The case of a positive logarithmically diverging interaction (which results in attractive forces between the anyons [@Dennis2002]) was recently discussed in Ref. [@Hamma2009].

Error models and simulations {#app:simulation}
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Error models
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We model the interaction of the system with a thermal environment by coupling each spin to a bath which can introduce $\sigma_x$-errors [@footnote01] in the initial state $|\Psi_0 \rangle$, assumed to be a ground state of Eq. (\[H0\]). From a standard master equation approach in the weak coupling limit [@Davies1974; @Alicki2009], we derive a rate equation for the probabilities $p_m$ of the system to be in state $|\Psi_m\rangle = \prod_{i\in m} \sigma_{x, i} |\Psi_0\rangle$, where $\{m\}$ is the set of all possible patterns of $\sigma_x$-errors. This rate equation reads $$\label{rate equation}
\dot p_m = \sum_{i}\left[\gamma(-\omega_i(m)) p_{x_i(m)} - \gamma(\omega_i(m)) p_m\right],$$ where we have defined $x_i(m)$ to be the state $m$ with an additional $\sigma_x$-error applied to spin $i$, and $\omega_i(m) = \epsilon_m - \epsilon_{x_i(m)}$ is the energy difference between the states $m$ and $x_i(m)$. The time evolution of the probabilities $p_{m}$ determines the decay of the expectation values $\langle Z_{(\rm ec)} \rangle = \sum_m p_m \langle \Psi_m |Z_{(\rm ec)}| \Psi_m \rangle$.

The rates $\gamma(\omega)$ describe the transition probabilities between 