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Register a new userUnveiling Hidden Orders: Magnetostriction as a Probe of Multipolar-Ordered States
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Broken symmetries in solids involving higher order multipolar degrees of freedom are historically referred to as hidden orders due to the formidable task of detecting them with conventional probes. Examples of such hidden orders include spin-nematic order in quantum magnets, and quadrupolar or higher multipolar orders in various correlated quantum materials. In this work, we theoretically propose that the study of magnetostriction provides a powerful and novel tool to directly detect higher-order multipolar symmetry breaking $-$ such as the elusive octupolar order $-$ by examining its scaling behaviour with respect to an applied magnetic field $h$. As an illustrative example, we examine such key scaling signatures in the context of Pr-based cage compounds with strongly correlated $f$-electrons, Pr(Ti,V,Ir)$_2$(Al,Zn)$_{20}$, whose low energy degrees of freedom are composed of purely higher-order multipoles: quadrupoles $mathcal{O}_{20,22}$ and octupole $mathcal{T}_{xyz}$. Employing a symmetry-based Landau theory of multipolar moments coupled to lattice strain fields, we demonstrate that a magnetic field applied along the [111] direction results in a length change with a distinct linear-in-$h$ scaling behaviour, accompanied by hysteresis, below the octupolar ordering temperature. We show that the resulting magnetostriction coefficient is directly proportional to the octupolar order parameter, providing the first clear access to this subtle order parameter. Along other field directions, we show that the field dependence of the magnetostriction provides a window into quadrupolar orders. Our work provides a springboard for future experimental and theoretical investigations of multipolar orders and their quantum phase transitions in a wide variety of systems.
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We demonstrate that the volume of the Fermi surface, measured very precisely using de Haas-van Alphen oscillations, can be used to probe changes in the nature and occupancy of localized electronic states. In systems with unconventional ordered states, this allows an underlying electronic order parameter to be followed to very low temperatures. We describe this effect in the field-induced antiferroquadrupolar (AFQ) ordered phase of PrOs4Sb12, a heavy fermion intermetallic compound. We find that the phase of de Haas-van Alphen oscillations is sensitively coupled, through the Fermi volume, to the configuration of the Pr f-electron states that are responsible for AFQ order. In particular, the beta-sheet of the Fermi surface expands or shrinks as the occupancy of two competing localized Pr crystal field states changes. Our results are in good agreement with previous measurements, above 300 mK, of the AFQ order parameter by other methods. In addition, the low temperature sensitivity of our measurement technique reveals a strong and previously unrecognized influence of hyperfine coupling on the order parameter below 300 mK within the AFQ phase. Such hyperfine couplings could provide insight into the nature of hidden order states in other systems.
In contrast to magnetic order formed by electrons dipolar moments, ordering phenomena associated with higher-order multipoles (quadrupoles, octupoles, etc.) are more difficult to characterize because of the limited choice of experimental probes that can distinguish different multipolar moments. The heavy-fermion compound CeB6 and its La-diluted alloys are among the best-studied realizations of the long-range-ordered multipolar phases, often referred to as hidden order. Previously the hidden order in phase II was identified as primary antiferroquadrupolar (AFQ) and field-induced octupolar (AFO) order. Here we present a combined experimental and theoretical investigation of collective excitations in the phase II of CeB6. Inelastic neutron scattering (INS) in fields up to 16.5 T reveals a new high-energy mode above 14 T in addition to the low-energy magnetic excitations. The experimental dependence of their energy on the magnitude and angle of the applied magnetic field is compared to the results of a multipolar interaction model. The magnetic excitation spectrum in rotating field is calculated within a localized approach using the pseudo-spin presentation for the Gamma8 states. We show that the rotating-field technique at fixed momentum can complement conventional INS measurements of the dispersion at constant field and holds great promise for identifying the symmetry of multipolar order parameters and the details of inter-multipolar interactions that stabilize hidden-order phases.
The hidden order developing below 17.5K in the heavy fermion material URu2Si2 has eluded identification for over twenty five years. This paper will review the recent theory of ``hastatic order, a novel two-component order parameter capturing the hybridization between half-integer spin (Kramers) conduction electrons and the non-Kramers 5f^2 Ising local moments, as strongly indicated by the observation of Ising quasiparticles in de Haas-van Alphen measurements. Hastatic order differs from conventional magnetism as it is a spinor order that breaks both single and double time-reversal symmetry by mixing states of different Kramers parity. The broken time-reversal symmetry simply explains both the pseudo-Goldstone mode between the hidden order and antiferromagnetic phases and the nematic order seen in torque magnetometry. The spinorial nature of the hybridization also explains how the Kondo effect gives a phase transition, with the hybridization gap turning on at the hidden order transition as seen in scanning tunneling microscopy. Hastatic order also has a number of new predictions: a basal-plane magnetic moment of order .01mu_B, a gap to longitudinal spin fluctuations that vanishes continuously at the first order antiferromagnetic transition and a narrow resonant nematic feature in the scanning tunneling spectra.
The nature of order in low-temperature phases of some materials is not directly seen by experiment. Such hidden orders (HO) may inspire decades of research to identify the mechanism underlying those exotic states of matter. In insulators, HO phases originate in degenerate many-electron states on localized f or d shells that may harbor high-rank multipole moments. Coupled by inter-site exchange, those moments form a vast space of competing order parameters. Here, we show how the ground state order and magnetic excitations of a prototypical HO system, neptunium dioxide NpO$_2$, can be fully described by a low-energy Hamiltonian derived by a many-body ab initio force-theorem method. Superexchange interactions between the lowest crystal-field quadruplet of Np$^{4+}$ ions induce a primary non-collinear order of time-odd rank-5 (triakontadipolar) moments with a secondary quadrupole order preserving the cubic symmetry of NpO$_2$. Our study also reveals an unconventional multipolar exchange-striction mechanism behind the anomalous volume contraction of the NpO$_2$ HO phase.
We review recent progress in point contact spectroscopy (PCS) to extract spectroscopic information out of correlated electron materials, with the emphasis on non-superconducting states. PCS has been used to detect bosonic excitations in normal metals, where signatures (e.g. phonons) are usually less than 1$%$ of the measured conductance. In the superconducting state, point contact Andreev reflection (PCAR) has been widely used to study properties of the superconducting gap in various superconductors. In the last decade, there have been more and more experimental results suggesting that the point contact conductance could reveal new features associated with the unusual single electron dynamics in non-superconducting states, shedding a new light on exploring the nature of the competing phases in correlated materials. We will summarize the theories for point contact spectroscopy developed from different approaches and highlight these conceptual differences distinguishing point contact spectroscopy from tunneling-based probes. Moreover, we will show how the Schwinger-Kadanoff-Baym-Keldysh (SKBK) formalism together with the appropriate modeling of the nano-scale point contacts randomly distributed across the junction leads to the conclusion that the point contact conductance is proportional to the {it effective density of states}, a physical quantity that can be computed if the electron self energy is known. The experimental data on iron based superconductors and heavy fermion compounds will be analyzed in this framework. These recent developments have extended the applicability of point contact spectroscopy to correlated materials, which will help us achieve a deeper understanding of the single electron dynamics in strongly correlated systems.
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