We propose exact expressions for the conformal anomaly and for three critical exponents of the tricritical O(n) loop model as a function of n in the range -2<or=n<or=3/2. These findings are based on an analogy with known relations between Potts and O(n) models and on an exact solution of a "tri-tricritical" Potts model described in the literature. We verify the exact expressions for the tricritical O(n) model by means of a finite-size scaling analysis based on numerical transfer-matrix calculations.
We propose exact expressions for the conformal anomaly and for three critical exponents of the tricritical O(n) loop model as a function of n in the range -2<or=n<or=3/2. These findings are based on an analogy with known relations between Potts and O(n) models and on an exact solution of a "tri-tricritical" Potts model described in the literature. We verify the exact expressions for the tricritical O(n) model by means of a finite-size scaling analysis based on numerical transfer-matrix calculations.
Using suitable Monte Carlo methods and finite-size scaling, we investigate critical and tricritical surface phenomena of two-dimensional Potts models. For the critical two- and three-state models, we determine a surface scaling dimension describing percolation properties of the so-called Potts clusters near the edges. On this basis, we propose an exact expression describing this exponent for the whole critical branch. For tricritical Potts models we find that varying the surface coupling constant or the surface magnetic field can induce a continuous phase transition. At bulk tricriticality and sufficiently strong surface couplings, spontaneous one-dimensional order occurs on the edges. We determine several critical exponents describing these edge transitions. On the basis of these results and conformal field theory, we conjecture exact expressions for these exponents.
Finite-size scaling and transfer-matrix techniques are used to determine the conformal anomaly and critical exponents of O(n) models on the square lattice. These calculations were performed on five branches of critical points parametrised by n. The results for two of the branches agree with the known universal properties of the O(n) model as derived for the honeycomb lattice. Finite-size convergence is poor for most of the third branch, so that in this case the classification in terms of known models is uncertain. The authors' data show that the fourth branch can be interpreted in terms of a combination of a low-temperature O(n) and a critical Ising model. Finally, the numerical results for the O(n) model on the remaining branch agree accurately with the known results for the low-temperature O(n+1) model. They prove this equivalence by means of an exact mapping.
The two-dimensional Q-state Potts model with real couplings has a first-order transition for Q>4. We study a loop-model realization in which Q is a continuous parameter. This model allows for the collision of a critical and a tricritical fixed point at Q=4, which then emerge as complex conformally invariant theories at Q>4, or even complex Q, for suitable complex coupling constants. All critical exponents can be obtained as analytic continuation of known exact results for Q≤4. We verify this scenario in detail for Q=5 using transfer-matrix computations.
We determine the backbone exponent X(b) of several critical and tricritical q-state Potts models in two dimensions. The critical systems include the bond percolation, the Ising, the q=2-sqrt[3], 3, and 4 state Potts, and the Baxter-Wu model, and the tricritical ones include the q=1 Potts model and the Blume-Capel model. For this purpose, we formulate several efficient Monte Carlo methods and sample the probability P2 of a pair of points connected via at least two independent paths. Finite-size-scaling analysis of P2 yields X(b) as 0.3566(2), 0.2696(3), 0.2105(3), and 0.127(4) for the critical q=2-sqrt[3], 1,2, 3, and 4 state Potts model, respectively. At tricriticality, we obtain X(b)=0.0520(3) and 0.0753(6) for the q=1 and 2 Potts model, respectively. For the critical q-->0 Potts model it is derived that X(b)=3/4. From a scaling argument, we find that, at tricriticality, X(b) reduces to the magnetic exponent, as confirmed by the numerical results.
Making complex CFTs real: The two-dimensional Potts model for Q>4 and complex Q
The two-dimensional Q-state Potts model with real couplings has a first-order transition for Q>4. Starting from a triangular-lattice Potts model with two- and three-spin interactions, we study an equivalent loop model in which Q is a continuous parameter. By a combination of analytical and numerical arguments, we show that this loop model allows for the collision of a critical and a tricritical fixed point at Q=4. These then emerge as a pair of complex conformally invariant theories at Q>4, or even complex Q, for suitable complex coupling constants. We conjecture that all conformal data (such as the central charge, critical exponents, and three-point structure constants) can be obtained by analytic continuation of known exact results for the loop model with Q≤4. This conjecture is checked, both for real Q>4 and for Q∈C, by extensive transfer-matrix computations and comparison to previous studies for Q=5.
We consider the 3-state Potts model in d ≥ 2 dimensions. For d less than the upper critical dimension d crit , the model has a critical and a tricritical fixed point. In d = 2 , these fixed points are described by minimal models, and so are exactly solvable. For d > 2 , however, strong coupling makes them difficult to study and there is no consensus on the value of d crit . We use the numerical conformal bootstrap to compute critical exponents of both the critical and tricritical fixed points for general d . In d = 2 our results match the expected values, and as we increase d we find that the critical exponents of each fixed point get closer until they merge near d crit ≲ 2.5 , which is an upper bound on d crit .