Abstract
Since the discovery of spin glasses in dilute magnetic systems, their study has been largely focused on understanding randomness and defects as the driving mechanism. The same paradigm has also been applied to explain glassy states found in dense frustrated systems. Recently, however, it has been theoretically suggested that different mechanisms, such as quantum fluctuations and topological features, may induce glassy states in defect-free spin systems, far from the conventional dilute limit. Here we report experimental evidence for the existence of a glassy state, that we call a spin jam, in the vicinity of the clean limit of a frustrated magnet, which shows unconventional spin glass properties, such as the quadratic low-temperature dependence of heat-capacity, a broad elastic neutron scattering peak centred around a non-zero wavevector (Q), and insensitivity to low-concentration of defects. We have mainly studied two isostructural frustrated magnets, SrCr9pGa12-9pO19 (SCGO(p)) and BaCr9pGa12-9pO19 (BCGO(p)), in which the magnetic Cr3+ (s=3/2) ions form a quasi-two-dimensional (Q2D) triangular system of bi-pyramids, in comparison to several other magnetic glasses ranging from dilute magnetic alloys to three-dimensional frustrated magnets.
We performed both bulk susceptibility and neutron scattering experiments on both SCGO(p) and BCGO(p) systems as a function of dilution p in the range of 0.4 ≲ p ≲0.97. Our bulk susceptibility data exhibit glassy behavior at much lower temperatures (T_f) than the absolute values of the Curie-Weiss temperature, |θ_CW | ≈ -504(2) K for SCGO(p = 0.968(6)) and -695(1) K for BCGO(p = 0.902(8)). The frustration index f=|θ_CW |/T_f is as high as 130 for SCGO(p = 0.968(6)) and 190 for BCGO(p = 0.902(8)) indicating strong frustration. Low-temperature susceptibility measurements of SCGO(p)/BCGO(p) reveals glass-like signatures such as zero-field cooled (ZFC) – field cooled (FC) hysteresis and frequency dependent AC susceptibility, but the low-temperature magnetic phases found in the dense limit of magnetic ions show distinct properties from conventional spin glasses.
Our inelastic neutron scattering data on SCGO(p = 0.968(6)) and BCGO(p = 0.902(8)) shows dispersionless magnetic excitations centered at ℏω = 18.6(1) meV and 16.5(1) meV respectively, due to singlet to triplet excitations of spin s = 3/2 dimers. The spin dimers are formed by Cr3+ ions in two 4fvi layers that lie between 12k-2a-12k (kagome–triangle–kagome) trilayers. Moreover, a continuum spectrum centered around Q≈1.5 Å^(-1), observed in time-of-flight (TOF) neutron scattering experiments for SCGO(p)/BCGO(p) magnets, confirm that the kagome−triangular−kagome trilayer is responsible for low-energy spin dynamics. The imaginary part of dynamical susceptibility (χ^'' (ω)) for SCGO(p = 0.968(6)) or BCGO(p = 0.902(8)) shows linear dependence at low-energies which is inconsistent with energy independent behavior of dilute spin glasses. Yet, the unconventional dynamics of these magnets can be explained by low-temperature Halprin and Saslow (HS) like modes. To investigate the low-energy excitations further, we performed neutron spin echo experiments on the BCGO(p = 0.902(8)) sample and observed slowly varying dynamics in the nanosecond time scale below the glass transition temperature which is inconsistent with conventional stretched exponential nature of spin glasses.
To confirm the spin jam nature in SCGO(p) magnets, we performed TOF neutron scattering experiments as a function of non-magnetic doping. As the nonmagnetic Ga3+ impurity concentration is changed, there are two distinct phases of glassiness: the spin jam which is insensitive to defects, for high magnetic concentration region (p > 0.8) and a cluster spin glass for lower magnetic concentration, (p < 0.8). This observation indicates that a spin jam is a unique vantage point from which glassy states in frustrated magnets, where the spins are densely packed, can be understood.
Furthermore, we performed a comparative study of three different magnetic glasses using various experimental techniques ranging from bulk susceptibility to inelastic neutron scattering. The systems we studied are a Q2D magnet BCGO(p = 0.902(8)), a three-dimensional frustrated magnet Y2Mo2O7 and a conventional spin glass CuMn2%. The magnetic elastic order parameter was observed as a function of energy resolution for each sample. The energy resolution dependent ordering temperature could be modeled by Vogel-Fulcher law, indicating glass-like freezing in all the three compounds, yet with different fitting parameters implying distinct glass phases. The magnetic field effects on bulk susceptibility have also been studied for all the three samples and a systematic method to distinguish different magnetic glasses is presented.
The notion of complex energy landscapes underpins the intriguing dynamical behaviors in many systems ranging from polymers to brain activity, to social networks and glass transitions. The spin glass state found in dilute magnetic alloys has been an exceptionally convenient laboratory frame for studying complex dynamics resulting from a hierarchical energy landscape with rugged funnels. Here, we show, by a bulk susceptibility and Monte Carlo simulation study, that densely populated frustrated magnets in a spin jam state exhibit much weaker memory effects than spin glasses, and the characteristic properties can be reproduced by a nonhierarchical landscape with a wide and nearly flat but rough bottom. Our memory effect results on SCGO(p = 0.97) and BCGO(p = 0.96) in comparison to CuMn2%, illustrate that the memory effects can be used to probe different slow dynamics of glassy materials, hence opening a window to explore their distinct energy landscapes.
Furthermore, we studied memory effects in various magnetic glasses including high-temperature superconductor-related materials, spin-orbit Mott insulators, frustrated magnets, and dilute magnetic alloys to characterize ubiquitous glassiness found in magnetic materials. Here, we show that scaling of magnetic memories with time can be used to classify magnetic glassy materials into two distinct classes. Our bulk magnetization measurements reveal that most densely populated magnets exhibit similar memory behavior characterized by a relaxation exponent of 1 - n ≈ 0.6(1). This exponent is different from 1 - n ≈ 1/3 of dilute magnetic alloys that were ascribed to their hierarchical and fractal energy landscape, and is also different from 1 - n = 1 of the conventional Debye relaxation expected for a spin solid, a state with long-range order. Furthermore, our systematic study of dilute magnetic alloys with varying magnetic concentration exhibits crossovers among the two glassy states and spin solid.
Finally, we have developed a numerical simulation technique to comprehend neutron scattering experiments on magnetic phases with short-range correlations using Landau-Lifshitz dynamics. Here, we study the spin-S Kitaev model in the classical (S → ∞) limit and compare against the dynamical structure factors of the spin-1/2. More interestingly, the low-temperature and low-energy spectrum of the classical model exhibits a finite energy peak, which is the precursor of the one produced by the Majorana modes of the S = 1/2 model. The classical peak is spectrally narrowed compared to the quantum result and can be explained by magnon excitations within fluctuating one-dimensional manifolds (loops). Hence the difference from the classical limit to the quantum limit can be understood by the fractionalization of magnons propagating in one-dimensional manifolds. Moreover, we show that the momentum space distribution of the low-energy spectral weight of the S = 1/2 model follows the momentum space distribution of zero modes of the classical model.
