Quantum Griffiths Phase Research Articles

Novel emergent phases in a two-dimensional superconductor

In this letter, we report our observation of an extraordinarily rich phase diagram of a LaScO3/SrTiO3 heterostructure. Close to the superconducting transition temperature, the system hosts a superconducting critical point of the Infinite-randomness type characterized by an effective dynamical exponent νz that diverges logarithmically. At lower temperatures, we find the emergence of a magnetic field-tuned metallic phase that co-exists with a quantum Griffiths phase (QGP). Our study reveals a previously unobserved phenomenon in 2D superconductors—an unanticipated suppression of the QGP below a crossover temperature in this system. This concealment is accompanied by the destruction of the superconducting quantum critical point (QCP) signaled by a power-law divergence (in temperature) of the effective dynamical exponent. These observations are entirely at odds with the predictions of the infinite-randomness scenario and challenge the very concept of a vanishing energy scale associated with a QCP. We develop and discuss possible scenarios like smearing of the phase transition that could plausibly explain our observations. Our findings challenge the notion that QGP is the ultimate ground state in two-dimensional superconductors.

Quantum Griffiths Singularity in Ordered Artificial Superconducting-Islands-Array on Graphene.

Quantum Griffiths phase (QGP) is a novel quantum phenomenon of quantum phase transition in two-dimensional (2D) superconductors, and the emergence of inhomogeneous superconducting rare regions immersed in a metallic matrix is theoretically related to the quantum Griffiths singularity (QGS). However, the theoretical proposal of superconducting rare regions still lacks intuitive experimental verification. Here, we construct an artificial ordered superconducting-islands-array on monolayer graphene with the aid of an anodic aluminum oxide (AAO) membrane. The QGS under both in-plane and out-of-plane magnetic fields is evidenced by the divergent dynamical critical exponent and is in compliance with the direct activated scaling behavior. The phase diagram clearly shows that the QGP is indeed bred in the rare superconducting regions within isolated superconducting islands with a vanished quantum coherence. Our results reveal the universal features of QGP in artificial heterostructured systems and provide a visualized platform for the theoretical proposal of QGS.

Quantum Griffiths phase in disordered Mn1−xFexSi

We show the presence of magnetic rare regions consistent with the quantum Griffiths phase in Fe-doped MnSi using detailed heat capacity, magnetization, and muon spin relaxation (\ensuremath{\mu}SR) measurements down to millikelvin temperatures. The slow dynamics of these rare regions at low temperatures leads to the non-Fermi-liquid behavior in heat capacity and magnetization. The \ensuremath{\mu}SR and magnetization results further indicate that the dynamics freezes into a cluster-glass state below ${T}_{f}\ensuremath{\sim}1.25$ K. The results are in agreement with theoretical models proposed in the literature for metallic systems with Heisenberg symmetry that exhibit the quantum Griffiths phase in the presence of strong disorder.

Effect of chemical disorder on Griffiths phase in weak itinerant ferromagnetic Ni92−xCuxCr8 alloy

We report magnetic and thermal properties of Cu substituted Ni92−xCuxCr8 polycrystalline alloys close to their critical concentration (xc~18%), where ferromagnetic order disappears. From the detailed analysis of field and temperature dependent magnetization data weak itinerant ferromagnetic character has been identified for the compositions below xc. Spin-fluctuations theory using the Ginzburg–Landau formalism further revealed the contributions of spin fluctuations, which increase near xc. Specific heat data shows unusual low-temperature variation with an enhanced Sommerfeld coefficient and non Fermi-liquid behavior due to strong magnetic fluctuations near xc. The magnetization data and the linear term of the specific heat follow the theoretical predictions of a ferromagnetic quantum critical point within the experimental uncertainties. The exponent of the power-law M(H) ∝Hλ fits to low-temperature magnetic isotherms and its variation with composition indeed suggests a possibility of quantum Griffiths phase in these alloy compositions. Finally, this study shows that the Cu substituted Ni-Cr system shows Griffiths phase at critical point even after the reduction of disorder.

Signatures of a Quantum Griffiths Phase Close to an Electronic Nematic Quantum Phase Transition.

In the vicinity of a quantum critical point, quenched disorder can lead to a quantum Griffiths phase, accompanied by an exotic power-law scaling with a continuously varying dynamical exponent that diverges in the zero-temperature limit. Here, we investigate a nematic quantum critical point in the iron-based superconductor FeSe_{0.89}S_{0.11} using applied hydrostatic pressure. We report an unusual crossing of the magnetoresistivity isotherms in the nonsuperconducting normal state that features a continuously varying dynamical exponent over a large temperature range. We interpret our results in terms of a quantum Griffiths phase caused by nematic islands that result from the local distribution of Se and S atoms. At low temperatures, the Griffiths phase is masked by the emergence of a Fermi liquid phase due to a strong nematoelastic coupling and a Lifshitz transition that changes the topology of the Fermi surface.

Observation of In-Plane Quantum Griffiths Singularity in Two-Dimensional Crystalline Superconductors.

Quantum Griffiths singularity (QGS) reveals the profound influence of quenched disorder on the quantum phase transitions, characterized by the divergence of the dynamical critical exponent at the boundary of the vortex glasslike phase, named as quantum Griffiths phase. However, in the absence of vortices, whether the QGS can exist under a parallel magnetic field remains a puzzle. Here, we study the magnetic field induced superconductor-metal transition in ultrathin crystalline PdTe_{2} films grown by molecular beam epitaxy. Remarkably, the QGS emerges under both perpendicular and parallel magnetic field in four-monolayer PdTe_{2} films. The direct activated scaling analysis with a new irrelevant correction has been proposed, providing important evidence of QGS. With increasing film thickness to six monolayers, the QGS disappears under perpendicular field but persists under parallel field, and this discordance may originate from the differences in microscopic processes. Our work demonstrates the universality of parallel field induced QGS and can stimulate further investigations on novel quantum phase transitions under parallel magnetic field.

Non-Fermi liquid behavior and signature of Griffiths phase in Ni–Cr binary alloy

Detailed magnetic, electrical, and thermal property measurements have been carried out on Ni100−xCrx binary alloys, mainly to study the effect of Cr. The following points emerge from this study: with the increase in Cr concentration, magnetic moment and Curie temperature linearly decreased and the ferromagnetic order is completely suppressed at the critical concentration (xcr ≈ 12.16 ± 0.03). The Rhodes–Wohlfarth ratio increases as the concentration approaches xcr, suggesting a weak itinerant ferromagnetic character of NiCr compositions (x < xcr). Analysis of low-temperature electrical resistivity and specific heat data suggests that the spin fluctuation’s contribution increases and the Fermi-liquid behavior breaks down as the concentration approaches xcr. For x ∼ xcr, the dc susceptibility χ(T) deviates from the Curie–Weiss law reminiscent to that of the Griffiths phase. The low-temperature magnetic isotherms of Ni–Cr follow power law, M(H)=M0+dλHλ, and the non-universal exponent (λ) shows a minimum at xcr ∼ 12. Further, temperature dependence of magnetization studies also support the presence of the quantum Griffiths phase, similar to that reported in the Ni–V alloy system. The temperature dependencies of the electrical resistivity, magnetization, and specific heat follow the theoretical predictions of a quantum critical point within experimental uncertainties.

Spin dynamics and Griffiths singularity in the random quantum Ising magnet PrTiNbO6

In crystalline magnets, interaction randomness is usually thought as a negative factor preventing interesting quantum phenomena to occur. However, intriguing interplay between randomness and quantumness can also leads to unique phenomena in the strongly correlated materials. Among others, the random transverse-field Ising spin chain (RTIC) hosts a renowned quantum Griffiths phase. Although the RTIC model has been regarded as a toy model for long, here we materialize this model with the compound PrTiNbO6, which has a disordered ground state with pronounced quantum fluctuations and continuous spin excitations. The observed anomalous spin dynamics of PrTiNbO6 can be accounted by the RTIC model with a consistent set of parameters determined from fitting the thermodynamic data, and it is ascribed to the quantum Griffiths rare regions in the system. Our results provide a concrete example of quantum Griffiths magnet, and offer an ideal experimental platform for investigating the dynamical properties of random many-body system.

Signatures of a quantum Griffiths phase close to an electronic nematic quantum phase transition

In the vicinity of a quantum critical point, quenched disorder can lead to a quantum Griffiths phase, accompanied by an exotic power-law scaling with a continuously varying dynamical exponent that diverges in the zero-temperature limit. Here, we investigate a nematic quantum critical point in the iron-based superconductor FeSe0.89S0.11 using applied hydrostatic pressure. We report an unusual crossing of the magnetoresistivity isotherms in the non-superconducting normal state which features a continuously varying dynamical exponent over a large temperature range. We interpret our results in terms of a quantum Griffiths phase caused by nematic islands that result from the local distribution of Se and S atoms. At low temperatures, the Griffiths phase is masked by the emergence of a Fermi liquid phase due to a strong nematoelastic coupling and a Lifshitz transition that changes the topology of the Fermi surface.

Comprehensive studies of the transformation between antiferromagnetic CeCoGe3 and heavy fermion CeFeGe3 compounds

We report extensive studies of a series of alloys CeCo1−xFexGe3 in which suppression of the antiferromagnetic order in CeCoGe3 with temperature decreasing to 0 K was observed as a result of substitution of Co with Fe. Unusual temperature dependences were observed of a number of parameters characterizing these compounds: magnetic susceptibility, specific heat, electrical resistivity, magnetoresistance and thermoelectric power, which may be indication towards non-Fermi liquid behaviour connected with quantum critical point (QCP) or can result from a quantum Griffiths phase (QGP) formation. The three AFM transitions found previously for a single crystal CeCoGe3 compound were also detected in our high quality polycrystalline samples, including the case of x>0. All measurements, especially those of magnetocaloric effect (MCE), suggest that for samples with x≥0.1 the contribution of ferromagnetic component, known already for x=0 from the neutron diffraction experiments, increases significantly, with the AFM contribution still present at low temperatures. Finally, the use of different complementary methods enabled construction of a comprehensive magnetic phase diagram of CeCo1−xFexGe3 system.

Quantum Griffiths phase near an antiferromagnetic quantum critical point: Muon spin relaxation study of Ce(Cu1−xCox)2Ge2

The antiferromagnetic order in the heavy-fermion compound ${\mathrm{CeCu}}_{2}{\mathrm{Ge}}_{2}$ can be suppressed by Co-doping, and at critical composition ${x}_{c}=0.6$ (${T}_{N}\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}0$ K) a quantum critical point has been observed. We have performed zero-field (ZF) and longitudinal-field muon spin relaxation ($\ensuremath{\mu}\mathrm{SR}$) measurements on polycrystalline samples of $\mathrm{Ce}{({\mathrm{Cu}}_{1\ensuremath{-}x}{\mathrm{Co}}_{x})}_{2}{\mathrm{Ge}}_{2}$ ($x=0,0.2,0.6,1$) over a temperature range of 100 mK to 10 K and in applied fields from 0 up to 3000 G. Above any ordering temperature, the muon relaxation spectra can be described by a Gaussian-Kubo-Toyabe times exponential line shape. Below the magnetic ordering temperature (i.e., for $x<0.6$), an additional Gaussian relaxation is observed. The zero-field muon relaxation rate suggests the presence of antiferromagnetic ordering below 4 and 0.8 K for $x=0$ and 0.2 samples, respectively. For $x=0.6$, the magnetic order is completely suppressed, and the quantum critical point is accompanied by non-Fermi-liquid behavior, manifested in the power-law divergence of exponential depolarization, i.e., $\ensuremath{\lambda}\phantom{\rule{0.16em}{0ex}}\ensuremath{\propto}\phantom{\rule{0.16em}{0ex}}{T}^{0.55}$. The relaxation rate of $x=0.6$ obeys the time-field scaling relation ${G}_{z}(t,H)={G}_{z}(t/{H}^{\ensuremath{\gamma}})$, which is considered to be a characteristic feature of quantum critical magnetic fluctuations. Furthermore, for $x=0.6$, the exponent of isotherm magnetization, $M\ensuremath{\sim}{H}^{\ensuremath{\eta}}$, and magnetization-field-temperature scaling is consistent with the ZF-$\ensuremath{\mu}\mathrm{SR}$ data. These results provide strong evidence for the formation of a quantum Griffiths phase near the antiferromagnetic quantum phase transition.

Smeared nematic quantum phase transitions due to rare-region effects in inhomogeneous systems

The concept of a vestigial nematic order emerging from a "mother" spin or charge density-wave state has been applied to describe the phase diagrams of several systems, including unconventional superconductors. In a perfectly clean system, the two orders appear simultaneously via a first-order quantum phase transition, implying the absence of quantum criticality. Here, we investigate how this behavior is affected by impurity-free droplets that are naturally present in inhomogeneous systems. Due to their quantum dynamics, finite-size droplets sustain long-range nematic order but not long-range density-wave order. Interestingly, rare droplets with moderately large sizes undergo a second-order nematic transition even before the first-order quantum transition of the clean system. This gives rise to an extended regime of inhomogeneous nematic order, which is followed by a density-wave quantum Griffiths phase. As a result, a smeared quantum nematic transition, separated from the density-wave quantum transition, emerges in moderately disordered systems.

Formation of gapless Z 2 spin liquid phase manganites in the (Sm1–y Gd y )0.55Sr0.45MnO3 system in zero magnetic field: Topological phase transitions to states with low and high density of 2D-vortex pairs induced by the magnetic field

The evolution of the ground state of the manganese spin ensemble in the (Sm1–y Gd y )0.55Sr0.45MnO3 in the case of isovalent substitution of rare-earth samarium ions with large radii with gadolinium ions with significantly smaller radii is studied. The measured temperature dependences of the ac magnetic susceptibility and the field dependences of the dc magnetizations are analyzed using the Heisenberg–Kitaev model describing the transition from the ordered spin state with classical isotropic AFM exchange to the frustrated spin state with quantum highly anisotropic FM exchange. A continuous transition from the 3D ferromagnetic state of manganese spins in the initial sample with y = 0 to zigzag AFM ordering of CE-type spins in ab planes for y = 0.5, coexisting in samples with y = 0.5, 0.6, and 0.7 at temperatures below T N ≅ 48.5 K with a disordered phase such as a quantum Griffiths phase is identified. As the gadolinium concentration further increases, the CE-type zigzag AFM structure is molten, which leads to the appearance of an unusual phase in Gd0.55Sr0.45MnO3 in the temperature range close to the absolute zero. This phase has characteristic features of a gapless Z 2 quantum spin liquid in zero external magnetic field. The step changes in the magnetization isotherms measured at 4.2 K in the field range of ±75 kOe are explained by quantum phase transitions of the Z 2 spin liquid to a phase with topological order in weak magnetic fields and a polarized phase in strong fields. The significant difference between critical fields and magnetization jumps in isotherms indicates the existence of hysteretic phenomena in quantum spin liquid magnetization–demagnetization processes caused by the difference between localization–delocalization of 2D vortex pairs induced by a magnetic field in a quantum spin liquid with disorder.

Quantum Griffiths Phase Inside the Ferromagnetic Phase of Ni_{1-x}V_{x}.

We study by means of bulk and local probes the d-metal alloy Ni_{1-x}V_{x} close to the quantum critical concentration, x_{c}≈11.6%, where the ferromagnetic transition temperature vanishes. The magnetization-field curve in the ferromagnetic phase takes an anomalous power-law form with a nonuniversal exponent that is strongly x dependent and mirrors the behavior in the paramagnetic phase. Muon spin rotation experiments demonstrate inhomogeneous magnetic order and indicate the presence of dynamic fluctuating magnetic clusters. These results provide strong evidence for a quantum Griffiths phase on the ferromagnetic side of the quantum phase transition.

Exploring quantum Griffiths phase in Ni1\u2212xVx nanoalloys

Metallic Ni1−xVx alloys exhibit a ferromagnetic to paramagnetic disordered quantum phase transition in bulk. Such a phase transition is accompanied by a quantum Griffiths phase (QGP), featuring fractional power-law temperature dependences of physical variables, like magnetic susceptibility and specific heat, at low temperatures. As nanoparticles (NP’s) usually exhibit properties significantly different from their bulk counterparts, it is intriguing to explore the occurrence of quantum Griffiths phase in Ni1−xVx nanoalloys. NP’s of Ni1−xVx (0 ≤ x ≤ 0.17) alloys are prepared by a chemical reflux method. The structure and composition of the nanoalloys are determined by X-ray diffraction, X-ray photoelectron spectroscopy and electron microscopy techniques. Metallicity of the samples has been ensured by electrical resistivity measurements. DC magnetization results suggest that ferromagnetism persists in the NP’s until x = 0.17. Low-temperature upturns in magnetic susceptibility and heat capacity hint at critical fluctuations evolving with V-doping. The fluctuations might stem from isolated Ni-clusters within the ferromagnetic NP, indicating a QGP region ranging from x = 0.085 to x ≫ 0.17.

Highly crystalline 2D superconductors

Recent technological advances in controlling materials have developed methods to produce idealized two-dimensional (2D) electron systems such as heterogeneous interfaces, molecular-beam-epitaxy (MBE) grown atomic layers, exfoliated thin flakes and field-effect devices. These 2D electron systems are highly-crystalline with less disorder in common, some of which indeed show sheet resistance more than one order of magnitude lower even in atomic layers or single layers than that of conventional amorphous/granular thin films. Here, we present a review on the recent developments of highly-crystalline 2D superconductors and a series of unprecedented physical properties discovered in these systems. In particular, we highlight the quantum metallic state (or possible metallic ground state), the quantum Griffiths phase in out-of-plane magnetic fields, and the superconducting state maintained in anomalously large in-plane magnetic fields, which were observed in exfoliated 2D materials, MBE-grown atomic-layer thin films and electric-double-layer (ion-gated) interfaces. These phenomena are discussed on the basis of weakened disorder and/or broken spatial inversion symmetry. These novel aspects suggest that highly-crystalline 2D systems are promising platforms for exploring new quantum physics and superconductors.

Effect of disorder on a pressure-inducedz=1magnetic quantum phase transition

Pressure-induced ordering close to a $z=1$ quantum critical point is studied in the presence of bond disorder in the quantum spin system (C$_4$H$_{12}$N$_2$)Cu$_2$(Cl$_{1-x}$Br$_{x}$)$_6$ (PHCX) by means of muon-spin rotation and relaxation. As for the pure system (C$_4$H$_{12}$N$_2$)Cu$_2$Cl$_6$, pressure allows PHCX with small levels of disorder ($x\leq 7.5\%$) to be driven through a quantum critical point separating a low-pressure quantum paramagnetic phase from magnetic order at high pressures. However, the pressure-induced ordered state is highly inhomogeneous for disorder concentrations $x>1\%$. This behavior might be related to the formation of a quantum Griffiths phase above a critical disorder concentration $7.5\%<x_{\rm c}<15\%$. Br-substitution increases the critical pressure and suppresses critical temperatures and ordered moment sizes.

Anomalous diffusion and griffiths effects near the many-body localization transition.

We explore the high-temperature dynamics of the disordered, one-dimensional XXZ model near the many-body localization (MBL) transition, focusing on the delocalized (i.e., "metallic") phase. In the vicinity of the transition, we find that this phase has the following properties: (i)local magnetization fluctuations relax subdiffusively; (ii)the ac conductivity vanishes near zero frequency as a power law; and (iii)the distribution of resistivities becomes increasingly broad at low frequencies, approaching a power law in the zero-frequency limit. We argue that these effects can be understood in a unified way if the metallic phase near the MBL transition is a quantum Griffiths phase. We establish scaling relations between the associated exponents, assuming a scaling form of the spin-diffusion propagator. A phenomenological classical resistor-capacitor model captures all the essential features.

Evidence for magnetic clusters in Ni1-xVx close to the quantum critical concentration

The d-metal alloy Ni1-xVx undergoes a quantum phase transition from a ferromagnetic ground state to a paramagnetic ground state as the vanadium concentration x is increased. We present magnetization, ac-susceptibility and muon-spin relaxation data at several vanadium concentrations near the critical concentration xc ~ 11.6 % at which the onset of ferromagnetic order is suppressed to zero temperature. Below xc, the muon data reveal a broad magnetic field distribution indicative of a long-range ordered ferromagnetic state with spatial disorder. We show evidence of magnetic clusters in the ferromagnetic phase and close to the phase boundary in this disordered itinerant system as an important generic ingredient of a disordered quantum phase transition. In contrast, the temperature dependence of the magnetic susceptibility above xc is best described in terms of a magnetic quantum Griffiths phase with a power-law distribution of fluctuation rates of dynamic magnetic clusters. At the lowest temperatures, the onset of a short-range ordered cluster-glass phase is recognized by an increase in the muon depolarization in transverse fields and maxima in ac-susceptibility.

Disorder-mediated Kondo effect in graphene

We study the emergence of strongly correlated states and Kondo physics in disordered graphene. Diluted short range disorder gives rise to localized midgap states at the vicinity of the system charge neutrality point. We show that long-range disorder, ubiquitous in graphene, allows for the coupling of these localized states to an effective (disorder averaged) metallic band. The system is described by an Anderson-like model. We use the numerical renormalization group (NRG) method to study the distributions of Kondo temperatures $P(T_K)$. The results show that disorder can lead to long logarithmic tails in $P(T_K)$, consistent with a quantum Griffiths phase.