The three-state Potts antiferromagnet on plane quadrangulations

Lv, Jian-Ping; Deng, Yong; Jacobsen, Jesper Lykke; Salas, José Valero

doi:10.1088/1751-8121/aad1fe

Cited by 11 publications

(22 citation statements)

References 101 publications

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“…This leaves room for a second order phase transition in a q = 5 antiferromagnet for which lattice candidates exist [8,9]. A line of fixed points with q = 3 and central charge 1 was also predicted for which a lattice realization has recently been found [10].A further, particularly remarkable feature of the scale invariant scattering formalism in two dimensions is that it extends to the problem of quenched disorder [11], i.e. to those "random" fixed points that had seemed out of reach for exact methods.…”

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confidence: 98%

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Critical lines in the pure and disordered O(N) model

Delfino¹,

Lamsen²

2019

J. Stat. Mech.

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We consider replicated O(N ) symmetry in two dimensions within the exact framework of scale invariant scattering theory and determine the lines of renormalization group fixed points in the limit of zero replicas corresponding to quenched disorder. A global pattern emerges in which the different critical lines are located within the same parameter space. Within the subspace corresponding to the pure case (no disorder) we show how the critical lines for non-intersecting loops (−2 ≤ N ≤ 2) are connected to the zero temperature critical line (N > 2) via the BKT line at N = 2. Disorder introduces two more parameters, one of which vanishes in the pure limit and is maximal for the solutions corresponding to Nishimori-like and zero temperature critical lines. Emergent superuniversality (i.e. N -independence) of some critical exponents in the disordered case and disorder driven renormalization group flows are also discussed.For the permutational symmetry S q , characteristic of the q-state Potts model, this exploration was performed in [6] and revealed new results. In particular, the maximal value of q allowing for a fixed point was found to be 1 5.56.., instead of the previously expected value 4. This leaves room for a second order phase transition in a q = 5 antiferromagnet for which lattice candidates exist [8,9]. A line of fixed points with q = 3 and central charge 1 was also predicted for which a lattice realization has recently been found [10].A further, particularly remarkable feature of the scale invariant scattering formalism in two dimensions is that it extends to the problem of quenched disorder [11], i.e. to those "random" fixed points that had seemed out of reach for exact methods. In particular, for the random bond Potts model it was then possible to see analytically for the first time the softening of the phase transition by disorder above q = 4 [11], a phenomenon expected on rigorous grounds [12] (see also [13]). Quite unexpected was instead the emergence of a mechanism allowing for the q-independence of the correlation length critical exponent ν, with the magnetization exponent β remaining q-dependent [11]. This phenomenon of partial superuniversality finally accounts for the evidence emerged from a variety of investigations [14,15,16,17,18,19,20,21,22] that ν does not show any appreciable deviation from the value 1 up to q = ∞.Recently [23] we have shown how the scale invariant scattering approach can be applied to the O(N ) vector model with quenched disorder and gives global and exact access to a pattern of fixed points that previously had been the object of perturbative (for weak disorder) [24] and numerical [25] studies. In this paper we give all the solutions of the fixed point equations (table 1 below) and discuss their physical properties. In particular, we show how the critical lines parametrized by N are located with respect to each other in the space of parameters. This is interesting already for the case without disorder ("pure"), for which we exhibit how the passage from the critical lines fo...

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confidence: 98%

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confidence: 99%

Critical lines in the pure and disordered O(N) model

Delfino¹,

Lamsen²

2019

J. Stat. Mech.

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“…2(b)]. As the qualitative behavior of the four lattices within each class turns out to be the same, we refrain from giving here all the details [28], and shall focus on one lattice of each type: Q(hextri) and Q(diced).…”

Section: Conjecture 1 For the 3-state Potts Af On A (Periodic) Plane mentioning

confidence: 99%

“…If the quadrangulation is of self-dual type, then we expect the number of ideal states [9,12] to be six: The system must choose which of the two sublattices to order, and in which of the three possible spin directions. It is therefore natural to expect (by using universality arguments) that, as for the square lattice, there will be a critical point at T = 0 characterized by a CFT with central charge c = 1 [28].…”

Section: Quadrangulations Of Self-dual Typementioning

confidence: 99%

Duality and the universality class of the three-state Potts antiferromagnet on plane quadrangulations

Deng

Jacobsen

et al. 2018

Phys. Rev. E

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We provide a criterion based on graph duality to predict whether the three-state Potts antiferromagnet on a plane quadrangulation has a zero- or finite-temperature critical point, and its universality class. The former case occurs for quadrangulations of self-dual type, and the zero-temperature critical point has central charge c=1. The latter case occurs for quadrangulations of non-self-dual type, and the critical point belongs to the universality class of the three-state Potts ferromagnet. We have tested this criterion against high-precision computations on four lattices of each type, with very good agreement. We have also found that the Wang-Swendsen-Kotecký algorithm has no critical slowing-down in the former case, and critical slowing-down in the latter.

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“…On the square lattice, it is critical at zero temperature, and disordered at positive temperatures [8][9][10][11]. On a set of planar lattices called quadrangulations the model either has a zero-temperature critical point, or it has three ordered coexisting phases, dependent on whether or not the quadrangulation is selfdual [12]. In view of this lattice-dependent behavior, AF Potts models have to be investigated case by case.…”

Section: Introductionmentioning

confidence: 99%

Three-state Potts model on the centered triangular lattice

Guo

Blöte

2020

Phys. Rev. E

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We study phase transitions of the Potts model on the centered-triangular lattice with two types of couplings, namely K between neighboring triangular sites, and J between the centered and the triangular sites. Results are obtained by means of a finite-size analysis based on numerical transfermatrix calculations and Monte Carlo simulations. Our investigation covers the whole (K, J) phase diagram, but we find that most of the interesting physics applies to the antiferromagnetic case K < 0, where the model is geometrically frustrated. In particular, we find that there are, for all finite J, two transitions when K is varied. Their critical properties are explored. In the limits J → ±∞ we find algebraic phases with infinite-order transitions to the ferromagnetic phase.

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The three-state Potts antiferromagnet on plane quadrangulations

Cited by 11 publications

References 101 publications

Critical lines in the pure and disordered O(N) model

Critical lines in the pure and disordered O(N) model

Duality and the universality class of the three-state Potts antiferromagnet on plane quadrangulations

Three-state Potts model on the centered triangular lattice

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