Quantum theory of bilayer quantum Hall smectics

Mean-field theory predicts that bilayer quantum Hall systems at odd integer total filling factors can have stripe ground states, in which the top Landau level is occupied alternately by electrons in one of the two layers. We report on an analysis of the properties of these states based on a coupled-Luttinger-liquid description that is able to account for quantum fluctuations of charge-density and position along each stripe edge. The soft modes associated with the broken symmetries of the stripe state lead to an unusual coupled-Luttinger-liquid system with strongly enhanced low-temperature heat capacity and strongly suppressed low-energy tunneling density of states. We assess the importance of the intralayer and interlayer backscattering terms in the microscopic Hamiltonian, which are absent in the Luttinger liquid description, by employing a perturbative renormalization group approach which rescales time and length along but not transverse to the stripes. With interlayer backscattering interactions present the Luttinger-liquid states are unstable either to an incompressible striped state that has spontaneous interlayer phase coherence and a sizable charge gap even at relatively large layer separations, or to Wigner crystal states. Our quantitative estimates of the gaps produced by backscattering interactions are summarized in Fig. 11 by a schematic phase diagram intended to represent predicted experimental findings in very high mobility bilayer systems at dilution refrigerator temperatures as a function of layer separation and bilayer density balance. We predict that the bilayer will form incompressible isotropic interlayer phase-coherent states for small layer separations, say $d<~1.5\mathcal{l}.$ At larger interlayer spacings, however, the bilayer will tend to form one of several different anisotropic states depending on the layer charge balance, which we parametrize by the fractional filling factor $\nu$ contributed by one of the two layers. For large charge imbalances $(\nu$ far from $1/2),$ we predict states in which anisotropic Wigner crystals form in each of the layers. For $\nu$ closer to $1/2,$ we predict stripe states that have spontaneous interlayer phase-coherence and a gap for charged excitations. These states should exhibit the quantum Hall effect for current flowing within the layers and also the giant interlayer tunneling conductance anomalies at low bias voltages that have been observed in bilayers when the $N=0\mathrm{}$ Landau level is partially filled. When the gaps produced by backscattering interactions are sufficiently small, the phenomenology observed at typical dilution fridge temperatures will be that of a smectic metal, anisotropic transport without a quantum Hall effect. For stripe states in the $N=2\mathrm{}$ Landau level, this behavior is expected over a range of bilayer charge imbalances on both sides of ν=1/2.