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 .
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.
Critical properties of the
Q-state Potts model for dimensions $3\ensuremath{\le}d\ensuremath{\le}6$ are calculated by means of Wilson's exact momentum-space renormalization-group equation. The scaling-field method of Golner and Riedel is used to approximate the functional differential equation by a set of 11 ordinary coupled differential equations. For $d=4\ensuremath{-}\ensuremath{\epsilon}$, lines of critical and tricritical Potts fixed points are found as functions of
Q that annihilate as
Q approaches a critical value ${Q}_{c}=\frac{2+{\ensuremath{\epsilon}}^{2}}{a+O({\ensuremath{\epsilon}}^{3})}$. For
Q>Qc, the Potts transition is first order. Along these fixed lines the critical and tricritical exponents (upper and lower sign, respectively) are to leading order: $\frac{1}{\ensuremath{\nu}}=2\ensuremath{-}\frac{1}{6}[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]$, $\frac{\ensuremath{\varphi}}{\ensuremath{\nu}}=\ensuremath{\mp}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}$, and $\ensuremath{\eta}=\frac{{[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]}^{2}}{216}+b\ensuremath{\delta}$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, $\ensuremath{\delta}=Q\ensuremath{-}2$, and $\ensuremath{\delta}\ensuremath{\le}{\ensuremath{\delta}}_{c}=\frac{{\ensuremath{\epsilon}}^{2}}{a}+O({\ensuremath{\epsilon}}^{3})$. While the form of the $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$ dependences is exact, the coefficients
a and
b cannot be obtained systematically by $\ensuremath{\epsilon}$ expansion, since the upper critical dimensionality of the Potts model is six when $Q\ensuremath{\ne}2$. In our truncation,
a=6.52 and
b=0.065. The results have been extended to dimensions $3.4\ensuremath{\le}d\ensuremath{\lesssim}4$ by solving the renormalization-group equations numerically. The percolation limit of the Potts model,
Q=1, is also investigated and the critical exponents ${\ensuremath{\nu}}^{P},{\ensuremath{\varphi}}^{P}$, and ${\ensuremath{\eta}}^{P}$ determined as functions of dimension for $3\ensuremath{\le}d\ensuremath{\le}6$.
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.
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.
Critical properties of the
q-state Potts and Potts-lattice-gas models are calculated by means of the Kadanoff variational renormalization group for dimensions 1.58, 2, and 2.32, and the Migdal approximation for arbitrary dimensions. Accurate results for the critical and tricritical exponents are obtained as well as for
qc, the largest value of
q for which the Potts transition is continuous. It is concluded unambiguously that the three-state Potts model in three dimensions undergoes a first-order phase transition.
Differential renormalization equation for the
q-state Potts model are proposed, and the critical behavior of the model near
q=qc discussed. The equations give rise to critical and tricritical fixed points which merge at
q=qc when a dilution field becomes marginal, to an essential singularity in the latent heat as a function of
q=qc, in accordance with the exact result of Baxter, and, for
q=qc, to a logarithm correction to the power-law behavior of the free energy as a function of $T\ensuremath{-}{T}_{c}$.