An SU(2) symmetric Semidefinite Relaxation for Ground States of Heisenberg Models*

The Heisenberg model plays a central role in understanding quantum materials, however, the task of obtaining the ground state of a given Heisenberg-type Hamiltonian is in general very difficult and is known to be QMA-complete. Here, we focus on the fact that for the (antiferromagnetic) Ising model, computational complexity theory provides strong evidence for the Goemans-Williamson algorithm, a semidefinite programming (SDP) based approach, to be *optimal* in the approximation ratio sense (for a particular convention of the base energy choice corresponding to the so-called "Max Cut" problem). We study the "quantum Max Cut" problem, a quantum generalization of this, which corresponds to the Heisenberg model with again a particular convention for the base energy. Following the spirit of valence-bond basis, by using an SU(2) symmetric basis for operators, we construct an SDP algorithm that is both theoretically most natural and practically implementable for the first time. We prove that it could be regarded as a first-order of a systematically improving sequence of approximations (i.e. a properly converging Lasserre/NPA-hierarchy), and furthermore give (in)exactness proofs for some families of graphs. Its potential use for understanding ground states of actual condensed matter stems as well as how it connects to "frustration-free" models known in the context of exact solvability will be discussed.