Abstract
We study complex CFTs describing fixed points of the two-dimensional QQ-state Potts model with Q≻ 4Q>4. Their existence is closely related to the weak first-order phase transition and the "walking" renormalization group (RG) behavior present in the real Potts model at QQ>4. The Potts model, apart from its own significance, serves as an ideal playground for testing this very general relation. Cluster formulation provides nonperturbative definition for a continuous range of parameter QQ, while Coulomb gas description and connection to minimal models provide some conformal data of the complex CFTs. We use one and two-loop conformal perturbation theory around complex CFTs to compute various properties of the real walking RG flow. These properties, such as drifting scaling dimensions, appear to be common features of the QFTs with walking RG flows, and can serve as a smoking gun for detecting walking in Monte Carlo simulations.The complex CFTs discussed in this work are perfectly well defined, and can in principle be seen in Monte Carlo simulations with complexified coupling constants. In particular, we predict a pair of S_5S5-symmetric complex CFTs with central charges c\approx 1.138 \pm 0.021 ic≈1.138±0.021i describing the fixed points of a 5-state dilute Potts model with complexified temperature and vacancy fugacity.
Highlights
The term ‘walking’ refers to renormalization group (RG) flows described by the beta-function β(λ) = − y − λ2, (1.1)with y > 0 a small fixed parameter
The term ‘walking’ refers to RG flows described by the beta-function β(λ) = − y − λ2, (1.1)
One example are 4d and 3d gauge theories, where the walking mechanism is realized, conjecturally, below the lower end of the conformal window. Another example is the Q-state Potts model which has a conformal phase at Q Qc(d), and the walking mechanism governs the physics of a weakly first-order transition just above Qc
Summary
Especially in d = 2 where Qc = 4, allows to firmly establish walking in the Potts model Another goal of [1] was to highlight the concept of ‘complex CFTs’, an unusual class of conformal field theories which describe fixed points of RG flow (1.1) occurring at complex coupling λ = ±i y. For integer Q = 2, 3, 4 the CFTs describing the tricritical and critical Potts model are unitary and exactly solvable (for Q = 2, 3 these are unitary minimal models, while for Q = 4 it’s an orbifold of compactified free boson, see appendix C).
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