Efficient density matrix renormalization group algorithm to study Y junctions with integer and half-integer spin

An efficient density matrix renormalization group (DMRG) algorithm is presented and applied to Y junctions, systems with three arms of $n$ sites that meet at a central site. The accuracy is comparable to DMRG of chains. As in chains, new sites are always bonded to the most recently added sites and the superblock Hamiltonian contains only new or once renormalized operators. Junctions of up to $N=3n+1\approx 500$ sites are studied with antiferromagnetic (AF) Heisenberg exchange $J$ between nearest-neighbor spins $S$ or electron transfer $t$ between nearest neighbors in half-filled Hubbard models. Exchange or electron transfer is exclusively between sites in two sublattices with $N_A\ne N_B$. The ground state (GS) and spin densities ${\rho }_r=\langle S_r^z\rangle$ at site $r$ are quite different for junctions with $S=1/2$, 1, 3/2, and 2. The GS has finite total spin $S_G=2S\left(S\right)$ for even (odd) $N$ and for $M_G=S_G$ in the $S_G$ spin manifold, ${\rho }_r>0(<0)$ at sites of the larger (smaller) sublattice. $S=1/2$ junctions have delocalized states and decreasing spin densities with increasing $N$. $S=1$ junctions have four localized $S_z=1/2$ states at the end of each arm and centered on the junction, consistent with localized states in $S=1$ chains with finite Haldane gap. The GS of $S=3/2$ or 2 junctions of up to 500 spins is a spin density wave with increased amplitude at the ends of arms or near the junction. Quantum fluctuations completely suppress AF order in $S=1/2$ or 1 junctions, as well as in half-filled Hubbard junctions, but reduce rather than suppress AF order in $S=3/2$ or 2 junctions.

Figure 1

Y junction of $N=10$ sites with three equal arms of $n=3$ sites. The numbering for the left (unprimed), up (primed), and down blocks (double primed) is used in the DMRG algorithm; the numbering in red is used for spin densities.

Figure 2

Schematic representation of the infinite DMRG algorithm for Y junctions with equal arms. At each step, the loop encloses the system and the superblock contains a new site shown as an open dot, and three arms.

Figure 3

Schematic representation of finite DMRG steps. Sites of the system block are enclosed in the loop; remaining sites are in the environmental block. The new site is the open dot. A, B, and C arm refer to the three different arms.

Figure 4

Total GS energy $E\left(100\right)-E\left(m\right)$ as a function of $m$ for 64-site Y junctions with equal arms, $J=1$, and the indicated $S$ per site in Eq. (1).

Figure 5

Spin densities ${\rho }_r$ along any two arms of 64-site Y junctions with $U=0$ or $4t$ in Eq. (2), or $S=1/2$ in Eq. (1). The junction is at $r=22$ with ${\rho }_J\le 0$.

Figure 6

Spin densities in one arm of a Y junction of $N=202$ sites, $S=1$, as a function of $r$ with $r=1$ at the first site and at $r=68$ at the junction.

Figure 7

Spin densities from Fig. 6 plotted as $\text{ln}|{\rho }_r|$ vs $r$ at an arm and at the junction. Circles (squares) represent negative (positive) spin density.

Figure 8

Spin densities ${\rho }_r$ in one arm of Y junctions with (a) $N=448$ and (b) $N=298$ spins $S=3/2$ and 2. Lines are based on parameters in Table 4.

Figure 9

Spin densities from Fig. 8 for $S=3/2, N=448$ plotted as $\text{ln}(|{\rho }_r|-0.336)$ vs $r$ at an arm and at the junction.

Figure 10

Spin densities from Fig. 8 for $S=2, N=448$ plotted as $\text{ln}(|{\rho }_r|-0.54)$ vs $r$ at an arm and at the junction.

Figure 11

Spin densities ${\rho }_r$ in half chains of (a) $N=150$ and (b) $N=300$ spins $S=3/2$ and 2 with antiferromagnetic Heisenberg exchange $J$ between neighbors.