Period 4 stripe in the extended two-dimensional Hubbard model

We study the competition between stripe states with different periods and a uniform $d$-wave superconducting state in the extended two-dimensional Hubbard model at 1/8 hole doping using infinite projected entangled-pair states (iPEPS). With increasing strength of negative next-nearest neighbor hopping $t^{\prime }$, the preferred period of the stripe decreases. For the values of $t^{\prime }$ predicted for cuprate high-$T_c$ superconductors, we find stripes with a period 4 in the charge order, in agreement with experiments. Superconductivity in the period 4 stripe is suppressed at $1/8$ doping. Only at larger doping, $0.18\lesssim \delta <0.25$, the period 4 stripe exhibits coexisting $d$-wave superconducting order. The uniform $d$-wave state is only favored for sufficiently large positive $t^{\prime }$.

Figure 1

Example stripe with a period 6 in the charge order and period 12 in the spin order obtained with iPEPS using a $12\times 2$ supercell, for $U/t=10, t^{\prime }/t=0, D=12$. The sizes of the red disks and the black arrows scale with the local hole density and the local magnetic moment, respectively, with average values indicated in the top and bottom rows. The width of a bond is proportional to the local singlet pairing amplitude on the corresponding bond, with different signs in $x$ and $y$ direction represented by the two different colors.

Figure 2

Phase diagram for $U/t=10, \delta =1/8$, for a fixed bond dimension $D=12$ as a function of the next-nearest neighbor hopping strength $t^{\prime }/t$. Each region in the phase diagram is defined by the corresponding lowest energy state, including stripe states with periods between 3 and 8 (W3–W8) or a uniform $d$-wave SC state (U).

Figure 3

(a) Phase boundaries of the period 4 (W4) stripe phase with its adjacent W5 and W3 stripe phases as a function of the inverse bond dimension. (b) Maximal local magnetic moment and amplitude of the hole density modulation of the W4 stripe for $t^{\prime }/t=-0.3$. (c-d) Energy per site of the W3, W4 and W5 stripes as a function of the inverse squared bond dimension at $t^{\prime }/t=-0.2$ (c) and $t^{\prime }/t=-0.3$ (d). Dashed lines are a guide to the eye.

Figure 4

(a) W4 stripe at $1/8$ doping and $t^{\prime }/t=-0.2$. The colored red bonds show the strength of the nearest-neighbor triplet pairing amplitude ${\mathrm{\Delta }}_{ij}^t$ along the stripe, obtained at finite $D=12$. (b) W4 stripe at larger doping $\delta \sim 0.2$ [same parameters as in (a)] coexisting with $d$-wave pairing. (c) Mean $d$-wave pairing (green data) and $p$-wave pairing (red data) as a function doping $\delta$ for different bond dimensions $D$. The extrapolated value (full symbols) of the $p$-wave pairing vanishes in the entire doping range, whereas the extrapolated $d$-wave pairing remains finite at large doping. (d) Difference of the energy per bond in $x$ and $y$ directions.

Figure 5

Local hole density, local magnetic moment, and various local energy contributions (cf. text) of the W7 and W4 stripes for $t^{\prime }/t=0$ (left panels) and $t^{\prime }/t=-0.3$ (right panels), for $U/t=10, \delta =1/8, D=12$ on each site (or bond) of the two stripes. The bottom panels show the average energy contributions. Each pair of bars has been shifted by the value indicated in the top row. All energies are in units of $t$.