Tree-graph based construction of quantum spin models with exact ground state

TL;DR

The paper addresses constructing quantum spin-\(\tfrac{1}{2}\) antiferromagnetic Heisenberg Hamiltonians with exact ground states by mapping exchange interactions to a tree graph. A block-spin analysis shows that when the hierarchy of couplings satisfies $J_v < J_{c_{\rm L}(v)}$ and $J_v < J_{\rm R}(v)$, the unique ground state is a singlet-dimer product, with an explicit form provided, and the approach extends to generic binary trees. Relaxing the coupling hierarchy produces massively degenerate ground-state manifolds, illustrating how degeneracy can be engineered through the tree structure. The method generalizes to $m$-ary trees and arbitrary spin, offering a graph-theoretic framework for exact ground states and suggesting routes for experimental realization and classification of spin Hamiltonians via tree graphs.

Abstract

We propose a protocol to generate an antiferromagnetic S=1/2 Heisenberg model with the exact ground state based on a tree graph. The generated model has a correspondence with a tree graph and possesses the product state of singlet dimers as its unique ground state. A procedure for constructing a model with exact, massively degenerate ground states is also introduced.

Figures (5)

• Figure 1: (a) Example of a generic binary tree graph for an eight-spin system. Squares and circles denote vertices and bare $S=1/2$ spins, respectively. The grey area represents the spin block assigned to the vertex $v$ indicated in the panel. (b) A vertex $v$ and its child vertices $c_{\rm L}(v)$ and $c_{\rm R}(v)$. The block spins $\tilde{S}_{c_{\rm L}(v)}$ and $\tilde{S}_{c_{\rm R}(v)}$ are coupled via an exchange interaction with the exchange constant $J_v$.
• Figure 2: Correspondence between tree graphs and quantum spin models. (a) Perfect-Binary Tree (PBT) with two layers, (b) PBT with three layers, and (c) a generic tree graph. In (a), (b), and (c), squares and circles represent the vertices and bare spins, respectively. (d), (e), and (f) depict the quantum spin models corresponding to (a), (b), and (c), respectively. In (d), (e), and (f), circles and lines denote the $S=1/2$ spins and exchange interactions, respectively.
• Figure 3: (a) A part of PBT at the first layer from the boundary. (b) A part of PBT in the two layers from the boundary.
• Figure 4: Correspondence between tree graphs and the ground states. (a) Tree graph containing only vertices whose spin block consists of an even number of spins. (b) Tree graph containing vertices whose spin block consists of an odd number of spins. The numbers indicated for vertices denote the magnitude of the block spin $\tilde{S}_v$ for each vertex in the ground state. Red lines represent the spin pairs forming the singlet state in the ground state.
• Figure 5: (a) A part of a tree graph representing a three-spin block consisting of ${\bm s}_1$, ${\bm s}_2$, and ${\bm s}_3$. (b) and (c) represent a part of a tree graph where two spin blocks of vertices $c_{\rm L}(v)$ and $c_{\rm R}(v)$ are merged at the vertex $v$. $\tilde{S}_v$, $\tilde{S}_{c_{\rm L}(v)}$, and $\tilde{S}_{c_{\rm R}(v)}$ denote the magnitude of the block spins in the ground state.