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Quantum brownian motion on a triangular lattice and c=2 boundary conformal field theory

We study a single particle diffusing on a triangular lattice and interacting with a heat bath, using boundary conformal field theory (CFT) and exact integrability techniques. We derive a correspondence between the phase diagram of this problem and that recently obtained for the 2 dimensional 3-state Potts model with a boundary. Exact results are obtained on phases with intermediate mobilities. These correspond to non-trivial boundary states in a conformal field theory with 2 free bosons which we explicitly construct for the first time. These conformally invariant boundary conditions are not simply products of Dirichlet and Neumann ones and unlike those trivial boundary conditions, are not invariant under a Heisenberg algebra.