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Graph--Theoretic Analysis of Phase Optimization Complexity in Variational Wave Functions for Heisenberg Antiferromagnets

Mahmud Ashraf Shamim, Moshiur Rahman, Mohamed Hibat-Allah, Paulo T Araujo

Abstract

Despite extensive study, the phase structure of the wavefunctions in frustrated Heisenberg antiferromagnets (HAF) is not yet systematically characterized. In this work, we represent the Hilbert space of an HAF as a weighted graph, which we term the Hilbert graph (HG), whose vertices are spin configurations and whose edges are generated by off-diagonal spin-flip terms of the Heisenberg Hamiltonian, with weights set by products of wavefunction amplitudes. Holding the amplitudes fixed and restricting phases to $\mathbb{Z}_2$ values, the phase-dependent variational energy can be recast as a classical Ising antiferromagnet on the HG, so that phase reconstruction of the ground state reduces to a weighted Max-Cut instance. This shows that phase reconstruction HAF is worst-case NP-hard and provides a direct link between wavefunction sign structure and combinatorial optimization.

Graph--Theoretic Analysis of Phase Optimization Complexity in Variational Wave Functions for Heisenberg Antiferromagnets

Abstract

Despite extensive study, the phase structure of the wavefunctions in frustrated Heisenberg antiferromagnets (HAF) is not yet systematically characterized. In this work, we represent the Hilbert space of an HAF as a weighted graph, which we term the Hilbert graph (HG), whose vertices are spin configurations and whose edges are generated by off-diagonal spin-flip terms of the Heisenberg Hamiltonian, with weights set by products of wavefunction amplitudes. Holding the amplitudes fixed and restricting phases to values, the phase-dependent variational energy can be recast as a classical Ising antiferromagnet on the HG, so that phase reconstruction of the ground state reduces to a weighted Max-Cut instance. This shows that phase reconstruction HAF is worst-case NP-hard and provides a direct link between wavefunction sign structure and combinatorial optimization.
Paper Structure (2 theorems, 12 equations, 1 figure)

This paper contains 2 theorems, 12 equations, 1 figure.

Key Result

Theorem 1

Let $G=(V_A\cup V_B,E)$ be a bipartite lattice and consider the spin--$\tfrac{1}{2}$ Hilbert space restricted to a fixed $S^z_{\mathrm{tot}}$ sector. Let $\Gamma(G)$ be the HG whose vertices are configurations $\sigma$ in this sector, and where $\{\sigma,\tau\}$ is an edge if and only if $\tau$ is o

Figures (1)

  • Figure 1: Left: A $2\times2$ open boundary square-lattice Heisenberg antiferromagnet with sites labeled by integers. Right: Corresponding bipartite HG $K_{2,4}$ in the $S^{z}_\mathrm{total}=0$ sector, with hub vertices $\{\sigma_3,\sigma_4\}$ corresponding to Néel configurations (blue) and leaf vertices $\{\sigma_1,\sigma_2,\sigma_5,\sigma_6\}$ corresponding to non-Néel configurations (purple). With phases $\phi_{3,4}=0$ and $\phi_{1,2,5,6}=\pi$, every NN edge crosses the Max-Cut, yielding the antiferromagnetic energy minimum. The states $\sigma_i$ are labeled such that, $\sigma_{1} = \uparrow_{1}\uparrow_{2}\downarrow_{3}\downarrow_{4}, \sigma_{2} = \downarrow_{1}\downarrow_{2}\uparrow_{3}\uparrow_{4}, \sigma_{5} = \uparrow_{1}\downarrow_{2}\downarrow_{3}\uparrow_{4}, \sigma_{6} = \downarrow_{1}\uparrow_{2}\uparrow_{3}\downarrow_{4}$ and $\sigma_{3} = \uparrow_{1}\downarrow_{2}\uparrow_{3}\downarrow_{4}, \sigma_{4} = \downarrow_{1}\uparrow_{2}\downarrow_{3}\uparrow_{4}$.

Theorems & Definitions (2)

  • Theorem 1: Bipartiteness inheritance
  • Theorem 2: PEC--bipartiteness