Critical spin chains and loop models with U (n) symmetry

Starting with the Ising model, statistical models with global symmetries provide fruitful approaches to interesting physical systems, for example percolation or polymers. These include the O (n) model (symmetry group O (n) ) and the Potts model (symmetry group SQ). Both models make sense for n, Q C and not just n, Q N, and both give rise to a conformal field theory in the critical limit. Here, we study similar models based on the unitary group U (n). We focus on the two-dimensional case, where the models can be described either as gases of non-intersecting orientable loops, or as alternating spin chains. This allows us to determine their spectra either by computing a twisted torus partition function, or by studying representations of the walled Brauer algebra. In the critical limit, our models give rise to a CFT with global U (n) symmetry, which exists for any n. Its spectrum is similar to those of the O (n) and Potts CFTs, but a bit simpler. We conjecture that the O (n) CFT is a Z₂ orbifold of the U (n) CFT, where Z₂ acts as complex conjugation.