Phenomenological theories of the low-temperature pseudogap: Hall number, specific heat, and Seebeck coefficient

Since its experimental discovery, many phenomenological theories successfully reproduced the rapid rise of the Hall number $n_H$, going from $p$ at low doping to $1+p$ at the critical doping $p^{*}$ of the pseudogap in superconducting cuprates. Further comparison with experiments is now needed in order to narrow down candidates. In this paper, we consider three previously successful phenomenological theories in a unified formalism—an antiferromagnetic mean field (AF), a spiral incommensurate antiferromagnetic mean field (sAF), and the Yang-Rice-Zhang (YRZ) theory. We find a rapid rise in the specific heat and a rapid drop in the Seebeck coefficient for increasing doping across the transition in each of those models. The predicted rises and drops are locked, not to $p^{*}$, but to the doping where antinodal electron pockets, characteristic of each model, appear at the Fermi surface shortly before $p^{*}$. While such electron pockets are still to be found in experiments, we discuss how they could provide distinctive signatures for each model. We also show that the range of doping where those electron pockets would be found is strongly affected by the position of the van Hove singularity.

Figure 1

Maximum gap-to-bandwidth ratio as a function of doping in the AF and sAF models (red) and in YRZ theory (blue). The ratio is computed as ${\mathrm{\Delta }}_{(0,\pi )}\left(p\right)/[{\xi }_{(\pi ,\pi )\left(p\right)}-{\xi }_{(0,0)}\left(p\right)]$. The gap vanishes at $p^{*}=0.2$. In the AF and sAF models, ${\mathrm{\Delta }}_{\mathrm{k}}\left(p\right)=12t(p^{*}-p)$ and the bandwidth is $8t$, which yields a ratio of $0.3-1.5p$. In YRZ theory, the $d$-wave gap is maximum for $\mathrm{k}=(0,\pi )$ with a value of ${\mathrm{\Delta }}_{(0,\pi )}\left(p\right)=3t(p^{*}-p)$ and the bandwidth changes with doping as $8t(\frac{2p}{1+p}+\frac{0.169}{{(1+p)}^2})$. The resulting gap-to-bandwidth ratio is thus $\frac{3}{8}(0.2-p)/(\frac{2p}{1+p}+\frac{0.169}{{(1+p)}^2})$.

Figure 2

Spectral weight at the Fermi level $A_{\mathrm{k}}(\epsilon =0)$ for band parameters $(t^{\prime },t^{″})=(-0.17,0.05)t$, at doping $p=0.24$ (just passed the van Hove singularity, at $p_{\text{vHs}}=0.23$). The comparison of the constant lifetime $\tau$ approximation (left) and the isotropic mean-free path $\ell$ approximation (right) shows how isotropic $\ell$ enhances the spectral weight near the antinode, compensating for the lower velocity of the states at the saddle points of the dispersion.

Figure 3

Hall number ($n_H$) and Fermi surfaces $\left[{\sum }_{\sigma }{A}_{n\mathrm{k}\sigma }(\omega =0)\right]$ in the antiferromagnetic model. The rise in Hall number starts at $p_e$, where the electron pockets at $(\pi ,0)$ and $(0,\pi )$ appear at the Fermi surface, and ends at $p^{*}=0.2$, where the gap vanishes. Three sets of band parameters (a) $(t^{\prime },t^{″})=(-0.3,0.2)t$, similar to YBCO, (b) $(t^{\prime },t^{″})=(-0.35,0)t$, similar to BSCCO, and (c) $(t^{\prime },t^{″})=(-0.17,0.05)t$ similar to LSCO, yield different values of $p_e$. The thick lines are the AF results and the thin lines are the bare band results; they merge together above $p^{*}=0.2$. Constant lifetime $\tau =5\hslash /t$ and isotropic mean-free path $\ell =10a$ approximations are identified. The dotted lines are guides to the eye following $p$ and $1+p$, while dashed lines identify $p_e,p^{*}$, and the doping $p_{\text{vHs}}$ where the van Hove singularity occurs. The latter is in the plot range only in case (c).

Figure 4

Specific heat ($C_V/T$) in the antiferromagnetic model, at temperature $\beta =500/t$, equivalent to $T\approx 6\mathrm{K}$, shown as a function of doping for band parameters (a) $(t^{\prime },t^{″})=(-0.3,0.2)t$, (b) $(t^{\prime },t^{″})=(-0.35,0)t$, and (c) $(t^{\prime },t^{″})=(-0.17,0.05)t$. The corresponding Fermi surfaces were shown in Fig. 3. Labels identify constant lifetime $\tau =5\hslash /t$ and isotropic mean-free path $\ell =10a$ approximations, along with the doping $p_e$ where electron pockets appear at the Fermi surface, the doping $p^{*}$ where the gap vanishes, and the doping $p_{\text{vHs}}$ where the van Hove singularity occurs. The thick lines are the AF results and the thin lines are the bare band results; they merge together above $p^{*}=0.2$.

Figure 5

Seebeck coefficient ($S_x/T$) in the antiferromagnetic model, at temperature $\beta =100/t$, equivalent to $T\approx 30\mathrm{K}$, shown as a function of doping for band parameters (a) $(t^{\prime },t^{″})=(-0.3,0.2)t$, (b) $(t^{\prime },t^{″})=(-0.35,0)t$, and (c) $(t^{\prime },t^{″})=(-0.17,0.05)t$. The corresponding Fermi surfaces were shown in Fig. 3. Labels identify constant lifetime $\tau =5\hslash /t$ and isotropic mean-free path $\ell =10a$ approximations, along with the doping $p_e$ where electron pockets appear at the Fermi surface, the doping $p^{*}$ where the gap vanishes, and the doping $p_{\text{vHs}}$ where the van Hove singularity occurs. The thick lines are the AF results and the thin lines are the bare band results; they merge together above $p^{*}=0.2$.

Figure 6

Hall number ($n_H$), specific heat ($C_V/T$), and Seebeck coefficient ($S_x/T$ and $S_y/T$) in the incommensurate spiral antiferromagnetic model. Band parameters are $(t^{\prime },t^{″})=(-0.35,0)t$. (a) The Fermi surfaces, (b) the Hall number $n_H$ at $T\to 0$, (c) the specific heat $C_V/T$ at $\beta =500/t$ ($T\approx 6\mathrm{K}$), and (d) the Seebeck coefficient $S_x/T$ at $\beta =100/t$ ($T\approx 30\mathrm{K}$) are all shown as a function of doping $p$. Labels identify constant lifetime $\tau =5\hslash /t$ and isotropic mean-free path $\ell =10a$ approximations, along with the doping $p_e$ where electron pockets appear at the Fermi surface, and the doping $p^{*}$ where the gap vanishes. The thick lines are the sAF results and the thin lines are the bare band results; they merge together above $p^{*}=0.2$. Labels $x$ and $y$ identify which lines are the $S_x/T$ and $S_y/T$ for the Seebeck coefficients.

Figure 7

Hall number ($n_H$), specific heat ($C_V/T$), and Seebeck coefficient ($S_x/T$) in the Yang-Rice-Zhang theory. The usual YRZ band parameters $(t^{\prime },t^{″})=(-0.3,0.2)t$ are renormalized by Gutzwiller factors, so even bare band results (thin lines) are different from the ones obtained in the AF and sAF models. (a) The Fermi surfaces, (b) the Hall number $n_H$ at $T\to 0$, (c) the specific heat $C_V/T$ at $\beta =500/t$ ($T\approx 6\mathrm{K}$), and (d) the Seebeck coefficient $S_x/T$ at $\beta =100/t$ ($T\approx 30\mathrm{K}$) are all shown as a function of doping $p$. Labels identify constant lifetime $\tau =5\hslash /t$ and isotropic mean-free path $\ell =10a$ approximations, along with the doping $p_e$ where electron pockets appear at the Fermi surface, the doping $p^{*}$ where the gap vanishes. The thick lines are the sAF results and the thin lines are the bare band results; they merge together above $p^{*}=0.2$.

Figure 8

Results for the bare band with band parameters $(t^{\prime },t^{″})=(-0.17,0.05)t$. (a) Fermi surfaces (${\xi }_{\mathrm{k}}=0$) as a function of doping, two on each side of the van Hove singularity (vHs) at $p=0.23$. [(b)–(d)] Comparison of (b) Hall numbers $n_H=1/e{R}_H$ in the limit of zero temperature, (c) specific heats $C_V/T$ at $\beta =500/t$, and (d) Seebeck coefficients $S\left(T\right)/T$ at $\beta =100/t$, as a function of doping. The isotropic mean-free path $\ell$ approximation is in red, while the constant lifetime $\tau$ approximation is in green and the legend identify the values of $\tau$ and $\ell$ used. $\tau \to \infty$ denotes the standard Boltzmann transport theory results, and ${\ell }_x\to \infty$ is the constant mean-free path approximation used in Refs. [18, 23], for comparison. In the latter case, it was possible to compute $S/T$ at $\beta =500/t$, revealing the divergence of the Seebeck coefficient at the van Hove singularity in the constant-$\ell$ approximation for lower temperatures.