Electronic Transport in the Coulomb Phase of the Pyrochlore Spin Ice

We investigate the transport properties of itinerant electrons interacting with a background of localized spins in a correlated paramagnetic phase of the pyrochlore lattice. We find a residual resistivity at zero temperature due to the scattering of electrons by the static dipolar spin-spin correlation that characterizes the metallic Coulomb phase. As temperature increases, thermally excited topological defects, also known as magnetic monopoles, reduce the spin correlation, hence suppressing electron scattering. Combined with the usual scattering processes in metals at higher temperatures, this mechanism yields a nonmonotonic resistivity, displaying a minimum at temperature scales associated with the magnetic monopole excitation energy. Our calculations agree quantitatively with resistivity measurements in ${\mathrm{Nd}}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ and ${\mathrm{Pr}}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$, shedding light on the origin of the resistivity minimum observed in metallic spin-ice compounds.

Figure 1

A fragment of the pyrochlore lattice. In the Coulomb phase, local spin arrangement on each individual tetrahedra obeys the so-called ice rules with two spins pointing in and two spins pointing out.

Figure 2

Lowest-order diagram of the electron self-energy ${\Sigma }_{\alpha \beta }(\mathbf{p},\omega )$. The solid and wavy lines denote the electron propagator $G_0$ and the spin correlator (3), respectively.

Figure 3

(a) Transverse and (b) longitudinal components of the electron scattering rate ${\tau }_e^{-1}(T)=2\mathrm{Im}\Sigma$ as a function of temperature. Both are normalized with respect to $\mathrm{Im}{\Sigma }_T(T=0)=g^2{\Lambda }^2/2\pi v_{F}K$. We have used ${\tau }_{m,0}D{p}_F^2=0.1$ and a Debye screening length ${\ell }_D(T=\Delta )=1.65\times {10}^3{\ell }_c$ in the calculations.

Figure 4

Dynamical conductivity $\sigma (\omega ,T)$ as a function of frequency $\omega$ at various temperatures. The conductivity is measured with respect to ${\sigma }_0={\omega }_p^2/4\pi \Delta$.

Figure 5

(a) Resistivity $\rho (T)$ (in units of ${\rho }_0$) as a function of temperature $T$ for varying dimensionless parameter $\eta ={\epsilon }_F{\tau }_{m,0}$. The Fermi energy is fixed at ${\epsilon }_F=100\Delta$ in the calculation. (b) The rescaled resistivity $\eta (\rho -{\rho }_0)$ collapses on a universal curve at low temperatures. Also shown is the rescaled experimental data from the measurements on ${\mathrm{Nd}}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ [14] and ${\mathrm{Pr}}_2{\mathrm{Ir}}_2{\mathrm{O}}_7$ [13]. Three fitting parameters: $\Delta$, ${\rho }_0$, and $\eta$, are used in the above scaling analysis.