There is growing experimental and theoretical evidence that very clean two dimensional electron systems form unidirectional charge density waves (UCDW) or "striped" states at low temperatures and at Landau level filling fractions of the form ν = M + x with 4 < M < 10 an integer and 0.4 < ∼ x < ∼ 0.6. Following previous work, we model the striped state using a Hartree Fock approach. We construct the low energy excitations of the system by making smooth deformations of the stripe edges analogous to the construction of edge state excitations of quantum Hall droplets. These low energy excitations are described as a coupled Luttinger liquid theory, as discussed previously by MacDonald and Fisher (Phys. Rev. B 61, 5724 (2000)). Here, we extend that work and explicitly derive all of the parameters of this low energy theory using a Hartree Fock approach. We also make contact with the equivalent low energy hydrodynamic approach of Fogler and Vinokur (Phys. Rev. Lett. 84, 5828 (2000)) and similarly derive the parameters of this theory. As examples of the use of these results, we explicitly calculate the low-energy excitation spectrum and study tunneling into the striped state.

We use the time-dependent Hartree Fock approximation to study the collective-mode spectra of = 2 quantum Hall bilayers in tilted magnetic fields, allowing for charge imbalance as well as tunneling between the two layers. In a previous companion paper to this work, we studied the zero-temperature global phase diagram of this system, which was found to include symmetric and ferromagnetic phases as well as a first-order transition between two canted phases with spontaneously broken U(1) symmetry. We further found that this first-order transition line ends in a quantum critical point within the canted region. In the current work, we study the excitation spectra of all of these phases and pay particular attention to the behavior of the collective modes near the phase transitions. We find, most interestingly, that the first-order transition between the two canted phases is signaled by a near softening of a magnetoroton minimum. Many of the collective-mode features explored here should be accessible experimentally in light-scattering experiments.

Variational Bayes (VB) is a critical method in machine learning and statistics, underpinning the recent success of Bayesian deep learning. The natural gradient is an essential component of efficient VB estimation, but it is prohibitively computationally expensive in high dimensions. We propose a hybrid quantum-classical algorithm to improve the scaling properties of natural gradient computation and make VB a truly computationally efficient method for Bayesian inference in highdimensional settings. The algorithm leverages matrix inversion from the linear systems algorithm by Harrow, Hassidim, and Lloyd [Phys. Rev. Lett. 103, 15 (2009)] (HHL). We demonstrate that the matrix to be inverted is sparse and the classical-quantum-classical handoffs are sufficiently economical to preserve computational efficiency, making the problem of natural gradient for VB an ideal application of HHL. We prove that, under standard conditions, the VB algorithm with quantum natural gradient is guaranteed to converge.

No abstract

Superfluidity and phase separation in 3 He-4 He mixtures immersed in a jungle-gym ͑nonrandom͒ aerogel are studied by renormalization-group theory. Phase diagrams are calculated for a variety of aerogel concentrations. Superfluidity at very low 4 He concentrations and a depressed tricritical temperature are found at the onset of superfluidity. A superfluid-superfluid phase separation, terminating at an isolated critical point, is found entirely within the superfluid phase. These phenomena and trends with respect to aerogel concentration are explained by the connectivity and tenuousness of a jungle-gym aerogel. ͓S0163-1829͑97͒02405-3͔ I. HELIUM MIXTURES IN AEROGEL3 He-4 He mixtures undergoing superfluid phase transitions immersed in the porous medium provided by aerogel 1-3 have recently been studied as experimental realizations of quenched random-bond systems, yielding striking new experimental results: 4,5 ͑1͒ The line of second-order phase transitions was found to reach zero temperature, as had been generally predicted for random-bond systems, 6-13 but did so by continuing to remarkably low ͑4%͒ 4 He concentrations. ͑2͒ A phase separation terminating at an isolated critical point was found entirely confined to within the superfluid phase. ͑3͒ Superfluidity order-parameter depletion, with increasing temperature, was found to be qualitatively different for 3 He-4 He mixtures in aerogel and for incomplete fillings of pure 4 He in aerogel. Subsequently, a microscopic model was constructed that recognized and incorporated the correlated ͑connected͒ nature of the randomness of aerogel. 14,15 Thus, all of the experimental observations ͑1͒-͑3͒ listed above were reproduced and explained by factorizing the effects of ͑a͒ the connectivity, ͑b͒ the randomness, and ͑c͒ the tenuousness of aerogel. This model was studied by Monte Carlo simulation.A renormalization-group study of the model for helium mixtures in aerogel is conducted in this paper, and the Monte Carlo results are confirmed and extended to a variety of aerogel concentrations. In factorizing the effects of the different characteristics of aerogel, we start with the nonrandom version of aerogel, dubbed ''jungle-gym aerogel,'' with the Hamiltonianwhere, at each site i of a cubic lattice, s i 2 ϭ0 or 1 represents the presence of 3 He or 4 He respectively; 16 in the latter case, s i ϭϮ1 represent the superfluid degrees of freedom; and ͗i j͘ indicates summation over all nearest-neighbor pairs of sites. In the system before any renormalization-group transformation is done, J i j ϭJ and K i j ϭK for all ͗i j͘, where JϾ0 is a potential favoring superfluidity and KϭK (33) ϩK (44) Ϫ2K (34) is the isotope differentiation in the van der Waals interactions K (␣␤) between ␣ He and ␤ He . Since the latter is a small effect, the unrenormalized value of K is taken as 0 in our calculations, but nonzero values are generated ͑and kept͒ under renormalization. A regular cubic superstructure of lines is distinguished on this lattice as the jungle-gym aerogel ͑Fig. 1͒. The 3 He-4 He chem...