Specific heat and validity of quasiparticle approximation in the half-filled Landau level

We calculate the specific heat of composite fermion system in the half-filled Landau level. Two different methods are used to examine validity of the quasiparticle approximation when the two-body interaction is given by $V(q) = V_0 / q^{2-\eta}$ ($1 \le \eta \le 2$). The singular part of the specific heat is calculated from the free energy of the gauge field, which is compared with the specific heat calculated from the quasiparticle approximation via the singular self-energy correction due to the gauge field fluctuations. It turns out that two results are in general different and they coincide only for the case of the Coulomb interaction ($\eta = 1$). This result supports the fact that the quasiparticle approximation is valid only for the case of the Coulomb interaction. It is emphasized that this result is obtained by looking at a gauge-invariant quantity -- the specific heat.Comment: 8 pages, Revte

Instantons and the spectral function of electrons in the half-filled Landau level

We calculate the instanton-anti-instanton action $S_{M {\bar M}} (\tau)$ in the gauge theory of the half-filled Landau level. It is found that $S_{M {\bar M}} (\tau) = (3 - \eta) \left [ \Omega_0 (\eta) \ \tau \right ]^{1 / (3 - \eta)}$ for a class of interactions $v ({\bf q}) = V_0 / q^{\eta} \ ( 0 \leq \eta < 2 )$ between electrons. This means that the instanton-anti-instanton pairs are confining so that a well defined `charged' composite fermion can exist. It is also shown that $S_{M {\bar M}} (\tau)$ can be used to calculate the spectral function of electrons from the microscopic theory within a semiclassical approximation. The resulting spectral function varies as $e^{ - \left [ \Omega_0 (\eta) / \omega \right ]^{1 / ( 2 - \eta ) } }$ at low energies.Comment: 13 pages, Plain Tex, MIT-CMT-APR-9

Beyond the random phase approximation in the Singwi-Sj\"olander theory of the half-filled Landau level

We study the $\nu=1/2$ Chern-Simons system and consider a self-consistent field theory of the Singwi-Sj\"olander type which goes beyond the random phase approximation (RPA). By considering the Heisenberg equation of motion for the longitudinal momentum operator, we are able to show that the zero-frequency density-density response function vanishes linearly in long wavelength limit independent of any approximation. From this analysis, we derive a consistency condition for a decoupling of the equal time density-density and density-momentum correlation functions. By using the Heisenberg equation of motion of the Wigner distribution function with a decoupling of the correlation functions which respects this consistency condition, we calculate the response functions of the $\nu=1/2$ system. In our scheme, we get a density-density response function which vanishes linearly in the Coulomb case for zero-frequency in the long wavelength limit. Furthermore, we derive the compressibility, and the Landau energy as well as the Coulomb energy. These energies are in better agreement to numerical and exact results, respectively, than the energies calculated in the RPA.Comment: 9 Revtex pages, 4 eps figures, typos correcte

Quantum Boltzmann equation of composite fermions interacting with a gauge field

We derive the quantum Boltzmann equation (QBE) of composite fermions at/near the $\nu = 1/2$ state using the non-equilibrium Green's function technique. The lowest order perturbative correction to the self-energy due to the strong gauge field fluctuations suggests that there is no well defined Landau-quasi-particle. Therefore, we cannot assume the existence of the Landau-quasi-particles {\it a priori} in the derivation of the QBE. Using an alternative formulation, we derive the QBE for the generalized Fermi surface displacement which corresponds to the local variation of the chemical potential in momentum space. {}From this QBE, one can understand in a unified fashion the Fermi-liquid behaviors of the density-density and the current-current correlation functions at $\nu = 1/2$ (in the long wave length and the low frequency limits) and the singular behavior of the energy gap obtained from the finite temperature activation behavior of the compressibility near $\nu = 1/2$. Implications of these results to the recent experiments are also discussed.Comment: 44 pages, Plain Tex, 5 figures (ps files) available upon reques

Surface acoustic wave attenuation by a two-dimensional electron gas in a strong magnetic field

The propagation of a surface acoustic wave (SAW) on GaAs/AlGaAs heterostructures is studied in the case where the two-dimensional electron gas (2DEG) is subject to a strong magnetic field and a smooth random potential with correlation length Lambda and amplitude Delta. The electron wave functions are described in a quasiclassical picture using results of percolation theory for two-dimensional systems. In accordance with the experimental situation, Lambda is assumed to be much smaller than the sound wavelength 2*pi/q. This restricts the absorption of surface phonons at a filling factor \bar{\nu} approx 1/2 to electrons occupying extended trajectories of fractal structure. Both piezoelectric and deformation potential interactions of surface acoustic phonons with electrons are considered and the corresponding interaction vertices are derived. These vertices are found to differ from those valid for three-dimensional bulk phonon systems with respect to the phonon wave vector dependence. We derive the appropriate dielectric function varepsilon(omega,q) to describe the effect of screening on the electron-phonon coupling. In the low temperature, high frequency regime T << Delta (omega_q*Lambda /v_D)^{alpha/2/nu}, where omega_q is the SAW frequency and v_D is the electron drift velocity, both the attenuation coefficient Gamma and varepsilon(omega,q) are independent of temperature. The classical percolation indices give alpha/2/nu=3/7. The width of the region where a strong absorption of the SAW occurs is found to be given by the scaling law |Delta \bar{\nu}| approx (omega_q*Lambda/v_D)^{alpha/2/nu}. The dependence of the electron-phonon coupling and the screening due to the 2DEG on the filling factor leads to a double-peak structure for Gamma(\bar{\nu}).Comment: 17 pages, 3 Postscript figures, minor changes mad

Influence of gauge-field fluctuations on composite fermions near the half-filled state

Taking into account the transverse gauge field fluctuations, which interact with composite fermions, we examine the finite temperature compressibility of the fermions as a function of an effective magnetic field $\Delta B = B - 2 n_e hc/e$ ($n_e$ is the density of electrons) near the half-filled state. It is shown that, after including the lowest order gauge field correction, the compressibility goes as ${\partial n \over \partial \mu} \propto e^{- \Delta \omega_c / 2 T} \left ( 1 + {A (\eta) \over \eta - 1} {(\Delta \omega_c)^{2 \over 1 + \eta} \over T} \right )$ for $T \ll \Delta \omega_c$, where $\Delta \omega_c = {e \Delta B \over mc}$. Here we assume that the interaction between the fermions is given by $v ({\bf q}) = V_0 / q^{2 - \eta} \ (1 \le \eta \le 2)$, where $A (\eta)$ is a $\eta$ dependent constant. This result can be interpreted as a divergent correction to the activation energy gap and is consistent with the divergent renormalization of the effective mass of the composite fermions.Comment: Plain Tex, 24 pages, 5 figures available upon reques

Gauge-invariant response functions of fermions coupled to a gauge field

We study a model of fermions interacting with a gauge field and calculate gauge-invariant two-particle Green's functions or response functions. The leading singular contributions from the self-energy correction are found to be cancelled by those from the vertex correction for small $q$ and $\Omega$. As a result, the remaining contributions are not singular enough to change the leading order results of the random phase approximation. It is also shown that the gauge field propagator is not renormalized up to two-loop order. We examine the resulting gauge-invariant two-particle Green's functions for small $q$ and $\Omega$, but for all ratios of $\Omega / v_F q$ and we conclude that they can be described by Fermi liquid forms without a diverging effective mass.Comment: Plain Tex, 35 pages, 5 figures available upon request, Revised Version (Expanded discussion), To be published in Physical Review B 50, (1994) (December 15 issue

Stability of the compressible quantum Hall state around the half-filled Landau level

We study the compressible states in the quantum Hall system using a mean field theory on the von Neumann lattice. In the lowest Landau level, a kinetic energy is generated dynamically from Coulomb interaction. The compressibility of the state is calculated as a function of the filling factor $\nu$ and the width $d$ of the spacer between the charge carrier layer and dopants. The compressibility becomes negative below a critical value of $d$ and the state becomes unstable at $\nu=1/2$. Within a finite range around $\nu=1/2$, the stable compressible state exists above the critical value of $d$.Comment: 4 pages, 4 Postscript figures, RevTe

Spin-singlet hierarchy in the fractional quantum Hall effect

We show that the so-called permanent quantum Hall states are formed by the integer quantum Hall effects on the Haldane-Rezayi quantum Hall state. Novel conformal field theory description along with this picture is deduced. The odd denominator plateaux observed around $\nu=5/2$ are the permanent states if the $\nu=5/2$ plateau is the Haldane-Rezayi state. We point out that there is no such hierarchy on other candidate states for $\nu=5/2$. We propose experiments to test our prediction.Comment: RevTex,4 pages, v2:typo,one reference adde

Numerical Study of a Mixed Ising Ferrimagnetic System

We present a study of a classical ferrimagnetic model on a square lattice in which the two interpenetrating square sublattices have spins one-half and one. This model is relevant for understanding bimetallic molecular ferrimagnets that are currently being synthesized by several experimental groups. We perform exact ground-state calculations for the model and employ Monte Carlo and numerical transfer-matrix techniques to obtain the finite-temperature phase diagram for both the transition and compensation temperatures. When only nearest-neighbor interactions are included, our nonperturbative results indicate no compensation point or tricritical point at finite temperature, which contradicts earlier results obtained with mean-field analysis.Comment: Figures can be obtained by request to [email protected] or [email protected]