Systems with separable many-particle interactions. I

Ouden, L.W.J. den; Capel, H.W.; Perk, Jacques H. H.; Tindemans, P.A.J.

doi:10.1016/0378-4371(76)90118-7

Cited by 29 publications

(4 citation statements)

References 31 publications

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“…Note added in proof. After the submission of the manuscript, it was pointed out that our Hamiltonians (4) and ( 6) were a special case of what was studied in [28][29][30].…”

Section: Acknowledgmentmentioning

confidence: 99%

Many-body transverse interactions in the quantum annealing of thep-spin ferromagnet

Seoane¹,

Nishimori²

2012

J. Phys. A: Math. Theor.

102

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Abstract. We study the performance of quantum annealing for the simple p-body infinite-range ferromagnetic Ising model. In particular, we generalize the transverse antiferromagnetic interactions proposed by Seki and Nishimori as a quantum driver to many-body transverse interactions to understand if the two-body interactions are essential to allow the system to avoid troublesome first-order quantum phase transitions. We conclude that the general many-body interactions are effective to let the system evolve only through second-order transitions as long as a few minor conditions are satisfied. It is also discussed whether the overlap of the ground-state wave function of the new driver term with the target ground state is an essential factor for the success.Many-body transverse interactions in the quantum annealing 2

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“…Note added in proof. After the submission of the manuscript, it was pointed out that our Hamiltonians (4) and ( 6) were a special case of what was studied in [28][29][30].…”

Section: Acknowledgmentmentioning

confidence: 99%

Many-body transverse interactions in the quantum annealing of thep-spin ferromagnet

Seoane¹,

Nishimori²

2012

J. Phys. A: Math. Theor.

102

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“…There is a large literature on models involving a mixture of infinite range (mean field) and short range interactions; we point to [15], [16] for the interested reader. Here we wish to point out a few works that are more specifically related to the present model.…”

Section: Introductionmentioning

confidence: 99%

Chains of mean-field models

Hassani¹,

Macris²,

Urbanke³

2012

J. Stat. Mech.

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We consider a collection of Curie-Weiss (CW) spin systems, possibly with a random field, each of which is placed along the positions of a one-dimensional chain. The CW systems are coupled together by a Kac-type interaction in the longitudinal direction of the chain and by an infinite range interaction in the direction transverse to the chain. Our motivations for studying this model come from recent findings in the theory of error correcting codes based on spatially coupled graphs. We find that, although much simpler than the codes, the model studied here already displays similar behaviors. We are interested in the van der Waals curve in a regime where the size of each Curie-Weiss model tends to infinity, and the length of the chain and range of the Kac interaction are large but finite. Below the critical temperature, and with appropriate boundary conditions, there appears a series of equilibrium states representing kink-like interfaces between the two equilibrium states of the individual system. The van der Waals curve oscillates periodically around the Maxwell plateau. These oscillations have a period inversely proportional to the chain length and an amplitude exponentially small in the range of the interaction; in other words the spinodal points of the chain model lie exponentially close to the phase transition threshold. The amplitude of the oscillations is closely related to a Peierls-Nabarro free energy barrier for the motion of the kink along the chain. Analogies to similar phenomena and their possible algorithmic significance for graphical models of interest in coding theory and theoretical computer science are pointed out.

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“…In section 7 we will see that a proper application of scaling theory near the tricritical point requires that the scaling parameters be defined in a curvilinear coordinate system, an idea proposed in [22] but, to our knowledge, not previously explored. Some of the results proved here are contained in the work of Capel and collaborators [6,7,8,9,19,20,21]. These papers introduce the mean-field B-C model and provide a general framework for studying mean-field models and obtaining the thermodynamic properties of these systems.…”

Section: Introductionmentioning

confidence: 87%

“…The formula for x( ) in (5. 19) is obtained by substituting a 2 , a 4 , and a 6 from (5.17) into the definition (A.1) of x(a 2 ).…”

Section: Asymptotic Behavior Of M(β N K N ) Near the Tricritical Pointmentioning

confidence: 99%

Ginzburg-Landau Polynomials and the Asymptotic Behavior of the Magnetization Near Critical and Tricritical Points

Ellis,

Machta,

Otto

2008

Preprint

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Systems with separable many-particle interactions. I

Cited by 29 publications

References 31 publications

Many-body transverse interactions in the quantum annealing of thep-spin ferromagnet

Many-body transverse interactions in the quantum annealing of thep-spin ferromagnet

Chains of mean-field models

Ginzburg-Landau Polynomials and the Asymptotic Behavior of the Magnetization Near Critical and Tricritical Points

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