Sharp spectroscopic fingerprints of disorder in an incompressible magnetic state

Abstract

Disorder significantly impacts the electronic properties of conducting quantum materials by inducing electron localization and thus altering the local density of states and electric transport. In insulating quantum magnetic materials, the effects of disorder are less understood and can drastically impact fluctuating spin states like quantum spin liquids. In the absence of transport tools, disorder is typically characterized using chemical methods or by semi-classical modeling of spin dynamics. This requires high magnetic fields that may not always be accessible. Here, we show that magnetization plateaus -- incompressible states found in many quantum magnets -- provide an exquisite platform to uncover small amounts of disorder, regardless of the origin of the plateau. Using optical magneto-spectroscopy on the Ising-Heisenberg triangular-lattice antiferromagnet K$_2$Co(SeO$_3$)$_2$ exhibiting a 1/3 magnetization plateau, we identify sharp spectroscopic lines, the fine structure of which serves as a hallmark signature of disorder. Through analytical and numerical modeling, we show that these fingerprints not only enable us to quantify minute amounts of disorder but also reveal its nature -- as dilute vacancies. Remarkably, this model explains all details of the thermomagnetic response of our system, including the existence of multiple plateaus. Our findings provide a new approach to identifying disorder in quantum magnets.

Figures (5)

• Figure 1: Magnetization plateau and far-infrared magneto-spectroscopy measurements of K$_2$Co(SeO$_3$)$_2$.a, Schematic representation of the normalized uniform magnetization $m=\langle S_{\rm tot}^z \rangle/NS$ of an ordered antiferromagnet described at the classical level (solid line) or including quantum corrections (dashed line). The spin-$S$ moments form a canted structure and become fully polarized at the saturation magnetic field $H_{\rm sat}$. b, Normalized uniform magnetization of an antiferromagnet with a field-induced quantized plateau state, which is stabilized between $H_{\rm c1}$ and $H_{\rm c2}$. The UUD state is a possible classical state realizing a $m=1/3$ magnetization plateau. c, Crystal structure of KCSO. Orange spheres represent oxygen, purple represents potassium, blue represents cobalt, and yellow represents selenium ions. The triangular lattice of magnetic Co$^{2+}$ ions is shown by blue bonds. d, Frequency and magnetic field dependence of the normalized magneto-transmission spectra $I(\omega; H)$ of KCSO measured in Faraday geometry at $T\!=\!5$ K with the magnetic field direction $z$ along the $c$-axis, see Methods and Supplementary Sec. IA-B for details of our measurement and data analysis protocols. e, Energy modes $E(H)$ extracted from Lorentzian fits to the spectra at each magnetic field. Colors encode the nature of the underlying magnetic state: green and yellow for the compressible Y-like and V-like supersolid phases, and blue, light blue, and purple for the incompressible UUD phase corresponding to a $m=1/3$ plateau. Black lines are linear fits to the data using the averaged $g_{z}$ values, see Supplementary Sec. IC for details. Dotted lines represent excitations originating from the Y and V phases, solid lines indicate dominant spin-flip modes, dot-dashed lines show the triple spin-flip modes, and dashed lines correspond to disorder-induced satellite modes of the UUD phase.
• Figure 2: Schematic interpretation of magnetic excitations in the UUD phase of K$_2$Co(SeO$_3$)$_2$.a, Magnetic excitations expected in the UUD phase with and without bond disorder. Solid lines (labeled #1 and #2) represent excitations in a clean system, while dashed lines (labeled #1a for above, and #1b, #2b for below) correspond to additional excitations in a disordered system. Blue and light blue indicate major spin-flip processes, while red and orange represent minor spin-flip processes. b, Spin structure of the ideal UUD phase for a $3\times3$ triangular-lattice supercell. c, Visualization of major and minor spin-flip processes in clean and disordered systems. The flipped spin in each process is shown with a green sphere. The green bond represents a disorder bond with modified exchange interactions $J_{zz}’ = J_{zz}(1-d_{zz})$ and $J_{ xy}’ = J_{xy}(1-d_{xy})$.
• Figure 3: Universality of the disorder-bond model and its correspondence with the vacancy model. Schematic UUD spin configurations and excitation modes for: a-b, the weakened-bond model, where for illustration the exchange interactions on green and dark green bonds are reduced with parameters $d_{zz}=d_{xy} =0.1$ (green) and $d_{zz}=d_{xy} = 0.2$ (dark green); c-d, the broken-bond model, where $d_{zz}=d_{xy} \rightarrow 1$ for the same broken bond; and e-f, the vacancy model, where two magnetic sites are removed. Throughout, red (blue) circles indicate up (down) spins. The line style for the spectra in b, d, and f follows the convention of Fig. \ref{['fig:2']}. The spectral intensity of the satellite modes is not shown and differs between panels d and f, which represent different defect densities.
• Figure 4: Simulated FIRMS spectra from our model of K$_2$Co(SeO$_3$)$_2$ with and without disorder.a-c, Frequency and magnetic field dependence of simulated $I(\omega; H)$ for a clean system simulated using LSWT, LLD, and ED, respectively. d-f, Similar simulation approaches but for a disordered system. Gray lines represent the linear fits to mode energies, identical to those reported in Fig. \ref{['fig:1']}e. Gray areas indicate the regime where the ground state of the system is not UUD. g, Same data as in Fig. \ref{['fig:1']}d, shown for better comparison with the simulation results.
• Figure 5: Magnetization and specific heat of our crystals of K$_2$Co(SeO$_3$)$_2$.a, Temperature dependence of the magnetization of KCSO in a $c$-axis magnetic field. b-c, Simulated isothermal magnetization curves using the classical MC method for clean and disordered systems, respectively. Magenta dot-dashed lines in each figure indicate the magnetization of the pre-plateau and UUD plateau phase with the vacancy concentration $n$. d, Schematic mechanism stabilizing an additional magnetization plateau around $\mu_0H =2.8$ T. The top and bottom panels show the spin configurations at 2 T and 4 T, respectively. Red, blue, and white circles indicate up spins, down spins, and vacancies, respectively. Orange circles highlight spin-flip sites that result in the additional plateau. e, Field derivative of the magnetization. f-g, Temperature and magnetic field dependence of the specific heat divided by temperature for KCSO in a $c$-axis magnetic field. The colorplot combines constant-field ($T\geq 0.8$ K) and constant-temperature ($T < 0.8$ K) measurements.