Competing States in the $t$-$J$ Model: Uniform $d$-Wave State versus Stripe State

Variational studies of the $t$-$J$ model on the square lattice based on infinite projected-entangled pair states confirm an extremely close competition between a uniform $d$-wave superconducting state and different stripe states. The site-centered stripe with an in-phase $d$-wave order has an equal or only slightly lower energy than the stripe with antiphase $d$-wave order. The optimal stripe filling is not constant but increases with $J/t$. A nematic anisotropy reduces the pairing amplitude and the energies of stripe phases are lowered relative to the uniform state with increasing nematicity.

Figure 1

Competing low-energy states in the $t$-$J$ model found with iPEPS simulations using different supercells ($J/t=0.4$). The diameter of the red dots (length of the arrows) is proportional to the local hole density (local magnetic moment) with average values given by the first (second) row of numbers below a panel. The width of a bond between two sites scales with the (singlet) pairing amplitude on the corresponding bond with a different sign in the horizontal and vertical direction indicated by the two different colors. Uniform (U) $d$-wave superconducting state with coexisting antiferromagnetic order ($\delta \sim 0.1$, $D=14$), where two different tensors for the two sublattices have been used. A site-centered vertical stripe state of width $W=5$ (W5) with in-phase $d$-wave order in a $5\times 2$ supercell ($\delta \sim 1/8$, $D=14$). A site-centered stripe state of width $W=5$ with antiphase $d$-wave order (W5AP) in a $10\times 2$ supercell ($\delta \sim 1/8$, $D=10$). A fully-doped (${\rho }_l=1$), insulating diagonal (Diag) stripe in a $L\times L$ cell using $L$ different tensors at a doping $\delta =1/L$ (here $L=5$, $D=14$). We considered sizes up to $L=11$.

Figure 2

(a) Energies of the competing states as a function of inverse bond dimension for $\delta =0.12$. The horizontal lines show the best fixed-node Monte Carlo result (with two Lanczos steps) from Ref. [52]. (b)–(c) Order parameters of the uniform $d$-wave state as a function of doping: (b) the pairing amplitude $\mathrm{\Delta }$ and (c) the local magnetic moment $m$. The extrapolated values have been obtained from a linear extrapolation of the finite $D$ data, which provides a rough estimate of the order parameters in the infinite $D$ limit. (d) Order parameters of the W5 stripe state as a function of inverse $D$ for $\delta =0.12$: the modulation strength of the local hole density $\mathrm{\Delta }n=n_{\max }-n_{\min }$ and of the local magnetic moment $\mathrm{\Delta }m=m_{\max }-m_{\min }$, where $n=⟨\widehat{n}⟩$ and $m=|⟨{\widehat{S}}_z⟩|$ are evaluated on each lattice site in the supercell. The filled squares show the maximal singlet pairing $|\mathrm{\Delta }|$. The order parameters decrease with increasing $D$, but remain finite in the infinite $D$ limit. The dashed lines are a guide to the eye.

Figure 3

(a) Pairing amplitude as a function of $J/t$ for $\delta =0.14$, $D=10$. (b) Optimal stripe filling ${\rho }_l=W\delta$ as a function of $J/t$ ($D=10$, $W=5$).

Figure 4

(a) Energies as a function of inverse bond dimension for $\delta =0.1$ in the nematic $t$-$J$ model with $J/t=0.4$ and $t_x=0.85t$. (b) The pairing amplitude of the uniform state decreases with increasing nematicity (here $(t_x+t_y)/2=1$ and $(J_x/t_x+J_y/t_y)/2=0.4$ are kept constant, $\delta =0.13$, $D=8$).