Stripe formation in doped Hubbard ladders

We investigate the formation of stripes in $7r\times 6$ Hubbard ladders with $4r$ holes doped away from half filling using the density-matrix renormalization group (DMRG) method. A parallelized code allows us to keep enough density-matrix eigenstates (up to $m=8000$) and to study sufficiently large systems (with up to $7r=21$ rungs) to extrapolate the stripe amplitude to the limits of vanishing DMRG truncation error and infinitely long ladders. Our work gives strong evidence that stripes exist in the ground state for strong coupling $(U=12t)$ but that the structures found in the hole density at weaker coupling $(U=3t)$ are an artifact of the DMRG approach.

Figure 1

(Color online) DMRG ground-state energy per site versus $m$ for the $21\times 6$ ladder at $U=12t$, using (circles) and not using (squares) block reflection in the middle of the lattice. Every data point corresponds to one sweep of the DMRG procedure.

Figure 2

(Color online) Hole (circle) and staggered spin (square) densities in the leg $\left[x\right]$ direction calculated on a $21\times 6$ Hubbard ladder with 12 holes for $U=3t$ (open symbols) and $U=12t$ (solid symbols).

Figure 3

(Color online) Normalized power spectrum of the hole (solid bar) and staggered spin (open bar) densities presented in Fig. 2 for a $21\times 6$ Hubbard ladder with $U=3t$ (upper panel) and $U=12t$ (lower panel).

Figure 4

(Color online) Maximum $H_{\max }$ of $\mid H(k_x,0)\mid$ in a $21\times 6$ system between $U=3$ (triangle) and $U=12$ (circle) as a function of the square root of the discarded weight $W_m$ for various numbers of density-matrix eigenstates $1800⩽m⩽8000$. Dashed lines are linear fits. Inset: Extrapolation $H_{\max }^0$ of $H_{\max }$ for $\sqrt{W_m}\to 0$ versus $U$.

Figure 5

(Color online) Maximum $S_{\max }$ of $\mid S(k_x,0)\mid$ for the $7\times 6$ system. Dashed line: linear fit, dotted line: quadratic fit, solid line: quadratic fit with constraint $S_{\max }\left(0\right)⩾0$ for $W_m\to 0$.

Figure 6

(Color online) Amplitude $H_{\max }∕\sqrt{R}\propto h_0$ of the hole density modulation for a fixed number $(6000⩽m⩽8000)$ of density-matrix eigenstates (square) and extrapolated to the limit $W_m\to 0$ (circle) as a function of the inverse ladder length $1∕R$ for $U=12t$ (solid symbols) and $U=3t$ (open symbols). Dashed lines are linear fits.