Quantum-Critical Resistivity of Strange Metals in a Magnetic Field

Resistivity in the quantum-critical fluctuation region of several metallic compounds such as the cuprates, the heavy fermions, Fe chalogenides and pnictides, Moiré bilayer graphene, and ${\mathrm{WSe}}_2$ is linear in temperature $T$ as well as in the magnetic field $H_z$ perpendicular to the planes. Scattering of fermions by the fluctuations of a time-reversal odd polar vector field $\mathbf{\Omega }$ has been shown to give a linear in $T$ resistivity and other marginal Fermi-liquid properties. An extension of this theory to an applied magnetic field is presented. A magnetic field is shown to generate a density of vortices in the field $\mathbf{\Omega }$ proportional to $H_z$. The elastic scattering of fermions from the vortices gives a resistivity linear in $H_z$ with the coefficient varying as the marginal Fermi-liquid susceptibility $\ln ({\omega }_c/T)$. Quantitative comparison with experiments is presented for cuprates and Moiré bilayer graphene.

Figure 1

Representation of the current distribution in the cu-o unit cell for (a) the vector field $\mathbf{\Omega }$, which has one of four possible angles ${\theta }_i$ in the unit cell $i$; and (b) the angular momentum ${\mathit{\ell }}_z$, which is a generator of rotations of $\mathbf{\Omega }$ and has a magnetization at its core. (c) Represents a fluctuation of $\mathbf{\Omega }$ over regions of many cells. A current represented by the green arrows runs at the boundary between any two orientations of $\mathbf{\Omega }$. At all corners of the variations in $\mathbf{\Omega }$ a vortex or ${\mathit{\ell }}_z$, represented by the black dot, is required to exist. At $H=0$, the vortices are of equally up and down orientations. But an applied finite $H$ leads to a net orbital angular momentum due to unequal density of vortices of different orientation.

Figure 2

(a) Elastic scattering of fermions by vortices of angular momentum $⟨L_z⟩$. (b) Inelastic scattering of fermions by fluctuations $\chi ”(\omega ,T,\mathit{q})$.

Figure 3

The temperature dependence of the linear in $H$ resistivity in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ for $x=0.19$. The data, shown as red dots, are taken from Fig. 1(b) of Ref. [16]. $\beta (T)$ is obtained from the fit to the resistivity (after subtracting a small residual value) $\rho (T,H)-{\rho }_0=\alpha T+\beta (T)H$ at $H=70$ Tesla. $A(T)=0.14\ln |{\omega }_c/\pi T|$, with ${\omega }_c\approx 1600K$.

Figure 4

The temperature dependence of the linear in $H$ resistivity in twisted bilayer graphene. The data is from Ref. [17] and replotted by the authors. $A_{B,1}\equiv \beta (T)$ is obtained just as for the cuprate compound in Fig. 3. The fit to experiments for the coefficient of the resistivity $\propto {\mu }_{B}H$ is given for the four band-fillings where the resistivity at zero magnetic field is most nearly linear in $T$.