Hall effect in cuprates with an incommensurate collinear spin-density wave

The presence of incommensurate spiral spin-density waves (SDW) has been proposed to explain the $p$ (hole doping) to $1+p$ jump measured in the Hall number $n_H$ at a doping $p^{*}$. Here we explore incommensurate collinear SDW as another possible explanation of this phenomenon, distinct from the incommensurate spiral SDW proposal. We examine the effect of different SDW strengths and wave vectors, and we find that the $n_H\sim p$ behavior is hardly reproduced at low doping. Furthermore, the calculated $n_H$ and Fermi surfaces give characteristic features that should be observed; thus, the lack of these features in experiment suggests that the incommensurate collinear SDW is unlikely to be a good candidate to explain the $n_H\sim p$ observed in the pseudogap regime.

Figure 1

(a) Local moment distributions for commensurate $(\pi ,\pi )$ collinear SDW (left) and incommensurate collinear SDW (right). The incommensurate SDW is the equivalent to a commensurate SDW modulated by $cos\left(2\pi \delta \widehat{\mathbf{y}}\right)$. Here, $L=9$ and $\delta =\frac{1}{18}$ (see Sec. 2b for definitions). (b) $\mathbf{Q}$ in the reciprocal space for commensurate collinear SDW (left) and incommensurate collinear SDW (right).

Figure 2

(bottom) Hall conductivity $n_H$ as a function hole doping $p$ for three different band parameters. (top) Fermi surfaces (${\sum }_n{A}_{\mathbf{k},n}\left(0\right)$) at $p_{\text{vHs}}$ of each of the three band parameters. The color in the legend specifies which curve and Fermi surface correspond to which band parameter. Yellow corresponds to the particle-hole symmetric case and black corresponds to an approximation to the band parameters of YBCO as calculated from density functional theory without interactions [19, 20]. Hence, brown corresponds to an intermediate case, to show the effect of neglecting $t^{″}$. We define the reference $t=1$, which corresponds to approximately 250 meV in YBCO. $p=1$ corresponds to an empty band (no electron), and $p=-1$ corresponds a completely full band. The triangles below the Fermi surface graphs show the doping corresponding to each Fermi surface, hence to each van Hove singularity. We see on the brown curve that this doping is not equal to the doping where $R_H=0$. The three dotted lines correspond to $p-1$, $p$, and $p+1$.

Figure 3

The middle two panels show the Hall number $n_H$ as a function of the hole doping $p$ relative to half-filling for different SDW amplitudes $M$ ($\delta =0$). The outer panels show the Fermi surface for five different dopings for $M=0.125$ (top blue Fermi surfaces) and for $M=4.0$ (bottom purple Fermi surfaces). The triangles below and above the Fermi surface graphs show the doping corresponding to each Fermi surface. The three dotted lines correspond to $p-1$, $p$, and $p+1$. Here, $t^{\prime }=-0.3$, $t^{″}=0.2$.

Figure 4

Same as Fig. 3 but with $\delta =1/6$.

Figure 5

Hall number $n_H$ as a function of the hole doping $p$ relative to half-filling for different SDW incommensurability $\delta$ with the same $M=1.0$. We separate small and big $\delta$ on two panels to lighten the plot. The $\delta$ used are 0, $\frac{1}{20}=0.050$, $\frac{1}{16}=0.063$, $\frac{1}{14}=0.071$, $\frac{1}{12}=0.083$, $\frac{1}{10}=0.100$, and $\frac{1}{8}=0.125$. $t^{\prime }=-0.3$ and $t^{″}=0.2$.

Figure 6

The top panel shows the chosen distribution of $\delta$ values as function of $p$. Each symbol of a given color corresponds to the same incommensurability $\delta$: 0, $\frac{1}{38}=0.026$, $\frac{1}{26}=0.038$, $\frac{1}{20}=0.050$, $\frac{1}{16}=0.063$, $\frac{1}{14}=0.071$, $\frac{1}{12}=0.083$, $\frac{1}{10}=0.100$, and $\frac{1}{8}=0.125$. The dotted line corresponds to the experimental reference $\delta \sim p-0.03$ [4]. In the middle panel, Hall number $n_H$ as a function of hole doping $p$ relative to half-filling for different SDW incommensurability $\delta$. We use $M=1.0$ below $p=0.2$ and $M=0.0$ above. Each curve corresponds to a different $\delta$. The symbols on the middle panel correspond to the symbols on the top panel. We only draw symbols on the portion of the curve that corresponds to the experimental reference $\delta \sim p-0.03$. We also show the corresponding curve for the whole $p$ range for each $\delta$ to highlight the deviations from $n_H\sim p$ close to the data points selected. In the bottom panels, we show Fermi surfaces for five different $p$ and $\delta$. The triangles above the Fermi surface graphs show the doping corresponding to each Fermi surface.