Superconductivity in the doped $\mathit{t}-\mathit{J}$ model: Results for four-leg cylinders

We report a density-matrix renormalization group study of the lightly doped $t\text{−}J$ model on a four-leg cylinder with doped hole concentrations per site $\delta =5\%\text{–}12.5\%$. By keeping an unusually large number of states and long system sizes, we are able to accurately document the interplay between superconductivity, spin, and charge-density-wave orders. The long-distance behavior is consistent with that of a Luther-Emery liquid with a spin gap and power-law charge-density-wave and superconducting correlations. This is the widest $t\text{−}J$ or Hubbard system in which power-law superconducting correlations have been established.

Figure 1

(a) Charge density profile $n\left(x\right)$ at doping levels $\delta =8.33\%$ and $\delta =12.5\%$ on a $L_x=96$ cylinder. The open squares and circles denote numerical data, while the red lines are fits to $n\left(x\right)=A_{\text{cdw}}cos(Qx+\theta )+n_0$, where $A_{\text{cdw}}$ and $Q$ are the CDW amplitude and ordering wave vector, respectively. Note that only the central-half region with rung indices $\frac{L_x}{4}<x\le \frac{3L_x}{4}$ are shown and used in the fitting to minimize the boundary effect. The red ovals label the “reference site” chosen to calculate the SC correlation in Eq. (3). (b) Finite-size scaling of $A_{\text{cdw}}$ as a function of $L_x$ and $\delta$ in a double-logarithmic plot.

Figure 2

Finite-size scaling of superconducting correlation ${\mathrm{\Phi }}_{yy}\left(\frac{L_x}{2}\right)$ as a function of $L_x$ and doping level $\delta$ in a double-logarithmic plot. The solid lines are fits to ${\mathrm{\Phi }}_{yy}\left(\frac{L_x}{2}\right)\sim {\left(\frac{L_x}{2}\right)}^{-K_{\mathrm{sc}}}$. Inset: ${\mathrm{\Phi }}_{yy}$ and $-{\mathrm{\Phi }}_{xy}$ on a $L_x=128$ cylinder.

Figure 3

Luttinger exponents $K_c,K_{\mathrm{sc}}$ and their product $K_c{K}_{\mathrm{sc}}$, as a function of $\delta$. The solid symbols represent the extracted values from the fits in Figs. 1 and 2. The lines are guides to the eyes.

Figure 4

Finite-size scaling of the spin-spin correlation $F\left(\frac{L_x}{2}\right)$ as a function of $L_x$ at $\delta =0\%$ to $12.5\%$ in the semilogarithemic plot. The inset shows the correlation length ${\xi }_s$ obtained from fits (solid lines) to data in the main panel using $F\left(\frac{L_x}{2}\right)\propto e^{-L_x/2{\xi }_s}$.

Figure 5

The charge density profile $n\left(x\right)$ at $\delta =12.5\%$ and ABC in the $\widehat{y}$ direction for $L_x=32$ and $L_x=160$ cylinders.

Figure 6

von Neumann entanglement entropy $S$ with $\delta =8.33\%$, $10\%$, and $12.5\%$. Inset: The extracted central charge $c$ as a function of $\delta$. Dashed line marks $c=1$.