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There is considerable evidence that the superconductivity of Sr2RuO4 has two components. Among this evidence is a jump in the shear elastic modulus $c_{66}$ at the critical temperature $T_c$, observed in ultrasound measurements. Such a jump is forbidden for homogeneous single-component order parameters, and implies that $T_c$ should develop as a cusp under the application of shear strain with $\langle 110 \rangle$ principal axes. This shear strain should split the onset temperatures of the two components, if they coexist, or select one component if they do not. Here, we report measurements of $T_c$ and the elastocaloric effect of Sr2RuO4 under uniaxial stress applied along the $[110]$ lattice direction. Within experimental resolution, we resolve neither a cusp in the stress dependence of $T_c$, nor any second transition in the elastocaloric effect data. We show that reconciling these null results with the observed jumps in $c_{66}$ requires extraordinarily fine tuning to a triple point of the Ginzburg-Landau parameter space. In addition, our results are inconsistent with homogeneous time reversal symmetry breaking at a temperature $T_2 \leq T_c$ as identified in muon spin relaxation experiments.

In the unconventional superconductor Sr$_2$RuO$_4$, uniaxial stress along the $[100]$ direction tunes the Fermi level through a Van Hove singularity (VHS) in the density of states, causing a strong enhancement of the superconducting critical temperature $T_\textrm{c}$. Here, we report measurements of the London penetration depth $\lambda$ as this tuning is performed. We find that the zero-temperature superfluid density, here defined as $\lambda(0)^{-2}$, increases by $\sim$15%, with a peak that coincides with the peak in $T_\textrm{c}$. We also find that the low temperature form of $\lambda(T)$ is quadratic over the entire strain range. Using scanning tunneling microscopy, we find that the gap increases from $\Delta_0 \approx 350~\mu$eV in unstressed Sr$_2$RuO$_4$ to $\Delta_0 \approx 600~\mu$eV in a sample strained to near the peak in $T_c$. With a nodal order parameter, an increase in the superconducting gap could bring about an increase in the superfluid density through reduced sensitivity to defects and through reduced non-local effects in the Meissner screening. Our data indicate that tuning to the VHS increases the gap throughout the Brillouin zone, and that non-local effects are likely more important than reduced scattering.

Uniaxial pressure provides an efficient approach to control the competition between charge density waves (CDWs) and superconductivity in underdoped YBa$_{2}$Cu$_{3}$O$_{\rm{y}}$. It can enhance the correlation volume of ubiquitous short-range 2D CDW correlations, and induces a long-range 3D CDW otherwise only accessible at large magnetic fields. Here, we use x-ray diffraction to study the strain and doping evolution of these CDWs. No signatures of discommensurations nor pair density waves are observed in the investigated strain-temperature parameter space, but direct evidence for a form of competition between 2D and 3D CDWs is uncovered. We show that the interplay between the 3D CDW, the 2D CDWs and superconductivity is qualitatively well described by including strain effects in simulations of a nonlinear sigma model of competing superconducting and CDW orders. From a broader perspective, our results underscore the potential of strain tuning as a powerful tool for probing and manipulating competing orders in quantum materials., Comment: I. Vinograd and S. M. Souliou contributed equally to this work

The interplay of electronic and structural degrees of freedom in solids is a topic of intense research. Experience and intuition suggest that structural changes drive conduction electron behavior, because the large number of valence electrons dominate the structural properties. As part of a seminal paper written over sixty years ago, Lifshitz discussed an alternative possibility: lattice softening driven by conduction electrons at topological Fermi surface transitions. The effect he predicted, however, was small, and has not been convincingly observed. Using measurements of the stress-strain relationship in the ultra-clean metal Sr$_{2}$RuO$_{4}$, we reveal a huge softening of the Young's modulus at a Lifshitz transition of a two-dimensional Fermi surface, and show that it is indeed entirely driven by the conduction electrons of the relevant energy band., Comment: 42 pages, 16 figures (including supplementary information)

More than two decades after the discovery of superconductivity in Sr$_2$RuO$_4$, it is still unclear whether the order parameter has a single component or two degenerate components. For any two-component scenario, application of uniaxial strain is expected to lift the degeneracy, generating two distinct phase transitions. The presence of a second (lower-temperature) transition may be observable by probes that are sensitive to changes in the London penetration depth, $\lambda$, as a function of temperature, $T$. Here, we use scanning SQUID microscopy combined with a uniaxial strain device to test for a second transition under strain. We only observe a single transition. Within the temperature range where a second transition has been suggested by $\mu$SR measurements, we further place a tight upper bound of less than 1% on the change in the zero temperature superfluid density $n_s\propto\lambda^{-2}(0)$ due to a second transition, suggesting that such a transition does not occur. These results constrain theories of the order parameter in Sr$_2$RuO$_4$.

Strontium ruthenate Sr$_2$RuO$_4$ is an unconventional superconductor whose pairing symmetry has not been fully clarified, despite more than two decades of intensive research. Recent NMR Knight shift experiments have rekindled the Sr$_2$RuO$_4$ pairing debate by giving strong evidence against all odd-parity pairing states, including chiral $p$-wave pairing that was for a long time the leading pairing candidate. Here, we exclude additional pairing states by analyzing recent elastocaloric measurements [YS. Li et al., Nature 607, 276--280 (2022)]. To be able to explain the elastocaloric experiment, we find that unconventional even-parity pairings must include either large $d_{x^2 - y^2}$-wave or large $\{d_{xz} \mid d_{yz}\}$-wave admixtures, where the latter possibility arises because of the body-centered point group symmetry. These $\{d_{xz} \mid d_{yz}\}$-wave admixtures take the form of distinctively body-centered-periodic harmonics that have horizontal line nodes. Hence $g_{xy(x^2-y^2)}$-wave and $d_{xy}$-wave pairings are excluded as possible dominant even pairing states., Comment: 22 pages, 8 figures

We study the response of spin and charge order in single crystals of La$_{1.875}$Ba$_{0.125}$CuO$_{4}$ to uniaxial stress, through $^{139}$La nuclear magnetic resonance (NMR) and $^{63}$Cu nuclear quadrupole resonance (NQR), respectively. In unstressed La$_{1.875}$Ba$_{0.125}$CuO$_{4}$, the low-temperature tetragonal structure onsets below $T_{\text{LTT}} = 57$K, while the charge order and the spin order transition temperatures are $T_\text{CO} = 54$K and $T_\text{SO} = 37$K, respectively. We find that uniaxial stress along the [110] lattice direction strongly suppresses $T_\text{CO}$ and $T_{\text{SO}}$, but has little effect on $T_\text{LTT}$. In other words, under stress along [110] a large splitting ($\approx 21$K) opens between $T_\text{CO}$ and $T_{\text{LTT}}$, showing that these transitions are not tightly linked. On the other hand, stress along [100] causes a slight suppression of $T_\text{LTT}$ but has essentially no effect on $T_\text{CO}$ and $T_{\text{SO}}$. Magnetic field $H$ along [110] stabilizes the spin order: the suppression of $T_\text{SO}$ under stress along [110] is slower under $H \parallel [110]$ than $H \parallel [001]$. We develop a Landau free energy model and interpret our findings as an interplay of symmetry breaking terms driven by the orientation of spins., Comment: 14 pages, 17 figures

We report the evolution of the electronic structure at the surface of the layered perovskite Sr$_2$RuO$_4$ under large in-plane uniaxial compression, leading to anisotropic $B_{1g}$ strains of ${\varepsilon_{xx}-\varepsilon_{yy}=-0.9\pm0.1\%}$. From angle-resolved photoemission, we show how this drives a sequence of Lifshitz transitions, reshaping the low-energy electronic structure and the rich spectrum of van Hove singularities that the surface layer of Sr$_2$RuO$_4$ hosts. From comparison to tight-binding modelling, we find that the strain is accommodated predominantly by bond-length changes rather than modifications of octahedral tilt and rotation angles. Our study sheds new light on the nature of structural distortions at oxide surfaces, and how targeted control of these can be used to tune density of states singularities to the Fermi level, in turn paving the way to the possible realisation of rich collective states at the Sr$_2$RuO$_4$ surface.

Under in-plane uniaxial stress, the largest Fermi surface sheet of the correlated metal Sr$_2$RuO$_4$ undergoes a Lifshitz transition from an electron-like to an open geometry. We investigate the effects of this transition on transport through measurement of the longitudinal resistivity $\rho_{xx}$ and the Hall coefficient $R_\text{H}$. At temperatures where scattering is dominated by electron-electron scattering, $R_\text{H}$ becomes more negative across the Lifshitz transition, opposite to expectations from the change in Fermi surface topology. We show that this change in $R_\text{H}$ is explainable only if scattering changes throughout the Brillouin zone, not just at the point in $k$-space where the Lifshitz transition occurs. In a model of orbital-dependent scattering, the electron-electron scattering rate on sections of Fermi surface with $xy$ orbital weight decreases dramatically. On the other hand, at temperatures where defect scattering dominates $\rho_{xx}$ and $R_\text{H}$ are essentially constant across the Lifshitz transition., Comment: 9 pages, 10 figures

Muon spin rotation/relaxation ($\mu$SR) and polar Kerr effect measurements provide evidence for a time-reversal symmetry breaking (TRSB) superconducting state in Sr$_2$RuO$_4$. However, the absence of a cusp in the superconducting transition temperature ($T_{\rm c}$) vs. stress and the absence of a resolvable specific heat anomaly at TRSB transition temperature ($T_{\rm TRSB}$) under uniaxial stress challenge a hypothesis of TRSB superconductivity. Recent $\mu$SR studies under pressure and with disorder indicate that the splitting between $T_{\rm c}$ and $T_{\rm TRSB}$ occurs only when the structural tetragonal symmetry is broken. To further test such behavior, we measured $T_\text{c}$ through susceptibility measurements, and $T_\text{TRSB}$ through $\mu$SR, under uniaxial stress applied along a $\langle 110 \rangle$ lattice direction. We have obtained preliminary evidence for suppression of $T_\text{TRSB}$ below $T_\text{c}$, at a rate much higher than the suppression rate of $T_\text{c}$., Comment: 9 pages, 4 figures and Appendix, accepted for publication in PRB

In-plane uniaxial pressure has been shown to strongly tune the superconducting state of Sr2RuO4 by approaching a Lifshitz transition and associated Van Hove singularity (VHS) in the density of states. At the VHS, $T_c$ and the in- and out-of-plane upper critical fields are all strongly enhanced, and the latter has changed its curvature as a function of temperature from convex to concave. However, due to strain inhomogeneity it has not been possible so far to determine how the upper critical fields change with strain. Here, we show the strain dependence of both upper critical fields, which was achieved due to an improved sample preparation. We find that the in-plane upper critical field is mostly linear in $T_c$. On the other hand, the out-of-plane upper critical field varies with a higher power in $T_c$, and peaks strongly at the VHS. The strong increase in magnitude and the change in form of $H_\mathrm{c2||c}$ occur very close to the Van Hove strain, and points to a strong enhancement of both the density of states and the gap magnitude at the Lifshitz transition.

The possibility of direct air capture of CO$_2$ in Antarctica is discussed. Because the concentration of H$_2$O in the atmosphere during the Antarctic winter is extremely low, an installation for direct air capture could employ a physisorption-based process, allowing, in principle, a very short adsorption/desorption cycle time. The lower required binding allows more options for materials for the sorbent. With a shorter cycle time, more resource could be spent on structuring the sorbent to improve energy efficiency, for example by improving its mass efficiency if desorption is driven thermally., Comment: This paper is built around a hypothesis that a one-second adsorption/desorption cycle time is possible. A justification for this hypothesis is not provided, and without it this paper should not be used as a basis for further work. I have been working to describe how such a cycle time might be achieved, and find that it is a difficult problem. This paper was posted prematurely

The Mn3X family of compounds was the first in which a large anomalous Hall effect (AHE) was predicted to arise from a purely antiferromagnetic structure, due to the Berry curvature in momentum space. Nearly simultaneously with this prediction, a large AHE was observed experimentally in one of the hexagonal members of this family, Mn3Sn. Aligning antiferromagnetic domains, a necessary step for observation of the AHE, is more challenging for the cubic members of the Mn3X family, due to a combination of a smaller spontaneous ferromagnetic moment and much stronger magnetic anisotropy. Here, we use a combination of uniaxial stress and applied magnetic field to align domains of bulk single-crystal Mn3Pt, and demonstrate for the first time a substantial AHE in a bulk sample of a cubic member of the Mn3X family. The AHE remains locked in with essentially no quantitative variation when the stress is ramped back to zero, which shows that it is not a consequence of any stress-induced ferromagnetic moment.

Conductive domain walls (CDWs) in insulating ferroelectrics have recently attracted considerable attention due to their unique topological, optical, and electronic properties, and offer potential applications such as in memory devices or re-writable circuitry. The electronic properties of domain walls (DWs) can be tuned by the application of strain, hence controlling the charge carrier density at DWs. In this work, we study the influence of uniaxial stress on the conductivity of DWs in the bulk single crystal lithium niobate (LiNbO$_3$). Using conductive atomic force microscopy (cAFM), we observe a large asymmetry in the conductivity of DWs, where only negatively screened walls, so called head-to-head DWs, are becoming increasingly conductive, while positively screened, tail-to-tails DWs, show a decrease in conductivity. This asymmetry of DW conductivity agrees with our theoretical model based on the piezoelectric effect. In addition, we observed that the current in the DW increases up to an order of magnitude for smaller compressive stresses of 100 MPa. This response of DWs remained intact for multiple stress cycles over 2 months, opening a path for future applications.

One of the main developments in unconventional superconductivity in the past two decades has been the discovery that most unconventional superconductors form phase diagrams that also contain other strongly correlated states. Many systems of interest are therefore close to more than one instability, and tuning between the resultant ordered phases is the subject of intense research1. In recent years, uniaxial pressure applied using piezoelectric-based devices has been shown to be a particularly versatile new method of tuning, leading to experiments that have advanced our understanding of the fascinating unconventional superconductor Sr$_2$RuO$_4$. Here we map out its phase diagram using high-precision measurements of the elastocaloric effect in what we believe to be the first such study including both the normal and the superconducting states. We observe a strong entropy quench on entering the superconducting state, in excellent agreement with a model calculation for pairing at the Van Hove point, and obtain a quantitative estimate of the entropy change associated with entry to a magnetic state that is observed in proximity to the superconductivity. The phase diagram is intriguing both for its similarity to those seen in other families of unconventional superconductors and for extra features unique, so far, to Sr$_2$RuO$_4$., Comment: 34 pages, 23 figures, published version, manuscript and supplementary information in one document

When Heisenberg spins interact antiferromagnetically on a triangular lattice and nearest-neighbor interactions dominate, the ground state is 120$^{\circ}$ antiferromagnetism. In this work, we probe the response of this state to lifting the triangular symmetry, through investigation of the triangular antiferromagnet PdCrO$_2$ under uniaxial stress by neutron diffraction and resistivity measurements. The periodicity of the magnetic order is found to change rapidly with applied stress; the rate of change indicates that the magnetic anisotropy is roughly forty times the stress-induced bond length anisotropy. At low stress, the incommensuration period becomes extremely long, on the order of 1000 lattice spacings; no locking of the magnetism to commensurate periodicity is detected. Separately, the magnetic structure is found to undergo a first-order transition at a compressive stress of $\sim$0.4 GPa, at which the interlayer ordering switches from a double- to a single-q structure., Comment: 9 pages, 5 figures, accepted by New Journal of Physics

Applying in-plane uniaxial pressure to strongly correlated low-dimensional systems has been shown to tune the electronic structure dramatically. For example, the unconventional superconductor Sr$_2$RuO$_4$ can be tuned through a single Van Hove singularity which results in a strong enhancement of both $T_\text{c}$ and $H_\text{c2}$. Out-of-plane ($c$ axis) uniaxial pressure is expected to tune the quasi-two-dimensional structure even more strongly, causing it to approach two Van Hove singularities simultaneously. Here we achieve a record value of $3.2$ GPa compression along the $c$ axis of Sr$_2$RuO$_4$. Although the rise in $H_\text{c2}$ shows that we are indeed approaching the van Hove points, $T_\text{c}$ is suppressed, a result that contradicts expectations based on simple two-dimensional models. As a first attempt to take the third dimension into account, we present three-dimensional calculations in the weak interaction limit, and discuss the extent to which they are consistent with observation. Our experimental results highlight the importance of out-of-plane effects in low-dimensional systems in general and provide new constraints on theories of the pairing interaction in Sr$_2$RuO$_4$.

We perform nuclear magnetic resonance (NMR) measurements of the oxygen-17 Knight shifts for Sr$_2$RuO$_4$, while subjected to uniaxial stress applied along [100] direction. The resulting strain is associated with a strong variation of the temperature and magnetic field dependence of the inferred magnetic response. A quasi-particle description based on density-functional theory calculations, supplemented by many-body renormalizations, is found to reproduce our experimental results, and highlights the key role of a van-Hove singularity. The Fermi liquid coherence scale is shown to be tunable by strain, and driven to low values as the associated Lifshitz transition is approached.

There is considerable evidence that the superconducting state of Sr$_2$RuO$_4$ breaks time reversal symmetry. In the experiments showing time reversal symmetry breaking its onset temperature, $T_\text{TRSB}$, is generally found to match the critical temperature, $T_\text{c}$, within resolution. In combination with evidence for even parity, this result has led to consideration of a $d_{xz} \pm id_{yz}$ order parameter. The degeneracy of the two components of this order parameter is protected by symmetry, yielding $T_\text{TRSB} = T_\text{c}$, but it has a hard-to-explain horizontal line node at $k_z=0$. Therefore, $s \pm id$ and $d \pm ig$ order parameters are also under consideration. These avoid the horizontal line node, but require tuning to obtain $T_\text{TRSB} \approx T_\text{c}$. To obtain evidence distinguishing these two possible scenarios (of symmetry-protected versus accidental degeneracy), we employ zero-field muon spin rotation/relaxation to study pure Sr$_2$RuO$_4$ under hydrostatic pressure, and Sr$_{1.98}$La$_{0.02}$RuO$_4$ at zero pressure. Both hydrostatic pressure and La substitution alter $T_\text{c}$ without lifting the tetragonal lattice symmetry, so if the degeneracy is symmetry-protected $T_\text{TRSB}$ should track changes in $T_\text{c}$, while if it is accidental, these transition temperatures should generally separate. We observe $T_\text{TRSB}$ to track $T_\text{c}$, supporting the hypothesis of $d_{xz} \pm id_{yz}$ order., Comment: 14 pages, 8 Figures

A key question regarding the unconventional superconductivity of Sr$_2$RuO$_4$ remains whether the order parameter is single- or two-component. Under a hypothesis of two-component superconductivity, uniaxial pressure is expected to lift their degeneracy, resulting in a split transition. The most direct and fundamental probe of a split transition is heat capacity. Here, we report measurement of heat capacity of samples subject to large and highly homogeneous uniaxial pressure. We place an upper limit on the heat-capacity signature of any second transition of a few per cent of that of the primary superconducting transition. The normalized jump in heat capacity, $\Delta C/C$, grows smoothly as a function of uniaxial pressure, favouring order parameters which are allowed to maximize in the same part of the Brillouin zone as the well-studied van Hove singularity. Thanks to the high precision of our measurements, these findings place stringent constraints on theories of the superconductivity of Sr$_2$RuO$_4$., Comment: This manuscript is an extended version of the scientific part of arXiv:1906.07597, which is being split into two separate papers. An extended discussion on the experimental methods can be found at arXiv:2009.03125. It includes supplemental material

The mechanism behind the nematicity of FeSe is not known. Through elastoresitivity measurements it has been shown to be an electronic instability. However, so far measurements have extended only to small strains, where the response is linear. Here, we apply large elastic strains to FeSe, and perform two types of measurements. (1) Using applied strain to control twinning, the nematic resistive anisotropy at temperatures below the nematic transition temperature Ts is determined. (2) Resistive anisotropy is measured as nematicity is induced through applied strain at fixed temperature above Ts. In both cases, as nematicity strengthens the resistive anisotropy peaks about about 7%, then decreases. Below ~40 K, the nematic resistive anisotropy changes sign. We discuss possible implications of this behaviour for theories of nematicity. We report in addition: (1) Under experimentally accessible conditions with bulk crystals, stress, rather than strain, is the conjugate field to the nematicity of FeSe. (2) At low temperatures the twin boundary resistance is ~10% of the sample resistance, and must be properly subtracted to extract intrinsic resistivities. (3) Biaxial inplane compression increases both in-plane resistivity and the superconducting critical temperature Tc, consistent with a strong role of the yz orbital in the electronic correlations., Comment: 18 pages, 13 figures. To be published in Phys. Rev. X

Hole-doped cuprate superconductors show a ubiquitous tendency towards charge order. Although onset of superconductivity is known to suppress charge order, there has not so far been a decisive demonstration of the reverse process, namely, the effect of charge order on superconductivity. To gain such information, we report here the dependence of the critical temperature $T_{\mathrm{c}}$ of YBa$_2$Cu$_3$O$_{6.67}$ on in-plane uniaxial stress up to 2 GPa. At a compression of about 1 GPa along the $a$ axis, 3D-correlated charge density wave (3D CDW) order appears. We find that $T_{\mathrm{c}}$ decreases steeply as the applied stress crosses 1 GPa, showing that the appearance of 3D CDW order strongly suppresses superconductivity. Through the elastocaloric effect we resolve the heat capacity anomaly at $T_{\mathrm{c}}$, and find that it does not change drastically as the 3D CDW onsets, which shows that the condensation energy of the 3D CDW is considerably less than that of the superconductivity.

We present a piezoelectric-driven uniaxial pressure cell that is optimized for muon spin relaxation and neutron scattering experiments, and that is operable over a wide temperature range including cryogenic temperatures. To accommodate the large samples required for these measurement techniques, the cell is designed to generate forces up to 1000 N, and to minimize the background signal the space around the sample is kept as open as possible. We demonstrate here that by mounting plate-like samples with epoxy, a uniaxial stress exceeding 1 GPa can be achieved in an active volume of 5 mm3. We show that for practical operation it is important to monitor both the force and displacement applied to the sample. Also, because time is critical during facility experiments, samples are mounted in detachable holders that can be rapidly exchanged. The piezoelectric actuators are likewise contained in an exchangeable cartridge., Comment: Review of Scientific Instruments accepted for publication Oct 2020, 9 pages, 8 figures

We present a method for measuring thermal expansion under tunable uniaxial stresses, and show measurements of the thermal expansion of Mn$_3$Sn, a room temperature antiferromagnet that exhibits a spontaneous Hall effect, under uniaxial stresses of up to 1.51 GPa compression. Measurement of thermal expansion provides thermodynamic data about the nature of phase transitions, and uniaxial stress provides a powerful tuning method that does not introduce disorder. Mn$_3$Sn exhibits an anomaly in its thermal expansion near $\sim$270 K, associated with a first-order change in its magnetic structure. We show this transition temperature is suppressed by 54.6 K by 1.51 GPa compression along [0001]. We find the associated entropy change at the transition to be $\sim$ 0.1 J mol$^{-1}$ K$^{-1}$ and to vary only weakly with applied stress.

Unambiguous identification of the superconducting order parameter symmetry of Sr$_2$RuO$_4$ has remained elusive for more than a quarter century. While a chiral $p$-wave ground state analogue to superfluid $^3$He-$A$ was ruled out only very recently, other proposed $p$-wave scenarios are still viable. Here, field-dependent $^{17}$O Knight shift measurements are compared to corresponding specific heat measurements, previously reported. We conclude that the shift results can be accounted for by the expected field-induced quasiparticle response only. An upper bound for the condensate magnetic response of $<10\%$ of the normal state susceptibility is sufficient to exclude odd-parity candidates., Comment: Main Article: 4 pages 3 figures Supplement: 6 pages 4 figures

Response to uniaxial stress has become a major probe of electronic materials. Tuneable uniaxial stress may be applied using piezoelectric actuators, and so far two methods have been developed to couple samples to actuators. In one, actuators apply force along the length of a free, beam-like sample, allowing very large strains to be achieved. In the other, samples are affixed directly to piezoelectric actuators, allowing study of mechanically delicate materials. Here, we describe an approach that merges the two: thin samples are affixed to a substrate, that is then pressurized uniaxially using piezoelectric actuators. Using this approach, we demonstrate application of large elastic strains to mechanically delicate samples: the van der Waals-bonded material FeSe, and a sample of CeAuSb$_2$ that was shaped with a focused ion beam., Comment: 11 pages, 9 figures

Sr$_2$RuO$_4$ has stood as the leading candidate for a spin-triplet superconductor for 26 years. Recent NMR experiments have cast doubt on this candidacy, however, and it is difficult to find a theory of superconductivity that is consistent with all experiments. What is needed are symmetry-based experiments that can rule out broad classes of possible superconducting order parameters. Here we use resonant ultrasound spectroscopy to measure the entire symmetry-resolved elastic tensor of Sr$_2$RuO$_4$ through the superconducting transition. We observe a thermodynamic discontinuity in the shear elastic modulus $c_{66}$, requiring that the superconducting order parameter is two-component. A two-component $p$-wave order parameter, such as $p_x+i p_y$, naturally satisfies this requirement. As this order parameter appears to be precluded by recent NMR experiments, we suggest that two other two-component order parameters, namely $\left\{d_{xz},d_{yz}\right\}$ or $\left\{d_{x^2-y^2},g_{xy(x^2-y^2)}\right\}$, are now the prime candidates for the order parameter of Sr$_2$RuO$_4$.

Among unconventional superconductors, Sr$_2$RuO$_4$ has become a benchmark for experimentation and theoretical analysis because its normal-state electronic structure is known with exceptional precision, and because of experimental evidence that its superconductivity has, very unusually, a spontaneous angular momentum, i.e. a chiral state. This hypothesis of chirality is however difficult to reconcile with recent evidence on the spin part of the order parameter. Measurements under uniaxial stress offer an ideal way to test for chirality, because under uniaxial stress the superconducting and chiral transitions are predicted to split, allowing the empirical signatures of each to be identified separately. Here, we report zerofield muon spin relaxation (ZF-$\mu$SR) measurements on crystals placed under uniaxial stresses of up to 1.05 GPa. We report a clear stress-induced splitting between the onset temperatures of superconductivity and time-reversal symmetry breaking, consistent with qualitative expectations for chiral superconductivity. We also report the appearance of unexpected bulk magnetic order under a uniaxial stress of ~ 1.0 GPa in clean Sr$_2$RuO$_4$., Comment: 28 pages, 12 figures

Uniaxial pressure applied along an Ru-Ru bond direction induces an elliptical distortion of the largest Fermi surface of Sr$_2$RuO$_4$, eventually causing a Fermi surface topological transition, also known as a Lifshitz transition, into an open Fermi surface. There are various anomalies in low-temperature properties associated with this transition, including maxima in the superconducting critical temperature and in resistivity. In the present paper, we report new measurements, employing new uniaxial stress apparatus and new measurements of the low-temperature elastic moduli, of the strain at which this Lifshitz transition occurs: a longitudinal strain $\varepsilon_{xx}$ of $(-0.44\pm0.06)\cdot10^{-2}$, which corresponds to a B$_{1g}$ strain $\varepsilon_{xx} - \varepsilon_{yy}$ of $(-0.66\pm0.09)\cdot10^{-2}$. This is considerably smaller than the strain corresponding to a Lifshitz transition in density functional theory calculations, even if the spin-orbit coupling is taken into account. Using dynamical mean-field theory we show that electronic correlations reduce the critical strain. It turns out that the orbital anisotropy of the local Coulomb interaction on the Ru site is furthermore important to bring this critical strain close to the experimental number, and thus well into the experimentally accessible range of strains.

In the triangular layered magnet PdCrO2 the intralayer magnetic interactions are strong, however the lattice structure frustrates interlayer interactions. In spite of this, long-range, 120$^\circ$ antiferromagnetic order condenses at $T_N = 38$~K. We show here through neutron scattering measurements under in-plane uniaxial stress and in-plane magnetic field that this occurs through a spontaneous lifting of the three-fold rotational symmetry of the nonmagnetic lattice, which relieves the interlayer frustration. We also show through resistivity measurements that uniaxial stress can suppress thermal magnetic disorder within the antiferromagnetic phase., Comment: 9 pages, 9 figures

We present a design for a piezoelectric-driven uniaxial stress cell suitable for use at ambient and cryogenic temperatures, and that incorporates both a displacement and a force sensor. The cell has a diameter of 46 mm and a height of 13 mm. It can apply a zero-load displacement of up to ~45 $\mu$m, and a zero-displacement force of up to ~245 N. With combined knowledge of the displacement and force applied to the sample, it can quickly be determined whether the sample and its mounts remain within their elastic limits. In tests on the oxide metal Sr$_2$RuO$_4$, we found that at room temperature serious plastic deformation of the sample onset at a uniaxial stress of ~0.2 GPa, while at 5 K the sample deformation remained elastic up to almost 2 GPa. This result highlights the usefulness of in situ tuning, in which the force can be applied after cooling samples to cryogenic temperatures.

Strontium ruthenate (Sr$_2$RuO$_4$) is a multiband superconductor that displays evidence of topological superconductivity, although a model of the order parameter that is consistent with all experiments remains elusive. We integrated a piezoelectric-based strain apparatus with a scanning superconducting quantum interference device (SQUID) microscope to map the diamagnetic response of single-crystal Sr$_2$RuO$_4$ as a function of temperature, uniaxial pressure, and position with micron-scale spatial resolution. We thereby obtained local measurements of the superconducting transition temperature $T_c$ vs. anisotropic strain $\epsilon$ with sufficient sensitivity for comparison to theoretical models that assume a uniform $p_x\pm ip_y$ order parameter. We found that $T_c$ varies with position and that the locally measured $T_c$ vs. $\epsilon$ curves are quadratic ($T_c\propto\epsilon^2$), as allowed by the C$_4$ symmetry of the crystal lattice. We did not observe the low-strain linear cusp ($T_c\propto \left| \epsilon \right|$) that would be expected for a two-component order parameter such as $p_x\pm ip_y$. These results provide new input for models of the order parameter of Sr$_2$RuO$_4$., Comment: 18 pages, 6 figures, 2 tables

We present results of measurements of resistivity of \CAS{} under the combination of $c$-axis magnetic field and in-plane uniaxial stress. In unstressed \CAS{} there are two magnetic phases. The low-field A phase is a single-component spin-density wave (SDW), with $\mathbf{q} = (\eta, \pm \eta, 1/2)$, and the high-field B phase consists of microscopically coexisting $(\eta, \eta, 1/2)$ and $(\eta, -\eta, 1/2)$ spin-density waves. Pressure along a $\langle 100 \rangle$ lattice direction is a transverse field to both of these phases, and so initially has little effect, however eventually induces new low- and high-field phases in which the principal axes of the SDW components appear to have rotated to the $\langle 100 \rangle$ directions. Under this strong $\langle 100 \rangle$ compression, the field evolution of the resistivity is much smoother than at zero strain: In zero strain, there is a strong first-order transition, while under strong $\langle 100 \rangle$ it becomes much broader. We hypothesize that this is a consequence of the uniaxial stress lifting the degeneracy between the (100) and (010) directions., Comment: 8 pages, 7 figures

We report the results of a combined study of the normal state resistivity and superconducting transition temperature $T_c$ of the unconventional superconductor Sr$_2$RuO$_4$ under uniaxial pressure. There is strong evidence that as well as driving $T_c$ through a maximum at $\sim$3.5 K, compressive strains $\varepsilon$ of nearly 1 % along the crystallographic [100] axis drive the $\gamma$ Fermi surface sheet through a Van Hove singularity, changing the temperature dependence of the resistivity from $T^2$ above and below the transition region to $T^{1.5}$ within it. This occurs in extremely pure single crystals in which the impurity contribution to the resistivity is $<$100 n$\Omega$cm, so our study also highlights the potential of uniaxial pressure as a more general probe of this class of physics in clean systems.

We study the response of the antiferromagnetism of CeAuSb$_2$ to orthorhombic lattice distortion applied through in-plane uniaxial pressure. The response to pressure applied along a $\langle 110 \rangle$ lattice direction shows a first-order transition at zero pressure, which shows that the magnetic order lifts the $(110)/(1\bar{1}0)$ symmetry of the unstressed lattice. Sufficient $\langle 100 \rangle$ pressure appears to rotate the principal axes of the order from $\langle 110 \rangle$ to $\langle 100 \rangle$. At low $\langle 100 \rangle$ pressure, the transition at $T_N$ is weakly first-order, however it becomes continuous above a threshold $\langle 100 \rangle$ pressure. We discuss the possibility that this behavior is driven by order parameter fluctuations, with the restoration of a continuous transition a result of reducing the point-group symmetry of the lattice., Comment: 6 pages, 7 figures

In this short review, we aim to provide a topical update on the status of efforts to understand the superconductivity of Sr2RuO4. We concentrate on the quest to identify a superconducting order parameter symmetry that is compatible with all the major pieces of experimental knowledge of the material, and highlight some major discrepancies that have become even clearer in recent years. As the pun in the title suggests, we have tried to start the discussion from scratch, making no assumptions even about fundamental issues such as the parity of the superconducting state. We conclude that no consensus is currently achievable in Sr2RuO4, and that the reasons for this go to the heart of how well some of the key probes of unconventional superconductivity are really understood. This is therefore a puzzle that merits continued in-depth study.

We report a combined experimental and theoretical study of the dependence of the superconductivity of the unconventional superconductor Sr$_2$RuO$_4$ on anisotropic strain. Novel piezoelectric apparatus is used to apply uniaxial pressures of up to $\sim$1~GPa along a $\langle 100 \rangle$ direction ($a$-axis) of the crystal lattice. $T_c$ increases from 1.5~K in unstrained material to 3.4~K at compression by $\approx$0.6\%, then falls steeply. The $c$-axis upper critical field for the strained $T_c$ = 3.4~K material is a factor of twenty larger than that of the unstrained crystal, whereas the in-plane ($a$-axis) critical field increases by only a factor of three. First-principles electronic structure calculations give evidence that the observed maximum $T_c$ occurs at or near a Lifshitz transition when the Fermi level passes through a van Hove singularity. Finally, we perform order parameter analyses using three-band renormalization group calculations. These, combined with the unexpectedly low in-plane critical field, open the possibility that the highly strained $T_c$=3.4~K Sr$_2$RuO$_4$ has an even- rather than an odd-parity order parameter. Potential implications such as a transition at nonzero strain between odd- and even-parity order parameters are discussed.

We report a field-temperature phase diagram and an entropy map for the heavy fermion compound CeAuSb$_2$. CeAuSb$_2$ orders antiferromagnetically below $T_N=6.6$~K, and has two metamagnetic transitions, at 2.8 and 5.6~T. The locations of the critical endpoints of the metamagnetic transitions, which may play a strong role in the putative quantum criticality of CeAuSb$_2$ and related compounds, are identified. The entropy map reveals an apparent entropy balance with Fermi liquid behavior, implying that above the N\'{e}el transition the Ce moments are incorporated into the Fermi liquid. High-field data showing that the magnetic behavior is remarkably anisotropic are also reported., Comment: 7 pages, 9 figures

Symmetry is a basic paradigm for analysis of problems in solid state physics, and to controllably break point-group symmetries is a powerful probe of novel forms of order. In this work, we apply two methods, controlled lattice strain and vector magnetic field, to study a well-known correlated electron phase that appears at low temperatures in the layered material Sr3Ru2O7. This phase responds strongly to C4-symmetry-breaking fields, suggesting spontaneous C4 symmetry breaking. However, in this work we find that the data are more easily explained with microscopic coexistence of two orthogonal order parameter components than spontaneous symmetry breaking. The techniques presented in this work will be relevant for other systems where spontaneous C4 symmetry breaking is suspected., Comment: 30 pages, 9 figures

We report quantum oscillation data on the metallic triangular antiferromagnet PdCrO$_2$. We find that, to very high accuracy, the observed frequencies of PdCrO$_2$ can be reproduced by reconstruction of the (nonmagnetic) PdCoO$_2$ Fermi surface into a reduced zone. The reduced zone corresponds to a magnetic cell containing six chromium sites, giving a $\sqrt{3} \times \sqrt{3}$ in-plane reconstruction, and $\times 2$ interplane reconstruction. The interplane ordering represents a reduction in lattice symmetry, possibly to monoclinic, and an associated lattice distortion is expected. In addition, we report a magnetic transition under an applied in-plane field that is probably equivalent to the spin-flop transition reported for CuCrO$_2$, and present data on its field-angle dependence. We also report measurements of the resistivity of PdCrO$_2$ up to 500 K., Comment: 9 pages, 10 figures

We report the design and construction of piezoelectric-based apparatus for applying continuously tuneable compressive and tensile strains to test samples. It can be used across a wide temperature range, including cryogenic temperatures. The achievable strain is large, so far up to 0.23% at cryogenic temperatures. The apparatus is compact and compatible with a wide variety of experimental probes. In addition, we present a method for mounting high-aspect-ratio samples in order to achieve high strain homogeneity., Comment: 8 pages, 8 figures

We present de Haas-van Alphen and resistivity data on single crystals of the delafossite PdCoO2. At 295 K we measure an in-plane resistivity of 2.6 \mu{\Omega}-cm, making PdCoO2 the most conductive oxide known. The low-temperature in-plane resistivity has an activated rather than the usual T^5 temperature dependence, suggesting a gapping of effective scattering that is consistent with phonon drag. Below 10 K, the transport mean free path is 20 \mum, approximately 10^5 lattice spacings and an astoundingly high value for flux-grown crystals. We discuss the origin of these properties in light of our data., Comment: accepted for publication in Phys. Rev. Lett

We measure the penetration depth $\lambda_{ab}(T)$ in Ba(Fe$_{1-x}$Co$_{x}$)$_2$As$_2$ using local techniques that do not average over the sample. The superfluid density $\rho_s(T)\equiv1/\lambda_{ab}(T)^2$ has three main features. First, $\rho_s(T=0)$ falls sharply on the underdoped side of the dome. Second, $\lambda_{ab}(T)$ is flat at low $T$ at optimal doping, indicating fully gapped superconductivity, but varies more strongly in underdoped and overdoped samples, consistent with either a power law or a small second gap. Third, $\rho_s(T)$ varies steeply near $T_c$ for optimal and underdoping. These observations are consistent with an interplay between magnetic and superconducting phases.