Study of spin symmetry in the doped $t-J$ model using infinite projected entangled pair states

We study the two-dimensional $t\text{−}J$ model on a square lattice using infinite projected entangled pair states (iPEPS). At small doping, multiple orders, such as antiferromagnetic order, stripe order and superconducting order, are intertwined or compete with each other. We demonstrate the role of spin symmetry at small doping by either imposing SU(2) spin symmetry or its U(1) subgroup in the iPEPS ansatz, thereby excluding or allowing spontaneous spin-symmetry breaking, respectively, in the thermodynamic limit. From a detailed comparison of our simulations, we provide evidence that stripe order is pinned by long-range antiferromagnetic order. We also find SU(2) iPEPS, enforcing a spin-singlet state, yields a uniform charge distribution and favors d-wave singlet pairing.

Figure 1

U(1) and SU(2) iPEPS results for the $t\text{−}J$ model at $J/t=0.4$. (a) The energy per hole $e_{\mathrm{h}}$ as a function of hole doping $\delta$ for $5\times 2$ and $2\times 2$ unit cells. (b)–(d) Spin, hole, and singlet pairing amplitude profiles at $\delta \simeq 0.1$, 0.2, and 0.3. U(1) iPEPS ($D=8$) yields stripes on $5\times 2$ clusters and charge-uniform states on $2\times 2$ clusters; SU(2) iPEPS ($D^{*}=6$) yields spin singlets. Symbols and linewidths are drawn to scale as indicated.

Figure 2

Averaged singlet pairing amplitude as a function of doping using SU(2) iPEPS. The error bar shows the mean absolute deviation of the pairing amplitudes among all bonds. For the $5\times 2$ unit cell, the error bars are smaller than the symbols.

Figure 3

Normalized staggered spin-spin correlation functions, computed on a $5\times 2$ unit cell along the long ($y$) direction (left column) and the short ($x$) direction (right column), using (a) and (b) U(1) iPEPS and (c)–(f) SU(2) iPEPS, on linear and semilogarithmic scales, respectively. Solid symbols indicate the variational state [U(1) or SU(2)] with the lower energy for a given $\delta$.

Figure 4

Comparison of normalized staggered spin-spin correlation functions for $\delta \simeq 0.2$ using U(1) iPEPS at $D=8$ along the long ($y$) direction and the short ($x$) direction on $L\times 2$ unit cells for $L=2,3,4,5$ in (a)–(d), respectively. The inset in (b) is a $\lambda 4$ stripe state obtained from different initialization.