Abstract |
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Using the planar Temperley-Lieb algebra, critical bond percolation on the square lattice can be reformulated as a loop model. In this form, it is incorporated as ${\cal LM}(2,3)$ in the Yang-Baxter integrable family of logarithmic minimal models ${\cal LM}(p,p')$. We consider this model of percolation in the presence of boundaries and with periodic boundary conditions. Inspired by Kuniba, Sakai and Suzuki, we rewrite the recently obtained infinite $Y$-system of functional equations. In this way, we obtain nonlinear integral equations in the form of a closed finite set of TBA equations described by a $D_3$ Dynkin diagram. Following the methods of Kl\"umper and Pearce, we solve the TBA equations for the conformal finite-size corrections. For the ground states of the standard modules on the strip, these agree with the known central charge $c=0$ and conformal weights $\Delta_{1,s}$ for $s\in {\mathbb Z_{\ge 1}}$ with $\Delta_{r,s}=\big((3r-2s)^2-1\big)/24$. For the periodic case, the finite-size corrections agree with the conformal weights $\Delta_{0,s}$, $\Delta_{1,s}$ with $s\in\frac{1}{2}{\mathbb Z_{\ge 0}}$. These are obtained analytically using Rogers dilogarithm identities. We incorporate all finite excitations by formulating empirical selection rules for the patterns of zeros of all the eigenvalues of the standard modules. We thus obtain the conformal partition functions on the cylinder and the modular invariant partition function (MIPF) on the torus. By applying $q$-binomial and $q$-Narayana identities, it is shown that our refined finitized characters on the strip agree with those of Pearce, Rasmussen and Zuber. For percolation on the torus, the MIPF is a non-diagonal sesquilinear form in affine $u(1)$ characters given by the $u(1)$ partition function $Z_{2,3}(q)=Z_{2,3}^{\text{Circ}}(q)$. The $u(1)$ operator content is ${\cal N}_{\Delta,\bar\Delta}=1$ for $\Delta=\bar\Delta=-\frac{1}{24}, \frac{35}{24}$ and ${\cal N}_{\Delta,\bar\Delta}=2$ for $\Delta=\bar\Delta=\frac{1}{8}, \frac{1}{3}, \frac{5}{8}$ and $(\Delta,\bar\Delta)=(0,1), (1,0)$. This result is compatible with the general conjecture of Pearce and Rasmussen, namely $Z_{p,p'}(q)=Z^{\text{Proj}}_{p,p'}(q)+n_{p,p'} Z^{\text{Min}}_{p,p'}(q)$ with $n_{p,p'}\in {\mathbb Z}$, where the minimal partition function is $Z^{\text{Min}}_{2,3}(q)=1$ and the lattice derivation fixes $n_{2,3}=-1$. |