Abstract
We study O(n)-symmetric two-dimensional conformal field theories (CFTs) for a continuous range of n below two. These CFTs describe the fixed point behavior of self-avoiding loops. There is a pair of known fixed points connected by an RG flow. When n is equal to two, which corresponds to the Kosterlitz-Thouless critical theory, the fixed points collide. We find that for n generic these CFTs are logarithmic and contain negative norm states; in particular, the O(n) currents belong to a staggered logarithmic multiplet. Using a conformal bootstrap approach we trace how the negative norm states decouple at n = 2, restoring unitarity. The IR fixed point possesses a local relevant operator, singlet under all known global symmetries of the CFT, and, nevertheless, it can be reached by an RG flow without tuning. Besides, we observe logarithmic correlators in the closely related Potts model.
Highlights
The critical O(n) model is one of the best studied examples of fixed points both in condensed matter and high energy physics, and yet it keeps supplying us with new ideas
We study O(n)-symmetric two-dimensional conformal field theories (CFTs) for a continuous range of n below two
We find that for n generic these CFTs are logarithmic and contain negative norm states; in particular, the O(n) currents belong to a staggered logarithmic multiplet
Summary
The critical O(n) model is one of the best studied examples of fixed points both in condensed matter and high energy physics, and yet it keeps supplying us with new ideas. In ordinary CFTs, given the scaling dimension and spin of an operator, two-point correlation functions are uniquely fixed by conformal invariance; as we already mentioned, the O(n) model is a logarithmic CFT (logCFT). Part of our motivation for studying the n 2 critical O(n) CFT is that for n > 2 this CFT becomes complex and, as explained in [1, 2], controls the walking RG behavior of a unitary massive theory for integer n 2 This type of RG flows is of interest for particle physics because they provide a natural way to generate a hierarchy of scales, and at the same time they control weakly-first-order phase transitions [31] in certain condensed matter systems. This is a result of highly non-trivial cancellations between operators
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