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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2017

Runaway electrons in tokamaks

The generation of runaway electrons is a complex and important phenomenon that impacts many areas of plasma physics. Due to the decrease of electron collision frequency with increasing velocity, electrons under strong electric field can experience unlimited “runaway” acceleration. In tokamaks, runaway electrons can be produced in disruptions, due to the strong inductive electric field formed as the thermal energy of plasma gets rapidly lost. This population of runaway electrons can undergo an exponential growth, denoted the runaway electron avalanche, due to hard collisions between relativistic runaway electrons and low energy electrons. It is predicted that in a large tokamak device like the International Thermonuclear Experimental Reactor (ITER), a runway electron beam generated in a disruption event can potentially cause severe damage to the device, which poses a significant challenge for ITER to achieve its mission. It is therefore extremely important to seek an effective mitigation mechanism for runaway electrons. Experimental efforts have been made to study the properties of runaway electrons in tokamaks, including their generation, diffusion, and radiation. In order to understand these experimental results, extensive theoretical and simulation studies of runaway electron physics are required. The main topic of this thesis is to study the wave particle interaction associated with runaway electron beams in tokamaks. The runaway electrons can emit and absorb electromagnetic waves through resonances, and can be diffused in momentum space by the waves. Initially, we address the Cherenkov radiation of runaway electrons, which originates from the polarization of the plasma medium. The energy and momentum loss of the Cherenkov radiation can be modeled by adding a correction to the Coulomb logarithm in the collisional drag force. Subsequently, we address pitch angle scattering caused by normal modes in the plasma, which are driven unstable by the anisotropicity of the runaway electron beam. The fluctuating electromagnetic fields are found to act as a seed for the unstable normal modes. Numerical simulations show that the pitch angle scattering effect from the normal modes, mainly whistler waves, can be significantly larger than that from collisional pitch angle scattering. Finally, we present a synthetic diagnostic tool we developed to calculate the electron cyclotron emission (ECE) from the runaway electrons, and successfully reproduce the prompt growth of the ECE signal observed in DIII-D quiescent runaway electron (QRE) experiments. Within the thesis, we also present the application of the adjoint method to runaway electron research, and show the calculations of the runaway probability function (RPF) and the expected loss time (ELT). These calculations not only help depict the dynamics of runaway electrons in momentum space, but also can be used to efficiently calculate experimentally relevant quantities such as the critical electric field for runaway electron avalanche and the avalanche growth rate.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Quantum beats in two-color photoionization to the spin-orbit split continuum of Ar

We report a study of the quantum beats in two-color photoionization of argon. An attosecond extreme ultraviolet pulse train prepares an electronic wave packet of definite odd parity, with total angular momentum J = 1, targeting the states between 14.0 and 14.5 eV from the ground state. Two-photon ionization of this wave packet with a tunable infrared probe pulse makes the constituent states interfere in both continuum channels, corresponding to the core angular momenta j c = 1/2 and 3/2, respectively. We analyze photoelectron spectrograms as a function of the time delay of the probe pulse and identify oscillations due to several pairs of states through Fourier decomposition. We observe phase differences between the corresponding beat signals in the two spin-orbit split continua. Comparison of theoretical simulations with the experimental measurements allows us to interpret the amplitudes and phases of ionization signals. Furthermore, we express the observed phase differences in terms of the off-diagonal elements of the short-range scattering matrix and the dipole matrix elements to the continuum eigenchannels.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2021

An Ab Initio Multiple Cloning Method for Non-Adiabatic Excited-State Molecular Dynamics in NWChem

The recently developed ab-initio multiple cloning (AIMC) approach based on the multicongurational Ehrenfest (MCE) method provides a powerful and accurate way of describing the excited-state dynamics of molecular systems. The AIMC method is a controlled approximation to non-adiabatic dynamics with a particular strength in the proper description of decoherence effects due to branching of vibrational wavepackets at a level crossing. In this work, we report a new implementation of the AIMC algorithm in the open source NWChem computational chemistry program. The framework combines linear-response time-dependent density functional theory with Ehrenfest mean-eld theory to determine the equations of motion for classical trajectories. The multi-dimensional wave function is decomposed into a superposition of Gaussian coherent states guided by Ehrenfest trajectories (i.e., MCE approach), which can clone with fully quantum mechanical amplitudes and phases. By using an efficient time-derivative based non-adiabatic coupling approach within the AIMC method, all observable are calculated on-the-y in the non-adiabatic molecular dynamics process. As a representative example, we apply our implementation to study the ultrafast photoinduced electronic and vibrational energy transfer in a pyridine molecule. The effects of the cloning procedure on electronic and vibrational coherence, relaxation and unidirectional energy transfer are discussed. This new AIMC implementation provides a high-level non-adiabatic molecular dynamics framework for simulating photoexcited dynamics in complex molecular systems and experimentally relevant ultrafast spectroscopic probes, such as nonlinear coherent optical and X-ray signals.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

KSSOLV 2.0: An efficient MATLAB toolbox for solving the Kohn-Sham equations with plane-wave basis set

KSSOLV (Kohn-Sham Solver) is a MATLAB toolbox for performing Kohn-Sham density functional theory (DFT) calculations with a plane-wave basis set. KSSOLV 2.0 preserves the design features of the original KSSOLV software to allow users and developers to easily set up a problem and perform ground-state calculations as well as to prototype and test new algorithms. Furthermore, it includes new functionalities such as new iterative diagonalization algorithms, k-point sampling for electron band structures, geometry optimization and advanced algorithms for performing DFT calculations with local, semi-local, and hybrid exchange-correlation functionals. It can be used to study the electronic structures of both molecules and solids. So, we describe these new capabilities in this work through a few use cases. We also demonstrate the numerical accuracy and computational efficiency of KSSOLV on a variety of examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Particle-hole mirror symmetries around the half-filled shell: Quantum numbers and algebraic structure of composite fermions

Composite fermions (CFs) of the fractional quantum Hall effect (FQHE) are described as spherical products of electron and vortex spinors, built from underlying 𝐿=1/2 ladder operators aligned so that the spinor angular momenta, 𝐿 𝑒 and 𝐿 𝑣 , are maximal. We identify the CF's quantum numbers as the angular momentum 𝐿 in (𝐿𝑒⁢𝐿𝑣)⁢𝐿, its magnetic projection 𝑚𝐿, the electron number 𝑁 [𝐿 𝑣 =(𝑁−1)/2], and magnetic 𝜈 spin, 𝑚 𝜈 = 𝐿 𝑒 −𝐿 𝑣 . Translationally invariant FQHE states are formed by fully filling 𝑝 subshells with their respective CFs, in order of ascending 𝐿 for fixed 𝐿 𝑒 and 𝐿 𝑣 , beginning with the lowest allowed value, 𝐿=|⁢𝑚 𝜈 |. CF subshells are contained entirely within the first Landau level (FLL). Alternatively, we provide an equivalent hierarchical wave function in which the underlying objects are vortices with 𝐿 𝑣 = $\frac{p}{2}$, correlated. Here, we show that CFs can be written as a valence operator carrying the angular momentum quantum numbers 𝐿, 𝑚 𝜈 , 𝑚 𝐿 acting on a scalar half-filled intrinsic state.

2-dimensional systems↗
DOE OSTI2025

Electronic and magnetic properties of hole-doped topological kagome Fe 1−𝑥 ⁢Mn 𝑥 ⁢Sn thin films

We have investigated the electronic and magnetic structures of topological kagome Fe 1−𝑥 ⁢Mn 𝑥 ⁢Sn (0 ≤ 𝑥 ≤ 0.3) thin films via neutron diffraction, electronic transport measurements, and ab initio density functional theory (DFT) to understand the interplay between hole doping, magnetism, and the electronic structures. Temperature-dependent neutron diffraction measurements on parent FeSn reveal the Néel temperature to be 𝑇 N ∼ 355 K and the underlying A-type antiferromagnetic ordering is associated with a wave vector 𝒒 = (001/2). Upon Mn doping to 𝑥 = 0.15, 𝑇 N decreases slightly while the magnetic ordering vector remains the same. Resistivity measurements show metallic characteristics and in-plane anisotropy down to 10 K for all the investigated samples. The effects of hole doping are mapped in terms of electronic ground state calculations via DFT which show that the Dirac point is moved closer to the Fermi level (𝐸 F ) and the flat bands get pushed away from 𝐸 F upon hole doping. However, a comparison between hole-doped Fe 1−𝑥⁢ Mn 𝑥⁢ Sn and electron-doped Fe 1−𝑥 ⁢Co 𝑥 ⁢Sn indicates that the Néel temperature does not scale with the position of 𝐸 F relative to the flat band. Furthermore, our results establish the antiferromagnetic state of FeSn and Fe 1−𝑥 ⁢Mn 𝑥⁢ Sn films at room temperature, laying the groundwork for future studies of magnetism in kagome heterostructures.

36 MATERIALS SCIENCE↗
DOE OSTI2015

Consistent structures and interactions by density functional theory with small atomic orbital basis sets

A density functional theory (DFT) based composite electronic structure approach is proposed to efficiently compute structures and interaction energies in large chemical systems. It is based on the well-known and numerically robust Perdew-Burke-Ernzerhoff (PBE) generalized-gradient-approximation in a modified global hybrid functional with a relatively large amount of non-local Fock-exchange. The orbitals are expanded in Ahlrichs-type valence-double zeta atomic orbital (AO) Gaussian basis sets, which are available for many elements. In order to correct for the basis set superposition error (BSSE) and to account for the important long-range London dispersion effects, our well-established atom-pairwise potentials are used. In the design of the new method, particular attention has been paid to an accurate description of structural parameters in various covalent and non-covalent bonding situations as well as in periodic systems. Together with the recently proposed three-fold corrected (3c) Hartree-Fock method, the new composite scheme (termed PBEh-3c) represents the next member in a hierarchy of “low-cost” electronic structure approaches. They are mainly free of BSSE and account for most interactions in a physically sound and asymptotically correct manner. PBEh-3c yields good results for thermochemical properties in the huge GMTKN30 energy database. Furthermore, the method shows excellent performance for non-covalent interaction energies in small and large complexes. For evaluating its performance on equilibrium structures, a new compilation of standard test sets is suggested. These consist of small (light) molecules, partially flexible, medium-sized organic molecules, molecules comprising heavy main group elements, larger systems with long bonds, 3d-transition metal systems, non-covalently bound complexes (S22 and S66×8 sets), and peptide conformations. For these sets, overall deviations from accurate reference data are smaller than for various other tested DFT methods and reach that of triple-zeta AO basis set second-order perturbation theory (MP2/TZ) level at a tiny fraction of computational effort. Periodic calculations conducted for molecular crystals to test structures (including cell volumes) and sublimation enthalpies indicate very good accuracy competitive to computationally more involved plane-wave based calculations. PBEh-3c can be applied routinely to several hundreds of atoms on a single processor and it is suggested as a robust “high-speed” computational tool in theoretical chemistry and physics.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2018

First-Principle Calculations of Structural, Elastic, and Electronic Properties of Intermetallic Rare Earth R{sub 2}Ni{sub 2}Pb (R = Ho, Lu, and Sm) Compounds

The structural, elastic, and electronic properties of rare earth intermetallic R{sub 2}Ni{sub 2}Pb (where R = Ho, Lu, and Sm) compounds were investigated with the density functional theory (DFT) calculations. The calculations are performed using the full potential-linearized augmented plane wave (FP-LAPW) method within the framework of local density approximation (LDA). The calculated values of the equilibrium lattice constants were in agreement with the available experimental values. The elastic constants (C{sub ij}) were also calculated to understand the mechanical properties and structural stability of the compounds. Furthermore, the density of states and the charge density distributions of the compounds were calculated to understand the nature of the bonding in the material. The calculated results are in accordance with the available data in the literature.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2014

In search of the X{sub 2}BO and X{sub 2}BS (X = H, F) free radicals: Ab initio studies of their spectroscopic signatures

The F{sub 2}BO free radical is a known, although little studied, species but similar X{sub 2}BY (X = H, D, F; Y = O, S) molecules are largely unknown. High level ab initio methods have been used to predict the molecular structures, vibrational frequencies (in cm{sup −1}), and relative energies of the ground and first two excited electronic states of these free radicals, as an aid to their eventual spectroscopic identification. The chosen theoretical methods and basis sets were tested on F{sub 2}BO and found to give good agreement with the known experimental quantities. In particular, complete basis set extrapolations of coupled-cluster single and doubles with perturbative triple excitations/aug-cc-pVXZ (X = 3, 4, 5) energies gave excellent electronic term values, due to small changes in geometry between states and the lack of significant multireference character in the wavefunctions. The radicals are found to have planar C{sub 2v} geometries in the X{sup ~2}B{sub 2} ground state, the low-lying A{sup ~2}B{sub 1} first excited state, and the higher B{sup ~2}A{sub 1} state. Some of these radicals have very small ground state dipole moments hindering microwave measurements. Infrared studies in matrices or in the gas phase may be possible although the fundamentals of H{sub 2}BO and H{sub 2}BS are quite weak. The most promising method of identifying these species in the gas phase appears to be absorption or laser-induced fluorescence spectroscopy through the allowed B{sup ~}-X{sup ~} transitions which occur in the visible-near UV region of the electromagnetic spectrum. The ab initio results have been used to calculate the Franck-Condon profiles of the absorption and emission spectra, and the rotational structure of the B{sup ~}-X{sup ~}0{sub 0}{sup 0} bands has been simulated. The calculated single vibronic level emission spectra provide a unique, readily recognizable fingerprint of each particular radical, facilitating the experimental identification of new X{sub 2}BY species in the gas phase.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2016

Inter-ribbon tunneling in graphene: An atomistic Bardeen approach

A weakly coupled system of two crossed graphene nanoribbons exhibits direct tunneling due to the overlap of the wavefunctions of both ribbons. We apply the Bardeen transfer Hamiltonian formalism, using atomistic band structure calculations to account for the effect of the atomic structure on the tunneling process. The strong quantum-size confinement of the nanoribbons is mirrored by the one-dimensional character of the electronic structure, resulting in properties that differ significantly from the case of inter-layer tunneling, where tunneling occurs between bulk two-dimensional graphene sheets. The current-voltage characteristics of the inter-ribbon tunneling structures exhibit resonance, as well as stepwise increases in current. Both features are caused by the energetic alignment of one-dimensional peaks in the density-of-states of the ribbons. Resonant tunneling occurs if the sign of the curvature of the coupled energy bands is equal, whereas a step-like increase in the current occurs if the signs are opposite. Changing the doping modulates the onset-voltage of the effects as well as their magnitude. Doping through electrostatic gating makes these structures promising for application towards steep slope switching devices. Using the atomistic empirical pseudopotentials based Bardeen transfer Hamiltonian method, inter-ribbon tunneling can be studied for the whole range of two-dimensional materials, such as transition metal dichalcogenides. The effects of resonance and of step-like increases in the current we observe in graphene ribbons are also expected in ribbons made from these alternative two-dimensional materials, because these effects are manifestations of the one-dimensional character of the density-of-states.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

A new generation of effective core potentials for correlated calculations

Here, we outline ideas on desired properties for a new generation of effective core potentials (ECPs) that will allow valence-only calculations to reach the full potential offered by recent advances in many-body wave function methods. The key improvements include consistent use of correlated methods throughout ECP constructions and improved transferability as required for an accurate description of molecular systems over a range of geometries. The guiding principle is the isospectrality of all-electron and ECP Hamiltonians for a subset of valence states. We illustrate these concepts on a few first- and second-row atoms (B, C, N, O, S), and we obtain higher accuracy in transferability than previous constructions while using semi-local ECPs with a small number of parameters. In addition, the constructed ECPs enable many-body calculations of valence properties with higher (or same) accuracy than their all-electron counterparts with uncorrelated cores. This implies that the ECPs include also some of the impacts of core-core and core-valence correlations on valence properties. The results open further prospects for ECP improvements and refinements.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

A light weight regularization for wave function parameter gradients in quantum Monte Carlo

The parameter derivative of the expectation value of the energy, ∂E/∂p, is a key ingredient in variational Monte Carlo (VMC) wave function optimization methods. In some cases, a naïve estimate of this derivative suffers from an infinite variance, which inhibits the efficiency of optimization methods that rely on a stable estimate of the derivative. In this work, we derive a simple regularization of the naïve estimator, which is trivial to implement in existing VMC codes, has finite variance, and a negligible bias, which can be extrapolated to zero bias with no extra cost. We use this estimator to construct an unbiased, finite variance estimation of ∂E/∂p for a multi-Slater–Jastrow trial wave function on the LiH molecule and in the optimization of a multi-Slater–Jastrow trial wave function on the CuO molecule. This regularized estimator is a simple and efficient estimator of ∂E/∂p for VMC optimization techniques.

36 MATERIALS SCIENCE↗
DOE OSTI2016

Non-neutral plasma diode in the presence of a transverse magnetic field

An analytical study of the plasma states in non-neutral plasma diodes in the presence of an external transverse magnetic field is presented for an arbitrary neutralization parameter γ. Considerations are restricted to the regime where no electrons are turned around by the magnetic field. The emitter electric field strength E{sub 0} is used as a characteristic function to investigate the existence of solutions depending on the diode length, the applied voltage, the neutralization parameter, and the magnetic field strength. The potential distribution has a wave form for small magnitudes of the external magnetic field, as well as for the case when magnetic field is absent. A new family of solutions appears along with the Bursian ones. On the other hand, as the Larmor radius becomes comparable with the beam Debye length, oscillations in the potential disappear, and only the Bursian branches remain. Unlike the vacuum diode, there are steady state solutions for the negative values of the emitter field strength. As the neutralization parameter (γ) increases, the emitter field strength relating to the SCL (space charge limit) bifurcation point diminishes, and at γ > 1, the value of the emitter's electric field strength at the space charge limit (E{sub 0,SCL}) turns out to be negative.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2014

Simulated two-dimensional electronic spectroscopy of the eight-bacteriochlorophyll FMO complex

The Fenna-Matthews-Olson (FMO) protein-pigment complex acts as a molecular wire conducting energy between the outer antenna system and the reaction center; it is an important photosynthetic system to study the transfer of excitonic energy. Recent crystallographic studies report the existence of an additional (eighth) bacteriochlorophyll a (BChl a) in some of the FMO monomers. To understand the functionality of this eighth BChl, we simulated the two-dimensional electronic spectra of both the 7-site (apo form) and the 8-site (holo form) variant of the FMO complex from green sulfur bacteria, Prosthecochloris aestuarii. By comparing the spectrum, it was found that the eighth BChl can affect two different excitonic energy transfer pathways: (1) it is directly involved in the first apo form pathway (6 → 3 → 1) by passing the excitonic energy to exciton 6; and (2) it facilitates an increase in the excitonic wave function overlap between excitons 4 and 5 in the second pathway (7 → 4,5 → 2 → 1) and thus increases the possible downward sampling routes across the BChls.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Order-Disorder Transitions in ( Ca x Sr 1 - x ) 3 Rh 4 Sn 13

The classification of structural phase transitions as displacive or order-disorder in character is usually based on spectroscopic data above the transition. Here, we use single crystal x-ray diffraction to investigate structural correlations in the quasiskutterudites, (Ca x Sr 1-x ) 3 Rh 4 Sn 13 , which have a quantum phase transition at x ~ 0.9. Three-dimensional pair distribution functions show that the amplitudes of local atomic displacements are temperature independent below the transition and persist to well above the transition, a signature of order-disorder behavior. The implications for the associated electronic transitions are discussed.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

Feasibility of a Novel Density Functional Method Outside the Kohn-Sham Framework for Modeling Global Potential Energy Surfaces of Molecular Chemical Reactions (Final Technical Report)

The project aimed towards the construction of computational methods capable of modeling the global potential energy surface (PES) of small molecules, molecular ions, and radicals — including the parts of the PES which correspond to chemical reactions, and the reaction paths connecting reactants, intermediates, and reaction products. This goal may seem humble at first glance, but for chemical systems with more than about six atoms in total, at the time the project was proposed, there were no established theoretical methods capable of simulating such systems reliably, not even for small molecules in the gas phase which are electronically benign. This restriction severely hampers our ability to model and control chemical processes under harsh conditions. With the goal of constructing a method capable of modeling such chemical systems, we proposed to pursue a novel approach towards a Multi-Configuration (MC) DFT outside the traditional frameworks of Kohn-Sham theory and other methods of coupling wave function theory with DFT. Rather than being a complete active space (CAS) method, the proposed DOCI-DFT would employ a special restricted form of the active space wave function, called Doubly-Occupied CI (DOCI)—this wave function form is sufficient to describe not only heterolytic, but also homolytic bond dissociation processes at the zeroth order (i.e., as active space wave function). There are no other standard mean field methods which can do so. In the original proposal, we also outlined strong formal and practical arguments speaking for this method. The project proposal was accepted by the Department of Energy and provided two years of seed funding for one graduate student, as well as two years of two weeks PI summer-salary for the PI. However, ultimately the project could not be effectively pursued due to severe interferences outside the context scientific problems and was cancelled. Apart from two weeks of PI salary (and associated fringe & overhead costs issues by the performing institution), no costs were charged against the award, with the entire rest of funding returned to the Department of Energy. This report reiterates the primary background information regarding the project, its original goals, and outlines the preliminary work performed during the two weeks of DOE funding charged.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Phase‐Dependent Band Gap Engineering in Alloys of Metal‐Semiconductor Transition Metal Dichalcogenides

Abstract Bandgap engineering plays a critical role in optimizing the electrical, optical and (photo)‐electrochemical applications of semiconductors. Alloying has been a historically successful way of tuning bandgaps by making solid solutions of two isovalent semiconductors. In this work, a novel form of bandgap engineering involving alloying non‐isovalent cations in a 2D transition metal dichalcogenide (TMDC) is presented. By alloying semiconducting MoSe 2 with metallic NbSe 2 , two structural phases of Mo 0.5 Nb 0.5 Se 2 , the 1T and 2H phases, are produced each with emergent electronic structure. At room temperature, it is observed that the 1T and 2H phases are semiconducting and metallic, respectively. For the 1T structure, scanning tunneling microscopy/spectroscopy (STM/STS) is used to measure band gaps in the range of 0.42–0.58 at 77 K. Electron diffraction patterns of the 1T structure obtained at room temperature show the presence of a nearly commensurate charge density wave (NCCDW) phase with periodic lattice distortions that result in an uncommon 4 × 4 supercell, rotated approximately 4° from the lattice. Density‐functional‐theory calculations confirm that local distortions, such as those in a NCCDW, can open up a band gap in 1T ‐Mo 0.5 Nb 0.5 Se 2 , but not in the 2H phase. This work expands the boundaries of alloy‐based bandgap engineering by introducing a novel technique that facilitates CDW phases through alloying.

Wang, Shuxi↗
DOE OSTI2021

Composite anyons on a torus

An adiabatic approach put forward by Greiter and Wilczek interpolates between the integer quantum Hall effects of electrons and composite fermions by varying the statistical flux bound to electrons continuously from zero to an even integer number of flux quanta, such that the intermediate states represent anyons in an external magnetic field with the same “effective” integer filling factor. We consider such anyons on a torus, and construct representative wave functions for their ground as well as excited states. These wave functions involve higher Landau levels in general, but can be explicitly projected into the lowest Landau level for many parameters. We calculate the variational energy gap between the first excited state and ground state and find that it remains open as the statistical phase is varied. Lastly, we obtain from these wave functions, both analytically and numerically, various topological quantities, such as ground-state degeneracy, the Chern number, and the Hall viscosity.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗