Exact Green's functions and Bosonization of a Luttinger liquid coupled to impedances

The exact Green's functions of a finite-size Luttinger Liquid (LL) connected to impedances are computed at zero and finite temperature. Bosonization for a LL with Impedance boundary conditions (IBC) is proven to hold. The LL with open boundary conditions (for both Neumann and Dirichlet cases) is explicitly recovered as a special limit when one has infinite impedances. Additionally when the impedances are equal to the characteristic impedance of the Luttinger liquid then the finite Luttinger liquid is shown to be effectively equivalent to an infinite Luttinger liquid

Convergence of the Lasserre Hierarchy of SDP Relaxations for Convex Polynomial Programs without Compactness

The Lasserre hierarchy of semidefinite programming (SDP) relaxations is an effective scheme for finding computationally feasible SDP approximations of polynomial optimization over compact semi-algebraic sets. In this paper, we show that, for convex polynomial optimization, the Lasserre hierarchy with a slightly extended quadratic module always converges asymptotically even in the face of non-compact semi-algebraic feasible sets. We do this by exploiting a coercivity property of convex polynomials that are bounded below. We further establish that the positive definiteness of the Hessian of the associated Lagrangian at a saddle-point (rather than the objective function at each minimizer) guarantees finite convergence of the hierarchy. We obtain finite convergence by first establishing a new sum-of-squares polynomial representation of convex polynomials over convex semi-algebraic sets under a saddle-point condition. We finally prove that the existence of a saddle-point of the Lagrangian for a convex polynomial program is also necessary for the hierarchy to have finite convergence.Comment: 17 page

Wavefunctions for the Luttinger liquid

Standard bosonization techniques lead to phonon-like excitations in a Luttinger liquid (LL), reflecting the absence of Landau quasiparticles in these systems. Yet in addition to the above excitations some LL are known to possess solitonic states carrying fractional quantum numbers (e.g. the spin 1/2 Heisenberg chain). We have reconsidered the zero modes in the low-energy spectrum of the gaussian boson LL hamiltonian both for fermionic and bosonic LL: in the spinless case we find that two elementary excitations carrying fractional quantum numbers allow to generate all the charge and current excited states of the LL. We explicitly compute the wavefunctions of these two objects and show that one of them can be identified with the 1D version of the Laughlin quasiparticle introduced in the context of the Fractional Quantum Hall effect. For bosons, the other quasiparticle corresponds to a spinon excitation. The eigenfunctions of Wen's chiral LL hamiltonian are also derived: they are quite simply the one dimensional restrictions of the 2D bulk Laughlin wavefunctions.Comment: 5 pages; accepted for publication in EPR B, Rapid Note

Fractional excitations in the Luttinger liquid

We reconsider the spectrum of the Luttinger liquid (LL) usually understood in terms of phonons (density fluctuations), and within the context of bosonization we give an alternative representation in terms of fractional states. This allows to make contact with Bethe Ansatz which predicts similar fractional states. As an example we study the spinon operator in the absence of spin rotational invariance and derive it from first principles: we find that it is not a semion in general; a trial Jastrow wavefunction is also given for that spinon state. Our construction of the new spectroscopy based on fractional states leads to several new physical insights: in the low-energy limit, we find that the $S_{z}=0$ continuum of gapless spin chains is due to pairs of fractional quasiparticle-quasihole states which are the 1D counterpart of the Laughlin FQHE quasiparticles. The holon operator for the Luttinger liquid with spin is also derived. In the presence of a magnetic field, spin-charge separation is not realized any longer in a LL: the holon and the spinon are then replaced by new fractional states which we are able to describe.Comment: Revised version to appear in Physical Review B. 27 pages, 5 figures. Expands cond-mat/9905020 (Eur.Phys.Journ.B 9, 573 (1999)

Semi Analytical Approach for Binary Mixture Conductivity in Hydraulic Fracturing

In hydraulic fracturing, a proppant injection schedule practice typically applies a binary proppant mixture (for example: 100 Mesh sand following by 40/70 Mesh sand in well MIP-3H, Marcellus Shale). The former injection agent is finer in size or less resistant to stress, whereas the latter injection agent is coarser in size or more resistant to stress. This practice creates a special region inside the fracture, in which two injected proppant types co-exist and is defined as the mixture zone. This research concentrates on the variability of the mixture zone under impact of different factors, introduces novel semi-analytical modelling approach to better estimate the hydraulic conductivity inside the mixture zone, and further applies this novel approach to prove its efficacy in conductivity estimation and cumulative production prediction. Variability of the mixture zone is studied by an OAT (One at A Time) sensitivity analysis to examine the percentage of the mixture zone’s area over the total propped area under variability of different parameters, including reservoir properties, geo-mechanics, and different design parameters in a proppant injection schedule. The novel semi-analytical model is derived by independent modelling for proppant pack’s permeability and width. Permeability model is an improvement from Carman Kozeny equation, in which Internal Specific Area is re-derived to differ a binary mixture from a single proppant type. Fracture width model is derived from Hertzian contact theory under assumptions of parabola distributed stress, elastic behavior and dual-layer schematic. Trust-region method is applied to determine all coefficients in the permeability model, which complies with a non-linear least square problem. Coefficients in the width model are determined by linear approximation from an in-house fracture width database. Based on satisfied validation results (7.98%-21.82% MRE), trust-region algorithm determines the novel model’s coefficients using lab data of only two Weight Concentration Ratios. The novel model is expanded to predict binary mixture’s conductivity at arbitrary confining stress and Weight Mixing Ratio values, which avoids misleading experimental outcomes. Proppant particle size distribution between two discrete Mesh numbers, deformation complexity between particles (quadratic form) and proppant crush effect are prospective improvements for better modelling validation results. LWAM is proved to overestimate conductivity compared to the novel model. Overestimation degree, which can exceed 70%, is separated into 3 overestimation zones (≤20%, 20-60% and ≥60%) and examined by the comparison matrices. When contrast in density and Mesh size between proppant types in a mixture is clear (for example: mixture 40/70 sand - 20/40 ceramic), overestimation is extreme, and when internal contrast between proppant types in a mixture is reduced (for example: mixture 20/40 sand - 20/40 ceramic), overestimation is dampened. The case study for Marcellus Shale applies comparison matrices to predict LWAM’s conductivity overestimation and conduct 1-year and 10-year cumulative production analyses. Reduction of confining stress axis in comparison matrices to 5400-6200 psi with a maximum difference of 200 psi between different depths (based on Marcellus Shale minimum horizontal stress data) allows predictive reasoning for conductivity overestimation from weight concentration ratio distribution. Overestimation for cumulative production data is observed to approach 10.73% (3.274 MMSCF in the early production time for a quarter fractured area). This suggests the level of risk caused by application of LWAM in reservoir simulation, depending on the intrinsic contrast between proppant types’ Mesh sizes and densities, in the selected proppant injection mixture