Edge States in Doped Antiferromagnetic Nanostructures

We study competition between different phases in a strongly correlated nanostructure with an edge. Making use of the self-consistent Green's function and density matrix renormalization group methods, we study a system described by the $t\mathrm{\text{−}}{J}_z$ and $t\mathrm{\text{−}}J$ models on a strip of a square lattice with a linear hole density $n_{||}$. At intermediate interaction strength $J/t$ we find edge stripelike states, reminiscent of the bulk stripes that occur at smaller $J/t$. We find that stripes attach to edges more readily than hole pairs, and that the edge stripes can exhibit a peculiar phase separation.

Figure 1

Phase diagram of a system with an edge, in the linear hole concentration ($n_{\parallel }$) vs attraction to the edge ($J$) plane. (a) Bandlike case; (b) case of complex bulk/edge states. The labels of the phases indicate where the holes reside.

Figure 2

(a) Energy of the system per hole as a function of hole density $n_{||}$ for the bulk and the edge stripe, and the bulk pair state in the $t\mathrm{\text{−}}{J}_z$ model (solid line, dashed line, and dotted line, respectively). Black circles are the DMRG data for the bulk stripe, Ref. 4. $J_z/t=0.35$, and $J_z/t=0.1$ (inset). Open symbols denote transitions: I, pair to edge stripe; II, edge to bulk stripe; III, pair to bulk stripe. (b) Constrained (see text) phase diagram of the $t\mathrm{\text{−}}{J}_z$ system with an edge. (A) bulk pairs; (B) bulk stripes; (C) edge stripes; (D) edge pairs.

Figure 3

DMRG results for the edge stripes in the $t\mathrm{\text{−}}{J}_z$ model in (a) $11\times 6$, (b) $11\times 8$, and (c) $20\times 5$ clusters. $J_z/t$ and the number of holes is as indicated. In (a) and (b), only the right half of the cluster containing all the holes is shown with the rightmost column being the open boundary. In (c) the bottom row is the “free” open boundary (see text). The arrows represent the on-site value of $⟨S_i^z⟩$ and the circles are for the hole density $⟨n_i⟩$. Scales for $⟨S_i^z⟩$ and $⟨n_i⟩$ in (b) and (c) are the same.