Quantum Ferromagnetism and Related Phenomena

I will review some of our efforts to study what happens when ferromagnets are squeezed until suppression of the magnetic order. Various interesting behaviors are observed experimentally in the temperature-pressure ($T$-$p$) phase diagrams of UGe$_2$, LaCrGe$_3$, and CeTiGe$_3$, LaCrSb$_3$. Namely, either the ferromagnetic-paramagnetic transition becomes of the first-order at a tricritical point before being suppressed such as in UGe$_2$ [1,2], or transitions to modulated magnetic phases appear such as in LaCrGe$_3$ [3,4] and CeTiGe$_3$[5]. We have shown that the addition of a magnetic field ($H$) can lead to new quantum critical points at the end of “wings” in the $T$-$p$-$H$ phase diagram in both UGe$_2$ and LaCrGe$_3$ [5] or to quantum tricritical points in CeTiGe$_3$. Our careful study of the “wings” near the tricritical point reveal new rules that apply to the $T$-$p$-$H$ phase diagram [6]. We discuss how our experimental $T$-$p$-$H$ phase diagrams of UGe$_2$, LaCrGe$_3$, and CeTiGe$_3$ illustrate different strength of quantum fluctuations based on recent theoretical results [7]. I will also discuss how new promising compounds to study quantum ferromagnetism can be identified. [1] V. Taufour et al. Phys. Rev. Lett. 105, 217201 (2010). [2] H. Kotegawa et al. J. Phys. Soc. Jpn., 80, 8, 083703 (2011). [3] V. Taufour et al. Phys. Rev. Lett. 117, 037207 (2016). [4] U. S. Kaluarachchi et al. Nature Communications 8, 546 (2017). [5] U. S. Kaluarachchi et al. Phys. Rev. B 97, 045139 (2018). [6] V. Taufour et al. Phys. Rev. B 94 060410 (2016). [7] Belitz et al. Phys. Rev. Lett. 119 267207 (2017).

Magnetic resonance is a very suitable microscopic tool for correlated matter at the verge of long range magnetic ordering and aims in particular to expose the real nature of the magnetic fluctuations [antiferromagnetic (afm) versus ferromagnetic (fm)] by temperature and field scaling. Among 3d magnets tunable ferromagnetic quantum criticality could be found in itinerant Fe-based systems like $NbFe_2$ and $(Ta,V)Fe_2$ but also in systems with more localized Fe moments like $YFe_2Al_{10}$ and $YbFe_2Al_{10}$. Signatures of Kondo type of correlations are also found in some Fe-based binary semimetals. FeSi, $FeSb_2$, and $FeGa_3$ attracted great attention because of their nonmagnetic ground state and their promising low temperature thermoelectric performance assigned to correlated in gap states. Metallic behavior and Fe-based magnetism could be introduced by controlled substitutions on the Fe- or the framework site. Here we present a NQR study on the electron doped systems $Fe(Sb,Te)_2$ and $Fe(Ga,Ge)_3$. Being a local probe at zero field NQR could capture both most relevant points: (a) the degree of disorder upon doping and (b) the evolution of electronic correlations and the onset of critical fluctuations at the verge of long range order. Whereas in $Fe(Sb,Te)_2$ the predominant electronic NQR broadening effect provides a microscopic evidence for the electronic Griffith phase formation in $Fe(Ga,Ge)_3$ an absence of induced disorder was found from NQR. Here a crossover from an insulator to an antiferromagnetic correlated local moment metal in the low-doping regime and the evolution of itinerant ferromagnetism upon further doping is found. For the nearly critical concentration at the threshold of ferromagnetic order, the spin lattice relaxation rate exhibits a pronounced T −4/3 power law over two orders of magnitude in temperature, which indicates three-dimensional quantum critical ferromagnetic fluctuations.

It is now well accepted that the suppression of ordered states, such as magnetism, can give rise to a novel state with highly anomalous metallic characteristics. It remains a challenge to understand the role of the quantum critical fluctuations associated with the T=0 phase transition in inducing this new state, and if there is feedback between the fluctuations and the essential properties of the quasiparticles in the non-Fermi liquid electronic state. A lack of detailed experimental results on suitable QC systems has slowed progress towards this understanding. The quasi-two dimensional metal YFe2Al10 is a very promising system, comprised of layers of nearly square nets of Fe atoms. Despite the strong divergence of the susceptibility,\chi^{''}(T)~T^{-1.4} there is no evidence for magnetic order above 0.02 K. Inelastic neutron scattering measurements find that the scattering has no indication of incipient magnetic order, rather there is no measureable wave vector dependence, beyond that of the form factor. However, the scattering displays a strong energy divergence, and the Kramers-Kronig analysis indicates that it is these quantum critical excitations that are responsible for the divergence in the magnetic susceptibility. The scattering is also temperature independent, evidence that the imaginary part of the dynamical susceptibility \chi^{''}”(E,T) displays E/T scaling, where the absence of any characteristic energy scale beyond temperature is the hallmark of quantum critical systems. The quantum critical fluctuations are very strong in YFe2Al10, signaling that it is very close to a T=0 phase transition. The neutron scattering measurements reveal that the critical fluctuations are completely local, with each moment fluctuating incoherently, each with the same spectrum of excitations. These findings rule out the possibility that YFe2Al10 is near magnetic ordering, and instead it seems likely that the phase transition corresponds to a purely electronic phase transition, possibly an orbital selective Mott transition, where localized magnetic moments first emerge.

Due to the low energy scales, the ground state of heavy fermion compounds can be readily tuned by parameters such as pressure, magnetic fields or doping. There have been numerous studies into antiferromagnetic quantum criticality and unconventional superconductivity, which have been revealed a possible role for spin fluctuations in the superconducting pairing [1, 2]. However, ferromagnetism in heavy fermion systems has been less frequently investigated [3, 4]. Here we report the resistivity and specific heat measurements of the ferromagnetic compound CeRh6Ge4 under pressure, which show the presence of a ferromagnetic critical point in this system. Similar results were also obtained by element substitutions. References: [1] N. D. Mathur et al., Nature 394, 39 (1998). [2] Z. F. Weng et al., Rep. Prog. Phys. 79, 094503 (2016) [3] A. Steppke et al. Science 339, 933 (2013). [4] M. Brando et al., Rev. Mod. Phys. 88, 025006 (2016).

Since the discovery of superconductivity in ferromagnet UGe$_2$ under pressure [1], U-based ferromagnetic (FM) superconductors have attracted much attention since spin-triplet superconductivity is anticipated. Within them, URhGe[2] and UCoGe[3] show superconductivity at ambient pressure, and the latter has the highest superconducting (SC) transition temperature $T_{\rm Super}$ = 0.57 K below FM ordering at $T_{\rm Curie}$ =2.5 K. We have studied single-crystal UCoGe with microscopic measurements of $^{59}$Co nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR). We showed that superconductivity occurs in the FM region [4] and that both phenomena originate from U 5-$f$ electrons[5], resulting in the microscopic coexistence of ferromagnetism and superconductivity realized in UCoGe. We also studied the spin-dynamic properties from the measurements of $1/T_1$ and Knight shift along the each crystalline axis. The results show that both static and dynamic susceptibilities possess the strong Ising anisotropy along the $c$ axis being the easy axis and that the FM fluctuations are predominant at low temperatures and persist even below $T_{\rm Curie}$ [6]. From the angle-resolved NMR measurements, we found that the magnetic field along the $c$ axis ($H \parallel c$) strongly suppresses both the FM Ising-type fluctuations and superconductivity in the same manner [7]. On the other hand, we found that the field along the $b$ axis ($H \parallel b$) above 5 T enhances the Ising FM fluctuations at low temperatures, and that the superconductivity also becomes robust in the same field region[8]. These results strongly suggest that the characteristic FM fluctuations tuned by external fields induce unique spin-triplet superconductivity in UCoGe. This scenario is also supported by the recent $^{59}$Co-NQR/NMR measurements under pressure[9]. We will introduce NMR results on single-crystal URhGe [10,11] and UGe$_2$ done by other groups, and discuss the similarity and difference in these superconductors. The NMR/NQR studies on single-crystal UCoGe were done in the collaboration with T. Hattori, M. Manago, S. Kitagawa, K. Karube, Y. Ihara, K. Deguchi, N. K. Sato, and T. Yamamura. References [1] S.S. Saxena et al., Nature 406, 587 (2000). [2] D. Aoki et al, Nature 413, 613 (2001). [3] N. T. Huy et al., Phys.Rev. Lett. 99, 067006 (2007). [4] T. Ohta et al., J. Phys. Soc. Jpn. 77, 023707 (2008). [5] K. Karube et al. J. Phys. Soc. Jpn. 80, 064711 (2011). [6] Y. Ihara et al., Phys. Rev. Lett. 105, 206403 (2010). [7] T. Hattori et al. Phys. Rev. Lett. 108, 066403 (2012). [8] T. Hattori et al. J. Phys. Soc. Jpn. 83, 073708 (2014). [9] M. Manago et al. submitted to PRL. [10] Y. Tokunaga et al. Phys. Rev. Lett. 114, 216401 (2015). [11] Y. Tokunaga et al. Phys. Rev. B 93, 201112(R) (2016)

One of the hot topic in the field of superconductivity is the effect of breaking inversion symmetry in the presence of strong spin orbit interaction. This leads to a spin-splitted Fermi surface with unique momentum-locked spin polarization even in the absence of an external or an exchange field. Superconductivity appearing from such polarized bands is robust against Zeeman pair-breaking effect and can host mixed-parity pairing providing exotic superconducting states. So far, this kind of phenomena has only been studied in a handful of non-centrosymmetric system such as CeRhSi3 under pressure and quasi-two-dimensional transition-metal dichalcogenide superconductors. Here, we report the discovery of heavy-fermion superconductivity in CeRh2As2 with Tc ~ 0.25 K at ambient pressure. This compounds crystallizes in the CaBe2Ge2 structure type where inversion symmetry is locally broken at the Ce site, but is still present for the global structure. We observe a huge upper critical field of ≳ 12 T for the out-of-plane direction surpassing the Pauli-paramagnetic limit of ~ 0.5 T. This provides a clear signature of a Rashba-type in-plane spin polarization arising from an alternating asymmetric potential due to the broken local inversion symmetry. In addition our results indicate this system to be very close to a quantum critical point (QCP) with a further transition at To ~ 0.4 K. The presence of a quasi-quartet crystal electric field ground state as well as a very unusual B-T phase diagram suggest this transition to be a quadrupolar one. Therefore CeRh2As2 is a promising candidate for studying how heavy-fermion superconductivity behaves under the influence of a Rashba-type interactions and a possible multipolar QCP.

I discuss the interplay between non-Fermi liquid behavior and superconductivity near a quantum-critical point (QCP) in a metal. It is widely thought that the tendency towards superconductivity and towards non-Fermi liquid behavior compete with each other, and if the pairing interaction is reduced below a certain threshold, the system displays a naked non-Fermi liquid QC behavior. I show that the situation is more complex. First, there is a difference between spin-triple and spin-singlet superconductivity. For spin-triplet pairing, thermal fluctuations are crucial and make superconducting transition first order. For spin-singlet pairing, they are essentially irrelevant, and the transition is second order. Second, I show that for spin-singlet pairing, there are multiple solutions with the same gap symmetry. For all solutions, except one, Tc vanishes when the pairing interaction drops below the threshold. For the one special solution, Tc remains finite even when the pairing interaction is arbitrary small, despite that there is no Cooper logarithm. I argue that superconductivity between this Tc and a lower T, when other solutions appear, is special, as it is entirely induced by fermions with the first Matsubara frequency. I show that this has specific implications for the observable quantities, such as the density of states and the spectral function.

Strongly correlated metals near magnetic quantum phase transitions (QPTs) are prime examples for quantum matter. For clean ferromagnetic metals, Belitz-Kirkpatrick-Vojta (BKV) theory shows in excellent agreement with experiment that ferromagnetic QPTs are generally first-order due to the coupling of the magnetization to electronic soft modes, in contrast to the classical analogue that is an archetypical second-order phase transition. Our recent extensive magnetization study on the disordered ferromagnetic metal $UCo_{1-x}Fe_xGe$ demonstrate that BKV theory even correctly predicts that the second order nature of the QPT is restored because the electronic soft modes change their nature from ballistic to diffusive [1]. Despite this significant progress in understanding ferromagnetic metals, to date, little information is available on the underlying low-energy spin fluctuations that drive exotic behavior such as partial magnetic order, and topological non-Fermi liquid behavior, and unconventional spin-triplet superconductivity that is frequently observed in the vicinity of ferromagnetic QPTs. Notably, the outstanding challenge is that the underlying characteristic energy scales that drive these quantum matter states are tiny compared to typical electronic energy scales in solids, and are, in turn, notoriously difficult to measure. Here we discuss some of our recent results to showcase how current advances in the resolution of neutron spectroscopy. Using the novel Modulated IntEnsity by Zero Effort (MIEZE) technique implemented at the neutron spectrometer RESEDA in Munich, we achieve ultra-high energy resolution of 1 μeV and reveal that the spin fluctuations in UGe2 exhibit a dual nature arising from the interplay of localized and itinerant electronic degrees of freedom in consistent with spin-triplet superconductivity proposed for this material [2]. [1] K. Huang, S. Eley, P. F. S. Rosa, L. Civale, E. D. Bauer, R. E. Baumbach, M. B. Maple, and M. Janoschek, Phys. Rev. Lett. 117, 237202 (2016). [2] F. Haslbeck, S. Säubert, M. Seifert, C. Franz, M. Schulz, A. Heinemann, T. Keller, P. Das, J.D. Thompson, E. D. Bauer, C. Pfleiderer, and M. Janoschek, Phys. Rev. B 99, 014429 (2019).

I will discuss newly-discovered spin-triplet superconductivity in UTe2. Although the normal state is paramagnetic, unusual magnetization-field scaling suggests strong magnetic correlations and proximity to a ferromagnetic instability. The superconducting order parameter arises from strong ferromagnetic fluctuations, yielding a very high upper critical field with similar anisotropy to that found in established ferromagnetic superconductors. There is a large residual specific heat that implies that while electrons with parallel spins pair, only half of the available electrons participate, yielding a spin-polarized condensate that coexists with a spin-polarized metal.

The nature of the quantum ferromagnetic phase transition in clean Dirac metals is studied. Generally it is shown that the transition is discontinuous one from a Dirac metal to a Weyl metal. Possible exceptions are pointed out. The results are somewhat surprising since strong spin-orbit scattering naively suppresses the mechanism that causes the first order transition ordinary metals. The important role of chirality is stressed.

3D Dirac semimetals have attracted wide interest for their unique magnetotransport properties, many of which are related to the topological properties of their low energy band structure. A recent key focus is on the exploration and potential exploitation of new phenomena through the introduction of magnetism to such systems, with only a few candidate materials known to date. Here I will present magnetotransport, thermodynamic and neutron scattering experiments combined with DFT band structure calculations establishing the phase diagram of the magnetic inverse perovskite Eu3PbO. Detailed analysis across the low temperature phases reveals that this material hosts not only one, but a multitude of different topological phases that are easily controllable by an external magnetic field. Examples are a ferromagnetic Weyl phase, an antiferromagnetic Dirac phase, and two canted antiferromagnetic phases with both Weyl points and nodal lines. I will discuss the key features of the unusual band dispersions as well as the implications for surface states and the Hall effect. This opens up the possibility to manipulate the interplay of band topology, magnetism, and transport and I will discuss pathways to quantum phase transitions in such systems.

I will discuss the behavior of collective excitations in a Fermi liquid near a Pomeranchuk transition towards a state with an order parameter characterized by angular momentum l. For each value of l, I will track the evolution of spin and/or charge collective modes with increasing magnitude of the Landau parameter F_l. I will argue that for each l there is one zero-sound mode, whose velocity vanishes at the Pomeranchuk instability, when F_l =-1. In a clean Fermi liquid the critical mode near the transition is purely relaxational in some cases; in others, it is almost propagating. This leads to qualitatively different time dependence of the susceptibility both near and at the transition. In a dirty Fermi liquid, the critical mode becomes purely relaxational in all cases. The situation is somewhat special for the l=1 order parameter coinciding with the spin or charge current. In this case, the residue of the critical mode vanishes at the Pomeranchuk transition, consistent with earlier results that the static susceptibility does not diverge at F_1 = -1. I will show, however, that the critical mode can be identified at any distance from the transition, and moves into the upper frequency half-plane at F_1 <-1, signaling that a Fermi surface gets distorted. The only peculiarity of a charge/spin current order parameter is that it takes longer to reach an equilibrium state at F_1 <-1.

Quantum phase transitions ferromagnetic metals impose long-standing questions including non-Fermi liquid behaviour with incompatible temperature dependencies in transport and thermodynamic properties. In most materials, ferromagnetic quantum critical points (QCPs) are avoided - most commonly through a change to 1st order as an initially 2nd order transition is tuned towards zero temperature. Here, we discuss our results on NbFe$_2$ in which the ferromagnetic quantum critical point is avoided through an intervening antiferromagnetic phase. We demonstrate that NbFe$_2$ is well described by a very general two-order-parameter model for a large set of samples with different compositions across the buried ferromagnetic QCP. This analysis establishes the presence of quantum tricritical points where both the uniform and finite wavelength susceptibility diverge. As our model is very general, we expect it to underlie a whole new class of quantum criticality in ferromagnets and other systems with two order parameters.

Non-Fermi liquids are exotic metallic states which do not support well defined quasiparticles. Due to strong quantum fluctuations and the presence of extensive gapless modes near the Fermi surface, it has been difficult to understand universal low-energy properties of non-Fermi liquids. In this talk, we discuss recent progress made on field theories for non-Fermi liquids. Based on a dimensional regularization scheme which tunes the co-dimension of Fermi surface, critical exponents that control universal scaling behaviors of physical observables can be computed in a controlled way. The systematic expansion also provides important insight into strongly interacting non-Fermi liquids. This allows us to find the non-perturbative solution for the strange metal realized at the antiferromagnetic quantum critical point in 2+1 dimensions, and predict the exact critical exponents that can be experimentally tested in layered compounds.

The layered oxide metal Sr$_3$Ru$_2$O$_7$ is close to ferromagnetic order at low temperatures. Thus it provides and model system to investigate the effect of a varying field induced exchange splitting on a 2D metal. For fields near 8T applied perpendicular to the RuO$_2$ planes spin density wave order is induced below 1K. I will describe our characterisation of the SDW ordered states using neutron scattering and show how the low-frequency magnetic quantum fluctuations can be controlled using a magnetic field. Our results suggest that the spin density wave order forms in a background of strong spin fluctuations. I will also show the magnetic excitations can be used to infer the low temperature specific heat.

CeTiGe$_3$ is a ferromagnetic Kondo system with a Curie temperature $T_C \approx 14$ K. The critical properties of pure CeTiGe3 follow the predictions of a classical 3D ferromagnet, despite a strong uniaxial anisotropy with the easy axis along the c-axis. This anisotropy is reduced with the substitution of Ti with V; CeVGe3 is an antiferromagnet with the ab-plane as easy plane. Furthermore the substitution of Ti with V results in a suppression of the ferromagnetic order with $T_C \rightarrow 0$ for $x \approx 0.4$. Around the critical concentration thermodynamic and NMR measurements point toward a mixture of ferromagnetic and antiferromagnetic spin-fluctuations. Overall our data indicate the existence of a quantum critical point in CeTi$_{1-x}$V$_x$Ge$_3$, which is, however, rather complex involving ferromagnetic and antiferromagnetic correlations.

The Belitz-Kirkpatrick-Vojta (BKV) theory has emerged as a leading framework within which to describe ferromagnetic quantum phase transitions, and there already are many systems that have been described in this context. This includes both $d$- and $f$-electron magnets, which often feature intertwining energy scales such as the magnetic exchange (both sign and strength), the spin orbit interaction, the Kondo interaction, magnetic ion valence stability, and other factors. Furthermore, chemical/structural disorder plays an important role. Thus, it remains important to probe the universality of this theory by studying even more examples that span different parts of the electronic phase space. In this talk I will report results for the isoelectronic chemical substitution series $Ce(Pd_{1-x}Ni_x)_2Si_2$, where a possible ferromagnetic quantum phase transition is uncovered in the temperature-concentration ($T−x$) phase diagram. This behavior results partly from the contraction of the unit cell volume, which (i) tunes the relative strengths of the Kondo and Ruderman–Kittel–Kasuya–Yosida (RKKY) interactions and (ii) modifies the $f$-electron valence stability. The behavior is also influenced by the introduction of disorder through alloying. Near the critical region at $x_{\rm{cr}}$ $\approx$ 0.7, the rate of contraction of the unit cell volume strengthens, indicating that the cerium $f$ valence crosses over from trivalent to a noninteger value. Consistent with this picture, x-ray absorption spectroscopy measurements reveal that while $CePd_2P_2$ has a purely trivalent cerium $f$ state, $CeNi_2P_2$ has a small (< 10 \%) tetravalent contribution. In a broad region around $x_{\rm{cr}}$, there is a breakdown of Fermi-liquid temperature dependence seen in the heat capacity, signaling the influence of quantum critical fluctuations and disorder effects. Electrical transport measurements of clean $CePd_2P_2$ furthermore show that applied pressure has an initial effect similar to alloying on the ferromagnetic order. From these results, $CePd_2P_2$ emerges as a keystone system to test the BKV model in $f$-electron metals, where distinct behaviors are expected in the dirty and clean limits.

In 2009, Kim, Stewart, and Samwer (PRB 79, 165119) investigated how the magnetic susceptibility, the magnetization, and the specific heat evolved with decreasing size in quantum critical CeRu0.8Rh1.2Si2 down to a particle size of 0.6 microns. They found evidence for Griffiths phase behavior with decreasing particle size, which below 3 microns began to be masked by uncompensated local moment defects seen in small particles of Ce compounds before. In order to avoid this masking behavior in small-sized CeRu0.8Rh1.2Si2, we report here on a similar study on samples of small particles of the undoped nFl compound U2Pt2In, also down to 0.6 microns in size. Uncompensated local moment defect peaks in the low temperature specific heat of CeRu0.8Rh1.2Si2 are avoided in our data for U2Pt2In, allowing a clearer look at the properties of this bulk nFl compound as a function of decreasing size. Properties of particles of U2Pt2In in the following size windows are reported: 10 - 20 microns; 3 - 10 microns; 1.2 - 3 microns; and 0.6 - 1.2 microns.

We study the impact of quenched disorder (random exchange couplings or site dilution) on easy-plane pyrochlore antiferromagnets. In the clean system, order-by-disorder selects a magnetically ordered state from a classically degenerate manifold. In the presence of randomness, however, different orders can be chosen locally depending on details of the disorder configuration. Using a combination of analytical considerations and classical Monte-Carlo simulations, we argue that any long-range-ordered magnetic state is destroyed beyond a critical level of randomness where the system breaks into magnetic domains due to random exchange anisotropies, becoming, therefore, a glass of spin clusters, in accordance with the available experimental data. These random anisotropies originate from off-diagonal exchange couplings in the microscopic Hamiltonian, establishing their relevance to other magnets with strong spin-orbit coupling.

The Ni-alloy, Ni$_{1-x}$V$_x$, presents the opportunity to study a ferromagnetic (FM) quantum phase transition (QPT) with strong ''disorder'' introduced by ''random'' chemical substitution. Using complementary methods, neutron scattering, muon spin relaxation ($\mu $ SR) and magnetization, we show evidence of disorder and probe signatures of quantum Griffiths singularities close to the quantum critical concentration $x_c \approx11.6\%$ where the FM magnetic order is suppressed [1]. The key findings in Ni-V are: A detailed pair density function (PDF) analysis confirms that Ni-V is one of the rare binary Ni-alloys that form a solid solution (with random atomic occupation on a fcc-lattice) in the relevant Ni-rich regime. Field-dependent magnetization data display power laws with non-universal exponents in an extended concentration region around $x_c$. This signals the first evidence of quantum Griffiths phase (QGP) in the FM phase. $\mu$SR data also recognize a remaining dynamic magnetic cluster distribution besides the FM ordered response in the FM phase close to $x_c$ [1]. New small angle neutron scattering (SANS) data reveal a short-range magnetic cluster contribution and evidence for long-range FM domains for the same samples. The experimental signatures of a QGP with its limits are discussed in this Ni-alloy with strong short range and weak long-range interaction. in collaboration with: A. Gebretsadik, S. Bhattarai, J.-G. Lussier, A. Alyami, R. Wang, Kent State University; T. Vojta, Missouri University of S & T; K. Page, L. Debeer-Schmitt, ORNL; K. Krycka, NIST; P. J. Baker, F. L. Pratt, ISIS, STFC RAL; S. J. Blundell, T. Lancaster, J. S. Möller, Oxford [1] R. Wang, A. Gebretsadik, S. Ubaid-Kassis et al., Phys. Rev. Lett. 118, 267202 (2017)

The dynamics of continuous phase transitions is governed by the dynamic scaling exponent relating the correlation length and correlation time. For transitions at finite temperature, thermodynamic critical properties are independent of the dynamic scaling exponent z. In contrast, at quantum phase transitions where the transition temperature becomes zero, static and dynamic properties are inherently entangled by virtue of the uncertainty principle. Consequently, thermodynamic scaling equations explicitly contain the dynamic exponent. Basic thermodynamic measurements of thr magnetization M and the specific heat C (as a function of temperature and magnetic field) for the itinerant ferromagnet $Sr{1−x}Ca_xRuO_3$ show that the transition temperature becomes zero for x = 0.7. For samples of this Ca concentraton, we find dynamic scaling of M and C with highly unusual quantum critical dynamics. We observe a small dynamic scaling exponent z of 1.76 strongly deviating from current models of ferromagnetic quantum criticality and likely being governed by strong disorder in conjunction with strong electron–electron coupling. Work done in collaboration with C. L. Huang, D. Fuchs, M. Wissinger, R. Schneider, M. C. Ling, M. S. Scheurer, and J. Schmalian

The intermetallic compound UCoGe has become a laboratory tool to study the unusual coexistence of itinerant ferromagnetism ($T_C$ = 3 K) and superconductivity ($T_s$ = 0.5 K) on the microscopic scale at ambient pressure [1]. Superconductivity in UCoGe is unconventional, as is demonstrated by the strongly anisotropic upper critical field and the extremely large $B_{c2}(0)$ values for magnetic field directions perpendicular to the ordered moment [2]. Here we report thermal expansion measurements in zero and applied magnetic fields on a single crystal of UCoGe around the magnetic and superconducting phase transitions [3]. The thermal expansion cell was mounted on a piezo-electric rotator in order to allow for a precise alignment of the magnetic field with the main crystal axes. The superconducting and magnetic phase diagram has been determined. With our bulk technique we confirm the S-shape of the upper-critical field for $B \parallel b$ and reinforcement of superconductivity above 6 T. At the same time the Curie point shifts towards lower temperatures when the field is applied along the $b$-axis. Our results lend further support to theoretical proposals of spin-fluctuation mediated reinforcement of superconductivity for $B \parallel b$. [1] N.T. Huy et al., Phys. Rev. Lett. 99, 067006 (2007). [2] D. Aoki et al., J. Phys. Soc. Jpn 78, 113709 (2009). [3] A.M. Nikitin et al., Phys. Rev. B 85, 115151 (2017).

We report new NMR results on UGe2 with focusing the electronic state near the critical point between FM1 and FM2 phases. At the phase boundary, NMR can evaluate the microscopic states in each phase separately and detect the magnetic fluctuations originated from the critical point. We will discuss the relationship between superconductivity and the observed fluctuations. In the latter part of the presentation, we will show the pressure-temperature-magnetic field phase diagrams of some f-electrons ferromagnets.

The threshold of metallic magnetism throws up fundamental questions that continue to challenge our understanding of correlated systems. Among these are the complex phase diagrams arising from the interplay of ferromagnetic and spin-density wave order, the non-Fermi liquid states observed near second order but in some cases also near first order quantum phase transitions, the consequences of disorder and inhomogeneity, the opportunities for topological defects to form, and the possibility of unconventional superconductivity. I will review progress with the layered iron-based superconductor YFe$_2$Ge$_2$ [1], which according to DFT [3] and neutron scattering studies [3] is closely balanced near the threshold of ferromagnetic and antiferromagnetic order. The high Sommerfeld ratio of order 100 mJ/molK$^2$ and a $T^{3/2}$ temperature dependence of the electrical resistivity at low temperature $T$ indicate that YFe$_2$Ge$_2$ is governed by strong electronic correlations. Our crystal growth study found that superconductivity in YFe$_2$Ge$_2$ is very sensitive to disorder scattering, pointing towards an unconventional pairing state. Only the most recent crystals, which reach resistance ratios of order 400, show a sharp superconducting heat capacity anomaly and enable a first examination of the superconducting state. 1. J. Chen et al. Phys. Rev. Lett. 116, 127001 (2016) and Phys. Rev. B 99, 020501 (2019). 2. D. J. Singh, Phys. Rev. B 89, 024505 (2014); A. Subedi Phys. Rev. B 89, 024504 (2014). 3. H. Wo, H. et al. cond-mat 180807262 (2018).

Motivated by the ongoing effort to search for high-resolution signatures of quantum spin liquids, we investigate the temperature dependence of the indirect resonant inelastic x-ray scattering (RIXS) response for the Kitaev honeycomb model. We find that, as a result of spin fractionalization, the RIXS response changes qualitatively at two well-separated temperature scales, $T_L$ and $T_H$, which correspond to the characteristic energies of the two kinds of fractionalized excitations, Z2 gauge fluxes and Majorana fermions, respectively. While thermally excited Z2 gauge fluxes at temperature TL lead to a general broadening and softening of the response, the thermal proliferation of Majorana fermions at temperature $T_H\sim 10$ $T_L$ results in a significant shift of the spectral weight, both in terms of energy and momentum. Due to its exclusively indirect nature, the RIXS process we consider gives rise to a universal magnetic response and, from an experimental perspective, it directly corresponds to the K-edge of Ru$^{3+}$ in the Kitaev candidate material $\alpha$-RuCl$_3$.

A seemingly ferromagnetic state can emerge in an Ising system with sufficiently large anisotropy even when effective exchange interactions are prohibited by strong dilution. A prime example is given by iron-doped lithium nitride, $Li_2(Li_{1-x}Fe_x)N$ with x << 1. The basic magnetic unit is not a cluster or a magnetic domain but the magnetic moment of single, isolated iron atoms, which are embedded in the non-magnetic lithium nitride matrix [1]. This novel model system shows an extremely large magnetic anisotropy and allows to study quantum tunneling of the magnetization in a rather simple, inorganic material. At the origin of the outstanding properties is the special geometry that the iron finds itself in: the linear coordination between two nitrogen atoms is not subject to a Jahn-Teller distortion and gives rise to an unquenched orbital moment. Accordingly, this rare-earth-free material shows a huge hysteresis with coercivity fields of more than 11 Tesla. Recently we have found an extreme field dependence of the spin reversal process in $Li_2(Li_{1-x}Fe_x)N$ [2]. The spin-flip probability strongly increases in transverse magnetic fields that proves the resonant character of this magnetic tunneling process. Applied Longitudinal fields, on the other hand, lift the ground-state degeneracy and destroy the tunneling condition. An increase of the relaxation time by four orders of magnitude in applied fields of only a few milliTesla reveals exceptionally sharp tunneling resonances. Therefore, it is possible either to freeze the orientation and mimic ferromagnetic ordering or to promote the flip of a spin-state by tiny applied fields. The up and down states of the iron atom's spin have been made switchable and provide an 'atomic quantum bit' at easily accessible liquid helium temperatures. [1] A. Jesche et al. Nat. Commun. 5:3333 (2014) doi: 10.1038/ncomms4333 [2] M. Fix, J. H. Atkinson, P. C. Canfield, E. del Barco & A. Jesche Phys. Rev. Lett. 120, 147202 (2018)

The composite heavy quasiparticles in Kondo lattice systems formed below the Kondo temperature T$_K$ hybridize with the conduction electrons. This leads to the formation of flat bands close to the Fermi level which alter the Fermi surface. Applying a magnetic field has two effects on this Fermi surface: First, it leads to Zeeman splitting of the flat bands, which can undergo Lifshitz transitions. Second, the composite quasiparticles eventually break up. The standard tool to determine Fermi surfaces are quantum oscillation measurements. However, they have to be performed at high magnetic fields, while their results are interpreted with the help of band structure calculations at zero field. The ability of quantum oscillation measurements to interpret zero field properties is therefore under intense discussion and a general understanding of the development of Kondo lattices in magnetic fields with an energy of the order of T$_K$ is needed. We studied the magnetic field development of the ferromagnetic Kondo lattice YbNi$_4$P$_2$ using low temperature thermopower and resistivity measurements. We observe several Zeeman driven Lifshitz transitions on top of a continuous suppression of the Kondo effect. The large number of Lifshitz transitions observed within a small energy window is the result of the flat renormalized band structure with strong 4f-electron character shaped by the Kondo lattice effect. Some of the Lifshitz transitions show a particularly strong signature in thermodynamic probes which can be a hint towards the involvement of the predicted quasi one-dimensional Fermi surface sheets.

In ferromagnetic superconductors, like URhGe, superconductivity coexists with magnetism near zero field, but then reappears in a finite field range, where the system also displays mass enhancement in the normal state. We present the theoretical understanding of this nonmonotonic behavior. We explore the multiband nature of URhGe and associate reentrant superconductivity and mass enhancement with the topological transition (Lifshitz) in one of the bands in a finite magnetic field. We find excellent agreement between our theory and a number of experimental results for URhGe, such as weakly first-order reentrant transition, the dependence of superconducting T$_c$ on a magnetic field, and the field dependence of the effective mass, the specific heat, and the resistivity in the normal state. Our theory can be applied to other ferromagnetic multiband superconductors.

Muon Spin Relaxation (MuSR) studies reveal several fundamental aspects of quantum and thermal phase transitions and Skyrmions in itinerant electron magnets MnSi, (Mn,Fe)Si and MnGe. In quantum phase transitions of MnSi tuned by hydrostatic pressure and in (Sr,Ca)RuO3 by (Sr,Ca) substitutions, we found clear signature of first order transition associated with phase separation [1]. With (Mn,Fe) substitutions of 15 %, (Mn,Fe)Si exhibits features expected in second order transitions in pressure-tuned quantum evolution, which can be related to the effect of disorder [2]. Both in bulk MnSi, (Mn,Fe)Si and thin film of MnSi, we detected significant spin dynamics in the Skyrmion state, associated with dynamic critical behavior in the thermal transition from paramagnetic state in bulk MnSi and (Mn,Fe)Si, but without such dynamic critical behavior in the thin film of MnSi. In MnGe, we found peaking of 1/T1 relaxation rate and dynamic critical behavior in the boundary between the field induced ferromagnetic phase and helical phase with a 3-d Skyrmion lattice. The absence of critical behavior at boundary between para and 2-d Skyrmion phase with different winding number in MnSi thin film, and the existence of critical behavior between ferro and 3-d Skyrmion lattice with the same zero winding number in MnGe are consistent with effect of winding-number topology on critical behavior. [1] Y.J. Uemura et al., Nature Physics 3 (2007) 29-35 [2] T. Goko et al., npj Quantum Materials 2 (2017) 44. Skyrmion work was performed in collaboration with the RIKEN/Tokyo group of Y. Tokura, N. Kanazawa and N. Nagaosa.

Recent discovery of the pressure-induced phase transition to an antiferromagnetic state in itinerant ferromagnet $LaCrGe_3$ [1] has been interpreted as a genuine manifestation of ferromagnetic quantum critical effects. I will present an alternative approach to a switching between ferromagnetic and antiferromagnetic groundstates in d-electron magnets [2], the Stoner-like microscopic mechanism, that relies on a very generic electronic structure properties rather than quantum criticality. Mechanism can be conveniently parametrized to mirror general energy scales and oxidation states of the real material. In particular, it provides not only an interpretation of the pressure-induced ferromagnetic to antiferromagnetic transition in $LaCrGe_3$, but also lack thereof in $LaV_{0.16}Cr_{0.84}Ge_3$ [3]. I will also demonstrate applicability of the mechanism to $ZrZn_2$ and CrAs. If time permits, I will additionally comment on the interpretation of the pressure-induced antiferromagnetic to ferromagnetic phase transition in $USb_2$ based on similar physical principles [4]. [1] V. Taufour, et al., Phys. Rev. Lett. 117, 037207 (2016) [2] M. M. Wysokiński, arXiv:1808.04109v2 [3] X. Lin, et al., Phys. Rev. B 88, 094405 (2013) [4] M. M. Wysokiński, Phys. Rev. B 96 201115(R) (2017)