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arXiv:2606.18125

Making complex CFTs real: The two-dimensional Potts model for $Q>4$ and complex $Q$

Jesper Lykke Jacobsen, Kay Jörg Wiese

Published Jun 16, 2026• 0 citations
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Abstract

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 $Q \le 4$. This conjecture is checked, both for real $Q>4$ and for $Q \in \mathbb{C}$, by extensive transfer-matrix computations and comparison to previous studies for $Q=5$.

DOI
10.48550/arxiv.2606.18125

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Making complex CFTs real: The two-dimensional Potts model for Q>4Q>4Q>4 and complex QQQ
The two-dimensional QQQ-state Potts model with real couplings has a first-order transition for Q>4Q>4Q>4. Starting from a triangular-lattice Potts model with two- and three-spin interactions, we study an equivalent loop model in which QQQ 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=4Q=4Q=4. These then emerge as a pair of complex conformally invariant theories at Q>4Q>4Q>4, or even complex QQQ, 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≤4Q \le 4Q≤4. This conjecture is checked, both for real Q>4Q>4Q>4 and for Q∈CQ \in \mathbb{C}Q∈C, by extensive transfer-matrix computations and comparison to previous studies for Q=5Q=5Q=5.
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 study the famous example of weakly first-order phase transitions in the (1+1)\left(1+1\right)(1+1)-dimensional quantum QQQ-state Potts model with Q>4Q>4Q>4. We numerically show that these weakly first-order transitions have approximately conformal invariance. Specifically, we find that entanglement entropy on considerably large system sizes fits perfectly with the universal scaling law of this quantity in conformal field theories (CFTs). This supports that the weakly first-order transitions are proximate to complex fixed points, which are described by recent conjectured complex CFTs. Moreover, the central charge extracted from this fitting is close to the real part of the complex central charge of these complex CFTs. We also study the conformal towers and the drifting behaviors of these conformal data (e.g., central charge and scaling dimensions).
The presence of nearby conformal field theories (CFTs) hidden in the complex plane of the tuning parameter was recently proposed as an elegant explanation for the ubiquity of "weakly first-order" transitions in condensed matter and high-energy systems. In this work, we perform an exact microscopic study of such a complex CFT (CCFT) in the two-dimensional O(n) loop model. The well-known absence of symmetry-breaking of the O(n>2) model is understood as arising from the displacement of the nontrivial fixed points into the complex temperature plane. Thanks to a numerical finite-size study of the transfer matrix, we confirm the presence of a CCFT in the complex plane and extract the real and imaginary parts of the central charge and scaling dimensions. By comparing those with the analytic continuation of predictions from Coulomb gas techniques, we determine the range of validity of the analytic continuation to extend up to n_{g}≈12.34, beyond which the CCFT gives way to a gapped state. Finally, we propose a beta function which reproduces the main features of the phase diagram and which suggests an interpretation of the CCFT as a liquid-gas critical point at the end of a first-order transition line.
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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.
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 .