On Ising's model of ferromagnetism

Peierls, Rudolf

doi:10.1017/s0305004100019174

Cited by 722 publications

(283 citation statements)

References 7 publications

Supporting

Mentioning

275

Contrasting

Unclassified

Order By: Relevance

“…Figure 7 gives the exact internal energy of the original Ising model on a finite lattice with periodic boundary conditions [39,40] and that of the dual model transcribed to the original one using Eq. (38). In the figure, also the data points obtained using the plaquette update are included and seen to indeed coincide with the dual curve.…”

Section: Monte Carlo Studymentioning

confidence: 65%

See 1 more Smart Citation

Geometrical vs. Fortuin–Kasteleyn clusters in the two-dimensional q-state Potts model

Janke

Schakel

2004

Nuclear Physics B

View full text Add to dashboard Cite

The tricritical behavior of the two-dimensional q-state Potts model with vacancies for 0 ≤ q ≤ 4 is argued to be encoded in the fractal structure of the geometrical spin clusters of the pure model. The known close connection between the critical properties of the pure model and the tricritical properties of the diluted model is shown to be reflected in an intimate relation between Fortuin-Kasteleyn and geometrical clusters: The same transformation mapping the two critical regimes onto each other also maps the two cluster types onto each other. The map conserves the central charge, so that both cluster types are in the same universality class. The geometrical picture is supported by a Monte Carlo simulation of the high-temperature representation of the Ising model (q = 2) in which closed graph configurations are generated by means of a Metropolis update algorithm involving single plaquettes.

show abstract

Section: Monte Carlo Studymentioning

confidence: 65%

“…By the well-known Kramers-Wannier duality [37], the HT graphs form Peierls domain walls [38] separating spin clusters of opposite orientation on the dual lattice. Each bond in a HT graph intersects a nearest neighbor pair on the dual lattice of unlike spins perpendicular to it.…”

Section: Monte Carlo Studymentioning

confidence: 99%

Geometrical vs. Fortuin–Kasteleyn clusters in the two-dimensional q-state Potts model

Janke

Schakel

2004

Nuclear Physics B

View full text Add to dashboard Cite

show abstract

“…Thermal stability in classical systems was first understood by the discovery of the Peierls argument (Peierls, 1936). It shows us that in statistical mechanics stability increases with dimensionality.…”

Section: Thermal Stability In High Dimensionsmentioning

confidence: 99%

“…The Peierls argument (Peierls, 1936), later refined by Griffiths (1964), shows that the critical phenomena of the Ising model are dependent on its dimensionality. For a modern overview of the Peierls argument, and other important topics relating to the Ising model, we refer the interested reader to Huang (1987) and McCoy and Wu (2014).…”

Section: A Stability In Classical Modelsmentioning

confidence: 99%

Quantum memories at finite temperature

et al. 2016

View full text Add to dashboard Cite

To use quantum systems for technological applications one first needs to preserve their coherence for macroscopic time scales, even at finite temperature. Quantum error correction has made it possible to actively correct errors that affect a quantum memory. An attractive scenario is the construction of passive storage of quantum information with minimal active support. Indeed, passive protection is the basis of robust and scalable classical technology, physically realized in the form of the transistor and the ferromagnetic hard disk. The discovery of an analogous quantum system is a challenging open problem, plagued with a variety of no-go theorems. Several approaches have been devised to overcome these theorems by taking advantage of their loopholes. The state-of-the-art developments in this field are reviewed in an informative and pedagogical way. The main principles of self-correcting quantum memories are given and several milestone examples from the literature of two-, three-and higher-dimensional quantum memories are analyzed.

show abstract

“…The method developed in (12) is a precursor of a now standard argument attributed to Simon (1980) and Lieb (1980) and is usually expressed as: finite susceptibility implies exponentially decaying correlations. In (13) Hammersley proved an upper bound for p c in terms of the boundary sizes of neighbourhoods of the origin, and he deduced by graphical duality that p c <1 for oriented and unoriented percolation on the square grid; this is the percolation equivalent of the Peierls argument for the Ising model (Peierls 1936). This general route to showing the existence of a phase transition is now standard for many models.…”

Section: Percolationmentioning

confidence: 97%

John Michael Hammersley. 21 March 1920 — 2 May 2004

Grimmett

Welsh²

2007

Biogr. Mems Fell. R. Soc.

View full text Add to dashboard Cite

John Hammersley was a pioneer among mathematicians, who defied classification as pure or applied; when introduced to guests at Trinity College, Oxford, he would say he did ‘difficult sums’. He believed passionately in the importance of mathematics with strong links to real–life situations and in a system of mathematical education in which the solution of problems takes precedence over the generation of theory. He will be remembered for his work on percolation theory, subadditive stochastic processes, self–avoiding walks and Monte Carlo methods, and, by those who knew him, for his intellectual integrity and his ability to inspire and to challenge. Quite apart from his extensive research achievements, for which he earned a reputation as an outstanding problem–solver, he was a leader in the movement of the 1950s and 1960s to rethink the content of school mathematics syllabuses.

show abstract

On Ising's model of ferromagnetism

Cited by 722 publications

References 7 publications

Geometrical vs. Fortuin–Kasteleyn clusters in the two-dimensional q-state Potts model

Geometrical vs. Fortuin–Kasteleyn clusters in the two-dimensional q-state Potts model

Quantum memories at finite temperature

John Michael Hammersley. 21 March 1920 — 2 May 2004

Contact Info

Product

Resources

About