Hidden Order Phase Research Articles

Direct observation of the hybridization gap in both the hidden order and large moment antiferromagnetic phases in URu2Si2

Despite extensive research on the heavy fermion superconductor $\mathrm{U}{\mathrm{Ru}}_{2}{\mathrm{Si}}_{2}$ in the past three decades, the nature of the hidden order (HO) phase transition occurring at 17.5 K remains ambiguous. Here we report a comparative scanning tunneling microscopy/spectroscopy (STM/STS) study on different terminations of the parent $\mathrm{U}{\mathrm{Ru}}_{2}{\mathrm{Si}}_{2}$ and Fe-doped samples. A small gap, which was ascribed to the HO parameter by previous STM/STS studies, emerges in both the HO and large moment antiferromagnetic phases on the U terminations, indicating it is not the unique hallmark of the HO parameter. Moreover, a peak-gap-peak structure is observed on the Si terminations. Variations of the two spectral features with Fe concentration and temperature show that they stem from the alteration of $f\text{\ensuremath{-}}c$ hybridization. The higher vanishing temperatures and larger sizes of the gap in the Fe-substituted samples indicate stronger $f\text{\ensuremath{-}}c$ hybridization strength compared to $\mathrm{U}{\mathrm{Ru}}_{2}{\mathrm{Si}}_{2}$. Our studies demonstrate hybridization is not the driving force for the HO phase transition.

In-Plane Anisotropic Response to Uniaxial Pressure in the Hidden Order State of URu2Si2

We investigate the uniaxial-pressure dependence of resistivity for URu2−x Fe x Si2 samples with x = 0 and 0.2, which host a hidden order (HO) and a large-moment antiferromagnetic (LMAFM) phase, respectively. For both samples, the elastoresistivity ζ shows a seemingly divergent behavior above the transition temperature T 0 and a quick decrease below it. We find that the temperature dependence of ζ for both samples can be well described by assuming the uniaxial pressure effect on the gap or certain energy scale except for ζ (110) of the x = 0 sample, which exhibits a nonzero residual value at 0 K. We show that this provides a qualitative difference between the HO and LMAFM phases. Our results suggest that there is an in-plane anisotropic response to the uniaxial pressure that only exists in the hidden order state without necessarily breaking the rotational lattice symmetry.

Cascade of magnetic-field-driven quantum phase transitions in Ce3Pd20Si6

Magnetically hidden order is a hypernym for electronic ordering phenomena that are visible to macroscopic thermodynamic probes but whose microscopic symmetry cannot be revealed with conventional neutron or x-ray diffraction. In a handful of f-electron systems, the ordering of odd-rank multipoles leads to order parameters with a vanishing neutron cross-section. Among them, Ce$_3$Pd$_{20}$Si$_6$ is known for its unique phase diagram exhibiting two distinct multipolar-ordered ground states (phases II and II'), separated by a field-driven quantum phase transition associated with a putative change in the ordered quadrupolar moment from $O_2^0$ to $O_{xy}$. Using torque magnetometry at subkelvin temperatures, here we find another phase transition at higher fields above 12 T, which appears only for low-symmetry magnetic field directions $\mathbf{B} \parallel \langle11L\rangle$ with $1 < L \leq 2$. While the order parameter of this new phase II'' remains unknown, the discovery renders Ce$_3$Pd$_{20}$Si$_6$ a unique material with two field-driven phase transitions between distinct multipolar phases. They are both clearly manifested in the magnetic-field dependence of the field-induced (111) Bragg intensities measured with neutron scattering for $\mathbf{B} \parallel [112]$. We also find from inelastic neutron scattering that the number of nondegenerate collective excitations induced by the magnetic field correlates with the number of phases in the magnetic phase diagram for the same field direction. Furthermore, the magnetic excitation spectrum suggests that the new phase II'' may have a different propagation vector, revealed by the minimum in the dispersion that may represent the Goldstone mode of this hidden-order phase.

From hidden order to antiferromagnetism: Electronic structure changes in Fe-doped URu2Si2

In matter, any spontaneous symmetry breaking induces a phase transition characterized by an order parameter, such as the magnetization vector in ferromagnets, or a macroscopic many-electron wave function in superconductors. Phase transitions with unknown order parameter are rare but extremely appealing, as they may lead to novel physics. An emblematic and still unsolved example is the transition of the heavy fermion compound [Formula: see text] (URS) into the so-called hidden-order (HO) phase when the temperature drops below [Formula: see text] K. Here, we show that the interaction between the heavy fermion and the conduction band states near the Fermi level has a key role in the emergence of the HO phase. Using angle-resolved photoemission spectroscopy, we find that while the Fermi surfaces of the HO and of a neighboring antiferromagnetic (AFM) phase of well-defined order parameter have the same topography, they differ in the size of some, but not all, of their electron pockets. Such a nonrigid change of the electronic structure indicates that a change in the interaction strength between states near the Fermi level is a crucial ingredient for the HO to AFM phase transition.

Collinear antiferromagnetic order in URu2Si2−xPx revealed by neutron diffraction

The hidden order phase in URu$_2$Si$_2$ is highly sensitive to electronic doping. A special interest in silicon-to-phosphorus substitution is due to the fact that it may allow one, in part, to isolate the effects of tuning the chemical potential from the complexity of the correlated $f$ and $d$ electronic states. We investigate the new antiferromagnetic phase that is induced in URu$_2$Si$_{2-x}$P$_x$ at $x\gtrsim0.27$. Time-of-flight neutron diffraction of a single crystal ($x=0.28$) reveals $c$-axis collinear $\mathbf{q}_\mathrm{m}=(\frac12,\frac12,\frac12)$ magnetic structure with localized magnetic moments ($\approx2.1\,\mu_\mathrm{B}$). This points to an unexpected analogy between the (Si,P) and (Ru,Rh) substitution series. Through further comparisons with other tuning studies of URu$_2$Si$_2$, we are able to delineate the mechanisms by which silicon-to-phosphorus substitution affects the system. In particular, both the localization of itinerant 5$f$ electrons as well as the choice of $\mathbf{q}_m$ appears to be consequences of the increase in chemical potential. Further, enhanced exchange interactions are induced by chemical pressure and lead to magnetic order, in which an increase in inter-layer spacing may play a special role.

Isoelectronic perturbations to f-d-electron hybridization and the enhancement of hidden order in URu2Si2

Electrical resistivity measurements were performed on single crystals of URu2-x Os x Si2 up to x = 0.28 under hydrostatic pressure up to P = 2 GPa. As the Os concentration, x, is increased, 1) the lattice expands, creating an effective negative chemical pressure Pch(x); 2) the hidden-order (HO) phase is enhanced and the system is driven toward a large-moment antiferromagnetic (LMAFM) phase; and 3) less external pressure Pc is required to induce the HO→LMAFM phase transition. We compare the behavior of the T(x, P) phase boundary reported here for the URu2-x Os x Si2 system with previous reports of enhanced HO in URu2Si2 upon tuning with P or similarly in URu2-x Fe x Si2 upon tuning with positive Pch(x). It is noteworthy that pressure, Fe substitution, and Os substitution are the only known perturbations that enhance the HO phase and induce the first-order transition to the LMAFM phase in URu2Si2 We present a scenario in which the application of pressure or the isoelectronic substitution of Fe and Os ions for Ru results in an increase in the hybridization of the U-5f-electron and transition metal d-electron states which leads to electronic instability in the paramagnetic phase and the concurrent formation of HO (and LMAFM) in URu2Si2 Calculations in the tight-binding approximation are included to determine the strength of hybridization between the U-5f-electron states and the d-electron states of Ru and its isoelectronic Fe and Os substituents in URu2Si2.

Griffiths phase-like exponents in the hidden-order state of URu2Si2

We report electrical resistivity, magnetic susceptibility, and specific heat measurements of high-quality URu2Si2 single crystals at ambient pressure, in the hidden-order (HO) region of the phase diagram. An amazing power-law behavior is observed, for both DC susceptibility and specific heat, establishing the Griffiths model for the HO phase. The Griffiths phase is characterized by residual short-range correlations on the collapse of the long-range large-moment antiferromagnetic (LMAFM) phase due to the dilution effects. The existence of cluster-like spins is intrinsic, as evidenced by the frequency dispersion of AC susceptibility and the relaxation of magnetoresistance. On account of the freezing of magnetic clusters, an unidirectional anisotropy of the resistivity in rotating magnetic fields is detected, which naturally explains any possible broken symmetries in tiny and clean crystals. In this manner, the HO phase of URu2Si2 is intimately related to the LMAFM phase. The proposal of Griffiths phase as an alternative of the HO phase is very promising, which may open a new route into the understanding of exotic electronic states in correlated matter.

Universal behavior in the Nernst effect of heavy fermion materials

We report the observation of a universal scaling of the Nernst coefficient over a wide intermediate temperature range in heavy fermion materials, including the superconductors CeCu$_2$Si$_2$, CeCoIn$_5$, and Ce$_2$PdIn$_8$, the ferromagnetic Kondo lattice Ce$_3$RhSi$_3$, the nonmagnetic CeRu$_2$Si$_2$, the intermediate valent YbAl$_3$, and the hidden order compound URu$_2$Si$_2$, that cover a broad spectrum of heavy fermion materials with different crystal, valence and ground state properties. The scaling formula follows exactly the magnetic susceptibility of emergent heavy quasiparticles as predicted in the two-fluid model. We give a tentative explanation of the scaling based on the skew scattering mechanism and the Boltzmann picture and argue that the Nernst effect is produced by the asymmetry of the quasiparticle density of states rather than that of the scattering rate. In URu$_2$Si$_2$, the giant Nernst signal in the hidden order phase is also found to follow the predicted scaling, indicating the potential involvement of hybridization physics. Our work suggests a candidate unified scenario for the Nernst effect in heavy fermion materials.

Electronic Nematicity in URu_{2}Si_{2} Revisited.

The nature of the hidden-order (HO) state in URu_{2}Si_{2} remains one of the major unsolved issues in heavy-fermion physics. Recently, torque magnetometry, x-ray diffraction, and elastoresistivity data have suggested that the HO phase transition at T_{HO}≈ 17.5 K is driven by electronic nematic effects. Here, we search for thermodynamic signatures of this purported structural instability using anisotropic thermal expansion, Young's modulus, elastoresistivity, and specific-heat measurements. In contrast to the published results, we find no evidence of a rotational symmetry breaking in any of our data. Interestingly, our elastoresistivity measurements, which are in full agreement with published results, exhibit a Curie-Weiss divergence, which we however attribute to a volume and not to a symmetry-breaking effect. Finally, clear evidence for thermal fluctuations is observed in our heat-capacity data, from which we estimate the HO correlation length.

Destabilization of hidden order in URu2Si2 under magnetic field and pressure

The mystery of the hidden-order phase in the correlated electron paramagnet URu2Si2 is still unsolved. To address this problem, one strategy is to search for clues in the subtle competition between this state and neighbouring magnetically ordered states. It is now well established that long-range antiferromagnetic order can be stabilized in this metal when it is under pressure and that a spin-density wave manifests when a magnetic field is applied along the easy magnetic axis c. However, the full boundaries of the hidden-order phase in the pressure–magnetic-field plane have not been determined so far. Here we present a systematic investigation of URu2Si2 under combined high pressures and intense magnetic fields. The boundaries of the hidden-order, antiferromagnetic and spin-density-wave phases are mapped out, indicating an intricate three-dimensional phase diagram. We show that the field-induced spin-density-wave and hidden-order phases disappear in favour of antiferromagnetism at high pressure. Interestingly, a large number of phase boundaries are controlled by the field and pressure dependences of a single parameter. This gives new constraints for theories that model the electronic correlations and ordered phases in URu2Si2. The nature of the hidden order in URu2Si2 is still unknown. Here detailed measurements of the phase diagram of this material produce constraints for theories that aim to describe that phase.

Effective model for the A2g Raman signal in URu2Si2

We propose an effective model to describe the $A_{2g}$ signal in Raman scattering experiments in the URu$_{2}$Si$_{2}$ compound. We follow the scheme proposed earlier by Khveshchenko and Wiegmann [Phys. Rev. Lett. 73, 500 (1994)] to calculate the $A_{2g}$ scattering vertex. We extract the imaginary part of a two-point current-current correlation function and compare it directly with the Raman response. We obtain an inelastic peak at $A_{2g}$ channel owing to the interplay between a local staggered ordering and a possible quantum spin liquid behavior. Our results offer an explanation for the electronic Raman scattering experiments at the hidden order phase in URu$_{2}$Si$_{2}$ compound [Phys. Rev. Lett. 113, 266405 (2014)].

Field-Angle-Resolved Magnetic Excitations as a Probe of Hidden-Order Symmetry in CeB6

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 and field-induced octupolar order. Here, we present a combined experimental and theoretical investigation of collective excitations in 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 a rotating field is calculated within a localized approach using the pseudospin representation for the Γ8 states. We show that the rotating-field technique at fixed momentum can complement conventional INS measurements of the dispersion at a constant field and holds great promise for identifying the symmetry of multipolar order parameters and the details of intermultipolar interactions that stabilize hidden-order phases.6 MoreReceived 6 January 2020Revised 20 February 2020Accepted 5 March 2020DOI:https://doi.org/10.1103/PhysRevX.10.021010Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Physical SystemsHeavy-fermion systemsStrongly correlated systemsTechniquesInelastic neutron scatteringCondensed Matter, Materials & Applied Physics

Intertwined dipolar and multipolar order in the triangular-lattice magnet TmMgGaO4

A phase transition is often accompanied by the appearance of an order parameter and symmetry breaking. Certain magnetic materials exhibit exotic hidden-order phases, in which the order parameters are not directly accessible to conventional magnetic measurements. Thus, experimental identification and theoretical understanding of a hidden order are difficult. Here we combine neutron scattering and thermodynamic probes to study the newly discovered rare-earth triangular-lattice magnet TmMgGaO4. Clear magnetic Bragg peaks at K points are observed in the elastic neutron diffraction measurements. More interesting, however, is the observation of sharp and highly dispersive spin excitations that cannot be explained by a magnetic dipolar order, but instead is the direct consequence of the underlying multipolar order that is “hidden” in the neutron diffraction experiments. We demonstrate that the observed unusual spin correlations and thermodynamics can be accurately described by a transverse field Ising model on the triangular lattice with an intertwined dipolar and ferro-multipolar order.

Instability of the f -electron state in URu2Si2−xPx probed using high magnetic fields

The widely studied hidden-order phase in ${\mathrm{URu}}_{2}{\mathrm{Si}}_{2}$ exemplifies a broad class of emergent macroscopic states originating from $d$- and $f$-electron orbitals and their collective modes. This naturally has resulted in a focus on the associated elements (i.e., U and Ru), but recently it was discovered that electronic tuning on the more passive $s$ and $p$ orbitals fundamentally alters the electronic behavior and thus opens opportunities to disentangle the hidden-order conundrum. Here we present a detailed magnetoresistivity study of ${\mathrm{URu}}_{2}{\mathrm{Si}}_{2\ensuremath{-}x}{\mathrm{P}}_{x}$, which focuses on two important aspects of this topic. First, we show that the anisotropy of the high field ordered states is preserved after hidden order is destroyed by Si $\ensuremath{\rightarrow}$ P chemical substitution, suggesting that this feature is mainly associated with the Kondo lattice, i.e., not hidden order itself. Second, Shubnikov--de Haas oscillations reveal that the observable parts of the Fermi surface topography are robust upon approaching the hidden-order phase boundary and the associated quasiparticle masses remain roughly constant. These observations place constraints on theories and prepare a path towards a resolution to the hidden-order problem.

Evolution of the propagation vector of antiferroquadrupolar phases in Ce3Pd20Si6 under magnetic field

Hidden-order phases that occur in a number of correlated f-electron systems are among the most elusive states of electronic matter. Their investigations are hindered by the insensitivity of standard physical probes, such as neutron diffraction, to the order parameter that is usually associated with higher-order multipoles of the f-orbitals. The heavy-fermion compound Ce3Pd20Si6 exhibits magnetically hidden order at subkelvin temperatures, known as phase II. Additionally, for magnetic field applied along the [001] cubic axis, another phase II' was detected, but the nature of the transition from phase II to phase II' remained unclear. Here we use inelastic neutron scattering to demonstrate that this transition is associated with a change in the propagation vector of the antiferroquadrupolar order from (111) to (100). Despite the absence of magnetic Bragg scattering in phase II', its ordering vector is revealed by the location of an intense magnetic soft mode at the (100) wave vector, orthogonal to the applied field. At the II-II' transition, this mode softens and transforms into quasielastic and nearly Q-independent incoherent scattering, which is likely related to the non-Fermi-liquid behavior recently observed at this transition. Our experiment also reveals sharp collective excitations in the field-polarized paramagnetic phase, after phase II' is suppressed in fields above 4 T.

Quasiparticle relaxation dynamics in URu2−xFexSi2 single crystals

We investigate quasiparticle relaxation dynamics in URu$_{2-x}$Fe$_{x}$Si$_{2}$ single crystals using ultrafast optical-pump optical-probe (OPOP) spectroscopy as a function of temperature ($T$) and Fe substitution ($x$), crossing from the hidden order (HO) phase ($x$ = 0) to the large moment antiferromagnet (LMAFM) phase ($x$ = 0.12). At low $T$, the dynamics for $x$ = 0 and $x$ = 0.12 are consistent with the low energy electronic structure of the HO and LMAFM phases that emerge from the high $T$ paramagnetic (PM) phase. In contrast, for $x$ = 0.1, two transitions occur over a narrow $T$ range (from ~15.5 - 17.5 K). A PM to HO transition occurs at an intermediate $T$ followed by a transition to the LMAFM phase at lower $T$. While the data at low $T$ are consistent with the expected coexistence of LMAFM and HO, the data in the intermediate $T$ phase are not, and instead suggest the possibility of an unexpected coexistence of HO and PM. Additionally, the dynamics in the PM phase reflect the presence of a hybridization gap as well as strongly interacting spin and charge degrees of freedom. OPOP yields insights into meV-scale electrodynamics with sub-Kelvin $T$ resolution, providing a complementary approach to study low energy electronic structure in quantum materials.

Rapid suppression of the energy gap and the possibility of a gapless hidden order state in URu2−xRexSi2

ABSTRACTWe investigated the energy gap associated with the hidden order (HO) phase and the Grüneisen ratio in the URuReSi system using a combination of thermal expansion coefficient and specific heat measurements. As the HO phase transition is suppressed to lower temperature, the ratio between the energy gap and the HO transition temperature decreases three-fold. This rapid suppression of the energy gap potentially leads to a scenario of a ‘gapless’ HO state, in which the energy gap of the HO phase vanishes before the HO transition temperature is suppressed to 0 K. We also investigated the Grüneisen ratio in the vicinity of the Re substituent composition where the HO is suppressed. The Grüneisen ratio shows divergent behaviour at x=0.12 and 0.15, providing evidence for the existence of a quantum critical point, which is consistent with our hypothesis of a ‘gapless’ HO state.

Pressure-induced rotational symmetry breaking in URu2Si2

Phase transitions and symmetry are intimately linked. Melting of ice, for example, restores translation invariance. The mysterious hidden order (HO) phase of URu$_2$Si$_2$ has, despite relentless research efforts, kept its symmetry breaking element intangible. Here we present a high-resolution x-ray diffraction study of the URu$_2$Si$_2$ crystal structure as a function of hydrostatic pressure. Below a critical pressure threshold $p_c\approx3$ kbar, no tetragonal lattice symmetry breaking is observed even below the HO transition $T_{HO}=17.5$ K. For $p>p_c$, however, a pressure-induced rotational symmetry breaking is identified with an onset temperatures $T_{OR}\sim 100$ K. The emergence of an orthorhombic phase is found and discussed in terms of an electronic nematic order that appears unrelated to the HO, but with possible relevance for the pressure-induced antiferromagnetic (AF) phase. Existing theories describe the HO and AF phases through an adiabatic continuity of a complex order parameter. Since none of these theories predicts a pressure-induced nematic order, our finding adds an additional symmetry breaking element to this long-standing problem.

Gaussian fluctuation corrections to a mean-field theory of complex hidden order in URu2Si2

Hidden-order phase transition in the heavy-fermion superconductor URu$_2$Si$_2$ exhibits the mean-field-like anomaly in temperature dependence of heat capacity. Motivated by this observation, here we explore the impact of the complex order parameter fluctuations on the thermodynamic properties of the hidden order phase. Specifically, we employ the mean-field theory for the hidden order which describes the hidden order parameter by an average of the hexadecapole operator. We compute the gaussian fluctuation corrections to the mean-field theory equations including both the fluctuations due to 'hidden order' as well as antiferromagnetic order parameters. We find that the gaussian fluctuations lead to the smearing of the second-order transition rendering it to become the first-order one. The strength of the first-order transition is weakly dependent on the strength of underlying antiferromagnetic exchange interactions.

A model for metastable magnetism in the hidden-order phase of URu[formula omitted]Si[formula omitted

We propose an explanation for the experiment by Schemm et al. (2015) where the polar Kerr effect (PKE), indicating time-reversal symmetry (TRS) breaking, was observed in the hidden-order (HO) phase of URu2Si2. The PKE signal on warmup was seen only if a training magnetic field was present on cool-down. Using a Ginzburg–Landau model for a complex order parameter, we show that the system can have a metastable ferromagnetic state producing the PKE, even if the HO ground state respects TRS. We predict that a strong reversed magnetic field should reset the PKE to zero.