Dichotomous dynamics of magnetic monopole fluids

Abstract

A recent advance in the study of emergent magnetic monopoles was the discovery that monopole motion is restricted to dynamical fractal trajectories [J. N. Hallén et al., Science 378, 1218 (2022)], thus explaining the characteristics of magnetic monopole noise spectra [R. Dusad et al., Nature 571, 234 (2019); A. M. Samarakoon et al., Proc. Natl. Acad. Sci. U.S.A. 119, e2117453119 (2022)]. Here, we apply this novel theory to explore the dynamics of field-driven monopole currents, finding them composed of two quite distinct transport processes: initially swift fractal rearrangements of local monopole configurations followed by conventional monopole diffusion. This theory also predicts a characteristic frequency dependence of the dissipative loss angle for AC field-driven currents. To explore these novel perspectives on monopole transport, we introduce simultaneous monopole current control and measurement techniques using SQUID-based monopole current sensors. For the canonical material Dy₂Ti₂O₇, we measure [Formula: see text], the time dependence of magnetic flux threading the sample when a net monopole current [Formula: see text] is generated by applying an external magnetic field [Formula: see text] These experiments find a sharp dichotomy of monopole currents, separated by their distinct relaxation time constants before and after t ~[Formula: see text] from monopole current initiation. Application of sinusoidal magnetic fields [Formula: see text] generates oscillating monopole currents whose loss angle [Formula: see text] exhibits a characteristic transition at frequency [Formula: see text] over the same temperature range. Finally, the magnetic noise power is also dichotomic, diminishing sharply after t ~[Formula: see text]. This complex phenomenology represents an unprecedented form of dynamical heterogeneity generated by the interplay of fractionalization and local spin configurational symmetry.

Keywords: dysprosium titanate; frustrated magnetism; magnetic dynamics; spin ice.

Fig. 1.

Simultaneous monopole current control and measurement spectrometer. (A) Conceptual representation of magnetic field–driven monopole current $J(t)$ passing through a superconducting loop (yellow). Positive charged monopoles (red) are driven to the right by an applied field $B$, and negatively charged (blue) to the left. These rapid monopole currents are occasionally terminated when the spin-flip rate is suppressed by specific local spin conformations, magnified within the smaller panels at Left and Right. Lower panel: Conceptual design of simultaneous monopole current control and measurement system based on direct high-precision SQUID sensing of the monopole current $J(t)=(d\mathrm{\Phi }/dt)/{\mu }_0$. (B) Monte Carlo simulation results of the magnetic response when an external magnetic field of strength 30 mT is suddenly applied at time $t=0$. $M_{\mathit{sat}}$ is the equilibrium value of the magnetization in the presence of the field. The main panel shows results for bSM dynamics at temperatures 1.7 K, 2.0 K, 2.4 K, 3.0 K, and 4.0 K (from upper to lower lines). The dashed gray lines are exponential fits (highlighting the longer polarization time scale). The Inset contrasts the behavior at shorter times and temperature 1.7 K between bSM (dark blue) and SM (orange) dynamics and the corresponding dashed line fit (highlighting the shorter polarization time scale in SM dynamics). The fast and slow contributions are plotted separately in SI Appendix, Fig. S19. (C) The loss angle $\theta (f)$ extracted from Monte Carlo simulations of spin-ice magnetization, $m\left(t\right)= m_0\sin [2\pi ft+\theta (f)],$ in response to an oscillating magnetic field, $B\left(t\right)= B_0\cos (2\pi ft)$, of amplitude $B_0=30$ mT and frequency $f$ applied along the [111] direction. The main figure shows the loss angle for bSM dynamics at temperatures 0.8 (dark blue), 1.0, 1.3, 1.7, 2.2, 3.0, and 4.0 (dark red) K. A bump-like feature is clearly evident at low temperatures, induced by a reconfiguration current contribution originating from the presence of termini in the emergent dynamical fractal, as explained in the main text. The Inset shows the loss angle at 0.8 K (dark blue) and 1.7 K (yellow) for both bSM (circles and solid lines) and SM (squares and dotted lines) dynamics; notice the absence of the characteristic loss-angle feature in the latter.

Fig. 2.

Magnetic monopole current dichotomy in Dy₂Ti₂O₇. (A) Typical example of monopole current control generation and detection system in operation. The green trace shows the magnetic field as a function of time while the blue curve is the time dependence of flux ${\mathrm{\Phi }}_{\mathrm{S}}(t)$ measured at the SQUID. Monopole current initiation (MCI) time marked by the blue sign is set to 0 for each current transient. (B) Typical example of $\log {\mathrm{\Phi }}_S(t)$ evolution beginning $200 \mathrm{\mu }\mathrm{s}$ after MCI at $t=0$. Since monopole current is $J(t)=(d\mathrm{\Phi }/dt)/{\mu }_0$ and ${\mathrm{\Phi }(t)\propto \mathrm{\Phi }}_S(t)$, these unprocessed data reveal two distinct monopole current regimes. At long times, there is a well-defined time constant ${\tau }_2$ for monopole current flow as indicated by the dashed straight line fit. At short times $t<600 \mathrm{\mu }\mathrm{s}$ from MCI, a transition occurs to a much shorter time constant ${\tau }_1$ for monopole current flow. (C) Measured $\log {\mathrm{\Phi }}_S(t)$ evolution beginning $200 \mathrm{\mu }\mathrm{s}$ after MCI for all temperatures studied and for both positive and negative magnetic field directions. For all transients at long times the time constant ${\tau }_2\left(T\right)$ is measured by a straight line fit. At short times $t<600 \mathrm{\mu }\mathrm{s}$ after MCI for all these transients, a transition occurs to a faster time constant ${\tau }_1(T)$ for monopole current decay. (D) Extracted ${\mathrm{\Phi }}_1(t)$ for fast-decaying currents and positive B-field direction. These fast-decaying currents have ceased for $t>600 \mathrm{\mu }\mathrm{s}$ from MCI. Simultaneous data in the lower panel show extracted ${\mathrm{\Phi }}_2(t)$ for slow-decaying currents and positive $B$-field direction. (E) Extracted ${\mathrm{\Phi }}_1(t)$ for fast-decaying currents and negative B-field direction. These fast-decaying currents have ceased for $t>600 \mathrm{\mu }\mathrm{s}$ from MCI. Simultaneous data in the lower panel show extracted ${\mathrm{\Phi }}_2(t)$ for slow-decaying currents and negative $B$-field direction. The phenomenology is indistinguishable from that in D.

Fig. 3.

AC magnetic monopole loss angle in Dy₂Ti₂O₇. (A) Typical example of sinusoidal monopole current generation and detection. The green trace shows the magnetic field $B_0\left(t\right)=B\cos \left(2\pi ft\right)$ as a function of time while the dark blue curve is the measured time dependence of flux ${\mathrm{\Phi }}_S(t)$ at the SQUID and the light blue curve is $d{\mathrm{\Phi }}_S(t)/dt$. This field modulation experiment is carried out for $10 \mathrm{Hz}\le f\le \mathrm{5,000}$ Hz. (B) From A and with $J\left(t\right)\propto \left(d{\mathrm{\Phi }}_S\right(t)/dt)/{\mu }_0$, monopole current $J\left(t\right)=Re{J}_f\cos \left(2\pi ft\right)+iIm{J}_f\sin \left(2\pi ft\right)$ is determined by using a lock-in amplifier to measure ${\mathrm{\Phi }}_S\left(t\right)=Re{\mathrm{\Phi }}_f\cos \left(2\pi ft\right)+iIm{\mathrm{\Phi }}_f\sin \left(2\pi ft\right)$ for all $f.$ The consequent in-phase current $Re{J}_f\equiv -\left(2\pi fIm{\mathrm{\Phi }}_f\right)/{\mu }_0$ and out-of-phase current $Im{J}_f\equiv \left(2\pi fRe{\mathrm{\Phi }}_f\right)/{\mu }_0$ are shown for all temperatures measured. The corresponding rate $\dot{N}$ of the number of monopoles passing through the superconducting loop is estimated by $\dot{N}=\dot{{\mathrm{\Phi }}_S}(t)/{\mu }_{0}m$. The practical rate $\dot{{\mathrm{\Phi }}_p}$ can be converted through the coefficient $M_{\mathrm{i}}/{(2L}_{\mathrm{p}}+L_{\mathrm{i}})$. (C) Measured ${\theta }_d\left(f\right)=\arctan (Im{J}_f/Re{J}_f)$ from all temperatures studied. The Inset shows that a dissipation transition occurs at $f_d\cong 1.8\mathrm{kHz}$.

Fig. 4.

Dichotomous monopole currents, dissipation, and noise. (A) The transition point ${\tau }_d={1/2\pi f}_d$ where the dissipative process alteration occurs at $f_d\approx 1.8 \mathrm{kHz}$ from all temperatures studied. (B) Ratio of monopole current–driven magnetization noise intensity ${\sigma }_1^2$ for fast-decaying currents $t<600 \mathrm{\mu }\mathrm{s}$ from MCI to magnetization noise intensity ${\sigma }_2^2$ for slow-decaying currents during $600 \mathrm{\mu }\mathrm{s}<t<\mathrm{1,200} \mathrm{\mu }\mathrm{s}$. This ratio is constant near 1.5 for all temperatures. (C) Experimentally determined monopole current relaxation time constants from all temperatures studied and both field application directions. Measured slow-decaying monopole current time constant ${\tau }_2$ in solid red; fast-decaying monopole current time constant ${\tau }_1$ in solid blue.