Deviation from Fermi-liquid transport behavior in the vicinity of a Van Hove singularity

Recent experiments revealed non-Fermi-liquid resistivity in the unconventional superconductor ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ when strain pushes one of the Fermi surfaces close to a van Hove singularity. The origin of this behavior and whether it can be understood from a picture of well-defined quasiparticles is unclear. We employ a Boltzmann transport analysis beyond the single relaxation-time approximation based on a single band which undergoes a Lifshitz transition where the Fermi surface crosses a van Hove singularity either due to uniaxial or epitaxial strain. First, analytically investigating impurity scattering, we clarify the role of the diverging density of states together with the locally flat band at the point of the Lifshitz transition. Additionally, including electron-electron scattering numerically, we find good qualitative agreement with resistivity measurements on uniaxially strained ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$, including the temperature scaling and the temperature dependence of the resistivity peak. Our results imply that, even close to the Lifshitz transition, a description starting from well-defined quasiparticles holds. To test the validity of Boltzmann transport theory near a van Hove singularity, we provide further experimentally accessible parameters, such as thermal transport, the Seebeck coefficient, and Hall resistivity and compare different strain scenarios.

Figure 1

Fermi surface tuned through the vHS at $(\pi ,0)$ or $(0,\pi )$ by (a) adjusting the chemical potential (doping) and (b) applying uniaxial stress. The dark areas (white diamond) denote regions where umklapp scattering by $(2\pi ,0)$ or $(0,2\pi )\left[\right(2\pi ,2\pi \left)\right]$ is possible.

Figure 2

Conductivity dip due to enhanced impurity scattering together with the switching sign of the Seebeck coefficient near the Lifshitz point normalized to the undoped resistivity ${\mu }_0=1.1t$. Inset (a) shows the density of states together with the energy intervals of the contributing states for $\mu /t=1.1$ (green), $\mu /t=1.2$ (red), and $\mu /t=1.3$ (blue). Inset (b) shows the contributing states for these three cases for the scattering phase-space $f_0\left(\mathbf{p}\right)[1-f_0\left(\mathbf{p}\right)]\le {10}^{-10}$.

Figure 3

Discretization of the first BZ adapted to the scattering phase space. The red curve corresponds to the Fermi-surface crossing the vHS. Blue curves denote the boundary equipotentials of the discretization, i.e., ${\varepsilon }_F-4T$ and ${\varepsilon }_F+4T$. Note that, for demonstration purposes, a high temperature of $\sim 100\mathrm{K}$ is shown.

Figure 4

Summary of numerical results for the electronic transport: Temperature evolution of the conductivity [(a) and (b)], normalized resistivity [(c) and (d)], temperature scaling coefficient $\alpha$ [(e) and (f)], as well as resistivity curves for various values of $\mu /t$ and ${\varepsilon }_{xx}$ comparing band filling (left panel) and uniaxial strain (right panel) scenarios. Subfigures (g) and (h) correspond to temperature cuts of the resistivity at values of interesting band fillings and values of uniaxial strains. Note the strongly nonlinear temperature dependence of ${\rho }_{xx}\left(T^2\right)$ considering $\mu /t=1.2$ [green curve in subfigure (g)] and similar dependence at both Lifshitz points ${\varepsilon }_{vH{S}^{\pm }}$ [orange and red curves in subfigure (h)].

Figure 5

Coefficient $A$ extracted by fitting the data used in Fig. 4 of $\rho \left(T\right)={\rho }_0+A{T}^2$ [subfigures (a) and (b)] together with the Kadowaki-Woods ratio $R_{\mathrm{KW}}=A/{\gamma }^2$ [subfigures (c) and (d)] as a function of band filling (left panel) and uniaxial strain (right panel). In the colored regions, fitting $\alpha$ yields a result $\notin [1.9,2.1]$, which we consider non-Fermi-liquid-like. Note that, outside of these regions, the density of states at the Fermi level (gray lines) remains almost constant.

Figure 6

Temperature evolution of the thermal conductivity [(a) and (b)] as well as the Seebeck coefficient divided by the temperature $Q/T$ [(c) and (d)] together with the detail at the low-temperature $Q/T$ [(e) and (f)] comparing the band filling and the uniaxial-strain scenarios.

Figure 7

Deviations from the Wiedemann-Franz law as a function of band filling (left panel) and uniaxial strain (right panel). The dashed lines mark the Lifshitz transitions.

Figure 8

Temperature dependence of the Hall resistivity for the band-filling as well as the uniaxial-strain scenarios. The line at $R_{\mathrm{H}}=0$ denotes where the character of the charge-carrier changes.