Hall number across a van Hove singularity

In the context of the relaxation time approximation to Boltzmann transport theory, we examine the behavior of the Hall number $n_H$ of a metal in the neighborhood of a Lifshitz transition from a closed Fermi surface to open sheets. We find a universal nonanalytic dependence of $n_H$ on the electron density in the high-field limit, but a nonsingular dependence at low fields. The existence of an assumed nematic transition produces a doping dependent $n_H$ similar to that observed in recent experiments in the high-temperature superconductor ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-x}$.

Figure 1

The Hall number $n_H$ in the $B\to 0$ and $B\to \infty$ limits vs density for tetragonal and orthorhombic systems. Insets show the Fermi surfaces (FS) at densities $n=0.2,0.5,$ and 0.8. When the FS is closed, $n_H$ is exactly equal to the carrier density (dashed lines) in the large $B$ limit, while for $B\to 0$, $n_H$ deviates strongly from the dashed line near the van Hove fillings where the topology of the FS changes. For orthorhombic systems, a sharp nonanalyticity exists in the high-field $n_H$ at the van Hove fillings. The red box shows the critical region examined in Fig. 2.

Figure 2

Asymptotic behavior of the high- and low-field Hall number across the van Hove singularity at ${\mu }_c=+2(t_y-t_x)=1$ (i.e., $t_y=1+\varphi$ and $t_x=1-\varphi$, with $\varphi =0.25$). For $\mu ={\mu }_c-\delta \mu$, the sharp nonanalyticity in the high-field limit is of the form (14), while in the low-field limit it is weaker, and of the form $\delta \mu ln|{\mu }_c/\delta \mu |$. Insets are schematics of the FS on either side of ${\mu }_c$.

Figure 3

The Hall number computed numerically, as a function of hole doping $p$ for a model in which a nematic phase transition occurs at $p_{\text{nem}}=20\%$. (a) The doping evolution of the nematic order parameter $\varphi$. (b) Fermi surfaces as a function of doping: the Lifshitz transition is generically separated from $p_{\mathrm{nem}}$; for ${\varphi }_0=1$, it occurs at $p_c<p_{\text{nem}}$, however, for the stronger onset (${\varphi }_0=3$), it is too close to $p_{\text{nem}}$ to be resolved. (c) and (d) The sharpness of the drop in $n_H$ in both the strong and weak field limits is controlled by ${\varphi }_0$. Longitudinal resistivities are shown in Appendix pp1.

Figure 4

A speculative zero-field phase diagram of an ideal cuprate with a nematic phase included. In a tetragonal cuprate, $p_{\text{nem}}$ is a nematic quantum critical point, which in YBCO would be rounded by weak orthorhombicity. Here, we have considered $p_c$ to be a Lifshitz transition, which generically occurs inside the nematic phase. There is then a continuous transition to a unidirectional CDW. SDW and AF represent different forms of magnetic order, although in the presence of disorder, the SDW is typically manifest as a spin-glass. All the ordered phases occur below a pseudogap crossover temperature $T^{*}$.

Figure 5

The chemical potential as a function of doping as the nematic order parameter onsets with differing strengths ${\varphi }_0$. While the Lifshitz transition from closed hole pockets to open sheets virtually coincides with $p_{\text{nem}}=20\%$ for strong nematic onset (green curves ${\varphi }_0=3$), it occurs at $p_c<p_{\text{nem}}$ when the nematic onset is weaker (blue curves, ${\varphi }_0=1$).

Figure 6

The longitudinal resistivities (units of $1/e^2\tau$, logarithmic scales) in the low-field (left) and high-field (right) limits as a function of doping. There is a large increase in ${\rho }_{x}x$ when the Fermi surface becomes open along the $x$ direction (i.e., the nematicity forces the system to become quasi-one-dimensional. Note that ${\rho }_{xx}$ technically diverges in the infinite-field limit due to the quadratic dependence, ${\rho }_{xx}\propto B^2$. The figure shows its value for large but not infinite fields.

Figure 7

The two sums, which occur in the high-field expression for the Hall number, shown logarithmically near the van Hove singularity.

Figure 8

The asymptotic expression for the Hall number, on logarithmic and linear scales. We have chosen $t_x=0.5$ and $t_y=1$, so that the critical point occurs at ${\mu }_c=1$. The constant $c({\mu }_0,{\mu }_c)$ is $0.2951$ for these parameters.

Figure 9

Contour used for finding Fourier coefficients of rational fractions of Jacobian elliptic functions, with the positions of the poles indicated.