Absence of static stripes in the two-dimensional $t-J$ model determined using an accurate and systematic quantum Monte Carlo approach

We examine the two-dimensional $t-J$ model by using variational approach combined with well established quantum Monte Carlo techniques [S. Sorella et al., Phys. Rev. Lett. 88, 117002 (2002).] that are used to improve systematically the accuracy of the variational ansatz. Contrary to recent density-matrix renormalization group and projected entangled-pair state calculations [P. Corboz et al., Phys. Rev. B 84, 041108(R) (2011).], a uniform phase is found for $J/t=0.4$, even when the calculation is biased with an ansatz that explicitly contains stripe order. Moreover, in the small hole doping regime, that is, $\delta \lesssim 0.1$, our results support the coexistence of antiferromagnetism and superconductivity.

Figure 1

Energy per hole as a function of the doping for $J/t=0.4$. Variational (left) and fixed-node (right) results are reported for $p=0$ and 1 ($p=0,1$ and 2) Lanczos steps for the wave function with (without) antiferromagnetism. The best variational DMRG and iPEPS energies[21] and the fixed-node with $p=2$ are connected by dashed lines for a better comparison.

Figure 2

Variational results for the variance extrapolation on a 162-site cluster for different numbers of holes: $p=0$ and 1 ($p=0,1$ and 2) Lanczos steps have been performed on the wave function with (without) antiferromagnetism. The best fixed-node results are also marked by arrows.

Figure 3

Upper panels: local density $n_i$ when a site-dependent chemical potential with $\delta \mu =1.6$ [see Eq. (4)] is added to the variational wave function; the cases with $l_s=12$ (a) and 8 (b) are reported. Lower panels: local density $n_i$ when a site-dependent potential [see Eq. (5)] is added to the $t-J$ Hamiltonian, with $l_s=12$ and $V=0.2$ (c) and $l_s=8$ and $V=0.4$ (d). Variational and fixed-node results are reported for a $12\times 12$ cluster and $\delta =1/8$. Insets: the difference between the largest and the smallest local density (at the fixed-node level) as a function of $V$.

Figure 4

Initial variational ansatz with stripe order (a) and fixed-node calculation (b) for charge and spin distributions in the $2\times 8$ unit cell of a $16\times 16$ lattice. The size of the circles and arrows is proportional to the electron density and spin along $z$, respectively. Largest symbols in the variational calculations: $\langle n_{R_i}\rangle =0.92$, $\langle S_{R_i}^z\rangle =\pm 0.09$.