====== $O(2)$ model ======

{{tag>3d}}

The $O(2)$ model is a well studied special case of the [[on_model]]. It also goes by the name "XY model".

===== Operator content =====

Three of the leading operators in the spectrum: $\phi$ (an $O(2)$ vector), $s$ (a singlet), and $t$ (a symmetric rank-2 tensor i.e. a charge 2 operator) have been determined precisely and rigorously. The scaling dimensions are
\[
\Delta_\phi = 0.519088(\mathbf{22}),~\Delta_s = 1.51136(\mathbf{22}),~\Delta_t = 1.23629(\mathbf{11}).
\]
Four of OPE coefficients between these leading operators have also been determined, albeit not rigorously:
\[
\lambda_{\phi\phi s} = 0.687126(27),~\lambda_{s s s} = 0.830914(32),~\lambda_{t t s} = 1.25213(14),~\lambda_{\phi\phi t} = 1.213408(65).
\]

Additionally, the flavor current and stress tensor central charges have been determined nonrigorously:

\[
C_J / C_J^{\text {free}}=0.904395(28),~C_T / C_T^{\text {free}}=0.944056(15).
\] 

Finally, four additional scalar operators: $s'$ (the next-to-leading singlet), $t'$ (the next-to-leading charge 2 operator), and the leading charge 3 and charge 4 operators, have been determined nonrigorously:

\[ 
\Delta_{s'} = 3.794(8),~\Delta_{t'} = 3.650(2),~\Delta_{3} = 2.1086(3),~\Delta_{4}=3.14(2).
\]

These data are from the bootstrap, computed here: [[https://arxiv.org/pdf/1912.03324.pdf]].
===== Physical realizations =====

Superfluid helium (both He-3 and H-4), apparently.

===== External links =====

  * [[https://en.wikipedia.org/wiki/Classical_XY_model|Wikipedia]]