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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.

The iron-based high-$T_c$ superconductors exhibit rich phase diagrams with intertwined phases, including magnetism, nematicity and superconductivity. The superconducting $T_c$ in many of these materials is maximized in the regime of strong nematic fluctuations, making the role of nematicity in influencing the superconductivity a topic of intense research. Here, we use the AC elastocaloric effect (ECE) to map out the phase diagram of Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ near optimal doping. The ECE signature at $T_c$ on the overdoped side, where superconductivity condenses without any nematic order, is quantitatively consistent with other thermodynamic probes that indicate a single-component superconducting state. In contrast, on the slightly underdoped side, where superconductivity condenses within the nematic phase, ECE reveals a second thermodynamic transition proximate to and below $T_c$. We rule out magnetism and re-entrant tetragonality as the origin of this transition, and find that our observations strongly suggest a phase transition into a multicomponent superconducting state. This implies the existence of a sub-dominant pairing instability that competes strongly with the dominant $s^\pm$ instability. Our results thus motivate a re-examination of the pairing state and its interplay with nematicity in this extensively studied iron-based superconductor, while also demonstrating the power of ECE in uncovering strain-tuned phase diagrams of quantum materials.

We measure the full elastic tensors of Mn$_3$Ge and Mn$_3$Sn as a function of temperature through their respective antiferromagnetic phase transitions. Large discontinuities in the bulk moduli at the N\'eel transitions indicate strong magnetoelastic coupling in both compounds. Strikingly, the discontinuities are nearly a factor of 10 larger in Mn$_3$Ge than in Mn$_3$Sn. We use the magnitudes of the discontinuities to calculate the pressure derivatives of the N\'eel temperature, which are 39 K/GPa 14.3 K/GPa for Mn$_3$Ge and Mn$_3$Sn, respectively. We measured in-plane shear modulus $c_{66}$, which couples strongly to the magnetic order, in magnetic fields up to 18 T and found quantitatively similar behavior in both compounds. Recent measurements have demonstrated strong piezomagnetism in Mn$_3$Sn: our results suggest that Mn$_3$Ge may be an even better candidate for this effect., Comment: 17 pages, 8 figures

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

Recent experiments suggest that the superconducting order parameter of Sr$_2$RuO$_4$ has two components. A two-component order parameter has multiple degrees of freedom in the superconducting state that can result in low-energy collective modes or the formation of domain walls -- a possibility that would explain a number of experimental observations including the smallness of the time reversal symmetry breaking signal at T$_\mathrm{c}$ and telegraph noise in critical current experiments. We perform ultrasound attenuation measurements across the superconducting transition of Sr$_2$RuO$_4$ using resonant ultrasound spectroscopy (RUS). We find that the attenuation for compressional sound increases by a factor of seven immediately below T$_\mathrm{c}$, in sharp contrast with what is found in both conventional ($s$-wave) and high-T$_\mathrm{c}$ ($d$-wave) superconductors. We find our observations to be most consistent with the presence of domain walls between different configurations of the superconducting state. The fact that we observe an increase in sound attenuation for compressional strains, and not for shear strains, suggests an inhomogeneous superconducting state formed of two distinct, accidentally-degenerate superconducting order parameters that are not related to each other by symmetry. Whatever the mechanism, a factor of seven increase in sound attenuation is a singular characteristic with which any potential theory of the superconductivity in Sr$_2$RuO$_4$ must be reconciled.

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$_4$.

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.

The unusual correlated state that emerges in URu$_2$Si$_2$ below T$_{HO}$ = 17.5 K is known as "hidden order" because even basic characteristics of the order parameter, such as its dimensionality (whether it has one component or two), are "hidden". We use resonant ultrasound spectroscopy to measure the symmetry-resolved elastic anomalies across T$_{HO}$. We observe no anomalies in the shear elastic moduli, providing strong thermodynamic evidence for a one-component order parameter. We develop a machine learning framework that reaches this conclusion directly from the raw data, even in a crystal that is too small for traditional resonant ultrasound. Our result rules out a broad class of theories of hidden order based on two-component order parameters, and constrains the nature of the fluctuations from which unconventional superconductivity emerges at lower temperature. Our machine learning framework is a powerful new tool for classifying the ubiquitous competing orders in correlated electron systems.

The ability to spatially modulate the electronic properties of solids has led to landmark discoveries in condensed matter physics as well as new electronic applications. Although crystals of strongly correlated metals exhibit a diverse set of electronic ground states, few approaches to spatially modulating their properties exist. Here we demonstrate spatial control over the superconducting state in mesoscale samples of the canonical heavy-fermion superconductor CeIrIn5. We use a focused ion beam (FIB) to pattern crystals on the microscale, which tailors the strain induced by differential thermal contraction into specific areas of the device. The resulting non-uniform strain fields induce complex patterns of superconductivity due to the strong dependence of the transition temperature on the strength and direction of strain. Electrical transport and magnetic imaging of devices with different geometry show that the obtained spatial modulation of superconductivity agrees with predictions based on finite element simulations. These results present a generic approach to manipulating electronic order on micrometer length scales in strongly correlated matter., Comment: 29 pages, 14 figures