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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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At least 991 records · Page 55
All sourcesNASA NTRSDOE OSTI
NASA NTRS1994

Wavelets, period-doubling, and time-frequency localization with application to organization of convection over the tropical western Pacific

In this paper, preliminary results in using orthogonal and continuous wavelet transform (WT) to identify period doubling and time-frequency localization in both synthetic and real data are presented. First, the Haar WT is applied to synthetic time series derived from a simple nonlinear dynamical system- a first-order quadratic difference equation. Second, the complex Morlet WT is used to study the time-frequency localization of tropical convection based on a high-resolution Japanese Geostationary Meteorological Satellite infrared (IR) radiance dataset. The Haar WT of the synthetic time series indicates the presence and distinct separation of multiple frequencies in a period-doubling sequence. The period-doubling process generates a multiplicity of intermediate frequencies, which are manifested in the nonuniformity in time with respect to the phase of oscillations in the lower frequencies. Wavelet transform also enables the detection of extremely weak signals in high-order subharmonics resulting from the period-doubling bifurcations. These signals are either undetected or considered statistically insignificant by traditional Fourier analysis. The Morlet WT of the IR radiance dataset indicates the presence of multiple timescales, which are localized in both frequency and time. There are two regimes in the variation of IR radiance, corresponding to the wet and dry periods. Multiple timescales, ranging from semidiurnal, diurnal, synoptic, to intraseasonal with embedding structures, are active in the wet regime. In particular, synoptic variability is more prominent during the wet phase of an intensive intraseasonal cycle. These are not only consistent with, but also show more details than, previous findings by using other techniques. The phase-locking relationships among the oscillations with different time-scales suggest that both synoptic and intraseasonal variations may be mixed oscillations due to the interaction of self-excited oscillations in the tropical atmosphere and external forcings such as annual and diurnal solar radiation variations. Both examples show that WT is a powerful tool for analysis of phenomena involving multiscale interactions that exhibit localization in both frequency and time. A discussion on the caveats in the use of WT in geophysical data analysis is also presented.

Weng, Hengyi↗
NASA NTRS1995

Formation Of the Giant Planets By Concurrent Accretion Of Solids And Gas

New numerical simulations of the formation of the giant planets are presented, in which for the first time both the gas and planetesimal accretion rates are calculated in a self-consistent, interactive fashion. The simulations combine three elements: 1) three-body accretion cross-sections of solids onto an isolated planetary embryo, 2) a stellar evolution code for the planet's gaseous envelope, and 3) a planetesimal dissolution code within the envelope, used to evaluate the planet's effective capture radius and the energy deposition profile of accreted material. Major assumptions include: The planet is embedded in a disk of gas and small planetesimals with locally uniform initial surface mass density, and planetesimals are not allowed to migrate into or out of the planet's feeding zone. All simulations are characterized by three major phases. During the first phase, the planet's mass consists primarily of solid material. The planetesimal accretion rate, which dominates that of gas, rapidly increases owing to runaway accretion, then decreases as the planet's feeding zone is depleted. During the second phase, both solid and gas accretion rates are small and nearly independent of time. The third phase, marked by runaway gas accretion, starts when the solid and gas masses are about equal. It is engendered by a strong positive feedback on the gas accretion rates, driven by the rapid contraction of the gaseous envelope and the rapid expansion of the outer boundary, which depends on the planet's total mass. The overall evolutionary time scale is generally determined by the length of the second phase. The actual rates at which the giant planets accreted small planetesimals is probably intermediate between the constant rates assumed in most previous studies and the highly variable rates that we have used. Within the context, of the adopted model of planetesimal accretion, the joint constraints of the time scale for dissipation of the solar nebula and the current high-Z masses of the giant planets lead to estimates of the initial surface density (sigma(sub init)) of planetesimals in the outer region of the solar nebula. The results show sigma(sub init) approx. = 10 g/sq cm near Jupiter's orbit and that sigma(sub init) proportional to alpha(sup -2), where alpha is the distance from the Sun. These values are a factor of 3 - 4 times as high as that of the "minimum mass" solar nebula at Jupiter's distance and a factor of 2 - 3 times as high it Saturn's distance. Our estimates for the formation time of Jupiter and Saturn are 1 - 10 million years while those for Uranus fall in the range of 2 - 16 million years. These estimates follow from the properties of our Solar System and do not necessarily apply to giant planets in other planetary systems.

Pollack, James B.↗
NASA NTRS2008

On-Wafer Measurement of a Silicon-Based CMOS VCO at 324 GHz

The world s first silicon-based complementary metal oxide/semiconductor (CMOS) integrated-circuit voltage-controlled oscillator (VCO) operating in a frequency range around 324 GHz has been built and tested. Concomitantly, equipment for measuring the performance of this oscillator has been built and tested. These accomplishments are intermediate steps in a continuing effort to develop low-power-consumption, low-phase-noise, electronically tunable signal generators as local oscillators for heterodyne receivers in submillimeter-wavelength (frequency > 300 GHz) scientific instruments and imaging systems. Submillimeter-wavelength imaging systems are of special interest for military and law-enforcement use because they could, potentially, be used to detect weapons hidden behind clothing and other opaque dielectric materials. In comparison with prior submillimeter- wavelength signal generators, CMOS VCOs offer significant potential advantages, including great reductions in power consumption, mass, size, and complexity. In addition, there is potential for on-chip integration of CMOS VCOs with other CMOS integrated circuitry, including phase-lock loops, analog- to-digital converters, and advanced microprocessors.

Samoska, Lorene↗
NASA NTRS2017

Valence State Partitioning of Cr and V Between Olivine-Melt and Pyroxene-Melt in Experimental Basalts of a Eucritic Composition

The partitioning of multivalent elements in basaltic systems can elucidate the oxygen fugacity (fO2) conditions under which basalts formed on planetary bodies (Earth, Moon, Mars, asteroids). Chromium and V are minor and trace elements in basaltic melts, partition into several minerals that crystallize from basaltic melts, exist in multiple valence states at differing fO2 conditions, and can therefore be used as oxybarometers for basaltic melts. Chromium is mostly 3+ in terrestrial basaltic melts at relatively high fO2 values (≥ IW+3.5), and mostly 2+ in melts at low fO2 values (≤ IW-1), such as those on the Moon and some asteroids. At intermediate fO2s, (i.e., IW-1 to IW+3.5), basaltic melts contain both Cr3+ and Cr2+. Vanadium in basaltic melts is mostly 4+ at high fO2, mostly 3+ at low fO2, and a mix of V3+ and V4+ at intermediate fO2 con-ditions. Understanding the partitioning of Cr and V into silicate phases with changing fO2 is therefore critical to the employment of Cr and V oxybarometers. In this abstract we examine the equilibrium partitioning of Cr and V between olivine/melt and pyroxene/melt in experimental charges of a eucritic composition produced at differing fO2 conditions. This study will add to the experimental data on DCr and DV (i.e., olivine/melt, pyroxene/melt) at differing fO2, and in turn these D values will be used to assess the fO2 of eucrite basalts and perhaps other compositionally similar planetary basalts.

Karner, J. M.↗
NASA NTRSDate not supplied

Rephasing and Loitering Strategies in the Gateway Near Rectilinear Halo Orbit

NASA's Gateway plans to operate in an L2 9:2 lunar synodic Near Rectilinear Halo Orbit (NRHO) for nominal operations. During the mission's lifetime, scenarios may arise where rephasing within the NRHO or loitering in an alternate trajectory are advantageous. This investigation discusses two strategies to perform this rephasing in the NRHO over a relatively short timespan. The first method uses two maneuvers to shift the phase along a single transfer arc. The second strategy employs four maneuvers, leveraging an intermediate orbit for rephasing. An optimization strategy is formulated to construct locally optimal solutions in the Circular Restricted Three-Body Problem (CR3BP) and verified in a higher-fidelity ephemeris model.

NRHO↗
DOE OSTI2024

Theoretical studies of chemical reactions related to the formation and growth of polycyclic aromatic hydrocarbons (PAH) and molecular properties of their key intermediates (Final Progress Report)

The formation mechanisms of polycyclic aromatic hydrocarbons, (PAHs) – organic molecules carrying fused benzene rings – are of great interest to scientists and engineers due to their importance in combustion chemistry and astrochemistry. On Earth, PAHs are largely produced in incomplete combustion of fossil fuel and are considered as critical precursors to unwanted soot particles leading to combustion inefficiency and causing air pollution along with detrimental health effects. Simple PAH molecules initially formed in the gas phase, are further involved in a build-up process in combustion flames leading to larger PAH, bowl-shaped nanostructures, fullerenes, and solid-phase species including carbonaceous dust, graphene particles, and soot. In deep space, PAH and their derivatives are potential key intermediates and nucleation sites leading eventually to carbonaceous nanoparticles (“interstellar grains”). Therefore, the understanding of the key processes in the synthesis of PAHs along with their precursors and their degradation mechanisms in combustion systems and in interstellar, circumstellar, and planetary atmospheric environments will provide critical insights into how complex aromatic structures, carbonaceous nanoparticles, and fullerenes are formed and destroyed. Achieving this understanding is an important step in the development of the efficient combustion processes and of the ecofriendly devices with reduced environmental pollution as well as technological strategies for the production of hydrogen and solid carbon through thermal or plasma-assisted pyrolysis of natural gas and biomass. Also, the understanding of the key processes of PAH and soot growth will help in our comprehension of chemical evolution in the universe. Detailed information on the mechanisms and reliable rate constants of the key elementary chemical reactions involved in PAH formation and destruction processes and in inception of soot particles is often missing, with the main deficiencies being the absence of temperature- and pressure-dependent rate constants for the broad range of conditions occurring in various terrestrial and interstellar processes and the lack of data on the reaction products and their branching ratios. Complementary to experimental studies, these gaps in knowledge can be filled by using quantum chemical calculations of reaction potential energy surfaces providing us with accurate energies of reaction products, intermediates, and transition states, revealing the reaction mechanism, and giving the molecular properties required to compute rate constants for relevant reaction steps and product branching ratios using the RRKM-Master Equation (ME) method. Molecular dynamics (MD) simulations can be used in cases when a reaction rate cannot be properly described by statistical theories. During the terminal renewal project period we employed these ab initio/RRKM-ME and MD approaches to complete our studies on several key reactions relevant to the formation/growth of PAH and inception of soot particles including (1) the reaction mechanism and kinetics of the resonance stabilized fulvenallenyl radical with propargyl and C 3 H 4 isomers; (2) the reaction mechanism and kinetics for the C + indene and C 2 + styrene reactions producing naphthyl or azulenyl radicals in low-temperature environments; (3) the MD study of non-equilibrium dimerization of acepyrene and coronene and its radical. The information derived from our theoretical calculations contributed to a better fundamental understanding of the reaction mechanisms and provide missing critical kinetic data to improve combustion models of hydrocarbon fuels and astrochemical models of the growth of carbonaceous molecules and particles in cold molecular clouds, circumstellar envelopes, and planetary atmospheres.

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

Selective Brookite Polymorph Formation Related to the Amorphous Precursor State in TiO 2 Thin Films

A wide variety of brookite TiO 2 synthesis methods have been published over the past several decades, but few studies discuss the underlying mechanism that stabilizes brookite over its stable counterparts, rutile and anatase. Here in this study, we investigate of the effect of pulsed laser deposition parameters on the as-deposited amorphous precursor titania thin films, which subsequently crystallize into stable and metastable TiO 2 polymorphs upon annealing. We find that oxygen pressure in the deposition chamber strongly influences the non-equilibrium state of the amorphous precursor, which ultimately allows for selective polymorph formation. Rutile forms as the dominant phase at low pO 2 < 0.1 mTorr, while anatase is favored at high pO 2 > 5 mTorr. Brookite forms primarily at intermediate pO 2 (0.5-1.0 mTorr). Controlling the amorphous structure (i.e. Ti - O bonding and polyhedral arrangement) of the precursors via oxygen deficiency is therefore likely for the selective formation of crystalline TiO 2 polymorphs from sub-stoichiometric amorphous precursors. Lastly, directing phase selectivity by manipulating the structure and internal energy of the precursor amorphous state may have tremendous potential for synthesis of metastable crystalline phases that exhibit more desirable properties in comparison to their stable counterparts.

36 MATERIALS SCIENCE↗
DOE OSTI2015

Implementation and Re nement of a Comprehensive Model for Dense Granular Flows

Dense granular ows are ubiquitous in both natural and industrial processes. They manifest three di erent ow regimes, each exhibiting its own dependence on solids volume fraction, shear rate, and particle-level properties. This research project sought to develop continuum rheological models for dense granular ows that bridges multiple regimes of ow, implement them in open-source platforms for gas-particle ows and perform test simulations. The rst phase of the research covered in this project involved implementation of a steady- shear rheological model that bridges quasi-static, intermediate and inertial regimes of ow into MFIX (Multiphase Flow with Interphase eXchanges - a general purpose computer code developed at the National Energy Technology Laboratory). MFIX simulations of dense granular ows in hourglass-shaped hopper were then performed as test examples. The second phase focused on formulation of a modi ed kinetic theory for frictional particles that can be used over a wider range of particle volume fractions and also apply for dynamic, multi- dimensional ow conditions. To guide this work, simulations of simple shear ows of identical mono-disperse spheres were also performed using the discrete element method. The third phase of this project sought to develop and implement a more rigorous treatment of boundary e ects. Towards this end, simulations of simple shear ows of identical mono-disperse spheres con ned between parallel plates were performed and analyzed to formulate compact wall boundary conditions that can be used for dense frictional ows at at frictional boundaries. The fourth phase explored the role of modest levels of cohesive interactions between particles on the dense phase rheology. The nal phase of this project focused on implementation and testing of the modi ed kinetic theory in MFIX and running bin-discharge simulations as test examples.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Electrical and structural properties of binary Ga–Sb phase change memory alloys

Material properties of Ga–Sb binary alloy thin films deposited under ultra-high vacuum conditions were studied for analog phase change memory (PCM) applications. Crystallization of this alloy was shown to occur in the temperature range of 180–264 °C, with activation energy >2.5 eV depending on the composition. X-ray diffraction (XRD) studies showed phase separation upon crystallization into two phases, Ga-doped A7 antimony and cubic zinc-blende GaSb. Synchrotron in situ XRD analysis revealed that crystallization into the A7 phase is accompanied by Ga out-diffusion from the grains. X-ray absorption fine structure studies of the local structure of these alloys demonstrated a bond length decrease with a stable coordination number of 4 upon amorphous-to-crystalline phase transformation. Mushroom cell structures built with Ga–Sb alloys on ø110 nm TiN heater show a phase change material resistance switching behavior with resistance ratio >100 under electrical pulse measurements. TEM and Energy Dispersive Spectroscopy (EDS) studies of the Ga–Sb cells after ∼100 switching cycles revealed that partial SET or intermediate resistance states are attained by the variation of the grain size of the material as well as the Ga content in the A7 phase. A mechanism for a reversible composition control is proposed for analog cell performance. These results indicate that Te-free Ga–Sb binary alloys are potential candidates for analog PCM applications.

Ume, Rubab (ORCID:0000000324766900)↗
DOE OSTI2017

Nitric oxide is an obligate bacterial nitrification intermediate produced by hydroxylamine oxidoreductase

Significance The enzymatic reactions that occur during nitrification are nature’s means to use ammonia as cellular fuel. Complete understanding of nitrification and related processes are vital to sustainable agriculture and renewable energy technologies. The prevailing view of the first phase of nitrification is that ammonia oxidizing bacteria use two enzymes, ammonia monooxygenase and hydroxylamine oxidoreductase, to oxidize ammonia to nitrite via hydroxylamine as an obligate intermediate. Our work reveals nitric oxide as an additional obligate intermediate. The presented findings necessitate revision of a key biogeochemical process, identify a new bioenergetic role for nitric oxide, predict participation of a third enzyme in the biological oxidation of ammonia to nitrite, and will inform models toward sustainable agriculture.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Epitaxial growth of conducting perovskite SrTiO3 on (0001) c-axis sapphire using a γ-Al2O3 intermediate layer

Epitaxial growth of complex oxides on large-area wafers, such as sapphire and silicon, represents a key step toward scalable oxide device production. Solid phase epitaxy allows the synthesis of γ-Al 2 O 3 on α-Al 2 O 3 and provides a template with a matched lattice constant and appropriate cubic symmetry for subsequent heteroepitaxial growth of perovskite complex oxides. Nb-doped SrTiO 3 thin films were deposited epitaxially on (111)-oriented γ-Al 2 O 3 intermediate layers on (0001) c-axis-oriented sapphire α-Al 2 O 3 crystals using pulsed laser deposition. The Nb:SrTiO 3 thin films with a thickness of 53 nm, grown at 700 °C on γ-Al 2 O 3 , reached fully relaxed lattice parameters and were epitaxially oriented with respect to the substrate. Nb:SrTiO 3 layers deposited using identical deposition conditions directly on α-Al 2 O 3 , without the γ-Al 2 O 3 intermediate layer, were polycrystalline. The sheet conductivity of Nb:SrTiO 3 grown on γ-Al 2 O 3 /α-Al 2 O 3 is more than ten times higher than that of Nb:SrTiO 3 grown directly on α-Al 2 O 3 without the γ-Al 2 O 3 layer. The results point to new directions for the integration of (111)-oriented pseudocubic perovskite complex oxides and the integration of epitaxial complex oxides over larger areas using α-Al 2 O 3 single-crystal substrates.

Atomic force microscopy↗
DOE OSTI2016

Morphological Expression of the Coherence and Relative Phase of Optical Inputs to the Photoelectrodeposition of Nanopatterned Se–Te Films

Highly anisotropic and ordered nanoscale lamellar morphologies can be spontaneously generated over macroscopic areas, without the use of a photomask or any templating agents, via the photoelectrodeposition of Se–Te alloy films. To form such structures, the light source can be a single, linearly polarized light source that need not necessarily be highly coherent. In this work, the variation in the morphologies produced by this deposition process was evaluated in response to differences in the coherence and relative phase between multiple simultaneous linearly polarized illumination inputs. Specifically, the morphologies of photoelectrodeposits were evaluated when two tandem same-wavelength sources with discrete linear polarizations, both either mutually incoherent or mutually coherent (with defined phase differences), were used. Additionally, morphologies were simulated via computer modeling of the interfacial light scattering and absorption during the photoelectrochemical growth process. The morphologies that were generated using two coherent, in-phase sources were equivalent to those generated using only a single source. In contrast, the use of two coherent, out-of-phase sources produced a range of morphological patterns. For small out-of-phase addition of orthogonal polarization components, lamellar-type patterns were observed. When fully out-of-phase orthogonal sources (circular polarization) were used, an isotropic, mesh-type pattern was instead generated, similar to that observed when unpolarized illumination was utilized. In intermediate cases, anisotropic lamellar-type patterns were superimposed on the isotropic mesh-type patterns, and the relative height between the two structures scaled with the amount of out-of-phase addition of the orthogonal polarization components. Similar results were obtained when two incoherent sources were utilized. In every case, the long axis of the lamellar-type morphology component aligned parallel to the intensity-weighted average polarization orientation. The observations consistently agreed with computer simulations, indicating that the observed morphologies were fully determined by the nature of the illumination utilized during the growth process. Here, the collective data thus indicated that the photoelectrodeposition process exhibits sensitivity toward the coherency, relative phase, and polarization orientations of all optical inputs and that this sensitivity is physically expressed in the morphology of the deposit.

77 NANOSCIENCE AND NANOTECHNOLOGY↗
DOE OSTI2024

Evaluation of phase change thermal storage in a cascade heat pump

Here this study proposes a low-global warming potential cascade heat pump system with integrated phase change material (PCM) for combined space heating and cooling. PCM is embedded in the intermediate heat exchanger that couples an outdoor R290 refrigerant circuit and an indoor CO 2 circuit. We used a clustering method to develop seasonally representative load and ambient temperature profiles for residential buildings in Minneapolis, Minnesota. We simulated the system peak shaving performance on the selected representative days, assuming three different PCM options including ice (0 °C), tetradecane (5.6 °C), and a commercially available PCM, SP9 (10.5 °C). The baseline was the same cascade system without thermal storage. We investigated the effect of both thermal energy storage capacity (kWh th ) and PCM transition temperature on system performance. The results show that higher thermal energy storage capacity shaves more electric energy until it reaches a maximum value set by the load characteristics of the building. SP9, with the highest transition temperature, shaves the most electric energy, while ice shaves the least. Moreover, ice has the highest charging cost, especially in the cooling season when the outdoor circuit has to operate at an evaporating temperature lower than 0 °C to make ice. Our results show that SP9 presents the lowest levelized cost of storage, closely followed by ice. A lower off-peak electricity rate benefits ice more than SP9 or tetradecane because ice has the largest charging cost penalty.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2025

Tetragonal BaCoO 3 : A Co 4+ Ferromagnetic Mott Insulator

We report the stabilization of the metastable body-centered tetragonal (BCT) phase of BaCoO 3 (BCT-BaCoO 3 ) under high-pressure (15 GPa) and hightemperature (1200 °C) conditions using a mixture precursor. This double perovskite adopts the EuTiO 3 -type structure (space group I4/mcm), as confirmed by powder Xray diffraction and high-resolution STEM. X-ray photoelectron spectroscopy indicates a predominant Co 4+ oxidation state without detectable oxygen vacancies. Magnetization and heat capacity measurements reveal ferromagnetic ordering at T C ∼ 107 K, attributable to the BCT-BaCoO 3 phase. Above this temperature, the mixed-phase sample exhibits Curie−Weiss paramagnetism, a low-spin to high-spin crossover upon cooling, and a possible intermediate-spin state at elevated temperatures. Resistivity data indicates insulating behavior with weak magnetoresistance. DFT and DFT + DMFT calculations suggest that the insulating state originates from an orbitally selective transition sensitive to the nominal valence of the Co-d shell. The metastable BCT-BaCoO 3 phase cannot be retained in pure form at ambient pressure but can be stabilized by embedding it in a disordered mixture, offering a potential route to discover and preserve other high-pressure phases under ambient conditions.

Magnetic properties↗
DOE OSTI2024

Pairing tendencies in the doped Kitaev-Heisenberg model

Here, we study the impact of hole doping on the Kitaev-Heisenberg model on the honeycomb lattice. We investigate the pairing tendencies and correlation functions in the framework of a t - J - K model using density matrix renormalization group calculations on three-leg cylinders. In the case of the pure Kitaev model, which realizes a quantum spin-liquid phase at half-filling, we find that binding of two holes only occurs at low values of the hopping, where the holes are slow. We have theoretically verified that pair formation occurs in the limit of immobile holes, where the pure Kitaev model remains exactly solvable. When we instead fix the hopping at an intermediate, more realistic, value, and vary the Heisenberg and Kitaev interaction strengths, we find pairing tendencies only in the Néel phase. This is in contrast to prior mean-field calculations, highlighting the importance of accounting for the kinetic energy of dopants in generalized Kitaev models. Interestingly, we also find signatures of pair-density wave formation over the studied range of model parameters, namely, a periodic modulation of the charge density as well as the spin-spin and pair-pair correlations in real space. Moreover, we present a comparative study of the different correlations as a function of doping. We finally discuss the potential for experimentally observing the studied physics in quantum materials and heterostructures.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Topological and magnetic phase transitions in the bilayer Kitaev-Ising model

We investigate the phase diagram of a bilayer Kitaev honeycomb model with Ising interlayer interactions, deriving effective models via perturbation theory and performing Majorana mean-field theory calculations. We show that a diverse array of magnetic and topological phase transitions occur, depending on the direction of the interlayer Ising interaction and the relative sign of Kitaev interactions. When two layers have the same sign of the Kitaev interaction, a first-order transition from a Kitaev spin liquid to a magnetically ordered state takes place. The magnetic order points along the Ising axis and it is (anti)ferromagnetic for (anti)ferromagnetic Kitaev interactions. However, when two layers have opposite signs of the Kitaev interaction, we observe a notable weakening of magnetic ordering tendencies and the Kitaev spin liquid survives up to a remarkably larger interlayer exchange. Our mean-field analysis suggests the emergence of an intermediate gapped ℤ 2 spin-liquid state, which eventually becomes unstable upon vison condensation. The confined phase is described by a highly frustrated 120° compass model. We furthermore use perturbation theory to study the model with the Ising axis pointing along the $\hat{z}$ axis or lying in the 𝑥⁢𝑦 plane. In both cases, our analysis reveals the formation of one-dimensional Ising chains, which remain decoupled in perturbation theory, resulting in a subextensive ground-state degeneracy. Our results highlight the interplay between topological order and magnetic ordering tendencies in bilayer quantum spin liquids.

Kitaev model↗