Stripes on a 6-Leg Hubbard Ladder

While density matrix renormalization group calculations find stripes on doped $n$-leg $t\mathrm{\text{−}}J$ ladders, little is known about the possible formation of stripes on $n$-leg Hubbard ladders. Here we report results for a $7\times 6$ Hubbard model with four holes. We find that a stripe forms for values of $U/t$ ranging from 6 to 20. For $U/t\sim 3–4$, the system exhibits the domain wall feature of a stripe, but the hole density is very broadened.

Figure 1

(a) Charge and spin distribution on a doped $17\times 6$, $t\mathrm{\text{−}}J$ lattice with 12 holes and $J/t=0.35$. All the lattices that we discuss have periodic boundary conditions in the vertical 6-site $y$ direction and have open boundary conditions in the $x$ direction. Here, on the $17\times 6$ lattice, three charge stripes have formed with four holes each separated by $\pi$-phase-shifted antiferromagnetic regions. (b) The same results, plotted differently. The open circles show the hole density $⟨n_h(l)⟩$ and the squares show the spin density $⟨S_z(l)⟩(-1{)}^l$. In addition, the asterisks show results for a $7\times 6$ system with four holes, shifted to the center of the system.

Figure 2

Expectation value of the spin $|⟨S_z(l)⟩|$ at a site in the center of an undoped $17\times 6$ $t\mathrm{\text{−}}J$ system, as a function of the error in the total energy per site. The error in the total energy per site is measured relative to a nearly fully converged DMRG calculation, accurate to about ${10}^{-6}$. Each symbol corresponds to a particular DMRG sweep, for which the number of states kept was $m$. Two sweeps were performed for each value of $m$. Although the exact ground state has $S_z(l)=0$, states with substantial static antiferromagnetic order can have energies only slightly above the ground state. The solid curve was generated by applying a staggered magnetic field to the system, and it represents the maximum value of $|⟨S_z(l)⟩|$ possible for a given error in the energy (see text).

Figure 3

(a) The ground state energy of a $7\times 6$ Hubbard model with $U/t=12$ and four holes versus the number of basis states kept in the DMRG calculation. The arrow indicates the approximate point at which the stripe spontaneously forms. (b) The charge distribution $⟨n_h(l)⟩$ seen in the DMRG calculation, labeled by the number of states kept per block $m$.

Figure 4

The charge and spin distribution for the system of Fig. 3.

Figure 5

(a) The hole density $⟨n_h(l)⟩$ of a $7\times 6$ Hubbard model as a function of the $x$ coordinate ${\ell }_x$ for various values of $U$. The solid lines refer to the list of $U$’s on the right, in order by peak height. The dotted line refers to $U=20$, which is out of order by peak height. The data are taken from the final sweep in each run, with $m$ values ranging from 3600 to 7500 (see text). Because of the different $m$ values, differences between curves of less than about 5% are not significant. (b) For the same set of systems the spin density $⟨S_z(l)⟩(-1{)}^l$, showing the $\pi$ phase shift of the stripe. Here, we estimate that differences of less than about 10% are not significant.