High-$T_c$ superconductors: A variational theory of the superconducting state

We use a variational approach to gain insight into the strongly correlated $d$-wave superconducting state of the high $T_c$ cuprates at $T=0$. We show that strong correlations lead to qualitatively different trends in pairing and phase coherence: the pairing scale decreases monotonically with hole doping while the superconducting order parameter shows a nonmonotonic dome. We obtain detailed results for the doping dependence of a large number of experimentally observable quantities, including the chemical potential, coherence length, momentum distribution, nodal quasiparticle weight and dispersion, incoherent features in photoemission spectra, optical spectral weight, and superfluid density. Most of our results are in remarkable quantitative agreement with existing data and some of our predictions, first reported in Phys. Rev. Lett. 87, 217002 (2001), have been recently verified.

Figure 1

(a) Doping dependence of the dimensionless variational parameter ${\tilde{\mathrm{\Delta }}}_{\mathrm{var}}$ (filled squares). (b) Doping dependence of the dimensionless variational parameter ${\tilde{\mu }}_{\mathrm{var}}\left(x\right)$ (filled squares) and the “BCS value” ${\tilde{\mu }}_{\mathrm{BCS}}\left(x\right)$ (open triangles) defined in the text, both plotted on the scale given on the left-hand $y$ axis. The physical chemical potential (in units of t) $\tilde{\mu }=\mu ∕t$ where $\mu =\partial ⟨\mathcal{H}⟩∕\partial N$ (filled triangles) is plotted on the right-hand $y$-axis scale.

Figure 2

Plot of the SC correlation function $F_{\alpha \beta }\left(\mathbf{r}\text{−}{\mathbf{r}}^{\prime }\right)$, with $\widehat{\alpha }$ and $\widehat{\beta }$ either $\widehat{x}$ or $\widehat{y}$ calculated on a ${15}^2+1$ system at $x\approx 0.07$, with $\mathbf{r}\text{−}{\mathbf{r}}^{\prime }$ along $\widehat{\mathbf{x}}$. The correlation function saturates to $\pm {\mathrm{\Phi }}^2$ as indicated by the dotted line, and defines the $d$-wave order parameter $\mathrm{\Phi }$.

Figure 3

Phase diagram obtained within our variational calculation is shown between panels (a) and (b). The phases are a spin-liquid Mott insulator at $x=0$, a correlated $d$-wave SC for $0<x<x_c$, and a Fermi liquid metal for $x>x_c$. (a) Doping dependence of the $d$-wave order parameter $\mathrm{\Phi }\left(x\right)$ showing a superconducting “dome” with optimal doping around $x\simeq 0.2$. (b) Doping dependence of various length scales: the “pair size” ${\xi }_{\text{pair}}={\overline{v}}_F^0∕{\mathrm{\Delta }}_{\mathrm{var}}$ is shown as open triangles; the average interhole separation $x^{-1∕2}$ is shown as a dashed line; and the SC coherence length ${\xi }_{\mathrm{sc}}⩾\min (x^{-1∕2},{\xi }_{\text{pair}})$.

Figure 4

Grayscale plots ($\text{black}\equiv 1$, $\text{white}\equiv 0$) of the momentum distribution $n\left(\mathbf{k}\right)$ at various dopings $x$. The left panel shows the results of the projected variational calculation. The dashed line marks the contour on which $n\left(\mathbf{k}\right)=1∕2$, which closely resembles the noninteracting Fermi surface. The right panel shows the $n\left(\mathbf{k}\right)$ of the unprojected BCS calculation at the corresponding doping value.

Figure 5

Schematic plot of the spectral function when gapless quasiparticles are dispersing across the chemical potential, $\omega =0$, with a quasiparticle weight $Z$ and velocity $v_F$. The solid (dashed) lines indicate the occupied (unoccupied) part of the spectral function.

Figure 6

(a), (b) The momentum distribution, $n\left(\mathbf{k}\right)$, along the nodal direction $(0,0)\to (\pi ,\pi )$ (black squares). The white squares are results for the tJ model, and correspond to ignoring $t∕U$ corrections in Eq. (A5). The nonmonotonic behavior of $n\left(\mathbf{k}\right)$ near $k_F$ is removed on including $\mathcal{O}(t∕U)$ terms. The discontinuity in $n\left(\mathbf{k}\right)$ at $k=k_F$ signals gapless quasiparticles with a weight $Z$ determined by the magnitude of the jump. (c) Doping dependence of $Z$ compared with $Z^{\mathrm{sb}}=x$ from slave-boson mean-field theory.

Figure 7

(a) The first moment of the occupied part of the spectral function $M_1\left(\mathbf{k}\right)$ along the zone diagonal $(0,0)\to (\pi ,\pi )$ showing discontinuity of $Z{v}_F$ in its slope, $d{M}_1\left(\mathbf{k}\right)∕dk$, at $k=k_F$. (b) Doping dependence of $v_F$, extracted from $M_1\left(\mathbf{k}\right)$. The error bars here are associated with fits to $M_1\left(\mathbf{k}\right)$ and $n\left(\mathbf{k}\right)$ and errors in $Z$. Also shown are the bare velocity $v_F^0$ (dashed line) and the QP velocity within slave-boson mean-field theory, $v_F^{\mathrm{sb}}$ (dotted line). The experimental QP velocity $\simeq 1.5\mathrm{eV}\mathrm{\AA }$ from ARPES data (see Ref. 51) in near optimal BSCCO, and experiments indicate that it is nearly doping independent (see Ref. 53).

Figure 8

The dimensionless variational parameter ${\mathrm{\Delta }}_{\mathrm{var}}\left(x\right)$ of Fig. 1a is converted to an energy scale $\alpha J{\mathrm{\Delta }}_{\mathrm{var}}\left(x\right)$, where $\alpha$ is a dimensionless prefactor of order unity. The corresponding energy scale (in meV) is plotted as a function of hole-doping $x$. For the choice $\alpha =0.8$ the energy scale (filled circles) agrees well with the ARPES energy gap in the SC state (open hexagons), while for $\alpha =3$ it (filled squares) agrees well with the “hump” scale (open triangles) in ARPES spectra at $\mathbf{k}=(\pi ,0)$. All the ARPES results are taken from Campuzano et al. (see Ref. 38).

Figure 9

(a) The momentum distribution $n\left(\mathbf{k}\right)$ along the $(\pi ,0)\to (\pi ,\pi )$ direction, compared with the unprojected BCS result at the same ${\tilde{\mathrm{\Delta }}}_{\mathrm{var}}$ and ${\mu }_{\mathrm{var}}$. These results imply that correlations leads to considerable broadening of $n\left(\mathbf{k}\right)$. (b) $n\left(\mathbf{k}\right)$ plotted for various $x$, showing increasing broadening as $x\to 0$, induced by correlations.

Figure 10

(a) The first moment of the occupied part of the spectral function $M_1\left(\mathbf{k}\right)$ along the $(\pi ,0)\to (\pi ,\pi )$ direction compared with the corresponding unprojected BCS result. $\mid M_1\left(\mathbf{k}\right)\mid$ is much larger than the BCS result from which we infer that strong correlations lead to very large incoherent linewidth. (b) Parametric plot of $⟨\omega ⟩\left(\mathbf{k}\right)\equiv \mid M_1\left(\mathbf{k}\right)\mid ∕n\left(\mathbf{k}\right)$ (see text) at $\mathbf{k}=(\pi ,0)$ vs ${\tilde{\mathrm{\Delta }}}_{\mathrm{var}}$ with doping $x$ as the implicit parameter. The linear relation indicates that ${\tilde{\mathrm{\Delta }}}_{\mathrm{var}}\left(x\right)$ is related to an incoherent energy scale in the spectral function at $(\pi ,0)$.

Figure 11

(a) Doping dependence of the total $\left(D_{\mathrm{tot}}\right)$ and Drude or low energy $\left(D_{\text{low}}\right)$ optical spectral weights. $D_{\text{low}}\sim x$ at low $x$, which implies a Mott insulating state at $x=0$. (b) Parametric plot of the Drude weight $D_{\text{low}}$ vs the nodal quasiparticle weight $Z$, with hole doping $x$ as the implicit parameter. We find $D_{\text{low}}\sim Z$ over $0<x<x_c$.

Figure 12

Left: Real space picture of the $L^2+1$ lattice for $L=5$, with periodic boundary conditions applied along the opposite edges of the tilted square indicated by dashed lines. Right: The $\mathbf{k}$-space Brillouin zone of the “tilted lattice” for $L=5$. In the calculations reported in this paper we used systems with $L=15$, 17, 19.