Magnetism and superconductivity in the $t\text{−}{t}^{\prime }\text{−}J$ model

We present a systematic study of the phase diagram of the $t\text{−}{t}^{\prime }\text{−}J$ model by using the Green’s function Monte Carlo (GFMC) technique, implemented within the fixed-node (FN) approximation and a wave function that contains both antiferromagnetism and $d$-wave pairing. This enables us to study the interplay between these two kinds of order and compare the GFMC results with the ones obtained by the simple variational approach. By using a generalization of the forward-walking technique, we are able to calculate true FN ground-state expectation values of the pair-pair correlation functions. In the case of $t^{\prime }=0$, there is a large region with a coexistence of superconductivity and antiferromagnetism that survives up to ${\delta }_c\sim 0.10$ for $J∕t=0.2$, and ${\delta }_c\sim 0.13$ for $J∕t=0.4$. The presence of a finite $t^{\prime }∕t<0$ induces a strong suppression of both magnetic (with ${\delta }_c\lesssim 0.03$ for $J∕t=0.2$ and $t^{\prime }∕t=-0.2$) and pairing correlations. In particular, the latter ones are depressed both in the low-doping regime and around $\delta \sim 0.25$, where strong size effects are present.

Figure 1

(Color online) Magnetization obtained from the spin-spin correlations at the maximum distance calculated for the $t\text{−}J$ model with $J∕t=0.2$ (upper panel) and $J∕t=0.4$ (lower panel). For the VMC calculations, the error bars are smaller than the symbol sizes. The VMC magnetization has been obtained from the isotropic correlations, whereas the FN one from the correlations along the $z$ axis (see text).

Figure 2

(Color online) Static spin-structure factor $S\left(q\right)$ for $L=16\times 16$ cluster and different hole concentrations for the $t\text{−}J$ model with $J∕t=0.2$. $\Gamma =(0,0)$, $X=(\pi ,\pi )$, and $M=(\pi ,0)$. Inset: $S\left(q\right)$ for the variational state (empty symbols) and for the FN approximation (full symbols).

Figure 3

(Color online) Spin-structure factor $S\left(q\right)$ for the $t\text{−}J$ with $J∕t=0.2$, doping $\delta =1∕8$, and different cluster sizes. The case of the Hubbard model for $U∕t=4$ and $L=16\times 16$ is also reported for comparison. Inset: Size scaling of the peak as a function of $1∕L$.

Figure 4

(Color online) The same as in Fig. 1 but for the $t\text{−}{t}^{\prime }\text{−}J$ model with $J∕t=0.2$ and $t^{\prime }∕t=-0.2$. For the VMC calculations, the error bars are smaller than the symbol sizes. The dashed line indicates a tentative estimation for the thermodynamic limit.

Figure 5

(Color online) Pair-pair correlations at the maximum distance as a function on the doping for $J∕t=0.4$ (upper panel) and $J∕t=0.2$ (middle panel). The results for the variational wave function (2) (empty symbols) and for the FN approximation (filled symbols) are reported. The results for the wave function without the Jastrow factors (both for spin and density) and magnetic order parameter are also reported (lower panel).

Figure 6

(Color online) The same as in Fig. 5 but for the $t\text{−}{t}^{\prime }\text{−}J$ model with $J∕t=0.2$ and $t^{\prime }∕t=-0.2$. The dashed line indicates a tentative estimation for the thermodynamic limit.

Figure 7

(Color online) Detail of the pair-pair correlations reported in Figs. 5, 6 at low doping.

Figure 8

Size scaling of the pair-pair correlations at the maximum distance for $t\text{−}{t}^{\prime }\text{−}J$ model with $J∕t=0.2$ and $t^{\prime }∕t=-0.2$ at $\delta \sim 0.3$.