One-component order parameter in URu ₂ Si ₂ uncovered by resonant ultrasound spectroscopy and machine learning

Abstract

† Authors contributed equally to this work.URu2Si2 exhibits a clear phase transition at THO = 17.5 K to a low-temperature phase known as "hidden order" (HO). Even the most basic information needed to construct a theory of this state-such as the number of components in the order parameter-has been lacking. Here we use resonant ultrasound spectroscopy (RUS) and machine learning to determine that the order parameter of HO is one-dimensional (singlet), ruling out a large class of theories based on twodimensional (doublet) order parameters. This strict constraint is independent of any microscopic mechanism, and independent of other symmetries that HO may break. Our technique is general for second-order phase transitions, and can discriminate between nematic (singlet) versus loop current (doublet) order in the high-Tc cuprates, and conventional (singlet) versus the proposed px + ipy (doublet) superconductivity in Sr2RuO4. The machine learning framework we develop should be readily adaptable to other spectroscopic techniques where missing resonances confound traditional analysis, such as NMR.

A proposal for reconciling diverse experiments on the superconducting state in Sr2RuO4

“…7There is a discontinuity at T c in the elastic modulus c 66 associated with the shear B 2g (ϵ xy ) strain, as inferred from ultrasound experiments 16,17 . While the elastic moduli associated with A 1g compressional strains exhibit discontinuities at all second-order phase transitions, such a discontinuity in a shear elastic constant is most straightforwardly associated with a two-component SC order parameter 25 . This follows from the requirement that a gauge-invariant bilinear can be constructed from the order parameter that transforms as xy under the point group operations-such a bilinear cannot be constructed for any one-component SC order parameter but can be constructed for a number of two-component order parameters.…”

Proximity to a critical point driven by electronic entropy in URu2Si2

“…Fluctuations of Δ within the pseudogap region are therefore likely to lead to a considerable softening of the crystalline lattice with respect to the A 1g mode. A lattice softening of this nature has recently been reported in resonant ultrasound spectroscopy experiments 80 , although the existing interpretation has assumed strictly localized 5f electrons-in effect, neglecting the extreme sensitivity of the f-electron core occupancy and hybridization strength to volume changes and lattice distortions. Because it preserves the fourfold symmetry of the lattice, the A 1g mode is the only one (other than a change in volume) that couples to the degree of delocalization of the 5f electrons (and thereby the strength of Δ) uniformly across all sites.…”

Diffraction by multipoles in a 5d2 rhenium double perovskite

“…Moreover, parity-odd magnetic (Dirac) multipoles have been observed in both materials by x-ray diffraction [5,6]. In addition to the examples of multipoles in the two 3 d-transition metal materials are multipoles observed in diffraction by rare earth and actinide ( f electron) compounds, e.g., neptunium dioxide NpO 2 [7] and URu 2 Si 2 [8], and like work is reviewed by Suzuki et al [9]. Neutron diffraction came late to the party with respect to gathering information on higher-order magnetic multipoles, although it is the technique of choice for determining motifs of conventional (axial) magnetic dipoles, beginning with a demonstration by Shull and Smart in 1949 of antiferromagnetic order in NaCltype MnO below 122 K [10].…”