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arXiv:2402.10732 · Physical Review Letters

Lattice Realization of Complex Conformal Field Theories: Two-Dimensional Potts Model with Q > 4 States

Jesper Lykke Jacobsen, Kay Jörg Wiese

Published Aug 16, 2024• 14 citations
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Abstract

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.

DOI
10.1103/physrevlett.133.077101

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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.
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 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.
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).
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 .
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 use the density matrix renormalization group (DMRG) to perform a detailed study of the critical properties of the two-dimensional QQQ state Potts model, including the magnetization and energy-density profiles, bulk and surface critical exponents, and the Casimir amplitudes. We apply symmetry breaking boundary conditions to a $L\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ strip and diagonalize the corresponding transfer matrix for a series of moderately large systems (L< 64)(L<~64)(L< 64) by the DMRG method. The numerically very accurate finite lattice results are then extrapolated by efficient sequence extrapolation techniques. The density profiles and the Casimir amplitudes at the critical point are found to follow precisely the conformal predictions for Q=2Q=2Q=2 and 333. Similarly, the bulk and surface critical exponents of the models are in very good agreement with the conformal and exact values: their accuracy has reached or even exceeded the accuracy of other available numerical methods. For the Q=4Q=4Q=4 model both the profiles and the critical exponents show strong logarithmic corrections, which are also studied.