Finite compressibility in the low-doping region of the two-dimensional $t\text{−}J$ model

We reexamine the important issue of charge fluctuations in the two-dimensional $t\text{−}J$ model by using an improved variational method based on a wave function that contains both antiferromagnetic and $d$-wave superconducting order parameters. In particular, we generalize the wave function introduced some time ago by Bouchaud, Georges, and Lhuillier [J. Phys. (Paris) 49, 553 (1988)] by considering also a long-range spin-spin Jastrow factor, in order to correctly reproduce the small-$q$ behavior of the spin fluctuations. We mainly focus our attention on the physically relevant region $J∕t\sim 0.4$ and find that, contrary to a previous variational ansatz, this state is stable against phase separation for small hole doping. Moreover, by performing projection Monte Carlo methods based on the so-called fixed-node approach, we obtain clear evidence that the $t\text{−}J$ model does not phase separate for $J∕t\lesssim 0.7$ and that the compressibility remains finite close to the antiferromagnetic insulating state.

Figure 1

(Color online) Results for the total spin $⟨S^2⟩$ at half filling as a function of the cluster size $L$ for the wave function of Eq. (2) defined by the mean-field Hamiltonian (5) and the two possible orientations of the magnetic field: i.e., Eq. (6), indicated by “Pfaff,” and Eq. (7), indicated by “$\mathrm{RVB}+\mathrm{AF}$.” The FN results for the former case are also shown.

Figure 2

(Color online) Spin-spin correlations at the maximum distance at half filling for the wave functions of Fig. 1. The exact value in the thermodynamic limit is marked by the arrow.

Figure 3

(Color online) Spin structure factor $S\left(q\right)$ at half filling for the variational wave function of Eq. (2) defined by the mean-field Hamiltonian of Eqs. (5, 6) with long-range and short-range (i.e., nearest-neighbor) Jastrow factors. Inset: detail for small momenta.

Figure 4

(Color online) FN results for the spin-spin correlations (along the $z$ direction) at the maximum distance as a function of the doping for $J∕t=0.2$ for different sizes of the cluster.

Figure 5

(Color online) Energy per hole $e_h\left(\delta \right)$ as a function of the doping $\delta$ for the 26-site cluster calculated by different approaches: the variational calculations for the Pfaffian wave function (circles), the FN approach of Eq. (11) (squares), and the expectation value of the Hamiltonian over the FN ground state given by Eq. (15) (triangles); the exact results are also shown (diamonds).

Figure 6

(Color online) Energy per hole $e_h\left(\delta \right)$ as a function of the doping $\delta$ for $J∕t=0.4$ and different sizes. The results are obtained by using the FN approach described in the text. Two different states are used as guiding function: The simple nonmagnetic state, denoted by “RVB,” and the state with pairing, antiferromagnetism in the $x\text{−}y$ plane, and the spin Jastrow factor, denoted by “Pfaff.” The expectation values of the Hamiltonian over the FN ground state are also shown for $L=162$ for the latter case. Inset: variational results of $e_h\left(\delta \right)$ for the Pfaffian wave function.

Figure 7

(Color online) The inverse compressibility of the half-filled Mott insulator for $J∕t=0.4$ calculated by extracting the second derivative of the polynomial fit of the FN energy. Inset: the chemical potential, defined through the difference of ground-state energies, as a function of the doping for different sizes of the cluster.

Figure 8

(Color online) The same as in Fig. 6 for $J∕t=0.6$.

Figure 9

(Color online) The same as in Fig. 6 for $J∕t=0.8$.

Figure 10

(Color online) FN results for the density correlation function for $8$ holes on 162 sites and different values of $J∕t$. The high-symmetry points are marked as $\Gamma =(0,0)$, $X=(\pi ,\pi )$, and $M=(\pi ,0)$.

Figure 11

(Color online) The same as in Fig. 10 but for 16 holes on 162 sites.

Figure 12

(Color online) Boundary for the phase separation (PS) instability. The results of previous works are also shown for comparison. The line is a guide to the eye.