Low-energy physical properties of high-$T_c$ superconducting Cu oxides: A comparison between the resonating valence bond and experiments

In a recent review by Anderson and co-workers, it was pointed out that an early resonating valence bond (RVB) theory is able to explain a number of unusual properties of high-temperature superconducting (SC) Cu oxides. Here we extend previous calculations to study more systematically the low-energy physical properties of the plain vanilla $d$-wave RVB state, and to compare the results with the available experiments. We use a renormalized mean-field theory combined with variational Monte Carlo and power Lanczos methods to study the RVB state of an extended $t\text{−}J$ model in a square lattice with parameters suitable for the hole-doped Cu oxides. The physical observable quantities we study include the specific heat, the linear residual thermal conductivity, the in-plane magnetic penetration depth, the quasiparticle energy at the antinode $(\pi ,0)$, the superconducting energy gap, the quasiparticle spectra, and the Drude weights. The traits of nodes (including $k_F$, the Fermi velocity $v_F$, and the velocity along Fermi surface $v_2$), and the SC order parameter are studied. Comparisons of the theory and the experiments in cuprates show an overall qualitative agreement, especially on their doping dependences.

Figure 1

(Color online) Illustration of the Fermi surface for LSCO (dashed square) and for YBCO (solid line), and the location of the gap nodes $k_F(\pm 1,\pm 1)$. The “ Fermi velocity” $v_F$ and the “gap velocity” $v_2$ are defined as the slopes of the quasiparticle energy along and perpendicular to the nodal direction. $v_F$ and $v_2$ specify the Dirac cone for the nodal quasiparticle dispersion.

Figure 2

(Color online) Comparison of the doping dependence of the Fermi wave vector $k_F$ obtained from our theoretical calculation with those obtained by the ARPES for (a) LSCO and (b) YBCO (Bi-2212). (Refs. 34, 35, 36, 37, 38) Theoretical results are obtained using parameters listed in Table for LSCO and YBCO (Bi-2212) with the in-plane lattice constant $a=3.8\mathrm{\AA }$. RMFT: renormalized mean field theory; VMC: variational Monte Carlo; PL1: Power Lanczos to the first order. Note that the values from VMC and PL1 may be very close, and get overlapped with each other in some plots following.

Figure 3

(Color online) Fermi velocity $v_F$ vs hole concentration $\delta$. The results from variational theory in Ref. 9 were obtained with the correction of order of $O(J∕t)$ included for the Hubbard model. $(1\mathrm{eV}\mathrm{\AA }\simeq 152\mathrm{km}∕\mathrm{s})$.

Figure 4

(Color online) (a) and (b) “ Gap velocity” $v_2$ vs hole concentration $\delta$. For optimal doped YBCO (Bi-2212), $v_2=10\sim 20\mathrm{km}∕\mathrm{s}$ was reported by various kinds of experiments (Refs. 35, 37, 46). The experimental data indicated in panel (a) is achieved by measuring the magnetic field dependence of the specific heat on LSCO at the zero temperature limit (Ref. 44). (c) and (d) the ratio $v_F∕v_2$ vs hole concentration $\delta$. The doping dependence of the ratio $v_F∕v_2$ is similar to that observed in the thermal conductivity experiments (see Fig. 6).

Figure 5

(Color online) The quadratic coefficient of the electronic specific heat: $\alpha =C_{\mathrm{el}}∕T^2$ vs hole concentration $\delta$.

Figure 6

(Color online) Linear residual thermal conductivity ${\kappa }_0∕T$ vs hole concentration $\delta$.

Figure 7

(Color online) In-plane magnetic penetration depth in LSCO vs hole concentration $\delta$.

Figure 8

(Color online) In-plane magnetic penetration depth in YBCO vs hole concentration $\delta$.

Figure 9

(Color online) In-plane magnetic penetration depth in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_{1-x}{\mathrm{Y}}_x{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ vs hole concentration $\delta$.

Figure 10

(Color online) Drude weight $D$ vs hole concentration $\delta$. The results from the variational theory given in Ref. 9 are for the Hubbard model with a finite cutoff to get rid of the contribution due to transitions from the ground state to the “upper Hubbard band.” In optical reflectivity measurements, the Drude weight is proportional to $({\omega }_p^{*}{)}^2$, for optimally doped YBCO $(\hslash {\omega }_p^{*}{)}^2\simeq 4.5\mathrm{eV}$, i.e., $D\simeq 145\mathrm{meV}$, along the $a$ axis (Ref. 67).

Figure 11

(Color online) The quasiparticle energy $E(\pi ,0)$ vs hole concentration $\delta$.

Figure 12

(Color online) Chemical potential shift $\tilde{\mu }$ vs hole concentration $\delta$. The experimental data were deduced from the shifts of photoemission and inverse-photoemission spectra of the core states of LSCO and Bi-2212 (Ref. 71, 72).

Figure 13

(Color online) Nodal quasiparticle weight $Z$ vs hole concentration $\delta$. $Z=1∕(1+\lambda )$ was estimated in Ref. 9, where $\lambda$ is the coupling constant estimated from the spectra function in ARPES (Ref. 75). The results from VMC simulation is presented for a complementary comparison (Ref. 9).

Figure 14

(Color online) (a) and (c) The maximal superconducting gap ${\Delta }_m$ vs hole concentration $\delta$. (b) and (d) Comparison of the superconducting order parameter ${\mathrm{\Delta }}_{\mathrm{SC}}=g_t\mathrm{\Delta }$ in our RMFT calculation with the gap ${\mathrm{\Delta }}_{\mathrm{BCS}}=2.14k_B{T}_c$, where $T_c$ is estimated from $T_c=T_c^{\max }[1-82.6(\delta -0.16{)}^2]$ ($T_c^{\max }=35\mathrm{K}$ for LSCO and $T_c^{\max }=95\mathrm{K}$ for Bi-2212) (Ref. 80).