Heisenberg antiferromagnet on Cayley trees: Low-energy spectrum and even/odd site imbalance

To understand the role of local sublattice imbalance in low-energy spectra of $s=\frac{1}{2}$ quantum antiferromagnets, we study the $s=\frac{1}{2}$ quantum nearest neighbor Heisenberg antiferromagnet on the coordination 3 Cayley tree. We perform many-body calculations using an implementation of the density matrix renormalization group (DMRG) technique for generic tree graphs. We discover that the bond-centered Cayley tree has a quasidegenerate set of a low-lying tower of states and an “anomalous” singlet-triplet finite-size gap scaling. For understanding the construction of the first excited state from the many-body ground state, we consider a wave function ansatz given by the single-mode approximation, which yields a high overlap with the DMRG wave function. Observing the ground-state entanglement spectrum leads us to a picture of the low-energy degrees of freedom being “giant spins” arising out of sublattice imbalance, which helps us analytically understand the scaling of the finite-size spin gap. The Schwinger-boson mean-field theory has been generalized to nonuniform lattices, and ground states have been found which are spatially inhomogeneous in the mean-field parameters.

Figure 1

(a) The bond-centered Cayley tree. (b) The site-centered Cayley tree. In both cases all sites, other than those on the boundary, have coordination 3. (c) The “Fibonacci Cayley tree” is constructed hierarchically and has some coordination 2 sites. The figure shows a generation 4 cluster constructed by connecting the roots (head sites) of the generation 2 and generation 3 trees to a common site (the root of the generation 4 tree). To have a globally balanced cluster we introduced a bond connecting the root of the generation 4 tree with the root of its mirror image. All clusters in (a), (b), and (c) are bipartite [the dark (red) and light (green) colors show the two sublattices] and have no loops.

Figure 2

Warm-up step of the DMRG involving multiple independent renormalization procedures on the tree utilizing energy-based truncation. The “army continues to march in” from all sides till one reaches the geometrically central point (often called the “focal point” or “root”). Here we show all stages for a given tree. The red dots represent renormalized blocks.

Figure 3

Division of the Cayley tree locally into site, system(s), and environment as required by the DMRG. One renormalization step consists of combining the site and system(s) into a new system, retaining the states governed by tracing out the environment degrees of freedom.

Figure 4

Ground-state energy per site for the bond-centered and site-centered Cayley trees for various lattice sizes. The Fibonacci Cayley tree energies are out of the scale considered. A fit to the bond-centered results given by Eq. (4.1) has been shown. Inset: Finite-size scaling of the energy gap $\Delta$ plotted on a log-log scale. The bond-centered and Fibonacci clusters appear gapless in the infinite lattice limit, with a finite-size scaling exponent of $\alpha \approx 2$ and $\alpha \approx 0.6$. However, the site-centered clusters have a finite $\Delta$ in the infinite lattice limit in concordance with the results of Ref. 22. The lines shown are fits to the DMRG data using Eqs. (4.2) and (4.3).

Figure 5

Ground-state spin-spin correlations $G_{a,i}$, as in Eq. (4.4), for the three kinds of Cayley tree. One (reference) spin $a$ is held fixed while the other spin $i$ is take at distance $a$ along the highlighted path. In (a) and (c), the reference spin is at a tip ($a\to \text{tip}$), and DMRG results are compared with numerical solutions of Schwinger-boson mean-field theory (SBMFT) from Sec. 6. In (b), the reference spin is at the central (“root”) site ($a\to 0$). (a) Bond-centered tree with $N_s=126$ sites. The SBMFT correlations shown have been scaled up by an overall factor of $1.8$ to take into account corrections beyond mean field (in the broken-symmetry phase). The DMRG and SBMFT results show good agreement, asymptoting to a constant. (b) Site-centered cluster with $N_s=190$ sites, in the maximum $S_z$ member of the ground-state multiplet ($S_z=S_0$). The magnetization $|\langle S_i^z\rangle |$ is also shown, as a function of distance from the center. Even though the connected correlation function decays to zero exponentially fast, the long-range order is encoded in the fact that that the magnetization is nonzero. (c) Fibonacci tree with $N_s=40$ sites. For the “quantum disordered” phase, the SBMFT correlations were scaled up by an overall factor of 3/2 (for details see Sec. 6b). Correlations appear to be decaying exponentially with distance.

Figure 6

Lowest energy level in every $S_z$ sector for the 108-site Fibonacci and the 126-site bond-centered Cayley trees. The range of $S$ from 0 to $S^{*}$ has been magnified and shown in the inset for the 126-site cluster. It shows a tower of states with a much larger moment of inertia than expected from the Anderson tower. This is seen as a sharp kinklike feature at $S^{*}$. In contrast, for the Fibonacci tree, the transition from the low to high $S$ behavior is less well defined.

Figure 7

Magnetization curves for bond-centered Cayley trees of various sizes obtained using DMRG. Inset: The rapid rise of the magnetization with application of a small magnetic field indicates a susceptibility diverging with system size.

Figure 8

Magnetization curves for sites on various shells of the 62-site bond-centered Cayley tree. The subscript refers to the shell on which the site is present; that is, 0 refers to the central two sites and the 4 refers to the boundary.

Figure 9

Amplitude of the SMA coefficients $u_i$ for various shells (normalized with respect to amplitude on the boundary) for the $N_s=$ 30, 62, and 126 site bond-centered lattices. Inset: The sign structure of $u_i$ is the same as Eq. (5.14). Dark (red) and light (green) indicates negative and positive $u_i$, respectively.

Figure 10

The entanglement spectrum for the bond-centered, site-centered, and Fibonacci trees as shown in the inset. $\lambda$ refers to the eigenvalue of the reduced density matrix of the shaded area. The ground state for the bond-centered and Fibonacci clusters was a singlet and only the $S_z>0$ sectors are shown. For the site-centered cluster we chose to work with the maximal $S_z$ sector (which is the $S_z=16$ for the 94-site cluster). $I_b$ and $I_s$ denote the “imbalance” metric defined in the introduction [refer to Eq. (2.3)]. For the bond-centered case, the largest degenerate eigenvalues of the reduced density matrix indicate a multiplet whose spin length exactly equals the imbalance $I_b$. For the site-centered case, the density matrix has largest weight in a state whose spin is $I_s$. For the Fibonacci case, a spin-1/2 state has the largest weight in the density matrix.

Figure 11

Schematic of the “giant spins”, which are the low-energy degrees of freedom for the bond- and site centered clusters (and which are absent for the Fibonacci tree).The numbering of sites shown here is used for the purpose of explaining our arguments in the text.

Figure 12

(a) Heuristic for computing the number of “dangling spins” as proposed by Wang and Sandvik. (b) The magnitude of the “flippability” as in Eq. (5.8) computed from DMRG and the condensate fraction $|{\beta }_i{|}^2$ computed from SBMFT on every shell of the bond-centered Cayley tree. Both quantities are qualitatively consistent with each other and confirm the “dangling” spin heuristic shown above.