# lattice model with compact boson CFT fixed point

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The free compact boson at some radius $R\neq \sqrt{p/q}$ is probably the simplest example of an irrational CFT. Does this theory (or any other irrational CFT) arise as the thermodynamic limit of any 1+1d quantum lattice model with finite dimensional Hilbert space?

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A one-dimensional Luttinger liquid should be an example. Concrete lattice realizations include XXZ model ($H=-\sum_i (S_i^+ S_{i+1}^- + S_i^- S_{i+1}^+ + \Delta S_i^z S_{i+1}^z)$ where $|\Delta|<1$ or Hubbard model. For XXZ, the Luttinger parameter, which determines the radius $R$ and all conformal dimensions, continuously varies with $\Delta$.

answered May 4, 2016 by (550 points)

Thanks Meng, this is exactly what I was looking for (:

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Can you not just discretize the target space? I mean, the compact boson is just a free field $\phi$ with the identification $\phi \sim \phi + \beta$. You could have a lattice with $L$ sites $x$ and at every site $\phi(x)$ can take values $0, \beta/N, \ldots, \beta$. You are truncating the spectrum of the theory, obviously: you only probe $\sim N$ vertex operators with low enough winding numbers and throw away the rest. The Hamiltonian would just be the kinetic term $(\partial_\mu \phi)^2$, and of course you need to take the identification $\beta \sim 0$ into account when computing its value. A nice feature is that this construction is obviously invariant under shifts $\phi \to \phi + \text{const}$. answered Apr 24, 2016 by Very Anonymous User
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Quite a lot of the logCFT literature is concerned with such examples.  The spin chains studied by Saleur's group, the loop models studied by Pearce, the sandpiles studied by Ruelle.  JPhysA had an entire special issue in 2013 devoted to logCFT with reviews from these folks. answered Apr 27, 2016 by anonymous

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