$d$-wave pairing in lightly doped Mott insulators

We define a suitable quantity $Z_c$ that measures the pairing strength of two electrons added to the ground-state wave function by means of the anomalous part of the one-particle Green’s function. $Z_c$ discriminates between systems described by one-electron states, like ordinary metals and band insulators, for which $Z_c=0$, and systems where the single particle picture does not hold, like superconductors and resonating valence bond insulators, for which $Z_c\ne 0$. By using a quantum Monte Carlo projection technique for the Hubbard model at $U∕t=4$, a finite $Z_c$, with $d$-wave symmetry, is found at half filling and in the lightly doped regime, thus emphasizing a qualitatively new feature coming from electronic correlation.

Figure 1

(Color online) Anomalous part of the Green’s function in real space at half filling for free electrons and for $U∕t=4$, obtained by the Monte Carlo projection technique described in Sec. 2.

Figure 2

(Color online) Size scaling of the $d$-wave pairing strength $Z_c$ for different wave functions at half filling: the uncorrelated BCS wave function with $\mathrm{\Delta }=0.05$, the free-electron wave function $(\mathrm{\Delta }\to 0)$, the wave function with a finite antiferromagnetic order parameter [obtained by adding to the $\mathrm{\Delta }\to 0$ BCS Hamiltonian a further antiferromagnetic mean-field term ${(-1)}^i{\mathrm{\Delta }}_{AF}(n_{i,\uparrow }-n_{i,\downarrow })$], the Gutzwiller wave function with $g=1$ (full squares) and the fully projected BCS wave function with $\mathrm{\Delta }=0.05$ (full circles). The same quantity for the exact ground state of the two-chain $t\text{−}J$ ladder at low doping is also shown (full triangles).

Figure 3

(Color online) The $d$-wave pairing strength $Z_c$ (circles) as a function of $U$ in an $L=18$ site cluster with $N^{*}=L$. The quasiparticle weight $Z_N$ for $N=16$ (squares) at the momentum $(2\pi ∕3,0)$ and the short-distance anomalous average $P$ (triangles) are also shown. Continuous $\left(Z_N\right)$, long dashed $\left(Z_c\right)$, and dotted $\left(P\right)$ lines correspond to the variational calculation with the variational wave function of Eq. (4) with an optimized ${\mathrm{\Delta }}_k$ and projected onto the subspaces with 16 and 18 electrons.

Figure 4

(Color online) The $d$-wave pairing strength $Z_c$ as a function of the projection time $\tau$ for different clusters at half filling. The inset shows the finite size scaling of $Z_c-Z_c^G$ for $\tau =6$.

Figure 5

(Color online) The $d$-wave pairing strength $Z_c-Z_c^G$ as a function of the projection time $\tau$ for different dopings. The arrows indicate the value of $Z_c$ corresponding to the variational wave function of Eq. (4) with an optimized ${\mathrm{\Delta }}_k$ for zero and eight holes. $Z_c$ for the $s$-wave pairing in one dimension is also shown. In this case, the value of the pairing $F_R$ at the nearest-neighbor distance is smaller than the corresponding on-site value, despite the repulsive interaction $U$.