Breakdown of the Wiedemann-Franz law at the Lifshitz point of strained ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$

Strain tuning ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ through the Lifshitz point, where the Van Hove singularity of the electronic spectrum crosses the Fermi energy, is expected to cause a change in the temperature dependence of the electrical resistivity from its Fermi liquid behavior $\rho \sim T^2$ to $\rho \sim T^2\log (1/T)$, a behavior consistent with experiments by Barber et al. [Phys. Rev. Lett. 120, 076602 (2018)]. This expectation originates from the same multiband scattering processes with large momentum transfer that were recently shown to account for the linear in $T$ resistivity of the strange metal ${\mathrm{Sr}}_3{\mathrm{Ru}}_2{\mathrm{O}}_7$. In contrast, the thermal resistivity ${\rho }_Q\equiv T/\kappa$, where $\kappa$ is the thermal conductivity, is governed by qualitatively distinct processes that involve a broad continuum of compressive modes, i.e., long-wavelength density excitations in Van Hove systems. While these compressive modes do not affect the charge current, they couple to thermal transport and yield ${\rho }_Q\propto T^{3/2}$. As a result, we predict that the Wiedemann-Franz law in strained ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ should be violated with a Lorenz ratio $L\propto T^{1/2}\log (1/T)$. We expect this effect to be observable in the temperature and strain regime where the anomalous charge transport was established.

Figure 1

Temperature dependence of the resistivity $\rho \left(T\right)\sim T^{2}log(D/T)$ (left) and thermal resistivity ${\rho }_Q\left(T\right)\sim T^{3/2}$ (right) divided by the Fermi liquid behavior $T^2$ at the Van Hove point (red curves). The more singular behavior of the thermal resistivity is clearly visible. $D$ is the effective bandwidth of the problem. As one moves away from the Van Hove point, one expects a temperature scale $T^{*}$ below which both transport coefficients recover ordinary Fermi liquid behavior following Eqs. (3) and (4). We use $T^{*}/D=0.01$ (green curves) and $T^{*}/D=0.03$ (blue curves). The inset shows the corresponding behavior for the Lorenz ratio $L\left(T\right)$. The right index shows the characteristic temperature scale $T^{*}$ for the crossover to Van Hove dominated scattering as the strain is varied. We used the parameters given in Appendix pp2. In this inset we also indicate the regimes where the Lorenz number $L$ tends to a finite value or vanishes as $T$ decreases.

Figure 2

Density of states of the $\gamma$ band at zero strain (blue) and at the strain value ${\epsilon }^{*}$ that corresponds to the Lifshitz point where one of the strain-split Van Hove singularities crosses the Fermi energy (red). The inset shows the Fermi energies at zero strain (left) and at the Lifshitz point (right), where the hot parts of the Fermi surface are indicated in red.

Figure 3

Scaling function $P(x,y)$ (left) of Eq. (13) (solid lines) determining the frequency and temperature dependence of the charge excitation spectrum near a Van Hove point as function of $x=\omega /|{\varepsilon }_{\mathrm{VH},\mathit{k}}|$, in comparison with the simplified version of Eq. (11) together with the substitution Eq. (14) (dashed lines) to include thermal effects. We used $m=1$ and show results for $y=0.1$ (red) $y=5$ (green), and $y=10$ (blue), where $y=T/|{\varepsilon }_{\mathrm{VH},\mathit{k}}|$. The right panel shows the numerical result for the density excitation spectrum of the $\gamma$ band at $T=0$ for $q=0.08\pi$ (dark red), $q=0.06\pi$ (red), and $q=0.04\pi$ (light red). For small $\omega /|{\varepsilon }_{\mathrm{VH},\mathit{k}}|$ the curves show the expected scaling behavior. As shown in the inset, the result deviates from the approximated result for large $\omega /|{\varepsilon }_{\mathrm{VH},\mathit{k}}|$.

Figure 4

A $\mathrm{cc}\to \mathrm{cc}$ process, where two electrons scatter from two cold initial states (purple) into two cold final states (orange) with umklapp scattering, leads to a resistivity of $\rho \propto T^2$. The solid arrows indicate the momentum transfer for each electron. An arrow starting at a cold momentum (blue end) can either end in another cold state (blue arrowhead) or in a Van Hove point (red arrowhead). The $\mathrm{cc}\to \mathrm{ch}$ process contributes by $\rho \propto T^{2}log(1/T)$ and dominates, if every point on the Fermi surface participates in this scattering. The $\mathrm{cc}\to \mathrm{hh}$ processes always conserve momentum, therefore they do not contribute to the resistivity.

Figure 5

When all bands are considered, every point on the cold Fermi surface can participate in $\mathrm{cc}\to \mathrm{ch}$ scattering, either by umklapp or inter band scattering.

Figure 6

$\mathrm{ch}\to \mathrm{ch}$ process dominating the thermal transport with two hot momenta near the Van Hove point (red) and two cold electrons with generic dispersion (blue). For small momentum transfer, this process is allowed everywhere on the Fermi surface.

Figure 7

Self-energy diagram determining the thermal resistivity with hot momenta ${\mathbf{k}}_2$ and ${\mathbf{k}}_2-\mathbf{q}$ with hyperbolic dispersion (red) and cold momenta ${\mathbf{k}}_1$ and ${\mathbf{k}}_1+\mathbf{q}$ with parabolic dispersion (blue).