Critical spin chains and loop models with $PSU(n)$ 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 $S_Q$). Both models make sense for $n,...
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| Format: | Article |
| Language: | English |
| Published: |
SciPost
2025-01-01
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| Series: | SciPost Physics |
| Online Access: | https://scipost.org/SciPostPhys.18.1.033 |
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| Summary: | 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 $S_Q$). Both models make sense for $n,Q∈ \mathbb{C}$ and not just $n,Q∈ \mathbb{N}$, and both give rise to a conformal field theory in the critical limit. Here, we study similar models based on the group $PSU(n)$. 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 $n∈\mathbb{C}$ and has a global $PSU(n)$ symmetry. 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 $\mathbb{Z}_2$ orbifold of the $PSU(n)$ CFT, where $\mathbb{Z}_2$ acts as complex conjugation. |
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| ISSN: | 2542-4653 |