Based on the spectrum identified in our earlier work [arXiv:1809.02191], we numerically solve the bootstrap to determine four-point correlation functions of the geometrical connectivities in the $Q$-state Potts model. Crucial in our approach is the existence of "interchiral conformal blocks", which arise from the degeneracy of fields with conformal weight $h_{r,1}$, with $r\in\mathbb{N}^*$, and are related to the underlying presence of the "interchiral algebra" introduced in [arXiv:1207.6334]. We also find evidence for the existence of "renormalized" recursions, replacing those that follow from the degeneracy of the field $Φ_{12}^D$ in Liouville theory, and obtain the first few such recursions in closed form. This hints at the possibility of the full analytical determination of correlation functions in this model.
We revisit in this paper the problem of connectivity correlations in the Fortuin-Kasteleyn cluster representation of the two-dimensional $Q$-state Potts model conformal field theory. In a recent work [M. Picco, S. Ribault and R. Santachiara, SciPost Phys. 1, 009 (2016); arXiv:1607.07224], results for the four-point functions were obtained, based on the bootstrap approach, combined with simple conjectures for the spectra in the different fusion channels. In this paper, we test these conjectures using lattice algebraic considerations combined with extensive numerical studies of correlations on i…
We discuss the q-state Potts models for q<=4, in the scaling regimes close to their critical or tricritical points. Starting from the kink S-matrix elements proposed by Chim and Zamolodchikov, the bootstrap is closed for the scaling regions of all critical points, and for the tricritical points when 4>q>=2. We also note a curious appearance of the extended last line of Freudenthal's magic square in connection with the Potts models.
The spectrum of conformal weights for the CFT describing the two-dimensional critical $Q$-state Potts model (or its close cousin, the dense loop model) has been known for more than 30 years. However, the exact nature of the corresponding $\hbox{Vir}\otimes\overline{\hbox{Vir}}$ representations has remained unknown up to now. Here, we solve the problem for generic values of $Q$. This is achieved by a mixture of different techniques: a careful study of "Koo--Saleur generators" [arXiv:hep-th/9312156], combined with measurements of four-point amplitudes, on the numerical side, and OPEs and the fou…
In this paper, we apply the form factor bootstrap approach to branch point twist fields in the $q$-state Potts model for $q\leq 3$. For $q=3$ this is an integrable interacting quantum field theory with an internal discrete $\mathbb{Z}_3$ symmetry and therefore provides an ideal starting point for the investigation of the symmetry resolved entanglement entropies. However, more generally, for $q\leq 3$ the standard Rényi and entanglement entropies are also accessible through the bootstrap programme. In our work we present form factor solutions both for the standard branch point twist field with …
We study four-point functions of critical percolation in two dimensions, and more generally of the Potts model. We propose an exact ansatz for the spectrum: an infinite, discrete and non-diagonal combination of representations of the Virasoro algebra. Based on this ansatz, we compute four-point functions using a numerical conformal bootstrap approach. The results agree with Monte-Carlo computations of connectivities of random clusters.
The "bootstrap determination" of the geometrical correlation functions in the two-dimensional Potts model proposed in a paper [arXiv:1607.07224] was later shown in [arXiv:1809.02191] to be incorrect, the actual spectrum of the model being considerably more complex than initially conjectured. We provide in this paper a geometrical interpretation of the four-point functions built in [arXiv:1607.07224], and explain why the results obtained by these authors, albeit incorrect, appeared so close to those of their numerical simulations of the Potts model. Our strategy is based on a cluster expansion …
We have considered clusters of like spin in the Q-Potts model, the spin Potts clusters. Using Monte Carlo simulations, we studied these clusters on a square lattice with periodic boundary conditions for values of Q in [1,4]. We continue the work initiated with Delfino and Viti (2013) by measuring the universal finite size corrections of the two-point connectivity. The numerical data are perfectly compatible with the 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 ene…
We build the Z$_{3}$ invariants fusion rules associated to the (D$_{4}$,A$_{6}$) conformal algebra. This algebra is known to describe the tri-critical Potts model. The 4-pt correlation functions of critical fields are developed in the bootstrap approach, and in the other hand, they are written in term of integral representation of the conformal blocks. By comparing both the expressions, one can determine the structure constantes of the operator algebra.
We consider the 3-state Potts model in $d\geq2$ dimensions. For $d$ less than the upper critical dimension $d_\text{crit}$, 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 $d_\text{crit}$. 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…
Using derivatives of primary fields (null or not) with respect to the conformal dimension, we build infinite families of non-trivial logarithmic representations of the conformal algebra at generic central charge, with Jordan blocks of dimension $2$ or $3$. Each representation comes with one free parameter, which takes fixed values under assumptions on the existence of degenerate fields. This parameter can be viewed as a simpler, normalization-independent redefinition of the logarithmic coupling. In the example of the vacuum module at central charge zero, this parameter characterizes a Jordan b…
Under the assumption that degenerate fields exist, diagonal CFTs such as Liouville theory can be solved analytically using the conformal bootstrap method. Here we generalize this approach to non-diagonal CFTs, i.e. CFTs whose primary fields have nonzero conformal spins. Assuming generic values of the central charge, we find that the non-diagonal sector of the spectrum must be parametrized by two integer numbers. We then derive and solve the equations that determine how three- and four-point structure constants depend on these numbers. In order to test these results, we numerically check crossi…
We explore the space of consistent three-particle couplings in $\mathbb Z_2$-symmetric two-dimensional QFTs using two first-principles approaches. Our first approach relies solely on unitarity, analyticity and crossing symmetry of the two-to-two scattering amplitudes and extends the techniques of [arXiv:1607.06110] to a multi-amplitude setup. Our second approach is based on placing QFTs in AdS to get upper bounds on couplings with the numerical conformal bootstrap, and is a multi-correlator version of [arXiv:1607.06109]. The space of allowed couplings that we carve out is rich in features, som…
In generic conformal field theories with $W_3$ symmetry, we identify a primary field $σ$ with rational Kac indices, which produces the full $\mathbb{Z}_3$ charged and neutral sectors by the fusion processes $σ\times σ$ and $σ\times σ^*$, respectively. In this sense, this field generalises the $\mathbb{Z}_3$ fundamental spin field of the three-state Potts model. Among the degenerate fields produced by these fusions, we single out a `parafermion' field $ψ$ and an `energy' field $\varepsilon$. In analogy with the Virasoro case, the exact curves for conformal dimensions $(h_σ,h_ψ)$ and $(h_σ,h_\va…
It has long been understood that non-trivial Conformal Field Theories (CFTs) with vanishing central charge ($c=0$) are logarithmic. So far however, the structure of the identity module -- the (left and right) Virasoro descendants of the identity field -- had not been elucidated beyond the stress-energy tensor $T$ and its logarithmic partner $t$ (the solution of the "$c\to 0$ catastrophe"). In this paper, we determine this structure together with the associated OPE of primary fields up to level $h=\bar{h}=2$ for polymers and percolation CFTs. This is done by taking the $c\to 0$ limit of $O(n)$ …
We study O(n)-symmetric two-dimensional conformal field theories (CFTs) for a continuous range of n below two. These CFTs describe the fixed point behavior of self-avoiding loops. There is a pair of known fixed points connected by an RG flow. When n is equal to two, which corresponds to the Kosterlitz-Thouless critical theory, the fixed points collide. We find that for n generic these CFTs are logarithmic and contain negative norm states; in particular, the O(n) currents belong to a staggered logarithmic multiplet. Using a conformal bootstrap approach we trace how the negative norm states deco…
