Stripes in the two-dimensional $t$-$J$ model with infinite projected entangled-pair states

We simulate the $t$-$J$ model in two dimensions by means of infinite projected entangled-pair states (iPEPS) generalized to arbitrary unit cells, finding results similar to those previously obtained by the density-matrix renormalization group (DMRG) for wide ladders. In particular, we show that states exhibiting stripes, that is, a unidirectional modulation of hole-density and antiferromagnetic order with a $\pi$-phase shift between adjacent stripes, have a lower variational energy than uniform phases predicted by variational and fixed-node Monte Carlo simulations. For a fixed unit-cell size the energy per hole is minimized for a hole density ${\rho }_l\sim 0.5$ per unit length of a stripe. The superconducting order parameter is maximal around ${\rho }_l\sim 0.75\text{--}0.8$.

Figure 1

(a) Diagrammatic representation of a PEPS with bond dimension $D$ on a $4\times 4$ lattice. (b) An MPS with bond dimension $m$ to represent a $4\times 4$ system. (c)–(e) Diagrams for the left move of the corner-transfer matrix method: (c) A plaquette of four reduced tensors $a$ embedded in the environment. The coordinates $[x,y]$ are to be understood relatively to the unit cell, that is, one has to take the coordinate $x$ ($y$) modulo $N_x$ (modulo $N_y$). Cutting through the lines as marked in the figure yields the tensor $Q^{[x,y+1]}$ in (d): a singular value decomposition of tensor $Q^{[x,y+1]}$ is performed, where only the $\chi$ largest singular values are kept. The resulting isometry $U^{[x,y+1]}$ and its conjugate are used as an approximate resolution of the identity $\mathrm{I}\approx U^{†[x,y+1]}{U}^{[x,y+1]}$. For a fixed $x=x_0$ one computes all isometries $U^{[x_0,y]}$ for all $y\in [1,N_y]$. These isometries are then used to obtain the renormalized corner tensors $C_1^{\prime [x+1,y]}$ and $C_4^{\prime [x+1,y]}$, and edge tensors $T_4^{\prime [x+1,y]}$ for all $y\in [1,N_y]$ and fixed $x=x_0$, shown in (e). This whole procedure is repeated $N_x$ times for $x_0\in [1,N_x]$ to complete an entire left move. (f) Initialization of a boundary tensor from a PEPS tensor and its conjugate, where crossings have been replaced by swap tensors (cf. Ref.9).

Figure 2

Energy per hole as a function of doping for $J/t=0.4$. For the $D=8$ results the estimated error due to a finite $\chi =100$ is $0.3\%$. For $D=10$ the values are extrapolated in $\chi$. Grey arrows mark stripes with 0.5 holes per unit length. The DMRG cylinders labeled with a shift are wrapped periodically with the indicated shift to connect the transverse stripes into one long continuous spiral stripe, allowing arbitrary filling.

Figure 3

Energy per hole at doping $\delta \approx 1/8$ for $J/t=0.4$ as a function of truncation error in DMRG (left panel) and as a function of $1/D$ with iPEPS using a $8\times 2$ unit cell (middle panel), with values obtained from extrapolation in the dimension $\chi$ (right panel). The quadratic fit in the middle panel is a guide to the eye.

Figure 4

Examples of stripes running in the vertical direction obtained from various iPEPS simulations. Each panel shows one unit cell of the infinite lattice. The diameter of the dots scales with the local hole density with average values given by the (upper) red numbers. The arrows represent the local magnetic moment with average magnitude given by the (lower) black numbers. There is a $\pi$-phase shift in the antiferromagnetic order between adjacent stripes. The width of a bond between two sites scales with the (singlet) pairing amplitude on the bond with a positive (green/dark gray) or negative (cyan/light gray) sign. A pattern with predominantly $d$-wave order is visible, with maximal pairing amplitude 0.01, 0.03, and 0.003 in the three examples, respectively. (a) Stripes of width 4 in a unit cell $8\times 2$ with hole density of ${\rho }_l\sim 0.5$ holes per unit length per stripe at a doping $\delta \sim 1/8$. (b) Stripes of width 6 in a unit cell $12\times 2$ with ${\rho }_l\sim 0.75$, $\delta \sim 1/8$, where the pairing is maximal. (c) Same as in (b) but with ${\rho }_l=1$, $\delta \sim 1/6$ where the pairing is suppressed.

Figure 5

Mean pairing amplitude as a function of linear hole density per unit length of a stripe ${\rho }_l$. The DMRG results are for one longitudinal stripe with a pairing field of 0.02 applied throughout, making the overall magnitude somewhat arbitrary.