Correlation inequalities for vector spin models

Correlation inequalities forn-vector spin models (n 2) are reviewed. A relatively simple and unified derivation of the inequalities is achieved, using duplicate variable methods, for spin dimensionalitiesn=2 (plane rotator model),n=3 (classical Heisenberg model), andn=4. Although correlation inequalities are lacking forn > 4, new proofs are presented for the comparison inequalities relating correlations for systems with arbitrary spin dimensionality to corresponding correlations for systems with low spin dimensionality (n = 1 or 2).Research supported by National Science Foundation under Grant DMR 76-23071.     

Correlation inequalities for Ising spin lattices

A change of spin representation is used to present expectation inequalities on Ising lattices directly as sums of terms of like sign. The technique is extended to correlation inequalities by introducing replica variables which convert correlations into expectations on a larger space. Second order correlations are analyzed in full from this viewpoint, recovering the FKG set, among others. Third order correlations are examined in some detail, and the sign of the multi-site Ursell correlationsF ₃,F ₄,F ₆ established under appropriate restrictions.     

Multipartite entanglement inequalities via spin vector geometry

We introduce inequalities for multipartite entanglement, derived from the geometry of spin vectors. The criteria are constructed iteratively from cross and dot products between the spins of individual subsystems, each of which may have arbitrary dimension. For qubit ensembles the maximum violation for our inequalities is larger than that for the Mermin-Klyshko Bell inequalities, and the maximally violating states are different from Greenberger-Horne-Zeilinger states. Our inequalities are violated by certain bound entangled states for which no Bell-type violation has yet been found.     

Inequalities between different classical spin models

We apply correlation inequalities to show that in a multicomponent spin model some quantities (e.g. magnetisation) decrease from a one-component model to a two-component, or from two to n > 2.     

The classical limit ofn-vector spin models

It is proved that the free energy of a system ofn-dimensional spins with Kac type potential is equal, in the infinite range zero strength limit, to the free energy of the corresponding Curie-Weiss system in which every spin interacts equally with every other spin.     

Optimal spin squeezing inequalities detect bound entanglement in spin models

We determine the complete set of generalized spin squeezing inequalities. These are entanglement criteria that can be used for the experimental detection of entanglement in a system of spin-1/2 particles in which the spins cannot be individually addressed. They can also be used to show the presence of bound entanglement in the thermal states of several spin models.     

The random-walk representation of classical spin systems and correlation inequalities

We use the random-walk representation to prove the first few of a new family of correlation inequalities for ferromagnetic ?⁴ lattice models. These inequalities state that the finite partial sums of the propagator-resummed perturbation expansion for the 4-point function form an alternating set of rigorous upper and lower bounds for the exact 4-point function. Generalizations to 2n-point functions are also given. A simple construction of the continuum ? _d ⁴ quantum field theory (d<4), based on these inequalities, is described in a companion paper.     

Phase transitions in classical vector models

The equation of state obtained earlier /6/ for 3-dimensional Ising ferromagnets in the first order in$\text{1}\text{z}$ (z is the number of the nearest neighbours) is generalized with regard to the classical D-component vector model. The results obtained are valid in a wide temperature range |τ|?[z²(D+2)]^?1 (τ=(T_c-T)/T_c) and become exact in the limit D → ∞.     

Vortex dynamics in classical non-Abelian spin models

We discuss the Abelian vortex dynamics in the Abelian projection approach to non-Abelian spin models. We show numerically that in the three-dimensional SU(2) spin model in the maximal Abelian projection the Abelian off-diagonal vortices, unlike the diagonal vortices, are not responsible for the phase transition. A generalization of the Abelian projection approach to SU(N) spin models is briefly discussed. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 8, 526–530 (25 April 1998) Published in English in the original Russian journal. Edited by Steve Torstveit.     

The random walk representation of classical spin systems and correlation inequalities

Ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion. By using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities. As an application of these inequalities we obtain a simple proof of the mass gap for the (⁴)₂ quantum field model. We also establish new upper bounds on critical temperatures.Partially supported by N.S.F. Grant No. 79-02490On leave from the University of VirginiaPartially supported by N.S.F. Grant No. DMR 81-00417     

Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems

We use Ginibre's general formulation of Griffiths' inequalities to derive new correlation inequalities for two-component classical and quantum mechanical systems of distinguishable particles interacting via two body potentials of positive type. As a consequence we obtain existence of the thermodynamic limit of the thermodynamic and correlation functions in the grand canonical ensemble at arbitrary temperatures and chemical potentials. For a large class of systems we show that the limiting correlation functions are clustering. (In a subsequent article these results are extended to the correlation functions of two-component quantum mechanical gases with Bose-Einstein statistics). Finally, a general construction of the thermodynamic limit of the pressure for gases which are not H-stable, above collapse temperature, is presented.Research supported in part by the U.S. National Science Foundation under grant MPS 75-11864A Sloan Foundation Fellow     

Correlation inequalities on some partially ordered sets

We prove that increasing function on a finite distributive lattice are positively correlated by positive measures satisfying a suitable convexity property. Applications to Ising ferromagnets in an arbitrary magnetic field and to the random cluster model are given.     

Energy-entropy inequalities for classical lattice systems

A new characterization of equilibrium states for classical lattice systems is given in terms of correlation inequalities. Their physical meaning is found to express thermodynamic stability. We demonstrate the applicability of the inequalities in specific models.Bevoegdverklaard navorser N.F.W.O.Aangesteld navorser N.F.W.O.     

Griffiths inequalities for noninteractingN-vector (classical heisenberg) models and applications to interacting systems

An order relation for tensors is defined. With this ordering it is shown that in noninteractingN-vector models 〈σ_Aσ_B〉?〈σ_A〉〈σ_B〉 is positive. Applications to interacting models include a proof for the alignment of spins and the subadditivity of the free energy.     

Spin squeezing inequalities for arbitrary spin

We determine the complete set of generalized spin squeezing inequalities, given in terms of the collective angular momentum components, for particles with an arbitrary spin. They can be used for the experimental detection of entanglement in an ensemble in which the particles cannot be individually addressed. We also present a large set of criteria involving collective observables different from the angular momentum coordinates. We show that some of the inequalities can be used to detect k-particle entanglement and bound entanglement.     

Correlation inequalities for antiferromagnets

We prove several correlation inequalities for a class of antiferromagnets with pair interaction and arbitrary fields. To do this we use the Ginibre-Percus method of double variables.     

New insights into quantum and classical correlations in XY spin models

We compute quantum dissonance Q (non-entangled quantum correlation), entanglement E, quantum discord D (total quantum correlation) and classical correlation C for spin pairs at any distance in the infinite XY spin-1/2 chains, i.e., the anisotropic XY model and the isotropic XY model with three-spin interactions. We obtain two simple dominance relations: C ≥ E and D ≥ E + Q Except this, there are no other simple ordering relations between them. We also show that Q can detect the special points of the system where the entanglement just appears or completely disappears. In addition, it is worthwhile to mention that dissonance and classical correlation can also clearly spotlight the critical points of quantum phase transitions in XY spin-1/2 chains.     

Quantum and classical inequalities for tomographic entropies

The entropies associated with tomographic-probability distributions describing quantum and classical states are discussed. The inequalities for the tomographic entropies of quantum states are reviewed. Examples of both discrete variables (qudits, spins) and continuous variables (photon quadratures, positions and momenta) are studied.     

Double potts chain and exact results for some two-dimensional spin models

An exact analytical solution for the q-state Potts model on a 2×∞ ladder with arbitrary two-, three-, and four-site interactions in a unit cell in presented is a closed form. This solution is used to show that the finite-size internal energy equation [6] yields an accurate value of the critical temperature for the triangular Potts lattice with three-site interactions in alternate triangular faces. It is argued that the above equation is exact at least for self-dual models on isotopic strips.