Exact Critical Point and Critical Exponents of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="normal">O</mml:mi><mml:mn /><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo><mml:mn /></mml:math>Models in Two Dimensions

Nienhuis, Bernard

doi:10.1103/physrevlett.49.1062

Cited by 972 publications

(960 citation statements)

References 27 publications

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“…Notice the global factor str 1 = 2n + m − 2n = m per closed loop, whether or not the loop is topologically nontrivial [the supertrace str here is in the vector representation V ′ ]. In general, the critical point of the model occurs at K c = 1/ 2 + √ 2 − m for −2 ≤ m ≤ 2 [56]. For m = 0, Z = 1, and at the critical value K c of K already known from the n = 0 case, this yields the theory of dilute polymers.…”

Section: Supersymmetric Modelsmentioning

confidence: 99%

Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions

2001

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We study two-dimensional nonlinear sigma models in which the target spaces are the coset supermanifolds U(n + m|n)/[U(1)×U(n + m − 1|n)] ∼ = CP n+m−1|n (projective superspaces) and OSp(2n + m|2n)/OSp(2n + m − 1|2n) ∼ = S 2n+m−1|2n (superspheres), n, m integers, −2 ≤ m ≤ 2; these quantum field theories live in Hilbert spaces with indefinite inner products. These theories possess non-trivial conformally-invariant renormalization-group fixed points, or in some cases, lines of fixed points. Some of the conformal fixed-point theories can also be obtained within LandauGinzburg theories. We obtain the complete spectra (with multiplicities) of exact conformal weights of states (or corresponding local operators) in the isolated fixed-point conformal field theories, and at one special point on each of the lines of fixed points. Although the conformal weights are rational, the conformal field theories are not, and (with one exception) do not contain the affine versions of their superalgebras in their chiral algebras. The method involves lattice models that represent the strong-coupling region, which can be mapped to loop models, and then to a Coulomb gas with modified boundary conditions. The results apply to percolation, dilute and dense polymers, and other statistical mechanics models, and also to the spin quantum Hall transition in noninteracting fermions with quenched disorder.

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Section: Supersymmetric Modelsmentioning

confidence: 99%

Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions

2001

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“…Note however that for the triangular lattice, we expect the regime q 0 (tri) < q ≤ 4 to be critical [54,52]. In particular the case q = 4 is equivalent to the three-coloring of the bonds of the hexagonal lattice [53], a critical model with c = 2 [55].…”

Section: Free Energy and Central Chargementioning

confidence: 99%

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models

2006

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We study the chromatic polynomial P G (q) for m × n square-and triangular-lattice strips of widths 2 ≤ m ≤ 8 with cyclic boundary conditions. This polynomial gives the zero-temperature limit of the partition function for the antiferromagnetic q-state Potts model defined on the lattice G. We show how to construct the transfer matrix in the Fortuin-Kasteleyn representation for such lattices and obtain the accumulation sets of chromatic zeros in the complex q-plane in the limit n → ∞. We find that the different phases that appear in this model can be characterized by a topological parameter. We also compute the bulk and surface free energies and the central charge.

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“…So we are left with configurations of coloured self-avoiding loops and dimers, edges in the loops being weighted with a factor β, and dimers with a factor γ/n, Summing over possible colourings reproduces the "topological" factor n per loop, and rescales the weight of marked (but uncoloured) dimers to γ. The case γ = 0 in the action (5.5), on a cubic lattice, corresponds to the loop-gas problem studied in the literature, in [10] and subsequent works.…”

Section: A Remark On the O(n) Loop-gas Modelmentioning

confidence: 99%

“…Integration over the spherical measure at each vertex leaves only with configurations of marked self-avoiding loops, weighted with a factor β per marked edge. Summing over loop colourings also produces the "topological" factor n per loop [10].…”

Section: A Remark On the O(n) Loop-gas Modelmentioning

confidence: 99%

“…The critical exponents of the n-vector model, at least on the range n ∈ [−2, 2], can be computed exactly, thanks to a mapping onto the solid-on-solid model [10] of a suitable choice of the weights, which, in the high-temperature expansion forbids loop crossing. We shall call this variant of the n-vector model the Nienhuis model, or Loop-gas model (as it is a 'hard-core lattice gas' in which the elementary objects are self-avoiding loops).…”

Section: Introductionmentioning

confidence: 99%

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Spanning Forests on Random Planar Lattices

Caracciolo

Sportiello

2009

J Stat Phys

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The generating function for spanning forests on a lattice is related to the q-state Potts model in a certain q → 0 limit, and extends the analogous notion for spanning trees, or dense self-avoiding branched polymers. Recent works have found a combinatorial perturbative equivalence also with the (quadratic action) O(n) model in the limit n → −1, the expansion parameter t counting the number of components in the forest. We give a random-matrix formulation of this model on the ensemble of degree-k random planar lattices. For k = 3, a correspondence is found with the Kostov solution of the loop-gas problem, which arise as a reformulation of the (logarithmic action) O(n) model, at n = −2. Then, we show how to perform an expansion around the t = 0 theory. In the thermodynamic limit, at any order in t we have a finite sum of finite-dimensional Cauchy integrals. The leading contribution comes from a peculiar class of terms, for which a resummation can be performed exactly.

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Exact Critical Point and Critical Exponents ofO(n)Models in Two Dimensions

Cited by 972 publications

References 27 publications

Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions

Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models

Spanning Forests on Random Planar Lattices

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