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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 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Interlaboratory Reproducibility of Contour Method Data in a High Strength Aluminum Alloy

The contour method for residual stress measurement has seen significant development, but an experimental reproducibility study utilizing physical samples has not been published. A double-blind reproducibly study is reported, having scope beginning with EDM cutting and ending with residual stress calculation. A reinforced I-beam sample geometry is identified for its unique residual stress profile when extracted from residual stress bearing quenched aluminum bar (7050-T74). Contour measurements are prescribed on a midplane of symmetry with dimensions 24.0 mm by 50.0 mm. Fourteen identically prepared samples are fabricated from a single long bar with well characterized and uniform residual stress. Five samples throughout the bar are identified for planning measurements to validate sample uniformity and overall suitability of the residual stress field. The planning measurements employ a range of techniques: contour method, neutron diffraction, and hole-drilling. Eight samples are distributed to an international group of participants to execute their standard measurement practice. A double-blind process is followed to provide anonymity. Results are provided by eight participants: six being self-similar and two being quite different, the latter set aside as outliers. An average residual stress field is established from non-outlying results and the spatial distribution of reproducibility standard deviation is determined. The average stress field ranges from -60 to 70 MPa and the reproducibility standard deviation averages 8.1 MPa on the measurement plane. The average reproducibility standard deviation is about 3 × larger for points within 1.0 mm of plane boundaries (17.6 MPa) than for the remaining points (6.1 MPa). Reproducibility standard deviation (among different labs) for contour method residual stress measurement is found to be very similar to repeatability standard deviation (in a single lab) reported in prior work. The reproducibility observed here, for the entire measurement process, is also similar to that found in a prior reproducibility study limited to contour method data analysis.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Possible surface magnetism in the topological Kondo insulator candidate FeSi

Our group has previously reported the existence of a conducting surface state (CSS) in FeSi, a candidate for a d-electron topological Kondo insulator (TKI), at low temperature. In this paper, we present the electrical transport properties of single crystals of FeSi studied in the phase space of temperature (T), magnetic field (B), and the angle (θ) between the electrical current and B. The normalized T-dependent electrical resistance (R) of a successively thinned FeSi crystal provides further confirmation of the existence of a CSS. We report that, in the CSS, the magnetoresistance (MR) exhibits a hysteresis loop bounded within ±0.5 T, suggesting two-dimensional magnetic ordering. The hysteretic MR is asymmetric in B and anisotropic with respect to θ. Further exploration of R(θ) at a fixed field of 9 T reveals an initial progressive rotation of the axis for twofold rotational symmetry from 2 to 10 K, then a stabilized axis for twofold symmetry until, at T > 40 K, the anisotropy vanishes, coincident with the disappearance of the CSS. Furthermore, these observations point to a possible magnetically ordered surface state that has been reported in similar systems such as FeSi nanofilms and bulk SmB 6 .

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

Atomic-Scale Andreev Probe of Unconventional Superconductivity

Recent emergence of low-dimensional unconventional superconductors and their exotic interface properties calls for new approaches to probe the pairing symmetry, a fundamental and frequently elusive property of the superconducting condensate. Here, we introduce the unique capability of tunneling Andreev reflection (TAR) to probe unconventional pairing symmetry, utilizing the sensitivity of this technique to specific Andreev reflections. Specifically, suppression of the lowest-order Andreev reflection due to quantum interference but emergence of the higher-order Andreev processes provides direct evidence of the sign-changing order parameter in the paradigmatic FeSe superconductor. Furthermore, TAR spectroscopy also reveals two superconducting gaps, points to a possibility of a nodal gap structure, and directly confirms that superconductivity is locally suppressed along the nematic twin boundary, with preferential and near-complete suppression of the larger energy gap. Furthermore, our findings therefore enable new, atomic-scale insight into microscopic, inhomogeneous, and interfacial properties of emerging quantum materials.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Twisted symmetric trilayer graphene: Single-particle and many-body Hamiltonians and hidden nonlocal symmetries of trilayer moiré systems with and without displacement field

Here, we derive the Hamiltonian for trilayer moiré systems with the Coulomb interaction projected onto the bands near the charge neutrality point. Motivated by the latest experimental results, we focus on the twisted symmetric trilayer graphene (TSTG) with a mirror symmetry with respect to the middle layer. We provide a full symmetry analysis of the noninteracting Hamiltonian with a perpendicular displacement field coupling the band structure made otherwise of the twisted bilayer graphene (TBG) and the high-velocity Dirac fermions, and we identify a hidden nonlocal symmetry of the problem. In the presence of this displacement field, we construct an approximate single-particle model, akin to the tripod model for TBG, capturing the essence of noninteracting TSTG. We also derive more quantitative perturbation schemes for the low-energy physics of TSTG with displacement field, obtaining the corresponding eigenstates. This allows us to obtain the Coulomb interaction Hamiltonian projected in the active band TSTG wave functions and derive the full many-body Hamiltonian of the system. We also provide an efficient parametrization of the interacting Hamiltonian. Finally, we show that the discrete symmetries at the single-particle level promote the U (2) × U (2) spin-valley symmetry to enlarged symmetry groups of the interacting problem under different limits. The interacting part of the Hamiltonian exhibits a large U (4) × U (4) × U (4) × U (4) symmetry in the chiral limit. Moreover, by identifying a symmetry which we dub spatial many-body charge conjugation, we show that the physics of TSTG is symmetric around charge neutrality.

2-dimensional systems↗
DOE OSTI2020

The inversion formula and 6j symbol for 3d fermions

In this work we study the 6 j symbol of the 3 d conformal group for fermionic operators. In particular, we study 4-point functions containing two fermions and two scalars and also those with four fermions. By using weight-shifting operators and harmonic analysis for the Euclidean conformal group, we relate these spinning 6 j symbols to the simpler 6 j symbol for four scalar operators. As one application we use these techniques to compute 3 d mean field theory (MFT) OPE coefficients for fermionic operators. We then compute corrections to the MFT spectrum and couplings due to the inversion of a single operator, such as the stress tensor or a low-dimension scalar. These results are valid at finite spin and extend the perturbative large spin analysis to include non-perturbative effects in spin.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Anomalous Hall effects in chiral superconductors

We report theoretical results for the electronic contribution to thermal and electrical transport for chiral superconductors belonging to even or odd-parity E 1 and E 2 representations of the tetragonal and hexagonal point groups. Chiral superconductors exhibit novel properties that depend on the topology of the order parameter and Fermi surface, and—as we highlight—the structure of the impurity potential. An anomalous thermal Hall effect is predicted and shown to be sensitive to the winding number, ν, of the chiral order parameter via Andreev scattering that transfers angular momentum from the chiral condensate to excitations that scatter off the random potential. For heat transport in a chiral superconductor with isotropic impurity scattering, i.e., point-like impurities, a transverse heat current is obtained for ν = ± 1, but vanishes for | ν | > 1. This is not a universal result. For finite-size impurities with radii of order or greater than the Fermi wavelength, R ≥ ℏ / p f, the thermal Hall conductivity is finite for chiral order with | ν | ≥ 2, and determined by a specific Fermi-surface average of the differential cross-section for electron-impurity scattering. Our results also provide quantitative formulae for analyzing and interpreting thermal transport measurements for superconductors predicted to exhibit broken time-reversal and mirror symmetries.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2020

Towards a UV model of kinetic mixing and portal matter

The nature of dark matter (DM) and how it might interact with the particles of the Standard Model (SM) is an ever-growing mystery. It is possible that the existence of new “dark sector” forces, yet undiscovered, are the key to solving this fundamental problem, and one might hope that in the future such forces might even be “unified” with the ones we already know in some UV-complete framework. In this paper, following a bottom-up approach, we attempt to take the first steps in the construction of such a framework. The much-discussed possibility of the kinetic mixing (KM) of the “dark photon” with the hypercharge gauge boson of the SM via loops of portal matter (PM) fields, charged in both sectors, offers an attractive starting point for these efforts. Given the anticipated finite strength of the KM in a UV-complete theory, the absence of anomalies, and the lifetime constraints on the PM fields arising from cosmic microwave background and nucleosynthesis constraints, PM must behave as vector-like copies of the known SM fermion fields, such as those which appear naturally in, e.g., E 6 -type models. Within such a setup, the SM and their corresponding partner PM fields would be related by a new S U ( 2 ) I gauge symmetry. With this observation as a springboard, we construct a generalization of these ideas where S U ( 2 ) I is augmented by an additional U ( 1 ) I Y factor so that the light dark photon is the result of a symmetry breaking analogous to the SM, i.e., S U ( 2 ) I × U ( 1 ) I Y → U ( 1 ) D , but with U ( 1 ) D now also broken at the ≲ GeV scale. While SM fields are U ( 1 ) D singlets, as in the conventional dark photon approach, they transform nontrivially under the full S U ( 2 ) I × U ( 1 ) I Y gauge group. This approach leads to numerous interesting signatures, both at low energies and at colliders, and can be viewed as an initial step in the construction of a more UV-complete framework.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Loops in 4+1d topological phases

2+1d topological phases are well characterized by the fusion rules and braiding/exchange statistics of fractional point excitations. In 4+1d, some topological phases contain only fractional loop excitations. What kind of loop statistics exist? We study the 4+1d gauge theory with 2-form \mathbb{Z}_2 ℤ 2 gauge field (the loop-only toric code) and find that while braiding statistics between two different types of loops can be nontrivial, the self “exchange” statistics are all trivial. In particular, we show that the electric, magnetic, and dyonic loop excitations in the 4+1d toric code are not distinguished by their self-statistics. They tunnel into each other across 3+1d invertible domain walls which in turn give explicit unitary circuits that map the loop excitations into each other. The SL(2, \mathbb{Z}_2 ℤ 2 ) symmetry that permutes the loops, however, cannot be consistently gauged and we discuss the associated obstruction in the process. Moreover, we discuss a gapless boundary condition dubbed the “fractional Maxwell theory” and show how it can be Higgsed into gapped boundary conditions. We also discuss the generalization of these results from the \mathbb{Z}_2 ℤ 2 gauge group to \mathbb{Z}_N ℤ N .

Chen, Xie↗
DOE OSTI2022

Hydrogen bonds and dispersion forces serving as molecular locks for tailored Group 11 bis(amidine) complexes

A flexible polydentate bis(amidine) ligand LH 2 , LH 2 = {CH 2 NH( t Bu)C=N-2-(6-MePy)} 2 , operates as a molecular lock for various coinage metal fragments and forms the dinuclear complexes [LH 2 (MCl) 2 ], M = Cu (1), Au (2), the coordination polymer [{(LH 2 ) 2 (py) 2 (AgCl) 3 }(py) 3 ] n (3), and the dimesityl-digold complex [LH 2 (AuMes) 2 ] (4) by formal insertion of MR fragments (M = Cu, Ag, Au; R = Cl, Mes) into the N–H∙∙∙N hydrogen bonds of LH 2 in yields of 43–95%. Complexes 1, 2, and 4 adopt C 2 -symmetrical structures in the solid state featuring two interconnected 11-membered rings that are locked by two intramolecular N–H∙∙∙R–M hydrogen bonds. QTAIM analyses of the computational geometry-optimized structures 1a, 2a, and 4a reveal 13, 11, and 22 additional bond critical points, respectively, all of which are related to weak intramolecular attractive interactions, predominantly representing dispersion forces, contributing to the conformational stabilization of the C2-symmetrical stereoisomers in the solid state. Variable-temperature 1 H NMR spectroscopy in combination with DFT calculations indicate a dynamic conformational interconversion between two C 2 -symmetrical ground state structures in solution (ΔG ‡ c = 11.1–13.8 kcal mol –1 ), which is accompanied by the formation of an intermediate possessing C i symmetry that retains the hydrogen bonds.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Stripe helical magnetism and two regimes of anomalous Hall effect in NdAlGe

Here, we report the magnetic and electronic transport properties of the inversion and time-reversal symmetry breaking Weyl semimetal NdAlGe. This material is analogous to NdAlSi, whose helical magnetism presents a rare example of a Weyl-mediated collective phenomenon, but with a larger spin-orbit coupling. Our neutron diffraction experiments revealed that NdAlGe, similar to NdAlSi, supports an incommensurate spin density wave (T inc =6.8 K) whose spins are predominantly pointing along the out-of-plane direction and have a small helical spin canting of 3°. The spin density wave has a long wavelength of ≈35 nm and transitions to a commensurate ferrimagnetic state below T com =5.1K. Using small-angle neutron scattering, we showed that the zero-field cooled ferrimagnetic domains form stripes in real space with characteristic length scales of 18 and 72 nm parallel and perpendicular to the [110] direction, respectively. Interestingly, for the transport properties, NdAlSi does not exhibit an anomalous Hall effect (AHE) that is commonly observed in magnetic Weyl semimetals. In contrast to NdAlSi, we identify two different AHE regimes in NdAlGe that are, respectively, governed by intrinsic Berry curvature and extrinsic disorders/spin fluctuations. Our paper suggests that Weyl-mediated magnetism prevails in this group of noncentrosymmetric magnetic Weyl semimetals NdAlX, but transport properties including AHE are affected by material-specific extrinsic effects such as disorders, despite the presence of prominent Berry curvature.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Candidate phases for SU(2) adjoint QCD$_4$ with two flavors from $\mathcal{N}=2$ supersymmetric Yang-Mills theory

We study four-dimensional adjoint QCD with gauge group SU(2) and two Weyl fermion flavors, which has an SU(2) R chiral symmetry. The infrared behavior of this theory is not firmly established. We explore candidate infrared phases by embedding adjoint QCD into N = 2 supersymmetric Yang-Mills theory deformed by a supersymmetry-breaking scalar mass M that preserves all global symmetries and 't Hooft anomalies. This includes 't Hooft anomalies that are only visible when the theory is placed on manifolds that do not admit a spin structure. The consistency of this procedure is guaranteed by a nonabelian spin-charge relation involving the SU(2) R symmetry that is familiar from topologically twisted N = 2 theories. Since every vacuum on the Coulomb branch of the N = 2 theory necessarily matches all 't Hooft anomalies, we can generate candidate phases for adjoint QCD by deforming the theories in these vacua while preserving all symmetries and 't Hooft anomalies. One such deformation is the supersymmetry-breaking scalar mass M itself, which can be reliably analyzed when M is small. In this regime it gives rise to an exotic Coulomb phase without chiral symmetry breaking. By contrast, the theory near the monopole and dyon points can be deformed to realize a candidate phase with monopole-induced confinement and chiral symmetry breaking. The low-energy theory consists of two copies of a CP 1 sigma model, which we analyze in detail. Certain topological couplings that are likely to be present in this CP 1 model turn the confining solitonic string of the model into a topological insulator. We also examine the behavior of various candidate phases under fermion mass deformations. We speculate on the possible large-M behavior of the deformed N = 2 theory and conjecture that the CP 1 phase eventually becomes dominant.

Córdova, Clay↗
DOE OSTI2020

Chiral algebra, localization, modularity, surface defects, and all that

We study the 2D vertex operator algebra (VOA) construction in 4D N = 2 superconformal field theories on S 3 × S 1 , focusing on both old puzzles and new observations. The VOA lives on a two-torus T 2 ⊂ S 3 × S 1 , it is 1 2 Z -graded, and this torus is equipped with the natural choice of spin structure (1,0) for the Z + 1 2 -graded operators, corresponding to the NS sector vacuum character. By analyzing the possible refinements of the Schur index that preserves the VOA, we find that it admits discrete deformations, which allows access to the remaining spin structures (1,1), (0,1), and (0,0), of which the latter two involve the inclusion of a particular surface defect. For Lagrangian theories, we perform the detailed analysis: we describe the natural supersymmetric background, perform localization, and derive the gauged symplectic boson action on a torus in any spin structure. In the absence of flavor fugacities, the 2D and 4D path integrals precisely match, including the Casimir factors. We further analyze the 2D theory: we identify its integration cycle and the two-point functions and interpret flavor holonomies as screening charges in the VOA. Next, we make some observations about modularity; the T-transformation acts on our four partition functions and lifts to a large diffeomorphism on S 3 × S 1 . More interestingly, we generalize the four partition functions on the torus to an infinite family labeled by both the spin structure and the integration cycle inside the complexified maximal torus of the gauge group. Members of this family transform into one another under the full modular group, and we confirm the recent observation that the S-transform of the Schur index in Lagrangian theories exhibits logarithmic behavior. Finally, we comment on how locally our background reproduces the Ω-background.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Topological field theory with Haagerup symmetry

Here, we construct a (1 + 1)d topological field theory (TFT) whose topological defect lines (TDLs) realize the transparent Haagerup $\mathscr{H}$ 3 fusion category. This TFT has six vacua, and each of the three non-invertible simple TDLs hosts three defect operators, giving rise to a total of 15 point-like operators. The TFT data, including the three-point coefficients and lasso diagrams, are determined by solving all the sphere four-point crossing equations and torus one-point modular invariance equations. We further verify that the Cardy states furnish a non-negative integer matrix representation under TDL fusion. While many of the constraints we derive are not limited to this particular TFT with six vacua, we leave open the construction of TFTs with two or four vacua. Finally, TFTs realizing the Haagerup $\mathscr{H}$ 1 and $\mathscr{H}$ 2 fusion categories can be obtained by gauging algebra objects. This article makes a modest offering in our pursuit of exotica and the quest for their eventual conformity.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Prediction of double-Weyl points in the iron-based superconductor CaKFe 4 As 4

Employing a combination of symmetry analysis, low-energy modeling, and ab initio simulations, we predict the presence of magnetic-field-induced Weyl points close to the Fermi level in CaKFe 4 As 4 . Depending on the relative strengths of the magnetic field and of the spin-orbit coupling, the Weyl fermions can carry a topological charge of ±1 or ±2, making CaKFe 4 As 4 a rare realization of a double-Weyl semimetal. Here, we further predict experimental manifestations of these Weyl points, both in bulk properties, such as the anomalous Hall effect, and in surface properties, such as the emergence of prominent Fermi arcs. Because CaKFe 4 As 4 displays unconventional fully gapped superconductivity below 30 K, our findings open a route to investigate the interplay between superconductivity and Weyl fermions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

Chemical Interpretation of Charged Point Defects in Semiconductors: A Case Study of Mg 2 Si

The electronic structures of charged point defects influence electrical and optical properties of semiconductors. Understanding the orbital interactions responsible for the electronic structures of defects therefore promotes a chemical intuition for defect-driven mechanisms in semiconductors. In this tutorial, we discuss a molecular orbital theory-based framework for understanding defect-induced electronic states based on local chemical interactions between the defect and the atoms surrounding the defect site. By using Mg 2 Si as a case study, we show how both the chemical interactions and molecular orbitals (i. e., wave functions) responsible for the charge state(s) of a defect can be understood from the bonding symmetry of the defect site. We anticipate that a chemistry-based perspective of charged defects will enrich defect engineering efforts for electronic and optical materials.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Tuning the Radius Ratio to Enhance Thermoelectric Properties in the Zintl Compounds AM 2 Sb 2 (A = Ba, Sr; M = Zn, Cd)

Five novel Zintl phase solid solutions in the Ba 1–x Sr x Zn 2–y Cd y Sb 2 (0 ≤ x ≤ 0.13(1); 0 ≤ y ≤ 0.32(2)) system were successfully synthesized by the molten Pb metal-flux method, and the powder X-ray diffraction and single-crystal X-ray diffraction analyses proved that all five title compounds adopted the BaCu 2 S 2 -type phase having the orthorhombic Pnma space group (Z = 4, Pearson code oP20) with five crystallographically independent atomic sites. The previously studied BaCu 2 S 2 -type antimonides demonstrated a limited tolerance for doping in contrast to the CaAl 2 Si 2 -type antimonides. To understand the relatively narrower phase width and limited dopability of the title BaCu 2 S 2 -type phase than the CaAl 2 Si 2 -type phase in the overall Ba 1–x Sr x Zn 2–y Cd y Sb 2 system, the radius ratio of cations and anionic elements r + /r – for two structure types were thoroughly investigated. For the first time, the r + /r – ratio was identified as a critical factor for the phase selectivity: (1) r + /r – > 1 favored the BaCu 2 S 2 -type phase, and (2) r + /r – < 1 favored the CaAl 2 Si 2 -type phase. Further, we also revealed the structural transformation mechanism from the more widely observed CaAl 2 Si 2 -type phase to the title BaCu 2 S 2 -type phase as the relatively larger cationic elements were introduced to the system. A series of DFT calculations using the three hypothetical models indicated that a resonance peak near EF in the density of states curves was descended from the relatively flat band structure at several special symmetry points rationalizing the enhanced Seebeck coefficients of Ba 0.94(1) Sr 0.06 Zn 1.86(3) Cd 0.14 Sb 2 and Ba 0.96(1) Sr 0.04 Zn 1.68(2) Cd 0.32 Sb 2 . Electron localization function analysis rationalized the correlation between the polarity change of anionic Zn/Cd–Sb bonds and the charge carrier mobility on the anionic frameworks. Temperature-dependent thermoelectric properties were studied for the four title compounds, and the results proved that the Sr and Cd doping in the title Ba 1–x Sr x Zn 2–y Cd y Sb 2 system successfully enhanced the ZT values through the increased Seebeck coefficients and the reduced total thermal conductivities.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Electromagnetic duality and D3-brane scattering amplitudes beyond leading order

We use on-shell methods to study the non-supersymmetric and supersymmetric low-energy S-matrix on a probe D3-brane, including both the 1-loop contributions of massless states as well as the effects of higher-derivative operators. Our results include: (1) A derivation of the duality invariance of Born-Infeld electrodynamics as the dimensional oxidation of the group of spatial rotations transverse to a probe M2-brane; this is done using a novel implementation of subtracted on-shell recursion. (2) The first explicit loop-level BCJ double-copy in a non-gravitational model, namely the calculation of the 4-point self-dual amplitude of non-supersymmetric Born-Infeld. (3) From previous results for n-point self-dual 1-loop BI amplitudes and the conjectured dimension-shifting relations in Yang-Mills, we obtain an explicit all-multiplicity, at all orders in E, expression for the 1-loop integrand of the MHV sector of N = 4 DBI. (4) For all n > 4, the explicitly integrated duality-violating 1-loop amplitudes (self-dual and next-to-self-dual in pure BI as well as MHV in N = 4 DBI) are shown to be removable at O(ε 0 ) by adding finite local counterterms; we propose that this may be true more generally at 1-loop order. (5) We find that in non-supersymmetric Born-Infeld, not all finite local counterterms needed to restore electromagnetic duality can be constructed using the double-copy with higher-derivative corrections, suggesting a fundamental tension between electromagnetic duality and color-kinematics duality at loop-level. Finally we comment on oxidation of duality symmetries in supergravity and the parallels it has to the M2-brane to D3-brane oxidation demonstrated in this paper.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Conformal conserved currents in embedding space

We study conformal conserved currents in arbitrary irreducible representations of the Lorentz group using the embedding space formalism. With the help of the operator product expansion, we first show that conservation conditions can be fully investigated by considering only two- and three-point correlation functions. We then find an explicitly conformally-covariant differential operator in embedding space that implements conservation based on the standard position space operator product expansion differential operator ∂μ, although the latter does not uplift to embedding space covariantly. The differential operator in embedding space that imposes conservation is the same differential operator $D_{ijA}$ used in the operator product expansion in embedding space. We provide several examples including conserved currents in irreducible representations that are not symmetric and traceless. With an eye on four-point conformal bootstrap equations for four conserved vector currents $\langle JJJJ\rangle$ and four energy-momentum tensors $\langle TTTT\rangle$, we mostly focus on conservation conditions for $\langle JJ\mathcal{O}\rangle$ and $\langle TT\mathcal{O}\rangle$. Finally, we reproduce and extend the consequences of conformal Ward identities at coincident points by determining three-point coefficients in terms of charges.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗