On-site attractive multiorbital Hamiltonian for $d$-wave superconductors

We introduce a two-orbital Hamiltonian on a square lattice that contains on-site attractive interactions involving the two $e_g$ orbitals. Via a canonical mean-field procedure similar to the one applied to the well-known negative-$U$ Hubbard model, it is shown that the model develops $d$-wave ($B_{1g}$) superconductivity with nodes along the diagonal directions of the square Brillouin zone. This result is also supported by exact diagonalization of the model in a small cluster. The expectation is that this relatively simple attractive model could be used to address the properties of multiorbital $d$-wave superconductors in the same manner that the negative-$U$ Hubbard model is widely applied to the study of the properties of $s$-wave single-orbital superconductors. In particular, we show that by splitting the $e_g$ orbitals and working at three-quarters filling, such that the $x^2-y^2$ orbital dominates at the Fermi level but the $3z^2-r^2$ orbital contribution is nonzero, the $d$-wave pairing state found here phenomenologically reproduces several properties of the superconducting state of the high $T_c$ cuprates.

Figure 1

Band dispersion for the tight-binding term of the MD model discussed here: (a) is the nonhybridized case ($t_3=0$) and (b) the hybridized case ($t_3=-\sqrt{3}{t}_0/8$); the dashed line indicates the chemical potential position that leads to $\langle n\rangle =1.5$; (c) Fermi surface corresponding to the tight-binding Hamiltonian of panel (b). In panels (a), (b), and (c) the colors indicate the orbital composition of the bands, with the scale ranging from 0 (blue, denoting $3z^2-r^2$) to 1 (red, denoting $x^2-y^2$). Panel (d) contains the orbital composition of the band that determines the Fermi surface of panel (c): The red (blue) circles indicate the weight of the $x^2-y^2$ ($3z^2-r^2$) orbital.

Figure 2

(a) Mean-field energy vs. the gap parameter $\mathrm{\Delta }$ parametric with the strength of the attraction $V$, at $\langle n\rangle =1.5$. The value of $\mathrm{\Delta }$ that minimizes the energy in each case is indicated with the dashed lines; (b) Band structure along the directions $\mathrm{\Gamma }-X-M-\mathrm{\Gamma }$ using the mean-field approximation for the $H_{\mathrm{MD}}$ model for the cases $V=2$ (thin orange lines) and $V=0$ (black line), with the continuous lines denoting the tight binding dispersion and the dashed lines the replicas mean-field “shadow” bands. The ellipses indicate the gaps that open below and above the FS due to the orbital mixing, the black rectangle shows the gap at the FS, and the magenta rectangle shows the $d$-wave node at the FS; (c) Detail corresponding to the black and magenta rectangles in panel (b).

Figure 3

The spectral function $\mathrm{A}(\mathbf{k},\omega )$ along the directions $\mathrm{\Gamma }-X-M-\mathrm{\Gamma }$ in the mean-field approximation for the $H_{\mathrm{MD}}$ model, working at $V=2$. The inset highlights the weak intensity “shadow” spectral weight below and above the gaps opened by the attraction $V$.

Figure 4

Exact diagonalization results using $2\times 2$ clusters. Shown is the Fourier transform at momentum zero, $D\left(0\right)$, of the $d$-wave pairing correlations for the MD model (with $\langle n\rangle =1.5$, i.e., $N=12$) vs the on-site attraction strength $V$ (squares). The diamonds denote $S\left(0\right)$, the Fourier transform of the on-site $s$-wave pairing correlations. For comparison, $S\left(0\right)$ for the negative $U$ Hubbard model is also shown varying the strength of the attraction $U$ (circles). The inset shows $D\left(0\right)$ for the MD model and $S\left(0\right)$ for the negative-$U$ Hubbard model in the weak coupling regime.

Figure 5

Exact diagonalization results corresponding to a $2\times 2$ cluster. Shown are the lowest energy levels for each possible value of $N$ as a function of the chemical potential for the MD model. The inset allows us to see that the states with $N$ = 14, 12, and 10 can be stabilized as ground states by the chemical potential for $V=2$.