arXiv subjects

Michal Vališka

Publications and source records attributed to Michal Vališka.

At least 19 recordsLinked to original sources

arXiv2026

Thermodynamic Evidence of Tetracritical Topology in the $H$-$T$ Phase Diagram of UTe$_2$ for $H \parallel b$

We report ultrasound velocity measurements on an ultraclean UTe$_2$ single crystal with $T_c>2$~K for $H \parallel b$, performed up to 18~T and down to 0.33~K. The measurements provide the missing bulk thermodynamic evidence for an additional high-field phase boundary near $μ_0H\sim14$--15~T. A distinct, non-hysteretic anomaly in the longitudinal $C_{33}$ mode, together with a weaker response in $C_{44}$, no resolvable anomaly in $C_{55}$, and a coincident kink in transverse magnetostriction, reveals a symmetry-selective coupling to lattice strain. This mode selectivity places constraints on the symmetry of the field-induced superconducting component. The phase line remains nearly field-constant near 14~T and meets three other phase boundaries at a tetracritical point near 13.5~T and 1.25~K. The results complete the local tetracritical topology of the $H$--$T$ phase diagram and support field-induced multicomponent superconductivity in UTe$_2$.

cond-mat.supr-con↗

arXiv2026

Evidence of intertwined pair density and charge density wave orders in UTe2

The strongly correlated spin-triplet superconductor UTe2 hosts an unusual landscape of magnetic-field-sensitive charge density wave (CDW) phases, positioning it as a compelling system for studying intertwined electronic orders. A central challenge is determining whether the observed charge modulations arise from a triplet pair density wave (PDW) order and, if so, how the anisotropic magnetic field response of triplet superconductivity is manifested in the CDW response. Here, using a scanning tunneling microscope equipped with a vector magnetic field, we systematically investigate the evolution and interrelation of distinct CDW orders. Complementing the previously identified incommensurate CDW peaks (qi=1,2,3), we resolve an additional set of nondispersive modulations (pi=1,2,3 and h1,2) with distinct temperature and magnetic field dependencies. The pi CDW peaks vanish near Tc, while the qi peaks survive well above Tc but are progressively suppressed by magnetic field in an anisotropic manner. The critical fields of the qi peaks mirror the directional hierarchy of Hc2, which suggests a PDW is present above the bulk Tc. This is consistent with a Landau free-energy picture where PDWs with wavevectors pi form above the bulk Tc, leading to composite CDW orders with wavevector qi. Below Tc, the coupling of PDWs and uniform superconductivity leads to the pi CDWs. Together, these findings establish UTe2 as a rare platform where both the parent PDW and descendant orders are directly resolved, enabling access to both the fundamental and emergent manifestations of PDW physics.

cond-mat.supr-con↗

arXiv2026

Topological surface states revealed by the Zeeman effect in superconducting UTe2

Intrinsic topological superconductors with protected boundary modes obeying non-Abelian statistics constitute a vanishingly small class of quantum materials. A defining spectroscopic signature of such phases is the presence of in-gap topological surface states (TSS). However, despite extensive theoretical proposals, their unambiguous experimental identification has remained elusive. Here we use vector magnetic-field scanning tunnelling microscopy to obtain direct spectroscopic evidence of TSS in the spin-triplet superconductor UTe2. Atomic-scale spectroscopy reveals striking site-dependent superconductivity: Te sites host a large in-gap density of states that nearly fills the superconducting gap, whereas neighboring atomic sites remain gapped. Upon application of a magnetic field, the in-gap states on the Te sites are selectively suppressed, yielding a spatially homogeneous superconducting state with a markedly deeper gap relative to zero field. This site-selective gap evolution is in quantitative agreement with theoretical predictions for TSS in UTe2 that possess dominant Te-orbital character. Spectral-function calculations incorporating the Zeeman coupling reproduce the observed magnetic-field response. Our results provide a spectroscopic fingerprint of the long-sought TSS in superconductors and establish UTe2 as a compelling system for exploring intrinsic topological superconductivity.

cond-mat.supr-con↗

arXiv2026

Crystal-Field--Driven Magnetoelectric Coupling in the Non-Kramers Hexaaluminate PrMgAl11O19

We report broadband dielectric spectra of the non-Kramers hexaaluminate PrMgAl\textsubscript{11}O\textsubscript{19}, revealing a pronounced interplay between permittivity and magnetization at cryogenic temperatures. The zero-field dielectric response follows a Barrett-type quantum-paraelectric form, while a broad dielectric anomaly near \SI{5}{K} shows a complex field dependence that mirrors the multi-hump behavior of the magnetic specific heat, evidencing robust magnetoelectric coupling. The inverse permittivity $\varepsilon'^{-1}(T,H)$ scales linearly with $M^2$, consistent with a biquadratic $P^2M^2$ term in a Landau framework. Fits yield temperature-dependent coupling constant $λ(T)$ that decreases with heating from ($1.07\pm0.01)\times10^{-4}\,μ_{\mathrm{B}}^{-2}$ (at 5\,K) to $(4.77\pm0.02)\times10^{-5}\,μ_{\mathrm{B}}^{-2}$ (at 10\,K), reflecting the thermal population of low-lying energy levels of Pr$^{3+}$. Consistently, the uniaxial thermal expansion develops an additional low-temperature hump below $\sim\SI{30}{K}$ that is progressively suppressed by magnetic field, recovering an approximately saturated response by \SI{9}{T}. These results identify PrMgAl\textsubscript{11}O\textsubscript{19} as a paradigmatic non-Kramers hexaaluminate where quantum paraelectricity and magnetoelectric interactions are intrinsically entangled, establishing hexaaluminates as a tunable platform for magnetoelectric physics in frustrated quantum materials.

cond-mat.str-el↗

arXiv2026

Observation of Kondo hybridization wave in UTe2

Condensed matter systems with strong electronic correlations often manifest a variety of intertwined ordered phases of charge, spin, orbital and other degrees of freedom. As a prototypical strongly correlated electronic system, the Kondo lattice provides fertile soil for many fascinating quantum states, including quantum criticality, unconventional superconductivity, hidden order and topological Kondo insulator/semimetal. The foundation of Kondo physics lies in the hybridization between localized moments and itinerant electrons. Generally, the evolution of Kondo hybridization is characterized as a broad crossover rather than a phase transition. Thus far, an ordered hybridization phase has not been observed. Here, we use scanning tunneling microscopy (STM) to identify a translational-symmetry-breaking order of Kondo hybridization wave(KHW) for the first time on the surface of the spin-triplet heavy-fermion superconductor UTe2. The unprecedented phase of KHW manifests as a periodically modulated Fano lattice, accompanied by a commensurate charge density wave (CDW) and a pronounced energy gap opening near the Fermi level. This KHW-imprinted CDW has an intriguing real-space texture of complementary occupation of the heavy f and conduction charges, thereby forming a Kondo superlattice. The KHW is coexistent with superconductivity in UTe2, which may provide valuable insight into its controversial spin-triplet pairing symmetry and the underlying mechanism. Our first experimental evidence for an ordered hybridization state potentially sheds new light on the strong correlation physics of Kondo lattice system.

cond-mat.str-el↗

arXiv2025

Magnetoelasticity - magnetic structure interrelation - tetragonal MnPt system study

Magnetic materials represent an essential ingredient for the contemporary industry. Apart from common material parameters such as magnetocrystalline anisotropy, coercivity, or saturation magnetization, magnetoelastic behavior is vital for applications serving in various devices, e.g., in acoustic actuators, transducers, or sensors providing a desirable fast response and high efficiency with respect to applied magnetic field. Magnetoelastic properties have been studied for ferromagnetic 3d elements, or especially in high symmetry systems containing rare-earth elements to achieve higher values. Since, unlike for rare earth Laves phases, in the transition metals or alloys, these effects are very weak. Here, in contrast, we analyze the magnetoelastic behavior of antiferromagnetic tetragonal system MnPt, explaining the experimentally measured data based on the theoretical calculations and discussing the influence of the magnetic structure. Particularly, we inspect the origin of magnetocrystalline anisotropy energy, as well as the size and source of the isotropic and anisotropic parts of magnetoelastic (magnetostriction) coefficients.

cond-mat.mtrl-sci↗

arXiv2024

Dramatic elastic response at the critical end point in UTe$_2$

The first-order transition line in the \textit{H-T} phase diagram of itinerant electron metamagnets terminates at the critical end point-analogous to the critical point on the gas-liquid condensation line in the \textit{p-T} phase diagram. To unravel the impact of critical magnetic fluctuations on the crystal lattice of a metamagnet at the critical end point, we performed an ultrasonic study of the itinerant electron metamagnet UTe$_2$ across varying temperatures and magnetic fields. At temperatures exceeding 9 K, a distinct V-shaped anomaly emerges, precisely centered at the critical field of the metamagnetic transition in the isothermal field dependence of elastic constants. This anomaly arises from lattice instability, triggered by critical magnetic fluctuations via strong magnetoelastic interactions. Remarkably, this effect is maximized precisely at the critical-end-point temperature. Comparative measurements of another itinerant metamagnet, UCoAl, reveal intriguing commonalities. Despite significant differences in the paramagnetic ground state, lattice symmetry, and the expected metamagnetic transition process between UTe$_2$ and UCoAl, both exhibit similar anomalies in elastic properties near the critical end point. These shared aspects may hold universality for other itinerant electron metamagnets.

cond-mat.str-el↗

arXiv2022

Tricritical fluctuations and elastic properties of the Ising antiferromagnet UIrSi$_3$

Elastic constants, thermal expansion, magnetostriction and heat capacity measurements were performed with and without applied magnetic field on a single crystal of UIrSi$_3$. The elastic properties were interpreted within the theory of the strain-exchange effect. The exchange-striction model together with the Ising model for the behavior of magnetic localized 5$f$ electrons and itinerant electrons of uranium correctly reproduces the main features of our magneto-acoustic experiments in UIrSi$_3$. Data on thermal expansion and magnetostriction confirm the conclusion that the dominant contribution of the measured temperature and field change in the speed of sound comes from the change in the elastic modulus itself. Based on the analysis of heat capacity measurements in the magnetic field, we explain the significant anomalies at the second-order branch of the phase transition boundary as a manifestation of the tricritical fluctuations, dominating the region below the tricritical point. The outcome confirms the 3D Ising model, at zero and low magnetic fields, with a crossover to the mean-field tricritical behavior at fields close to the tricritical point, where tricritical fluctuations dominate the temperature evolution of the given property.

cond-mat.str-el↗

arXiv2020

Evidence of Fermi-Surface Reconstruction at the Metamagnetic Transition of the Strongly Correlated Superconductor UTe2

Thermoelectric power ($S$) and Hall effect ($R_\mathrm{H}$) measurements on the paramagnetic superconductor UTe$_2$ with magnetic field applied along the hard magnetization $b$-axis are reported. The first order nature of the metamagnetic transition at $H_\mathrm{m}=H^b_\mathrm{c2}=35$~T leads to drastic consequences on $S$ and $R_\mathrm{H}$. In contrast to the field dependence of the specific heat in the normal state through $H_\mathrm{m}$, $S(H)$ is not symmetric with respect to $H_\mathrm{m}$. This implies a strong interplay between ferromagnetic (FM) fluctuations and a Fermi-surface reconstruction at $H_\mathrm{m}$. $R_\mathrm{H}$ is very well described by incoherent skew scattering above the coherence temperature $T_\mathrm{m}$ corresponding roughly to the temperature of the maximum in the susceptibility $T_{χ_\mathrm{max}}$ and coherent skew scattering at lower temperatures. The discontinuous field dependence of both, $S(H)$ and the ordinary Hall coefficient $R_0$, at $H_\mathrm{m}$ and at low temperature, provides evidence of a change in the band structure at the Fermi level.

cond-mat.str-el↗

arXiv2020

Anisotropy of the Upper Critical Field in the Heavy-Fermion Superconductor UTe2 under Pressure

We studied the anisotropy of the superconducting upper critical field $H_{\rm c2}$ in the heavy-fermion superconductor UTe$_2$ under hydrostatic pressure by magnetoresistivity measurements. In agreement with previous experiments we confirm that superconductivity disappears near a critical pressure $p_{\rm c} \approx 1.5$~GPa, and a magnetically ordered state appears. The unusual $H_{\rm c2}(T)$ at low temperatures for $H \parallel a$ suggests that the multiple superconducting phases which appear under pressure have quite different $H_{\rm c2}$. For a field applied along the hard magnetization $b$ axis $H_{\rm c2} (0)$ is glued to the metamagnetic transition $H_{\rm m}$ which is suppressed near $p_{\rm c}$. The suppression of $H_{\rm m}$ with pressure follows the decrease of temperature $T_χ^{\rm max}$, at the maximum in the susceptibility along $b$. The strong reinforcement of $H_{\rm c2}$ at ambient pressure for $H \parallel b$ above 16~T is rapidly suppressed under pressure due to the increase of $T_{\rm sc}$ and the decrease of $H_{\rm m}$. The change in the hierarchy of the anisotropy of $H_{\rm c2}(0)$ on approaching $p_{\rm c}$ points out that the $c$ axis becomes the hard magnetization axis.

cond-mat.str-el↗

arXiv2019

Field-reentrant superconductivity close to a metamagnetic transition in the heavy-fermion superconductor UTe$_2$

We present a study of the upper critical field of the newly discovered heavy fermion superconductor UTe$_2$ by magnetoresistivity measurements in pulsed magnetic fields up to 60~T and static magnetic fields up to 35~T. We show that superconductivity survives up to the metamagnetic transition at $H_{\rm m} \approx 35$~T at low temperature. Above $H_{\rm m}$ superconductivity is suppressed. At higher temperature superconductivity is enhanced under magnetic field leading to reentrance of superconductivity or an almost temperature independent increase of $H_{\rm c2}$. By studying the angular dependence of the upper critical field close to the $b$ axis (hard magnetization axis) we show that the maximum of the reentrant superconductivity temperature is depinned from the metamagnetic field. A key ingredient for the field-reinforcement of superconductivity on approaching $H_{\rm m}$ appears to be an immediate interplay with magnetic fluctuations and a possible Fermi-surface reconstruction.

cond-mat.str-el↗

arXiv2019

Magnetic-Field-Induced Phenomena in the Paramagnetic Superconductor UTe$_{2}$

We present magnetoresistivity measurements on the heavy-fermion superconductor UTe$_{2}$ in pulsed magnetic fields $μ_0H$ up to 68~T and temperatures $T$ from 1.4 to 80~K. Magnetic fields applied along the three crystallographic directions $\mathbf{a}$ (easy magnetic axis), $\mathbf{b}$, and $\mathbf{c}$ (hard magnetic axes), are found to induce different phenomena - depending on the field direction - beyond the low-field suppression of the superconducting state. For $\mathbf{H}\parallel\mathbf{a}$, a broad anomaly in the resistivity is observed at $μ_0H^*\simeq10$~T and $T = 1.4$~K. For $\mathbf{H}\parallel\mathbf{c}$, no magnetic transition nor crossover are observed. For $\mathbf{H}\parallel\mathbf{b}$, a sharp first-order-like step in the resistivity indicates a metamagnetic transition at the field $μ_0H_m \simeq 35$~T. When the temperature is raised signature of first-order metamagnetism is observed up to a critical endpoint at $T_{CEP}\simeq7$~K. At higher temperatures a crossover persists up to 28~K, i.e., below the temperature $T_χ^{max} = 35$~K where the magnetic susceptibility is maximal. A sharp maximum in the Fermi-liquid quadratic coefficient $A$ of the low-temperature resistivity is found at $H_m$. It indicates an enhanced effective mass associated with critical magnetic fluctuations, possibly coupled with a Fermi surface instability. Similarly to the URhGe case, we show that UTe$_{2}$ is a candidate for field-induced reentrant superconductivity in the proximity of $H_m$.

cond-mat.str-el↗

arXiv2018

Antiferromagnetism and phase transitions in non-centrosymmetric UIrSi$_3$

Magnetization and specific heat measurements on a UIrSi3 single crystal reveal Ising-like antiferromagnetism below T$_N$ = 41.7 K with easy magnetization direction along the c-axis of tetragonal structure. The antiferromagentic ordering is suppressed by magnetic fields > H$_c$ (μ$_0$H$_c$ = 7.3 T at 2 K) applied along the c-axis. The first-order metamagnetic transition at H$_c$ exhibits asymmetric hysteresis reflecting a slow reentry of the complex ground-state antiferromagnetic structure with decreasing field. The hysteresis narrows with increasing temperature and vanishes at 28 K. A second-order metamagnetic transition is observed at higher temperatures. The point of change of the order of transition in the established H-T magnetic phase diagram is considered as the tricritical point (at T$_{tc}$ = 28 K and μ$_0$H$_{tc}$ = 5.8 T). The modified-Curie-Weiss-law fits of temperature dependence of the a- and c-axis susceptibility provide opposite signs of Weiss temperatures, Θ$_p^a$ ~ -51 K and Θ$_p^c$ ~ +38 K, respectively. This result and the small value of μ$_0$H$_c$ contrasting to the high T$_N$ indicate competing ferromagnetic and antiferromagnetic interactions responsible for the complex antiferromagnetic ground state. The simultaneous electronic-structure calculations focused on the total energy of ferromagentic and various antiferromagnetic states, the U magnetic moment and magnetocrystalline anisotropy provide results consistent with experimental findings and the suggested physical picture of the system.

cond-mat.str-el↗

arXiv2017

Neutron Diffraction Study on Single-crystalline UAu${_2}$Si$_2$

Magnetic structure of tetragonal UAu$_2$Si$_2$ was investigated by single-crystal neutron diffraction experiments. Below $T_{\rm N}$ = 20 K it orders antiferromagnetically with a propagation vector of $k = (2/3, 0, 0)$ and magnetic moments of uranium ions pointing along the tetragonal $c$-axis. Weak signs of the presence of a ferromagnetic component of magnetic moment were traced out.Taking into account a group theory calculation and experimental results of magnetization and $^{29}$Si-NMR, the magnetic structure is determined to be a squared-up antiferromagnetic structure, with a stacking sequence ($+ + -$) of the ferromagnetic $ac$-plane sheets along the $a$-axis. This result highlights similar magnetic correlations in UAu$_2$Si$_2$ and isostructural URu$_2$Si$_2$.

cond-mat.str-el↗

arXiv2017

Effect of lattice distortion on U magnetic moments in $\mathrm{U_{4}Ru_{7}Ge_{6}}$ studied by polarized neutron diffraction

Small spontaneous lattice distortion of a cubic ferromagnet at temperatures below $T_{\mathrm{C}}$ is expected, but usually a neglected phenomenon. This study is focused on such effect on the example of $\mathrm{U_{4}Ru_{7}Ge_{6}}$. We propose the lattice distortion from the cubic space group to a lower symmetry as a result of our DFT calculations. The strong spin-orbit coupling on the U site plays the most essential role, that should lower the symmetry of the system and give rise to the two different U sites (U1 and U2). We bring convincing experimental evidence for this splitting resulting from our polarized neutron diffraction experiment. Proper treatment of the flipping ratios collected in the ordered state and in the paramagnetic state reveals dramatically different magnetic moments on the U1 and U2 sites. We have reached good agreement of the results from the MAXENT method and from direct fitting of the data. The ratio of the orbital and spin component on the U2 site is lower than for the free $\mathrm{U^{3+}}$ or $\mathrm{U^{4+}}$ ion and points to a strong hybridization between the U $5f$ and Ru $4d$ wave functions. The same ratio on the U1 site is unexpectedly low. This would mean that its orbital component is almost quenched. We have observed the absence of a magnetic moment on the Ru1 site, but a rather large induced moment on the Ru2 site. Very similar results were obtained also in the paramagnetic state in the regime induced by the magnetic field. It points to the intimate coupling of the magnetic ordering and structural distortion as it is possible to split the U site in to two different polarizing paramagnetic U moments above $T_{\mathrm{C}}$ by a strong magnetic field. We propose that the difference between the magnetic moment on the U sites is caused by the change of local point symmetry of the sites that is tightly bound to the role of the strong spin-orbit interaction.

cond-mat.str-el↗

arXiv2016

Uranium ferromagnet with negligible magnetocrystalline anisotropy - $\mathrm{U_{4}Ru_{7}Ge_{6}}$

Strong magnetocrystalline anisotropy is a well-known property of uranium compounds. The almost isotropic ferromagnetism in $\mathrm{U_{4}Ru_{7}Ge_{6}}$ reported in this paper represents a striking exception. We present results of magnetization, AC susceptibility, thermal expansion, specific heat and electrical resistivity measurements performed on a $\mathrm{U_{4}Ru_{7}Ge_{6}}$ single crystal at various temperatures and magnetic fields and discuss them in conjunction with results of first-principles electronic-structure calculations. $\mathrm{U_{4}Ru_{7}Ge_{6}}$ behaves as an itinerant 5$f$-electron ferromagnet ($T_{\mathrm{C}}= 10.7 K$, $μ_{\mathrm{S}}= 0.85 μ_{\mathrm{B}}/f.u.$ at 1.9 K. The ground-state easy-magnetization direction is along the [111] axis of the cubic lattice. The anisotropy field $μ_{0}H_{\mathrm{a}}$ along the [001] direction is only of $\sim0.3 T$, which is at least 3 orders of magnitude smaller value than in other U ferromagnets. At $T_{r}=5.9 K$ the easy magnetization direction changes for [001] which holds at temperatures up to $T_{\mathrm{C}}$. The magnetoelastic interaction induces a rhombohedral (tetragonal) distortion of the paramagnetic cubic crystal lattice in case of the [111]([001]) easy-magnetization direction. The rhombohedral distortion is connected with two crystallographically inequivalent U sites. The ab initio calculated ground-state magnetic moment of $1.01 μ_{\mathrm{B}}/f.u.$ is oriented along [111]. The two crystallographically inequivalent U sites are a consequence of spin-orbit coupling of the U 5$f$-electrons. In the excited state which is only 0.9 meV above the ground state the moment points to the [001] direction in agreement with experiment.

cond-mat.str-el↗

arXiv2016

Peculiar Magnetism of UAu$_{2}$Si$_{2}$

Single-crystalline UAu$_2$Si$_2$ has been grown by a floating-zone melting method, and its magnetic, thermal and transport properties have been investigated through measurements of magnetization, specific heat and electrical resistivity to reveal its peculiar magnetism. It is shown that UAu$_2$Si$_2$ undergoes a second-order phase transition at \tm = 19 K, which had been believed to be ferromagnetic ordering in the literature, from a paramagnetic phase to an uncompensated antiferromagnetic phase with spontaneous magnetization along the tetragonal $c$-axis (the easy magnetization direction). The magnetic entropy analysis points to the itinerant character of 5f electrons in the magnetic ordered state of UAu$_2$Si$_2$ with large enhancement of the electronic specific heat coefficient of $γ$ $\sim$ 150 mJ/K$^2$mol at 2 K. It also reveals the relatively isotropic crystalline electric field effect of this compound, with contrast to the other relative isostructural compounds. The observed magnetization curves strongly suggest that there is a parasitic ferromagnetic component developing below $\sim$ 50 K in high coercivity with the easy axis along the tetragonal $c$-axis. The results are discussed in the context of evolution of magnetism within the entire family of isostructural U$T_2$Si$_2$ compounds.

cond-mat.str-el↗

arXiv2015

Evolution of ferromagnetic and non-Fermi liquid state with doping:the case of Ru doped UCoGe

We have investigated the impact of Ru substitution for Co on the behavior of the ferromagnetic superconductor UCoGe by performing x-ray diffraction, magnetization, specific heat and electrical resistivity measurements on polycrystalline samples of the $\mathrm{UCo}_{1-x}\mathrm{Ru}_{x}\mathrm{Ge}$ series ($0\geq x\leq0.9$). The initial Ru substitution up to $x\approx0.1$ leads to a simultaneous sharp increase of the Curie temperature and spontaneous magnetization up to maximum values of $T_{\mathrm{C}}=8.6 K$ and $M_{\mathrm{S}}=0.1 μ_{\mathrm{B}}$ per formula unit, respectively, whereas superconductivity vanishes already for $x\approx0.03$. Further increase of the Ru content beyond $x\approx0.1$ leads to a precipitous decrease of both, $T_{\mathrm{C}}$ and $M_{\mathrm{S}}$ towards a ferromagnetic quantum critical point (QCP) at $x_{\mathrm{cr}}=0.31$. Consequently the $T-x$ magnetic phase diagram consists of a well-developed ferromagnetic dome. We discuss the evolution of ferromagnetism with $x$ on the basis of band structure changes due to varying 5$f$-ligand hybridization. This scenario is supported by the results of electronic structure calculations and consideration of the simplified periodic Anderson model. The analysis of the temperature dependencies of the electrical resistivity and heat capacity at low temperatures of the samples in the vicinity of the QCP reveals a non-Fermi liquid behavior and assigns the ferromagnetic quantum phase transition to be most likely of a continuous Hertz-Millis type.

cond-mat.str-el↗