limits of large and small Zeeman spin splitting. However, the excitation spectra are different. At $\ensuremath{\nu}=2$, we find spin domains at half-polarization (P=12)(P=\frac{1}{2})(P=21​), while antiferromagnetic order seems most favorable in the CF systems. The transition between P=0P=0P=0 and 1, as seen when, e.g., the magnetic field is tilted, is also studied by exact diagonalization in toroidal and spherical geometries. The essential role of an effective CF-CF interaction is discussed, and the experimentally observed incompressible half-polarized state is found in some models.