Modeling the instantaneous normal mode spectra of liquids as that of unstable elastic media

We study the instantaneous normal mode (INM) spectrum of a simulated soft-sphere liquid at different equilibrium temperatures T. We find that the spectrum of eigenvalues ρ(λ) has a sharp maximum near (but not at) λ=0 and decreases monotonically with |λ| on both the stable and unstable sides of the spectrum. The spectral shape strongly depends on temperature. It is rather asymmetric at low temperatures (close to the dynamical critical temperature) and becomes symmetric at high temperatures. To explain these findings we present a mean-field theory for ρ(λ), which is based on a heterogeneous elasticity model, in which the local shear moduli exhibit spatial fluctuations, including negative values. We find good agreement between the simulation data and the model calculations, done with the help of the self-consistent Born approximation (SCBA), when we take the variance of the fluctuations to be proportional to the temperature T. More importantly, we find an empirical correlation of the positions of the maxima of ρ(λ) with the low-frequency exponent of the density of the vibrational modes of the glasses obtained by quenching to T=0 from the temperature T. We discuss the present findings in connection to the liquid to glass transformation and its precursor phenomena.

The investigation of the potential energy surface (PES) V(r1(t)…rN(t)) of a liquid (made up of N particles with positions r1(t)…rN(t) at a time instant t) and the corresponding instantaneous normal modes (INMs) of the (Hessian) matrix of curvatures has been a focus of liquid and glass science since the appearance of Goldstein’s seminal article (1) on the relation between the PES and the liquid dynamics in the viscous regime above the glass transition (2–27).The PES has been shown to form a rather ragged landscape in configuration space (8, 28, 29) characterized by its stationary points. In a glass these points are minima and are called “inherent structures.” The PES is believed to contain important information on the liquid–glass transformation mechanism. For the latter a complete understanding is still missing (28, 30, 31). The existing molecular theory of the liquid–glass transformation is mode-coupling theory (MCT) (32, 33) and its mean-field Potts spin version (28, 34). MCT predicts a sharp transition at a temperature TMCT>Tg, where T_g is the temperature of structural arrest (glass transition temperature). MCT completely misses the heterogeneous activated relaxation processes (dynamical heterogeneities), which are evidently present around and below T_MCT and which are related to the unstable (negative-λ) part of the INM spectrum (28, 30).Near and above T_MCT, apparently, there occurs a fundamental change in the PES. Numerical studies of model liquids have shown that minima present below T_MCT change into saddles, which then explains the absence of activated processes above T_MCT (2–24). Very recently, it was shown that T_MCT is related to a localization–delocalization transition of the unstable INM modes (25, 26).The INM spectrum is obtained in molecular dynamic simulations by diagonalizing the Hessian matrix of the interaction potential, taken at a certain time instant t:Hijαβ(t)=∂2∂xi(α)∂xj(β)V{r1(t)…rN(t)},[1]with ri=(xi(1),xi(2),xi(3)). For large positive values of the eigenvalues λ_j (j=1…N, N being the number of particles in the system) they are related to the square of vibrational frequencies λj=ωj2, and one can consider the Hessian as the counterpart of the dynamical matrix of a solid. In this high-frequency regime one can identify the spectrum with the density of vibrational states (DOS) of the liquid viag(ω)=2 ωρ(λ(ω))=13N∑jδ(ω−ωj) .[2]For small and negative values of λ this identification is not possible. For the unstable part of the spectrum (λ<0) it has become common practice to call the imaginary number λ=iω˜ and define the corresponding DOS asg(ω˜)≡2 ω˜ρ(λ(ω˜)).[3]This function is plotted on the negative ω axis and the stable g(ω), according to [2], on the positive axis. However, the (as we shall see, very interesting) details of the spectrum ρ(λ) near λ = 0 become almost completely hidden by multiplying the spectrum with |ω|. In fact, it has been demonstrated by Sastry et al. (6) and Taraskin and Elliott (7) already 2 decades ago that the INM spectrum of liquids, if plotted as ρ(λ) and not as g(ω) according to [2] and [3], exhibits a characteristic cusp-like maximum at λ = 0. The shape of the spectrum changes strongly with temperature. This is what we find as well in our simulation and what we want to explore further in our present contribution.In the present contribution we demonstrate that the strong change of the spectrum with temperature can be rather well explained in terms of a model, in which the instantaneous harmonic spectrum of the liquid is interpreted to be that of an elastic medium, in which the local shear moduli exhibit strong spatial fluctuations, which includes a large number of negative values. Because these fluctuations are just a snapshot of thermal fluctuations, we assume that they are obeying Gaussian statistics, the variance of which is proportional to the temperature.Evidence for a characteristic change in the liquid configurations in the temperature range above T_g has been obtained in recent simulation studies of the low-frequency vibrational spectrum of glasses, which have been rapidly quenched from a certain parental temperature T*. If T* is decreased from high temperatures toward T_MCT, the low-frequency exponent of the vibrational DOS of the daughter glass (quenched from T* to T = 0) changed from Debye-like g(ω)∝ω2 to g(ω)∝ωs with s > 2. In our numerical investigation of the INM spectra we show a correlation of some details of the low-eigenvalue features of these spectra with the low-frequency properties of the daughter glasses obtained by quenching from the parental temperatures.The stochastic Helmholtz equations (Eq. 7) of an elastic model with spatially fluctuating shear moduli can be readily solved for the averaged Green’s functions by field theoretical techniques (35–37). Via a saddle point approximation with respect to the resulting effective field theory one arrives at a mean-field theory (self-consistent Born approximation [SCBA]) for the self-energy of the averaged Green’s functions. The SCBA predicts a stable spectrum below a threshold value of the variance. Restricted to this stable regime, this theory, called heterogeneous elasticity theory (HET), was rather successful in explaining several low-frequency anomalies in the vibrational spectrum of glasses, including the so-called boson peak, which is an enhancement at finite frequencies over the Debye behavior of the DOS g(ω)∝ω2 (37–41). We now explore the unstable regime of this theory and compare it to the INM spectrum of our simulated soft-sphere liquid.^*We start Results by presenting a comparison of the simulated spectra of the soft-sphere liquid with those obtained by the unstable version of HET-SCBA theory. We then concentrate on some specific features of the INM spectra, namely, the low-eigenvalue slopes and the shift of the spectral maximum from λ = 0. Both features are accounted for by HET-SCBA. In particular, we find an interesting law for the difference between the slopes of the unstable and the stable parts of the spectrum, which behaves as T−2/3, which, again, is accounted for by HET-SCBA.In the end we compare the shift of the spectral maximum with the low-frequency exponent of the DOS of the corresponding daughter glasses and find an empirical correlation. We discuss these results in connection with the saddle to minimum transformation near T_MCT.     

First-order antiferromagnetic transitions of SrMn2P2 and CaMn2P2 single crystals containing corrugated-honeycomb Mn sublattices

SrMn₂P₂ and CaMn₂P₂ are insulators that adopt the trigonal CaAl₂Si₂-type structure containing corrugated Mn honeycomb layers. Magnetic susceptibility χ and heat capacity versus temperature T data reveal a weak first-order antiferromagnetic (AFM) transition at the Néel temperature TN=53(1) K for SrMn₂P₂ and a strong first-order AFM transition at TN=69.8(3) K for CaMn₂P₂. Both compounds exhibit isotropic and nearly T-independent χ(T≤TN), suggesting magnetic structures in which nearest-neighbor moments are aligned at ≈120° to each other. The ³¹P NMR measurements confirm the strong first-order transition in CaMn₂P₂ but show critical slowing down above TN for SrMn₂P₂, thus also evidencing second-order character. The ³¹P NMR measurements indicate that the AFM structure of CaMn₂P₂ is commensurate with the lattice whereas that of SrMn₂P₂ is incommensurate. These first-order AFM transitions are unique among the class of (Ca, Sr, Ba)Mn₂ (P, As, Sb, Bi)₂ compounds that otherwise exhibit second-order AFM transitions. This result challenges our understanding of the circumstances under which first-order AFM transitions occur.

The Mn-based 122-type pnictides AMn2Pn2 (A= Ca, Sr, Ba; Pn = P, As, Sb, Bi) have received attention owing to their close stoichiometric 122-type relationship to high-Tc iron pnictides. The undoped Mn pnictides are local-moment antiferromagnetic (AFM) insulators like the high-Tc cuprate parent compounds (1–3). The BaMn2Pn2 compounds crystallize in the body-centered tetragonal ThCr2Si2 structure as in AFe2As2 (A = Ca, Sr, Ba, Eu), whereas the (Ca,Sr)Mn2Pn2 compounds crystallize in the trigonal CaAl2Si2-type structure (4). Recently, density-functional theory (DFT) calculations for the 122 pnictide family have suggested that the trigonal 122 transition-metal pnictides that have the CaAl2Si2 structure might compose a new family of magnetically frustrated materials in which to study the potential superconducting mechanism (5, 6). It had previously been suggested on theoretical grounds that CaMn₂Sb₂ is a fully frustrated classical magnetic system arising from proximity to a tricritical point (7–9).The electrical resistivity ρ and heat capacity Cp versus temperature T of single-crystal CaMn₂P₂ were reported in ref. 10. The compound is an insulator at T = 0 and undergoes a first-order transition of some type at 69.5 K. The Raman spectrum of CaMn₂P₂ at T = 10 K showed new peaks compared to the spectrum at 300 K, whereas the authors’ single-crystal X-ray diffraction measurements showed no difference in the crystal structure at 293 and 40 K. They suggested that the results of the two types of measurements could be reconciled if a superstructure formed below 69.5 K (10). The authors’ magnetic susceptibility χ(T) measurements below 400 K revealed no evidence for a magnetic transition.Here we report the detailed properties of trigonal CaMn₂P₂ and SrMn₂P₂ (11) single crystals. We present the results of single-crystal X-ray diffraction (XRD), electrical resistivity ρ in the ab plane (hexagonal unit cell) versus temperature T, isothermal magnetization versus applied magnetic field M(H), magnetic susceptibility χ(T), heat capacity Cp(H,T), and ³¹P NMR measurements. We find from Cp(T), χ(T), and NMR that CaMn₂P₂ exhibits a strong first-order AFM transition at TN=69.8(3) K whereas SrMn₂P₂ shows a weak first-order transition at TN=53(1) K but with critical slowing down on approaching TN from above as revealed from NMR, a characteristic feature of second-order transitions. Thus, remarkably, the AFM transition in SrMn₂P₂ has characteristics of both first- and second-order transitions. The χ(T) data also reveal the presence of strong isotropic AFM spin fluctuations in the paramagnetic (PM) state above TN up to our maximum measurement temperatures of 900 and 350 K for SrMn₂P₂ and CaMn₂P₂, respectively. This behavior likely arises from spin fluctuations associated with the quasi–two-dimensional nature of the Mn spin layers (12) together with possible contributions from magnetic frustration. Our single-crystal XRD data at room temperature and high-resolution synchrotron XRD data at T = 20 K for SrMn₂P₂ and CaMn₂P₂ demonstrate conclusively that there is no structure change of either compound on cooling below their respective TN.Our studies of SrMn₂P₂ and CaMn₂P₂ thus identify the only known members of the class of materials with general formula AMn2Pn2 containing Mn2+ spins S = 5/2 that exhibit first-order AFM transitions, where A = Ca, Sr, or Ba and the pnictogen Pn= P, As, Sb, or Bi. In particular, only second-order AFM transitions are found in CaMn2As2 (13), SrMn2As2 (13–15), CaMn2Sb2 (8, 9, 16–19), SrMn2Sb2 (16, 19), and CaMn2Bi2 (20).     

Dynamic rupture initiation and propagation in a fluid-injection laboratory setup with diagnostics across multiple temporal scales

Fluids are known to trigger a broad range of slip events, from slow, creeping transients to dynamic earthquake ruptures. Yet, the detailed mechanics underlying these processes and the conditions leading to different rupture behaviors are not well understood. Here, we use a laboratory earthquake setup, capable of injecting pressurized fluids, to compare the rupture behavior for different rates of fluid injection, slow (megapascals per hour) versus fast (megapascals per second). We find that for the fast injection rates, dynamic ruptures are triggered at lower pressure levels and over spatial scales much smaller than the quasistatic theoretical estimates of nucleation sizes, suggesting that such fast injection rates constitute dynamic loading. In contrast, the relatively slow injection rates result in gradual nucleation processes, with the fluid spreading along the interface and causing stress changes consistent with gradually accelerating slow slip. The resulting dynamic ruptures propagating over wetted interfaces exhibit dynamic stress drops almost twice as large as those over the dry interfaces. These results suggest the need to take into account the rate of the pore-pressure increase when considering nucleation processes and motivate further investigation on how friction properties depend on the presence of fluids.

The close connection between fluids and faulting has been revealed by a large number of observations, both in tectonic settings and during human activities, such as wastewater disposal associated with oil and gas extraction, geothermal energy production, and CO₂ sequestration (1–11). On and around tectonic faults, fluids also naturally exist and are added at depths due to rock-dehydration reactions (12–15) Fluid-induced slip behavior can range from earthquakes to slow, creeping motion. It has long been thought that creeping and seismogenic fault zones have little to no spatial overlap. Nonetheless, growing evidence suggests that the same fault areas can exhibit both slow and dynamic slip (16–19). The existence of large-scale slow slip in potentially seismogenic areas has been revealed by the presence of transient slow-slip events in subduction zones (16, 18) and proposed by studies investigating the physics of foreshocks (20–22).Numerical and laboratory modeling has shown that such complex fault behavior can result from the interaction of fluid-related effects with the rate-and-state frictional properties (9, 14, 19, 23, 24); other proposed rheological explanations for complexities in fault stability include combinations of brittle and viscous rheology (25) and friction-to-flow transitions (26). The interaction of frictional sliding and fluids results in a number of coupled and competing mechanisms. The fault shear resistance τres is typically described by a friction model that linearly relates it to the effective normal stress σ^n via a friction coefficient f:τres=f σ^n=f (σn−p),[1]where σn is the normal stress acting across the fault and p is the pore pressure. Clearly, increasing pore pressure p would reduce the fault frictional resistance, promoting the insurgence of slip. However, such slip need not be fast enough to radiate seismic waves, as would be characteristic of an earthquake, but can be slow and aseismic. In fact, the critical spatial scale h* for the slipping zone to reach in order to initiate an unstable, dynamic event is inversely proportional to the effective normal stress (27, 28) and hence increases with increasing pore pressure, promoting stable slip. This stabilizing effect of increasing fluid pressure holds for both linear slip-weakening and rate-and-state friction; it occurs because lower effective normal stress results in lower fault weakening during slip for the same friction properties. For example, the general form for two-dimensional (2D) theoretical estimates of this so-called nucleation size, h*, on rate-and-state faults with steady-state, velocity-weakening friction is given by:h*=(μ*DRS)/[F(a,b)(σn−p)],[2]where μ*=μ/(1−ν) for modes I and II, and μ*=μ for mode III (29); DRS is the characteristic slip distance; and F(a, b) is a function of the rate-and-state friction parameters a and b. The function F(a, b) depends on the specific assumptions made to obtain the estimate: FRR(a,b)=4(b−a)/π (ref. 27, equation 40) for a linearized stability analysis of steady sliding, or FRA(a,b)=[π(b−a)2]/2b, with a/b>1/2 for quasistatic crack-like expansion of the nucleation zone (ref. 30, equation 42).Hence, an increase in pore pressure induces a reduction in the effective normal stress, which both promotes slip due to lower frictional resistance and increases the critical length scale h*, potentially resulting in slow, stable fault slip instead of fast, dynamic rupture. Indeed, recent field and laboratory observations suggest that fluid injection triggers slow slip first (4, 9, 11, 31). Numerical modeling based on these effects, either by themselves or with an additional stabilizing effect of shear-layer dilatancy and the associated drop in fluid pressure, have been successful in capturing a number of properties of slow-slip events observed on natural faults and in field fluid-injection experiments (14, 24, 32–34). However, understanding the dependence of the fault response on the specifics of pore-pressure increase remains elusive. Several studies suggest that the nucleation size can depend on the loading rate (35–38), which would imply that the nucleation size should also depend on the rate of friction strength change and hence on the rate of change of the pore fluid pressure. The dependence of the nucleation size on evolving pore fluid pressure has also been theoretically investigated (39). However, the commonly used estimates of the nucleation size (Eq. 2) have been developed for faults under spatially and temporally uniform effective stress, which is clearly not the case for fluid-injection scenarios. In addition, the friction properties themselves may change in the presence of fluids (40–42). The interaction between shear and fluid effects can be further affected by fault-gauge dilation/compaction (40, 43–45) and thermal pressurization of pore fluids (42, 46–48).Recent laboratory investigations have been quite instrumental in uncovering the fundamentals of the fluid-faulting interactions (31, 45, 49–57). Several studies have indicated that fluid-pressurization rate, rather than injection volume, controls slip, slip rate, and stress drop (31, 49, 57). Rapid fluid injection may produce pressure heterogeneities, influencing the onset of slip. The degree of heterogeneity depends on the balance between the hydraulic diffusion rate and the fluid-injection rate, with higher injection rates promoting the transition from drained to locally undrained conditions (31). Fluid pressurization can also interact with friction properties and produce dynamic slip along rate-strengthening faults (50, 51).In this study, we investigate the relation between the rate of pressure increase on the fault and spontaneous rupture nucleation due to fluid injection by laboratory experiments in a setup that builds on and significantly develops the previous generations of laboratory earthquake setup of Rosakis and coworkers (58, 59). The previous versions of the setup have been used to study key features of dynamic ruptures, including sub-Rayleigh to supershear transition (60); rupture directionality and limiting speeds due to bimaterial effects (61); pulse-like versus crack-like behavior (62); opening of thrust faults (63); and friction evolution (64). A recent innovation in the diagnostics, featuring ultrahigh-speed photography in conjunction with digital image correlation (DIC) (65), has enabled the quantification of the full-field behavior of dynamic ruptures (66–68), as well as the characterization of the local evolution of dynamic friction (64, 69). In these prior studies, earthquake ruptures were triggered by the local pressure release due to an electrical discharge. This nucleation procedure produced only dynamic ruptures, due to the nearly instantaneous normal stress reduction.To study fault slip triggered by fluid injection, we have developed a laboratory setup featuring a hydraulic circuit capable of injecting pressurized fluid onto the fault plane of a specimen and a set of experimental diagnostics that enables us to detect both slow and fast fault slip and stress changes. The range of fluid-pressure time histories produced by this setup results in both quasistatic and dynamic rupture nucleation; the diagnostics allows us to capture the nucleation processes, as well as the resulting dynamic rupture propagation. In particular, here, we explore two injection techniques: procedure 1, a gradual, and procedure 2, a sharp fluid-pressure ramp-up. An array of strain gauges, placed on the specimen’s surface along the fault, can capture the strain (translated into stress) time histories over a wide range of temporal scales, spanning from microseconds to tens of minutes. Once dynamic ruptures nucleate, an ultrahigh-speed camera records images of the propagating ruptures, which are turned into maps of full-field displacements, velocities, and stresses by a tailored DIC) analysis. One advantage of using a specimen made of an analog material, such as poly(methyl meth-acrylate) (PMMA) used in this study, is its transparency, which allows us to look at the interface through the bulk and observe fluid diffusion over the interface. Another important advantage of using PMMA is that its much lower shear modulus results in much smaller nucleation sizes h* than those for rocks, allowing the experiments to produce both slow and fast slip in samples of manageable sizes.We start by describing the laboratory setup and the diagnostics monitoring the pressure evolution and the slip behavior. We then present and discuss the different slip responses measured as a result of slow versus fast fluid injection and interpret our measurements by using the rate-and-state friction framework and a pressure-diffusion model.     

Nonlinear nanoelectrodynamics of a Weyl metal

Chiral Weyl fermions with linear energy-momentum dispersion in the bulk accompanied by Fermi-arc states on the surfaces prompt a host of enticing optical effects. While new Weyl semimetal materials keep emerging, the available optical probes are limited. In particular, isolating bulk and surface electrodynamics in Weyl conductors remains a challenge. We devised an approach to the problem based on near-field photocurrent imaging at the nanoscale and applied this technique to a prototypical Weyl semimetal TaIrTe₄. As a first step, we visualized nano-photocurrent patterns in real space and demonstrated their connection to bulk nonlinear conductivity tensors through extensive modeling augmented with density functional theory calculations. Notably, our nanoscale probe gives access to not only the in-plane but also the out-of-plane electric fields so that it is feasible to interrogate all allowed nonlinear tensors including those that remained dormant in conventional far-field optics. Surface- and bulk-related nonlinear contributions are distinguished through their “symmetry fingerprints” in the photocurrent maps. Robust photocurrents also appear at mirror-symmetry breaking edges of TaIrTe₄ single crystals that we assign to nonlinear conductivity tensors forbidden in the bulk. Nano-photocurrent spectroscopy at the boundary reveals a strong resonance structure absent in the interior of the sample, providing evidence for elusive surface states.

Nonlinear optical effects emerge when a material’s polarization density reacts to second- or higher-order powers of the electric field of light (1). For example, the second-harmonic generation (SHG) describes the phenomenon where the incoming light frequency ω is transformed to 2ω after interacting with the nonlinear medium, characterized by the SHG susceptibility χ(2)(2ω;ω,ω). A close counterpart of the SHG is the bulk photovoltaic effect (BPVE), where a direct current (dc) is generated in the bulk of the sample with finite frequency excitation at ω, described by the nonlinear conductivity σ(2)(0;ω,−ω). Optical nonlinearities play a preeminent role in modern photonics. Applications aside, nonlinear effects are connected to geometrical properties of the electronic wavefunctions (2–5) often categorized under the notions of Berry curvature/connection and have emerged as key probes of topological effects in semimetals (6–12).Along with band topologies, crystal symmetries play a preeminent role in optical nonlinearities. Nonmagnetic Weyl semimetals break the inversion symmetry and therefore fulfill the requirement for the observation of various second-order nonlinear responses. Extraordinarily strong SHG has been observed in the Weyl semimetal TaAs (6). However, the energy scale associated with the large SHG (≈1.5 eV) exceeds that of the Weyl bands by at least an order of magnitude (13, 14) and contributions irrelevant to Weyl physics have to be considered. Since the Berry curvature Ω associated with the Weyl points is divergent in the momentum space (3, 5) as Ω∼±1/k2, low-energy midinfrared and terahertz photons are best suited to probe the electrodynamics associated with Weyl points. Indeed, giant BPVE [also known as shift current (15, 16)] has been observed in TaAs in the midinfrared at λ=10.6 μm (ω≈943 cm^{– 1}) (7) and in TaIrTe₄ at λ=4 μm (ω=2,500 cm−1) (8). The shift current tensor σ(2) encodes the nonlinear light–matter interaction in a steady-state dc Ji=σijk(2)Ej(ω)Ek(ω)*, where Ej(ω)=E0,jeiωt (j=a,b or c) is the light electric field. Beyond bulk Weyl points, another defining aspect of the Weyl semimetal is the unique surface states connecting the projection of the bulk Weyl points (13, 14). Due to the metallic nature of the bulk, such surface conducting channels are extremely challenging to disentangle from bulk response using linear transport and conventional optical measurements (17).Nonlinear responses offer a symmetry-based approach to distinguish the surface and bulk responses. Indeed, both SHG (18) and photocurrent (19) are adept at isolating the surface states response of a three-dimension topological insulator from that of the bulk because the latter vanishes by virtue of inversion symmetry. However, this elegant approach is inept for nonmagnetic Weyl semimetals where inversion symmetry is broken both in the bulk and at the surfaces. Novel nonlinear observables with the bulk and surface selectivity are therefore highly desirable. In this work, we establish near-field nano-photocurrent imaging as a powerful symmetry-sensitive approach for investigating nonlinear optical responses with a metallic tip. Different nonlinear conductivity tensors are differentiated by their distinct real-space photocurrent patterns. The net effect is that the bulk and surface contributions of Weyl semimetals are disentangled by their inherent “symmetry fingerprint” in real space.We utilized a near-field optical microscope for nano- photocurrent measurements in the midinfrared range (11 to 4.5 µm) with ≈30-nm spatial resolution limited by the metallic tip apex (Fig. 1A). We focused on the type-II Weyl semimetal TaIrTe₄ (20, 21) and the measured nano-photocurrent pattern shows intriguing reversals in sign that depend on the crystal axis. This pattern was successfully reproduced using the Shockley–Ramo (SR) formalism adapted for nonlocal photocurrent response (22). We augmented this analysis approach for experimental settings with an optical antenna: the metallic tip in our near-field apparatus. The antenna simultaneously provides the in-plane (E_a, E_b) and the out-of-plane (E_c) electric fields and generates the nonlinear shift current via Ji=σiic(2)Ei(ω)Ec(ω)* (i = a or b). The photocurrent patterns in the interior of the crystals reveal strong directionality in TaIrTe₄. The spectral characteristics of the nano-photocurrent in the interior include a broad maximum around 1,400 cm^{– 1} that is accounted for by the density functional theory (DFT) calculations. We also discovered robust photocurrent responses near the edges of the crystal that can be understood with a nonlinear tensor forbidden in the bulk. The spectral response of edges reveals an additional resonance structure compared to the interior photocurrent. We discuss the spectroscopic features of these resonances in relation to the expected surface states in TaIrTe₄.Open in a separate windowFig. 1.Near-field nano-optical and nano-photocurrent experiments on TaIrTe₄. (A) Schematic for nano-infrared and nano-photocurrent measurements where the scattering (S) and zero-bias photocurrent (IPC) signals are collected simultaneously. (B) Simulation of the field enhancement for the z-component of the electric field (E_z) for a metallic tip (≈ 30-nm radius) close to TaIrTe₄ surface with p-polarized incident light. (C) Crystal structure of TaIrTe₄ for the side view (Upper) and top view (Lower). (D) Nano-infrared image of the scattering amplitude at ω=1,600 cm−1 (λ=6.25 μm) for a 12-nm-thick sample (Sample 1) at ambient conditions. (Lower) The linecuts along the a-axis (red solid line) and b-axis (red dashed line). (E) Nano-photocurrent map of the same region as in D, showing clear direction-switching pattern near the Au contact. (Lower) The linecuts of IPC taken along the same paths as in D. (Scale bars in D and E, 600 nm.)We exfoliated the bulk crystal and deposited gold electrical contacts on the thin TaIrTe₄ flakes, as shown schematically in Fig. 1A (see Methods for details). A midinfrared laser was focused on a metallic atomic force microscope tip that raster-scans across the sample in the tapping mode. Local near-field signal is scattered out by the tip and detected in the far field while the photocurrent signal is amplified and detected via one of the Au contacts. Both the scattering signal and the photocurrent signal are demodulated at high harmonics of the tip-tapping frequency to suppress the far-field background (see Methods). The deep-subdiffractional mapping concurs with strong field enhancement under the metallic tip (Fig. 1B), allowing access of the nonlinear shift current tensor components with z axis electric field.We begin with a direct comparison of the scattering and photocurrent signals for a 12-nm-thick TaIrTe₄ nanocrystal (Sample 1). In Fig. 1 D and E we display the local near-field scattering amplitude and photocurrent signal collected at laser frequency ω=1,600 cm−1 at room temperature, respectively. At ω=1,600 cm−1, the reflectance contrast along a- and b-axis is around R^a (ω)/R^b (ω) = 1.7 (SI Appendix, Fig. S1). However, this anisotropy is not readily captured in antenna-based nano-infrared experiments (23, 24). Indeed, Fig. 1D shows a homogeneous scattering amplitude along the a- and b-axis. Remarkably, the nano-photocurrent map in Fig. 1E evinces a completely different picture. We observe strong “hot spots” near the Au contact and the polarity of the photocurrent at these hot spots depends on the crystal axes. The photocurrent signal also vanishes close to the sample edge (black dashed line) aligned with the a-axis of the crystal. We emphasize that in Fig. 1E the wavelength of laser light (λ=6.25 μm) is larger than the entire image and the direction-switching behavior will be smeared out with conventional far-field scanning photocurrent measurements. Furthermore, through the analysis of signal decay with the increasing tip-tapping harmonic, we confirmed that the observed nano-photocurrent signal originates from the local evanescent fields underneath the tip (see SI Appendix, Figs. S2 and S3).The characteristic direction-switching photocurrent pattern is also robust with changing laser frequencies (SI Appendix, Fig. S17). In Fig. 2A, the nano-photocurrent at λ=4.5 μm (ω≈2,222.2 cm−1) for Sample 1 is shown, with direction-switching pattern on both the collecting and the ground contacts. To gain insights into the spatial nanotextures of the photocurrent, we utilized the standard SR formalism (22, 25) to model the nano-photocurrent map. The SR theorem provides a basis for understanding complex spatial patterns of photocurrent where current collection is nonlocal. Specifically, the measured current (I_PC) between electrical contacts is related to the local source of photocurrent jloc viaIPC=C∫jloc(r)·∇ψ(r)d2r,[1]where C is a prefactor governed by the device configuration. ∇ψ(r) is an auxiliary weighting field determined by solving the Laplace equation for a suitable potential ψ(r) and boundary conditions. Modeling based on the SR formalism has been successfully applied to graphene (26, 27) and topological insulators (28) to interpret experimental real-space photocurrent patterns. The SR analysis involves first solving the auxiliary weighting field ∇ψ(r) and then collecting the product of local photocurrent jloc(r) and ∇ψ(r). Among the seven symmetry-allowed shift current tensors in TaIrTe₄ (σaac=σaca, σbbc=σbcb, σ_caa, σ_cbb, σ_ccc), only σ_aac and σ_bbc give rise to in-plane photocurrent (8), provided out-of-plane electric fields are present. In our experiment and simulations, the tip provided both the in-plane and out-of-plane electric fields that generate local in-plane photocurrent in the sample via jloc(r)=(σaacEaa^+σbbcEbb^)Ec. We remark that while the c-axis electric field (E_c) originating from the tip is rotationally symmetric and has a definite sign (Fig. 1B), the in-plane components (E_a and E_b) have a dipole-like pattern, as shown in Fig. 2B. Both the c-axis (Fig. 1B) and in-plane tip electric fields (Fig. 2B) were simulated based on the experimental configurations taking into account the dielectric functions of TaIrTe₄ (SI Appendix, Fig. S1). The calculated electric field distribution can be well-approximated with the simple point-charge model Ea,b(r)∝r/(r2+d2)3/2 and Ec(r)∝d/(r2+d2)3/2 for the tip, where d is the height of the charge and r is the in-plane distance away from it (see SI Appendix, Figs. S5 and S6).Open in a separate windowFig. 2.Nano-photocurrent experiment and modeling based on the SR theorem. (A) Nano-photocurrent map of Sample 1 at λ=4.5 μm, showing similar direction-switching behavior as in Fig. 1E. (B) Tip-mediated nonlinear photocurrent generation in the interior of the sample. Color plots are numerical calculation of the tip electric fields in the sample. (C) Magnitude and direction of the jloc (red arrows) on a 100-nm-diameter circle centered at the tip position according to jloc(r)=(σaacEaa^+σbbcEbb^)Ec. Green arrows are schematics of the auxiliary field distribution ∇ψ. (D) Model simulation of the nano-photocurrent pattern with the jloc(r) profile in C, showing good agreement with the experiment. (E) Model simulation using jb=σbaaEaEa, showing no direction-switching pattern near the contact. (Inset) The simulated EaEa distribution and the corresponding photocurrent (red arrows). (F) Model simulation using ja=σaabEaEb, showing more sign changes near the contacts compared to the experiment in A. (Inset) The simulated EaEb distribution and the corresponding photocurrent (red arrows). The magnitudes of the simulated photocurrent in E and F are scaled by 0.05 and 100 times, respectively. a.u., arbitrary units.In general, the two shift current tensors σ_aac and σ_bbc in an anisotropic material are different and may have opposite signs (3, 5). Accordingly, the direction of jloc deviates from the direction of the local field produced by the tip (Eaa^+Ebb^). Therefore, both the magnitude and the direction of jloc varies as a function of in-plane angle with respect to the tip. In Fig. 2C, we illustrate an example of the jloc calculated using the point-charge model of the optical antenna with σaac:σbbc=−4:1. The magnitude of the generated photocurrent jloc(r) follows the decay of the tip electric fields away from the tip center position rtip. As a result, jloc(r) from a finite area of the sample will contribute to the final collected IPC(rtip) at each tip location rtip (see SI Appendix, section III). We emphasize that jloc(r) reverses sign when r is reversed, as shown by the two red arrows connected with a green dashed arrow in Fig. 2C. This symmetry of jloc(r) is imposed by the tip electric fields as Ea,b(r) is odd in r (Fig. 2B) while Ec(r) is even in r (Fig. 1B). Because IPC depends both on jloc(r) and ∇ψ(r), finite difference in ∇ψ(r) (green solid arrows in Fig. 2C) will lead to finite photocurrent even in the interior of the sample. This nanoscale asymmetry in ∇ψ(r) combined with the nonlinear mechanism of jloc(r) leads to the observed direction-switching behavior around the contact.The combination of SR formalism (Eq. 1) and the tip-mediated shift current generation (Fig. 2 B and C) determine the IPC at any point inside the sample, thus facilitating detailed comparisons with experiments. Indeed, the real-space simulation shown in Fig. 2D exhibits an excellent agreement with the experimental photocurrent map (Fig. 2A) using jloc(r)=(σaacEaa^+σbbcEbb^)Ec. The direction-switching behavior near the contact is captured in the simulation using σaac:σbbc=−4:1 (Fig. 2C). We remark that direction-switching pattern is only weakly sensitive to the precise ratio of σaac:σbbc (SI Appendix, Fig. S11) but is strongly influenced by the dc anisotropy ratio σ0,a/σ0,b. With increasing σ0,a/σ0,b, the “zero-crossing” photocurrent direction (green dashed line in Fig. 2A) will bend toward the a-axis of the crystal (SI Appendix, Fig. S9). We therefore conclude that our observed photocurrent pattern is dominated by the bulk TaIrTe₄ through shift current tensors σ_aac and σ_bbc.The tip-promoted shift current generation mechanism yields verifiable predictions for other nonlinear tensors. For example, the surface of the TaIrTe₄ breaks the glide mirror symmetry (Mb˜) (10) and allows σ_baa and σ_aab to appear at the surface and contribute to jloc(r). Since jb=σbaaEaEa depends on even order of the in-plane electric field E_a, it always points to the same direction (Fig. 2E inset). Therefore, the direction-switching photocurrent pattern near the contact is absent for σ_baa (Fig. 2E). On the other hand, EaEb has a quadruple-like pattern (Fig. 2F, Inset) and leads to additional sign reversals for ja=σaabEaEb (Fig. 2F). Furthermore, we also considered the photothermoelectric (PTE) effect [reported in nano-photocurrent experiments on graphene (29–31)] and the direction-switching pattern is also absent in the PTE simulation (SI Appendix, Fig. S10). We refer to these distinct photocurrent patterns (Fig. 2 D–F) as the inherent “symmetry fingerprints” of the underlying nonlinear conductivities.By analyzing the photocurrent along the perimeter of the TaIrTe₄ crystal we gain a complimentary access to the surface response of a Weyl semimetal. To explore the photocurrent at the boundary, an additional device with contacts along both a- and b-axis was fabricated and investigated: Sample 2 in Fig. 3A. The nano-photocurrent map demonstrates direction-switching behavior near the Au contacts similar to that in Figs. 1E and 2A (Sample 1). Remarkably, we also observe prominent photocurrent localized near the boundary of the sample, in stark contrast to Sample 1 with the edge oriented along the a-axis. These contrasting trends are understood from the symmetry of the edges within the SR formalism (26, 32). In TaIrTe₄, the mirror symmetry M_a (Fig. 1C) will force any local boundary photocurrent to be perpendicular to the a-axis. Since the boundary condition ensures that only tangential components of ∇ψ(r) exist near the sample boundaries (Fig. 3B), no photocurrent will be collected if mirror symmetries force jloc to be perpendicular to the edge. The observation of boundary photocurrent in Sample 2 (Fig. 3A) therefore suggests local photocurrent flowing parallel to the mirror-symmetry-breaking edges (j∥ in Fig. 3B). Among the six shift current tensors allowed at the boundaries (SI Appendix, Fig. S12), we find that σ_aaa and σ_abb give a consistent boundary photocurrent pattern as observed in the experiment. In Fig. 3C the simulation using jloc(r)=(σaacEaa^+σbbcEbb^)Ec for the interior and ja=σaaaEaEa at the edge is shown, again exhibiting good agreement with the data (Fig. 3A). The mirror-symmetry-braking edges therefore offer an additional photocurrent response devoid of bulk contributions.Open in a separate windowFig. 3.Boundary photocurrent in TaIrTe₄. (A) Experimental photocurrent map in a four-terminal device (Sample 2), displaying direction-switching real-space pattern near the ground (top left) and collecting (bottom left) contacts. The two contacts on the right are floated. (Inset) An optical image of the device. (B) Simulation of the auxiliary field (green arrow) for Sample 2 under the SR scheme. Red arrows indicate local photocurrent jloc generated near the boundary of the sample. (C) Model simulation of the nano-photocurrent pattern with the jloc(r) profile in Fig. 2C for the interior and additional boundary photocurrent contribution (Inset), showing good agreement with the experiment. (D) Band structure of TaIrTe₄ showing two of the four Weyl points at around 0.1 eV above the Fermi energy (E_F). (Bottom Inset) The finite projection of Fermi-arc states on a mirror-symmetry breaking edge (black line). (E) Photocurrent (red) and scattering amplitude (black) along the black dashed line in A. The photocurrent is peaked at the physical edge (L≈200 nm). The gray solid line is the corresponding topography profile. (F) Numerical simulation of the in-plane electric field (E_a) in the ac-plane (side view) for tip position located 40 nm away from the sample edge.The boundary photocurrent signals described by σ_aaa and σ_abb may entail the surface state response in TaIrTe₄. In Fig. 3D, the band structure of TaIrTe₄ near the Weyl points obtained from DFT calculations is shown. The bottom schematic illustrates the nonvanishing projection of the Fermi-arc surface state (orange curve) on the mirror-symmetry-breaking edge and surface. We examine the photocurrent and the concurrent near-field scattering amplitude through a linecut across the edge (black dashed line in Fig. 3A), shown in Fig. 3E. The scattering amplitude (S₄) is enhanced at the boundary of the sample (L≈200 nm) and keeps increasing gradually toward the interior of the sample. In contrast, the photocurrent signal is peaked at the boundary and slowly decays in the interior. Such a slow decay of boundary photocurrent can be understood by the slow decay of in-plane electric field E_a. Physically, the photocurrent in Fig. 3E contain contributions from both the one-dimensional boundary in the ab-plane and the side surface of the sample due to its finite thickness. In Fig. 3F (SI Appendix, Fig. S6) the simulated in-plane electric fields on the side (top) surface remain prominent even as the tip is 40 nm (≈100 nm) away from the edge. As a result, the photocurrent due to the boundary and side surface could still be collected as the tip is located a few hundred nanometers away from the edge.Complementary to spatial patterns, spectroscopic measurements offer additional insights on the origin of the photocurrent. We begin with the linear optical conductivity spectra extracted from the broadband infrared reflectance spectra (SI Appendix, Fig. S1). In Fig. 4A, the real part of the experimental optical conductivity (σ(ω)=σ1(ω)+iσ2(ω)) at T = 10 K is compared with DFT calculations (0 K) for both the a- and b-axis. The overall agreement between the experimental σ1(ω) and DFT calculations for TaIrTe₄ is excellent among topological semimetals (33, 34). The dramatically enhanced optical conductivity along the a-axis compared to the b-axis is consistent with the quasi-one-dimensional Ta and Ir chains along the a-axis of the crystal (Fig. 1C).Open in a separate windowFig. 4.Linear and nonlinear spectroscopy of TaIrTe₄. (A) Comparison of experimental linear optical conductivity at 10 K (black) with DFT calculations (blue solid and dashed lines). Red-shaded region corresponds to the frequency range where the photocurrent spectroscopy measurements were performed. (B) Photocurrent spectroscopy for the interior of the sample (red dots), showing a narrow peak due to the SiO₂ phonon and a broad peak near 1,400 cm^{– 1}. Gray dotted line is the DFT calculation of σeff(2) at 300 K. Red dotted line is the calculated near-field scattering amplitude square using experimental dielectric function at 300 K (SI Appendix, Fig. S1). (C) Photocurrent spectroscopy of the edge response normalized by the interior response at different tip-tapping harmonics (n = 1, 2, 3). The normalized spectra show asymmetric behaviors that are fitted by a Fano line shape (dotted line). (Inset) A schematic of the extent of the tip electric field into the sample for different tip-tapping harmonics.Near-field photocurrent spectroscopy augments the nano-photocurrent pattern with crucial frequency-domain information. We performed spectroscopic nanoimaging along the gray line in Fig. 3A, which contains both the boundary and interior regions. The photocurrent amplitude at the boundary and in the interior are spatially averaged to compare their frequency dependence. In Fig. 4B, the interior photocurrent spectrum is represented as red dots, with a peak near 1,130 cm^{– 1} that comes from the SiO₂ phonon (35). Interestingly, this phonon feature from the substrate can be well-described by the square of the near-field scattering amplitude (|S|2 red dotted line) calculated via the lightning-rod model (36) (see SI Appendix, section V). The scattering amplitude S encodes the local tip–sample interaction but is still a linear function of the dielectric function (or equivalently the optical conductivity σ(ω)). As a result, large deviation appears near 1,400 cm^{– 1}, where a broad peak appears in the photocurrent yet is absent in the |S|2 calculation based on experimental σ(ω). The DFT calculation for the effective shift current tensor σeff(2)(ω)=|σaac+σbbc| (dotted gray line) reproduces the photocurrent peak near 1,400 cm^{– 1}, further attesting to the nonlinear shift current mechanism observed through real-space patterns (Figs. 2 and 3).The agreement of the interior photocurrent with DFT calculations of bulk shift current tensors provide a basis for understanding the photocurrent at the boundary. We normalized the boundary photocurrent by the interior photocurrent as depicted in Fig. 4C for different tip-tapping harmonics. Interestingly, a clear resonance near 1,240 cm^{– 1} (≈ 0.15 eV) appears in all three harmonics. Such a resonance is distinct from the SiO₂ resonance (≈ 1,130 cm^{– 1}) and is also not predicted by the bulk shift current calculations (Fig. 4B). As we have demonstrated from spatial modeling, the photocurrent at the boundary originates from σ_aaa and/or σ_abb allowed at mirror-symmetry-breaking edges. It is likely that this additional resonance comes from σ_aaa and σ_abb where surface-state-related optical transitions around 0.15 eV can contribute (21, 37, 38). Such a scenario is consistent with the tip-modulation pattern of the normalized photocurrent spectrum and also with the asymmetric spectral shape of the resonance. As demonstrated in near-field studies of topological insulator surface states (39), higher tip-tapping harmonics are dominated by response of the surface since the electric fields are more confined spatially (Fig. 4C, Inset). Compared to n = 1, the normalized IPC,n at n = 2 and 3 indeed show less influence from the SiO₂ resonance at 1,130 cm^{– 1} and stronger resonance near 1,240 cm^{– 1}. Furthermore, the asymmetric lineshape of IPC,2 and IPC,3 can be well-described by a fit with a Fano-type line shape (40): IPC∝[(ω−ω0)/γ+q]21+(ω−ω0)2/γ2+b (fitting parameters are listed in SI Appendix, Table S2). This formula describes the interaction of a resonant transition at frequency ω₀ and a continuum of background transitions, where γ is the width of the resonance, b is an offset, and q is the Fano parameter. The good agreement of the Fano-type asymmetric lineshape and the harmonic dependence of the boundary photocurrent provide evidence for the elusive surface states in TaIrTe₄.In this work, we investigated the Weyl semimetal TaIrTe₄ with near-field nano-photocurrent imaging and spectroscopy. Unique direction-switching patterns appear in the photocurrent map and are successfully identified as tip-mediated nonlinear shift current through extensive spatial modeling. Surface- and bulk-related nonlinear contributions are further distinguished through their distinct “symmetry fingerprints” in the photocurrent maps. We also discovered robust boundary photocurrent features that are not explained by the bulk nonlinear response. Asymmetrical spectral line shape and the harmonic dependence of the nano-photocurrent spectra further support the surface-state origin of the photocurrent at the boundary. Applying near-field probes to topological semimetals constitutes a new paradigm of research on nanoscale nonlinearity. Our combined experimental and theoretical approach also paves the way for investigating more complex correlated electronic phenomena in real space (41, 42).     

On randomly changing conformity bias in cultural transmission

Humans and nonhuman animals display conformist as well as anticonformist biases in cultural transmission. Whereas many previous mathematical models have incorporated constant conformity coefficients, empirical research suggests that the extent of (anti)conformity in populations can change over time. We incorporate stochastic time-varying conformity coefficients into a widely used conformity model, which assumes a fixed number n of “role models” sampled by each individual. We also allow the number of role models to vary over time (nt). Under anticonformity, nonconvergence can occur in deterministic and stochastic models with different parameter values. Even if strong anticonformity may occur, if conformity or random copying (i.e., neither conformity nor anticonformity) is expected, there is convergence to one of the three equilibria seen in previous deterministic models of conformity. Moreover, this result is robust to stochastic variation in nt. However, dynamic properties of these equilibria may be different from those in deterministic models. For example, with random conformity coefficients, all equilibria can be stochastically locally stable simultaneously. Finally, we study the effect of randomly changing weak selection. Allowing the level of conformity, the number of role models, and selection to vary stochastically may produce a more realistic representation of the wide range of group-level properties that can emerge under (anti)conformist biases. This promises to make interpretation of the effect of conformity on differences between populations, for example those connected by migration, rather difficult. Future research incorporating finite population sizes and migration would contribute added realism to these models.

Cavalli-Sforza and Feldman (1) studied the finite population dynamics of a trait whose transmission from one generation to the next depended on the mean value of that trait in the population. This “group transmission” constrained the within-group variability but could lead to increasing variance in the average trait value between groups. Other analyses of cultural transmission biases have incorporated characteristics of trait variation, such as the quality, and characteristics of transmitters, including success and prestige (2). Another class of transmission biases is couched in terms of the frequencies of the cultural variants in the population (3). These “frequency-dependent” biases include conformity and anticonformity, which occur when a more common variant is adopted at a rate greater or less than its population frequency, respectively (4).Humans have exhibited conformity in mental rotation (5), line discrimination (6), and numerical discrimination tasks (7). Anticonformity has been exhibited by young children performing numerical discrimination (7). Unbiased frequency-dependent transmission, known as random copying (8), has been suggested to account for choices of dog breeds (9), Neolithic pottery motifs, patent citations, and baby names (10, 11). However, baby name distributions appear more consistent with frequency-dependent (8, 12) and/or other (13, 14) biases.In nonhuman animals, conformity has been observed in nine-spined sticklebacks choosing a feeder (15) and great tits solving a puzzle box (16, 17) (but see ref. 18). Fruit flies displayed both conformist and anticonformist bias with respect to mate choice (19) (but these authors used a different definition of anticonformity from that of ref. 4, which we use, and therefore did not consider these behaviors to be anticonformist).Asch (20, 21) used a different definition of conformity from ref. 4, namely “the overriding of personal knowledge or behavioral dispositions by countervailing options observed in others” (ref. 22, p. 34). Aschian conformity (22) has been observed in chimpanzees (23, 24), capuchin monkeys (25, 26) (but see ref. 27), vervet monkeys (28), and great tits (16). It has also been empirically tested in at least 133 studies of humans and, in the United States, has declined from the 1950s to the 1990s (29).Temporal variation may also occur in forms of conformity other than Aschian. In ref. 12, popular US baby names from 1960 to 2010 show a concave turnover function indicative of negative frequency-dependent bias, but male baby names from earlier decades (1880 to 1930) show a convex turnover indicative of positive frequency-dependent or direct bias. However, most previous mathematical models of conformity have incorporated constant, rather than time-dependent, conformity coefficients.Cavalli-Sforza and Feldman (ref. 3, chap. 3) and Boyd and Richerson (ref. 4, chap. 7) studied models of frequency-dependent transmission of a cultural trait with two variants. Boyd and Richerson (4) incorporated conformist and anticonformist bias through a conformity coefficient denoted by D. In their simplest model, if the frequency of variant A is p and that of variant B is 1−p, then the frequency of variant A in the offspring generation, p′, isp′=p+Dp(1−p)(2p−1),[1]where D>0 entails conformity (A increases if its frequency is p>12), D<0 entails anticonformity, D=0 entails random copying, and −2<D<1. In this model, each offspring samples the cultural variants of n=3 members of the parental generation (hereafter, role models). Sampling n>3 role models requires different constraints and, if n>4, there are multiple conformity coefficients (Eq. 19).Many subsequent models have built upon Boyd and Richerson’s (4) simplest model (Eq. 1). These have incorporated individual learning, information inaccuracy due to environmental change (30–34), group selection (35), and other transmission biases, including payoff bias (36), direct bias, and prestige bias (37). Other models, which include a single conformity coefficient and preserve the essential features of Eq. 1, incorporate individual learning, environmental variability (32, 38), group selection (39), and multiple cultural variants (38).In agent-based statistical physics models, the up and down spins of an electron are analogous to cultural variants A and B (40, 41). Individuals are nodes in a network and choose among a series of actions with specified probabilities, such as independently acquiring a spin, or sampling neighboring individuals and adopting the majority or minority spin in the sample. The number of sampled role models can be greater than three (42, 43). (Anti)conformity may occur if all (42–47), or if at least r (40, 48), sampled individuals have the same variant. In contrast, Boyd and Richerson’s (4) general model (Eq. 19) allows, for example, stronger conformity to a 60% majority of role models and weaker conformity or anticonformity to a 95% majority (in humans, this might result from a perceived difference between “up-and-coming” and “overly popular” variants).In Boyd and Richerson’s (4) general model, individuals sample n role models, which is more realistic than restricting n to 3 (as in Eq. 1); individuals may be able to observe more than three members of the previous generation. With n>4, different levels of (anti)conformity may occur for different samples j of n role models with one variant. In addition to the example above with 60 and 95% majorities, other relationships between the level of conformity and the sample j of n are possible. For example, the strength of conformity might increase as the number of role models with the more common variant increases. In a recent exploration of Boyd and Richerson’s (4) general model, we found dynamics that departed significantly from those of Eq. 1 (49). If conformity and anticonformity occur for different majorities j of n role models (i.e., j>n2), polymorphic equilibria may exist that were not possible with Eq. 1. In addition, strong enough anticonformity can produce nonconvergence: With as few as 5 role models, stable cycles in variant frequencies may arise, and with as few as 10 role models, chaos is possible. Such complex dynamics may occur with or without selection.Here, we extend both Boyd and Richerson’s (4) simplest (Eq. 1) and general (Eq. 19) models to allow the conformity coefficient(s) to vary randomly across generations, by sampling them from probability distributions. Although some agent-based models allow individuals to switch between “conformist” and “non-” or “anticonformist” states over time (40, 42, 47, 50, 51), to our knowledge, random temporal variation in the conformity coefficients themselves has not been modeled previously. In reality, the degree to which groups of individuals conform may change over time, as illustrated by the finding that young children anti-conformed while older children conformed in a discrimination task (7); thus, it seems reasonable to expect that different generations may also exhibit different levels of conformity. Indeed, generational changes have occurred for Aschian conformity (29) and possibly in frequency-dependent copying of baby names (12). Our stochastic model may therefore produce more realistic population dynamics than previous deterministic models, and comparisons between the two can suggest when the latter is a reasonable approximation to the former.We also allow the number of role models, nt, to vary over time. Agent-based conformity models have incorporated temporal (43) and individual (43, 45, 46) variation in the number of sampled individuals, whereas here, all members of generation t sample the same number nt of role models. Causes of variation in nt are not explored here, but there could be several. For instance, different generations of animals may sample different numbers of role models due to variation in population density. In humans, changes in the use of social media platforms or their features may cause temporal changes in the number of observed individuals. For example, when Facebook added the feature “People You May Know,” the rate of new Facebook connections in a New Orleans dataset nearly doubled (52).In the stochastic model without selection, regardless of the fluctuation in the conformity coefficient(s), if there is conformity on average, the population converges to one of the three equilibria present in Boyd and Richerson’s (4) model with conformity (D(j)>0 for n2<j<n in Eq. 19). These are p*=1 (fixation on variant A), p*=0 (fixation on variant B), and p*=12 (equal representation of A and B). However, their stability properties may differ from those in the deterministic case. In Boyd and Richerson’s (4) model with random copying, every initial frequency p0 is an equilibrium. Here, with random copying expected and independent conformity coefficients, there is convergence to p*=0,12, or 1. In this case, and in the case with conformity expected, convergence to p*=0,12, or 1 also holds with stochastic variation in the number of role models, nt. With either stochastic or constant weak selection in Boyd and Richerson’s (4) simplest model (Eq. 1) and random copying expected, there is convergence to a fixation state (p*=0 or 1). Finally, with anticonformity in the deterministic model or anticonformity expected in the stochastic model, nonconvergence can occur.     

Brief Report: Lattices in Tate modules

Refining a theorem of Zarhin, we prove that, given a g-dimensional abelian variety X and an endomorphism u of X, there exists a matrix A∈M2g(ℤ) such that each Tate module TℓX has a ℤℓ-basis on which the action of u is given by A, and similarly for the covariant Dieudonné module if over a perfect field of characteristic p.     

Magnetoelastic standing waves induced in UO2 by microsecond magnetic field pulses

Magnetoelastic dilatometry of the piezomagnetic antiferromagnet UO₂ was performed via the fiber Bragg grating method in magnetic fields up to 150 T generated by a single-turn coil setup. We show that in microsecond timescales, pulsed-magnetic fields excite mechanical resonances at temperatures ranging from 10 to 300 K, in the paramagnetic as well as within the robust antiferromagnetic state of the material. These resonances, which are barely attenuated within the 100-µs observation window, are attributed to the strong magnetoelastic coupling in UO₂ combined with the high crystalline quality of the single crystal samples. They compare well with mechanical resonances obtained by a resonant ultrasound technique and superimpose on the known nonmonotonic magnetostriction background. A clear phase shift of π in the lattice oscillations is observed in the antiferromagnetic state when the magnetic field overcomes the piezomagnetic switch field Hc=−18 T. We present a theoretical argument that explains this unexpected behavior as a result of the reversal of the antiferromagnetic order parameter at Hc.

The antiferromagnetic (AFM) insulator uranium dioxide UO₂ has been the subject of extensive research during the last decades predominantly due to its widespread use as nuclear fuel in commercial power reactors (1). Besides efforts to understand the unusually poor thermal conductivity of UO₂, which impacts its performance as nuclear fuel (2), a recent magnetostriction study in pulsed magnetic fields up to 92 T uncovered linear magnetostriction in UO₂ (3), a hallmark of piezomagnetism.Piezomagnetism is characterized by the induction of a magnetic polarization by application of mechanical strain, which, in the case of UO₂, is enabled by broken time-reversal symmetry in the 3-k AFM structure that emerges below TN=30.8 K (4–7) and is accompanied by a Jahn–Teller distortion of the oxygen cage (8–11). This also leads to a complex hysteretic magnetoelastic memory behavior where magnetic domain switching occurs at fields around ±18 T at T=2.5 K. Interestingly, the very large applied magnetic fields proved unable to suppress the AFM state that sets in at TN (3). These earlier results provide direct evidence for the unusually high energy scale of spin-lattice interactions and call for further studies in higher magnetic fields.Here we present axial magnetostriction data obtained in a UO₂ single crystal in magnetic fields to 150 T. These ultrahigh fields were produced by single-turn coil pulsed resistive magnets (12, 13) and applied along the [111] crystallographic axis at various temperatures between 10 K and room temperature. At all temperatures, we observe a dominant negative magnetostriction proportional to H² accompanied by unexpectedly strong oscillations that establish a mechanical resonance in the sample virtually instantly upon delivery of the 102 T/μs pulsed magnetic field rate of change. The oscillations are long-lasting due to very low losses and match mechanical resonances obtained with a resonant ultrasound spectroscopy (RUS) technique (14). Mechanical resonances were suggested to explain anomalies in magnetostriction measurements during single-turn pulses (15, 16); however, their potential to elucidate magnetic dynamics was not explored so far. When the sample is cooled below room temperature, the frequencies soften, consistent with observations in studies of the UO₂ elastic constant c₄₄ as a function of temperature (17, 18).In the AFM state, i.e., T<30.8 K, the characteristic magnetic field sign switch in our single-turn coil magnet (a feature of destructive magnets) results in applied field values in excess of the UO₂ AFM domain switch field of Hc≈−18 T. This field sign switch exposes yet another unexpected result, namely, a π (180°) phase shift in the magnetoelastic oscillations. We use a driven harmonic oscillator and an analytical model to shed light on the origin of the observed phase shift.     

Taylor’s law of fluctuation scaling for semivariances and higher moments of heavy-tailed data

We generalize Taylor’s law for the variance of light-tailed distributions to many sample statistics of heavy-tailed distributions with tail index α in (0, 1), which have infinite mean. We show that, as the sample size increases, the sample upper and lower semivariances, the sample higher moments, the skewness, and the kurtosis of a random sample from such a law increase asymptotically in direct proportion to a power of the sample mean. Specifically, the lower sample semivariance asymptotically scales in proportion to the sample mean raised to the power 2, while the upper sample semivariance asymptotically scales in proportion to the sample mean raised to the power (2−α)/(1−α)>2. The local upper sample semivariance (counting only observations that exceed the sample mean) asymptotically scales in proportion to the sample mean raised to the power (2−α2)/(1−α). These and additional scaling laws characterize the asymptotic behavior of commonly used measures of the risk-adjusted performance of investments, such as the Sortino ratio, the Sharpe ratio, the Omega index, the upside potential ratio, and the Farinelli–Tibiletti ratio, when returns follow a heavy-tailed nonnegative distribution. Such power-law scaling relationships are known in ecology as Taylor’s law and in physics as fluctuation scaling. We find the asymptotic distribution and moments of the number of observations exceeding the sample mean. We propose estimators of α based on these scaling laws and the number of observations exceeding the sample mean and compare these estimators with some prior estimators of α.

Heavy-tailed nonnegative random variables with infinite moments, such as nonnegative stable laws with index α in (0,1), have theoretical and practical importance [e.g., Carmona (1), Feller (2), Resnick (3), and Samorodnitsky and Taqqu (4)]. Heavy-tailed nonnegative random variables with some or all infinite moments have been claimed to arise empirically in finance [operational risks in Nešlehová et al. (5)], economics [income distributions in Campolieti (6) and Schluter (7); returns to technological innovations in Scherer et al. (8) and Silverberg and Verspagen (9)], demography [city sizes in Cen (10)], linguistics [word frequencies in Bérubé et al. (11)], and insurance [economic losses from earthquakes in Embrechts et al. (12) and Ibragimov et al. (13)]. Partial reviews are in Carmona (1) and Ibragimov (14).Brown et al. (15) (hereafter BCD) showed that when a random sample is drawn from a nonnegative stable law with index α∈(0,1), the sample variance is asymptotically (as the sample size n goes to ∞) proportional to the sample mean raised to a power that is an explicit function of α (Eqs. 11 and 13). This relationship generalizes to stable laws with infinite moments a widely observed power-law relationship between the variance and the mean in families of distributions with finite population mean and finite population variance. This power-law relationship is commonly known as Taylor’s law in ecology [Taylor (16, 17)] and as fluctuation scaling in physics [Eisler et al. (18)].To the two ingredients combined by BCD (nonnegative stable laws with infinite moments and Taylor’s law), this paper adds two more ingredients. We establish scaling relationships that generalize the usual Taylor’s law, for light-tailed distributions, to many functions of the sample in addition to the variance, including all positive absolute and central moments, upper and lower semivariances, and several measures of risk-adjusted investment performance such as the Sortino, Sharpe, and Farinelli–Tibiletti ratios. In addition, based on these scaling relationships, we propose several estimators of the index α of a nonnegative stable law with infinite first moment.Section 1 defines most of the sample functions studied here. Section 2 gives background on Taylor’s law, semivariances, and nonnegative stable laws, including key prior results from BCD. Section 3 establishes that the lower sample semivariance, the upper sample semivariance, the local lower sample semivariance, and the local upper sample semivariance are asymptotically each a power of the sample mean with explicitly given exponents. These results are the core of the paper. When investment returns obey a nonnegative heavy-tailed law with index α∈(0,1), these results reveal the asymptotic behavior of the Sharpe ratio, the Sortino ratio, and the Farinelli–Tibiletti ratio. Section 4 extends these results to higher central and noncentral moments and various indices of volatility. Section 5 analyzes the number of observations from a stable law or an approximately stable (i.e., regularly varying) law that exceed the sample mean. Section 6 proposes and compares estimators of α by simulation. SI Appendix gives all proofs of results stated in the text and additional numerical simulations.     

Mechanism of shaping membrane nanostructures of endoplasmic reticulum

Recent advances in super-resolution microscopy revealed the previously unknown nanoscopic level of organization of endoplasmic reticulum (ER), one of the most vital intracellular organelles. Membrane nanostructures of 10- to 100-nm intrinsic length scales, which include ER tubular matrices, ER sheet nanoholes, internal membranes of ER exit sites (ERES), and ER transport intermediates, were discovered and imaged in considerable detail, but the physical factors determining their unique geometrical features remained unknown. Here, we proposed and computationally substantiated a common concept for mechanisms of all ER nanostructures based on the membrane intrinsic curvature as a primary factor shaping the membrane and ultra-low membrane tensions as modulators of the membrane configurations. We computationally revealed a common structural motif underlying most of the nanostructures. We predicted the existence of a discrete series of equilibrium configurations of ER tubular matrices and recovered the one corresponding to the observations and favored by ultra-low tensions. We modeled the nanohole formation as resulting from a spontaneous collapse of elements of the ER tubular network adjacent to the ER sheet edge and calculated the nanohole dimensions. We proposed the ERES membrane to have a shape of a super flexible membrane bead chain, which acquires random walk configurations unless an ultra-low tension converts it into a straight conformation of a transport intermediate. The adequacy of the proposed concept is supported by a close qualitative and quantitative similarity between the predicted and observed configurations of all four ER nanostructures.

Endoplasmic reticulum (ER) is the largest and one of the central membrane-bound organelles of eukaryotic cells (1–3) whose crucial functions include the protein and lipid synthesis, exchanging the produced molecules with other intracellular organelles, and wrapping the nucleus (1). Related to its intracellular tasks, ER membrane is organized into highly intricate and inhomogeneous networks of interlinked nanotubules and sheets. ER networks exhibit a multiscale structure, which include macroscopic and nanoscopic levels of organization.The nanoscopic ER structures were discovered only recently by super-resolution microscopy and are characterized by the internal length scales in the range of 10 to 100 nm (4, 5). The four types of the ER nanostructures are the ER tubular matrices (4), the nanoholes in ER sheets (5), the internal membrane arrangements of ER exit sites (ERES), and the ER transport intermediates (6).The ER matrices are arrays of tightly packed intertubular three-way junctions connected by short tubules (4). The characteristic spacing between the junctions is of the order of 100 nm going, in some cases, down to about 50 nm (4). The ER matrices undergo rapid shape fluctuations, which include fast interconversions between the extremely dense and the relatively loose packing of the three-way junctions and strong undulations of the network plane. While, for the fluctuation reasons, the spatial arrangements of the junctions within the matrices look very irregular in most of the images, some snapshots reveal the junction packing in nearly regular nanoarrays (see Fig. 2A) (4).Open in a separate windowFig. 2.Equilibrium configurations of a fragment of hexagonal tubular network. (A) Image of ER matrix showing a regular hexagonal lattice with typical unit cell size of ∼50 nm as shown by the white bar within one of the unit cells. (Scale bar, 100 nm.) Adopted from ref. 4. (B and C) The computed configurations of an element or hexagonal tubular network for zero (B) and one (C) peristaltic blocks between neighboring junctions. Scale bars are represented by the box sides and are equal. (Scale bars, B, 16⋅ J0−1; C, 40⋅J0−1.)The sheet nanoholes are circular dynamic pores of about 100 nm diameter formed in the ER sheet plane across the 30- to 50-nm-thick sheet lumen (5, 7, 8) (see Fig. 3A).Open in a separate windowFig. 3.Generation and structure of ER sheet nanoholes. (A) ER sheet nanohole image from Schroeder et al. (5) showing confocal (Left) and zoomed-in stimulated emission depletion microscopy (STED) (Right) images of ER sheets and tubules in mammalian cells with ∼100 nm diameter nanoholes within the sheets. (B, 1–4) Computed equilibrium configurations of the interface region between a tubular network and a sheet: 1 and 2 for vanishing tension γ=0, 3 and 4 for the ultra-low tension γ=2.5μNm. The sheet thickness and the tubule intrinsic curvature are taken equal, respectively, to 50 nm andJ0=150 nm−1 according to data from Schroeder et al. (5), and the membrane-bending rigidity is κ=20⋅kBT. Scale bars are represented by the box sides. (Scale bars, 16⋅J0−1.) (C) Computed shape of a nanohole for vanishing tension γ=0 with the luminal diameter ∼65 nm (for J0=150 nm−1). (D) Left axis and blue graph represent the difference in the elastic energy between the equilibrium configurations without and with nanohole as a function of tension, γ. Right axis and red graph represent the luminal diameter of the nanohole as a function of the tension. The parameter values are κ=20⋅kBT and J0=150 nm−1, and the sheet thickness is 50 nm.The ERES internal nanostructures appear as about 300-nm large grape-like ensembles of contiguous round membrane swellings of approximately 60 nm diameter (6) (see Fig. 4E). Upon transport initiation, ERES produces ER intermediate structures, which exhibit shapes of micrometer long, few tens of nanometers thick, pearled membrane tubes with periodic varicosities (6) (see Fig. 4G).Open in a separate windowFig. 4.ERES internal membrane organization and ER transport intermediate. (A) Bead chain–like equilibrium shape of a continuous membrane whose area, A, strongly exceeds the internal length scale set by the intrinsic curvature A>J0−2; the edge is constricted to a circle of a small radius, r=0.5 J0−1, and the tension vanishes, γ=0. In the computations, the radii of the membrane necks were assumed to exceed the membrane thickness, which corresponds to r≥0.1 J0−1. The dashed shape represents the possible continuity of the chain to any arbitrary number of beads. (B) Schematic representation (solid line) of one chain element consisting of a bead with the necks connecting it to the adjacent beads (dashed lines). The angle ϕ characterizes a kink of the chain axis mediating the chain bending. (C) The chain-bending energy per one chain element as a function of the kink angle ϕ. The black dots are the calculated energies. The red curve is a fit of Eq. 6 to the computational results in the angle range between 0 and 53°, which corresponds to the assumption of small curvatures, J<l−1. (Insets) Computed shapes of the chain elements with different kink angles. (D) An example of a high-probability random orientation of a bead chain–like membrane configuration consisting of 15 elements and corresponding to a self-avoiding random walk. (Scale bars, 20⋅J0−1.) Quantitative agreement with the observed configurations (E, Left and Right) is reached for J0= 0.05nm−1. (E) Observed configuration of ERES internal structure from Weigel et al. (6). (Left) The box dimension is ∼1.2 μm. (Right) The mean dimension of a chain element is ∼60 nm, and the mean size of the entire ERES is ∼360 nm. (F) Bead chain–like configurations under different membrane tensions. The chain length equals 26⋅J0−1. (Scale bar, 5⋅J0−1(100 nm for J0=120 nm−1.) (G) Focused ion beam scanning electron microscopy (FIB-SEM) and three-dimensional rendering of transport intermediate from Weigel et al. (6).Due to the major effort of biological research over the last decade, the key proteins responsible for the generation of local curvatures of ER membranes have been largely identified (2, 9–11). Yet, the physical mechanisms determining the unique architecture of ER nanostructures have never been considered and represent the subject of this study.Here, we proposed a common mechanism of shaping of all four ER nanostructures based on the homogeneous intrinsic curvature of the membrane as the major factor determining the membrane configurations. Our computations revealed a common structural motif underlying the architecture of most of ER nanostructures and having the origin in the geometry of surfaces of constant mean curvature. The ultra-low membrane tensions are suggested to serve as a shape-modulating factor. We obtained, computationally, the membrane shapes reproducing the experimentally observed geometrical features of the ER matrices, ER sheet nanoholes, ERES internal membranes, and ER transport intermediates. We predicted the existence of the energetically degenerated conformations of the ER matrices and described their architecture. We proposed the ultra-low tensions to support the most compact configuration of ER matrices, to promote formation the ER sheet nanoholes, and to drive the transformation of ERES membranes into the ER transport intermediates.     

Non-Fermi liquid behavior below the Néel temperature in the frustrated heavy fermion magnet UAu2

The term Fermi liquid is almost synonymous with the metallic state. The association is known to break down at quantum critical points (QCPs), but these require precise values of tuning parameters, such as pressure and applied magnetic field, to exactly suppress a continuous phase transition temperature to the absolute zero. Three-dimensional non-Fermi liquid states, apart from superconductivity, that are unshackled from a QCP are much rarer and are not currently well understood. Here, we report that the triangular lattice system uranium diauride (UAu₂) forms such a state with a non-Fermi liquid low-temperature heat capacity C/T∼log (1/T) and electrical resistivity ρ(T)−ρ(0)∝T1.35 far below its Néel temperature. The magnetic order itself has a novel structure and is accompanied by weak charge modulation that is not simply due to magnetostriction. The charge modulation continues to grow in amplitude with decreasing temperature, suggesting that charge degrees of freedom play an important role in the non-Fermi liquid behavior. In contrast with QCPs, the heat capacity and resistivity we find are unusually resilient in magnetic field. Our results suggest that a combination of magnetic frustration and Kondo physics may result in the emergence of this novel state.

The understanding of metals and superconductors rests on the paradigm that their low-energy excitations are governed by quasiparticle excitations around a stable Fermi surface. This paradigm breaks down at quantum critical points (QCPs) in a metal (1, 2), where the emergent spatial- and temporal-scale invariance results in spectral functions that no longer exhibit quasiparticle poles. As the temperature is increased, a region opens up about a QCP in which various quantities, such as the heat capacity and electrical resistivity, change with temperature in different ways from a Fermi liquid. A prominent example of such a non-Fermi liquid (NFL) is the “strange metal” in the copper-oxide superconductors (3), which has a T-linear resistivity instead of a leading T² dependence characteristic of a Fermi liquid. A QCP is in this case hidden under a superconducting dome (4). Very recently, angle-resolved photoemission spectroscopy in the high-T_c cuprate family Bi2212 has challenged this picture. It revealed an abrupt recovery of quasiparticles above a temperature-independent doping level (5) coinciding with the discontinuous vanishing of the pseudogap. This suggests that the strange metal might not emerge from a QCP but could instead be a distinct NFL state of matter, and this begs the question as to whether such states occur more widely.Itinerant magnets and heavy-fermion metals exhibit QCPs and NFLs that are easier to study than the copper oxides owing to lower characteristic temperatures and the ease with which they can be tuned with pressure and magnetic fields (6). However, very few NFL states (not counting superconductivity) have been found that do not emanate from a QCP (1, 7). One rare example occurs in the high-pressure partially ordered phase of the helimagnet MnSi. It is stable over a wide pressure range and has a T3/2 electrical resistivity power law below ∼6 K down to millikelvin temperatures (8, 9). The presence of a large topological Hall signal suggests that a partially ordered magnetic structure with nontrivial topology may underlie the NFL (10, 11).β-YbAlB₄ is magnetically unordered and provides an example of NFL behavior without tuning in zero field. This NFL may be the result of a very unusual QCP at exactly zero field or instead, represent a unique quantum critical phase that is driven into a Fermi liquid state by an infinitesimal magnetic field (12). However, resistivity measurements have indicated that the NFL may still be attached to a QCP that can be reached with pressure (13).There are two well-known theoretical ways to get a three-dimensional (3D) NFL not emanating from QCPs, both based on local physics. The first is the two-channel Kondo effect, requiring an exact balance between two screening channels (14), that like a QCP, could be thought of as a point of precise tuning. NFL behavior for the two-channel Kondo model has recently been seen above quadrupole ordering temperatures in a series of cubic Pr materials with non-Kramers doublet ground states ([15]) and in related dilute systems with small concentrations of Pr replacing Y (16), as well as in earlier studies of Y1−x U_x Pd₃ (17). These local NFLs are predicted to have a temperature dependence of their electrical resistivity ρ(T)−ρ(0)∝T and be sensitive to magnetic field (18). The second way requires a range of single-channel Kondo energy scales extending to zero energy, generated by disorder (19), and results in an approximately T-linear resistivity. The NFL state we report in UAu₂ is robust over a wide range of magnetic fields and has signatures that are too strong to be attributed to impurities. UAu₂ is also crystallographically well ordered with a low residual electrical resistivity, ruling out explanations based solely on chemical disorder. We also do not see the entropy release in magnetic field that would be expected for the usual two-channel Kondo effect.For MnSi, reported thermodynamic measurements in the NFL state at high pressure are restricted to the lattice constant (20, 21). The measurements show that the thermal expansion has a low temperature dependence α∝T (20), which follows the same power law as a conventional Fermi-liquid. Here, the NFL we report in UAu₂ is manifest in thermodynamic measurements extending to very low temperature, and unlike for β-YbAlB₄, this is in a fully magnetically ordered state.UAu₂ is a little-studied heavy fermion metal known to order magnetically at 43 K (22). It has a simple AlB₂-type hexagonal crystal structure (23) comprising a vertical stacking of flat sheets of a triangular lattice of U atoms, separated by sheets of gold atoms (Fig. 1A). The calculated paramagnetic band structure is shown in Fig. 1B (calculation details and band dispersions are given in the SI Appendix). The magnetic phase diagram and magnetic structure we find are shown in Fig. 1C. After discussing magnetic and charge ordering (Fig. 2), we present our thermodynamic and transport measurements showing NFL behavior (Figs. 3 to 5), and then, we discuss these and present our conclusions.Open in a separate windowFig. 1.(A) The crystal structure of UAu₂ viewed along the c axis showing 2 × 2 unit cells in the ab plane. Au atoms (colored gold) are at c = 1/2, and U atoms (blue) are at c = 0. There is only a single crystallographic inequivalent site for each atom. The U atoms lie on a triangular lattice in the plane. In the magnetic unit cell, U atoms in the plane labeled A, B, and C are no longer equivalent to each other. Geometric frustration occurs for Ising moments with antiferromagnetic nearest neighbor interactions. To see this, consider the case where the A sites have spin ↑ and C sites have spin ↓; the energy is then the same for either choice of spin direction at every B site. (B) The calculated Fermi surface. The “crown” in blue is a hole band (band 1), and the “button” (maroon) is an electron band (band 3). Band 2 has two electron pockets: the “propeller” at the zone center and the “doughnut” at the zone edge. The orange arrows connecting menisci of the crown are nesting vectors (0,0,0.14) responsible for a peak in the static Lindhard function. Band dispersions and the density of states, decomposed into orbital contributions, are shown in SI Appendix, Fig. S1. (C) The temperature–magnetic field phase diagram determined from our measurements of intermediate- and high-quality single crystals. The transitions are detected in magnetization measurements (M) and resistivity (ρ) and with neutron scattering (SI Appendix has details). In the ferrimagnetic state (F state), A=↑, B=↓, and C is aligned with the magnetic field with no modulation along the c axis. The ordered state that occupies most of the figure is incommensurately modulated along the c direction (δ state). This magnetic structure is shown in D, which depicts seven–unit cell lengths along the c direction (vertical). The moments (red arrows) are m→∝c→cos (2πδ.z+θ). The phase angle θ differs by ±2π/3 between adjacent columns. In the region labeled δ state, a small trace of the F state was still found. For high-quality crystals, the corresponding volume fraction was below 0.6%.Open in a separate windowFig. 2.(A) The zero-field temperature dependence of the SDW and CDW amplitudes squared, normalized to their values at low temperature proportional to the integrated neutron and X-ray scattering at Bragg peaks with modulation vectors δ (neutrons, SDW) and 2δ (X-rays, CDW). Both neutron and X-ray measurements were performed on the same high-quality single crystals that contained less than 0.6% of the F state (SI Appendix). The lines through the points are guides to the eye. The SDW amplitude clearly saturates at low temperatures below 15 K and has a globally convex shape, which are conventional features. In contrast, the CDW amplitude continues to grow with decreasing temperature at the lowest temperatures and has a concave shape. The dashed line shows the square of the SDW intensity, which is the expected form for a charge modulation due to magnetostriction. The CDW in UAu₂ clearly has a very different temperature dependence, showing that it is not principally caused by magnetostriction. B shows that the value of δ is the same for both the CDW and SDW (the line is a guide to the eye).Open in a separate windowFig. 3.Low-temperature macroscopic properties of UAu₂ in zero field. A–C show (A) the heat capacity divided by temperature C / T, (B) the susceptibility χ, and (C) the resistivity ρ plotted against temperature on a logarithmic scale. There are clear signatures at the magnetic ordering temperature TN=43 K and a cross-over in behavior around T*≈20 K. In A, the solid line shows a fit to A−Blog (T) between 0.5 and 4 K. In B, the solid line is a fit to A−Blog (T) for T  <  9 K. In C, the resistivity is normalized to the value at 100 K. Inset shows the low-temperature resistivity against T1.35. The straight dashed lines indicate that the T1.35 dependence persists to ∼300 mK. A gradual increase in the exponent of resistivity to T² occurs below 300 mK. All measurements were performed on high-quality single crystals that contain less than 0.6% of the F state (SI Appendix).Open in a separate windowFig. 5.The relative change in the a and c lattice parameter as a function of temperature in zero field deduced from X-ray diffraction for the (3,0,0) and (3,0,1) diffraction peaks of high-quality single crystals that contain less than 0.6% of the F state. The solid lines are fits to Δa∝Δc∝∫C(T)dT, with C/T∼log (69/T) (the only free parameter is a constant of proportionality for each direction). This analysis is valid for a constant Grüneisen parameter.