Mapping Phase Diagrams of Quantum Spin Systems through Semidefinite-Programming Relaxations

Identifying quantum phase transitions poses a significant challenge in condensed matter physics, as this requires methods that both provide accurate results and scale well with system size. In this work, we demonstrate how relaxation methods can be used to generate the phase diagram for one- and two-dimensional quantum systems. To do so, we formulate a relaxed version of the ground-state problem as a semidefinite program, which we can solve efficiently. Then, by taking the resulting vector of moments for different model parameters, we identify all phase transitions based on their cosine similarity. Furthermore, we show how spontaneous symmetry breaking is naturally captured by bounding the corresponding observable. Using these methods, we reproduce the phase transitions for the one-dimensional transverse field Ising model and the two-dimensional frustrated bilayer Heisenberg model. We also illustrate how the phase diagram of the latter changes when a next-nearest-neighbor interaction is introduced. Overall, our work demonstrates how relaxation methods provide a novel framework for studying and understanding quantum phase transitions.

💡 Research Summary

In this paper the authors introduce a novel computational framework for mapping phase diagrams of quantum spin systems based on semidefinite programming (SDP) relaxations. The central idea is to reformulate the ground‑state energy minimization as a polynomial optimization over non‑commuting spin operators, then replace the exact positivity constraint on the density matrix with a positive‑semidefinite constraint on a moment matrix built from a selected set of Pauli monomials. This relaxation defines a convex superset of the true set of physical moments, guaranteeing that the SDP yields a lower bound on the ground‑state energy (and, with additional constraints, upper bounds as well). Because the moment matrix size and the number of linear constraints scale polynomially with system size, the approach remains tractable for larger lattices where exact diagonalization fails.

The method is first benchmarked on the one‑dimensional transverse‑field Ising (TFI) model with 30 sites. For each value of the transverse field h/J the SDP is solved, producing a moment vector ŷ(h). By fixing a reference vector ŷ_fix in one phase and computing the cosine similarity S_C(ŷ_fix, ŷ_j) across the whole parameter range, the authors observe that S_C stays close to unity within the same phase and drops sharply at the known second‑order transition at h/J = 1. This unsupervised similarity measure thus cleanly identifies the critical point. Moreover, by solving auxiliary SDPs that bound specific observables (⟨σ_z⟩, ⟨σ_x⟩) they obtain certified upper and lower bounds. The difference between these bounds peaks at the transition, and the bounds correctly capture the spontaneous Z₂ symmetry breaking for h/J < 1, whereas the raw moment value may not.

Next, the authors apply the technique to the two‑dimensional frustrated bilayer Heisenberg (FBH) model on a 6‑site (3×3 per layer) lattice, characterized by inter‑layer coupling J⊥, intra‑layer coupling J∥, and a diagonal coupling Jx. The known phases are a dimer‑singlet (DS), a dimer‑triplet antiferromagnet (DT_AF), and a bilayer antiferromagnet (BAF), with a first‑order DS↔DT_AF line and a second‑order DT_AF↔BAF line. By selecting three reference moment vectors deep inside each phase and scanning the full (J⊥/J∥, Jx/J∥) plane, the cosine similarity again reveals the three distinct regions and the location of the phase boundaries. While the similarity alone distinguishes the two types of transitions qualitatively, the authors note that a reliable classification of transition order requires complementary information from observable bounds and, optionally, variational calculations.

Finally, the framework is extended to a frustrated bilayer model with an added next‑nearest‑neighbor (NNN) inter‑layer interaction (FBH‑NNN) with J2/J∥ = 0.5. Using the same three reference vectors obtained from the J2 = 0 case, the similarity analysis shows that the DT_AF and DS phases persist with high overlap, whereas the overlap with the BAF phase is dramatically reduced, indicating a substantial reshaping of the phase diagram. Further analysis in the supplementary material confirms a shift of the first‑order line and suggests the possible emergence of an intermediate phase.

Overall, the paper demonstrates that SDP relaxations provide (i) certified energy bounds tighter than those from standard variational ansätze, (ii) a scalable way to generate moment vectors that encode phase information, (iii) an unsupervised similarity metric capable of locating phase transitions without prior knowledge, and (iv) a natural mechanism to bound order parameters and detect spontaneous symmetry breaking. The authors argue that higher levels of the SDP hierarchy could systematically tighten the bounds, and that integrating machine‑learning clustering techniques could automate the identification of transition order. This work opens a promising avenue for studying complex quantum many‑body systems where conventional methods face sign problems or exponential scaling.