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arXiv:cond-mat/9803098 · Physical review. B, Condensed matter

Laughlin-Jastrow-correlated Wigner crystal in a strong magnetic field

Hangmo Yi, H. A. Fertig

Published Aug 15, 1998• 65 citations
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Abstract

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 $m$ 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.

DOI
10.1103/physrevb.58.4019

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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.
The two-dimensional Wigner crystals are studied with the variational quantum Monte Carlo method. The close relationship between the ground-state wave function and the collective excitations in the system is illustrated, and used to guide the construction of the ground-state wave function of the strongly correlated solid. Exchange, correlation, and magnetic-field effects all give rise to distinct physical phenomena. In the absence of any external magnetic field, interesting spin orderings are observed in the ground state of the electron crystal in various two-dimensional lattices. In particular, two-dimensional bipartite lattices are shown not to lead necessarily to an antiferromagnetic ground state. In the quantum Hall effect regime, a strong magnetic field introduces energy and length scales. The magnetic field quenches the kinetic energy and poses constraints on how the electrons may correlate with each other. Care is taken to ensure the appropriate translational properties of the wave function when the system is in a uniform magnetic field. We have examined the exchange, intra-Landau-level correlation as well as Landau-level-mixing effects with various variational wave functions. We also determine their dependences on the experimental parameters such as the carrier effective mass at a modulation-doped semiconductor heterojunction. Our results, when combined with some recent calculations for the energy of the fractional quantum Hall liquid including Landau-level-mixing, show quantitatively that in going from n doping to p doping in GaAs/Alx{\mathrm{Al}}_{\mathit{x}}Alx​${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As heterojunction systems, the crossover filling factor from the fractional quantum Hall liquid to the Wigner crystal changes from filling factor \ensuremath{\nu}\ensuremath{\sim}1/5 to \ensuremath{\nu}\ensuremath{\sim}1/3. This lends strong support to the claim that the observed reentrant insulating phases around \ensuremath{\nu}=1/5 for n-doped and around \ensuremath{\nu}=1/3 for p-doped high-mobility samples are primarily caused by electron-electron interaction effects. We discuss the possible implications of our theoretical results for some recent experiments carried out in the quantum Hall regime in search of the electron solid.
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.
The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx=e2/3h, in the vicinity of the Landau level filling factor ν=1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N≤7 electrons continuously as a function of magnetic field in the range 1/4≲ν<1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν=1/3 Laughlin state for N≤8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.
Finite-size two-dimensional systems are investigated by numerical diagonalization of the Hamiltonian. Both fermion systems and boson systems are investigated. The energy of the liquidlike ground state of the fermion system shows a downward cusp at 13\frac{1}{3}31​ filling of the lowest Landau level. This behavior is consistent with the observed fractional quantum Hall effect. The short-range behavior of the pair distribution function of the ground state is analyzed and good agreement with Laughlin's trial wave function is obtained.
The interplay between strongly correlated liquid and crystal phases for two-dimensional electrons exposed to a high transverse magnetic field is of fundamental interest. Through the nonperturbative fixed-phase diffusion Monte Carlo method, we determine the phase diagram of the Wigner crystal in the ν-κ plane, where ν is the filling factor and κ is the strength of Landau-level (LL) mixing. The phase boundary is seen to exhibit a striking ν dependence, with the states away from the magic filling factors ν=n/(2pn+1) being much more susceptible to crystallization due to Landau-level mixing than those at ν=n/(2pn+1). Our results explain the qualitative difference between the experimental behaviors observed in n- and p-doped gallium arsenide quantum wells and, in particular, the existence of an insulating state for ν<1/3 and also for 1/3<ν<2/5 in low-density p-doped systems. We predict that, in the vicinity of ν=1/5 and ν=2/9, increasing LL mixing causes a transition not into an ordinary electron Wigner crystal, but rather into a strongly correlated crystal of composite fermions carrying two vortices.