This review is written by non-expert and used for personal study only.
Dynamical fractal and anomalous noise in a clean magnetic crystal
- Spin Ice
Researchers are exploring $\textbf{topological matter}$ to achieve a deep understanding similar to that of conventional systems. They focus on the $\textbf{dynamical properties}$ of systems with unique excitations, known as $\textbf{exotic excitations}$. Recent experiments on $\textbf{spin ice}$ have uncovered unusual power law behavior in $\textbf{magnetic noise spectral density}$. This behavior cannot be explained using the standard spin ice dynamics model due to the restrictions imposed by the $\textbf{emergent gauge field}$ and the $\textbf{local field distribution}$.
The spin ice’s $\textbf{fractal structure}$ near a percolation transition is characterized, and anomalous exponents in magnetic noise are attributed to $\textbf{subdiffusive monopole movement}$. A $\textbf{fractal}$ is a complex pattern that repeats itself at different scales, and a $\textbf{monopole}$ is a hypothetical particle with a single magnetic charge. The dynamic fractal’s sparse and structured nature accounts for the faster-than-anticipated increase in $\textbf{macroscopic relaxation time}$ upon cooling, without leaving thermodynamic signatures.
The existing spin ice dynamics model faces challenges in accounting for the large energy scale, as evidenced by susceptibility and magnetic noise experiments. Improved dynamics, which go beyond the standard model, consider the bimodal internal field distribution on spins, with some flipping at a lower rate and others at a finite transverse field. The time scale $t_{\text{fast}}$, the only fitting parameter, sets the unit of time in the analysis.
An extension of the $\textbf{dipolar spin ice Hamiltonian}$ is discussed, incorporating long-range dipolar interactions and first, second, and third nearest-neighbor exchange terms with specific strengths. $\textbf{Magnetic noise measurements}$ on a single Dy$_2$Ti$_2$O$_7$ crystal are displayed in terms of $\textbf{power spectral density}$ (PSD), offering insight into weakly interacting quasiparticle behavior above the spin ice freezing point.
Simulated $\textbf{beyond Standard Model (bSM) dynamics}$ closely match experimental results across a broad frequency and noise power range, with only one fitting parameter. The relaxation time obtained from bSM dynamics aligns well with experimental data, unlike the puzzling $\textbf{Standard Model (SM) dynamics}$ that have confounded the community for years.
Isolated monopole motion in spin ice with bSM dynamics, providing zero to three site movement options, generates a fractal cluster that modifies the system’s relaxation properties, elucidating the anomalous behavior in magnetic noise and susceptibility.
To comprehend magnetization fluctuations, monopole motion’s statistical properties must be examined. Monopole movement occurs on a real-space dynamical cluster, defining a percolation problem near the critical point. The cluster’s properties are accurately replicated by a random walker on a random percolation cluster within the diamond lattice at a filling fraction $p = 0.43$, just above its critical value $p_c \approx 0.39$.
Multiple monopoles’ behavior can influence local spin and transverse field configurations, causing slow dynamics in the percolation cluster. Higher monopole density lowers the crossover value. Monopole PSD exhibits anomalous decay at high frequencies and conventional decay at low frequencies, with $s = 0.50 \pm 0.01$ in $d = 3$, clarifying the experimentally observed anomalous power law near $n^{-1.5}$.
Scientists have identified a fractal object within a uniform, stoichiometric, and disorder-free bulk crystal, generated by the motion of magnetic mon opoles subject to dual constraints. This peculiar noise phenomenon can be accessed via magnetization response and fluctuations, validating a detailed microscopic dynamics model.
Manipulating different spin ice compounds offers promising prospects for future experimentation. The effective percolation problem is at an optimal point near criticality, and introducing quenched or dynamical disorder could illuminate glassy physics below 650 mK. $\textbf{Glassy physics}$ refers to the study of disordered materials that exhibit slow dynamics and a lack of crystalline order.
Highly tunable $\textbf{noisy intermediate-scale quantum}$ (NISQ) platforms unveil uncharted avenues, including inquiries into quantum monopole diffusion and the role of increasingly coherent many-body quantum dynamics. NISQ platforms are quantum devices with a limited number of qubits and noise levels that prevent them from achieving error correction, but they can still be used for various quantum computing tasks.
Researchers are studying topological matter, specifically spin ice, to understand the unusual behavior observed in magnetic noise spectral density and relaxation properties. By analyzing monopole motion and fractal structures, they can shed light on these anomalous properties, paving the way for further experiments and investigations in the field of condensed matter physics.
Required Additional Study Materials
- P. M. Chaikin, T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge Univ. Press, 1995).
- R. Moessner, J. E. Moore, Topological Phases of Matter (Cambridge Univ. Press, 2021).
- A. Stern, Anyons and the quantum Hall effect—A pedagogical review. Ann. Phys. 323, 204–249 (2008).
- M. Udagawa, L. Jaubert, Spin Ice (Springer, 2021).
- C. Castelnovo, R. Moessner, S. Sondhi, Spin ice, fractionalization, and topological order. Annu. Rev. Condens. Matter Phys. 3, 35–55 (2012).
- D. Stauffer, A. Aharony, Introduction to Percolation Theory (Taylor & Francis, 2018).
- B. D. Hughes, in Complex Media and Percolation Theory, M. Sahimi, A. G. Hunt, Eds. (Springer, 2021), pp. 191–235.
- S. T. Bramwell, M. J. Gingras, Spin ice state in frustrated magnetic pyrochlore materials. Science 294, 1495–1501 (2001).
Introductory material
- “Fractals” by Jens Feder
- “Fractals in Physics” 1st Edition by L. Pietronero and E. Tosatti
Reference
Hallen, J. et al. Dynamical fractal and anomalous noise in a clean magnetic crystal. Science 378, 1218 - 1221 (2022).
Hyperspectral imaging of exciton confinement within a moiré unit cell with a subnanometer electron probe
- Quantum Materials
Moiré superlattices, formed by stacking van der Waals crystals, are being utilized to explore fundamental physical phenomena such as the formation of bound electron-hole pairs, or $\textbf{excitons}$. These excitons can be harnessed for quantum simulations and technologies. Recent studies using electron microscopy have investigated the nanoscale confinement and localization of excitons in transition metal dichalcogenide (TMDC) heterostructures, revealing the potential for periodic arrays of well-localized quantum excitations.
Electron microscopy, equipped with high-sensitivity direct electron detectors, allows for simultaneous probing of weak exciton states and structural reconstruction at subnanometer scales. This has been demonstrated through annular dark-field scanning transmission electron microscopy (ADF-STEM) and low-loss scanning transmission electron microscopy-electron energy loss spectroscopy (STEM-EELS) mapping. These techniques image the in-plane structural reconstruction and oscillator strength of moiré excitons within the moiré unit cell of a WS$_2$-WSe$_2$ superlattice.
Researchers prepared an R-stacked heterostructure consisting of WS$_2$ and WSe$_2$ monolayers encapsulated within hexagonal boron nitride (hBN) to protect the sample and narrow excitonic linewidths. The moiré periodicity, influenced by lattice mismatch, affects the local energy landscape. Among different stackings, AA stacking has the highest energy.
ADF-STEM images were analyzed to obtain structural information at subnanometer scales. Quantitative ADF-STEM has been previously utilized to study in-plane reconstructions in large-period, lattice-matched moiré systems. Simulations of relaxed moiré heterostructures reveal clear distinctions between the sizes of bright and dark regions. The observed in-plane structural reconstruction in R-stacked van der Waals interfaces aligns with theoretical predictions and contributes to the formation of flat electronic bands and spatially modulated excitonic states.
The impact of in-plane structural reconstruction on moiré excitonic states was examined using ensemble optical spectroscopy and localized low-loss STEM-EELS. The electron energy loss (EEL) signal measures the local probability of exciton creation, and the split A and B exciton peak positions correspond well with optical data. The largest spectral contribution originates from moiré exciton peak I, consistent with the maximum oscillator strength predicted using ab initio GW plus Bethe-Salpeter equation calculations. The exciton associated with peak I forms a spatially modulated Wannier exciton with the highest charge density at the AA stacking in the moiré superlattice.
Hyperspectral imaging was utilized by researchers to determine the spatial extent of exciton modulation within a moiré unit cell. The intralayer exciton was discovered to be confined to AA sites, forming a triangular arrangement in the moiré pattern. In-plane structural reconstruction leads to spatially modulated electronic bands, observed using cryogenic correlated ADF-STEM and STEM-EELS techniques.
The exciton wave function is localized to the AA site within a radius of approximately 2 nm, indicating the potential for a triangular lattice of excitons in moiré heterostructures. Strong atomic relaxations within each moiré cell result in exciton confinement at specific stacking sites, enabling nanoscopic engineering of bosonic lattices.
Required Additional Study Materials
- D. M. Kennes, M. Claassen, L. Xian, A. Georges, A. J. Millis, J. Hone, C. R. Dean, D. N. Basov, A. N. Pasupathy, A. Rubio, Moiré heterostructures as a condensed-matter quantum simulator. Nat. Phys. 17, 155–163 (2021).
- L. Rangel DaCosta, H. G. Brown, P. M. Pelz, A. Rakowski, N. Barber, P. O’Donovan, P. McBean, L. Jones, J. Ciston, M. C. Scott, C. Ophus, Prismatic 2.0 - Simulation software for scanning and high resolution transmission electron microscopy (STEM and HRTEM). Micron 151, 103141 (2021).
- R. F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope (Springer, ed. 3, 2011).
Introductory material
- “Hyperspectral Imaging: Techniques for Spectral Detection and Classification” by Chein-I Chang
- “Hyperspectral Imaging Remote Sensing: Physics, Sensors, and Algorithms” by Dimitris G. Manolakis, Ronald B. Lockwood, and Thomas W. Cooley
Reference
Sandhya, S. et al. Hyperspectral imaging of exciton confinement within a moiré unit cell with a subnanometer electron probe. Science 378, 1235-1239 (2022).
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