Kagome metals, particularly those in the AV₃Sb₅ family (A = K, Rb, Cs), are attracting significant interest due to their unique structural motif and the interplay of complex phenomena like charge order (CO) and superconductivity (SC). CsV₃Sb₅, in particular, presents an unusual double superconducting dome within the temperature-pressure phase diagram, a region where charge order remains present. This coexistence of SC and CO, and the resulting double-dome structure, contrasts with the behavior of its Rb and K counterparts, which exhibit single superconducting domes. The microscopic mechanisms driving this unusual behavior in CsV₃Sb₅ are not fully understood, prompting the need for further investigation. The coexistence of SC and CO in CsV₃Sb₅ raises fundamental questions regarding their interplay. Does the CO compete with or promote SC? How does the nature of the CO itself evolve under pressure, and how does this evolution affect the superconducting properties? Addressing these questions requires a detailed understanding of the microscopic properties of the material under various pressure conditions. Hydrostatic pressure serves as an effective tuning knob, allowing for systematic study of the phase diagram and the interplay between CO and SC. Previous studies have utilized various techniques such as scanning tunneling microscopy (STM), polar Kerr rotation, and muon spin relaxation/rotation (µSR) to explore the properties of CsV₃Sb₅, revealing evidence of time-reversal symmetry breaking (TRSB) chiral charge order below 94 K. However, a comprehensive understanding that links the microscopic details of the CO with the unusual pressure dependence of the superconducting properties remains elusive. This study aims to bridge this gap by employing a combined experimental and theoretical approach, providing deeper insights into the interplay between CO and SC in CsV₃Sb₅.

The discovery of superconductivity in the kagome metals AV₃Sb₅ (A = K, Rb, Cs) has spurred intensive research efforts. These materials exhibit a unique kagome lattice structure, which leads to complex electronic properties and intriguing phenomena. Several studies have reported the presence of charge density wave (CDW) order in these materials, often coexisting with superconductivity. In CsV₃Sb₅, a double superconducting dome has been observed, with the higher dome appearing after the suppression of the CDW, indicating a competition between the two phases. Previous work has also suggested the possibility of time-reversal symmetry breaking (TRSB) in the superconducting phase, though its precise nature remains an open question. Numerous theoretical models have attempted to explain the electronic properties and phase transitions in these kagome materials. First-principles calculations predict the existence of phonon instabilities, leading to different types of CDW order. The specific CO configuration observed can significantly impact the superconducting properties, as it can affect the density of states at the Fermi level and the effective mass of the charge carriers.

This research employed a multifaceted approach combining high-pressure muon spin relaxation/rotation (µSR) experiments with first-principles density functional theory (DFT) calculations. The µSR measurements, performed at the Swiss Muon Source (SµS) at the Paul Scherrer Institute (PSI), provided crucial insights into the microscopic properties of CsV₃Sb₅ under hydrostatic pressure. Both transverse-field (TF) and zero-field (ZF) µSR techniques were used. TF-µSR is a highly sensitive probe to measure the magnetic penetration depth (λ) in the vortex state of type-II superconductors. The penetration depth is directly related to the superfluid density (n_s), providing information about the superconducting state. Measurements were performed at various temperatures and pressures to determine the temperature and pressure dependence of λ and thus n_s. A two-gap s + s-wave model was used to fit the obtained data, reflecting the multi-gap nature of the superconducting state reported in previous studies. The pressure was achieved using a low-background double wall pressure cell, ensuring hydrostatic pressure conditions. ZF-µSR measurements are essential for probing spontaneous magnetic fields indicative of time-reversal symmetry breaking (TRSB). These measurements were also carried out at various temperatures and pressures to explore the possibility of TRSB in the superconducting state. The data were analyzed using the Gaussian-Kubo-Toyabe (GKT) depolarization function and an exponential decay function. The obtained electronic relaxation rate provides insights into the internal field distribution within the sample. The experimental results were complemented by first-principles DFT calculations. These calculations were used to investigate the possible evolution of the charge-ordered state under pressure. Different charge-ordered structures were considered, and their enthalpies were computed as a function of pressure. The DFT calculations were performed using the Vienna Ab initio Simulation Package (VASP) with the PBEsol exchange-correlation functional. The density of states (DOS) for the different CO phases was also calculated, allowing for a comparison between the calculated DOS and the experimentally observed superconducting properties. Analysis of the phonon instabilities and the resulting lattice distortions contributed to the understanding of CO transitions under pressure.

The key findings of this study are summarized as follows: 1. **Double-peak behavior in superfluid density:** The TF-µSR measurements revealed a double-peak feature in the pressure dependence of the superfluid density at zero temperature (λ^-2(0)), mirroring the double-dome behavior of the superconducting critical temperature (T_c). This indicates a strong correlation between the superfluid density and T_c, suggesting a significant impact of charge order on the superconducting properties. 2. **Three distinct regions in the phase diagram:** The pressure-temperature phase diagram exhibits three distinct regions. Region I exhibits a strong positive correlation between λ^-2(0) and T_c and it shows coexistence of SC and CO. Region II shows a weak correlation between these parameters and also shows the coexistence of both SC and CO. Region III shows a fully suppressed CO and a weak to negligible correlation between λ^-2(0) and T_c. The transition between these regions is marked by sharp changes in both T_c and λ^-2(0), suggesting first-order phase transitions. 3. **Pressure-induced enhancement of superfluid density:** A nearly threefold enhancement of the superfluid density is observed at the highest pressure (1.74 GPa), indicating a significant increase in the density of superconducting charge carriers. The increase is attributed to the suppression of charge order, which partially gaps the Fermi surface in the lower pressure regions. 4. **Evolution of charge order patterns:** DFT calculations support the interpretation that the observed double-dome behavior arises from an evolution in the charge order pattern with pressure. A transition is suggested from a superimposed tri-hexagonal Star-of-David phase at low pressures (Region I) to a staggered tri-hexagonal phase at intermediate pressures (Region II). The staggered tri-hexagonal phase is predicted to compete more strongly with superconductivity, leading to lower T_c and λ^-2(0) in Region II compared to Region I. The transition occurs in the pressure range of 0.63 to 1.74 GPa. 5. **Time-reversal symmetry breaking in the pure superconducting state:** ZF-µSR measurements indicate time-reversal symmetry breaking (TRSB) in the superconducting state once the charge order is completely suppressed (Region III). This TRSB is not attributed to the charge order itself but is instead an intrinsic property of the superconducting state in the absence of charge order.

The results of this study provide compelling evidence for a complex interplay between charge order and superconductivity in CsV₃Sb₅. The observed double-dome structure, characterized by a non-monotonic pressure dependence of both T_c and the superfluid density, highlights the competition between the charge order and the superconducting state. The DFT calculations provide a plausible explanation for the pressure-induced evolution of charge order and its impact on the superconducting properties. The transition from a superimposed tri-hexagonal Star-of-David phase at low pressure to a staggered tri-hexagonal phase at higher pressure offers a compelling microscopic picture for understanding the changes in T_c and λ^-2(0) as pressure increases. The observation of time-reversal symmetry breaking (TRSB) in the pure superconducting state (Region III) further underscores the unconventional nature of superconductivity in CsV₃Sb₅. This adds to the growing body of evidence suggesting that the TRSB is an intrinsic property of the superconducting state in this family of kagome materials, rather than a consequence of charge order. This discovery opens new avenues for research on unconventional superconductors with spontaneous TRSB and further emphasizes the importance of investigating the intricate relationship between competing electronic orders in these systems.

This study reveals a rich phase diagram for CsV₃Sb₅, highlighting the complex interplay between charge order and superconductivity. Two distinct types of charge order, identified through DFT calculations, exhibit different degrees of competition with superconductivity, leading to the observed double-dome behavior. The pressure-induced suppression of charge order results in a threefold enhancement of the superfluid density and the emergence of a time-reversal symmetry-breaking superconducting state. Future research should focus on more detailed investigations of the microscopic mechanisms driving the TRSB and the nature of the superconducting gap in CsV₃Sb₅ and its relationship to the unique kagome lattice structure. Further theoretical and experimental investigations exploring the role of various types of charge order and other forms of electron correlations are warranted.

While this study provides significant insights into the interplay of charge order and superconductivity in CsV₃Sb₅, certain limitations should be acknowledged. The DFT calculations, while informative, cannot fully capture the complex interplay of many-body interactions and subtle effects such as fluctuations. The analysis of the pressure dependence relies on the assumption of hydrostatic pressure conditions within the pressure cell, which might not be perfectly achieved in practice. Furthermore, the two-gap model used to analyze the superconducting penetration depth might be a simplification of a more complex multi-band superconducting state. These considerations imply a need for more complex theoretical modeling and refined experimental techniques in future investigations.