Session L48: Frustrated Magnetism: Spin Ice

Long-range Coulomb interactions and monopole population inversion following thermal quenches in spin ice

In spin ice materials, such as rare earth titanate pyrochlores Dy2Ti2O7 and Ho2Ti2O7, magnetic frustration leads to a highly degenerate ground state that can be seen as a topologically ordered magnetic vacuum whose elementary excitations take the form of point-like magnetic charges. Numerical simulations have predicted the emergence of monopole-rich metastable states following a thermal quench, which have been recently observed in experiments. This metastability arises from a population inversion between what are called free monopoles and noncontractible pairs. In this work we investigate in detail the origin of the population inversion by simulating an effective spin ice model with explicit Coulomb interactions between the monopoles. Our results demonstrate that the long-range tail of the Coulomb interaction is responsible for the population inversion, and thence for the dynamical arrest. We also build a mean field analytic model of the monopole population dynamics to explain the different regimes observed.   

Comparison of the Supercooled Spin Liquid States in the Pyrochlore Magnets Dy$_{\mathrm{2}}$Ti$_{\mathrm{2}}$O$_{\mathrm{7}}$ and Ho$_{\mathrm{2}}$Ti$_{\mathrm{2}}$O$_{\mathrm{7}}$

Despite a well-ordered crystal structure and strong magnetic interactions between the Dy or Ho ions, no long-range magnetic order has been detected in the pyrochlore titanates Ho$_{\mathrm{2}}$Ti$_{\mathrm{2}}$O$_{\mathrm{7}}$ and Dy$_{\mathrm{2}}$Ti$_{\mathrm{2}}$O$_{\mathrm{7}}$ [1]. The low temperature state in these materials is governed by spin-ice rules. These constrain the Ising like spins in the materials, yet does not result in a global broken symmetry state. To explore the actual magnetic phases, we simultaneously measure the time- and frequency-dependent magnetization dynamics of Dy$_{\mathrm{2}}$Ti$_{\mathrm{2}}$O$_{\mathrm{7}}$ and Ho$_{\mathrm{2}}$Ti$_{\mathrm{2}}$O$_{\mathrm{7}}$ using toroidal, boundary-free magnetization transport techniques. We demonstrate a distinctive behavior of the magnetic susceptibility of both compounds, that is indistinguishable in form from the permittivity of supercooled dipolar liquids. Moreover, we show that the microscopic magnetic relaxation times for both materials increase along a super-Arrhenius trajectory also characteristic of supercooled glass-forming liquids. Both materials therefore exhibit characteristics of a supercooled spin liquid. Strongly-correlated dynamics of loops of spins is suggested as a possible mechanism which could account for these findings. Potential connections to many-body spin localization will also be discussed. [1] J. S. Gardner, et al., Rev. Mod. Phys., 82, 53 (2010).   

How muons alter spin ice and what they probe

Muons implanted into spin ice sit at a well-defined site that is now determined using DFT+$\mu$ calculations. It is also known [Foronda et al., PRL {\bf 114}, 017602 (2015)) that the muon-induced distortion in Pr-containing pyrochlores leads to a dominating effect on the muon-induced response. We show that the effect in Dy- and Ho-containing systems is quite different, even though the muon site and muon-induced distortions are similar. By using further DFT+$\mu$ calculations and in experiments on Dy$_2$Ti$_2$O$_7$, Ho$_2$Ti$_2$O$_7$, as well as Sc-doped samples, we show that this picture can be understood quantitatively and allows us to understand the relationship between the muon-induced response and the magnetic properties. We assess the extent to which muon measurements in these systems can be sensitive to monopole behaviour.   

Emergent Order in the Kagome Ising Magnet Dy$_{3}$Mg$_{2}$Sb$_{3}$O$_{14}$

The 3:1 cation ordered pyrochlore Dy$_{3}$Mg$_{2}$Sb$_{3}$O$_{14}$ forms a magnetic lattice of layered two-dimensional (2D) Kagome sheets, comprised of corner sharing triangles. As in the 3D spin-ice analogue, Dy$_{2}$Ti$_{2}$O$_{7}$, the Dy$^{3+}$ spins have Ising like behaviour. Here, we show that the layered Ising magnet Dy$_{3}$Mg$_{2}$Sb$_{3}$O$_{14}$ hosts an emergent order predicted theoretically for individual kagome layers of in-plane Ising spins. Neutron-scattering and bulk thermomagnetic measurements, supported by Monte Carlo simulations, reveal a phase transition at $T$* $=$ 0.3 K from a disordered spin-ice like regime to an "emergent charge ordered" state in which emergent charge degrees of freedom exhibit three-dimensional order while spins remain partially disordered. Our results establish Dy$_{3}$Mg$_{2}$Sb$_{3}$O$_{14}$ as a tuneable system to study interacting emergent charges arising from kagome Ising frustration.   

From pyrochlore to the tripod kagome lattice

Finding new kagome lattice-containing compounds with spin-type variability has been an experimental challenge for realizing the exotic states predicted theoretically. Recently, we discovered such a new kagome compound family, A$_{2}$RE$_{3}$Sb$_{3}$O$_{14}$ (A = Mg, Zn; RE = rare earth element), by partial ion substitution in the pyrochlore lattice. These compounds feature a hitherto unstudied structure, namely the ``tripod kagome lattice". In this talk, I shall demonstrate that due to the unique tripod-like spin anisotropies and a large variability of the rare earth spin sets, the complex interplay between crystal field splitting and spin-spin interactions in the tripod kagome lattice leads to various exotic states. These include a dipolar spin order, a magnetic charge order, a quantum kagome ice, a quantum spin liquid, and a possible Kosterlitz-Thouless transition, as evidenced by our susceptibility, specific heat, and neutron scattering measurements. We hope our works will stimulate both experimental and theoretical studies on these exciting compounds.\\ References: [1] Z. L. Dun, J. Trinh, K. Li, M. Lee, K. W. Chen, R. Baumbach, Y. F. Hu, Y. X. Wang, E. S. Choi, B. S. Shastry, A. P. Ramirez, and H. D. Zhou, Phys. Rev. Lett. 116, 157201 (2016). [2] Z. L. Dun, J. Trinh, M. Lee, E. S. Choi, K. Li, Y. F. Hu, Y. X. Wang, N. Blanc, A. P. Ramirez, H. D. Zhou, arXiv:1610.08396.   

Spin and emergent magnetic charge ordering in single and stacked layers of dipolar kagome spin ice

Kagome Ising systems with long range (dipolar) interactions can host interesting phases with unusual ordering features driven by frustration. These include intermediate-temperature states where order and disorder coexist, arising from spin ``fragmentation'', and low-temperature states with novel spin textures, that are sometimes accompanied by pronounced freezing. Often these ordering phenomena are best understood in terms of emergent magnetic charges and the effective interactions between them. In this work, by means of analytical and phenomenological arguments combined with Monte Carlo simulations, we investigate a family of models on the kagome lattice in two and three dimensions, where the Ising easy axes of the spins can be tuned from in-plane to out-of-plane. In two dimensions, we elucidate low-temperature ordering and freezing phenomena with an effective charge picture and appropriate order parameter. In three dimensions, emergent charges order in a sodium-chloride-like structure in the intermediate temperature regime. The low-temperature ordering behaviour depends on the level of canting, including a novel spin texture. We discuss the relevance of our results for experiments on artificial arrays and the new tripod kagome lattice compounds (which include Dy3Mg2Sb3O14).   

A new spinel selenide spin ice material

Rare earth pyrochlores such as the rare earth stannates and titanates are well known for having a variety of exotic magnetic ground states. The variety stems from the unique combination of the highly frustrated lattice geometry of corner sharing tetrahedra that forms the pyrochlore lattice, and from how each rare earth ion behaves in the local crystal electric field. In this talk we extend the discussion to a new class of materials with rare earth spins on the pyrochlore sublattice of the spinel structure: MgRE$_2$Se$_4$ (RE = rare earth). Since the local crystal field is vastly different in the spinels, the materials have distinctly different ground states from the pyrochlores with the same rare earth. In this talk, we focus on the RE = Er case, and present data demonstrating spin ice behavior. Heat capacity data shows the expected $1/2ln(3/2)$ residual entropy. We show magnetic diffuse scattering from neutron powder diffraction, and with reverse Monte Carlo fits show it is consistent with expectations for a spin ice. Lastly, we present measurements of the crystal electric field levels via inelastic neutron spectroscopy, and subsequent Monte Carlo fits of that data which confirm the local Ising nature of the material.   

Field-induced ordering in dipolar spin ice

We present numerical studies of dipolar spin ice in the presence of a magnetic field slightly tilted away from the [111] axis. We find a first-order transition from a kagome ice to a $\mathbf{q}=$X state when the external field is tilted toward the $[11\bar{2}]$ direction. This is consistent with the anomalous critical scattering previously observed in the neutron scattering experiment on the spin ice material $\textrm{Ho}_{2}\textrm{Ti}_{2}\textrm{O}_{7}$ in a tilted field [Nat. Phys. \textbf{3}, 566 (2007)]. We show that this ordering originates from the antiferromagnetic alignment of spin chains on the kagome planes. The residual entropy of the kagome ice is fully recovered. Our result captures the features observed in the experiments and points to the importance of the dipolar interaction in determining ordered states in the spin ice materials. We place our results in the context of recent susceptibility measurements on $\textrm{Dy}_{2}\textrm{Ti}_{2}\textrm{O}_{7}$, showing two features for a [111] field.   

Monopole Quasiparticle Dynamics in Spin Ice via SQUID Magnetometry

The spin ice pyrochlore class of frustrated magnets exhibits thermally-activated quasiparticle Dirac monopoles. We use SQUID magnetometry to measure magnetization fluctuations induced by the nucleation and diffusion of monopole/anti-monopole pairs in Dy2Ti2O7 single crystals and thin films grown by pulsed laser ablation. To test aspects of the monopole picture, we also probe the non-equilibrium magnetization from the imbalance of opposite-polarity monopoles induced by the simultaneous application of a temperature gradient that creates a gradient in the monopole pair density and a magnetic field that drives monopole polarities in opposite directions. Under these conditions we seek to observe the monopolar distribution, dynamics, and relaxation in and out of the spin ice regime.   

Anomalous temperature dependence of electron conductivity in pyrochlore spin ice

The strong interaction between the conduction electron and local moments has been suggested to be the reason behind the anomalous behavior of the resistivity at low temperatures. Indeed, resistivity minimum observed in several itinerant magnetic systems has been attributed to the formation of a singlet state between conduction electrons and the magnetic impurity, a phenomena known as the Kondo effect. Recently, a non-Kondo mechanism has been proposed to explain a similar resistivity minimum observed in metallic spin ice [1,2]. In these theories, the resistivity upturn at low temperature is related to the nontrivial temperature dependence of spin correlation in the ice phase. Here we address this issue using the Ziman formula that relates the electron scattering to the spin correlation function. We calculate the correlation function analytically using the spherical approximation and high-$T$ expansion approaches, the results are also compared with that obtained from direct Monte Carlo simulations. [1] M. Udagawa, H. Ishizuka, and Y. Motome. Phys. Rev. Lett. {\bf 108}, 066406 (2012). [2] G.-W. Chern, S. Maiti, R. Fernandes, and P. Wolfle, Phys. Rev. Lett. {\bf 110}, 146602 (2013).   

Time dependent Diffuse Scattering experiments on Spin Ice

A recent investigation into the spin ice material Dy2Ti2O7 has shown evidence of a very slow equilibration time constant. Such slow equilibration could be a consequence of either modifications of the spin ice ground state or a result of structural defects in the sample. Neutron scattering provides perhaps the most accurate way of measuring spin correlations and can subsequently be used to constrain the Hamiltonian of the spin ice system. Here we present detailed neutron scattering measurements, in combination with in-situ ac susceptibility measurements to measure the spin correlations as a function of time at 350mK for 10\textasciicircum 6 s. By measuring the spin temperature using susceptibility we can accurately comment on the thermal stability of the sample as a function of time and also characterize the inherent beam heating of the sample. We have investigated the diffuse scattering with an oxygen annealed single crystal, isotopically enriched sample and we will present the results of the thermal annealing experiment concentrating on the diffuse scattering centered around (0,0,3) and (3/2,3/2,3/2).