Lectures on the spin and loop \(O(n)\) models. (English) Zbl 1446.82019
Sidoravicius, Vladas (ed.), Sojourns in probability theory and statistical physics. I. Spin glasses and statistical mechanics, a festschrift for Charles M. Newman. Singapore: Springer; Shanghai: NYU Shanghai. Springer Proc. Math. Stat. 298, 246-320 (2019).
Summary: The classical spin \(O(n)\) model is a model on a d-dimensional lattice in which a vector on the \((n-1)\)-dimensional sphere is assigned to every lattice site and the vectors at adjacent sites interact ferromagnetically via their inner product. Special cases include the Ising model \((n=1)\), the XY model \((n=2)\) and the Heisenberg model \((n=3)\). We discuss questions of long-range order and decay of correlations in the spin \(O(n)\) model for different combinations of the lattice dimension d and the number of spin components \(n\).
The loop \(O(n)\) model is a model for a random configuration of disjoint loops. We discuss its properties on the hexagonal lattice. The model is parameterized by a loop weight \(n\geq 0\) and an edge weight \(x\geq 0\). Special cases include self-avoiding walk \((n=0)\), the Ising model \((n=1)\), critical percolation \((n=x=1)\), dimer model (\(n=1\), \(x=\infty\)), proper 4-coloring (\(n=2\), \(x=\infty\)), integer-valued \((n=2)\) and tree-valued (integer \(n \geq 3)\) Lipschitz functions and the hard hexagon model \((n=\infty)\). The object of study in the model is the typical structure of loops. We review the connection of the model with the spin \(O(n)\) model and discuss its conjectured phase diagram, emphasizing the many open problems remaining.
For the entire collection see [Zbl 1429.60003].
The loop \(O(n)\) model is a model for a random configuration of disjoint loops. We discuss its properties on the hexagonal lattice. The model is parameterized by a loop weight \(n\geq 0\) and an edge weight \(x\geq 0\). Special cases include self-avoiding walk \((n=0)\), the Ising model \((n=1)\), critical percolation \((n=x=1)\), dimer model (\(n=1\), \(x=\infty\)), proper 4-coloring (\(n=2\), \(x=\infty\)), integer-valued \((n=2)\) and tree-valued (integer \(n \geq 3)\) Lipschitz functions and the hard hexagon model \((n=\infty)\). The object of study in the model is the typical structure of loops. We review the connection of the model with the spin \(O(n)\) model and discuss its conjectured phase diagram, emphasizing the many open problems remaining.
For the entire collection see [Zbl 1429.60003].
MSC:
82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
82D40 | Statistical mechanics of magnetic materials |
82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |
82B43 | Percolation |
82B26 | Phase transitions (general) in equilibrium statistical mechanics |
82B27 | Critical phenomena in equilibrium statistical mechanics |
81R40 | Symmetry breaking in quantum theory |
82B30 | Statistical thermodynamics |
60J67 | Stochastic (Schramm-)Loewner evolution (SLE) |
Keywords:
spin \(\mathrm{O}(n)\) model; loop \(\mathrm{O}(n)\) model; Ising model; XY model; Heisenberg model; phase transitions; spontaneous magnetization; symmetry breaking; decay of correlations; Mermin-Wagner; Berezinskii-Kosterlitz-Thouless transition; reflection positivity; chessboard estimate; infra-red bound; Gaussian domination; graphical representation; conformal loop ensemble; Schramm-Loewner evolution; self-avoiding walk; dilute Potts model; Lipschitz functions; hard hexagon model; critical percolation on the triangular lattice; dimer model; proper 4-coloring of the triangular lattice; macroscopic loops; microscopic loopsReferences:
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