Critical spin chains and loop models with $PSU(n)$ symmetry

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Abstract
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) O ( n ) model (symmetry group O(n) O ( n ) ) and the Potts model (symmetry group S_Q S Q ). Both models make sense for n,Q∈ \mathbb{C} n , Q ∈ ℂ and not just n,Q∈ \mathbb{N} n , Q ∈ ℕ , and both give rise to a conformal field theory in the critical limit. Here, we study similar models based on the group PSU(n) P S 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 that exists for any n∈\mathbb{C} n ∈ ℂ and has a global PSU(n) P S U ( n ) symmetry. Its spectrum is similar to those of the O(n) O ( n ) and Potts CFTs, but a bit simpler. We conjecture that the O(n) O ( n ) CFT is a \mathbb{Z}_2 ℤ 2 orbifold of the PSU(n) P S U ( n ) CFT, where \mathbb{Z}_2 ℤ 2 acts as complex conjugation.
Funding Information
  • Agence Nationale de la Recherche (ANR-21-CE40-0003)
  • European Research Council

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