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arXiv:0904.1213 · Physical Review B

Separability and ground-state factorization in quantum spin systems

S. M. Giampaolo, Gerardo Adesso, Fabrizio Illuminati

Published Jun 29, 2009• 86 citations

Abstract

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.

DOI: 10.1103/physrevb.79.224434

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We introduce a general analytic approach to the study of factorization points and factorized ground states in quantum cooperative systems. The method allows us to determine rigorously the existence, location, and exact form of separable ground states in a large variety of, generally nonexactly solvable, spin models belonging to different universality classes. The theory applies to translationally invariant systems, irrespective of spatial dimensionality, and for spin-spin interactions of arbitrary range.

The existence of definite orders in frustrated quantum systems is related rigorously to the occurrence of fully factorized ground states below a threshold value of the frustration. Ground-state separability thus provides a natural measure of frustration: strongly frustrated systems are those that cannot accommodate for classical-like solutions. The exact form of the factorized ground states and the critical frustration are determined for various classes of nonexactly solvable spin models with different spatial ranges of the interactions. For weak frustration, the existence of disentangling transitions determines the range of applicability of mean-field descriptions in biological and physical problems such as stochastic gene expression and the stability of long-period modulated structures.

The exact factorized ground state of a heterogeneous (ferrimagnetic) spin model which is composed of two spins (ρ, σ) has been presented in detail. The Hamiltonian is not necessarily translational invariant and the exchange couplings can be competing antiferromagnetic and ferromagnetic arbitrarily between different sublattices to build many practical models such as dimerized and tetramerized materials and ladder compounds. The condition to get a factorized ground state is investigated for non-frustrated spin models in the presence of a uniform and a staggered magnetic field. According to the lattice model structure we have categorized the spin models in two different classes and obtained their factorization conditions. The first class contains models in which their lattice structures do not provide a single uniform magnetic field to suppress the quantum correlations. Some of these models may have a factorized ground state in the presence of a uniform and a staggered magnetic field. However, in the second class there are several spin models in which their ground state could be factorized whether a staggered field is applied to the system or not. For the latter case, in the absence of a staggered field the factorizing uniform field is unique. However, the degrees of freedom for obtaining the factorization conditions are increased by adding a staggered magnetic field.

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.

We examine the existence of completely separable ground states (GS) in finite spin-

s

arrays with anisotropic

XYZ

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

{S}_{z}

-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 determine the conditions for the existence of nontransverse factorizing magnetic fields in general spin arrays with anisotropic

XYZ

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.