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arXiv:1606.08554 · Physical Review A

Factorization in spin systems under general fields and separable ground-state engineering

Marco Cerezo, R. Rossignoli, N. Canosa

Published Oct 24, 2016• 13 citations
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Abstract

We discuss ground-state factorization schemes in spin $S$ arrays with general quadratic couplings under general magnetic fields, not necessarily uniform or transverse. It is shown that, given arbitrary spin alignment directions at each site, nonzero $XYZ$ couplings between any pair and fields at each site always exist such that the ensuing Hamiltonian has an exactly separable eigenstate with the spins pointing along the specified directions. Furthermore, by suitable tuning of the fields this eigenstate can always be cooled down to a nondegenerate ground state. It is also shown that in open one-dimensional systems with fixed arbitrary first-neighbor couplings at least one separable eigenstate compatible with an arbitrarily chosen spin direction at one site is always feasible if the fields at each site can be tuned. We demonstrate as well that in the vicinity of factorization, i.e., for small perturbations in the fields or couplings, pairwise entanglement reaches full range. Some noticeable examples of factorized eigenstates are unveiled. The present results open the way for separable ground-state engineering. A notation to quantify the complexity of a given type of solution according to the required control on the system couplings and fields is introduced.

DOI
10.1103/physreva.94.042335

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We discuss ground-state factorization schemes in spin SSS arrays with general quadratic couplings under general magnetic fields, not necessarily uniform or transverse. It is shown that, given arbitrary spin alignment directions at each site, nonzero XYZXYZXYZ couplings between any pair and fields at each site always exist such that the ensuing Hamiltonian has an exactly separable eigenstate with the spins pointing along the specified directions. Furthermore, by suitable tuning of the fields this eigenstate can always be cooled down to a nondegenerate ground state. It is also shown that in open one-dimensional systems with fixed arbitrary first-neighbor couplings at least one separable eigenstate compatible with an arbitrarily chosen spin direction at one site is always feasible if the fields at each site can be tuned. We demonstrate as well that in the vicinity of factorization, i.e., for small perturbations in the fields or couplings, pairwise entanglement reaches full range. Some noticeable examples of factorized eigenstates are unveiled. The present results open the way for separable ground-state engineering. A notation to quantify the complexity of a given type of solution according to the required control on the system couplings and fields is introduced.
We determine the conditions for the existence of nontransverse factorizing magnetic fields in general spin arrays with anisotropic XYZXYZXYZ couplings of arbitrary range. It is first shown that a uniform, maximally aligned, completely separable eigenstate can exist just for fields ${\mathbit{h}}_{s}$ parallel to a principal plane and forming four straight lines in the field space, with the alignment direction different from that of ${\mathbit{h}}_{s}$ and determined by the anisotropy. Such a state always becomes a nondegenerate ground state for sufficiently strong (yet finite) fields along these lines, in both ferromagnetic and antiferromagnetic-type systems. In antiferromagnetic chains, this field coexists with the nontransverse factorizing field ${\mathbit{h}}_{s}^{\ensuremath{'}}$ associated with a degenerate N\'eel-type separable ground state, which is shown to arise at a level crossing in a finite chain. It is also demonstrated for arbitrary spin that pairwise entanglement reaches full range in the vicinity of both ${\mathbit{h}}_{s}$ and ${\mathbit{h}}_{s}^{\ensuremath{'}}$, vanishing at ${\mathbit{h}}_{s}$ but approaching small yet finite side limits at ${\mathbit{h}}_{s}^{\ensuremath{'}}$, which are analytically determined. The behavior of the block entropy and entanglement spectrum in their vicinity is also analyzed.
We determine the conditions for the existence of a pair of degenerate parity breaking separable eigenstates in general arrays of arbitrary spins connected through XYZXYZXYZ couplings of arbitrary range and placed in a transverse field, not necessarily uniform. Sufficient conditions under which they are ground states are also provided. It is then shown that in finite chains, the associated definite parity states, which represent the actual ground state in the immediate vicinity of separability, can exhibit entanglement between any two spins regardless of the coupling range or separation, with the reduced state of any two subsystems equivalent to that of pair of qubits in an entangled mixed state. The corresponding concurrences and negativities are exactly determined. The same properties persist in the mixture of both definite parity states. These effects become specially relevant in systems close to the XXZXXZXXZ limit. The possibility of field induced alternating separable solutions with controllable entanglement side limits is also discussed. Illustrative numerical results for the negativity between the first and the jth{j}^{\text{th}}jth spin in an open spin sss chain for different values of sss and jjj are as well provided.
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 examine the existence of completely separable ground states (GS) in finite spin-sss arrays with anisotropic XYZXYZXYZ couplings, immersed in a nonuniform magnetic field along one of the principal axes. The general conditions for their existence are determined. The analytic expressions for the separability curve in field space, and for the ensuing factorized state and GS energy, are then derived for alternating solutions, valid for any spin and size. They generalize results for uniform fields and show that nonuniform fields can induce GS factorization also in systems which do not exhibit this phenomenon in a uniform field. It is also shown that such a curve corresponds to a fundamental Sz{S}_{z}Sz​-parity transition of the GS, present for any spin and size, and that two different types of GS parity diagrams can emerge, according to the anisotropy. The role of factorization in the magnetization and entanglement of these systems is also analyzed, and analytic expressions at the borders of the factorizing curve are provided. Illustrative examples for spin pairs and chains are as well discussed.
We analyze ground state (GS) factorization in general arrays of spins s_{i} with XXZ couplings immersed in nonuniform fields. It is shown that an exceptionally degenerate set of completely separable symmetry-breaking GSs can arise for a wide range of field configurations, at a quantum critical point where all GS magnetization plateaus merge. Such configurations include alternating fields as well as zero-bulk field solutions with edge fields only and intermediate solutions with zero field at specific sites, valid for d-dimensional arrays. The definite magnetization-projected GSs at factorization can be analytically determined and depend only on the exchange anisotropies, exhibiting critical entanglement properties. We also show that some factorization-compatible field configurations may result in field-induced frustration and nontrivial behavior at strong fields.
We examine the entanglement of cyclic spin-1/2 chains with anisotropic XYZXYZXYZ Heisenberg couplings of arbitrary range at transverse factorizing magnetic fields. At these fields the system exhibits a degenerate symmetry-breaking separable ground state (GS). It is shown, however, that the side limits of the GS pairwise entanglement at these fields are actually nonzero in finite chains, corresponding such fields to a GS spin-parity transition. These limits exhibit universal properties like being independent of the pair separation and interaction range, and are directly related to the magnetization jump. Illustrative exact results are shown for chains with (I) full range and (II) nearest-neighbor couplings. Global entanglement properties at such points are also discussed.