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arXiv:2605.03022

General method for obtaining the energy minimum of spin Hamiltonians for separable states

Géza Tóth, József Pitrik

quant-phmath-phSubmitted May 4, 2026• Announced May 6, 2026• 0 citations
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Abstract

We present a general method to determine the energy minimum of spin Hamiltonians over separable states when the single-particle reduced density matrices are fixed. For ferromagnetic Ising and Ising-like models with nearest-neighbor interactions on lattices of any dimension and on a fully connected graph in an external field, this minimum is given by a compact analytic formula involving the quantum Fisher information. For the ferromagnetic Heisenberg chain of spin-1/2 particles, the minimum is expressed via the Uhlmann-Jozsa fidelity. These relations enable the direct extraction of both the quantum Fisher information and the fidelity from correlation measurements on the ground states of suitably engineered spin models.

Comments
9 pages including 3 figures, revtex 4.2

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We present a general method to determine the energy minimum of spin Hamiltonians over separable states when the single-particle reduced density matrices are fixed. For ferromagnetic Ising and Ising-like models with nearest-neighbor interactions on lattices of any dimension and on a fully connected graph in an external field, this minimum is given by a compact analytic formula involving the quantum Fisher information. For the ferromagnetic Heisenberg chain of spin-1/2 particles, the minimum is expressed via the Uhlmann-Jozsa fidelity. These relations enable the direct extraction of both the quantum Fisher information and the fidelity from correlation measurements on the ground states of suitably engineered spin models.
We construct entanglement witnesses using fundamental quantum operators of spin models which contain two-particle interactions and have a certain symmetry. By choosing the Hamiltonian as such an operator, our method can be used for detecting entanglement by energy measurement. We apply this method to the Heisenberg model in a cubic lattice with a magnetic field, the XYXYXY model, and other familiar spin systems. Our method provides a temperature bound for separable states for systems in thermal equilibrium. We also study the Bose-Hubbard model and relate its energy minimum for separable states to the minimum obtained from the Gutzwiller ansatz.
We investigate the existence and the properties of fully separable (fully factorized) ground states in quantum spin systems. Exploiting techniques of quantum information and entanglement theory, we extend a recently introduced method and construct a general, self-contained theory of ground state factorization in frustration-free quantum spin models defined on lattices in any spatial dimension and for interactions of arbitrary range. We show that, quite generally, nonexactly solvable translationally invariant models in presence of an external uniform magnetic field can admit exact, fully factorized ground state solutions. Unentangled ground states occur at finite values of the Hamiltonian parameters satisfying well-defined balancing conditions between the applied field and the interaction strengths. These conditions are analytically determined together with the type of magnetic orderings compatible with factorization and the corresponding values of the fundamental observables such as energy and magnetization. The method is applied to a series of examples of increasing complexity, including translationally invariant models with short, long, and infinite ranges of interaction, as well as systems with spatial anisotropies, in lower and higher dimensions. We also illustrate how the general method, besides yielding a large series of exact results for complex models in any dimension, recovers, as particular cases, the results previously achieved on simple models in low dimensions exploiting direct methods based on factorized mean-field ansatz.
We consider an axial Ising model with competing interactions up to third neighbors within a low-temperature approximation in terms of interacting and nonreversing domain walls. Assuming small third-neighbor coupling, we find a close relation between the transfer matrix of this classical model and the Hamiltonian of an S=1/2 anisotropic Heisenberg chain in a uniform magnetic field. This connection allows us to obtain the phase boundaries of the ferromagnetic and other low-temperature modulated states. The ground-state energy for arbitrary magnetization values of the Heisenberg chain is shown to provide information about the density of walls or wave number. In contrast with the two-dimensional axial next-nearest-neighbor Ising model, this quantity displays plateaus and reentrant behavior for some relations among the coupling constants. An alternative way to evaluate correlation functions in the antiferromagnetic Heisenberg chain is presented.
We investigate superpositions of quantum Ising ground states generated through a quantum interaction, in which the nearest-neighbor coupling strength is promoted to an operator acting on a central auxiliary spin. We prepare the central spin in a superposition, which allows the chain to evolve coherently under different effective interaction strengths. The states are prepared by adiabatic evolution of the joint system comprising the central spin and the many-body spin chain, followed by a measurement on the central spin and post-selection of a specific outcome. It is possible to prepare superpositions of transverse-field quantum Ising ground states belonging to the three magnetic phases, namely, the paramagnetic, ferromagnetic, and anti-ferromagnetic phases. For a spin-1/2 central system, it is possible to prepare superpositions of ground states belonging to any two phases, while for the spin-1 case, the superpositions may contain three ground states, each belonging to one of the three phases. We analytically derive expressions for the magnetization, spin correlations, and entanglement of such states. The magnetization and spin correlations show distinct features in different superposition sectors. The nearest-neighbor entanglement is largest where the transverse field and Ising interaction compete, and its behavior differs from that of the magnetic observables. We also derive the quantum Fisher information relevant to estimating the Ising interaction strength. And it exhibits multiple peaks near the critical points of the constituent ground states, with the maximum value showing Heisenberg scaling, for the parameters and system sizes examined.