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Physical review. B, Condensed matter

Cooperative ring exchange and the fractional quantum Hall effect

Steven Allan Kivelson, Catherine Kallin, Daniel P. Arovas, J. R. Schrieffer

Published Jul 15, 1987• 92 citations
Publisher

Abstract

We study the transition of the interacting two-dimensional electron gas at high magnetic field from a low-density Wigner crystal to a higher-density correlated state which exhibits the fractionally quantized Hall effect. The phase transition is precipitated by a condensation of ring-exchange processes, which by nature of their sensitivity to enclosed magnetic flux allow for especially low-energy exchange condensates at certain values of the Landau-level filling factor, \ensuremath{\nu}=${\ensuremath{\nu}}_{i}$. In the condensate, the exchanges mediate a logarithmic potential between local-density fluctuations, leading to a cusp in the ground-state energy as a function of \ensuremath{\Vert}\ensuremath{\nu}-${\ensuremath{\nu}}_{i}$\ensuremath{\Vert} in the vicinity of a preferred state. In addition, we derive quasiparticle excitations of sharp fractional charge which are analogous to the excitations derived by Laughlin.

DOI
10.1103/physrevb.36.1620

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•Jan 20, 2018•0 citations
We study the transition of the interacting two-dimensional electron gas at high magnetic field from a low-density Wigner crystal to a higher-density correlated state which exhibits the fractionally quantized Hall effect. The phase transition is precipitated by a condensation of ring-exchange processes, which by nature of their sensitivity to enclosed magnetic flux allow for especially low-energy exchange condensates at certain values of the Landau-level filling factor, \ensuremath{\nu}=\ensuremathνi{\ensuremath{\nu}}_{i}\ensuremathνi​. In the condensate, the exchanges mediate a logarithmic potential between local-density fluctuations, leading to a cusp in the ground-state energy as a function of \ensuremath{\Vert}\ensuremath{\nu}-\ensuremathνi{\ensuremath{\nu}}_{i}\ensuremathνi​\ensuremath{\Vert} in the vicinity of a preferred state. In addition, we derive quasiparticle excitations of sharp fractional charge which are analogous to the excitations derived by Laughlin.
Electrons in two dimensions in a high magnetic field can condense into an incompressible quantum fluid state when the electron density is equal to certain rational fractions of the flux density. Much progress has been made in understanding the nature of this state based on the ansatz wavefunction of Laughlin. Here, we show that the incompressible nature of the quantum fluid can be derived from a semiclassical evaluation of the imaginary time propagator. The dominant classical paths consist of cooperative ring exchange events. When these condense, they mediate a logarithmic interaction between density fluctuations, and produce a precise fractional quantization condition for the quasi-particle charges. We further show that the essential long wavelength properties of Laughlin's ground state and quasi-particle wavefunctions can be derived constructively from this approach. We conclude with some speculations concerning the nature of the off-diagonal order in the quantum Hall liquid.
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.
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.
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.
By using a bosonic Chern-Simons gauge theory the ground state of the planar electron system is analyzed in the presence of an external magnetic field. The distinctive feature of our theory is that a renormalization counter term (|α|/ M )|ψ| 4 which represents explicitly a hard-core condition plays important roles, where α and M are the statistics parameter and the mass of the electron. The following results are obtained systematically in the semiclassical approximation. When ν=1/ k ( k odd) with ν being the Landau level filling factor, the ground state is incompressible and described by the Laughlin wavefunction. On the other hand, when ν∼1/ k , the ground state is compressible and it is given by the Wigner crystal made of quasiholes (antivortices). The wavefunction of the Wigner crystal is also calculated, which is found to coincide with that of the Laughlin quasiholes. In these analyses the Coulomb interaction between electrons plays a crucial role.
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 non-perturbative fixed phase diffusion Monte Carlo method, we determine the phase diagram of the Wigner crystal in the ν−κ\nu-\kappaν−κ plane, where ν\nuν is the filling factor and κ\kappaκ is the strength of Landau level mixing. The phase boundary is seen to exhibit a striking ν\nuν dependence, with the states away from the magic filling factors ν=n/(2pn+1)\nu=n/(2pn+1)ν=n/(2pn+1) being much more susceptible to crystallization due to Landau level mixing than those at ν=n/(2pn+1)\nu=n/(2pn+1)ν=n/(2pn+1). Our results explain the qualitative difference between the experimental behaviors observed in n-doped and p-doped GaAs quantum wells, and, in particular, the existence of an insulating state for ν<1/3\nu<1/3ν<1/3 and also for 1/3<ν<2/51/3 <\nu< 2/51/3<ν<2/5 in low density p-doped systems. We predict that in the vicinity of ν=1/5\nu=1/5ν=1/5 and ν=2/9\nu=2/9ν=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.