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Infinite-layer nickelates $R_{1-x}$Sr$_{x}$NiO$_{2}$ ($R$ = La, Pr, Nd) are a class of superconductors with structural similarities to cuprates. Although long-range antiferromagnetic order has not been observed for these materials, magnetic effects such as antiferromagnetic spin fluctuations and spin-glass behavior have been reported. Different experiments have drawn different conclusions about whether the pairing symmetry is $s$- or $d$ wave. In this paper, we applied a scanning superconducting quantum interference device (SQUID) to probe the magnetic behavior of film samples of three infinite-layer nickelates (La$_{0.85}$Sr$_{0.15}$NiO$_2$, Pr$_{0.8}$Sr$_{0.2}$NiO$_2$, and Nd$_{0.775}$Sr$_{0.225}$NiO$_2$) grown on SrTiO$_3$ (STO), each with a nominal thickness of 20 unit cells. In all three films, we observed a ferromagnetic background. We also measured the magnetic susceptibility above the superconducting critical temperature in Pr$_{0.8}$Sr$_{0.2}$NiO$_2$ and La$_{0.85}$Sr$_{0.15}$NiO$_2$ and identified a non-Curie-Weiss dynamic susceptibility. Both magnetic features are likely due to NiO$_x$ nanoparticles. Additionally, we investigated superconductivity in Pr$_{0.8}$Sr$_{0.2}$NiO$_2$ and Nd$_{0.775}$Sr$_{0.225}$NiO$_2$, which exhibited inhomogeneous diamagnetic screening. The superfluid density inferred from the diamagnetic susceptibility in relatively homogeneous regions shows $T$-linear behavior in both samples. Finally, we observed superconducting vortices in Nd$_{0.775}$Sr$_{0.225}$NiO$_2$. We determined a Pearl length of 330 $\upmu$m for Nd$_{0.775}$Sr$_{0.225}$NiO$_2$ at 300 mK both from the strength of the diamagnetism and from the size and shape of the vortices. These results highlight the importance of considering NiO$_x$ particles when interpreting experimental results for these films.

Progress in the study of infinite-layer nickelates has always been highly linked to materials advances. In particular, the recent development of superconductivity via hole-doping was predicated on the controlled synthesis of Ni in a very high oxidation state, and subsequent topotactic reduction to a very low oxidation state, currently limited to epitaxial thin films. Here we demonstrate a process to combine these steps with a heterostructure which includes an epitaxial soluble buffer layer, enabling the release of freestanding membranes of (Nd,Sr)NiO2 encapsulated in SrTiO3, which serves as a protective layer. The membranes have comparable structural and electronic properties to that of optimized thin films, and range in lateral dimensions from millimeters to ~100 micron fragments, depending on the degree of strain released with respect to the initial substrate. The changes in the superconducting transition temperature associated with membrane release are quite similar to those reported for substrate and pressure variations, suggestive of a common underlying mechanism. These membranes structures should provide a versatile platform for a range of experimental studies and devices free from substrate constraints.

The dynamics of Higgs mode in superconductors, manifested as coherent oscillations of the superconducting order parameter amplitude, provides vital insights into the nature of the superconducting gap structure and symmetry. Here we utilize two-dimensional terahertz coherent spectroscopy to investigate Higgs dynamics of a newly discovered infinite-layer nickelate superconductor. While we observe distinct nonlinear terahertz responses from the superconducting state, well-defined long-lived Higgs modes, as commonly observed in $s$-wave superconductors, are entirely absent in the nickelate film. Instead, we find the coherent nonlinear terahertz response is dominated by the quasiparticle excitations. These observations strongly indicate that the Higgs mode in infinite-layer nickelates is heavily damped by the quasiparticle excitations at arbitrarily low energies, which is a characteristic of $d$-wave pairing symmetry. Additionally, by examining the temperature dependence of the nonlinear terahertz response, we discover short-range superconducting fluctuations in the vicinity of $T_\mathrm{c}$. Our findings provide proof of a new $d$-wave system and establish a foundation for investigating the unconventional superconductivity in nickelates., Comment: 10 pages, 5 figures

The discovery of superconductivity in infinite-layer nickelates establishes a new category of unconventional superconductors that share structural and electronic similarities with cuprates. Despite exciting advances, such as the establishment of a cuprate-like phase diagram and the observation of charge order and short-range antiferromagnetic fluctuation, the key issues of superconducting pairing symmetry, gap amplitude, and superconducting fluctuation remain elusive. In this work, we utilize static and ultrafast terahertz spectroscopy to address these outstanding problems. We demonstrate that the equilibrium terahertz conductivity and nonequilibrium terahertz responses of an optimally Sr-doped nickelate film ($T_c$ = 17 K) are in line with the electrodynamics of $d$-wave superconductivity in the dirty limit. The gap-to-$T_c$ ratio 2$\Delta/k_\mathrm{B}T_\mathrm{c}$ is extracted to be 3.4, indicating the superconductivity falls in the weak-coupling regime. In addition, we observed significant superconducting fluctuation near $T_\mathrm{c}$, while it does not extend into the deep normal state as optimally hole-doped cuprates. Our result highlights a new $d$-wave system which closely resembles the electron-doped cuprates, expanding the family of unconventional superconductivity in oxides., Comment: 8 pages, 4 figures

We report measurements of the Nernst and Seebeck effects in Nd$_{1-x}$Sr$_x$NiO$_2$ thin films near the superconducting transition temperature, $T$ = 6.5 - 15 K. Our main result is the observation of a vortex-Nernst signal $S_{yx}(T,H)$ with a maximum at $\mu_0H$ = 5 T and a tail that extends to $\mu_0H$ $\approx$ 15 T, which we identify as the upper-critical field $H_{c2}$. At $T > T_{\rm c} =$ 6.1 K, $H_{c2}$ remains large (15 T), up to the highest temperature we can resolve from $S_{yx}$ (11 K). These results indicate the existence of a vortex-liquid state over a wide range of finite-resistance temperatures, as in the high-$T_{\rm c}$ cuprates., Comment: 10 pages, 6 figures

The notion of twisted sectors play a crucial role in orbifold Gromov-Witten theory. We introduce the notion of dihedral twisted sectors in order to construct Lagrangian Floer theory on symplectic orbifolds and discuss related issues., Comment: In honour of Professor Jean-Pierre Bourguignon on the occasion of his 75th birthday

The consequences of varying the rare-earth element in the superconducting infinite-layer nickelates have been much debated. Here we show striking differences in the magnitude and anisotropy of the superconducting upper critical field across the La-, Pr-, and Nd-nickelates. These 5 distinctions originate from the 4f electron characteristics of the rare-earth ions in the lattice: they are absent for La3+, nonmagnetic for the Pr3+ singlet ground state, and magnetic for the Nd3+ Kramer's doublet. The unique polar and azimuthal angle-dependent magnetoresistance found in the Nd-nickelates can be understood to arise from the magnetic contribution of the Nd3+ 4f moments. In the absence of rare-earth effects, we find that the nickelates broadly violate the Pauli limit. Such robust and tunable superconductivity suggests potential in future high-field applications., Comment: 8 pages, 4 figures, 1 supplementary materials

For any compact connected Lie group $G$, we study the Hamiltonian sum of two compact Hamiltonian group $G$-manifolds $(X^+,\omega^+,\mu^+)$ and $(X^-,\omega^-,\mu^-)$ with a common codimension 2 Hamiltonian submanifold $Z$ of the opposite equivariant Euler classes of the normal bundles. We establish that the symplectic reduction of the Hamiltonian sum agrees with the symplectic sum of the reduced symplectic manifolds. We also compare the equivariant first Chern class of the Hamiltonian sum with the equivariant first Chern classes of $X^\pm$., Comment: 25 pages

The occurrence of superconductivity in proximity to various strongly correlated phases of matter has drawn extensive focus on their normal state properties, to develop an understanding of the state from which superconductivity emerges. The recent finding of superconductivity in layered nickelates raises similar interests. However, transport measurements of doped infinite-layer nickelate thin films have been hampered by materials limitations of these metastable compounds - in particular, a relatively high density of extended defects. Here, by moving to a substrate (LaAlO$_{3}$)$_{0.3}$(Sr$_{2}$TaAlO$_{6}$)$_{0.7}$ which better stabilizes the growth and reduction conditions, we can synthesize the doping series of Nd$_{1-x}$Sr$_{x}$NiO$_{2}$ essentially free from extended defects. This enables the first examination of the 'intrinsic' temperature and doping dependent evolution of the transport properties. The normal state resistivity exhibits a low-temperature upturn in the underdoped regime, linear behavior near optimal doping, and quadratic temperature dependence for overdoping. This is strikingly similar to the copper oxides, despite key distinctions - namely the absence of an insulating parent compound, multiband electronic structure, and a Mott-Hubbard orbital alignment rather than the charge-transfer insulator of the copper oxides. These results suggest an underlying universality in the emergent electronic properties of both superconducting families.

Infinite-layer nickelates present a new family of potential unconventional superconductors. A key open question is the superconducting pairing symmetry. We present low-temperature measurements of the London penetration depth in optimally doped La_{0.8}Sr_{0.2}NiO_{2}, Pr_{0.8}Sr_{0.2}NiO_{2}, and Nd_{0.8}Sr_{0.2}NiO_{2}. For La and Pr-nickelates, the superfluid density shows a quadratic temperature dependence, indicating nodal superconductivity in the presence of disorder. Nd-nickelate exhibits complex low-temperature behavior, which we attribute to magnetic impurities. These results are consistent with d-wave pairing., Comment: 5 pages, 4 figures, 1 table, 1 supplementary material

The discovery of superconductivity in Nd$_{0.8}$Sr$_{0.2}$NiO$_2$ [1] introduced a new family of layered nickelate superconductors that has now been extended to include a range of Sr-doping [2, 3], Pr or La in place of Nd [4-6], and the 5-layer Nd$_6$Ni$_5$O$_{12}$ [7]. A number of studies indicate that electron correlations are strong in these materials [8-14], and hence a central question is whether or not magnetism is present as a consequence of these interactions. Here we report muon spin rotation/relaxation studies of a series of superconducting infinite-layer nickelates. In all cases we observe an intrinsic magnetic ground state, regardless of the rare earth ion or doping, arising from local moments on the nickel sublattice. The coexistence of magnetism - which is likely to be antiferromagnetic and short-range ordered - with superconductivity is reminiscent of some iron pnictides [15] and heavy fermion compounds [16], and qualitatively distinct from the doped cuprates [17].

Most existing Graph Neural Networks (GNNs) are proposed without considering the selection bias in data, i.e., the inconsistent distribution between the training set with test set. In reality, the test data is not even available during the training process, making selection bias agnostic. Training GNNs with biased selected nodes leads to significant parameter estimation bias and greatly impacts the generalization ability on test nodes. In this paper, we first present an experimental investigation, which clearly shows that the selection bias drastically hinders the generalization ability of GNNs, and theoretically prove that the selection bias will cause the biased estimation on GNN parameters. Then to remove the bias in GNN estimation, we propose a novel Debiased Graph Neural Networks (DGNN) with a differentiated decorrelation regularizer. The differentiated decorrelation regularizer estimates a sample weight for each labeled node such that the spurious correlation of learned embeddings could be eliminated. We analyze the regularizer in causal view and it motivates us to differentiate the weights of the variables based on their contribution on the confounding bias. Then, these sample weights are used for reweighting GNNs to eliminate the estimation bias, thus help to improve the stability of prediction on unknown test nodes. Comprehensive experiments are conducted on several challenging graph datasets with two kinds of label selection biases. The results well verify that our proposed model outperforms the state-of-the-art methods and DGNN is a flexible framework to enhance existing GNNs., Comment: Accepted by TNNLS;12 pages

Nickel-based superconductors provide a long-awaited experimental platform to explore possible cuprate-like superconductivity. Despite similar crystal structure and $d$ electron filling, these systems exhibit several differences. Nickelates are the most polar layered oxide superconductor, raising questions about the interface between substrate and thin film -- thus far the only sample geometry to successfully stabilize superconductivity. We conduct a detailed experimental and theoretical study of the prototypical interface between Nd$_{1-x}$Sr$_x$NiO$_2$ and SrTiO$_3$. Atomic-resolution electron energy loss spectroscopy in the scanning transmission electron microscope reveals the formation of a single intermediate Nd(Ti,Ni)O$_3$ layer. Density functional theory calculations with a Hubbard $U$ term show how the observed structure alleviates the strong polar discontinuity. We explore effects of oxygen occupancy, hole doping, and cation structure to disentangle the contributions of each for reducing interface charge density. Resolving the nontrivial interface structure will be instructive for future synthesis of nickelate films on other substrates and in vertical heterostructures.

A defining signature of strongly correlated electronic systems is the existence of competing phases with similar ground state energies, resulting in a rich phase diagram. While in the recently discovered nickelate superconductors, a high antiferromagnetic exchange energy has been reported, which implies the existence of strong electronic correlations, signatures of competing phases have not yet been observed. Here, we uncover a charge order (CO) in infinite-layer nickelates La1-xSrxNiO2 using resonant x-ray scattering across the Ni L-edge. In the parent compound, the CO arranges along the Ni-O bond direction with an incommensurate wave vector (0.344+/-0.002, 0) r.l.u., distinct from the stripe order in other nickelates which propagates along a direction 45 degree to the Ni-O bond. The CO resonance profile indicates that CO originates from the Ni 3d states and induces a parasitic charge modulation of La electrons. Upon doping, the CO diminishes and the ordering wave vector shifts toward a commensurate value of 1/3 r.l.u., indicating that the CO likely arises from strong correlation effects and not from Fermi surface nesting., Comment: 24 pages, 8 figures

Graph Neural Networks (GNNs) are proposed without considering the agnostic distribution shifts between training and testing graphs, inducing the degeneration of the generalization ability of GNNs on Out-Of-Distribution (OOD) settings. The fundamental reason for such degeneration is that most GNNs are developed based on the I.I.D hypothesis. In such a setting, GNNs tend to exploit subtle statistical correlations existing in the training set for predictions, even though it is a spurious correlation. However, such spurious correlations may change in testing environments, leading to the failure of GNNs. Therefore, eliminating the impact of spurious correlations is crucial for stable GNNs. To this end, we propose a general causal representation framework, called StableGNN. The main idea is to extract high-level representations from graph data first and resort to the distinguishing ability of causal inference to help the model get rid of spurious correlations. Particularly, we exploit a graph pooling layer to extract subgraph-based representations as high-level representations. Furthermore, we propose a causal variable distinguishing regularizer to correct the biased training distribution. Hence, GNNs would concentrate more on the stable correlations. Extensive experiments on both synthetic and real-world OOD graph datasets well verify the effectiveness, flexibility and interpretability of the proposed framework., Comment: IEEE TPAMI 2023

The discovery of infinite-layer nickelate superconductors has spurred enormous interest. While the Ni$^{1+}$ cations possess nominally the same 3$d^9$ configuration as Cu$^{2+}$ in cuprates, the electronic structure variances remain elusive. Here, we present a soft x-ray photoemission spectroscopy study on parent and doped infinite-layer Pr-nickelate thin films with a doped perovskite reference. By identifying the Ni character with resonant photoemission and comparison to density functional theory + U (on-site Coulomb repulsion energy) calculations, we estimate U ~5 eV, smaller than the charge transfer energy $\Delta$ ~8 eV, confirming the Mott-Hubbard electronic structure in contrast to charge-transfer cuprates. Near the Fermi level ($E_F$), we observe a signature of occupied rare-earth states in the parent compound, which is consistent with a self-doping picture. Our results demonstrate a correlation between the superconducting transition temperature and the oxygen 2$p$ hybridization near $E_F$ when comparing hole-doped nickelates and cuprates., Comment: 28 pages, 10 figures

The observation of superconductivity in infinite layer nickelate (Nd,Sr)NiO$_{2}$ thin films has led to rapid theoretical and experimental investigations of these copper-oxide-analogue systems [1-15]. Superconductivity has also been found in (Pr,Sr)NiO$_{2}$ [16,17], but not previously in (La,Sr)NiO$_{2}$ [2], raising a fundamental question whether superconductivity is associated with the presence of rare-earth moments [18,19]. Here we show that with significant materials optimization, substantial portions of the La$_{1-x}$Sr$_{x}$NiO$_{2}$ phase diagram can enter the regime of coherent low-temperature transport ($x$ = 0.14 - 0.20), with subsequent superconducting transitions and a maximum onset of ~ 9 K at $x$ = 0.20. Additionally, we observe the unexpected indication of a superconducting ground state in undoped LaNiO$_{2}$, which likely reflects the self-doped nature of the electronic structure. Combining the results of (La/Pr/Nd)$_{1-x}$Sr$_{x}$NiO$_{2}$ reveals a generalized superconducting dome, characterized by systematic shifts in the unit cell volume and in the relative electron-hole populations across the lanthanides., Comment: 23 pages, 4 figures

We report a systematic magnetotransport study of superconducting infinite-layer nickelate thin films Nd$_{1-x}$Sr$_x$NiO$_2$ with $0.15 \leq x \leq 0.225$. By suppressing superconductivity with out-of-plane magnetic fields up to 37.5 T, we find that the normal state resistivity of Nd$_{1-x}$Sr$_x$NiO$_2$ is characterized by a crossover from a metallic $T^2$-behavior to an insulating log(1/$T$)-behavior for all $x$ except $x = 0.225$, at which the resistivity is predominantly metallic. The log(1/$T$)-behavior is found to be robust against magnetic fields, inconsistent with scenarios involving localization or Kondo scattering, and points to an anomalous insulating state possibly driven by strong correlations. In the metallic state, we find no evidence for non-Fermi-liquid behavior arising from proximity to a putative quantum critical point located inside the superconducting dome.

We give a complete solution, for discrete countable groups, to the problem of defining and computing a geometric pairing between the left hand side of the Baum-Connes assembly map, given in terms of geometric cycles associated to proper actions on manifolds, and cyclic periodic cohomology of the group algebra. Indeed, for any such group $\Gamma$ (without any further assumptions on it) we construct an explicit morphism from the Left-Hand side of the Baum-Connes assembly map to the periodic cyclic homology of the group algebra. This morphism, called here the Chern-Baum-Connes assembly map, allows to give a proper and explicit formulation for a Chern-Connes pairing with the periodic cyclic cohomology of the group algebra. Several theorems are needed to formulate the Chern-Baum-Connes assembly map. In particular we establish a delocalised Riemann-Roch theorem, the wrong way functoriality for periodic delocalised cohomology for $\Gamma$-proper actions, the construction of a Chern morphism between the Left-Hand side of Baum-Connes and a delocalised cohomology group associated to $\Gamma$ which is an isomorphism once tensoring with $\mathbb{C}$, and the construction of an explicit cohomological assembly map between the delocalised cohomology group associated to $\Gamma$ and the homology group $H_*(\Gamma,F\Gamma)$. We then give an index theoretical formula for the above mentioned pairing (for any $\Gamma$) in terms of pairings of invariant forms, associated to geometric cycles and given in terms of delocalized Chern and Todd classes, and currents naturally associated to group cocycles using Burghelea's computation. As part of our results we prove that left-Hand side group used in this paper is isomorphic to the usual analytic model for the left-hand side of the assembly map., Comment: This article replaces a previous version whose title was the same plus part I. Originally we planned to work out a second part but we finally extended and completed the first part in a single paper. We thank the colleague that encouraged us to complete part I into a single piece and gave us invaluable references that were key to the present work

The recent observation of superconductivity in thin film infinite-layer nickelates$^{1-3}$ offers a different angle to investigate superconductivity in layered oxides$^{4}$. A wide range of candidate models have been proposed$^{5-10}$, emphasizing single- or multi-orbital electronic structure, Kondo or Hund's coupling, and analogies to cuprates. Clearly, further experimental characterization of the superconducting state is needed to develop a full understanding of the nickelates. Here we use magnetotransport measurements to probe the superconducting anisotropy in Nd$_{0.775}$Sr$_{0.225}$NiO$_{2}$. We find that the upper critical field is surprisingly isotropic at low temperatures despite the layered crystal structure. In a magnetic field the superconductivity is strongly Pauli-limited, such that the paramagnetic effect dominates over orbital de-pairing. Underlying this isotropic response is a substantial anisotropy in the superconducting coherence length, which is at least four times longer in-plane than out-of-plane. A prominent low-temperature upturn in the upper critical field indicates the presence of an unconventional ground state., Comment: 9 pages, 4 figures, 1 supplementary info

We report the phase diagram of infinite layer Pr$_{1-x}$Sr$_{x}$NiO$_2$ thin films synthesized via topotactic reduction from the perovskite precursor phase using CaH$_2$. Based on the electrical transport properties, we find a doping-dependent superconducting dome extending between $x$ = 0.12 and 0.28, with a maximum superconducting transition temperature $T_{\rm{c}}$ of 14 K at $x$ = 0.18, bounded by weakly insulating behavior on both sides. In contrast to the narrower dome observed in Nd$_{1-x}$Sr$_{x}$NiO$_2$, a local $T_{\rm{c}}$ suppression near $x$ = 0.2 was not observed for the Pr$_{1-x}$Sr$_{x}$NiO$_2$ system. Normal state Hall effect measurements indicate mixed carrier contributions of both electrons and holes, and show a sign change in the Hall coefficient as functions of temperature and $x$, quite similar to that in Nd$_{1-x}$Sr$_{x}$NiO$_2$. Also similar is the observation of a minimum in the normal state resistivity associated with the superconducting compositions. These findings indicate an infinite layer nickelate phase diagram that is relatively insensitive to the rare-earth element, but suggest that disorder arising from the variations of the ionic radii on the rare-earth site affects the superconducting dome., Comment: 17 pages, 3 figures

Anomalous metallic behavior, marked by a saturating finite resistivity much lower than the Drude estimate, has been observed in a wide range of two-dimensional superconductors. Utilizing the electrostatically gated LaAlO3/SrTiO3 interface as a versatile platform for superconductor-metal quantum phase transitions, we probe variations in the gate, magnetic field, and temperature to construct a phase diagram crossing from superconductor, anomalous metal, vortex liquid, to Drude metal states, combining longitudinal and Hall resistivity measurements. We find that the anomalous metal phases induced by gating and magnetic field, although differing in symmetry, are connected in the phase diagram and exhibit similar magnetic field response approaching zero temperature. Namely, within a finite regime of the anomalous metal state, the longitudinal resistivity linearly depends on field while the Hall resistivity diminishes, indicating an emergent particle-hole symmetry. The universal behavior highlights the uniqueness of the quantum bosonic metallic state, distinct from bosonic insulators and vortex liquids.

Most of existing clustering algorithms are proposed without considering the selection bias in data. In many real applications, however, one cannot guarantee the data is unbiased. Selection bias might bring the unexpected correlation between features and ignoring those unexpected correlations will hurt the performance of clustering algorithms. Therefore, how to remove those unexpected correlations induced by selection bias is extremely important yet largely unexplored for clustering. In this paper, we propose a novel Decorrelation regularized K-Means algorithm (DCKM) for clustering with data selection bias. Specifically, the decorrelation regularizer aims to learn the global sample weights which are capable of balancing the sample distribution, so as to remove unexpected correlations among features. Meanwhile, the learned weights are combined with k-means, which makes the reweighted k-means cluster on the inherent data distribution without unexpected correlation influence. Moreover, we derive the updating rules to effectively infer the parameters in DCKM. Extensive experiments results on real world datasets well demonstrate that our DCKM algorithm achieves significant performance gains, indicating the necessity of removing unexpected feature correlations induced by selection bias when clustering., Comment: Accepted by main track of IJCAI2020;SOLE copyright holder is IJCAI (international Joint Conferences on Artificial Intelligence), all rights reserved

A variety of nickel oxide compounds have long been studied for their manifestation of various correlated electron phenomena. Recently, superconductivity was observed in nanoscale infinite layer nickelate thin films of Nd$_{0.8}$Sr$_{0.2}$NiO$_2$, epitaxially stabilized on SrTiO$_3$ substrates via topotactic reduction from the perovskite precursor phase. Here we present the synthesis and properties of PrNiO$_2$ thin films on SrTiO$_3$. Upon doping in Pr$_{0.8}$Sr$_{0.2}$NiO$_2$, we observe superconductivity with a transition temperature of 7-12 K, and robust critical current density at 2 K of 334 kA/cm$^2$. These findings indicate that superconductivity in the infinite layer nickelates is relatively insensitive to the details of the rare earth 4$f$ configuration. Furthermore, they motivate the exploration of a broader family of compounds based on two-dimensional NiO$_2$ planes, which will enable systematic investigation of the superconducting and normal state properties and their underlying mechanisms., Comment: 19 pages, 4 figures

Consider a proper, isometric action by a unimodular locally compact group $G$ on a Riemannian manifold $M$ with boundary, such that $M/G$ is compact. Then an equivariant Dirac-type operator $D$ on $M$ under a suitable boundary condition has an equivariant index $\operatorname{index}_G(D)$ in the $K$-theory of the reduced group $C^*$-algebra $C^*_rG$ of $G$. This is a common generalisation of the Baum-Connes analytic assembly map and the (equivariant) Atiyah-Patodi-Singer index. In part I of this series, a numerical index $\operatorname{index}_g(D)$ was defined for an element $g \in G$, in terms of a parametrix of $D$ and a trace associated to $g$. An Atiyah-Patodi-Singer type index formula was obtained for this index. In this paper, we show that, under certain conditions, $\tau_g(\operatorname{index}_G(D)) = \operatorname{index}_g(D)$, for a trace $\tau_g$ defined by the orbital integral over the conjugacy class of $g$. This implies that the index theorem from part I yields information about the $K$-theoretic index $\operatorname{index}_G(D)$. It also shows that $\operatorname{index}_g(D)$ is a homotopy-invariant quantity., Comment: 44 pages. The first version of the preprint 1904.11146 was split into two parts, this is the second part

Advances in complex oxide heteroepitaxy have highlighted the enormous potential of utilizing strain engineering via lattice mismatch to control ferroelectricity in thin-film heterostructures. This approach, however, lacks the ability to produce large and continuously variable strain states, thus limiting the potential for designing and tuning the desired properties of ferroelectric films. Here, we observe and explore dynamic strain-induced ferroelectricity in SrTiO$_3$ by laminating freestanding oxide films onto a stretchable polymer substrate. Using a combination of scanning probe microscopy, optical second harmonic generation measurements, and atomistic modeling, we demonstrate robust room-temperature ferroelectricity in SrTiO$_3$ with 2.0% uniaxial tensile strain, corroborated by the notable features of 180{\deg} ferroelectric domains and an extrapolated transition temperature of 400 K. Our work reveals the enormous potential of employing oxide membranes to create and enhance ferroelectricity in environmentally benign lead-free oxides, which hold great promise for applications ranging from non-volatile memories and microwave electronics.

The recent observation of superconductivity in Nd$_{0.8}$Sr$_{0.2}$NiO$_2$ has raised fundamental questions about the hierarchy of the underlying electronic structure. Calculations suggest that this system falls in the Mott-Hubbard regime, rather than the charge-transfer configuration of other nickel oxides and the superconducting cuprates. Here, we use state-of-the-art, locally-resolved electron energy loss spectroscopy to directly probe the Mott-Hubbard character of Nd$_{1-x}$Sr$_x$NiO$_2$. Upon doping, we observe emergent hybridization reminiscent of the Zhang-Rice singlet via the oxygen-projected states, modification of the Nd 5$d$ states, and the systematic evolution of Ni 3$d$ hybridization and filling. These results clearly evidence the multiband nature of this system and the distinct electronic landscape for infinite-layer nickelates despite their formal similarity to the cuprates.

We report the phase diagram of Nd$_{1-x}$Sr$_x$NiO$_2$ infinite layer thin films grown on SrTiO$_3$. A superconducting dome spanning $0.125 < x < 0.25$ is found, remarkably similar to cuprates, albeit over a narrower doping window. However, while cuprate superconductivity is bounded by an insulator for underdoping and a metal for overdoping, here we observe weakly insulating behavior on either side of the dome. Furthermore, the normal state Hall coefficient is always small and proximate to a continuous zero crossing in doping and in temperature, in contrast to the $\sim 1/x$ dependence observed for cuprates. This suggests the presence of both electron- and hole-like bands, consistent with band structure calculations., Comment: 18 pages (including Supplemental Material), 7 figures (including 4 supplemental figure)

The recent observation of superconductivity in Nd$_{0.8}$Sr$_{0.2}$NiO$_{2}$ calls for further investigation and optimization of the synthesis of this metastable infinite-layer nickelate structure. Here, we present our current understanding of important aspects of the growth of the parent perovskite compound via pulsed laser deposition on SrTiO$_{3}$ (001) substrates, and the subsequent topotactic reduction. We find that to achieve single-crystalline, single-phase superconducting Nd$_{0.8}$Sr$_{0.2}$NiO$_{2}$, it is essential that the precursor perovskite Nd$_{0.8}$Sr$_{0.2}$NiO$_{3}$ thin film is stabilized with high crystallinity and no impurity phases; in particular, a Ruddlesden-Popper-type secondary phase is often observed. We have further investigated the evolution of the soft-chemistry topotactic reduction conditions to realize full transformation to the infinite-layer structure with no film decomposition or formation of other phases. We find that capping the nickelate film with a subsequent SrTiO$_{3}$ layer provides an epitaxial template to the top region of the nickelate film, much like the substrate. Thus, for currently optimized growth conditions, we can stabilize superconducting single-phase Nd$_{0.8}$Sr$_{0.2}$NiO$_{2}$ (001) epitaxial thin films up to ~ 10 nm.