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arXiv:2111.03846 · Journal of Statistical Mechanics Theory and Experiment

On the CFT describing the spin clusters in 2 d Potts model

Marco Picco, Raoul Santachiara

Published Feb 1, 2022• 6 citations

Abstract

Abstract We have considered clusters of like spins in the Q -Potts model, the spin Potts clusters (S clusters). Using Monte Carlo simulations, we studied these clusters on a square lattice with periodic boundary conditions for values of Q ∈ [1, 4]. We continue the work initiated by Delfino et al (2013 J. Stat. Mech. P11011) by measuring the universal finite size corrections of the two-point connectivity. The numerical data are perfectly compatible with the conformal field theory (CFT) prediction, thus supporting the existence of a consistent CFT, still unknown, describing the connectivity Potts spin clusters. We provided in particular new insights on the energy field of such theory. For Q = 2, we found a good agreement with the prediction that the Ising spin clusters behave as the Fortuin–Kasteleyn ones at the tri-critical point of the dilute one-Potts model. We show that the structure constants are likely to be given by the imaginary Liouville structure constants, consistently with the results by Delfino et al (2013 J. Stat. Mech. P11011); Ang and Sun (2021 arXiv:2107.01788). For Q ≠ 2 instead, the structure constants we measure do not correspond to any known bootstrap solutions. The validity of our analysis is backed up by the measures of the spin Potts cluster wrapping probability for Q = 3. We evaluate the main critical exponents and the correction to the scaling. A new exact and compact expression for the torus one-point of the Q -Potts energy field is also given.

DOI: 10.1088/1742-5468/ac4c3d

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Abstract We have considered clusters of like spins in the Q -Potts model, the spin Potts clusters (S clusters). Using Monte Carlo simulations, we studied these clusters on a square lattice with periodic boundary conditions for values of Q ∈ [1, 4]. We continue the work initiated by Delfino et al (2013 J. Stat. Mech. P11011) by measuring the universal finite size corrections of the two-point connectivity. The numerical data are perfectly compatible with the conformal field theory (CFT) prediction, thus supporting the existence of a consistent CFT, still unknown, describing the connectivity Potts spin clusters. We provided in particular new insights on the energy field of such theory. For Q = 2, we found a good agreement with the prediction that the Ising spin clusters behave as the Fortuin–Kasteleyn ones at the tri-critical point of the dilute one-Potts model. We show that the structure constants are likely to be given by the imaginary Liouville structure constants, consistently with the results by Delfino et al (2013 J. Stat. Mech. P11011); Ang and Sun (2021 arXiv:2107.01788). For Q ≠ 2 instead, the structure constants we measure do not correspond to any known bootstrap solutions. The validity of our analysis is backed up by the measures of the spin Potts cluster wrapping probability for Q = 3. We evaluate the main critical exponents and the correction to the scaling. A new exact and compact expression for the torus one-point of the Q -Potts energy field is also given.

For the two-dimensional

Q

-state Potts model at criticality, we consider Fortuin-Kasteleyn and spin clusters and study the average number ${N}_{\ensuremath{\Gamma}}$ of clusters that intersect a given contour $\ensuremath{\Gamma}$. To leading order, ${N}_{\ensuremath{\Gamma}}$ is proportional to the length of the curve. Additionally, however, there occur logarithmic contributions related to the corners of $\ensuremath{\Gamma}$. These are found to be universal and their size can be calculated employing techniques from conformal field theory. For the Fortuin-Kasteleyn clusters relevant to the thermal phase transition, we find agreement with these predictions from large-scale numerical simulations. For the spin clusters, on the other hand, the cluster numbers are not found to be consistent with the values obtained by analytic continuation, as conventionally assumed.

Abstract In a recent paper, we considered the effects of the torus lattice topology on the two-point connectivity of Q- Potts clusters. These effects are universal and probe non-trivial structure constants of the theory. We complete here this work by considering the torus corrections to the three- and four-point connectivities. These corrections, which depend on the scale invariant ratios of the triangle and quadrilateral formed by the three and four given points, test other non-trivial structure constants. We also present results of Monte Carlo simulations in good agreement with our predictions.

We study the three-point correlation function of the backbone in the two-dimensional Q-state Potts model using the Fortuin-Kasteleyn (FK) representation. The backbone is defined as the biconnected skeleton of an FK cluster after removing all dangling ends and bridges. To circumvent the severe critical slowing down in direct Potts simulations for large Q, we employ large-scale Monte Carlo simulations of the O(n) loop model on the hexagonal lattice, which is regarded to correspond to the Potts model with Q=n^{2}. Using a highly efficient cluster algorithm, we compute the universal three-point amplitude ratios for the backbone (R_{BB}) and FK clusters (R_{FK}). Our computed R_{FK} exhibits excellent agreement with exact conformal field theory predictions, validating the reliability of our numerical approach. In the critical regime, we find that R_{BB} is systematically larger than R_{FK}. Conversely, along the tricritical branch, R_{BB} and R_{FK} coincide within numerical accuracy, strongly suggesting that R_{BB}=R_{FK} holds throughout this regime. This finding mirrors the known equality of the backbone and FK cluster fractal dimensions at tricriticality, jointly indicating that both structures share the same geometric universality.

Abstract We investigate the operator content of the Q -state Potts model in arbitrary dimension, using the representation theory of the symmetric group. In particular we construct all possible tensors acting on N spins, corresponding to given symmetries under S Q and S N , in terms of representations involving any Young diagram. These operators transform non-trivially under the group of spatial rotations, with a definite conformal spin. The two-point correlation functions are then computed, and their physical interpretation is given in terms of Fortuin–Kasteleyn clusters propagating between two neighbourhoods of each N spins. In two dimensions, we obtain analytically the critical exponent corresponding to each operator. In the simplest and physically most relevant cases, we confirm the values of the critical exponent and the conformal spin by numerical measurements, using both Monte Carlo simulations and transfer matrix diagonalisations. Our classification partially provides the structure of Jordan cells of the dilatation operator in arbitrary dimensions, which in turn gives rise to logarithmic correlation functions.

We address the geometrical critical behavior of the two-dimensional Q -state Potts model in terms of the spin clusters (i.e. connected domains where the spin takes a constant value). These clusters are different from the usual Fortuin–Kasteleyn clusters, and are separated by domain walls that can cross and branch. We develop a transfer matrix technique enabling the formulation and numerical study of spin clusters even when Q is not an integer. We further identify geometrically the crossing events which give rise to conformal correlation functions. This leads to an infinite series of fundamental critical exponents , valid for 0 ⩽ Q ⩽ 4, that describe the insertion of ℓ 1 thin and ℓ 2 thick domain walls.

We present a Monte Carlo study of the backbone and the shortest-path exponents of the two-dimensional

Q

-state Potts model in the Fortuin-Kasteleyn bond representation. We first use cluster algorithms to simulate the critical Potts model on the square lattice and obtain the backbone exponents

{d}_{B}=1.732\phantom{\rule{0.16em}{0ex}}0(3)

and 1.794(2) for

Q=2,3

, respectively. However, for large

Q

, the study suffers from serious critical slowing down and slowly converging finite-size corrections. To overcome these difficulties, we consider the

\mathrm{O}(n)

loop model on the honeycomb lattice in the densely packed phase, which is regarded to correspond to the critical Potts model with

Q={n}^{2}

. With a highly efficient cluster algorithm, we determine from domains enclosed by the loops

{d}_{B}=1.643\phantom{\rule{0.16em}{0ex}}39(5),1.732\phantom{\rule{0.16em}{0ex}}27(8),1.793\phantom{\rule{0.16em}{0ex}}8(3),1.838\phantom{\rule{0.16em}{0ex}}4(5),1.875\phantom{\rule{0.16em}{0ex}}3(6)

for

Q=1,2,3,2+\sqrt{3},4

, respectively, and

{d}_{\mathrm{min}}=1.094\phantom{\rule{0.16em}{0ex}}5(2),1.067\phantom{\rule{0.16em}{0ex}}5(3),1.047\phantom{\rule{0.16em}{0ex}}5(3),1.032\phantom{\rule{0.16em}{0ex}}2(4)

for

Q=2,3,2+\sqrt{3},4

, respectively. Our estimates significantly improve over the existing results for both

{d}_{B}

and

{d}_{\mathrm{min}}

. Finally, by studying finite-size corrections in backbone-related quantities, we conjecture an exact formula as a function of

n

for the leading correction exponent.